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ISBN 978-951-42-8621-6 (Paperback)ISBN 978-951-42-8622-3 (PDF)ISSN 0355-3213 (Print)ISSN 1796-2226 (Online)

U N I V E R S I TAT I S O U L U E N S I SACTAC

TECHNICA

OULU 200

C 284

Marian Codreanu

MULTIDIMENSIONAL ADAPTIVE RADIO LINKSFOR BROADBAND COMMUNICATIONS

FACULTY OF TECHNOLOGY,DEPARTMENT OF ELECTRICAL AND INFORMATION ENGINEERING, CENTRE FOR WIRELESS COMMUNICATIONS,UNIVERSITY OF OULU

C 284

ACTA

Marian C

odreanu

C234etukansi.fm Page 1 Thursday, October 18, 2007 9:55 AM

A C T A U N I V E R S I T A T I S O U L U E N S I SC Te c h n i c a 2 8 4

MARIAN CODREANU

MULTIDIMENSIONAL ADAPTIVE RADIO LINKS FOR BROADBAND COMMUNICATIONS

Academic dissertation to be presented, with the assent ofthe Faculty of Technology of the University of Oulu, forpublic defence in Auditorium IT116, Linnanmaa, onNovember 16th, 2007, at 12 noon

OULUN YLIOPISTO, OULU 2007

Copyright © 2007Acta Univ. Oul. C 284, 2007

Supervised byProfessor Matti Latva-aho

Reviewed byProfessor Vincent PoorDoctor Wei Yu

ISBN 978-951-42-8621-6 (Paperback)ISBN 978-951-42-8622-3 (PDF)http://herkules.oulu.fi/isbn9789514286223/ISSN 0355-3213 (Printed)ISSN 1796-2226 (Online)http://herkules.oulu.fi/issn03553213/

Cover designRaimo Ahonen

OULU UNIVERSITY PRESSOULU 2007

Codreanu, Marian, Multidimensional adaptive radio links for broadbandcommunicationsFaculty of Technology, University of Oulu, P.O.Box 4000, FI-90014 University of Oulu, Finland,Department of Electrical and Information Engineering, Centre for Wireless Communications,University of Oulu, P.O. Box 4500, FI-90014 University of Oulu, Finland Acta Univ. Oul. C 284, 2007Oulu, Finland

AbstractAdvanced multiple-input multiple-output (MIMO) transceiver structures which utilize the knowledgeof channel state information (CSI) at the transmitter side to optimize certain link parameters (e.g.,throughput, fairness, spectral efficiency, etc.) under different constraints (e.g., maximum transmittedpower, minimum quality of services (QoS), etc.) are considered in this thesis. Adaptive transmission schemes for point-to-point MIMO systems are considered first. A robust linkadaptation method for time-division duplex systems employing MIMO-OFDM channel eigenmodebased transmission is developed. A low complexity bit and power loading algorithm which requireslow signaling overhead is proposed.

Two algorithms for computing the sum-capacity of MIMO downlink channels with full CSIknowledge are derived. The first one is based on the iterative waterfilling method. The convergenceof the algorithm is proved analytically and the computer simulations show that the algorithmconverges faster than the earlier variants of sum power constrained iterative waterfilling algorithms.The second algorithm is based on the dual decomposition method. By tracking the instantaneous errorin the inner loop, a faster version is developed.

The problem of linear transceiver design in MIMO downlink channels is considered for a casewhen the full CSI of scheduled users only is available at the transmitter. General methods for jointpower control and linear transmit and receive beamformers design are provided. The proposedalgorithms can handle multiple antennas at the base station and at the mobile terminals with anarbitrary number of data streams per scheduled user. The optimization criteria are fairly general andinclude sum power minimization under the minimum signal-to-interference-plus-noise ratio (SINR)constraint per data stream, the balancing of SINR values among data streams, minimum SINRmaximization, weighted sum-rate maximization, and weighted sum mean square error minimization.Besides the traditional sum power constraint on the transmit beamformers, multiple sum powerconstraints can be imposed on arbitrary subsets of the transmit antennas.This extends the applicabilityof the results to novel system architectures, such as cooperative base station transmission usingdistributed MIMO antennas. By imposing per antenna power constraints, issues related to thelinearity of the power amplifiers can be handled as well.

The original linear transceiver design problems are decomposed as a series of remarkably simpleroptimization problems which can be efficiently solved by using standard convex optimizationtechniques. The advantage of this approach is that it can be easily extended to accommodate varioussupplementary constraints such as upper and/or lower bounds for the SINR values and guaranteedQoS for different subsets of users. The ability to handle transceiver optimization problems where anetwork-centric objective (e.g., aggregate throughput or transmitted power) is optimized subject touser-centric constraints (e.g., minimum QoS requirements) is an important feature which must besupported by future broadband communication systems.

Keywords: adaptive radio link, beamforming, convex optimization, downlink multiple-input multiple-output channel sum-capacity

To my wife

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Preface

The research related to this thesis has been carried out at the Centre for Wireless Com-munications (CWC), University of Oulu, Finland, during the years 2002-2007. I joinedCWC in August 2002 to start my postgraduate studies after I have worked for threeyears at University Politehnica Bucharest, Romania. I wish to thank the director ofCWC at the time of my arrival, Professor Matti Latva-aho, for giving me the opportu-nity to work in such a joyful and inspiring research unit. Dr. Ian Oppermann and Lic.Tech. Ari Pouttu, the following directors of CWC, are acknowledged for creating thefriendly working atmosphere which I enjoyed during these years.

Having Professor Matti Latva-aho as my supervisor was a truly privilege. His con-stant encouragement, meticulous guidance, and unreserved help are invaluable, and Iwish to thank him for that. I am grateful to Professor Markku Juntti and Dr. DjordjeTujkovic for the support in the preparation of some of the publications related to mythesis. Their research standards and technical precision will always remain my profes-sional goals. A special thank goes to Professor Cristian Negrescu for convincing me tochoose a research career and for guiding my firsts steps in the wonderful world of sig-nals, circuits and systems. The special thank extends to Professor Keijo Ruotsalainenfor the fruitful discussions, especially those related to the optimization theory. I wishto thank the reviewers of the thesis, Professor H. Vincent Poor from the Princeton Uni-versity, New Jersey, United States, and Professor Wei Yu from University of Toronto,Canada. Their comments significantly improved the quality of the thesis. I am gratefulto Ms. Anna Shepherd for proofreading the manuscript.

Most of the work presented in the thesis was conducted in the Future Radio Access(FUTURA), Packet Access Networks with Flexible Spectrum Use (PANU), and FixedWireless Systems (FIXWIRE) projects. I would like to thank managers of these projects,Professor Jari Iinatti, Dr. Kari Hooli, Dr. Kimmo Kansanen, Dr. Pekka Pirinen and Mr.Markus Myllylä, and all my colleagues that have participated for directly or indirectlyinfluencing my work. My special thanks are due to Mr. Antti Tölli for being a joyfuland inspiring room mate for years. The cooperation and joint work with him provedto be very fruitful and I hope it will continue in the future. I am grateful to Mr. EsaKunnari and Lic. Tech. Mikko Vehkaperä for providing the fading channel simulatorand the turbo coding package, respectively.

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I thank all personnel in the CWC and Telecommunication Laboratory for providinga pleasant working environment. My appreciation goes to Lic. Tech. Raffaello Tesi, Dr.Honglei Miao, Dr. Zexian Li, Lic. Tech. Mikko Vehkaperä, Dr. Nenad Veselinovic,Mr. Juha Karjalainen and Mr. Petri Komulainen for all the lively discussions through-out my postgraduate studies at the CWC. The administrative support of Hanna Saarela,Elina Komminaho, Sari Luukkonen, Laila Kuhalampi, Mikko Ojala, Timo Äikäs, An-tero Kangas and the computer support of Jari Silanpää are gratefully acknowledged.The financial support provided by the Finnish Funding Agency for Technology andInnovation, the National Technology Agency of Finland, the Finnish Defence Forces,Infotech Oulu Graduate School, Nokia Ltd., Elektrobit Ltd., Instrumentointi Ltd., aswell as the Nokia Foundation, HPY:n Tutkimussäätiö Foundation enabled this workand is thus gratefully acknowledged.

I want to express my deepest gratitude to my loving parents Maria and Ioan forthe love and support they gave me throughout my life and hard years of study. Theirpositive attitude towards education has been an important driving force in my studies. Iwish to thank my parents-in-law Geta and Gheorghe for the support and understandingthey had for me during these years. Also, I wish to thank my dear friends Valeriu Gâlaand Corina Badulescu, and the small Romanian community from Oulu for the nicemoments we shared throughout these years.

My warmest thanks belong to my lovely wife Monica for her sincere love and un-derstanding she had for me during these years. Without her support and continuousencouragement this work would not have been finished. Hence, this thesis is dedicatedto her.

Oulu, October 4th, 2007 Marian Codreanu

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Abbreviations

A normalized cross-coupling matrix (S × S)Aj variable in the auxiliary optimization problem used to prove the mono-

tonicity of the IWF-RUP algorithm (CRk(i)j× CR

k(i)j

)Anew

j updated value for the matrix Aj (CRk(i)j× CR

k(i)j

)An nth subset of transmit antennas at the BSB nonnegative matrix used in the derivation of minimum SINR maximiza-

tion algorithm (S + 1× S + 1)Bj variable in the auxiliary optimization problem used to prove the mono-

tonicity of the IWF-RUP algorithm (CRk(i)j× CR

k(i)j

)bm effective rate provided by the mth MCSbc vector of bits associated to the complex symbols contained in dc

c subcarrier indexC number of subcarriersCsum sum-capacity of MIMO downlink channelC

(SEE)erg ergodic capacity of a point-to-point MIMO system with SEE power al-

locationC

(WF)erg ergodic capacity of a point-to-point MIMO system with WF power allo-

cationC

(NO CSI)inst instantaneous capacity of a point-to-point MIMO system with no CSI

knowledge at TX sideC

(WF)inst instantaneous capacity of a point-to-point MIMO system with WF

power allocationC

(NO CSI)x%outage outage capacity of a point-to-point MIMO system with no CSI knowl-

edge at TX sideC cross-coupling matrix (S × S)C arbitrary subset of the set of subcarriers indicesds data symbol associated to the sth stream in the downlink channeld′s data symbol associated to the sth stream in the dual uplink channelds estimated data symbol at the output of the sth linear combiner in the

downlink channel

9

d′s estimated data symbol at the output of the sth linear combiner in thedual uplink channel

d vector of transmitted data symbols in the downlink channel (S × 1)d′ vector of data symbols transmitted in the dual uplink channel (S × 1)dc vector of transmitted data symbols at the cth subcarrier (r × 1)d vector of estimated data symbols at the output of linear combiners in the

downlink channel (S × 1)d′ vector of estimated data symbols at the output of linear combiners in the

dual uplink channel (S × 1)dc vector of linear post-combiner output at the cth subcarrier (r × 1)D diagonal matrix containing the normalized SINR constraints for the data

streams (S × S)D diagonal matrix containing the normalized weights for the SINR values

of data streams (S × S)e(·) instantaneous error of the subproblem in the sum-rate maximization via

dual decomposition methode1(·) rate error of the subproblem in the sum-rate maximization via dual de-

composition methode2(·) sum power error of the subproblem in the sum-rate maximization via

dual decomposition methodemax(·) upper bound of the instantaneous error of the subproblem in the sum-

rate maximization via dual decomposition methodf(·) nondecreasing function of SINR values used to express a generic system

performance criteriaf0(·) concave function used to express the dependency of the sum-rate versus

a randomly selected pair of transmit covariance matrices in the IWF-RUP sum-rate maximization algorithm

f1(·) objective function of the auxiliary optimization problem used to provethe monotonicity of IWF-RUP algorithm

fbs(·) bit-to-symbol mapping functionfdl(·) function describing the updating of power allocation vector in the down-

link channelfsub(·) objective function of the subproblem in the sum-rate maximization via

dual decomposition methodful(·) function describing the updating of the power allocation vector in the

10

dual uplink channelg(λ) optimal value of the subproblem for a fixed waterfilling level in the sum-

rate maximization via dual decomposition methodgs,k cross-coupling coefficient between data streams s and k

G average SNR gap for a given MCS setGw weighted sum MSE between transmitted data and the output of LMMSE

receiver; the weights are specified by the vector w º 0

Gs MSE of the sth data streamH downlink channel matrix (

∑Ss=1 Rks × T )

Hc channel matrix at the cth subcarrier of the point-to-point MIMO system(R× T )

Hk,c channel matrix of the kth user at the cth subcarrier in the downlink chan-nel (Rk × T )

Hk channel matrix of the kth user in the downlink channel; in case ofOFDM systems Hk = diag{Hk,1, . . . ,Hk,C} (CRk × CT )

Hk channel matrix of the kth user in the dual multiple-access channel(CT × CRk)

Hc estimated channel matrix at the cth subcarrier of the point-to-pointMIMO system (R× T )

i vector of indices of the scheduled users, sorted in ascending order(U × 1)

jc carrier index user to sort the eigenvalues from the ith cluster in descend-ing order, i.e., λi,j1 ≥ λi,j2 ≥ · · · ≥ λi,jC

k user indexks index of the user associated with the sth data streamk

(i)1 first user randomly selected at the ith iteration of the IWF-RUP algo-

rithmk

(i)2 second user randomly selected at the ith iteration of the IWF-RUP algo-

rithmK number of usersK(i) pair of users randomly selected at the ith iteration of the IWF-RUP al-

gorithmlk path-loss of the kth userlmin minimum length of a codeword in the point-to-point MIMO systemsl(·) Lagrange dual function of the subproblem in the sum-rate maximization

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via dual decomposition methodL number of OFDM symbols in one frameL(·) Lagrangian of the subproblem in the sum-rate maximization via dual

decomposition methodLk Cholesky factor of Rk (CRk × CRk)m(·) general monomial functionm?

i index of the optimum MCS at the ith clusterms transmit beamformer for the sth data stream in the downlink channel

(T × 1)m[n]

s part of the transmit beamformer for the sth data stream that contains theweights of antennas belonging to An (|An| × 1)

M number of MCS in the BPLAM transmit beamformer matrix in the downlink channel, i.e., M =

[m1, . . . ,mS ] (T × S)M[n] transmit beamformer matrix in the downlink channel that contains the

weights of antennas belonging to An (|An| × S)N number of subsets of transmit antennasN0 two-sided power spectral density of the noise processNCP length of the cyclic prefix in the OFDM symbolNc error of the channel matrix estimation at the cth subcarriern noise vector at the output of linear combiners of the downlink channelnc noise vector at the cth subcarrier of the point-to-point MIMO system

(R× 1)nk,c vector of interference-plus-noise signals received by the the kth user at

the cth subcarrier (Rk × 1)nk vector of all interference-plus-noise signals received by the kth user; in

case of OFDM systems nk = [nTk,1, . . . ,n

Tk,C ]T (CRk × 1)

nk vector of interference-plus-noise signals at the output of the whiteningfilter of the kth user (CRk × 1)

n noise vector in the dual multiple-access channel (CT × 1)p(·) general posynomial functionps power allocated to the sth data stream in the downlink channelp(i)s power allocated to the sth data stream at the ith iteration in the downlink

channelpk sum power constraint of the kth user in the dual multiple-access channel

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p power allocation vector in the downlink channel (S × 1)p(i) power allocation vector at the ith iteration in the downlink channel

(S × 1)Pi total power allocate to the ith clusterPi,c power allocated to the ith spatial sub-channel from the cth subcarrier of

the point-to-point MIMO systemPmax transmit sum power constraintPn sum power constraint for the nth subset of transmit antennasP

(NO CSI)i,c power allocated to the ith spatial sub-channel from the cth subcarrier of

the point-to-point MIMO system with no CSI at the TX sideP

(SEE)i,c SEE based power allocated to the ith spatial sub-channel from the cth

subcarrier of the point-to-point MIMO systemP

(MWF)i,c MWF based power allocated to the ith spatial sub-channel from the cth

subcarrier of the point-to-point MIMO systemP

(WF)i,c WF based power allocated to the ith spatial sub-channel from the cth

subcarrier of the point-to-point MIMO systemPi,c(γ) power required by the ith spatial sub-channel from the cth subcarrier to

achieve a target SNR value, γ

Pc diagonal matrix controlling the power allocation at the cth subcarrier ofa point-to-point MIMO system (r × r)

Pk diagonal matrix containing the eigenvalues of the covariance matrix Qk

(CRk × CRk)P1 subset of data streams with guaranteed minimum SINR valuesP2 subset of data streams with SINR values maintained in some fixed ratiosqs power allocated to the sth data stream in the dual uplink channelq(i)s power allocated to the sth data stream at the ith iteration in the dual

uplink channelq power allocation vector in the dual uplink channel (S × 1)q(i) power allocation vector at the ith iteration in the dual uplink channel

(S × 1)Qk,c transmit covariance matrix of the kth user at the cth subcarrier in the

dual multiple-access channel (Rk ×Rk)Qk transmit covariance matrix of the kth user in the dual multiple-access

channel (CRk × CRk)Q?

k sum-rate optimal transmit covariance matrix of the kth user in the dual

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multiple-access channel (CRk × CRk)Q(i)

k transmit covariance matrix of the kth user after the ith iteration in thedual multiple-access channel sum-rate maximization (CRk × CRk)

Qk auxiliary covariance matrix used in the IWF-RUP sum rate maximiza-tion algorithm (CRk × CRk)

Qk auxiliary covariance matrix used in the IWF-RUP sum rate maximiza-tion algorithm (CRk × CRk)

r minimum between the number of transmit and receive antennas in apoint-to-point MIMO system

rs rate of the sth data streamrmax maximum available rate in a given set of MCSR number of receive antennas in the case of point-to-point MIMO systemsRk number of receive antennas at the kth userRsum sum-rate of the dual multiple-access channelRw weighed sum of the individual rates of data streams; the weights are

specified by the vector w º 0

Rk covariance matrix of the interference-plus-noise signal at the kth user(CRk × CRk)

Rk,c covariance matrix of the interference-plus-noise signal at the cth subcar-rier of the kth user (Rk ×Rk)

R covariance matrix of the interference-plus-noise signal in the dualmultiple-access channel (CT × CT )

sj number of streams allocated to the jth scheduled usersi,m maximum number of eigenmodes that can be selected at the ith clusters vector of numbers of streams allocated to the scheduled users (U × 1)S number of transmitted data streams in the downlink channelS set of scheduled usersSk Hermitian nonnegative definite matrix variable used in the sum-rate

maximization via the dual decomposition method (CRk × CRk)t vector used to store temporary values in the BPLA (C × 1)tc temporary variable used in BPLAT number of transmit antennasTc equivalent channel matrix at the cth subcarrier of the point-to-point

MIMO system (R×R)u1(·) function describing the covariance update performed at the first step of

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the IWF-RUP algorithmu2(·) function describing the covariance update performed at the second step

of the IWF-RUP algorithmus normalized receive beamformer for the sth data stream in the downlink

channel (Rks × 1)u(i)

s normalized receive beamformer for the sth data stream at the ith itera-tion in the downlink channel (Rks × 1)

us LMMSE combiner for the sth data stream in the downlink channel(Rks × 1)

U number of scheduled usersU block diagonal matrix containing the normalized receive beamformers

of the downlink channel, i.e., UH = diag{uH1 , . . . ,uH

S} (∑

k Rk × S)Uc unitary matrix comprising the left singular vectors of the channel matrix

at the cth subcarrier of the point-to-point MIMO systems (R×R)vs normalized transmit beamformer for the sth data stream in the downlink

channel (T × 1)v[n]

s part of the normalized transmit beamformer for the sth data stream thatcontains the weights of the antennas belonging to An (|An| × 1)

v(i)s normalized transmit beamformer for the sth data stream at the ith itera-

tion in the downlink channel (T × 1)vs LMMSE combiner for the sth data stream in the dual uplink channel

(T × 1)V normalized transmit beamformer matrix in the downlink channel, i.e.,

V = [v1, . . . ,vS ] (T × S)Vc unitary matrix comprising the right singular vectors of the channel ma-

trix at the cth subcarrier of the point-to-point MIMO systems (R×R)Vc normalized transmit beamforming matrix at the cth subcarrier of point-

to-point MIMO systems (T × r)Vc unitary matrix comprising the right singular vectors of the estimated

channel matrix at the cth subcarrier of the point-to-point MIMO systems(R×R)

ˆVc normalized transmit beamforming matrix at the cth subcarrier of point-to-point MIMO systems (T × r)

Vk unitary matrix containing the eigenvectors of the covariance matrix Qk

(CRk × CRk)

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ws generic weight associated to the sth data streamw generic weight vector (S × 1)W auxiliary Hermitian positive semidefinite matrix variable used in the

sum-rate maximization via dual decomposition (CT × CT )Wk composite channel matrix of the kth user (CRk × CT )xi generic variable for an optimization problemx vector of all signals transmitted by the BS; in the case of OFDM systems

x = [xT1, . . . ,x

TC ]T (CT × 1)

xc vector of signals transmitted at the cth subcarrier (T × 1)xk,c vector of signals transmitted by the kth user at the cth subcarrier in the

dual multiple-access channel (Rk × 1)xk vector of signals transmitted by the kth user in the dual multiple-access

channel (CRk × 1)X matrix of all transmitted signals of the point-to-point MIMO-OFDM

system, i.e., X = [x1, . . . ,xC ] (T × C)yc vector of signals received at the cth subcarrier of the point-to-point

MIMO system (T × 1)yk,c vector of signals received by the kth user at the cth subcarrier (Rk × 1)yk vector of all signals received by the kth user; in case of OFDM systems

yk = [yTk,1, . . . ,y

Tk,C ]T (CRk × 1)

y vector of signals received in the dual multiple-access channel (CT × 1)yk vector of signals at the output of the whitening filter of the kth user

(CRk × 1)Y matrix of all received signals in the point-to-point MIMO-OFDM sys-

tem, i.e., Y = [y1, . . . ,yC ] (R× C)zc vector of signals at the input of linear pre-combiner in point-to-point

MIMO systems (T × 1)α parameter of the signomial program; it controls the size of the trust

regionαs weight associated with SINR of the sth data stream used in the SINR

balancing algorithmβ power reduction factor used in downlink linear transceiver design with

power constraints per antenna groupsγs SINR of the sth data streamγk,s SINR of the kth user at the sth data stream in the downlink channel

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γmax SNR required to select the most spectrally efficient MCS from a givenMCS set

γdls SINR of the sth data stream in the downlink channel

γuls SINR of the sth data stream in the dual uplink channel

γ specified target SNR valueγm SNR required by the mth MCS to achieve the target FER in AWGN

channelγs minimum SINR value for the sth data streamγs initial SINR guess for the sth data streamγdl vector of SINR values in the downlink channel (S × 1)γ

(i)dl vector of SINR values at the ith iteration in the downlink channel (S×1)

γul vector of SINR values in the dual uplink channel (S × 1)γ

(i)ul vector of SINR values at the ith iteration in the dual uplink channel

(S × 1)γ vector of minimum SINR values (S × 1)γ vector of initial SINR guesses (S × 1)Γ SNR gap to the channel capacityΓ Lagrange dual variable in the sum-rate maximization via the dual de-

composition method (CT × CT )Γ feasible Lagrange dual variable in the sum-rate maximization via the

dual decomposition method (CT × CT )δ square of the power reduction factor used in downlink linear transceiver

design with power constraints per antenna groups∆Csum(i) mean normalized sum-capacity deviation after the ith covariance update∆R

(i)sum increase of the sum-rate at the ith iteration of the IWF-RUP algorithm

∆Rsum(N) increase of the sum-rate after N iterations of the IWF-RUP algorithm∆f

(i)1 increase of the f1(·) at the ith iteration of the IWF-RUP algorithm

ε accuracy required for the subproblem in the sum-rate maximization viathe dual decomposition method

ε′ accuracy required for the sum power in the sum-rate maximization viathe dual decomposition method

εγ accuracy required for the optimum SINR value of a data stream in asignomial program

εµ accuracy required for the waterfiling level in the sum-rate maximizationvia the dual decomposition method

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ηi,c equivalent power gain of the eigenmode (i, c)Θ auxiliary Hermitian positive semidefinite matrix variable used for deriv-

ing the lower bound of the sum MSE in the dual uplink channel (T ×T )λ Lagrange dual variable associated to the sum power constraint in the

sum-rate maximization via the dual decomposition methodλi,c ith eigenvalue of Hermitian positive semidefinite matrix HcHH

c

λ normalization factor used to generate a feasible Lagrange dual variablein the sum-rate maximization via the dual decomposition method

λi,c ith eigenvalue of Hermitian positive semidefinite matrix HcHHc

λM,c largest eigenvalue of Hermitian positive semidefinite matrix HcHHc

Λc matrix of singular values obtained by SVD of channel matrix (R× T )Λc square matrix containing the singular values of the channel matrix on

the main diagonal (R×R)µ generic waterfilling levelµmax maximum waterfilling levelµmin minimum waterfilling levelνk Lagrange dual variable associated with the sum power constraint of

the kth user in the sum-rate maximization via the dual decompositionmethod

ξ auxiliary variable used in the min-max characterization of the Perron-Frobenius eigenvalue

ρ square root of the power reduction factor% common weighted SINR value used in the SINR balancing algorithmσ2

N variance of the channel estimation errorυ parameter of the sum-rate maximization via the dual decomposition al-

gorithm; it controls the reduction of the interval in which the optimumwaterfilling level is contained

υ auxiliary vector variable used in the min-max characterization of thePerron-Frobenius eigenvalue (S + 1× 1)

Φc matrix describing the angles between the estimated and the true singularvectors of the channel matrix (R×R)

χi,c normalization factor for the power allocated to the ith spatial sub-channel from the cth subcarrier

Ψk Lagrange dual variable associated with the positive semidefinitenessconstraint of the transmit covariance matrix for the kth user in the sum-

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rate maximization via the dual decomposition method (CRk × CRk)AWGN additive white Gaussian noiseBER bit error rateBPLA bit and power loading algorithmBS base stationCP cyclic prefixCS covariance scalingCSI channel state informationDD dual decompositionDL downlinkDPC dirty-paper codingFDD frequency-division duplexFEE full eigenmode excitationFER frame error rateFFT fast Fourier transformGP geometric programGSM Global System for Mobile communicationsHH Hughes-HartogsIFFT inverse fast Fourier transformIS-95 Interim Standard 95ISI inter-symbol interferenceIWF iterative waterfillingIWF-RUP iterative waterfilling for random user pairsKKT Karush-Kuhn-TuckerOFDM orthogonal frequency division multiplexingQAM quadrature-amplitude modulationQoS quality of serviceLCC low correlated channelLMMSE linear minimum mean square errorMCS modulation and coding schemeMIMO multiple-input multiple-outputMISO multiple-input single-outputMSE mean square errorM-QAM multilevel quadrature-amplitude modulationRX receiver

19

SCC strong correlated channelSDP semidefinite programSEE strongest eigenmode excitationSER symbol error rateSIMO single-input multiple-outputSINR signal-to-interference-plus-noise ratioSISO single-input single-outputSNR signal-to-noise ratioSOC second-order coneSOCP second-order cone programSPC-IWF sum power constraint iterative waterfillingSTBC space-time block codingSVD singular value decompositionTC turbo codesTCM trellis-coded modulationTDD time-division duplexTDMA time-division multiple accessTX transmitterUL uplinkWF waterfillingZF zero forcing(·)∗ complex conjugation(·)? a solution of an optimization problem|X| determinant of matrix X

|S| cardinality of set SXT transpose of matrix X

XH conjugate transpose (Hermtian) of matrix X

X−1 inverse of matrix X

X−1/2 Hermitian square root of the Hermitian matrix X, i.e., X−1/2X−1/2 =X

CN (m,C) complex circularly symmetric Gaussian vector distribution with meanm and covariance matrix C1

I identity matrix; a subscript can be used to indicate the dimension

1A complex Gaussian random vector z = x + y is circularly symmetric (also termed proper Gaussian) if x

and y have identical autocovariance matrices and their crosscovariance matrix is skey-symmetric [153].

20

diag(x) diagonal matrix with the elements of vector x on the main diagonaldiag({Dk}) block-diagonal matrix with blocks given by the set Dk; in particular, if

the Dk’s are scalars, it reduces to a diagonal matrixE(·) expectationIm(·) imaginary partinf(·) largest lower bound (infimum)ln(·) natural logarithmlog(·) logarithm in base 2max(·) maximummin(·) minimumO(·) big O notationrank(·) rankRe(·) real partsup(·) smallest upper bound (supremum)vec(·) vec-operator: if X = [x1, . . . ,xn], then vec(X) = [xT

1, . . . ,xTn]T

[X]i,j element at the ith row and jth column of matrix X

[x]n nth element of vector x

(x)+ positive part of x, i.e., max(0, x)|x| absolute value (magnitude) of the scalar x

‖x‖2 Euclidian norm of vector x

‖X‖2 Frobenius norm of matrix X

tr(X) trace of matrix X

∇xf(x) gradient of function f(x) with respect to x∆= defined as≈ approximative equal∼ distributed according toº generalized inequality in a proper cone [26, Sect. 2.4]; between real

vectors, it represents componentwise inequality; between Hermitian ma-trices, A º B means that A−B is positive semidefinite

IR the set of real numbersIR+, IR++ the set of nonnegative real and positive real numbersCm×n the set of m× n complex matricesIRn

+ the cone of nonnegative n-dimensional real vectors (the n-dimensionalnonnegative orthant)

21

22

Contents

AbstractPreface 7Abbreviations 9Contents 221 Introduction 25

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.2 Review of earlier and parallel work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.2.1 SISO transmission with CSI at TX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.2.2 Single-user MIMO transmission with CSI at TX . . . . . . . . . . . . . . . . . . 31

1.2.3 MIMO broadcast channel capacity with CSI at TX . . . . . . . . . . . . . . . . 34

1.2.4 Linear processing for MIMO downlink channels with CSI atTX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.3 Aims and outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1.4 The author’s contribution to the publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2 Point-to-point adaptive MIMO-OFDM transmission 432.1 System model and capacity for MIMO-OFDM unbalanced antennas

systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2 Bit and power loading algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50

2.3 The compensation of CSI errors at TX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.5 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3 Sum-capacity computation in MIMO downlink channels 653.1 Sum-capacity for MIMO-OFDM downlink channel . . . . . . . . . . . . . . . . . . . . . 66

3.2 Iterative waterfilling for random user pairs algorithm . . . . . . . . . . . . . . . . . . . . 68

3.2.1 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.3 Sum-rate maximization via dual decomposition method . . . . . . . . . . . . . . . . . . 79

3.3.1 Derivation of the error upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.3.2 Algorithm derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84

3.3.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.4 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

23

4 Linear transceiver design for MIMO downlink channels 914.1 Linear transceiver system model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.2 Linear transceiver design under a sum power constraint . . . . . . . . . . . . . . . . . . 95

4.2.1 Heuristic SINR vector maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.2.2 General algorithm for joint beamformer and power

optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.2.3 Computing the power allocation vectors . . . . . . . . . . . . . . . . . . . . . . . . . 984.2.4 A sum MSE lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.2.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.3 Linear transceiver design with power constraints per antenna groups . . . . . 1234.3.1 Weighted sum-rate maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.3.2 Generalization to other system performance criteria . . . . . . . . . . . . . . 1274.3.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.4 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315 Conclusions and future work 133References 137Appendices 153

24

1 Introduction

”To those who do not know mathematics it is difficult to get across a real

feeling as to the beauty, the deepest beauty, of nature ... If you want to

learn about nature, to appreciate nature, it is necessary to understand the

language that she speaks in.” – Richard Feynman

1.1 Motivation

The next generation of mobile communication systems should provide a large range ofservices, such as voice, data and multimedia, at a reasonable cost and quality of service(QoS), comparable to competing wire-line technologies. However, the dispersive natureof the radio propagation [6, 167, 227] makes it more difficult to achieve these goalsover a wireless channel. The multipath propagation between the transmitter (TX) andreceiver (RX) induces large power fluctuations in the received signal [197, 198]. Thisphenomenon, known as multipath fading, is the main impediment to achieving reliabletransmission at high data rates over wireless channels [80, 91].

When the channel state information (CSI) is not known at the TX side, the com-munication system must be designed to maintain an acceptable performance level forall possible channel conditions. Such systems are effectively designed for the worst ex-pected channel conditions, resulting in an inefficient use of the available resources andchannel capacity [15, 192]. This has been the case, e.g., in designing second generationmobile phone systems, such as GSM and IS-95 [175, 177]. In the case of communica-tion systems operating over slowly fading channels2, some level of CSI knowledge canbe acquired at the TX side and adaptive link techniques can be used to increase the chan-nel usage efficiency [44, 128, 291]. In the case of time-division duplex (TDD) systems,the radio channel reciprocity can be exploited to estimate the forward channel based onthe pilot symbols received over the reverse channel [32, 127]. In frequency-division du-plex (FDD) the CSI estimated at the receiver side can be conveyed to the transmitter viaa feedback channel [92]. Alternatively, the receiver itself can decide on a set of trans-mission parameters and communicate them to the transmitter via a feedback channel,

2A channel is characterized as slowly fading if the symbol time period is smaller than the channel coherencetime [197].

25

resulting in so called closed-loop adaptation schemes [4, 91].Adaptive link techniques consist of adapting the transmission parameters according

to channel variations in order to optimize some link quality parameters (e.g., through-put, delay, etc.) subject to some constraints imposed by the application (e.g., maximumtransmitted power, minimum QoS, etc.). In general, the adaptive schemes may adaptany combination of the following parameters: transmit power, symbol rate, modulationlevel, coding rate, puncturing pattern for convolutional or turbo codes, transmit beam-formers, scheduled users [44, 91]. The third generation mobile cellular systems [99]are the first major commercial wireless application utilizing TX/RX parameter adapta-tion based on channel characteristics. In the case of multiuser systems, if the receivingconditions of the users are known in advance, the presence of fading can be turned intoan advantage. The channel resources can be allocated to users who can use them themost effectively, resulting in a multiuser diversity gain [193, 227, 246]. These types ofapproaches are gradually being adopted into mobile cellular systems, e.g., the 3G-LTEconcept and the 4G concept proposal being developed within the European Winnerproject [145].

1.2 Review of earlier and parallel work

1.2.1 SISO transmission with CSI at TX

The first adaptive transmission technique was proposed by Hayes in the late 1960’s [92]for a fixed rate (i.e., binary) transmission through a single-input single-output (SISO)multipath fading channel. By assuming that the receiver conveys the CSI to the trans-mitter through a delay-less noise-free feedback, Hayes proposed a variable power trans-mission where the transmitter adapts its power according to the instantaneous CSI tominimize the average bit error rate (BER) subject to an average transmit power con-straint. He derived the optimum power allocation strategy and showed that large powersaving is possible, especially at low target BER values. An alternative solution, wherethe transmit power is fixed and the symbol rate is adapted according to CSI, was pro-posed by Cavers in [31]. The effect of feedback delay and rate quantization has alsobeen analyzed, and the overall power saving relative to a nonadaptive transmission was14–17 dB for a frequency-flat3 Rayleigh fading channel. However, due to limitations re-

3A channel is characterized as frequency-flat fading if the fading process affects equally all the spectralcomponents of the transmitted signal [197].

26

lated to practical implementations at that time, interest in adaptive transmission schemeswas marginal until the early 1990’s.

The main drawback of the Cavers scheme is that by changing the symbol dura-tion the bandwidth of the transmitted signal changes accordingly [172]. This com-plicates the application of the scheme in practical systems. Alternative variable rateschemes which adapt the coding rate according to the channel quality were investigatedin [2, 251]. Convolutional codes with a variable puncturing pattern [89] were proposedby Vucetic in [251], and adaptive trellis-coded modulation (TCM) [235, 236] was usedby Alamouti and Kallel in [2] to increase spectral efficiency. The principle of adaptivemultilevel quadrature-amplitude modulation (M-QAM) for flat fading channels was in-troduced by Webb and Steele in [253]. Similar schemes were investigated by Otsuki,Sampei and Morinaga [157], Kamio et al. [122] as well as Torrance and Hanzo [221].These schemes exploit the fluctuating nature of the received signal power by selectinghigher/lower level constellation when the channel conditions are good/poor.

In [253], the signal constellation is adapted in such a way that the average data rateand the short-term BER (i.e., averaged over the fast fading variations) are kept con-stant, but the long-term BER varies according to the large-scale fading variations. Thisscheme provides a constant throughput, avoids burst errors, and provides a 5 dB gainover fixed modulation schemes. Adaptation schemes which maintain the BER undera target limit and adapt the long-term average throughput according to the large-scalefading variations were studied in [219, 220]. Dynamic channel assignment techniquesfor time-division multiple access (TDMA) systems with adaptive M-QAM transmissionwere considered in [103–106, 224]. The latency induced by changing the rate accordingto the channel fading variations was studied in [222–224]. Frequency hopping [222] orstatistical multiplexing [223] were proposed to mitigate the latency effect.

The above mentioned schemes were mostly designed based on engineering intuitionrather than a rigorous theoretical analysis of the achievable performance bounds. Anoptimal adaptive scheme which achieves the capacity of a flat-fading channel with anaverage power constraint was developed by Goldsmith and Varaiya in [84]. When CSIis known both at the transmitter and the receiver, an optimal transmission scheme uses awaterfilling (WF) power allocation [62] and a variable-rate multiplexed coding scheme.However, Caire and Shamai [27] showed that variable-rate multiplexed coding is notnecessary to achieve the capacity. A simple constant-rate Gaussian code is sufficientfor achieving the capacity, provided that the code symbols are dynamically scaled byan appropriate power-allocation function before transmission [27].

27

Inspired by the information theoretic results of [84], Goldsmith and Chua proposedin [81] an adaptive M-QAM transmission scheme where both the transmission rate andpower are jointly optimized to maximize spectral efficiency, while satisfying averagepower and instantaneous BER constraints. It was found that the variable-rate variable-power scheme provides a power gain of 5-10 dB over the variable-power fixed-ratescheme and up to 20 dB power saving compared to fixed modulation schemes. Theimpact of practical limitations, such as CSI estimation errors, delay, and finite granu-larity of the rate adaptation, has also been evaluated. In particular, it was shown [81]that by using only six different constellations, the adaptive scheme achieves a spec-tral efficiency within 1 dB of the efficiency achieved with continuous rate adaptation.Different suboptimal adaptation policies, such as fixed-power variable-rate, channel in-version and truncated channel inversion, were studied in [3] for adaptive transmissionin conjunction with a diversity combining receiver.

It is important to remark that the spectral efficiency for both the capacity achiev-ing scheme [84] and the adaptive uncoded M-QAM transmission scheme [81] has thesame expression, with an effective power loss in the latter case. The power loss, alsoknown as the power-gap between the Shannon capacity and the spectral efficiency of anuncoded M-QAM scheme [45], is a function of the target BER and is independent onthe fading distribution. A similar constant power gap has been observed by Kalet forthe multitone channel [120]. The power-gap can be also interpreted as an upper-boundof the achievable coding gain when an outer channel encoder is superimposed on anadaptive M-QAM scheme.

Adaptive coded modulation schemes were proposed and analyzed in [78, 82, 98,156, 240]. The separability of the code and modulation design of the coset-codes [237]was exploited in [82] to superimpose coset-codes designed for additive white Gaussiannoise (AWGN) channels over a general class of adaptive modulation schemes. It hasbeen shown that the adaptive coding schemes of [82] achieve approximatively the samecoding gain as in the case of a AWGN channel. In order to increase the coding gain,adaptive turbo-coded modulation schemes were proposed in [240]. A spectral efficiencywithin 3 dB of the Rayleigh fading channel capacity was obtained by adapting both theturbo encoder rate and the transmit power. The impact of fading channel variationon the optimal design of adaptive coded modulation was studied in [78]. Adaptivetransmission schemes based on channel prediction were considered in [69, 155].

A unified study on the tradeoff in adapting combinations of the transmission rate,power and instantaneous BER to maximize the spectral efficiency subject to an average

28

power and BER constraints is provided in [44]. Both continuous-rate and discrete-rateadaptation are considered, as well as average and instantaneous BER constraints. It hasbeen found that close spectral efficiencies are achievable under average and instanta-neous BER constraint, as well as under a continuous-rate and discrete-rate adaptation.Moreover, a negligible spectral efficiency penalty is encountered when either the trans-mit power or the rate is restricted to be constant. These results suggest that close to op-timum performance can be achieved by exploiting only a subset of all available degreesof freedom in adaptive transmission. Closed form expressions for the outage proba-bility, spectral efficiency, and the average BER of a fixed-power variable-rate adaptiveM-QAM scheme operating in Nakagami [150] fading channel are provided in [4].

An adaptive scheme for simultaneous voice and data transmission was proposedin [5]. The voice and data are multiplexed between inphase and quadrature channels.The voice channel uses a fixed-rate variable-power adaptation scheme, whilst the datachannel uses a variable-rate variable-power one. The power is balanced between thetwo channels to provide a high average spectral efficiency for data while meeting theBER requirements for the voice channel.

Single-carrier based adaptive M-QAM schemes for frequency-selective fading chan-nels were proposed and analyzed in [180, 181, 183, 230–233]. The transmitted rate ismodified by simultaneously adapting the constellation according to the strength of thechannel and the effective symbol rate according to the delay spread of the channel. Thetransmitted data symbols are repeated as many times as needed to keep the delay spreadsmaller than a fixed fraction of the effective symbol period in order to attenuate theinter-symbol interference (ISI). Adaptive modulation in conjunction with the decision-feedback equalization [172] was studied in [182, 260, 261].

As the symbol rate increases, the delay spread of the channel becomes much largerthan the symbol rate and the channel exhibits strong frequency selectivity. For suchchannel conditions, the orthogonal frequency division multiplexing (OFDM) [255]emerged as an attractive transmission scheme because it requires low-complexity or noequalizer to combat ISI. The OFDM modulation converts the frequency-selective chan-nel into a set of parallel frequency-flat fading sub-channels, where each sub-channelcorresponds to a subcarrier [17, 238, 252]. Consequently, the equalization task is dras-tically simplified and the flexibility of the link adaptation is greatly increased sincethe power and rate allocated to each sub-channel can be jointly adapted according tothe channel conditions [63, 120, 128, 257]. Comparisons between OFDM based sys-tems and traditional single-carrier systems with linear [64, 65, 120] or decision feed-

29

back [257] equalizers showed that OFDM modulation can provide a significant improve-ment at low-to-moderate signal-to-noise ratio (SNR) values or in the case of channelswith deep nulls in the transfer function.

The concept of optimizing the power and rate allocation among the subcarriers to ex-ploit the frequency selectivity of the channel was introduced by Kalet in [120]. Assum-ing adaptive M-QAM modulation with constant symbol error rate (SER) among the sub-carriers, Kalet showed that the aggregate rate under a transmit power constraint is max-imized by the well-known waterfilling power allocation among the subcarriers. Similarto the frequency-flat fading channel, the power-gap to the channel capacity [121] is in-dependent of the channel transfer function and depends only on the target SER. Thissimilarity comes from the fact [84] that the frequency-flat fading channel is capacityequivalent to a set of parallel sub-channels, where each sub-channel corresponds to apossible fading state.

In his derivation, Kalet assumed a continuous rate allocation among the subcarriers.However, in practice the set of modulation and coding schemes (MCS) available forrate adaptation is always finite and often limited to a few MCS. Therefore, the rate mustbe distributed among the subcarriers with a finite granularity [45]. Such algorithms,commonly known as bit and power loading algorithms (BPLA) or discrete loading al-gorithms, were initially developed for discrete multitone modulation over twisted-pairchannels [29, 30, 41, 70, 102]. The first to be proposed, and perhaps the most well-known BPLA, is the Hughes-Hartogs (HH) algorithm [102]. The algorithm uses agreedy bit loading method to maximize the throughput subject to a maximum error rateand a maximum transmit power. It starts with a zero rate allocated to all sub-channels,and each increment of additional information is placed on the sub-channel that wouldrequire the least incremental energy for its transport. However, the extensive searchingand sorting required by the HH algorithm makes it impractical for high data rate applica-tions. The consequent contributions of [30, 40, 41, 70, 132] provide more computation-ally efficient methods to find the optimum [30, 131] or a close to optimum [40, 41, 70]bit and power allocation. Adaptive tracking of the optimal bit and power allocation intime varying channels was studied in [134]. A detailed treatment of the discrete loadingalgorithms for parallel channels can be found in [45].

One of the first adaptive OFDM schemes for wideband radio channels was proposedby Czylwik in [63]. He considered an M-QAM scheme where the power and rateallocated to each subcarrier can be jointly optimized, and he developed a BPLA whichminimizes the BER for a fixed throughput transmission. Czylwik showed that for a BER

30

of 10−3, power savings of up to 15 dB are possible compared to a fixed OFDM scheme.In addition, it was found that close results can be obtained by using fixed-power andadapting only the rate among the subcarriers. Similar observation holds also when onemaximizes the data rate subject to a maximum error rate constraint [40, 44, 278, 280].

In [133], an adaptive TCM scheme was proposed, where the power and MCS areadapted at a subcarrier basis to minimize the total transmitted power subject to fixedthroughput and BER constraints. The BPLA used to find the optimum allocation isessentially a fast version of the HH algorithm. The scheme provides a power saving ofover 7 dB compared to a fixed TCM OFDM transmission. Adaptive modulation andcoding for maximizing the throughput of bit-interleaved coded OFDM packet transmis-sions was investigated in [199], and adaptive OFDM based on partial CSI was addressedin [178, 271].

Unlike the single-carrier based adaptive transmission schemes, which employ thesame MCS for all data symbols in a transmission packet [224, 225], adaptive OFDMsystems can adapt the modulation parameters on a subcarrier-by-subcarrier basis. How-ever, the signaling overhead required to inform the receiver about the MCS chosen ateach subcarrier may become prohibitive in certain conditions. Blockwise loading algo-rithms, where a certain number of adjacent subcarriers are clustered in blocks whichare allocated the same MCS, were proposed in [87, 124, 126] to reduce the signalingoverhead. Alternative solutions, where the receiver employs blind detection of the MCSallocated to each block were studied in [125].

1.2.2 Single-user MIMO transmission with CSI at TX

Historically, multiple antennas were used first only at the receiver side to achieve ar-ray gain and spatial diversity [110, 259]. In the late 1980’s, the pioneering work ofWinters [258] suggested that a much higher gain is available when multiple anten-nas are used at both sides of the link. Almost a decade later, Telatar, Foschini, andGans [73, 211, 212] showed that in a reach scattering environment, multiple-inputmultiple-output (MIMO) systems provide enormous capacity gains compared to tra-ditional SISO and single-input multiple-output (SIMO) systems. They proved that thecapacity of a MIMO flat fading channel increases with the number of antennas, propor-tional to the minimum number of transmit and receive antennas. The MIMO channelcapacity studies were extended to multipath channels in [173, 174] and to correlatedfading channels in [36, 42, 228]. An extensive overview of the recent results on MIMO

31

channels capacity is provided in [83].The above information-theoretic results triggered a large amount of research in chan-

nel coding and signal processing algorithms to develop practical schemes that achievea large fraction of the promised capacity [14, 169, 206]. In general, the MIMO sys-tem design is strongly influenced by the CSI knowledge available at the transmitterside [76]. With no channel knowledge at the transmitter side, space-time coding tech-niques which aim to maximize the diversity gain were proposed in [1, 97, 149, 208–210], and spatial multiplexing schemes which aim to maximize the data rate were con-sidered in [16, 22, 23, 71, 74, 79]. The optimal tradeoff between the multiplexing anddiversity gains has been studied in [286]. Switching schemes between transmit diver-sity and spatial multiplexing mode based on a low rate feedback from the receiver totransmitter were proposed in [93, 95].

If the channel is perfectly known at the TX side, the MIMO channel can be con-verted into a set of parallel, noninterfering SISO channels through a singular valuedecomposition (SVD) of the channel matrix [211]. The gains of these elementary spa-tial subchannels, commonly denoted as eigenmodes, are equal to the singular valuesof the MIMO channel matrix. To maximize the mutual information, the total availablepower should be distributed to eigenmodes based on the WF principle [62, 163]. Inthe case of a multipath fading channel, OFDM modulation can be used to convert thefrequency selective MIMO channel into a set of frequency flat fading MIMO channelswhich can be individually processed in space domain [173, 174]. The resulting MIMO-OFDM system achieves asymptotically the MIMO channel capacity, as the number ofsubcarriers grows to infinity [173].

The capacity-optimal signaling strategy described above assumes optimal channelcoding (i.e., Gaussian codebooks [62]) with continuous rate allocation among the eigen-modes. In practice, the Gaussian code is substituted by a simple signal constellation anda practical coding scheme [172]. Therefore, various design criteria adapted to practicalsystem architectures have been considered in the literature [158]. The problem of jointdesign of a linear precoder at the transmitter and equalizer at the receiver to minimizethe sum of the mean square errors (MSE) of all eigenmodes (i.e., the trace of the MSEmatrix) was studied in [136, 185, 187, 269]. The sum MSE minimization criterion wasgeneralized to the weighted sum MSE minimization in [179]. Remarkably, if the sys-tem is designed to minimize the determinant (instead of the trace) of the MSE matrix,the classical capacity-achieving waterfilling result is obtained [187, 268]. In [186], amaximum signal to interference-plus-noise ratio (SINR) criterion with a zero-forcing

32

(ZF) constraint was also considered. SINR maximization for MIMO frequency selec-tive fading channels with cochannel interference was addressed in [263]. Transmitpower minimization under different QoS requirements for the eigenmodes was studiedin [165]. The power and constellation adaptation was considered in [160, 264] and inthe author’s contributions [56–59, 213]. The minimization of the average BER under apower constraint was addressed in [67, 161]. A single beamforming approach, whereonly the strongest spatial subchannel is used, was studied in [164]. A general frame-work based on convex optimization techniques was developed in [159, 162] to handle awide range of design criteria, including MSE-, SINR-, and BER-based criteria.

Perfect CSI at the transmitter side is often difficult to obtain in a realistic wire-less environment [10, 249]. Therefore, transmission schemes capable of exploitingthe partial CSI at the transmitter, such as the mean or the covariance of the MIMOchannel [83, 249], gained a large interest in the research community. The capacity ofsuch schemes was first analyzed assuming a multi-input single-output (MISO) systemin [147, 151, 152, 244], and then extended to the MIMO systems in [108, 109, 118, 119,196, 229, 272]. It has been found that, unlike the single antenna systems, the capacityimprovement through even partial channel knowledge can be substantial. Addition-ally, it has been proved that the optimal covariance matrix in the covariance feedbackcase [83] has the same eigenvectors as the known channel covariance matrix. However,the power allocation is different from the traditional WF power allocation. Numericalmethods for computing the optimal power allocation are provided in [119, 229]. Whenthe transmitter knows the mean of the channel matrix, the eigenvectors of the optimal co-variance matrix are given by the right singular vectors of the channel’s mean [195, 249].

Hybrid schemes, composed by an orthogonal space-time block code (STBC) whoseoutput is passed through a linear precoder before launching it into the channel, areable to exploit different levels of CSI at the transmitter [114, 154, 168, 250, 267, 287–289]. The basic idea of such schemes is to enhance the performance of a standardSTBC by combining it with transmit beamforming, for which the partial CSI can beexploited. In [115], the authors studied how a partial CSI available at the transmittercan be incorporated into the design of unstructured space-time block codes. Robustbeamforming schemes, where the channel uncertainty is explicitly taken into accountto obtain a robust beamformer design, were studied in [88] and [158, Chap.7].

When only a low rate feedback from the receiver is available, feeding back even thechannel statistics may become impractical. In such cases, the limited feedback can beused to choose one precoder from a finite cardinality set of possible linear precoding

33

matrices that is designed off-line and is known to both the transmitter and receiver [37,115, 139–141, 148, 151]. The transmit antenna subset selection schemes [86, 94, 96]can also be viewed as a particular case of the above precoder selection method.

1.2.3 MIMO broadcast channel capacity with CSI at TX

The study of the broadcast channel, where a single transmitter communicates with manyreceivers, is of great interest as it models a downlink transmission environment in a cel-lular wireless system [83]. The general broadcast channel has been introduced in [61].Its capacity region is given by the set of rates at which all users can simultaneouslyreceive with an arbitrary small probability of error. In the case of a degraded broad-cast channel4(e.g., scalar broadcast channel), where the users can be ordered accord-ing to their channel quality, the capacity region has been fully understood since the1970’s [11]. Using a superposition of random Gaussian codes at the transmitter com-bined with interference substraction at the receivers is the optimal strategy [62]. How-ever, the broadcast channel with more than one transmit antenna is not degraded [83],and its capacity region remained unknown until recently [254].

The pioneering work of Caire and Shamai [28] was a decisive step in understand-ing the capacity region of the MIMO broadcast channel. They were the first to applyCosta’s result [60] and Sato’s cooperative upper bound [184] to analyze the capacityregion of the multiple-antenna broadcast channel. Costa considered a channel affectedby both additive Gaussian noise and additive Gaussian interference, where the interfer-ence is noncausally known at the transmitter but unknown at the receiver. He showedthat, under an input power constraint, the capacity of this channel is the same as ifthe interference did not exist [60], i.e., the effect of the interference can be completelycanceled out. Sato’s upper bound is based on the two following ideas. First, since thereceivers cooperation can only increase the capacity, the capacity of the broadcast chan-nel must be upper bounded by the capacity of the cooperative channel. Second, sincethe receivers can not cooperate, the sum-capacity of the broadcast channel depends onlyon the marginal distributions of the noise and not on the correlation between noise atdifferent receivers. Thus, an upper bound of the sum-capacity of the broadcast channelcan be obtained by minimizing the capacity of the cooperative channel over all possiblejoint distributions of the noise. A noise distribution which minimizes the capacity of

4A broadcast channel is degraded if the channel inputs and the outputs form a Markov chain [62].

34

the cooperative channel is called the least-favorable noise [60].In [28], Caire and Shamai applied the above results to the broadcast channel. They

proposed a successive encoding strategy, known as dirty-paper coding (DPC), wherethe interference caused by the previously encoded users (which can be computed at thetransmitter) is canceled out by using Costa’s coding scheme. For the special case oftwo single-antenna receivers, they showed that the DPC scheme is optimal in achievingthe sum-capacity. The optimality was proved by showing that the sum-capacity of theDPC scheme achieves Sato’s upper bound [184]. However, their proof involves a directcalculation and does not hold for more than two users.

The DPC result [28] was independently extended to the case of any number of usersand an arbitrary number of antennas at each receiver in [241, 245, 279]. In [279],Yu and Cioffi used the DPC techniques to convert the generalized decision feedbackequalizer of the cooperative channel into an equivalent precoder and decoder scheme,which, for the least favorable noise, do not require cooperation between receivers. Sincethe generalized decision feedback equalizer archives the capacity of the cooperativechannel, their scheme achieves the sum capacity of the broadcast channel.

The extensions in [241, 245] are based on the notion of a dual (or reciprocal)multiple-access channel, which is obtained from the original broadcast channel by re-versing the role of transmitters and receivers. The authors observed that under a sumpower constraint, the DPC achievable region of the MIMO broadcast channel is ex-actly the capacity region of the dual multiple-access channel. This result, called uplink-downlink duality, is in fact the multiple-antenna extension of the previously establishedduality between the scalar broadcast and multiple-access channels [112, 113]. Basedon this duality, the sum-capacity optimality of the DPC was proved by showing that thesum-capacity of the dual multiple-access achieves Sato’s upper bound.

The above results proved only that the DPC scheme achieves the sum-capacity ofthe MIMO broadcast channel, but the optimality of the entire capacity region is formu-lated only as a conjecture. Another important step towards the characterization of thewhole capacity region was reported in [226, 242]. The authors constructed a degradedversion of the MIMO broadcast channel by ordering the users and allowing them toaccess the signal received by all users ranked below them. They further showed that ifGaussian coding is optimal for the degraded broadcast channel, then the DPC regionis the capacity region. Finally, the optimality of the Gaussian coding has been provedin [254].

The uplink-downlink duality is very useful from a numerical point of view be-

35

cause a direct computation of the DPC region in a broadcast channel leads to non-concave rate functions of the covariances matrices [83, 241]. In contrast, the ratesin the dual multiple-access channel are concave functions of the covariance matrices.Thus, the optimal covariance matrices which achieve the boundary of the capacity re-gion of the dual multiple-access channel can be found by using a standard convex pro-gram [13, 26, 142, 144]. Then, they can be transformed to the corresponding optimalbroadcast covariance matrices. The explicit transformations between the optimal covari-ances of the dual multiple-access channel and the optimal covariances of the originalbroadcast channel are given in [241]. In [277, 281], the uplink-downlink duality hasbeen generalized to MIMO broadcast channels with arbitrary linear covariance con-straints by using the framework of Lagrange duality in minimax Gaussian mutual infor-mation optimization [279]. When multiple linear constraints are present, the dual of thebroadcast channel becomes a multiple-access channel with an uncertain noise, and thesum-capacity is the saddle-point of a minimax problem.

Numerical solutions for computing the sum-capacity of the MIMO broadcast chan-nel as well as the optimal covariance matrices have been considered in [111, 247, 274,276] as well as in the author’s original publications [46–49]. A gradient based ascentmethod was proposed in [247]. It can be used to compute the entire capacity regionbut has a rather slow convergence. A sum power constraint iterative waterfilling (SPC-IWF) algorithm was developed in [111]. It has been introduced as a natural extensionto the original iterative waterfilling (IWF) algorithm [282], used to optimize the trans-mit covariance matrices of the MIMO multiple-access channel with individual per-userpower constraints. Since the SPC-IWF algorithm does not always converge for morethan two users, an averaging step is introduced between two successive covariance up-dates. The averaging step reestablishes the algorithm convergence, but slows down itsrate of convergence, especially for large numbers of users.

The author’s contributions [46, 48] provide a provably convergent algorithm whichovercomes the previously mentioned problems and has clearly lower computationalcomplexity. In [111], all covariance matrices are updated in each iteration to maintain aconstant waterfilling level for all users. In contrast, the algorithm proposed in [46, 48],denoted as iterative waterfilling for random user pairs (IWF-RUP), does not impose thisconstraint at each step, but the common waterfilling level is asymptotically achievedafter the convergence period. At each step, the IWF-RUP algorithm increases the sum-rate by updating just two randomly selected covariance matrices under their sum powerconstraint. Hence, there is no power exchange between the updated pair of covariance

36

matrices and the other ones. This ensures that the sum power constraint is satisfied aftereach update step given that the initial set of covariance matrices satisfies the sum powerconstraint. The numerical results show that the IWF-RUP algorithm converges fasterthan all variants of the SPC-IWF algorithm proposed in [111].

A sum-capacity maximization algorithm based on dual decomposition method wasproposed in [274]. It consists of two nested loops: the inner loop is an IWF-like al-gorithm which optimizes the covariance matrices for a fixed water level, and the outerloop finds the optimum water level by using a bisection search. Since it requires suc-cessive optimizations of the inner loop, the global convergence is rather slow. A fasterimplementation has been proposed in [276] which shows better asymptotic convergencebehavior than [111], especially in the case of a large numbers of users. The basic ideaof the improved method [276] is not to run the inner loop until convergence, but onlyto a certain tolerance that still guarantees the algorithm convergence. However, thisrequires tracking the instantaneous error of the inner loop, and this issue has not beenaddressed in [276]. The author’s contributions [47, 49] provide an upper bound of theinstantaneous error of the inner loop which can be used to construct a non-heuristicstopping criterion. The upper bound is obtained by deriving the Lagrange dual functionof the inner optimization problem and providing a method for obtaining a feasible dualvariable.

1.2.4 Linear processing for MIMO downlink channels withCSI at TX

The MIMO downlink transmission based on linear transmit and receive processing,commonly known as beamforming, has gained an increased interest during the lastyears. While these schemes are suboptimal, they lead to simpler transmitter and re-ceiver structures and allow for a reasonable tradeoff between performance and com-plexity [18, 200, 273]. When multiple users are served simultaneously without em-ploying DPC at the transmitter, the signals sent to different users interfere which eachother generating inter-user interference [20]. Therefore, the optimal designs for single-user MIMO systems are not directly applicable to a multiuser scenario [8, 270]. Thepower control [35, 72, 90, 234, 284] and the transmit and receive beamforming di-rections [77, 135, 138, 248] have to be jointly optimized according to certain systemperformance criteria, such as rate maximization, power minimization, and worst SINR

37

maximization [21, 188, 205, 256, 266].Conceptually simple ZF beamforming schemes [38, 39, 166, 201, 202, 265] form

a class of suboptimal beamforming strategy where the transmit and receive beamform-ers are designed to avoid inter-user interference. This strategy enables to decouplethe beamforming design and power control problems, which is a significant benefit inpractical system design. The MIMO downlink channel matrix is diagonalized in [265]by using a space-time processing technique based on successive Jacobi rotations [100].In [38], inter-user interference is nulled out by constraining the transmit beamform-ers of each user to lie in the null space of the other users’ channels. This precodingmethod is commonly referred to as block-diagonalization, since it decomposes the mul-tiuser MIMO channel into a set of parallel single user channels to which any single-userMIMO transmission scheme [162, 165, 173, 179, 187] can be applied. However, theblock-diagonalization method of [38] is applicable only when the number of transmitantennas is larger than the total number of receive antennas. This restriction was elim-inated in [166, 201, 202] by using coordinated transmitter-receiver processing. Thesum-capacity optimality of zero forcing beamforming schemes for channels with largenumbers of users have been analyzed in [273]. While being asymptotically optimalwhen the number of users goes to infinity, the ZF techniques suffer from a power penaltyin the case of a moderate number of users.

By allowing a limited amount of inter-user interference, the set of potential beam-former solutions is extended and substantial power saving is possible, especially at lowto medium SNR values. In [170], a regularized channel inversion scheme was proposed,where the regularization parameter was chosen to maximize the SINR. In the case of sin-gle antenna terminals, the rate maximization problem was considered in [18, 205, 256],and the transmit power minimization was considered in [256]. The beamformer designproblem is more difficult when the terminals are also equipped with multiple antennas,since it often leads to nonconvex optimization problems [281].

A key technique, which provides a substantial reduction of the computational com-plexity employed in the joint design of the transmit and the receive beamformers, con-sists of exploiting the duality between the downlink and uplink channels. The uplink-downlink SINR duality has been discovered in [176, 243] in the context of single an-tenna terminals. It has been shown therein that the minimum sum power required toachieve a set of SINR values in the downlink channel is equal to the minimum sumpower required to achieve the same set of SINR values in the dual uplink channel. Fur-thermore, the optimum transmit beamformers for the downlink channel which minimize

38

the transmitted power under the minimum SINR constraint per data stream are the sameas the optimum receive beamformers for the dual uplink channel up to a scaling factor.Based on this duality the authors [176, 243] proposed iterative algorithms to computeefficiently the optimum downlink beamformers and the optimum power allocation viathe dual uplink channel. The uplink-downlink SINR duality has been extended to arbi-trary beamformer vectors in [19, 245] and to multiple antenna terminals in [226]. It wasshown therein that the achievable SINR region is closely related to the Perron-Frobeniuseigenvalue of the cross-coupling matrix between the users. This duality result has beenexploited in [33, 188, 189, 266] to jointly design the downlink beamformers and powerallocation under minimum SINR constraints.

The problem of minimizing the sum MSE between the transmitted and estimatedsymbols has been considered, e.g., in a single-user setup [179, 187] and in a multiusersetup [116, 117, 190, 191, 194]. In the recent work of [190], the uplink-downlinkduality of the MSE region has been used to state the sum MSE minimization prob-lem as a semidefinite program [239]. General methods for joint design of the lineartransmit and receive beamformers according to different optimization criteria, includ-ing weighted sum-rate maximization, weighted sum mean square error minimization,SINR balancing, minimum SINR maximization and sum power minimization under aminimum SINR constraint, has been proposed by the author in [51–54]. Transmit pre-coder design via conic optimization for fixed MIMO receivers was proposed in [256].Transmitter optimization algorithms with per-antenna or per groups of antennas powerconstraints were proposed in [281] and in the author’s contributions [55, 214, 217].

1.3 Aims and outline of the thesis

Novel MIMO transmission techniques which utilize the knowledge of CSI at the trans-mitter side to optimize certain link parameters (e.g., throughput, fairness, spectral effi-ciency, etc.) under different constraints (e.g., maximum transmitted power, minimumQoS, etc.) are investigated. The aim of the thesis is to develop and validate novellink adaptation schemes for single- and multiuser broadband communication systemsoperating in MIMO wireless channels. All results presented in this thesis were pub-lished or accepted for publication in peer reviewed journals. For the sake of clarity anduniformity, the thesis is presented as monograph.

Chapter 2, the results of which have been presented in [50, 56–59, 213], considersadaptive transmission schemes for point-to-point MIMO systems. A low complexity

39

bit and power loading algorithm, which requires low signaling overhead and appliesto coded systems, is proposed. Its performance is compared to the performance of theuniversally accepted Hughes-Hartogs algorithm. Methods for compensating the CSIerrors at the TX side are also discussed.

Chapter 3, the results of which have been documented in [46–49], is devoted tonumerical methods for computing the sum-capacity of MIMO downlink channels. Byexploiting the broadcast to multiple access channel duality, two efficient algorithmsare derived and their performance is compared to the performance of other existingmethods. Issues related to the convergence and to the tracking of the instantaneouserror are also considered.

Chapter 4, the results of which are presented in [51–55, 214, 217], considers theproblem of joint power control and linear transmit and receive beamformers designfor MIMO downlink channels. General algorithms for linear transceiver optimizationaccording to different system performance criteria are provided. The optimization cri-teria include sum power minimization under the minimum SINR constraint per datastream, the balancing of SINR values among data streams, minimum SINR maximiza-tion, weighted sum-rate maximization, and weighted sum mean square error minimiza-tion. Besides the traditional sum power constraint on the transmit beamformers, so-lutions for transceiver optimization under a more general power constrain model areprovided. The performance of the proposed algorithms is evaluated by computer simu-lations.

Chapter 5 concludes the thesis. The main results are summarized and some openproblems for future research are pointed out.

1.4 The author’s contribution to the publications

The thesis is based in part on four journal papers [48, 49, 52, 59] and ten conferencepapers [46, 47, 50, 51, 53–58]. The author has had the main responsibility in performingthe analysis, developing the simulation software, generating the numerical results, andwriting all the papers [46–59]. Other authors provided help, comments, and criticismduring the process.

In addition to the papers [46–59], the author contributed to three other journal pa-pers [213, 214, 217] where he is not the first author. In [213], the author providedthe bit and power loading algorithm and much of the mutual information analysis pre-sented in Section III. In [214, 217], the idea of constructing an ascent algorithm by

40

combining the precoder design via conic optimization and the power control via signo-mial programming was proposed by the author. The first author further developed theideas, developed the simulation software, and generated all the numerical simulations.In [214], the author also wrote large parts of Section III, which describes the weightedsum-rate maximization algorithm.

The results and analysis presented in this thesis have been produced by the author.

41

42

2 Point-to-point adaptive MIMO-OFDMtransmission

By applying OFDM modulation to a MIMO system, the frequency selective MIMOchannel is turned into a set of frequency-flat fading MIMO channels which can beindividually processed [23]. If the CSI is known at the TX side, the channel matrixcan be decomposed for each subcarrier using SVD. As a result, a set of orthogonalsub-channels is obtained in the space domain [173, 179, 187, 262]. These elementarysub-channels will be referred to as eigenmodes in the sequel.

To maximize the throughput, the total available power should be distributed to theeigenmodes based on the waterfilling (WF) principle [173, 179, 187]. However, suchan optimal signaling strategy would require Gaussian codebooks [62] with continuousrate allocation among the eigenmodes, which is not feasible in practice. Therefore,a near optimal solution is required which takes into account the finite rate allocationgranularity imposed by the limited number of MCSs. The most accepted amongst suchschemes is the HH algorithm [102]. Its reduced complexity versions [41, 70, 278]mainly rely on the fact that it is more important to avoid using the most attenuated sub-channels than to find exactly the optimum amount of power to be allocated to the strongsub-channels. For example, it has been shown in [278] that an on/off power distribution,as long as it uses nearly the same transmission sub-channels as WF, exhibits negligiblecapacity loss with respect to exact WF shape.

All the above algorithms were mainly developed for very slowly varying channels,like asymmetric digital subscriber lines. Therein, the signaling overhead to the receiverabout the chosen MCS for each eigenmode can be neglected. Such an assumptionis too strong for wireless communication systems, where the transmission parametershave to be signaled to the receiver with a rate proportional to the channel’s Dopplerspread [127, 128].

In this chapter, a logarithm-free bit and power loading algorithm (BPLA) is pro-posed, where the signaling overhead is C times reduced (C is the number of subcarri-ers) compared to the HH algorithm. This is achieved by grouping the eigenmodes into anumber of clusters and assigning a fixed MCS for each cluster. The first cluster containsthe strongest eigenmodes from each subcarrier, the second cluster contains the secondstrongest eigenmodes from each subcarrier, and so on.

43

A cluster based BPLA, which minimizes the BER subject to a constant throughputand a constant transmitted power, has been proposed in [75]. However, the cluster posi-tions are not predefined, and, therefore, the grouping helps to reduce only the computa-tional complexity, but not to reduce the signaling overhead. Moreover, the optimizationmethod, based on the Fischer-Huber [70] algorithm, is applicable for an uncoded sys-tem only and requires logarithm operations. In contrast, the proposed logarithm-freeBPLA can be applied to coded systems as well.

In the presence of CSI errors, the eigenmode orthogonality is lost and a spatialequalizer is used at each subcarrier to mitigate the inter-eigenmode interference. Byanalyzing the noise enhancement at the linear equalizer output, a simple CSI error com-pensation is developed. It maintains the target error rate by increasing the instantaneoustransmitted power in such a way that the average SNR at the equalizer output is keptconstant.

The simulation results show that for a constant frame error rate (FER), the through-put degradation compared to the optimum HH algorithm is negligible. For an unbal-anced MIMO system with a larger number of transmit antennas, the spectral efficiencyachieved at low and medium SNR is higher than the outage capacity with unknown CSIat the TX. Consequently, the proposed scheme constitutes a practical solution for highdata rate wireless communication systems.

The remainder of this chapter is organized as follows. The adaptive MIMO-OFDMeigenmode based system model is revised in Section 2.1, and the proposed bit and powerloading algorithm is presented in Section 2.2. The CSI error compensation schemeis described in Section 2.3, the simulation results are presented in Section 2.4, andSection 2.5 concludes the chapter.

2.1 System model and capacity for MIMO-OFDMunbalanced antennas systems

The block diagram of the adaptive MIMO-OFDM system considered with C subcarri-ers, T transmit, and R receive antennas is depicted in Figure 1. At each symbol interval,the preprocessor, which comprises channel coding, bit loading of information data bitstream, and linear pre-combining, generates a T × C dimensional matrix X. The tthrow of matrix X is associated with the OFDM modulator at the transmit antenna t.

44

Bit & Power Loading Algorithm

Channel Estimation

SVD&

Eigenmode's Gain

Estimation

Turbo EncoderCluster 1

Turbo EncoderCluster r

Interleav. &Bit to Symbol

Conv. 1

Interleav. &Bit to Symbol

Conv. r

Power Control

Subcarr 1

Linear Pre-comb.

1

P1 PC

Eigenmode

SNR Gain

MCSrMCS1

Initial Power Dividing between Clusters - Modified WF

Per Cluster MCS Optimizization& Eigenmodes Power Allocation

P1 Pr

DMUX

OFDM MOD.

#1

#TxC

z1

zC

d1

dC

Inputbits

Power Control

Subcarr C

Linear Pre-comb.

C

x1IFFT

1

IFFTT

ICPP/S1

ICPP/ST

#1

#R

FFT1

FFTR

S/PRCP

1

S/PRCP

R

Linear Post-comb.

1

Linear Post-comb.

C

SoftDemod.

1

Soft Demod.

C

Deinterleav.1

Deinterleav.r

Turbo DecoderCluster 1

Turbo DecoderCluster r

MUX

Outbits

yC

y1d1

dC

~

~

V1 VC

Hc

OFDM DEM.

PRE - PROCESOR

POST - PROCESOR

σN2

X

Y

Fig 1. The block diagram of the point-to-point TDD based adaptive MIMO-OFDMsystem.

The OFDM modulator comprises inverse fast Fourier transformer (IFFT), cyclicprefix (CP) insertion, and parallel-to-serial conversion. The CP is chosen to be longerthan the maximum excess delay of the channel to avoid inter-symbol and inter-carrierinterference. At the receiver side, the CP is removed, and the resulting signals areserial-to-parallel converted and passed through the fast Fourier transformer (FFT). Ateach symbol interval, the OFDM demodulator generates an R× C dimensional matrixY, each row in the matrix corresponding to one receive antenna.

Assuming perfect frequency and sample clock synchronization between the trans-mitted and the received signals, the input-output relation for the OFDM modulator/de-modulator chain is given by

yc = Hcxc + nc, c = 1, . . . , C (1)

where c is the subcarrier index, xc ∈CT×1 and yc ∈CR×1 denotes for the cth columnsin the matrices X and Y respectively, nc ∼ CN (0, N0IR) represents the received noise

45

vector, and Hc ∈CR×T is the channel matrix. The entry [Hc]r,t ∼ CN (0, 1) representsthe complex channel gain between the TX antenna t and the RX antenna r, at the sub-carrier c. The channel matrix was generated according to the stochastic MIMO channelmodel [129] where two types of spatial fading correlations were introduced. The lowcorrelated channel (LCC) is typical for an indoor radio link, while the strong correlatedchannel (SCC) is used to reflect an outdoor-to-indoor link with the base station antennalocated above surrounding scatterers. The power delay profile was generated accordingto the ETSI BRAN Channel A [68] described in Table 1. The frame length L is chosento be smaller than the coherence time of the channel. Therefore, the channel matrix isassumed to be constant during one coded frame.

For capacity analysis, it is assumed that vectors zc at the input of linear pre-combinercomprise T independent Gaussian components, i.e., zc ∼ CN (0, diag{P1,c, . . . , PT,c}),and the linear pre-combining and post-combining are performed with unitary matri-ces Vc ∈ CT×T and UH

c ∈ CR×R (for the virtue of maximization of mutual in-formation [62]). The instantaneous capacity of the MIMO-OFDM system is givenby [23, 173]

Cinst =1C

C∑c=1

min(T,R)∑

i=1

log(

1 + λi,cPi,c

N0

)(2)

where λi,c denotes for the eigenvalues of the Hermitian positive semidefinite matrixHcHH

c .In the case of perfect prior knowledge CSI at the TX side, Vc and Uc are obtained

by SVD of the channel matrix

Hc = UcΛcVHc (3)

where Λc ∈ CR×T has the elements λ1/21,c , . . . , λ

1/2min(T,R),c on the main diagonal and

all other entries are zeros. In this way, a set of r = min(T,R) orthogonal spatial sub-channels, denoted as eigenmodes, are obtained at each subcarrier. The instantaneouscapacity (2) is maximized by the WF power allocation [62]

P(WF)i,c =

(µ− N0

λi,c

)+

(4)

where the waterfilling level µ is calculated from the constraint on the total transmittedpower, i.e.,

∑Cc=1

∑ri=1 Pi,c = Pmax.

Without the knowledge of CSI at the transmitter, Uc and Vc reduce to identitymatrices, and the optimal power shaping assumes an equal power allocation among the

46

transmit antennasP

(NO CSI)i,c =

Pmax

CT. (5)

This chapter focusses on an unbalanced antenna system with T ≥ R. It is wellunderstood that for a balanced antenna system, i.e., T = R, the difference between thechannel capacity with CSI at the transmitter (C(WF)

inst ) and the channel capacity withoutCSI at the transmitter side (C(NO CSI)

inst ) decreases as the SNR ∆= Pmax/N0 increases,and at high SNR values both capacities tend to become equal [42]. However, in the caseof an unbalanced antenna system, the WF power allocation at high SNR values tends tothe equal power allocation over the nonzero eigenmodes, i.e.,

limSNR→∞

P(WF)i,c =

Pmax

CR. (6)

By comparing (6) and (5) one should remark that a power penalty equal to T/R isencountered in the case of unknown CSI at the transmitter. Equivalently, the differencebetween the two capacities converges to

limSNR→∞

(C

(WF)inst − C

(NO CSI)inst

)=

1C

C∑c=1

R∑

i=1

limSNR→∞

log1 + λi,c

SNRCR

1 + λi,cSNRCT

= R log(

T

R

). (7)

From the concavity of the capacity in variables Pi,c [26], it follows that there is anadditional penalty at low and medium SNR values due to the equal power allocation.This degradation depends on both the channel frequency selectivity and the space selec-tivity reflected through the variation of λi,c with respect to indices c and i, respectively.

A suboptimal transmission scheme, where only the strongest eigenmode at eachsubcarrier is excited, is also considered. Such a transmission scheme is denoted asstrongest eigenmode excitation (SEE). As will be shown in the sequel, at low SNR val-ues such method offers close to optimal performance with reduced receiver complexity.The vectors zc reduce to scalars, and the matrices Vc and Uc are replaced by theircolumns corresponding to the largest eigenvalues, λ1,c. The optimal power allocationis given by

P(SEE)i,c =

(µ− N0

λ1,c

)+

if i = 1

0 if i 6= 1(8)

where the waterfilling level µ is chosen to satisfy the total transmitted power constraint,i.e.,

∑Cc=1 P

(SEE)1,c = Pmax.

47

It is important to remark that in the case of optimal channel coding (i.e., a Gaussiancodebook [62]) with perfect CSI knowledge at the TX side, the two-dimensional eigen-beamformer combined with Alamouti’s code proposed in [267, 287, 289] reduces to thepower allocation (8). Obviously, when the channel matrix has a rank one, (e.g. MISOsystems, or in the case of MIMO channels experiencing the keyhole effect [36]) the SEEis the optimal solution. When the channel matrix has rank larger than one, the capacitydegradation due to the use of only the strongest eigenmode for each subcarrier willstrongly depend on the average SNR, channel correlation, and the number of antennas.

Figure 2 shows the three signaling capacities, C(NO CSI)1%outage , C

(WF)ergodic, and C

(SEE)ergodic

[169], for different channel correlations and different antenna setups. C(WF)ergodic repre-

sents the upper-bound of the spectral efficiency achievable with the optimal channelcoding by adapting the transmission rate and the power according to each instanta-neous channel realization. C

(SEE)ergodic has similar explanation, except that it uses only the

strongest eigenmode at each subcarrier. C(NO CSI)x%outage represents the maximum spectral

efficiency that can be guaranteed for (100 − x)% of channel realizations without theknowledge of CSI at the TX [169]. Notice that in this case the transmission rate cannot be adjusted according to the instantaneous channel realization. A more detaileddescription of the ergodic and outage capacity for fading MIMO channels and their re-lationship to the degree of CSI knowledge at the TX side can be found in [169, SectionIV.C ].

For an unbalanced antenna system, a remarkable capacity gain is obtained when thechannel is known at the transmitter, even at high SNR values. At low SNR values or forhighly correlated channels, waterfilling over all available eigenmodes and waterfillingonly over the largest eigenmodes at each subcarrier produces almost equal results, i.e.,for (T,R) = (4, 2) system in SCC, C

(WF)ergodic and C

(SEE)ergodic are almost equal for SNR

values lower than 10 dB. When the SNR increases or the antenna correlation decreases,the difference between the optimum WF and the suboptimal SEE strategy increases.Motivated by these results, in Section 2.2, we propose an adaptive transmission algo-rithm which automatically decides the best signaling strategy in terms of performanceand complexity for a given scenario.

48

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

Sig

nal

ling

cap

acit

y [b

its/

sec/

Hz]

SNR [dB]

LCC, ergodic cap, WFLCC, ergodic cap., SEELCC, 1% outage cap., no CSI at TXSCC, ergodic cap., WFSCC, ergodic cap., SEESCC, 1% outage cap., no CSI at TX

(a)

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

Sig

nal

ling

cap

acit

y [b

its/

sec/

Hz]

SNR [dB]

LCC, ergodic cap, WFLCC, ergodic cap., SEELCC, 1% outage cap., no CSI at TXSCC, ergodic cap., WFSCC, ergodic cap., SEESCC, 1% outage cap., no CSI at TX

(b)

Fig 2. Comparison between CCC(NO CSI)(NO CSI)(NO CSI)

1%outage1%outage1%outage, CCC

(WF)(WF)(WF)ergodicergodicergodic and CCC

(SEE)(SEE)(SEE)ergodicergodicergodic for low respectively

strong correlated Rayleigh fading MIMO channel; (a) T = 4T = 4T = 4, R = 2R = 2R = 2 antennas, (b)T = 8T = 8T = 8, R = 2R = 2R = 2 antennas.

49

2.2 Bit and power loading algorithm

For the sake of clarity, the bit and power loading algorithm (BPLA) is presented in thissection assuming perfect CSI knowledge at the TX. Issues related to CSI estimationerrors are discussed in Section 2.3. It is assumed that M distinct modulation and codingschemes (MCS) are available for bit loading, and bm, m = 1, . . . ,M , denotes theeffective rate (including the coding rate) provided by the mth MCS.

Assuming the optimal linear combining with matrices Uc and Vc given by (3),the power required by the ith spatial sub-channel from the cth subcarrier (denoted aseigenmode (i, c) in the sequel) to achieve a given target SNR value, γ, is given by[206, 262]

Pi,c(γ) = γN0

λi,c. (9)

The main idea behind the proposed algorithm is to exploit the intrinsic spatial di-versity achieved by using the eigenmode based transmission. The RC available eigen-modes are divided into R clusters. The first cluster contains the strongest eigenmodesfrom each subcarrier, the second cluster contains the second strongest eigenmodes, andso forth. As a result, only a small difference between the eigenmode gains belongingto the same cluster is experienced due to spatial diversity, except within the most fadedsubcarriers. Consequently, by skipping the most faded eigenmodes, the same MCScan be used for all selected eigenmodes belonging to the same cluster with a negligi-ble throughput degradation. In this way, the signaling overhead required to inform thereceiver about the selected MCS at each eigenmode is C times reduced.

The powerful turbo codes (TC) [12] are used for channel coding due their largecoding gain. The encoding is performed jointly in time and frequency domains, so thata code word covers the selected eigenmodes from a given cluster during L consecu-tive OFDM symbols belonging to one transmission frame. In this way, the codewordis enlarged to achieve TC’s interleaving gain [7], while the adaptability between theclusters is still preserved. According to the channel conditions, the resulted codewordswill have variable lengths, which makes the control of the instantaneous BER very diffi-cult. Therefore, a constant FER design will be considered because for sufficiently longframes, FER is less sensitive than BER to the variations of the codeword length [7].

The eigenmode power allocation algorithm is divided into two steps to avoid exces-sive complexity. For each transmitted frame, at the first step, the total power is dividedbetween clusters based on modified WF power allocation. At the second step, the MCS

50

and eigenmode selection is performed independently for each cluster to maximize thethroughput. This is done subject to the constraint that all selected eigenmodes withinthe cluster have equal SNR values. Such a constraint produces a major complexity re-duction in the implementation of the BPLA with a negligible throughput degradation. Ithelps to maintain the target FER by using a look-up-table containing the SNR requiredby each MCS to maintain the target FER in an AWGN channel. These SNR values areobtained off line by preliminary simulations. In the sequel, γm, m = 1, . . . ,M denotesthe SNR required by the mth MCS to achieve the target FER. Without the loss of gener-ality, it can be assumed that effective rates bm and theirs corresponding SNR values γm

are given in an ascending order, i.e., b1 < b2 < . . . < bM and γ1 < γ2 < . . . < γM .In practice, the rate allocated to an eigenmode is limited not only by the noise

power, but also by the MCS set to bM , i.e., the rate provided by the most spectrallyefficient MCS. Therefore, the power allocated to an eigenmode should also be limitedto Pi,c(γM ). Consequently, for the initial power division between the clusters, theclassical WF power allocation is modified to take into account the eigenmode’s SNRsaturation and the average SNR gap as follows

P(MWF)i,c = min

[(µ− N0

λi,cG

)+

,N0

λi,cγM

](10)

where G represents the average SNR gap [41, 45] between the Shannon capacity andthe spectral efficiency provided by the MCS set, and µ comes from the total transmittedpower constraint, i.e.,

∑Ri=1

∑Cc=1 P

(MWF)i,c = Pmax.

Figure 3 shows the graphical illustration of the relation (10). As shown in the fig-ure, the power allocated to some eigenmodes is lower than the one which would beallocated by the classical WF policy. These eigenmodes are called saturated becausethe amount of information transmitted by these eigenmodes has reached the maximumlimit bM , although the power allocated could be increased. Obviously, for an unlimitedconstellation size and optimum channel coding case (i.e., γ → ∞ and G → 1), (10)becomes equivalent to the classical WF power allocation.

Based on the initial power allocation (10), the total power Pi allocated to the ithcluster is given by

Pi =C∑

c=1

P(MWF)i,c . (11)

In the second step of the BPLA, each cluster is independently optimized by search-ing for the MCS that provides the maximum throughput when the total power Pi is dis-

51

Fig 3. The modified waterfilling power allocation.

tributed to the eigenmodes according to the target FER, under equal SNR constraints.This is equivalent to the maximization over the whole MCS set of the product betweenthe maximum number of eigenmodes that can be selected by using a certain MCS andthe spectral efficiency provided by that MCS.

Let us denote by m?i the index of the optimum MCS at cluster i, i.e.,

m?i = arg max

m=1,...,Mbmsi,m (12)

where si,m represents the maximum number of the eigenmodes that can be selected inthe ith cluster by using the mth MCS, i.e.,

si,m = maxC⊂{1,...,C}:∑c∈C Pi,c(γm)≤Pi

|C| (13)

By ordering the eigenmodes from each cluster according to their strength, the valuesof si,m can be computed by using the following algorithm.

Algorithm 2.2.1. Computation of the maximum number of eigenmodes in each cluster

for different MCSs

1. Let i = 1.

2. Organize the eigenvalues λi,c in descending order, i.e., find the vector of indices

[j1, j2, . . . , jC ] such that λi,j1 ≥ λi,j2 ≥ · · · ≥ λi,jC.

3. Compute the vector t = [t1, . . . , tC ], with each entry given by

tc =c∑

n=1

N0

λi,jn

. (14)

52

4. Define si,0 = C. From m = 1 to M do

a) Let si,m = si,m−1.

b) If tsi,m γm > Pi, decrease the number of selected eigenmodes si,m = si,m − 1until the power constraint tsi,m γm ≤ Pi is satisfied.

c) Save the corresponding value si,m.

5. If i < R, let i = i + 1 and go to Step 1. Otherwise STOP.

Using the values si,m, the optimum MCS for each cluster is found by using (12).The powers allocated to the selected eigenmodes are given by Pi,c(γm?

i).

The following remarks are important for practical implementation. In the case ofstrong correlation between antennas and/or very low SNR values, it is possible thatthe number of selected eigenmodes at the weakest cluster is very small, resulting ina very short code word. In such cases, the performance of turbo codes decreases andthe target FER cannot be maintained. This problem is avoided by setting a minimumcodeword length, lmin, and deactivating the weakest cluster when its codeword lengthis smaller than lmin. Note that, as shown in Figure 2, the capacity loss in the SEE modeis negligible for such channel conditions. The power allocated initially to the weakestcluster is reused by the stronger clusters and their throughput is increased. The γm

values stored in the look-up-table are determined by setting a codeword length equal tolmin.

2.3 The compensation of CSI errors at TX

The CSI errors model introduced in [267, 287, 289] is assumed in this section. Theestimated channel matrix at TX side is given by

Hc = Hc + Nc (15)

where Nc ∈ CR×T models the channel estimation error. It has i.i.d. elements, i.e.,[Nc]r,c∼ CN (0, σ2

N ), with σ2N known at the TX. The linear pre-combining matrix is

obtained based on the SVD of the estimated channel matrix, i.e., Hc = UcΛcVHc . The

transmitted vector at the cth subcarrier is given by

xc = ˆVcP1/2c dc (16)

where dc = [d1,c, . . . , dR,c]T is the vector of complex data symbols transmitted at

subcarrier c, the matrix ˆVc ∈ CT×R contains the first R columns of V, and the ma-trix Pc = diag (P1,c, . . . , PR,c) controls the power allocated to each eigenmode (the

53

constellation power is normalized to E{|di,c|2} = 1). By substituting (16) in (1), thereceived signal at the cth subcarrier becomes

yc = UcΛcVHc

ˆVcP1/2c dc + nc = Tcdc + nc (17)

where Tc = UcΛcVHc

ˆVcP1/2c represents the equivalent channel matrix at the subcar-

rier c. It expresses the cumulative effect of signal processing at the TX side and channelpropagation on the transmitted data signal. It can be obtained at the receiver side byinserting pilot symbols for channel estimation in lieu of dc. In the case of a large num-ber of transmit antennas, this matrix is easier to estimate than the channel matrix, Hc,due to its reduced number of elements. Furthermore, no other signaling is required toinform the receiver about the power adaptation at the TX side. The estimation of Tc isbeyond the scope of the thesis, and it will be assumed in the following that it is perfectlyknown at the receiver side [267].

Following the approach of [203], the log-likelihood of the individual bits at thesubcarrier c can be computed from the received signal yc as

LLR {[bc]l} = log

∑dc: dc={fbs(bc)|[bc]l=1}

R∏r=1

exp

−

∣∣∣∣[yc]r−R∑

t=1[Tc]r,t[dc]t

∣∣∣∣2

N0

∑dc: dc={fbs(bc)|[bc]l=0}

R∏r=1

exp

−

∣∣∣∣[yc]r−R∑

t=1[Tc]r,t[dc]t

∣∣∣∣2

N0

(18)where bc represents the binary vector associated with the complex vector dc, and fbs(·)represents the bit-to-symbol mapping function, e.g., dc = fbs(bc).

In some applications, the exhaustive search required by maximum likelihood softdemodulator (18) is too complex, especially for high level constellations, and it is prefer-able to use some suboptimal soft demodulators at each subcarrier comprising a spatiallinear equalizer followed by a soft demodulator for each eigenmode. The spatial equal-izer can be implemented, e.g., by zero forcing (ZF) or linear minimum mean squarederror (LMMSE) structure [158]. In the sequel is analyzed the case in which the lin-ear post-combiner is implemented by a ZF equalizer structure. However, as will beshown by simulations, essentially the same results are obtained with an LMMSE equal-izer. The covariance matrix of the decision variables dc generated by the linear post-

54

combiner (see Figure 1) is given by [123]

CZFdc

= N0

(TH

c Tc

)−1= N0P−1/2

c

(ˆV

H

c VcΛHc UcUH

c ΛcVHc

ˆVc

)−1

P−1/2c

= N0P−1/2c

(ΦcΛ2

cΦHc

)−1

P−1/2c (19)

where Φc = ˆVH

c Vc, the matrix Vc ∈ CT×R contains the first R columns of Vc, andΛc = diag(λ1/2

c,1 , . . . , λ1/2c,R).

Closed form expression for the statistics of Φc cannot be found [204], even inthe case of small CSI errors, due to the unseparability of subspaces correspondingto close singular values. Therefore, an approximation will be used. For channel re-alizations with close singular values the approximation is not accurate, but the prob-ability of such a case is small enough. Clearly, in the case of perfect CSI, the ma-trix Φc becomes an identity matrix. Hence, one can expect that in the case of fairlysmall CSI errors it will have a strong diagonal structure as well. This was confirmedby numerical evaluations of the empirical cumulative distribution function of the ratioζ =

(maxi 6=j

∣∣[Φc]i,j∣∣)/(

mini

∣∣[Φc]i,i∣∣) for different values of CSI errors. For ex-

ample, in the case of σ2N = 0.01, T = 8, and R = 2 antennas it has been obtained

Pr{ζ < −15dB} = 95%. By neglecting the off diagonal elements in matrix Φc, theSNR at the equalizer output for the eigenmode (i, c) can be approximated by

γi,c ≈ Pi,cλi,c

N0

∣∣∣[Φc]i,i∣∣∣2

(20)

where∣∣[Φc]i,i

∣∣2 express the SNR loss due to the CSI errors.Let us assume for a moment that λi,c can be estimated accurately enough. In such

a case, the average SNR can be maintained by increasing the instantaneous power allo-cated to the eigenmodes (i, c) with a factor χi,c ≥ 1 such that E

{∣∣[Φc]i,i∣∣2χi,c

}= 1.

With diagonal Φc, it follows from (15) that

E

{∣∣∣[Φc]i,i∣∣∣2 λi,c

λi,c

}= 1 +O

(σ2

N

λi,c

)(21)

where λi,c are the eigenvalues of HHc Hc. Therefore, the instantaneous power allocation

at the eigenmode (i, c) required to maintain a desired average SNR at the post-combinercan be approximated as

Pi,c(γ) ≈ N0

ηi,cγ (22)

55

−20 −18 −16 −14 −12 −10 −8 −6 −40

1

2

3

4

5

6

7

8

9

10

11

σN2 [dB]

Ave

rag

e S

INR

at

equ

aliz

er o

utp

ut

[db

]

strongest eigenmode target SNR = 10 dB

weakest eigenmode target SNR = 4 dB

LMMSE equal. with CSI errors compensationZF equal. with CSI errors compensationLMMSE equal. without CSI errors compensationZF equal. without CSI errors compensation

Fig 4. Average SINR at the output of the LMMSE and ZF equalizers.

where ηi,c = λ2i,c

/λi,c represents the equivalent power gain of the eigenmode (i, c) in

the presence of CSI errors at the TX.Figure 4 illustrates the accuracy of approximation (22). The average SINR obtained

at the output of the post-combiner is plotted as a function of the variance of channelestimation error σ2

N for both linear equalizer structures (ZF and LMMSE). The simu-lated system had T = 8 and R = 2 antennas with different target SNR values for thetwo clusters. Specifically, the target SNR was 10 dB for the strongest cluster and 4 dBfor the weakest one. For comparison, the obtained SINR values when the CSI errorsare neglected are also shown. The simulation results show that by using the proposedCSI error compensation method, the targeted SNR is well approximated by (22) for alarge range of channel estimation errors, i.e., σ2

N ∈ (0.01÷ 0.25). In the case in whichthe TX does not compensate for the CSI errors, a large gap between the targeted andobtained SINR values results.

It is important to remark that the power allocation (22) can be used to maintainthe average SINR only, but nothing can be said about the instantaneous SINR at eacheigenmode. This means that (22) can be used to control the average error rate onlyin conjunction with a channel encoder that covers multiple eigenmodes from different

56

subcarriers, the averaging process being realized by the decoder itself.The power allocation (22) is applied to a practical system by replacing the channel

eigenvalues λi,c with some reliable estimates λi,c. In the case of R = 2, a closed formexpression can be obtained for λi,c, and it allows to evaluate analytically the effect ofCSI errors and to design a simple algorithm which increases the estimation accuracy bypartial compensation of the bias between λi,c and λi,c. Let us denote the entries in the2× 2 Hermitian matrices HcHH

c , respectively HcHHc as follows

HcHHc =

[h1,1,c h1,2,c

h∗1,2,c h2,2,c

]; HcHH

c =

[h1,1,c h1,2,c

h∗1,2,c h2,2,c

](23)

and by solving the characteristic equations, the eigenvalues can be expressed as

λ1,2,c =12

(h1,1,c + h2,2,c ±

√(h1,1,c − h2,2,c)

2 + 4|h1,2,c|2)

=12

(Tr{HcHH

c } ±√

(h1,1,c − h2,2,c)2 + 4|h1,2,c|2

). (24)

This equation holds also for λi,c where h1,1,c, h1,2,c, h2,2,c and Hc are replaced byestimated values h1,1,c, h1,2,c, h2,2,c and Hc. The bias between λi,c and λi,c containstwo components: a linear one, which can be easily evaluated as the bias between thetraces of the matrices, and a nonlinear one given by the bias of the terms under thesquare root. Since the square root does not have a convergent Taylor series around zero,the effect of the nonlinear part of the bias cannot be evaluated and will be neglected.However, as will be shown in the simulation results, its effect is not critical for a largerange of channel estimation errors and SNR values of practical interest.

Since the channel matrix Hc and the estimation error Nc in (15) are statisticallyindependent, the linear part of the bias is given by

12E

{Tr{HcHH

c } − Tr{HcHHc }

}=

12E

{Tr{NcNH

c }}

= Tσ2N

and the estimated eigenvalues used to compute the equivalent power gains, ηi,c in (22),are given by

λi,c =(λi,c − Tσ2

N

)+

(25)

where the eigenmodes corresponding to the eigenvalues which become negative afterbias compensation are discarded from the bit and power loading algorithm.

57

2.4 Numerical examples

The performance of the proposed scheme is evaluated over a correlated Rayleigh fadingMIMO channel by using Monte Carlo computer simulations. The OFDM system wasassumed to operate at a sampling rate of 20 MHz using C = 64 subcarriers. The lengthof the OFDM symbols is 4 microseconds with a 0.8 microseconds CP. A block fadingchannel model was assumed, where the fading coefficients were kept constant over onecoded frame and independent from frame to frame. One coded frame consists of L = 16OFDM symbols.

The channel’s delay taps were considered independent of each other with the powerdelay profile specified by ETSI BRAN Channel A [68] and reproduced in Tabel 1. Foreach tap, the spatial fading correlation was generated according to the stochastic MIMOchannel model [129]. The spatial covariance matrices for the LCC are given by [129,eq.12], and the covariance matrices for the SCC are given by [129, eq.13].

The MCSs used for the adaptation process are 4QAM, 16QAM, and 64QAM, allturbo encoded [12] and punctured to rate = 1/2. By using an iterative receiver [12]with 8 iterations per frame, the SNR values required by each MCS to achieve the tar-get FER=10−2 in an AWGN channel are {γ1, γ2, γ3} = {2.61dB, 8.04dB, 12.68dB}for a minimum codeword length lmin = 1000 bits, and {γ1, γ2, γ3} = {2.76dB,

8.29dB, 12.93dB} for lmin = 500 bits, respectively.Figure 5 shows the spectral efficiency achieved by the proposed BPLA for T = 8,

R = 2 antennas, lmin = 500 bits, and perfectly known CSI. Each figure shows alsothe ergodic and outage capacities, C

(WF)ergodic and C

(NO CSI)1%outage . In addition, the HH BPLA

provides an upper bound on the spectral efficiency achievable by using the given setof MCSs. For comparison, the spectral efficiency provided by the adaptive MIMO-OFDM system [267, 287, 289] (using the same set of MCSs) is also plotted, whichin the case of perfectly known CSI reduce to Alamouti’s STBC cascaded with linearpre-combiner working in the SEE regime. In order to allow a direct comparison, therequired signaling overhead was neglected for all spectral efficiency curves. The SNRis defined as the average transmitted power per subcarrier normalized with the noisespectral density N0. In each figure, the SNR is kept constant after the saturation pointin order to have constant FER. Thus, the SNR axis values do not hold after saturation,and therefore dotted lines are used for those regions.

As Figure 5 shows, the spectral efficiency of the proposed BPLA follows closelythe upper bound given by the HH algorithm for all SNR points and outperforms the

58

Table 1. ETSI BRAN Channel A power delay profile.

Tap Number 1 2 3 4 5 6 7 8 9

Delay[ns] 0 10 20 30 40 50 60 70 80Relative Power[dB] 0.0 -0.9 -1.7 -2.6 -3.5 -4.3 -5.2 -6.1 -6.9

Tap Number 10 11 12 13 14 15 16 17 18

Delay[ns] 90 110 140 170 200 240 290 340 390Relative Power[dB] -7.8 -4.7 -7.3 -9.9 -12.5 -13.7 -18.0 -22.4 -26.7

STBC-SEE system for SNR > 1.8 dB in LCC and SNR > 2.4 dB in SCC. With a lowcomplexity SISO channel turbo-encoder and a rather small codeword length, the spec-tral efficiency achieved at low and medium SNR is significantly higher than C

(NO CSI)1%outage .

As the SNR increases, the gap to C(WF)ergodic increases due to the limited spectral efficiency

provided by the most efficient MCS (i.e., 3 b/s/Hz) which produces the eigenmode sat-uration effect described in Section 2.2.

Figure 6 shows the robustness of the proposed scheme against the CSI errors at theTX side. For each simulated case, the spectral efficiency achieved with the estimatedCSI is compared to the spectral efficiency with perfectly known CSI. It has been as-sumed that the equivalent channel matrix Tc is perfectly known at the receiver sideand the CSI error compensation method described in Section 2.3 has been used at theTX side. The resulted FER for both LMMSE and ZF equalizer structures is shown.In addition, the probability of selecting both clusters, denoted as full eigenmode ex-citation (FEE) mode, is also shown. The channel has been estimated at the TX sideby using a low complexity least square channel estimator based on an orthogonal pi-lot symbol appended at the end of the reverse link. It contains NCP = 16 (NCP isthe number of samples in the CP) pilot tones equidistantly distributed over the OFDMsymbol which are encoded using Alamouti’s STBC. The channel matrix, Hc, is esti-mated first at the subcarriers corresponding to the pilot tones and then interpolated inthe frequency domain over the remaining subcarriers. For a channel with maximumexcess delay shorter than the CP length, the resulting normalized MSE is given byσ2

N = −SNR − 10 log10

(NCP

/C

)dB. The minimum codeword length was increased

to lmin = 1000 bits in order to provide a good SNR averaging for each coded frame.The results show that the proposed scheme is robust against the CSI errors, and the

spectral efficiency degradation compared to the case of perfectly known CSI is negligi-

59

ble at low SNR values where the system operates mainly in the SEE mode. As SNRincreases, the system changes to the FEE mode and the SNR degradation due to imper-fect CSI is in the range of 1dB. The targeted FER can be maintained in the presence ofCSI errors for all SNR values of practical interest ( SNR > 2dB). However, at SNR = 0dB, a small FER degradation is encountered due to extremely high channel estimationerrors, σ2

N = 0.25. Another interesting result is that the LMMSE and ZF equalizershave essentially the same performance. This is due to the fact that the columns ofthe equivalent channel matrix are close to orthogonal. Furthermore, due to the powercontrol mechanism, the selected eigenmodes operate at high SNR values.

Finally, Figure 7 presents all spectral efficiency results obtained with estimated CSIfor different antenna setups and channel spatial correlations. As expected from thecapacity analysis in Section 2.1 (see Figure 2), the spatial correlation has small influ-ence at low SNR values where the system operates in the SEE mode, but produces asevere spectral efficiency degradation at high SNR values where the system switchesto the FEE mode. For example, the 4x2 MIMO system in a low correlated channeloutperforms the 8x2 MIMO system operating in a strong correlated channel. The sameobservation holds when we compare the 2x2 MIMO system with the 4x2 one. In theSEE mode, the proposed system can achieve the full array gain, i.e., a 3dB SNR gain isobtained when the number of transmit antennas is doubled.

60

0 2 4 6 8 10 12 14 16 18 202

2.5

3

3.5

4

4.5

5

5.5

6

SNR [dB]

Sp

ectr

al e

ffic

ien

cy [

bit

s/s/

Hz]

ergodic capacity with CSI at TX1% outage capacity with no CSI at TXspectral efficiency uper bound (HH)proposed low signalling overhead BPLASTBC−SEE (HH BPLA)

(a)

0 2 4 6 8 10 12 14 16 18 202

2.5

3

3.5

4

4.5

5

5.5

6

SNR [dB]

Sp

ectr

al e

ffic

ien

cy [

bit

s/s/

Hz]

ergodic capacity with CSI at TX1% outage capacity with no CSI at TXspectral efficiency uper bound (HH)proposed low signalling overhead BPLASTBC−SEE (HH BPLA)

(b)

Fig 5. The spectral efficiency for correlated Rayleigh fading MIMO channel withperfectly known CSI, T = 8T = 8T = 8 and R = 2R = 2R = 2 antennas; (a) LCC, (b) SCC.

61

0 2 4 6 8 10 12 14 16 18 201.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Sp

ectr

al e

ffic

ien

cy [

bit

s/se

c/H

z]

SNR [dB]

low correlatedchannel

strong correlatedchannel

perfect CSIestimated CSI

0 2 4 6 8 10 12 14 16 18 20

10−2

10−1

100

SNR [dB]

LCCSCCFER ZF equal.FER LMMSE equal.FEE probability

(a)

0 2 4 6 8 10 12 14 16 18 201.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Sp

ectr

al e

ffic

ien

cy [

bit

s/se

c/H

z]

SNR [dB]

low correlatedchannel

strong correlatedchannel

perfect CSIestimated CSI

0 2 4 6 8 10 12 14 16 18 20

10−2

10−1

100

SNR [dB]

LCCSCCFER ZF equal.FER LMMSE equal.FEE probability

(b)

Fig 6. The influence of CSI errors at the TX side; (a) T = 4T = 4T = 4 and R = 2R = 2R = 2 antennas, (b)T = 8T = 8T = 8 and R = 2R = 2R = 2 antennas.

62

0 2 4 6 8 10 12 14 16 18 201

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

SNR [dB]

Sp

ectr

al e

ffic

ien

cy [

bit

s/se

c/H

z]

low correlated channelstrong correlated channelT × R = 8 × 2 antennasT × R = 4 × 2 antennasT × R = 2 × 2 antennas

Fig 7. Spectral efficiency with estimated CSI at the TX side for correlated Rayleighfading MIMO channel for different antenna setups and channel spatial correla-tions.

63

2.5 Summary and discussion

The problem of link adaptation in point-to-point MIMO systems was considered inthis chapter. The capacity analysis suggests that large throughput gains are availableespecially in the case of unbalanced antenna systems, where the number of transmitantennas is larger than the number of receive antennas. A new bit and power loadingalgorithm for MIMO-OFDM systems was proposed . With a negligible degradation inthroughput, the proposed algorithm reduces the signaling overhead by a factor of C

compared to the universally accepted HH algorithm.The effect of fading correlation on the achieved spectral efficiency was evaluated by

simulations for different antenna setups, including (T×R) = {(2×2), (4×2), (8×2)}.By using a simple scalar channel encoder in an unbalanced antenna MIMO system, thespectral efficiency achieved at low and medium SNR values is significantly higher thanthe outage MIMO capacity with unknown CSI at the TX side.

Powerful turbo codes [12] have been used for channel coding due to their large cod-ing gain. The proposed scheme can be easily generalized to other coding schemes bymodifying the look-up-table’s entries {γ1, . . . , γM}. As a result, the spectral efficiencycurves presented in the Figures 5-7 will be shifted to the right with the difference be-tween the coding gains achieved by the chosen coding scheme and turbo codes.

The effect of CSI errors at the TX side has also been considered. A simple methodwas proposed to maintain a constant FER by increasing the instantaneous transmittedpower in such a way that the average SNR at the equalizer output is kept constant.Promising results were obtained by using a low complexity least square channel esti-mator at the TX side and a linear equalizer at the receiver. Thus, the proposed schemeconstitutes a practical solution for high data rate wireless systems.

64

3 Sum-capacity computation in MIMOdownlink channels

Using the broadcast to multiple-access channel duality [241, 245, 279], the nonconvexbroadcast channel sum-capacity computation problem can be converted into a convexand well structured sum-rate maximization for the dual multiple-access channel. Theduality result can be stated as follows. Under a sum power constraint, the originalbroadcast and the associated dual multiple-access channel have the same sum-capacity.Furthermore, there is a one-to-one correspondence between the optimum transmit co-variance matrices of the original broadcast channel and those of the dual multiple-accesschannel.

Since the sum-rate maximization for the dual multiple-access channel is a convexoptimization problem, it can be solved by using a standard interior point convex pro-gram [26]. A gradient based ascent method has been considered in [247]. However,such general algorithms are not able to exploit the particular structure of the problem,and their computational complexity does not scale well when there are large numbersof users [111]. Inspired by the particular form of the Karush-Kuhn-Tucker (KKT) opti-mality conditions, specialized iterative algorithms with lower computational complexitycan be developed. With separate power constraints for each user, the sum-rate max-imization problem can be efficiently solved by using the iterative waterfilling (IWF)algorithm [282]. Since the dual multiple-access channel has a sum power constraintacross all the transmitters rather than a set of power constraints on each transmitter, theIWF algorithm cannot be directly applied.

An extension based on the dual decomposition method was proposed in [274]. Thealgorithm consists of two nested loops: the inner loop is an IWF-like algorithm whichoptimizes the transmit covariance matrices for a fixed waterfilling level, and the outerloop finds the optimum waterfilling level by using a bisection search. Recently, a fasterconverging implementation has been proposed in [276]. The basic idea of this improvedmethod, denoted as Algorithm 2 in [276], is not to run the inner loop until convergence,but only to a certain tolerance that still guarantees the convergence of the global algo-rithm. However, this requires tracking the instantaneous error of the inner loop, and,obviously, the optimum value of the inner loop is not known during the optimizationprocess. This issue has not been addressed in [276].

65

A sum power constraint iterative waterfilling (SPC-IWF) algorithm was consideredin [111]. Since the algorithm does not always converge for more than two users, twomodifications were further proposed in [111]. The first one, referred to as Algorithm 1in [111], uses all covariance matrices computed in the previous K − 1 iterations (K isthe number of users) to construct a cyclic coordinate ascent algorithm. However thismethod slows down the convergence speed and requires a large memory of K(K − 1)covariance matrices. The second one, referred to as Algorithm 2 in [111], eliminatesthe memory requirement increase by introducing an averaging step after each iteration.However, this slows down the convergence even more.

The rest of this chapter is organized as follows. The sum-capacity computationproblem for MIMO-OFDM downlink channels is formulated in Section 3.1. A newalgorithm, named iterative waterfilling for random user pairs (IWF-RUP), is proposedin Section 3.2. Section 3.3 presents an improved version of the dual decompositionbased sum-rate maximization Algorithm 2 introduced in [276]. By using the Lagrangeduality, a tight upper bound of the instantaneous error of the inner loop is derived in Sec-tion 3.3.1. The algorithm derivation is presented in Section 3.3.2, and its performanceis analyzed in Section 3.3.3. Finally, Section 3.4 concludes the chapter.

3.1 Sum-capacity for MIMO-OFDM downlink channel

A single-cell MIMO-OFDM downlink system with K users and C subcarriers is con-sidered. The base station has T transmit antennas and user k, k = 1, . . . ,K is equippedwith Rk receive antennas. The discrete-time MIMO-OFDM downlink (or broadcast5)channel model is mathematically described as

yk,c = Hk,cxc + nk,c , c = 1, . . . , C, k = 1, . . . , K (26)

where c is the subcarrier index, Hk,c ∈ CRk×T is the channel matrix between BS andthe user k, xc ∈CT is the signal vector transmitted by the BS, yk,c ∈CRk is the signalvector received by the kth user, and nk,c ∈CRk models the thermal noise and the effectof other sources of interference6 at the kth terminal. In this chapter, it is assumed thatthe channel matrices are fixed and known to the BS and to each user.

To express the MIMO-OFDM broadcast sum-capacity, the equivalence between

5The terms "broadcast" and "downlink" will be used interchangeably in this chapter.6Typical examples are the intercell interference and other coexisting communication systems sharing thesame bandwidth.

66

OFDM tones and virtual antennas [146] is used. By stacking on top of each otherthe transmitted, received, and the noise vector from each subcarrier, i.e., yk =[yT

k,1, . . . ,yTk,C ]T, x = [xT

1, . . . ,xTC ]T, nk = [nT

k,1, . . . ,nTk,C ]T, the broadcast channel

model (26) can be rewritten in a more compact form as

yk = Hkx + nk , k = 1, . . . , K (27)

where Hk = diag{Hk,1, . . . ,Hk,C}. The covariance matrix of the interference-plus-noise signal at the kth user is denoted by Rk = E{nknH

k}, and Rk = LkLHk is its

Cholesky factorization [85]. The whitened downlink channel matrix for the kth user isdefined as Wk = L−1

k Hk, i.e., the composite channel comprising the physical down-link channel and the whitening filter. Since no information is lost in the whiteningprocess [62], the original downlink system can be equivalently described in the capac-ity sense as

yk = Wkx + nk, k = 1, . . . ,K (28)

where nk = L−1k nk and E{nknH

k} = I. Using the broadcast to multiple-access chan-nels duality [241], the dual multiple-access channel associated with (28) is defined as

y =K∑

k=1

Hkxk + n (29)

where xk represents the signal transmitted by the kth user, Hk = WHk is the uplink

channel matrix for the kth user, and the covariance matrix of the interference-plus noiseat the BS is R = E{nnH} = I. Under a power constraint E{‖x‖22} ≤ Pmax, theoriginal broadcast channel and the dual multiple-access channel have the same sum-capacity, given by [241]

Csum = log

∣∣∣∣∣K∑

k=1

HkQ?kH

Hk + I

∣∣∣∣∣ (30)

where the Hermitian positive semidefinite matrices {Q?k}K

k=1 solve the sum-rate maxi-mization problem for the dual multiple-access channel, i.e.,

maximize Rsum(Q1, . . . ,QK) ∆= log∣∣∣∑K

k=1 HkQkHHk + I

∣∣∣subject to Qk º 0, k = 1, . . . ,K∑K

k=1 tr{Qk} ≤ Pmax

. (31)

The objective of (31) represents the achievable sum-rate in the dual multiple-accesschannel (29) when the variables {Qk}K

k=1 are used as transmit covariance matrices,

67

i.e., Qk = E{xkxHk}, k = 1, . . . , K. Note that the dual multiple-access channel has

a sum power constraint across all the transmitters rather than a set of individual powerconstraints on each transmitter. The optimal broadcast covariance matrices can be com-puted directly from the {Q?

k}Kk=1, and the explicit transformation is given in [241, Sec-

tion IVB ].The KKT optimality conditions of problem (31) imply that the optimal covariance

matrices {Q?k}K

k=1 must satisfy the waterfilling conditions [111]. Specifically, the eigen-vectors of each user’s covariance matrix must be aligned to the single-user optimumsignaling direction when signals of the others users are treated as noise, and the powerallocation is obtained by waterfilling across all K users. Nevertheless, finding the co-variance matrices which satisfy these conditions still requires solving the sum-rate max-imization problem (31).

3.2 Iterative waterfilling for random user pairs algorithm

This section presents a new provably convergent algorithm for solving the sum-ratemaximization problem (31). It overcomes the previously mentioned problems of theSPC-IWF algorithms and has clearly lower computational complexity. The convergencewith probability one is proved analytically.

Inspired by the original IWF method [282], at each iteration i the SPC-IWF algo-rithm updates all the K covariance matrices according to

{Q(i+1)k }K

k=1 = arg max{Qk}K

k=1: Qkº0,

∑Kk=1 tr{Qk}≤Pmax

K∑

k=1

log∣∣∣∣HkQkHH

k+K∑

j=1,j 6=k

HjQ(i)j HH

j +I∣∣∣∣ .

(32)The maximization problem in (32) is equivalent to a classical point-to-point capacity

maximization over the K parallel non-interfering channels with a common sum-powerconstraint. Thus, the optimum set of covariances is easily computed in closed form [111,Section V]. Each user treats the other users’ signals as colored noise by aligning theeigenvectors of its covariance matrix to the single-user optimum signaling directions,and the power allocation is obtained by waterfilling across all K users to maintain aconstant waterfilling level.

The IWF-RUP algorithm proposed in this section does not impose this supple-mentary constraint at each step, but the common water-filling level is asymptotically

68

achieved after a convergence period. At each step, the IWF-RUP algorithm increasesthe sum capacity by updating just two randomly selected covariance matrices undertheir sum power constraint. Therefore, there is no power exchange between the updatedpair of covariance matrices and the other ones. This ensures that the sum power con-straint is satisfied after each update step given that the initial set of covariance matricessatisfies the sum power constraint.

The following lemma is useful in the description of the proposed IWF-RUP sum-rate maximization algorithm.

Lemma 3.2.1. Let {Q(i)k }K

k=1 be a feasible set for (31) and denote byK(i) = {k(i)1 , k

(i)2 }

a set of two different randomly selected users. Let {Qk(i)1

, Qk(i)2} be the solution of the

following optimization problem, i.e.,

{Q

k(i)1

, Qk(i)2

}= arg max

{Qk}k∈K(i) : Qkº0,

∑k∈K(i) tr{Qk}≤

∑k∈K(i) tr{Q(i)

k }

∑

k∈K(i)

log∣∣∣∣HkQkHH

k

+K∑

j=1,j 6=k

HjQ(i)j HH

j + I∣∣∣∣ . (33)

The following covariance update step:

Q(i+1)k =

Q(i)k if k /∈ K(i)

Qk if k ∈ K(i) and f0

(Q

k(i)1

, Qk(i)2

) ≥ f0

(Q

k(i)1

, Qk(i)2

)

Qk if k ∈ K(i) and f0

(Q

k(i)1

, Qk(i)2

)< f0

(Q

k(i)1

, Qk(i)2

) (34)

where

Qk =12Qk +

12Q(i)

k , k ∈ K(i) (35)

and

f0(Q1,Q2) = log∣∣∣∣Hk

(i)1

Q1HHk(i)1

+Hk(i)2

Q2HHk(i)2

+K∑

k=1,k/∈K(i)

HkQ(i)k HH

k +I∣∣∣∣ (36)

is an ascent step for the sum-rate maximization problem (31).

Proof. One should recognize that the covariance update (34) is an ascent step for thesum-rate maximization problem (31), where {Qk}k/∈K(i) are fixed to Q(i)

k , and themaximization is performed with respect to {Qk}k∈K(i) under their own sum powerconstraint,

∑k∈K(i) tr{Qk} ≤ ∑

k∈K(i) tr{Q(i)k }. The procedure is similar to the

69

proof of [111, Theorem 1] and is based on analyzing a single step of a block coordinateascent algorithm performed on the following auxiliary optimization problem:

maximize f1 (A1,B1,A2,B2)∆=

12f0 (A1,B2) +

12f0 (B1,A2)

subject to A1 º 0, A2 º 0tr{A1}+ tr{A2} ≤

∑k∈K(i) tr{Q(i)

k }B1 º 0, B2 º 0tr{B1}+ tr{B2} ≤

∑k∈K(i) tr{Q(i)

k }

(37)

with separable constraints on {A1,A2} and {B1,B2}. Starting from the feasible pointA1 = B1 = Q(i)

k(i)1

, A2 = B2 = Q(i)

k(i)2

, a block coordinate ascent algorithm applied to

(37) updates the variables {A1,A2} according to

{Anew1 ,Anew

2 }= arg max

A1º0, A2º0,

tr{A1+A2}≤∑

k∈K(i) tr{Q(i)k }

f1

(A1,Q

(i)

k(i)1

,A2,Q(i)

k(i)2

)(38)

= arg maxA1º0, A2º0,

tr{A1+A2}≤∑

k∈K(i) tr{Q(i)k }

12f0

(A1,Q

(i)

k(i)2

)+

12f0

(Q(i)

k(i)1

,A2

)

(39)

By substituting (36) in (39) it is easy to observe that the optimization problems (33)and (39) are equivalent7, i.e.,

Anew1 = Q

k(i)1

, Anew2 = Q

k(i)2

(40)

From (36) and (34) it follows that the variation of the objective function in problem(31) at the ith covariance update step can be expressed as

∆R(i)sum

∆= Rsum

(Q(i+1)

1 , . . . ,Q(i+1)K

)−Rsum

(Q(i)

1 , . . . ,Q(i)K

)

= f0

(Q(i+1)

k(i)1

,Q(i+1)

k(i)2

)− f0

(Q(i)

k(i)1

,Q(i)

k(i)2

)(41)

7Note that the additional multiplication by 12

in (39) does not affect the solution.

70

and it can be bounded as follows

∆R(i)sum ≥ f0

(Q

k(i)1

, Qk(i)2

)− f0

(Q(i)

k(i)1

,Q(i)

k(i)2

)(42)

= f0

(12Q

k(i)1

+12Q(i)

k(i)1

,12Q

k(i)2

+12Q(i)

k(i)2

)− f0

(Q(i)

k(i)1

,Q(i)

k(i)2

)

≥ 12f0

(Q

k(i)1

,Q(i)

k(i)2

)+

12f0

(Q(i)

k(i)1

, Qk(i)2

)− f0

(Q(i)

k(i)1

,Q(i)

k(i)2

)(43)

= f1

(Q

k(i)1

,Q(i)

k(i)1

, Qk(i)2

,Q(i)

k(i)2

)− f1

(Q(i)

k(i)1

,Q(i)

k(i)1

,Q(i)

k(i)2

,Q(i)

k(i)2

)(44)

≥ 0 (45)

where the first inequality follows from (34), the second inequality follows from theconcavity [26] of the function f0(Q1,Q2), and the last one follows by substituting (40)in (38).

The proof is completed by observing that for any {Qk(i)1

, Qk(i)2} given by (33),

∑

k∈K(i)

tr{Qk} =12

∑

k∈K(i)

tr{Qk}+12

∑

k∈K(i)

tr{Q(i)k } ≤

∑

k∈K(i)

tr{Q(i)k } (46)

thus, Q(i+1)k given by (34) is feasible for the sum-rate maximization problem (31).

For conciseness, it is useful to denote the covariance update steps described by (32)and (34) as {Q(i+1)

k }Kk=1 = u1

({Q(i)k }K

k=1

)and {Q(i+1)

k }Kk=1 = u2

({Q(i)k }K

k=1

), re-

spectively. Lemma 3.2.1 suggests that the optimum set of covariance matrices {Q?k}K

k=1

for the sum-rate maximization problem (31) can be found by using the following itera-tive algorithm:

Algorithm 3.2.1. Iterative waterfilling for random user pairs (IWF–RUP)

1. Initialize {Q(0)k }K

k=1 = 0 and compute {Q(1)k }K

k=1 = u1

({Q(0)k }K

k=1

)to obtain a

initial feasible set.

2. Repeat {Q(i+1)k }K

k=1 = u2

({Q(i)k }K

k=1

)for randomly generated user pairs K(i) un-

til the sum-capacity converges.

The following remarks provide some insight into the proposed IWF-RUP algorithm.Since the sum power constraint in (32) is always satisfied with equality [111], it followsthat

∑Kk=1 tr{Q(1)

k } = Pmax. Similar to the SPC-IWF algorithm, the maximizationin (33) is also equivalent to the classical point-to-point capacity maximization over thetwo parallel non-interfering channels corresponding to the randomly selected user pair

71

K(i). Therefore, the optimum covariance matrices can be easily computed in closedform as follows. For each user from the selected pair K(i), the algorithm treats theother users’ signals as noise by aligning the eigenvectors of the covariance matrices tothe single-user optimum signaling directions, i.e.,

Qk = VkPkVHk , k ∈ K(i) (47)

where the columns of Vk contains the eigenvectors of the positive semidefinite matrixG(i)

k∆= HH

k

( ∑Kj=1,j 6=k HjQ

(i)j HH

j + I)−1

H, see e.g., [111, Section IV ]. Pk is a di-agonal matrix which controls the power allocation. It is obtained by waterfilling acrossthe selected users k ∈ K(i), i.e.,

[Pk]j,j =

(µ− 1

λj

(G(i)

k

))+

(48)

and their common waterfilling level µ is determined by their own sum power constraint,i.e.,

∑k∈K(i) tr{Pk} ≤

∑k∈K(i) tr{Q(i)

k }. Thus, there is no power exchange betweenthe updated pair of covariance matrices and the others. Consequently, the second stepof Algorithm 3.2.1 does not change the sum power, i.e.,

∑Kk=1 tr{Q(i)

k } = Pmax forany i.

The last line of the covariance matrices update step (34) ensures that the objec-tive function of (31) is at least as large as f0(Qk

(i)1

, Qk(i)2

) after ith iteration, which isenough to prove the following convergence theorem.

Theorem 3.2.2. Algorithm 3.2.1 (IWF–RUP) converges with probability one to an op-

timal set of covariance matrices8 for the sum-rate maximization problem (31), i.e.,

Pr{{Q(i)

k }Kk=1

i→∞−→ {Q?k}K

k=1

}= 1 . (49)

Proof. The objective of (31) is monotonically increased at each step, therefore, the totalincrease after N steps is given by

∆Rsum(N) ∆= Rsum

(Q(N+1)

1 , . . . ,Q(N+1)K

)−Rsum

(Q(1)

1 , . . . ,Q(1)K

)(50)

=N∑

i=1

∆R(i)sum (51)

8Note that the sum-capacity is unique, but the optimal set of covariance matrices that maximizes (31) maynot be unique.

72

and based on (44), it can be bounded as

∆Rsum(N) ≥N∑

i=1

∆f(i)1 (52)

where

∆f(i)1

∆= f1

(Q

k(i)1

,Q(i)

k(i)1

, Qk(i)2

,Q(i)

k(i)2

)− f1

(Q(i)

k(i)1

,Q(i)

k(i)1

,Q(i)

k(i)2

,Q(i)

k(i)2

)(53)

Let N → ∞, and since the objective of (31) is bounded, the series with positiveterms

∑Ni=1 ∆f

(i)1 must be convergent, thus, for any ε > 0 there exists an index t so

that ∆f(i)1 < ε for any i > t. Therefore, the algorithm converges to a set of covariance

matrices that solves optimization problem (38), and, therefore satisfy its KKT optimal-ity conditions.

Since the users are randomly selected with uniform probability, the resulted se-quence of user pairs, {K(i)}i≥1, is typical with probability one [62]. Thus, the KKTconditions of (38) are satisfied with probability one for all distinct user pairs. Now itcan be observed that this set also maximizes the sum capacity (31). This follows fromthe fact that the KKT conditions for the optimization problem (38) taken for all distinctuser pairs are identical with the KKT conditions of (31). Since (31) is a convex problem,the KKT conditions are sufficient for optimality, and the proof is complete.

Regarding an OFDM system, the following remarks are made. To preserve thegenerality in the algorithm description, it was considered that covariance matrix of nk

can have any Hermitian positive definite structure. However, if nk is uncorrelated in thefrequency domain, i.e., Rk = diag{Rk,1, . . . ,Rk,C} where Rk,c = E{nk,cnH

k,c}, thenWk has a block diagonal structure, and the optimal covariance matrices have a blockdiagonal structure as well, i.e., Qk = diag{Qk,1, . . . ,Qk,C}with Qk,c = E{xk,cxH

k,c},for k = 1, . . . ,K. Therefore, at each iteration, the waterfilling type optimization isapplied just over the T ×Rk dimensional matrices.

Clearly, if nk contains also a frequency structured interference component in addi-tion to thermal noise, the composite channel matrix has no block diagonal structure dueto the whitening filter and the capacity achieving signaling requires inducing of correla-tion between subcarriers (i.e., cooperative subcarrier transmission) and joint frequency-space processing at the receiver side.

73

3.2.1 Numerical examples

It is shown in [111] that the family of SPC-IWF algorithms proposed therein outper-form the previously proposed broadcast sum capacity maximization methods [247, 274].Therefore, in this section, the performance of the proposed IWF-RUP algorithm is com-pared to the algorithms in [111]. Note that all K covariance matrices are updated in eachiteration in [111], while the proposed algorithm updates just two randomly selected co-variance matrices per iteration, and the computational complexity per covariance updateis the same for all the considered algorithms. Therefore, for a fair comparison betweenthe different algorithms, the dependence of mean normalized sum capacity deviation

∆Csum(i) = E

Csum −Rsum

(Q(i)

1 , . . . ,Q(i)K

)

Csum

(54)

versus the number of covariance updates is chosen as the performance metric. Theexpectation in (54) is taken with respect to channel realization.

In [111], two hybrid algorithms have also been developed. They use the originalSPC-IWF algorithm for the first few iterations to generate a good starting point for themodified algorithms, and then switch to [111, Alg.1] or [111, Alg.2] to ensure the con-vergence. For each simulated scenario, the convergence behaviors of the original SPC-IWF and the hybrid SPC-IWF/[111, Alg.1] are also presented. As proposed in [111],the hybrid algorithm uses the original SPC-IWF algorithm for the first five iterations.

Figure 8 shows the convergence behavior of the considered algorithms for a ten userchannel with four transmit and four receive antennas and SNR = 10 dB. In Figure 8(a),an OFDM system with C = 64 subcarriers and uniform power delay profile with 16channel taps is considered, while Figure 8(b) corresponds to a single carrier frequency-flat channel. The results show that the proposed IWF-RUP algorithm converges fasterthan all algorithms developed in [111], and the difference is more accentuated as thenumber of subcarriers increases. Specifically, in the case of 64 subcarriers (Figure 8(a)),the proposed IWF-RUP algorithm requires just less than 150 covariance updates toachieve an average normalized sum capacity error of 10−4, whilst [111, Alg.2] requiresmore than 1000 covariance updates, and [111, Alg.1] requires about 600 covarianceupdates for the same accuracy. Note that [111, Alg.1] requires also a K − 1 timesthe memory of the IWF-RUP algorithm. The hybrid SPC-IWF/[111, Alg.1] performsslightly better than the proposed IWF-RUP in the case of a frequency-flat channel ifthe tolerable normalized sum capacity deviation is larger than 10−3. Nevertheless, for

74

higher accuracy the proposed IWF-RUP algorithm converges the fastest.Figure 9 uses the same channel setup as that in Figure 8(a), i.e., C = 64 subcarriers

and T × Rk = 4 × 4 antennas, but the number of users has been increased to K =30. The SNR was 10 dB in Figure 9(a) and 20 dB in Figure 9(b). As the simulationresults show, the proposed IWF-RUP algorithm converges significantly faster than thealgorithms proposed in [111], and the difference is more accentuated as the number ofusers increases. Since hybrid SPC-IWF/[111, Alg.1] performs first a fixed number ofSPC-IWF type iterations, it can show even worse convergence than [111, Alg.1] if itdoes not switch to [111, Alg.1] before SPC-IWF starts to diverge. This effect can beobserved in both Figures 9(a) and 9(b). By comparing Figures 9(a) and 9(b), it canbe seen that the SNR has a negligible effect on the speed of convergence for all theconsidered algorithms.

Figure 10 illustrates the influence of the antenna setup on the convergence behavior.It is considered a ten user channel with Rk = 2 antennas, C = 64 subcarriers, uniformpower delay profile with 16 channel taps and SNR = 10 dB. In Figure 10(a) there areT = 4 transmit antennas, while Figure 8(b) corresponds to T = 8 transmit antennas.The results show that the proposed IWF-RUP algorithm has the fastest convergence andthe difference increases as T increases, e.g., it is four times faster than [111, Alg.1] andeight times faster than [111, Alg.2] for a normalized sum capacity deviation of 10−6 inFigure 10(b).

75

0 200 400 600 800 1000 120010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

# Covariance updates

Ave

rage

nor

mal

ized

sum

−cap

acity

dev

iatio

n

original SPC−IWFJindal et al, Alg.1Jindal et al, Alg.2Jindal et al, hybrid SPC−IWF/Alg.1proposed IWF−RUP

(a)

0 200 400 600 800 1000 120010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

# Covariance updates

Ave

rage

nor

mal

ized

sum

−cap

acity

dev

iatio

n

original SPC−IWFJindal et al, Alg.1Jindal et al, Alg.2Jindal et al, hybrid SPC−IWF/Alg.1proposed IWF−RUP

(b)

Fig 8. Average normalized sum capacity deviation vs. number of covariance up-dates for K = 10K = 10K = 10 users, T ×Rk = 4× 4T ×Rk = 4× 4T ×Rk = 4× 4 antennas and SNR = 10 dB; (a) C = 64C = 64C = 64

subcarriers, (b) C = 1C = 1C = 1 carrier (frequency-flat channel). "Jindal et al" stands forreference [111].

76

0 200 400 600 800 1000 1200

10−4

10−3

10−2

10−1

100

# Covariance updates

Ave

rage

nor

mal

ized

sum

−cap

acity

dev

iatio

n

original SPC−IWFJindal et al, Alg.1Jindal et al, Alg.2Jindal et al, hybrid SPC−IWF/Alg.1proposed IWF−RUP

(a)

0 200 400 600 800 1000 1200

10−4

10−3

10−2

10−1

100

# Covariance updates

Ave

rage

nor

mal

ized

sum

−cap

acity

dev

iatio

n

original SPC−IWFJindal et al, Alg.1Jindal et al, Alg.2Jindal et al, hybrid SPC−IWF/Alg.1proposed IWF−RUP

(b)

Fig 9. Average normalized sum capacity deviation vs. number of covariance up-dates for K = 30K = 30K = 30 users, T ×Rk = 4× 4T ×Rk = 4× 4T ×Rk = 4× 4 antennas, C = 64C = 64C = 64 subcarriers; (a) SNR = 10dB, (b) SNR = 20 dB. "Jindal et al" stands for reference [111].

77

0 200 400 600 800 1000 120010

−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

# Covariance updates

Ave

rage

nor

mal

ized

sum

−cap

acity

dev

iatio

n

original SPC−IWFJindal et al, Alg.1Jindal et al, Alg.2Jindal et al, hybrid SPC−IWF/Alg.1proposed IWF−RUP

(a)

0 200 400 600 800 1000 120010

−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

# Covariance updates

Ave

rage

nor

mal

ized

sum

−cap

acity

dev

iatio

n

original SPC−IWFJindal et al, Alg.1Jindal et al, Alg.2Jindal et al, hybrid SPC−IWF/Alg.1proposed IWF−RUP

(b)

Fig 10. Average normalized sum capacity deviation vs. number of covarianceupdates for K = 10K = 10K = 10 users, SNR = 10 dB and C = 64C = 64C = 64 subcarriers; (a) T ×Rk = 4× 2T ×Rk = 4× 2T ×Rk = 4× 2

antennas, (b) T ×Rk = 8× 2T ×Rk = 8× 2T ×Rk = 8× 2 antennas. "Jindal et al" stands for reference [111].

78

3.3 Sum-rate maximization via dual decompositionmethod

In general, when the main difficulty encountered in solving an optimization problem iscaused by a small set of constraints, called complicating constraints, the dual decompo-sition method can be used to break the original problem into a subproblem and a masterproblem [13, 24]. The subproblem is obtained by dropping out the complicating con-straints and replacing the original objective by the partial Lagrangian associated withthem. The master problem is the dual problem associated with the partial Lagrangian,and it is usually solved by using a subgradient method [13, 24]. At each iteration asubgradient is computed by solving the subproblem and the master problem adjusts thedual variables accordingly. In this section, it is shown how the above method can beapplied to solving the sum-rate maximization problem (31). Detailed treatments of thegeneral dual decomposition method can be found in [13, 24].

To reveal the complicating constraints of the sum-rate maximization problem (31),it is first reformulated in the following equivalent form

maximize log∣∣∣∑K

k=1 HkQkHHk + I

∣∣∣subject to Qk º 0, k = 1, . . . , K

tr{Qk} ≤ pk, k = 1, . . . , K∑Kk=1 pk ≤ Pmax

(55)

where {pk}Kk=1 are auxiliary variables introduced to decouple the sum power constraint.

Note that without the last constraint, problem (55) would be equivalent to the sum-ratemaximization problem for the MIMO multiple-access channel with individual powerconstraints pk on each user [282], and it could be solved by using the standard IWFalgorithm. Therefore, the last constraint is a complicating constraint.

By following the approach of [274], problem (55) can be solved via dual decompo-sition by iteratively solving the following subproblem (i.e., the inner loop)

maximize log∣∣∣∑K

k=1 HkSkHHk + I

∣∣∣− λ(∑K

k=1 pk − Pmax

)

subject to Sk º 0, k = 1, . . . , K

tr{Sk} ≤ pk, k = 1, . . . , K

(56)

with the variables {Sk}Kk=1, {pk}K

k=1 for different fixed values of λ. The value of λ canbe interpreted as the inverse of the waterfilling level, and it is the optimization variable

79

of the master problem (i.e., the outer loop) [274]

minimize g(λ)subject to λ ≥ 0

(57)

where g(λ) denotes the optimal value of subproblem (56) for a given λ, i.e.,

g(λ) = log

∣∣∣∣∣K∑

k=1

HkS?kH

Hk + I

∣∣∣∣∣− λ

(K∑

k=1

p?k − Pmax

)(58)

where {S?k}K

k=1, {p?k}K

k=1 is the solution of subproblem (56). The optimal value of themaster problem (57) is the sum-capacity, i.e., Csum = g(λ?). Furthermore, when λ insubproblem (56) is equal to λ?, the solution satisfies

∑Kk=1 p?

k = Pmax and {S?k}K

k=1 isthe solution of the sum-rate maximization problem (31), i.e., Q?

k = S?k, k = 1, . . . ,K.

Subproblem (56) is efficiently solved by using an IWF-like algorithm, and masterproblem (57) finds the optimum waterfilling level µ? = 1/λ? by using a bisectionsearch [274]. Recently, it has been shown in [276] that the dual decomposition methodcan be sped up by partially solving subproblem (56) to a certain tolerance level ε, whichstill guarantees the convergence of the global algorithm. The choice of ε has beendiscussed in [276], but the issue of tracking the instantaneous error of subproblem (56)has not been addressed. In the sequel, an upper bound of the instantaneous error willbe derived, and it will then be used to construct a non-heuristic stopping criterion forsubproblem (56).

3.3.1 Derivation of the error upper bound

In this section, the Lagrange dual function of subproblem (56) is derived and a methodto obtain a feasible dual variable is provided. Then, the duality gap theory [26] is usedto provide an upper bound of the instantaneous error. Finally, it is shown that the upperbound is attained at an optimal point.

In order to derive the Lagrange dual function, subproblem (56) is first reformulatedin the following equivalent form

minimize − log |W|+ λ∑K

k=1 pk − λPmax

subject to W ¹ ∑Kk=1 HkSkHH

k + I

Sk º 0, k = 1, . . . , K

tr{Sk} ≤ pk, k = 1, . . . , K

(59)

80

where W is an auxiliary Hermitian positive semidefinite matrix variable. The equiv-alence between problems (56) and (59) follows from the matrix-monotonicity of thelog | · | function [26], which implies that at the optimal point of (59) the first constraintholds with equality (see Appendix 1 for a formal proof).

The Lagrangian of problem (59) can be expressed as

L(W, {Sk}K

k=1, {pk}Kk=1,Γ, {Ψk}K

k=1, {νk}Kk=1

)

= − log |W|+ λK∑

k=1

pk − λPmax + tr

[Γ

(W −

K∑

k=1

HkSkHHk − I

)]

−K∑

k=1

tr(ΨkSk) +K∑

k=1

νk[tr(Sk)− pk]

= − log |W|+ tr(ΓW)− tr(Γ) +K∑

k=1

(λ− νk)pk

+K∑

k=1

tr[(

νkI− HHkΓHk −Ψk

)Sk

]− λPmax (60)

where Γ º 0,Ψk º 0, νk ≥ 0 are the dual variables associated with the inequalityconstraints. The second expression in (60) was obtained by using the trace property,tr(ΓHkSkHH

k ) = tr(HH

kΓHkSk

).

The Lagrange dual function of problem (59) is given by

l(Γ,{Ψk}K

k=1, {νk}Kk=1

)

= infW,{Sk}K

k=1,{pk}Kk=1

L(W, {Sk}K

k=1, {pk}Kk=1,Γ, {Ψk}K

k=1, {νk}Kk=1

). (61)

The Lagrangian is a convex function of the primal variables. Thus, for any dualfeasible variables9, the Lagrangian’s gradient vanishes at the optimum point of (61),i.e., ∇WL = 0, ∇Sk

L = 0, ∇pkL = 0. This implies that

Γ = W−1

HHkΓHk = νkI−Ψk, k = 1, . . . , K

νk = λ, k = 1, . . . , K .

(62)

Therefore, for any Γ º 0,Ψk º 0, νk ≥ 0, the Lagrange dual function (61) can be

9The set of dual feasible variables is defined as{Γ º 0, {Ψk º 0}K

k=1, {νk ≥ 0}Kk=1

∣∣ l(Γ, {Ψk}K

k=1,

{νk}Kk=1

)> −∞}

, i.e., it contains all the dual variables for which the Lagrangian is bounded from below.

81

expressed as

l(Γ,{Ψk}K

k=1, {νk}Kk=1

)

=

log |Γ| − tr(Γ) + T − λPmax if HHkΓHk = λI−Ψk,

k = 1, . . . ,K

−∞ otherwise

(63)

and by eliminating the slack variables Ψk º 0, the Lagrange dual problem of (59) canbe formulated as

maximize log |Γ| − tr(Γ) + T − λPmax

subject to HHkΓHk ¹ λI, k = 1, . . . , K

Γ º 0 .

(64)

By denoting the objective function of subproblem (56) as fsub({Sk}K

k=1, {pk}Kk=1

),

the instantaneous error of the inner loop can be expressed as

e({Sk}K

k=1, {pk}Kk=1

)= g(λ)− fsub

({Sk}Kk=1, {pk}K

k=1

). (65)

Problem (59) is equivalent to minimizing the objective function −fsub({Sk}K

k=1,

{pk}Kk=1

)under the constraints of problem (56). Thus, its optimal value is −g(λ). A

basic result of the duality gap theory [26] is that for any dual feasible variables, theobjective of dual problem (64) is a lower bound of the optimal value of primal problem(59), i.e.,

−g(λ) ≥ log |Γ| − tr(Γ) + T − λPmax (66)

for any Γ º 0 that satisfies HHkΓHk ¹ λI for k = 1, . . . ,K. Therefore, the error (65)

can be bounded as

e({Sk}K

k=1,{pk}Kk=1

)

≤ − log |Γ|+ tr(Γ)− T + λPmax − fsub({Sk}K

k=1, {pk}Kk=1

)

= emax

(Γ, {Sk}K

k=1, {pk}Kk=1

). (67)

In order to construct a stopping criterion for subproblem (56) based on the upperbound (67), a systematic way to construct a dual feasible variable Γ for any given set ofprimal variables {Sk}K

k=1 is required. Such a procedure is described by the followinglemma.

82

Lemma 3.3.1. Let {Sk}Kk=1 be a set of primal variables of problem (59). The following

Hermitian positive semidefinite matrix

Γ({Sk}K

k=1

)=

λ

λ({Sk}K

k=1

)(

K∑

k=1

HkSkHHk + I

)−1

(68)

where

λ({Sk}K

k=1

)= max

k=1,...,Kλmax

HH

k

(K∑

k=1

HkSkHHk + I

)−1

Hk

(69)

is a dual feasible variable.

Proof. From (69) it follows that

HHk

(K∑

k=1

HkSkHHk + I

)−1

Hk ¹ λ({Sk}K

k=1

)I, k = 1, . . . , K (70)

which further implies that

HHk Γ

({Sk}Kk=1

)Hk ¹ λI, k = 1, . . . , K (71)

Thus, Γ({Sk}K

k=1

)given by (68) satisfies the constraints of the Lagrange dual

problem (64) and, therefore, it is a dual feasible variable.

By substituting (68) and the expression of fsub({Sk}K

k=1, {pk}Kk=1

)in (67), the

upper bound of the instantaneous error of the inner loop can be expressed as

emax

[Γ

({Sk}Kk=1

), {Sk}K

k=1, {pk}Kk=1

]

= − log∣∣Γ({Sk}K

k=1

)∣∣ + tr[Γ

({Sk}Kk=1

)]− T

− log

∣∣∣∣∣K∑

k=1

HkSkHHk + I

∣∣∣∣∣ + λK∑

k=1

pk

= T

[log

(λ({Sk}K

k=1

)

λ

)− 1

]+ λ

K∑

k=1

pk + tr[Γ

({Sk}Kk=1

)]

(72)

Note that the problem (59) satisfies Slater’s condition [26, Section 5.2.3], i.e., itis strictly feasible. Thus, strong duality holds between the convex problems (59) and(64). Furthermore, in Appendix 1 it is shown that for {Sk}K

k=1 = {S?k}K

k=1 the dual

83

variable defined in (68) becomes optimal for dual problem (64). Therefore, the errorupper bound (67) is attained at the optimum point of (59), i.e.,

e({S?

k}Kk=1, {p?

k}Kk=1

)= emax

[Γ

({S?k}K

k=1

), {S?

k}Kk=1, {p?

k}Kk=1

]= 0 (73)

This result ensures that the error upper bound (67) can be used as a stopping crite-rion for subproblem (56) for any desired accuracy ε > 0.

3.3.2 Algorithm derivation

The algorithm described in this section differs from the one presented in [276, SectionIIB] in two aspects: a covariance scaling step is introduced between two successiveouter loops, and the stopping criterion of the inner loop is based on the error upperbound derived in Section 3.3.1. In the sequel, these two aspects are described in detail.

The optimum set of covariance matrices for the dual multiple-access channel sat-isfies the sum power constraint of problem (31) with equality [111]. However, thecovariance matrices obtained by solving subproblem (56) do not necessarily satisfythis condition. In fact, it can be shown that the solution of subproblem (56) satisfies∑K

k=1 tr{S?k} = Pmax if and only if λ is the solution of the master problem (57), i.e.,

λ = λ?. Inspired by this observation, a covariance scaling step is introduced betweentwo successive outer iterations. Specifically, the covariance matrices obtained by solv-ing subproblem (56) are multiplied by Pmax

/∑Kk=1 tr{Sk} so that the sum power con-

straint is satisfied with equality. Then, the resulting scaled covariance matrices are usedas a starting point in the next inner loop optimization. Note that the convergence of theglobal algorithm is still guaranteed, since the convergence conditions do not depend onthe specific initialization of the inner problem [276]. As will be shown in Section 3.3.3,this covariance scaling greatly improves the rate of convergence of the global algorithm.

In the following, it is shown how the error upper bound derived in Section 3.3.1 canbe used to construct a stopping criterion for the inner loop. The instantaneous error ofsubproblem (56) can be expressed as

e({Sk}K

k=1, {pk}Kk=1

)= e1

({Sk}Kk=1

)+ λe2

({pk}Kk=1

)(74)

where

e1

({Sk}Kk=1

)= log

∣∣∣∣∣K∑

k=1

HkS?kH

Hk + I

∣∣∣∣∣− log

∣∣∣∣∣K∑

k=1

HkSkHHk + I

∣∣∣∣∣ (75)

84

and

e2

({pk}Kk=1

)=

K∑

k=1

pk −K∑

k=1

p?k . (76)

The main idea of [276, Algorithm 2] is to stop the IWF-like algorithm in the innerloop when

∣∣e2

({pk}Kk=1

)∣∣ ≤ ε′, where ε′ is a certain tolerance level that still guaran-tees the algorithm convergence. However, since the exact value of e2

({pk}Kk=1

)cannot

be computed without solving the inner problem, an approximation is always needed totest the stopping criterion. In [276], the error e2

({pk}Kk=1

)was approximated by the

difference between the resulting sum power∑K

k=1 pk in two successive covariance up-dates [275]. However, a better approximation can be obtained as follows. Via extensivenumerical examples, it was observed that the upper bound (72) is not only larger thanthe total error e

({Sk}Kk=1, {pk}K

k=1

), but it is also larger than the second term of the

instantaneous error (74), i.e.,

λ∣∣e2

({pk}Kk=1

)∣∣ ≤ emax

[Γ

({Sk}Kk=1

), {Sk}K

k=1, {pk}Kk=1

]. (77)

Therefore, the condition emax

[Γ

({Sk}Kk=1

), {Sk}K

k=1, {pk}Kk=1

]/λ < ε′ can be used

as a stopping criterion for the inner loop.Based on the above discussions, the modified version of [276, Algorithm 2] can be

summarized as follows.

Algorithm 3.3.1. Sum-rate maximization via dual decomposition

1. Initialize {Sk = 0, pk = 0}Kk=1, µmin = 0, and µmax = Pmax + 1/λmax(HH

1 H1).2. Let µ = (µmin + µmax)/2.

3. Set ε′ = (µmax − µmin) · υ · rank(H)/4, where H = [H1, . . . , HK ].4. Solve for {Sk, pk}, k = 1, . . . ,K in subproblem (56) with λ = 1/µ, by iteratively

optimizing each of {Sk, pk} using [276, (19)-(23)] while keeping all other {Sj , pj},

j 6= k fixed. The iterations cycle through k = 1, . . . ,K, 1, . . . , K, . . . until

emax

[Γ

({Sk}Kk=1

), {Sk}K

k=1, {pk}Kk=1

]/λ < ε′ .

5. If∑K

k=1 pk > Pmax, then set

µmax = µmax − (µmax − µmin) · (0.5− υ) .

else set

µmin = µmin + (µmax − µmin) · (0.5− υ) .

85

6. Set Sk = Sk · Pmax

/∑Kk=1 tr{Sk}.

7. If µmax − µmin ≤ εµ, STOP. Othewise, go to Step 2.

Step 4 is the inner loop which iteratively updates variables {Sk, pk} of subproblem(56) by using a modified IWF algorithm, where the waterfilling level is fixed to µ. Step 3selects the required tolerance of subproblem (56) as described in [276, Section II.B].The outer loop, comprising steps 2 and 5, searches for the optimum waterfilling level byusing a modified bisection search. Note that the search domain [µmin, µmax] is reducedby a factor of (0.5 − υ) at each iteration. The value of υ can be selected within theinterval [0, 0.5], and a typical value is υ = 0.1. The covariance scaling is performed atstep 6. The value of εµ represents the tolerance of the final waterfilling level.

3.3.3 Numerical examples

For the numerical results, the same simulation setup as in [276] was considered, i.e.,a frequency-flat channel model, where each entry of the users’ channel matrices is ani.i.d. complex Gaussian random variable with zero mean and unitary variance. Thetransmitter has T = 10 antennas and each user terminal is equipped with Rk = 2antennas. The sum power constraint is Pmax = 10.

Figure 11 shows the convergence behavior for different variants of the dual decom-position (DD) based sum-rate maximization algorithm. In [276, Alg.2], there is nocovariance scaling (CS) step, and, for testing the stopping criterion of the inner loop,the sum power error e2

({pk}Kk=1

)is approximated by the difference between the result-

ing sum power in two successive covariance updates. To reveal a deeper insight, somemodified versions were also considered. The first one corresponds to a hypotheticalimplementation where the true sum power error could be computed and is used to testthe stopping criterion. Clearly, this requires the optimal value of (56) to be known inadvance, which is impossible in practice. The second one approximates e2

({pk}Kk=1

)

for testing the stopping criterion, but employs the CS step. The convergence behaviorof the proposed Algorithm 4.2.1 described in Section 3.3.2 is also presented. The re-sults clearly show that the proposed covariance scaling step greatly improves the rateof convergence, and the proposed stopping criterion is superior to the one used in [276].Moreover, when the covariance scaling is used, the performance degradation due to theusage of the upper bound instead of the true sum power error becomes negligible.

Figure 12 compares the convergence behaviors of the proposed IWF-RUP Algo-

86

rithm 3.2.1 and DD based Algorithm 4.2.1 to those of the modified SPC-IWF [111,Alg.1] and DD based [276, Alg.2]. The results show that for a moderate number ofusers, e.g., K = 10 in Figure 12(a), the IWF-RUP algorithm clearly outperforms thepreviously proposed [111, Alg.1] and [276, Alg.2]. Furthermore, despite its simplic-ity, the proposed IWF-RUP algorithm converges almost as fast as the DD based Al-gorithm 4.2.1. Therefore, it can be concluded that for a moderate number of users itprovides the best tradeoff between complexity and speed of convergence. As the num-ber of users increases, e.g., K = 50 in Figure 12(b), the DD based Algorithm 4.2.1provides substantial performance improvement over the others algorithms. However, itmust be noted that this comes at the expense of increased complexity of the inner loop,since the complexity of finding a feasible dual variable increases also linearly with K.Remarkably, the IWF-RUP algorithm has a convergence speed similar to that of [276,Alg.2] even in the case of a large number of users. Furthermore, it clearly outperforms[111, Alg.1] in all simulated conditions. This is intuitively expected, since the pro-posed IWF-RUP does not have a memory in the covariance updating process, whilst[111, Alg.1] has a memory of K − 1 previous steps.

87

0 200 400 600 800 1000 120010

−15

10−10

10−5

100

# Covariance updates

Nor

mal

ized

sum

−cap

acity

dev

iatio

n

Yu, Alg.2Yu, Alg.2 w known sum power errorYu, Alg.2 w CSproposed DD (CS and new stopping crit.)DD w known sum power error and CS

(a)

0 1000 2000 3000 4000 5000 600010

−15

10−10

10−5

100

# Covariance updates

Nor

mal

ized

sum

−cap

acity

dev

iatio

n

Yu, Alg.2Yu, Alg.2 w known sum power errorYu, Alg.2 w CSproposed DD (CS and new stopping crit.)DD w known sum power error and CS

(b)

Fig 11. Different dual decomposition based sum-rate maximization algorithms:normalized sum-capacity deviation versus number of covariance updates forC = 1C = 1C = 1 carrier (frequency-flat channel), T ×Rk = 10× 2T ×Rk = 10× 2T ×Rk = 10× 2 antennas, SNR = 10 dB;(a) K = 10K = 10K = 10 users, (b) K = 50K = 50K = 50 users. "Yu" stands for reference [276].

88

0 200 400 600 800 1000 120010

−15

10−10

10−5

100

# Covariance updates

Nor

mal

ized

sum

−cap

acity

dev

iatio

n

Jindal et al, Alg.1Yu, Alg.2proposed IWF−RUPproposed DD

(a)

0 2000 4000 6000 8000 10000 1200010

−15

10−10

10−5

100

# Covariance updates

Nor

mal

ized

sum

−cap

acity

dev

iatio

n

Jindal et al, Alg.1Yu, Alg.2proposed IWF−RUPproposed DD

(b)

Fig 12. Comparison between different sum-rate maximization algorithms: normal-ized sum-capacity deviation versus number of covariance updates for C = 1C = 1C = 1 car-rier (frequency-flat channel), T ×Rk = 10× 2T ×Rk = 10× 2T ×Rk = 10× 2 antennas, SNR = 10 dB; (a) K = 10K = 10K = 10

users, (b) K = 50K = 50K = 50 users. "Jindal et al" stands for reference [111] and "Yu" standsfor [276].

89

3.4 Summary and discussion

Numerical methods for computing the sum-capacity of MIMO-OFDM downlink chan-nel were considered in this chapter. The broadcast to multiple-access channel dualitywas used to convert the original downlink problem into a convex and well-structuredsum-rate maximization for the dual multiple-access channel. Two new iterative algo-rithms were proposed, and their performance was analyzed by computer simulations.

The first one, named IWF-RUP algorithm, increases iteratively the sum-rate by up-dating at each step just two randomly selected covariance matrices. The eigenvectorsof the updated covariance matrices are aligned to the single-user optimum direction ob-tained by treating the other users’ signals as noise. The power allocation is obtained bywaterfilling across the selected users, and their common waterfilling level is determinedby their own sum power constraint. Thus, there is no power exchange between theupdated pair of covariance matrices and the other ones, and, therefore, the sum poweris preserved after each iteration. Unlike the recently proposed SPC-IWF and its modi-fied versions [111], the IWF-RUP algorithm has lower computational complexity andrequires no additional precautions to ensure the convergence. It was proved analyticallythat the IWF-RUP algorithm is convergent with probability one, and the computer simu-lations clearly show that it converges faster than all variants of the SPC-IWF algorithmproposed in [111].

The second algorithm is based on a dual decomposition method which breaks theoriginal optimization problem into two nested loops. The inner loop is an IWF-like al-gorithm which optimizes the transmit covariance matrices for a fixed waterfilling level,and the outer loop finds the optimum waterfilling level by using a bisection search. Theproblem of tracking the instantaneous error of the inner loop was considered and anupper bound was derived by using the Lagrange theory. Specifically, the Lagrange dualfunction of the inner loop was derived, and a systematic method to obtain the dual feasi-ble variables was provided. Then, the duality gap theory was used to provide an upperbound of the instantaneous error. Finally, it was shown that the upper bound is attainedat an optimal point, and, therefore, can be used to test a stopping criterion for any re-quired accuracy. The simulation results showed that the proposed algorithm based onthe derived upper bound performs very close to a hypothetical implementation, wherethe true sum power error could be computed. It was also shown that the rate of conver-gence of the global algorithm can be improved substantially by introducing a covariancenormalization step between two successive outer iterations.

90

4 Linear transceiver design for MIMOdownlink channels

As discussed in Chapter 1, the knowledge of CSI at the transmitter side can dramaticallyimprove the performance of MIMO communications systems. However, especially inthe case of multiuser systems, it is difficult to achieve full CSI knowledge of every userat the transmitter side.

An efficient method to exploit the multiuser diversity gain with limited CSI feed-back is the opportunistic beamforming technique [193, 246]. It has been shown that inthe case of systems with a large numbers of users, opportunistic beamforming performsvery close to the optimum beamforming [43, 193, 246]. However, the performance de-teriorates rapidly for systems with moderate or small numbers of users [130]. For suchcases, the use of a supplementary CSI feedback has been proposed in [130] to allowbeamforming and, thus, to reduce the detrimental effect at small to moderate numbersof users. Typically, these schemes attempt to design the transmit and receive beamform-ers such that the SINR at the output of the linear receiver filter is maximized. However,the problem of linear transceiver design for the MIMO downlink channel is still anactive research area.

In this chapter, the case of full CSI for scheduled users only is considered. Generalmethods for joint design of the linear transmit and receive beamformers according to dif-ferent optimization criteria are proposed. The treatment is restricted to linear processingboth at the transmitter and receiver, since this is simple to implement and, thus, an im-portant solution in practical system design. The main contribution of this chapter is thegeneralization of the known schemes [18, 116, 130, 176, 188–190, 194, 205, 243, 256]to handle multiple antennas at the BS and at the mobile users with an arbitrary numberof data streams per scheduled user. Furthermore, the optimization criteria are fairlygeneral and include sum power minimization under the minimum SINR constraint perdata stream, the balancing of SINR values among data streams, minimum SINR maxi-mization, weighted sum-rate maximization, and weighted sum MSE minimization. Be-sides the traditional sum power constraint on the transmit beamformer, solutions fortransceiver optimization under a more general power constrain model are provided,which cover a large range of practical applications. Specifically, sum power constraintscan be imposed for any number of arbitrary subsets of the transmit antennas. Finally, the

91

advantage of the proposed algorithms is that they can be easily modified to accommo-date supplementary constraints (e.g., upper and / or lower bounds for the SINR valuesof data streams) and the feasibility of the resulting optimization problems can be easilyverified.

The optimization problems employed in the beamformer design are not convex ingeneral. Therefore, in some cases, the problem of finding the global optimum is in-trinsically non-tractable, and suboptimal methods must be employed. The original op-timization problems are decomposed as a series of remarkably simpler optimizationproblems which can be efficiently solved by using standard convex optimization tech-niques [25, 34]. Even though each subproblem is optimally solved, there is no guaranteethat the global optimum has been found due to the nonconvexity of the problem. How-ever, the simulations show that the algorithms converge rapidly to a solution, which canbe a local optimum, but is still efficient.

Convex optimization theory [13, 26] and standard optimization programs such assecond-order cone programming [137], semidefinite programming [239], and geometricprogramming [25], are powerful tools which allow for efficient numerical solution forvarious signal processing and digital communications problems [8–10, 34, 35, 116, 142,144, 158, 160, 161, 168, 194, 207, 243, 248, 256, 270, 276, 277, 281]. In particular,they have been used to solve a wide range of optimal transmit and receive beamformerdesign problems [8–10, 161, 243, 256, 270]. Power allocation algorithms via geometricprogramming have been considered in [25, 34, 35]. Robust design methods based onimperfect CSI were provided in [158, 168, 248]. More examples of communicationproblems solved via convex optimization systems can be found in the tutorials [25, 34,142, 144], while a more detailed mathematical treatment is provided in [13, 25, 26, 137,239].

The rest of this chapter is organized as follows. The MIMO multiuser system modelis described in Section 4.1. The problem of linear transceiver design under a sum powerconstraints is addressed in Section 4.2. Solutions for linear transceiver design withpower constraints per antenna groups are presented in Section 4.3. Simulation resultsare presented in Sections 4.2.5 and 4.3.3. Finally, Section 4.4 concludes the chapter.

4.1 Linear transceiver system model

A single-cell MIMO downlink system with K decentralized users operating in frequency-flat fading channel is considered. The base station has T transmit antennas and user k,

92

k = 1, . . . , K is equipped with Rk receive antennas. The discrete-time MIMO down-link channel is mathematically described as

yk = Hkx + nk , k = 1, . . . , K (78)

where Hk ∈ CRk×T is the channel matrix between BS and the user k, x ∈ CT isthe signal vector transmitted by the BS, yk ∈ CRk is the signal vector received by thekth user, and nk ∼ CN (0,Rk) models the thermal noise and inter-cell interference atthe kth terminal. Without loss of generality it is assumed that Rk = I. Note that thisassumption is not a restriction, since colored additive noise can be viewed as white afteran appropriate whitening transform on the channel matrix [226, Section 1]. Thus, it isassumed in (78) that the whitening filter10 is contained in Hk.

A linear transmission and reception strategy is considered. The base station trans-mits S ≤ T independent data streams, by generating the antenna signal vector

x = Vdiag(p)1/2d (79)

where d ∈ CS×1, E{ddH} = I, contains the current transmitted data symbols, V =[v1, . . . ,vS ] ∈ CT×S , ‖vs‖2 = 1, is the normalized beamforming matrix, and thevector p = [p1, . . . , pS ]T controls the power allocated to each stream.

For each data stream s, s = 1, . . . , S, the base station’s scheduler unit associatesan intended user ks. Note that more than one stream can be associated to a user, and,therefore, the cardinality of the set of scheduled users, S = {ks|s = 1, . . . , S}, isless or equal to S. Let us , ‖us‖2 = 1 be the normalized receive beamformer (orantenna combiner vector) used by the ksth user to generate the decision variables forthe sth data stream, ds = uH

s yks . By collecting all the decision variables into the vectord = [d1, . . . , dS ]T, the downlink linear processing model is given by

d = UHHVdiag(p)1/2d + n (80)

where

UH ∆=

uH1 0 . . . 00 uH

2 . . . 0...

......

0 0 . . . uHS

, H ∆=

Hk1

Hk2

...HkS

, n ∆=

uH1nk1

uH2nk2

...uH

SnkS

. (81)

10The whitening filter is defined up to a scale factor. The assumption Rk = I implies that the scale factor ischosen such that the output noise power is normalized to 1.

93

Note that E{|[n]s|2} = ‖us‖2 = 1. Therefore, the SINR of the sth stream, γdls , is

given by [245]

γdls =

ps|uHs Hksvs|2

1 +∑S

i=1,i6=s pi|uHs Hksvi|2

. (82)

Following the approach of [245, Section II], the dual (or reciprocal) uplink linearprocessing model is defined as

d′ = VHHHUdiag(q)1/2d′ + n (83)

where the vector q = [q1, . . . , qS ]T controls the uplink power allocated to each stream,us is the transmit beamformer used by the ksth user, vs is the receive beamformer usedat the BS to generate the decision variable for the sth stream, d′s. The SINR of the sthstream, γul

s , is given by [245]

γuls =

qs|vHs HH

ksus|2

1 +∑S

i=1,i 6=s qi|vHs HH

kiui|2

. (84)

The following two lemmas are useful in the description of the proposed lineartransceiver design algorithms.

Lemma 4.1.1. Consider the downlink channel (80), where {v1, . . . ,vS} and p are

fixed and given. The normalized maximum SINR receiver is given by the set of beam-

formers, {u?1, . . . ,u

?S}, that solves the following multi-objective optimization problem

(the beamformers are normalized such that their Euclidian norms are equal to unity)

maximize (w.r.t.IRS+) γdl

∆= [γdl1 , . . . , γdl

S ]T

subject to ‖us‖2 = 1, s = 1, . . . , S(85)

where the maximization is with respect to the nonnegative orthant, IRS+ [26]. The ob-

jectives of (85) are noncompeting and the optimum set {u?1, . . . ,u

?S} is obtained by

normalizing the combiner vectors of the linear minimum mean square error (LMMSE)

receiver [158], i.e.,

u?s =

us

‖us‖2, uH

s =√

psvHs HH

ks

(S∑

i=1

piHksvivH

i HHks

+ I

)−1

. (86)

Lemma 4.1.2. Consider the uplink channel (83), where {u1, . . . ,uS} and q are fixed

and given. The normalized maximum SINR receiver is given by the set of beamformers,

94

{v?1, . . . ,v

?S}, that solves the following multi-objective optimization problem

maximize (w.r.t.IRS+) γul

∆= [γul1 , . . . , γul

S ]T

subject to ‖vs‖2 = 1, s = 1, . . . , S.(87)

The optimum set {v?1, . . . ,v

?S} is obtained by normalizing the LMMSE receiver’s com-

biner vectors [158], i.e.,

v?s =

vs

‖vs‖2, vH

s =√

qsuHs Hks

(S∑

i=1

qiHHki

uiuHi Hki + I

)−1

. (88)

In this chapter, a simple scheduling algorithm11 based on the opportunistic beam-forming strategy [193, 246] is considered. It provides an efficient beamformer initial-ization based only on a limited CSI feedback from the mobile terminals. The schedul-ing algorithm can be summarized as follows. Let S = min{T,

∑Kk=1 Rk}, p(0) =

1Pmax/S and form S orthogonal beams by generating randomly the orthogonal set{v(0)

1 , . . . ,v(0)S } with (v(0)

i )Hv(0)j = 0 for 1 ≤ i 6= j ≤ S. Each user measures

the SINR achieved in each12 beam by using the LMMSE receiver and sends it back tothe BS. Based on this information, the BS’s scheduler unit makes the beam-to-usersassociation described by the following set of active users

S = {ks|ks = arg maxk=1,...,K

γk,s, s = 1, . . . , S} (89)

where ks is the user associated to sth beam (or data stream) and γk,s denotes the SINRof the kth user in the sth beam. For further derivations, let us define {u(0)

1 , . . . ,u(0)S } as

the optimum set of receive beamformers given by (86) and γ(0)dl = [γk1,1, . . . , γkS ,S ]T

as the SINR values of the scheduled users. In the following, it is assumed that theselected users feed back their instantaneous CSI to the BS.

4.2 Linear transceiver design under a sum powerconstraint

In this section, the problem of maximizing a system performance criterion under atransmit sum power constraint is considered. The proposed iterative algorithm is based11For simplicity, we only consider a best effort type of scheduling algorithm. Fairness or other cross-layerissues can be taken into account as well by properly modifying the scheduling algorithm, but this falls outsidethe scope of this thesis.12In the case of a large number of users, close to optimum performance can be achieved if each user feedsback only the maximum (over the beams) SINR value and the index of the corresponding beam [246].

95

on the following uplink-downlink SINR duality theorem.

Theorem 4.2.1. For any given (fixed) set of normalized beamformers {u1, . . . ,uS}and {v1, . . . ,vS}, the set of SINR values, γ = [γ1, . . . , γS ]T, γ Â 0, is achievable in

the downlink channel (80) under the sum power constraint 1Tp ≤ Pmax if and only if

the same set of SINR values is achievable in the uplink channel (83) under the same

sum power constraint, 1Tq ≤ Pmax.

Theorem 4.2.1 was independently proved in [19, 245] under the assumption thatthe cross-coupling matrix between the users is irreducible. An alternative proof viaLagrange duality, which holds for arbitrary cross-coupling matrices, is given in Ap-pendix 2.

The uplink-downlink SINR duality theorem suggests that (86) and (88) can be usedto iteratively adjust the sets of normalized beamformers, {u1, . . . ,uS} and {v1, . . . ,vS},by sequentially switching between the downlink and uplink channels and by updatingthe power allocation vectors, p and q, according to a certain optimization criterionbefore each channel switching. A simple heuristic procedure that increases the SINRvector at each iteration is described in the following.

4.2.1 Heuristic SINR vector maximization

Consider a downlink channel specified by the transmit beamformers {v(0)1 , . . . ,v(0)

S }and the initial power allocation p(0), 1Tp(0) = Pmax. The optimum set of receive beam-formers, {u(0)

1 , . . . ,u(0)S }, that maximizes13 the downlink SINR vector γdl is given by

(86) and the optimum SINR vector, γ(0)dl

∆= γdl({u(0)s ,v(0)

s , p(0)s }S

s=1, is given by (82).Let us first consider the dual uplink channel (83), where the users’ transmit beamform-ers are {u(0)

1 , . . . ,u(0)S } and the BS’s combining vectors are {v(0)

1 , . . . ,v(0)S }. Theo-

rem 4.2.1 ensures the existence of a power allocation q(1), 1Tq(1) = Pmax such thatγul({u(0)

s ,v(0)s , q

(1)s }S

s=1) = γ(0)dl . The key point is to observe that for this new power

allocation, the BS’s combiners {v(0)1 , . . . ,v(0)

S } are not necessarily optimal, and, there-fore, the SINRs can be increased by updating them to a new value {v(1)

1 , . . . ,v(1)S }

given by (88). Lemma 4.1.2 ensures that γ(1)ul º γ

(0)dl , where γ

(1)ul is the new SINR vec-

tor given by (84), i.e., γ(1)ul

∆= γul({u(0)s ,v(1)

s , q(1)s }S

s=1). Secondly, for the downlinkchannel (80), which uses {v(1)

1 , . . . ,v(1)S } as transmit beamformers, a new power allo-

cation vector p(1), 1Tp(1) = Pmax can be found such that γdl({u(0)s ,v(1)

s , p(1)s }S

s=1) =13The vector maximization is with respect to the nonnegative orthant, IRS

+ [26].

96

γ(1)ul . By updating the users’ receive beamformers to the new values {u(1)

1 , . . . ,u(1)S }

given by (86), Lemma 4.1.1 ensures that γ(1)dl

∆= γdl({u(1)s ,v(1)

s , p(1)s }S

s=1) º γ(1)ul .

The scheme continues switching between the uplink and downlink channels untilthe SINR values converge or a satisfactory solution is found. Note that at each iterationthe sum power is kept fixed, i.e., 1Tq(i) = 1Tp(i) = Pmax, and the achieved SINRvalues are monotonically increasing, i.e.,

γ(i+1)dl º γ

(i+1)ul º γ

(i)dl º γ

(i)ul º . . . º γ

(0)dl (90)

where γ(i)dl

∆= γdl({u(i)s ,v(i)

s , p(i)s }S

s=1), γ(i+1)ul

∆= γul({u(i)s ,v(i+1)

s , q(i+1)s }S

s=1) fori ≥ 0. Since the achievable SINR region under the sum power constraint is bounded,the algorithm’s convergence to an equilibrium point is guaranteed.

4.2.2 General algorithm for joint beamformer and poweroptimization

By modifying the power update policies before each channel switching14, the previousmethod can be further generalized to include different optimality criteria, e.g., weightedsum-rate maximization, weighted sum-MSE minimization, maximization of the mini-mum SINR among the data streams, and minimization of the sum power required tosatisfy some target SINR values. The general iterative optimization algorithm can besummarized as follows.

Algorithm 4.2.1. Joint beamformer and power optimization under a sum power con-

straint

1. For a given stream-to-users association described by the set of active users S ={ks|s = 1, . . . , S}, an initial beamformer configuration {v(0)

s }Ss=1, and an initial

power allocation p(0) = 1Pmax/S compute the optimum set of receive beamformers

{u(0)s }S

s=1 given by (86). Let i = 0 and go to Step 2.

2. Consider the uplink channel (83) specified by {u(i)s }S

s=1 and {v(i)s }S

s=1. Update the

power allocation vector q(i+1) by solving a certain optimization problem which will

be specified in detail in Section 4.2.3, i.e.,

q(i+1) = ful

({u(i)

s ,v(i)s }S

s=1

). (91)

14The iterative method, based on the uplink-downlink SINR duality, is a "virtual" technique, and the algorithmis implemented at the BS only. It requires no extra communication between BS and mobile terminals.

97

Update the base station’s beamformers to {v(i+1)s }S

s=1 given by (88). Compute the

achieved SINR vector γ(i+1)ul given by (84) and go to Step 3.

3. Consider the downlink channel (80) specified by {u(i)s }S

s=1 and {v(i+1)s }S

s=1. Up-

date the power allocation vector p(i+1) by solving a certain optimization problem

which will be specified in Section 4.2.3, i.e.,

p(i+1) = fdl

({u(i)

s ,v(i+1)s }S

s=1

). (92)

Update the users’ beamformers to {u(i+1)s }S

s=1 given by (86). Compute the achieved

SINR vector γ(i+1)dl given by (82) and test a stopping criterion. If it is not satisfied,

let i = i + 1 and go to Step 2, otherwise STOP.

The main idea of Algorithm 4.2.1 is clarified in the following example. Consider theproblem of maximizing a system performance criterion under a sum power constraint,and assume that the performance criterion can be expressed as a nondecreasing functionof the SINR values of the data streams, i.e., f(γ) such that γ1 º γ2 ⇒ f(γ1) ≥ f(γ2),where γ contains the SINR values of the data streams. For such a problem, (91) and(92) compute an optimum uplink and downlink power allocation that maximize f(γul)and f(γdl), respectively. Following the reasoning of Section 4.2.1, it is easy to observethat Theorem 4.2.1 ensures that f(γ) is monotonically increased at each iteration ofAlgorithm 4.2.1, i.e.,

f(γ

(i+1)dl

) ≥ f(γ

(i+1)ul

) ≥ f(γ

(i)dl

) ≥ f(γ

(i)ul

) ≥ . . . ≥ f(γ

(0)dl

). (93)

4.2.3 Computing the power allocation vectors

In this section, the optimization problems (91) and (92), which provide the power allo-cation vectors q(i+1) and p(i+1) in Algorithm 4.2.1, are derived in detail. The objectivefunctions are either a performance criterion of the system that has to be maximized (e.g.,SINR, rate, etc.) or a cost (e.g., sum power) that has to minimized. Since the objectivefunctions and the constraints depend only on the streams’ SINRs and the sum power,the uplink-downlink SINR duality theorem allows us to formulate similar optimizationproblems for computing the power allocation vectors, p(i+1) for the downlink channeland q(i+1) for the uplink channel.

98

Sum power minimization

Consider the problem of minimizing the total transmitted power under a constraint onthe minimum SINR of each data stream. According to (82) and (84), this problem canbe formulated as the following linear program

minimize∑S

k=1 xk

subject toxsgs,s

1 +∑S

k=1,k 6=s xkgs,k

≥ γs, s = 1, . . . , S

xs ≥ 0, s = 1, . . . , S

(94)

where the variables are x1, . . . , xS , γs is the minimum SINR required for the sth datastream, and the problem data, gs,k, s, k = 1, . . . , S, are defined according to which(uplink or downlink) power allocation is solved, i.e.,

gs,k∆=

∣∣∣(v(i)s )HHH

kku(i)

k

∣∣∣2

for solving (91)∣∣∣(u(i)

s )HHksv(i+1)k

∣∣∣2

for solving (92). (95)

The solution of the linear program (94), x? = [x?1, . . . , x

?S ]T, provides the power

allocation vectors required at Step 2 and Step 3 of Algorithm 4.2.1, i.e., q(i+1) = x?

when solving (91) or p(i+1) = x? when solving (92).In the following it is shown that x? can be expressed in a closed form. First, the

problem (94) is written in a more compact form, as

minimize 1Tx

subject to (I−DC)x º D1

x º 0

(96)

where x ∆= [x1, . . . , xS ]T, and the matrices C and D are defined as

[C]s,k∆=

{gs,k if s 6= k

0 if s = k; D ∆= diag

{γ1

g11, . . . ,

γS

gSS

}. (97)

Note that D1 Â 0 and the cross-coupling matrix A ∆= DC is nonnegative. Assum-ing that A is primitive [100, Definition 8.5.0], the problem (96) is feasible if and only ifthe Perron-Frobenius eigenvalue of A is strictly less than 1 [19, 72, 90, 171, 284]. Notethat this assumption does not necessarily hold in general, since after a few iterations ofAlgorithm 4.2.1, any of gs,k, s 6= k in (95) may become zero. In fact, the computer

99

simulations have confirmed that this is often the case. Therefore, an alternative proof,which holds for any nonnegative A, is given in Appendix 2. An independent and differ-ent proof of the result can be found in the parallel work of [20]. When the problem isfeasible, the optimum solution is given by (see Appendix 2)

x? = (I−DC)−1D1 . (98)

It is worthwhile to remark that the heuristic SINR vector maximization presentedin Section 4.2.1 can be interpreted as a modified version of the sum power minimiza-tion algorithm, where the SINR constraint vector γ

∆= γ1, . . . , γS is updated at eachiteration, according to

γ =

{γ

(i)dl for solving (91)

γ(i+1)ul for solving (92)

. (99)

In some special cases, it is possible that certain SINR constraints cannot be satis-fied at the first iteration of Algorithm 4.2.1 for any initial power allocation. This oc-curs when the spatial subchannels created by the initial beamformers {u(0)

1 , . . . ,u(0)S }

and {v(0)1 , . . . ,v(0)

S }, obtained at the first step of Algorithm 4.2.1, are not "separableenough". Therefore, the system is in the interference limited region. Obviously, whenthe spatial subchannels are not interfering each other, i.e., gs,k = 0 for s 6= k, anySINR requirements γ are achievable since A = 0. Therefore, a simple method thatalways finds a feasible starting point consists of replacing the receive beamformers forthe virtual uplink channel {v(0)

1 , . . . ,v(0)S } by the zero-forcing solution.

Weighted sum-rate maximization

Consider the problem of maximizing the weighted sum of the rates of the individual datastreams under a sum power constraint, Pmax. The weight vector, w ∆= [w1, . . . , wS ]T,w º 0, is used to prioritize differently the data streams and can be chosen based on dif-ferent criteria, e.g., the states of queues or buffers in the case of cross-layer optimizationschemes. The proposed weighted sum-rate maximization algorithm can also be used tocompute the achievable rate region under the linear processing constraint. Obviously,w = 1 corresponds to the usual sum-rate maximization or best effort.

Assuming optimal channel coding (i.e., Gaussian codebook [62]) for each data

100

stream, the weighted sum-rate can be expressed as

Rw =S∑

s=1

wsrs =S∑

s=1

ws log(1 + γs) = − logS∏

s=1

(1 + γs)−ws (100)

where rs and γs are the rate and the SINR of the sth data stream. Since Rw increaseswith respect to each γs and − log(·) is a decreasing function, the weighted sum-ratemaximization problem can be formulated as follows

minimize∏S

s=1(1 + γs)−ws

subject to γs ≤ xsgs,s

1 +∑S

k=1,k 6=s xkgs,k

, s = 1, . . . , S

∑Sk=1 xk ≤ Pmax ; xs ≥ 0, s = 1, . . . , S

(101)

where the variables are x1, . . . , xS , γ1, . . . , γS , and the problem data, gs,k, s, k =1, . . . , S are given in (95).

The optimization problem (101) is a signomial optimization problem [25]. Thus,it is not convex in general. In high SINR region γs À 1, it can be accurately ap-proximated by a geometric program [26]. This approach means that the objective isreplaced by the monomial function

∏Ss=1 γ−ws

s . Clearly, the condition γs À 1 cannotbe controlled (e.g., imposed as a constraint) without affecting the solution since γs is aproblem variable. Therefore, a method applicable to the whole SINR range is needed.Such a procedure consists of searching for a close local minimum by solving a sequenceof geometric programs which locally approximate the original problem. This procedure,which converges very fast [25] (in a few iterations), is presented in the following.

Lemma 4.2.2. Let m(γ1, . . . , γS) = c∏S

s=1 γass be a monomial function [25] used to

approximate the objective of (101), f(γ1, . . . , γS) =∏S

s=1(1 + γs)−ws , near the point

γ = (γ1, . . . , γS). The parameters c and as of the best monomial local approximation

are given by

as = −wsγs

1 + γs, c =

f(γ1, . . . , γS)∏Ss=1 γas

s

. (102)

Proof. The monomial function m is the best local approximation of f near the pointγ = (γ1, . . . , γS) if [25]

{m(γ1, . . . , γS) = f(γ1, . . . , γS)∇m(γ1, . . . , γS) = ∇f(γ1, . . . , γS)

(103)

101

and by replacing the expressions of m and f in (103) we obtain the following systemof equations

{c∏S

s=1 γass = f(γ1, . . . , γS)

c∏S

s=1 γass asγ

−1s = −wsf(γ1, . . . , γS)(1 + γs)−1, s = 1, . . . , S

(104)

which have the solution given by (102).

By using the local approximation given by Lemma 4.2.2 in the objective functionof problem (101), and ignoring the multiplicative constant c which does not affect theproblem solution, the following iterative algorithm is obtained.

Algorithm 4.2.2. Power optimization for weighted sum-rate maximization

1. Let the initial SINR guess, γ = (γ1, . . . , γS), be

γ =

{γ

(i)dl for solving (91)

γ(i+1)ul for solving (92)

(105)

2. Solve the following geometric program,

minimize∏S

s=1 γ−ws

γs1+γs

s

subject to (1− α)γs ≤ γs ≤ (1 + α)γs , s = 1, . . . , S

g−1s,sx−1

s γs +∑S

k=1,k 6=s gs,kxkg−1s,sx−1

s γs ≤ 1 , s = 1, . . . , S∑Sk=1 xk ≤ Pmax

(106)with positive variables x1, . . . , xS , γ1, . . . , γS . Denote the solution by x?

1, . . . , x?S ,

γ?1 , . . . , γ?

S . If maxs |γ?s − γ| > εγ set γ = (γ?

1 , . . . , γ?S) and go to Step 2, otherwise

STOP.

The optimal points x?1, . . . , x

?S are the power allocation vectors required at Step 2

and Step 3 of Algorithm 4.2.1, i.e., q(i+1) = [x?1, . . . , x

?S ]T when solving (91) or

p(i+1) = [x?1, . . . , x

?S ]T when solving (92). Problem (106) is a geometric program

which approximates the original signomial problem (101) around the point γ =(γ1, . . . , γS). The first set of inequality constraints of problem (106) are called trustregion constraints [25] and they limit the domain of variables γs to a region wherethe monomial approximation is accurate enough. The small constant α < 1 controlsthe desired approximation accuracy, and a typical value is α = 0.1 [25]. Note thatthe approximation error vanishes as α goes to zero. Thus, monotonicity (93) can beguaranteed by a proper control of α.

102

Let us now consider a more general weighted sum-rate maximization problem,where the services provided by a subset of the data streams, P1 ⊆ {1, . . . , S}, re-quire some minimum SINR values, i.e., γs ≥ γs for s ∈ P1. Furthermore, a moreaccurate model for the rate provided by the sth data stream can be considered, i.e., rs =min{log(1 + Γγs), rmax}, where Γ is the SNR gap to the channel capacity and rmax isthe maximum data stream rate (Γ and rmax are used to describe the modulation and cod-ing schemes set [41]). Note that rs increases with γs for γs ≤ γmax

∆= (2rmax − 1)/Γand remains constant for γs > γmax. Thus, it is easy to observe that any sum-rate op-timal design must satisfy γs ≤ γmax for s = 1, . . . , S. Therefore, this problem can besolved by modifying the power allocation problem (101) as follows

minimize∏S

s=1(1 + Γγs)−ws

subject to γs ≤ xsgs,s

1 +∑S

k=1,k 6=s xkgs,k

, s = 1, . . . , S

γs ≤ γmax, s = 1, . . . , S

γs ≥ γs, s ∈ P1∑Sk=1 xk ≤ Pmax ; xs ≥ 0, s = 1, . . . , S .

(107)

Assuming that problem (107) is feasible under the initial beamforming configura-tion, {u(0)

1 , . . . ,u(0)S } and {v(0)

1 , . . . ,v(0)S }, obtained at the Step 1 of Algorithm 4.2.1,

it can be solved by following the same approach as in the case of problem (101). Asufficient condition for the feasibility is γs ≤ [γ(0)

dl ]s for any s ∈ P1. The method canalso be extended to handle general minimum SINR constraints, which are not neces-sarily feasible under the initial beamforming configuration. This is possible by solvingfirst the sum power minimization problem under γs ≥ γs, s ∈ P1 constraints, usingthe method described in Section 4.2.3. Recall that this problem is always feasible. Ifthe obtained minimum sum power is larger than Pmax, optimization problem (107) isinfeasible, otherwise the resulting beamformer configuration can be used as a feasiblestarting point for the original optimization problem (107). Note that the sum powerminimization problem needs to be solved with high accuracy only to certify the infeasi-bility of the minimum SINRs constraints under the sum power constraint Pmax. In thecase of feasible SINR constraints, the sum power minimization process can be stoppedafter any iteration that produces an initial feasible point for problem (107), i.e., a sumpower lower than Pmax. Starting from this beamformer configuration, Algorithm 4.2.1is used to solve the weighted sum-rate maximization problem by updating the powerallocation vectors q(i+1) and p(i+1) according to (107).

103

SINR balancing

Suppose now that the system has to provide different services which can be translatedinto minimum SINR requirements γs for the corresponding data streams, i.e., γs ≥ γs

for s ∈ P1, where P1 ⊆ {1, . . . , S} is a subset of data streams and γs is the SINR ofthe sth data stream. Furthermore, the SINR values of the remaining streams have to bekept in some fixed ratios, i.e., γs/αs = %, s ∈ P2 = {1, . . . , S} \ P1 for some fixedvalues αs > 0, and % has to be maximized under a total transmitted power, Pmax. Thisoptimization problem can be formulated as follows

maximize mins∈P2

α−1s xsgs,s

1 +∑S

k=1,k 6=s xkgs,k

subject toxsgs,s

1 +∑S

k=1,k 6=s xkgs,k

≥ γs, s ∈ P1

∑Sk=1 xk ≤ Pmax ; xs ≥ 0, s = 1, . . . , S

(108)

where the variables are x1, . . . , xS and the problem data, gs,k, s, k = 1, . . . , S, aregiven by (95). A particular case of this problem, with no minimum SINR requirements(i.e., P1 = {∅}), has been studied in [188, 256] in the case of single-antenna terminals.

The objective function of problem (108) is quasiconcave, since it is the minimumof a family of linear fractional functions [26]. Thus, it can be solved by using thebisection method [26]. In the following it is shown that it can be reformulated as ageometric program [26] and, therefore, solved more efficiently. First, problem (108) istransformed by inverting its objective and the first set of constraints, as follows:

minimize maxs∈P2

αsg−1s,sx−1

s +∑S

k=1,k 6=s αsgs,kg−1s,sxkx−1

s

subject to g−1s,sx−1

s +∑S

k=1,k 6=s gs,kg−1s,sxkx−1

s ≤ γ−1s , s ∈ P1∑S

k=1 xk ≤ Pmax ; xs ≥ 0, s = 1, . . . , S.

(109)

By expressing the problem (109) in the epigraph form15, the following equivalentgeometric program is obtained

minimize t

subject to αsg−1s,sx−1

s +∑S

k=1,k 6=s αsgs,kg−1s,sxkx−1

s ≤ t, s ∈ P2

g−1s,sx−1

s +∑S

k=1,k 6=s gs,kg−1s,sxkx−1

s ≤ γ−1s , s ∈ P1∑S

k=1 xk ≤ Pmax

(110)

15The epigraph form [26, Section 4.1.3] of the optimization problem minimize f(x) is obtained by intro-ducing an auxiliary scalar variable t and reformulating the original problem as minimize t subject to theadditional constraint t ≥ f(x).

104

with positive variables t, x1, . . . , xS , i.e., t ≥ 0, xs ≥ 0, s = 1, . . . , S, are im-plicit constraints. The optimal points x?

1, . . . , x?S are the power allocation vectors, i.e.,

q(i+1) = [x?1, . . . , x

?S ]T when solving (91) or p(i+1) = [x?

1, . . . , x?S ]T when solving

(92).Let us now discuss some particular cases which do not require solving the geometric

program (110). Without the minimum SINR requirements, i.e., P1 = {∅}, the problem(108) can be expressed as

maximize %

subject toα−1

s xsgs,s

1 +∑S

k=1,k 6=s xkgs,k

≥ %, s = 1, . . . , S

∑Sk=1 xk ≤ Pmax ; xs ≥ 0, s = 1, . . . , S

(111)

which can be written in a more compact form as

maximize %

subject to %DCx + %D1 ¹ x

1Tx ≤ Pmax ; x º 0

(112)

where C is given by (97) and D ∆= diag {α1/g11, . . . , αS/gSS}.Since all SINR constraints are active at the optimal point, the sum power constraint

can be relaxed to %1TDCx + %1TD1 ≤ Pmax. Furthermore, from the first constraintsof (111) follows that for any % > 0, x must be strictly positive. By changing the variable%−1 = ξ, problem (112) is equivalent to

minimize ξ

subject to

[DC D1

P−1max1

TDC P−1max1

TD1

][x

1

]¹ ξ

[x

1

]

x  0

(113)

which is the epigraph form [26] of the following equivalent problem:

minimize maxs∈{1,...,S+1}

[Bυ]s[υ]s

subject to υ Â 0 ; [υ]S+1 = 1(114)

where υ∆= [xT, 1]T and the matrix B ∈ IRS+1×S+1 is given by

B ∆=

[DC D1

P−1max1

TDC P−1max1

TD1

]. (115)

105

Since B is a nonnegative matrix, the problem (114) represents the min-max char-acterization of the Perron-Frobenius eigenvalue [100]. Thus the optimal value, ξ?, isthe Perron-Frobenius eigenvalue of matrix B and υ? is its corresponding eigenvector,scaled such that [υ?]S+1 = 1. The optimal values [υ?]1, . . . , [υ?]S represent the opti-mum power allocation vectors, i.e., q(i+1) when solving (91) or p(i+1) when solving(92), and the achieved SINRs are given by γs/αs = 1/ξ?. The same result has beenobtained in [188] for the case of single-antenna terminals.

When the values αs are equal for all s, problem (111) reduces to the classical min-

imum SINR maximization problem. Since the rate is an increasing function of SINR,it can be also interpreted as a sum-rate maximization problem under equal SINR con-straint. The MSE of the sth data stream at the output of optimum LMMSE receiver Gs

is related to the stream SINR, by Gs = (1 + γs)−1 [158] and, thus, it is a decreasingfunction of γs. Therefore, the minimization of the maximum MSE problem reducesalso to the minimum SINR maximization problem.

Weighted sum MSE minimization

Consider the problem of minimizing the weighted sum of the MSEs of the individualdata streams under a sum power constraint Pmax. The weight vector, w ∆= [w1, . . . , wS ]T,w º 0, is used to prioritize differently the data streams or to compute the achievableMSE region. Obviously, w = 1 corresponds to the usual minimum MSE optimizationcriterion. The weighted sum MSE at the output of the optimum LMMSE receiver canbe expressed as [158]

Gw =S∑

s=1

ws

γs + 1(116)

where γs is the SINR of the sth data stream. Since Gw is a decreasing function of vari-ables γs, the weighted sum MSE minimization problem can be formulated as follows,

minimize∑S

s=1 ws(1 + γs)−1

subject to γs ≤ xsgs,s

1 +∑S

k=1,k 6=s xkgs,k

, s = 1, . . . , S

∑Sk=1 xk ≤ Pmax ; xs ≥ 0, s = 1, . . . , S

(117)

where the variables are x1, . . . , xS , γ1, . . . , γS , and the problem data, gs,k, s, k =1, . . . , S are given by (95).

Optimization problem (117) is a signomial problem [25]. In the high SINR region,γs À 1, it can be accurately approximated by a geometric program by replacing the

106

objective by the posynomial function [25]∑S

s=1 wsγ−1s . Following the approach in-

troduced in Section 4.2.3, a solving method applicable to the whole SINR range ispresented in the sequel. All the constraints can be handled by a standard geometricprogram solver, but the objective function has to be replaced by a local posynomialapproximation. Since the objective function is separable16, we search for a separableposynomial approximation, i.e., p(γ1, . . . , γS) =

∑Ss=1 csγ

ass , where the coefficients

cs and the exponents as are given by the following lemma.

Lemma 4.2.3. Let p(γ1, . . . , γS) =∑S

s=1 csγass be a posynomial function [25] used

to approximate the objective of (117), f(γ1, . . . , γS) =∑S

s=1 ws(1 + γs)−1, near

the point γ = (γ1, . . . , γS). The parameters cs and as of the best posynomial local

approximation are given by

as = − γs

1 + γs, cs = ws

γγs

1+γss

1 + γs(118)

Proof. The posynomial function p is the best local approximation of f near the pointγ = (γ1, . . . , γS) if [25]

{p(γ1, . . . , γS) = f(γ1, . . . , γS)∇p(γ1, . . . , γS) = ∇f(γ1, . . . , γS)

(119)

and by replacing the expressions of p and f in (119) we obtain the following system ofequations { ∑S

s=1 csγass =

∑Ss=1 ws(1 + γs)−1

csasγas−1s = −ws(1 + γs)−2, s = 1, . . . , S

(120)

which is satisfied by cs and as given by (118).

By using the local approximation given by Lemma 4.2.3 in the objective functionof problem (117), the following iterative algorithm is obtained.

Algorithm 4.2.3. Power optimization for weighted sum MSE minimization

1. Let the initial SINR guess, γ = (γ1, . . . , γS), be

γ =

{γ

(i)dl for solving (91)

γ(i+1)ul for solving (92)

(121)

16The objective function f(x1, . . . , xn) is called separable [26, Section 5.5.5] if it can be expressed as a sumof functions of the individual variables x1, . . . , xn, i.e., f(x1, . . . , xn) =

∑ni=1 fi(xi).

107

2. Solve the following geometric program,

minimize∑S

s=1 wsγγs

1+γss (1 + γs)−1γ

− γs1+γs

s

subject to (1− α)γs ≤ γs ≤ (1 + α)γs , s = 1, . . . , S

g−1s,sx−1

s γs +∑S

k=1,k 6=s gs,kxkg−1s,sx−1

s γs ≤ 1 , s = 1, . . . , S∑Sk=1 xk ≤ Pmax

(122)with positive variables x1, . . . , xS , γ1, . . . , γS . Denote the solution by x?

1, . . . , x?S ,

γ?1 , . . . , γ?

S . If maxs |γ?s − γ| > εγ set γ = (γ?

1 , . . . , γ?S) and go to Step 2, otherwise

STOP.

The optimal points x?1, . . . , x

?S are the power allocation vectors required at Step 2

and Step 3 of Algorithm 4.2.1, i.e., q(i+1) = [x?1, . . . , x

?S ]T when solving (91) or

p(i+1) = [x?1, . . . , x

?S ]T when solving (92). It is worthwhile mentioning that the

weighted sum MSE minimization method can be also modified to accommodate sup-plementary constraints following the same approach as presented in Section 4.2.3.

4.2.4 A sum MSE lower bound

In this section, a lower bound for the sum MSE is derived. The result will be used inSection 4.2.5 to evaluate the convergence of the proposed algorithm. Let i be the vectorof the indices of the scheduled users, sorted in ascending order, i.e., i = [i1, . . . , iU ]T

where U is the cardinality of set S defined in (89), ij ∈ S for j = 1, . . . , U , andi1 < i2 < · · · < iU . Furthermore, let us define the vector s = [s1, . . . , sU ]T where sj ,j = 1, . . . , U is the number of data streams allocated to the user ij .

The sum MSE for the dual uplink channel (83) can be expressed as [142, 190]

G1 = 1Ts− T + tr{(∑Uj=1 HH

ijQijHij + I)−1} (123)

where 1Ts represents the total number of data streams and Qij is the transmit covariancematrix of the user ij and has rank sj . The sum MSE minimization problem can beformulated as follows:

minimize tr{(∑Uj=1 HH

ijQij

Hij+ I)−1}

subject to Qij º 0, j = 1, . . . , U∑Pj=1 tr{Qij

} ≤ Pmax

rank{Qij} = sj , j = 1, . . . , U

(124)

108

where the variables are the covariance matrices Qij , j = 1, . . . , U . The objective andthe first two constraints are convex, but the rank constraints are nonconvex. In otherwords, once the stream to user allocation has been fixed by the scheduling unit, the sumMSE minimization problem becomes nonconvex. Nevertheless, a lower bound for thesum MSE can be obtained by relaxing the rank constraints. By following the approachof [142, 143], the relaxed problem can be transformed into the following equivalentsemidefinite program [26, 239]:

minimize tr{Θ}

subject to

[Θ I

I∑P

j=1 HHijQijHij + I

]º 0

Qij º 0, j = 1, . . . , U∑Pj=1 tr{Qij} ≤ Pmax

(125)

where Θ is an auxiliary Hermitian positive semidefinite matrix variable. Note thatthe sum MSE lower bound is achievable whenever the covariance rank constraints areredundant for problem (124), i.e., the ranks of the optimal covariance matrices Q?

ijob-

tained by solving the relaxed problem (125) satisfy rank{Q?ij} = sj for j = 1, . . . , U .

4.2.5 Numerical examples

A MIMO downlink channel with T = 4 antennas at the BS and Rk = 2 antennasat each user terminal has been considered for the numerical results. A frequency-flatfading channel with uncorrelated antennas was assumed, i.e., vec(Hk) ∼ CN (0, I).

Figure 13 shows the sum-rate performance of the weighted sum-rate maximiza-tion algorithm for the case w = [1, 1, 1, 1]T, i.e., all S = 4 data streams are equallyweighted. Specifically, the sum rate achieved after different number of iterations of Al-gorithm 4.2.1 is presented. The power allocation vectors q(i+1) and p(i+1) at Step 1 and2, respectively have been updated according to Algorithm 4.2.2 of Section 4.2.3. Figure13(a) shows the sum-rate versus SNR for K = 20 users and Figure 13(b) shows the sum-rate versus the number of users for SNR = 10 dB. Each figure presents also the sum-rateachieved when only opportunistic beamforming is used, and the MIMO broadcast sumcapacity computed for the following two distinct cases: when all K users are consid-ered and when only the scheduled users are considered. The sum-rate achieved by usingonly the opportunistic beamforming is the starting point of the proposed optimizationmethod (i.e., iteration i = 0), while the sum-capacity computed for the scheduled users

109

is an upper bound of the sum-rate. The upper bound is not achievable with any lineartransmission strategy, since the sum-capacity achieving schemes require nonlinear pre-coding based on DPC [245]. However, it can be still used as an ultimate performancebenchmark. As the figures show, the sum-rate provided by the proposed optimizationmethod is within 0.5 – 1.5 bits/second/Hz close to the sum-capacity, which representsmore than 90 percent of the sum-capacity. The sum rate improvement relative to thesimple opportunistic beamforming increases dramatically with SNR (Figure 13(a)) andremains large even in the case of K = 256 users (Figure 13(b)). The proposed methodconverges relatively fast, i.e., depending on the SNR point and K, it requires about30–70 iterations to arrive at a sum-rate stationary point. Note that most of the gain isachieved in the first few iterations. Therefore, the proposed algorithm is attractive forpractical implementation, where an implementation with a fixed number of iterationsmay be preferable due to the complexity reasons.

In Figure 14, the sum-rate provided by the true sum-rate maximization algorithm(i.e., Algorithm 4.2.2) is compared to the resulting sum-rate when the power allocationvectors p(i+1) and q(i+1) are updated based on different optimization criteria, suchas sum MSE minimization, minimum SINR maximization, and heuristic SINR vectormaximization. The minimum SINR maximization criterion is equivalent to imposing asupplementary constraint of equal SINR values for all data streams, but, as discussed inSection 4.2.3, each power control step of Algorithm 4.2.1 can be solved as an eigende-composition problem. Therefore, it has reduced complexity. The results show that forK > 8 users, all algorithms considered achieve a sum-rate within 0.5 bits/sec/Hz closeto the sum-rate maximization algorithm. In other words, the resulting sum-rate is justslightly decreased if all data streams are constrained to have equal SINR values, e.g.,due to some fairness requirements. Moreover, the heuristic SINR vector maximizationalgorithm provides essentially the same sum rate as the minimum SINR maximization(obviously, the SINR values of the individual data streams are not necessarily equal).From a practical perspective this is an important remark, since the heuristic SINR vec-tor maximization has the least complexity. The sum MSE minimization algorithm ap-pears to be the closest to the true sum-rate maximization algorithm. This confirms theintuition that, under linear processing constraint, a rate efficient design can be obtainedby minimizing the sum MSE. Interestingly, as will be shown later, the converse is nottrue.

Figure 14 shows also the sum-rate provided by the ZF solutions proposed in [202].Specifically, for identical stream-to-user allocation as in case of the proposed algo-

110

rithms, the coordinated transmit-receive channel block diagonalization method com-bined with the WF power allocation was used, as described in [202, Section V]. Theresults show that the proposed sum-rate maximization method provides a SNR gainof more than 4 dB (Figure 14(a)) and a spectral efficiency improved by up to 3.5 bit-s/sec/Hz (Figure 14(b)) compared to the ZF solution. By comparing Figures 13 and14, it is also interesting to remark that only one iteration of the proposed method isenough to substantially outperform the zero forcing based solution. However, it mustbe noted that the considered opportunistic beamforming based scheduling algorithm isnot particularly well suited for the ZF applications. It selects the most appropriate usersfor the randomly generated transmit beamformers by assuming the optimum linear in-terference suppression at the receiver side (i.e., the LMMSE receiver). In contrast, theblock diagonalization method proposed in [202] assumes fixed receive beamformers(i.e., the dominant singular vectors of the users’ channels), and eliminates completelythe multiuser interference by linear processing at the transmitter side. This approachcauses a severe performance degradation, since the interference rejection capabilitiesof the receiver are not fully exploited when the transmit beamformers are designed. ZFbased schemes may work substantially better when full CSI knowledge of every useris available at the transmitter, and, consequently, the users with the most orthogonalchannels can be scheduled at the same time. However, this kind of CSI knowledge levelis often difficult to achieve and falls outside the scope of the thesis.

Figure 15 shows the weighted sum-rate performance of the algorithms consideredin Figure 14, for a different weight vector, w = [4/3, 4/3, 2/3, 2/3]T, i.e., the first twostreams have double weights compared to the last two streams. The results show thatthe performance gap between different algorithms increases as the spread of the weightsincreases. Therefore, to accurately compute the entire rate region achievable with lineartransmission strategy, the weighted sum-rate maximization algorithm is required.

Figure 16 shows the sum MSE performance of the weighted sum MSE minimizationalgorithm for the case w = [1, 1, 1, 1]T. Specifically, the sum MSE resulted after differ-ent numbers of iterations of Algorithm 4.2.1 is presented. The power allocation vectorsp(i+1) and q(i+1) have been updated according to Algorithm 4.2.3 of Section 4.2.3. Thesum MSE achieved after iteration i = 0 (i.e., when only opportunistic beamforming isused), and the sum MSE lower bound obtained in Section 4.2.4 by relaxing the rankconstraints on the users’ covariance matrices, are also presented. As the figures show,the sum MSE improvement relative to the simple opportunistic beamforming increasesdramatically with SNR (Figure 16(a)) and remains large even in the case of K = 256

111

users (Figure 16(b)). The proposed method converges fast, i.e., depending on the SNRpoint and K, it requires about 20–50 iterations to achieve a sum MSE stationary point,which is very close to the sum MSE lower bound. Similarly to the sum-rate maximiza-tion, most of the sum MSE improvement is achieved after the first few iterations.

In Figure 17, the sum MSE provided by the true sum MSE minimization algorithm(i.e., Algorithm 4.2.3) is compared to the resulting sum MSE when the power allocationvectors p(i+1) and q(i+1) are updated based on different optimization criteria. The re-sults show that for K > 20, all algorithms considered achieve a close to minimum sumMSE. In particular, the minimum SINR maximization algorithm achieves essentiallythe same minimum sum MSE as the true sum MSE minimization algorithm in all thesimulation setups. Recall that the minimum SINR maximization criterion is equivalentto imposing the supplementary constraint of equal MSE values (or SINR values) for alldata streams, and, as discussed in Section 4.2.3, it has reduced complexity. Therefore,the resulting sum MSE is almost unaltered if all data streams are constrained to haveequal SINR values. More surprisingly, the least complex heuristic SINR vector maxi-mization algorithm performs well even in the case of a small number of users. Figure17 reveals an interesting and probably non-intuitive fact: the sum-rate maximizationalgorithm provides the largest sum MSE among the algorithms considered. Conversely,recall that, as shown in Figure 14, the sum MSE minimization algorithm provides theclosest sum rate to the true sum-rate maximization algorithm, Algorithm 4.2.2.

Figures 16 and 17 show that the minimum sum MSE achieved by the sum MSEminimization algorithm is very close to the sum MSE lower bound obtained in Section4.2.4. A natural question is whether the remaining small gap is due to the non-tightnessof the lower bound or due to the sub-optimality of the solution provided by the optimiza-tion algorithm. As discussed in Section 4.2.4, the lower bound is achievable wheneverthe covariance rank constraints are redundant for the problem (124). Therefore, a par-tial answer can be obtained if we look at the sum MSE gap obtained for those particularchannel realizations. Figure 18 shows the sum MSE gap averaged over all channel real-izations for which the ranks of the optimal covariance matrices Q?

ijobtained by solving

the relaxed problem (125) satisfy rank{Q?ij} = sj for j = 1, . . . , U . First, it can be

seen that in such conditions, the lower bound is asymptotically attained. The algorithmhas a fast initial convergence, i.e., the initial gap is decreased by 2 decades in the first10 iterations, and the speed of convergence decreases as the algorithm approaches theoptimal point.

112

Figure 19 shows the weighted sum MSE performance of the algorithms consideredin Figure 17, for a different weight vector, w = [4/3, 4/3, 2/3, 2/3]T, i.e., the first twostreams have double weights compared to the last two streams. The results show thatthe performance gap between the different algorithms increases as the spread of thestreams’ weights increases. Therefore, the weighted sum MSE minimization algorithmhas to be used to accurately compute the entire MSE region.

Figure 20 considers the case where the users experience different path-losses. Inorder to avoid the scheduler to select only the users with large path gains, a scenariowith K = 4 users and a single data stream per user has been considered. The to-tal transmitted power is fixed to Pmax and the path-loss of the kth user is modeled aslk = −(k − 1) × ∆SNR [dB]. Thus, the SNR of the kth user is given by SNRk =SNR0 + (k − 1) × ∆SNR dB, where SNR0 = 5 dB. The initial transmit beamform-ers {v(0)

1 , . . . ,v(K)1 } were obtained by applying the Gram-Schmidt orthogonalization

to the set of dominant right singular vectors of all users’ channels. The Gram-Schmidtprocedure started from the dominant singular vector with the largest singular value andcontinued towards the one corresponding to the smallest singular value. As shown inFigure 20(a), the above initialization method produces an efficient starting point, e.g.,95% of the final sum-rate is achieved after only one iteration of the sum-rate maxi-mization algorithm. The results show that the performance gap between the differentalgorithms increases as ∆SNR increases. Figure 20(a) also shows that the resultingsum-rate decreases considerably if all data streams are constrained to have the sameSINR (i.e., the minimum SINR maximization algorithm is used) in case of large ∆SNRvalues. It is interesting to remark that, as already observed in Figures 14 and 17, thesum MSE minimization algorithm provides a fairly large sum-rate. In contrast, thesum-rate maximization algorithm provides the largest sum MSE among the consideredalgorithms. This is due to the fact that the sum-rate maximization algorithm allocatesmore power to the data streams with large channel gains and less power to the weakerone. Consequently, the MSE of the weakest data streams dominate the resulting sumMSE.

The same scenario with a random initialization of the transmit beamformers wasalso simulated. It was observed that the initialization method has a negligible effecton the final results of all the considered algorithms except for the heuristic SINR vectormaximization. Recall from Section 4.2.1 that it can only increase the initial SINR valuesof all data streams. Therefore, the initialization has a higher influence on the final result.However, the results show that, when the heuristic SINR vector maximization algorithm

113

is properly initialized, it performs reasonably well according to both rate maximizationand MSE minimization criteria, even for large ∆SNR values. For example, when thetransmit beamformers are initialized based on the channels’ dominant singular vectors,the heuristic SINR vector maximization algorithm achieves a slightly higher sum-ratethan the sum MSE minimization algorithm. With a random initialization, it achievesalmost the same sum MSE as the minimum SINR maximization algorithm.

114

2 4 6 8 10 12 14 16 18 204

6

8

10

12

14

16

18

20

22

24

26

SNR

Sum

−rat

e [b

its/s

ec/H

z]

1,2,4,8,16,32,64,100iterations

sum−cap. of all users (DPC)sum−cap. of only the scheduled users (DPC)sum−rate with proposed rate maximization alg.sum−rate with opportunistic beamforming

(a)

101

102

6

8

10

12

14

16

18

20

Number of users

Sum

−rat

e [b

its/s

ec/H

z]

1,2,4,8,16,32,100iterations

sum−cap. of all users (DPC)sum−cap of only the scheduled users (DPC)sum−rate with proposed rate maximization alg.sum−rate with opportunistic beamforming

(b)

Fig 13. Sum-rate performance of Algorithm 4.2.2 after different numbers of iter-ations for T = 4T = 4T = 4, Rk = 2Rk = 2Rk = 2 antennas and w = [1, 1, 1, 1]= [1, 1, 1, 1]= [1, 1, 1, 1] T; (a) K = 20K = 20K = 20 users, SNR =2÷ 202÷ 202÷ 20 dB, (b) SNR = 10 dB, K = 2÷ 256K = 2÷ 256K = 2÷ 256 users.

115

2 4 6 8 10 12 14 16 18 204

6

8

10

12

14

16

18

20

22

24

26

SNR

Sum

−rat

e [b

its/s

ec/H

z]

sum−cap. of only the scheduled users (DPC)sum−rate maximizationsum MSE minimizationminimum SINR maximizationheuristic SINR vector maximizationzero forcingopportunistic beamforming only

(a)

101

102

6

7

8

9

10

11

12

13

14

15

16

Number of users

Sum

−rat

e [b

its/s

ec/H

z]

sum−cap. of scheduled users (DPC)sum−rate maximizationsum MSE minimizationminimum SINR maximizationheuristic SINR vector maximizationzero forcingopportunistic beamforming only

(b)

Fig 14. Sum-rate performance of different algorithms for T = 4T = 4T = 4, Rk = 2Rk = 2Rk = 2 anten-nas and w = [1, 1, 1, 1]= [1, 1, 1, 1]= [1, 1, 1, 1] T; (a) K = 20K = 20K = 20 users, SNR = 2÷ 202÷ 202÷ 20 dB, (b) SNR = 10 dB,K = 2÷ 256K = 2÷ 256K = 2÷ 256 users.

116

2 4 6 8 10 12 14 16 18 204

6

8

10

12

14

16

18

20

22

24

SNR

Wei

ghte

d su

m−r

ate

[bits

/sec

/Hz]

weighted sum−rate maximizationweighted sum MSE minimizationminimum SINR maximizationheuristic SINR vector maximizationopportunistic beamforming only

(a)

101

102

6

7

8

9

10

11

12

13

14

15

16

Number of users

Wei

ghte

d su

m−r

ate

[bits

/sec

/Hz]

weighted sum−rate maximizationweighted sum MSE minimizationminimum SINR maximizationheuristic SINR vector maximizationopportunistic beamforming only

(b)

Fig 15. Weighted sum-rate performance of different algorithms for T = 4T = 4T = 4, Rk = 2Rk = 2Rk = 2

antennas and w = [4/3, 4/3, 2/3, 2/3]= [4/3, 4/3, 2/3, 2/3]= [4/3, 4/3, 2/3, 2/3] T; (a) K = 20K = 20K = 20 users, SNR = 2÷ 202÷ 202÷ 20 dB, (b) SNR= 10 dB, K = 2÷ 256K = 2÷ 256K = 2÷ 256 users.

117

2 4 6 8 10 12 14 16 18 20

10−1

100

SNR

Su

m M

SE

1,2,4,8,16,32,64,100iterations

opportunistic beamforming onlyproposed sum MSE minimization alg.lower bound rank constraint relaxed

(a)

101

102

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Number of users

Su

m M

SE

1,2,4,8,16,100iterations

opportunistic beamforming onlyproposed sum MSE minimization alg.lower bound rank constraint relaxed

(b)

Fig 16. Sum MSE performance of Algorithm 4.2.3 after different numbers of iter-ations for T = 4T = 4T = 4, Rk = 2Rk = 2Rk = 2 antennas and w = [1, 1, 1, 1]= [1, 1, 1, 1]= [1, 1, 1, 1] T; (a) K = 20K = 20K = 20 users, SNR =2÷ 202÷ 202÷ 20 dB, (b) SNR = 10 dB, K = 2÷ 256K = 2÷ 256K = 2÷ 256 users.

118

2 4 6 8 10 12 14 16 18 20

10−1

100

SNR

Su

m M

SE

opportunistic beamforming onlysum−rate maximizationheuristic SINR vector maximizationminimum SINR maximizationsum MSE minimizationlower bound rank constraint relaxed

(a)

101

102

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Number of users

Su

m M

SE

opportunistic beamforming onlysum−rate maximizationheuristic SINR vector maximizationminimum SINR maximizationsum MSE minimizationlower bound rank constraint relaxed

(b)

Fig 17. Sum MSE performance of different algorithms for T = 4T = 4T = 4, Rk = 2Rk = 2Rk = 2 anten-nas and w = [1, 1, 1, 1]= [1, 1, 1, 1]= [1, 1, 1, 1] T; (a) K = 20K = 20K = 20 users, SNR = 2÷ 202÷ 202÷ 20 dB, (b) SNR = 10 dB,K = 2÷ 256K = 2÷ 256K = 2÷ 256 users.

119

0 10 20 30 40 50 60 70 80 90 100

10−4

10−3

10−2

10−1

100

Number of iterations

Sum

MS

E −

sum

MS

E lo

wer

bou

nd

Fig 18. Average sum MSE gap versus iteration for T = 4T = 4T = 4, Rk = 2Rk = 2Rk = 2, K = 2÷ 256K = 2÷ 256K = 2÷ 256 andSNR = 10 dB. Only the channel realizations for which the sum MSE lower bound isachievable have been considered.

120

2 4 6 8 10 12 14 16 18 20

10−1

100

SNR

Wei

gh

ted

su

m M

SE

opportunistic beamforming onlyweighted sum−rate maximizationheuristic SINR vector maximizationminimum SINR maximizationweighted sum MSE minimization

(a)

101

102

0.2

0.4

0.6

0.8

1

1.2

Number of users

Wei

gh

ted

su

m M

SE

opportunistic beamforming onlyweighted sum−rate maximizationheuristic SINR vector maximizationminimum SINR maximizationweighted sum MSE minimization

(b)

Fig 19. Weighted sum MSE performance of different algorithms for T = 4T = 4T = 4, Rk = 2Rk = 2Rk = 2

antennas and w = [4/3, 4/3, 2/3, 2/3]= [4/3, 4/3, 2/3, 2/3]= [4/3, 4/3, 2/3, 2/3] T; (a) K = 20K = 20K = 20 users, SNR = 2÷ 202÷ 202÷ 20 dB, (b) SNR= 10 dB, K = 2÷ 256K = 2÷ 256K = 2÷ 256 users.

121

0 1 2 3 4 5 6 74

6

8

10

12

14

16

18

20

22

24

∆ SNR

Sum

−rat

e [b

its/s

ec/H

z]

sum−capacity (DPC)sum−rate maximizationsum−rate maximization − one iterationheuristic SINR vector maximization (a)sum MSE minimizationheuristic SINR vector maximization (b)minimum SINR maximizationzero forcing

(a)

0 1 2 3 4 5 6 7

10−0.5

10−0.4

10−0.3

10−0.2

10−0.1

100

100.1

∆ SNR

Su

m M

SE

sum−rate maximizationheuristic SINR vector maximization (a)heuristic SINR vector maximization (b)minimum SINR maximizationsum MSE minimization

(b)

Fig 20. Comparison between different algorithms for T = 4T = 4T = 4, Rk = 2Rk = 2Rk = 2 antennas,w = [1, 1, 1, 1]= [1, 1, 1, 1]= [1, 1, 1, 1] T, K = 4K = 4K = 4 users and a single data stream per user. The SNR of kkkthuser is SNRk = SNR0 + (k − 1)×∆SNRSNRk = SNR0 + (k − 1)×∆SNRSNRk = SNR0 + (k − 1)×∆SNR dB, where SNR0 = 5SNR0 = 5SNR0 = 5 dB; (a) sum-rate per-formance, (b) Sum MSE performance.

122

4.3 Linear transceiver design with power constraints perantenna groups

The linear transceiver optimization Algorithm 4.2.1, derived in Section 4.2, was basedon the uplink-downlink SINR duality Theorem 4.2.1. Therefore, the method is re-stricted to the case in which the transmit beamformers are subject to a single sum powerconstraint (see Theorem 4.2.1). However, there are certain applications where moregeneral transmit power constraints must be considered during the linear transceiver op-timization.

Such power constraints are required, e.g., in the case of distributed MIMO antennaarchitectures, where a BS is connected via high speed links to multiple antenna headswhich are distributed over a large geographical area [66, 107, 283, 285, 290]. The BScan perform cooperative antenna heads transmission, but an individual sum power con-straint must be applied for each antenna head [214–218]. Specific practical applicationsmay also require a per antenna power constraint. This is the case when each transmit an-tenna is equipped with its own power amplifier, and the linearity of the power amplifieris the limiting performance factor [281].

The transmitter optimization for MIMO downlink channel with per antenna or pergroup of antennas power constraints via its dual uplink problem with uncertain noisecovariance was investigated in [281]. This section describes an alternative design basedon the recent results in [25, 34, 256]. The precoder design via conic optimization forfixed linear MIMO receivers [256] and the power allocation via signomial program-ming [25, 34] are combined to provide an iterative ascent algorithm.

The transmit power constraint model considered is fairly general and covers a largerange of practical applications, including those discussed above. Specifically, sumpower constraints can be imposed for any number of arbitrary subsets of the transmitantennas as follows. LetAn be an arbitrary subset of transmit antennas. The part of thetransmit beamformer vs that contains the weights of the antennas belonging to An isdenoted by v[n]

s . If Pn is the maximum transmit power for the antenna setAn, then thesum power constraint can be expressed as

∑Ss=1 ps

∥∥v[n]s

∥∥2

2≤ Pn according to (79).

4.3.1 Weighted sum-rate maximization

Consider first the problem of maximizing the weighted sum-rate Rw (100) subjectto N sum power constraints P1, . . . , PN on arbitrary subsets of transmit antennas,

123

A1, . . . ,AN . Since Rw increases with respect to each γs and log(·) is a increasingfunction, the weighted sum-rate maximization problem can be formulated as

maximize∏S

s=1(1 + γs)ws

subject to γs ≤ps

∣∣uHs Hksvs

∣∣2

1 +∑S

k=1,k 6=s pk

∣∣uHs Hksvk

∣∣2 , s = 1, . . . , S

∑Ss=1 ps

∥∥v[n]s

∥∥2

2≤ Pn, n = 1, . . . , N

‖us‖2 = 1, ‖vs‖2 = 1, ps ≥ 0, s = 1, . . . , S

(126)

where the variables are vs ∈ CT , us ∈ CRks , ps ∈ IR, γs ∈ IR, s = 1, . . . , S.The variables vs, ps, and us represent the normalized transmit beamformer, the powerallocated, and the normalized receive beamformer for data stream s. It is easy to observethat at the optimal point of (126), the first constraint holds with equality17, and, thus,the optimal value of γs represents the SINR of the sth data stream.

Problem (126) is not convex, and, hence, the problem of finding the global optimumis intrinsically non-tractable. However, it can be maximized with respect to differentsubsets of variables by considering the others fixed. For instance, the maximum SINRreceiver given by (86) is optimal for any fixed vs and ps. Furthermore, by fixing thebeamformers vs and us, (126) becomes a signomial problem [25, 34] and can be solvedsimilarly to problem (101). Finally, for fixed receive beamformers us and γs values, amaximum power reduction factor, common for all per antenna group power constraints,can be found such that the SINR values are preserved. It is given by the optimum β?

that solves the following problem:

minimize β

subject to γs ≤ps

∣∣uHs Hksvs

∣∣2

1 +∑S

k=1,k 6=s pk

∣∣uHs Hksvk

∣∣2 , s = 1, . . . , S

∑Ss=1 ps

∥∥v[n]s

∥∥2

2≤ βPn, n = 1, . . . , N∥∥vs

∥∥2

= 1, ps ≥ 0, s = 1, . . . , S

(127)

where the variables are β ∈ IR, ps ∈ IR, vs ∈CT , s = 1, . . . , S.Note that the solutions v?

s and p?s do not directly increase the objective of the orig-

inal problem (126). However, they increase the power margin for a fixed value of theobjective, and, hence, the saved power can be used to increase the objective. This isrealized by updating vs and ps in (126) to the new values v?

s and p?s/β?, respectively,

17It follows from the fact that Rw increases with respect to each γs and there are no other constraints on γs

124

and increasing all γs until all SINR constraints become tight. Note that this is an ascentstep, since β? ≤ 1 for any us and γs that are feasible for the problem (126).

The above observations suggest the following iterative ascent algorithm.

Algorithm 4.3.1. Weighted sum-rate maximization with power constraints per antenna

groups

1. Initialization: start with a given stream-to-users association described by the set of

active users S = {ks|s = 1, . . . , S}, an initial beamformer configuration {v(0)s }S

s=1,

{u(0)s }S

s=1 and a feasible initial power allocation p(0). Let i = 1 and go to Step 2.

2. Solve the problem (126) for the variables ps and γs, by fixing us = u(i−1)s and

vs = v(i−1)s , s = 1, . . . , S. Denote the solutions by p?

s and γ?s . Update the achieved

SINR values γtmps = γ?

s , s = 1, . . . , S.

3. Solve the problem (127), where γs = γtmps and us = u(i−1)

s , s = 1, . . . , S. Denote

the solutions by β?, p?s and v?

s . Update p(i)s = p?

s/β? and v(i)s = v?

s , s = 1, . . . , S.

4. Update the receive beamformers u(i)s according to normalized maximum SINR re-

ceive given by (86). Update the achieved SINR values γ(i)dl = [γdl

1 , . . . , γdlS ]T accord-

ing to (82) and test a stopping criterion. If it is not satisfied, let i = i + 1 and go to

Step 2, otherwise STOP.

The set of active users S, the initial beamformers {v(0)s }S

s=1 and {u(0)s }S

s=1, andthe power allocation p(0) can be initialized by using the scheduling algorithm based onthe opportunistic beamforming strategy described in Section 4.1. In the following, theoptimization problems solved at Steps 2 and 3 of Algorithm 4.3.1 are derived in detail.

The second step of Algorithm 4.3.1, which solves problem (126) for the variables{ps}S

s=1 and {γs}Ss=1 by fixing the beamformers {us}S

s=1 and {vs}Ss=1, results in the

following optimization problem:

minimize∏S

s=1(1 + γs)−ws

subject to γs ≤ psgs,s

1 +∑S

k=1,k 6=s pkgs,k

, s = 1, . . . , S

∑Ss=1 ps

∥∥v[n]s

∥∥2

2≤ Pn, n = 1, . . . , N

ps ≥ 0, s = 1, . . . , S

(128)

where gs,k =∣∣uH

s Hksvk

∣∣2, s, k = 1, . . . , S. Note that this problem is identical to(101), except that the sum power constraint has been replaced by N distinct powerconstraints. Since all power constraints are linear, the same signomial programmingmethod described in Section 4.2.3 can be used to solve problem (128). The procedureis summarized in the following algorithm.

125

Algorithm 4.3.2. Power allocation for weighted sum-rate maximization with power

constraints per antenna groups

1. Let the initial SINR guess be γ = γ(i−1)dl .

2. Solve the following geometric program,

minimize∏S

s=1 γ−ws

γs1+γs

s

subject to (1− α)γs ≤ γs ≤ (1 + α)γs, s = 1, . . . , S

g−1s,sp−1

s γs +∑S

k=1,k 6=s gs,kpkg−1s,sp−1

s γs ≤ 1, s = 1, . . . , S∑Ss=1 ps

∥∥v[n]s

∥∥2

2≤ Pn, n = 1, . . . , N

(129)

with positive variables {ps}Ss=1 and {γs}S

s=1. Denote the solution by p?s and γ?

s . If

maxs |γ?s − γ| > εγ set γ = (γ?

1 , . . . , γ?S) and go to Step 2, otherwise STOP.

Let us now focus on Step 3 of Algorithm 4.3.1 and describe in detail how the op-timization problem (127) is solved. First, observe that the change of variable ms =√

psvs defines a bijective mapping between the sets {(ps,vs) |ps > 0, ‖vs‖2 = 1, ps ∈IR,vs ∈ CT } and CT . Thus, one can solve problem (127) for the variables ms ∈ CT ,and then recover the optimal ps and vs. Furthermore, by replacing the positive variableβ by δ2, the following equivalent reformulation of problem (127) is obtained:

minimize δ2

subject to γs ≤∣∣uH

s Hksms

∣∣2

1 +∑S

k=1,k 6=s

∣∣uHs Hksmk

∣∣2 , s = 1, . . . , S

∑Ss=1

∥∥m[n]s

∥∥2

2≤ δ2Pn, δ ≥ 0, n = 1, . . . , N

. (130)

Note that the constraints of problem (130) can be expressed as generalized inequal-ity with respect to the second-order cone [256]. Thus, by following the approach of[256, Section IV.B], problem (130) can be further reformulated as the following second-order cone program (SOCP):

minimize δ

subject to

√1 + 1

γsuH

s Hksms

MHHHks

us

1

ºSOC 0, s = 1, . . . , S

[δ√

Pn

vec(M[n])

]ºSOC 0, n = 1, . . . , N

(131)

where M = [m1, . . . ,mS ], M[n] = [m[n]1 , . . . ,m[n]

S ], and ºSOC denotes the gener-alized inequality with respect to the second-order cone [26, 256], i.e., for any x ∈ IR

126

and y ∈ Cn, [x,yT]T ºSOC 0 is equivalent to x ≥∥∥y

∥∥2. Note that the objective

function of problem (130) has been replaced by δ in problem (131), since for any δ ≥ 0,minimizing δ2 is equivalent to minimizing δ. Problem (131) can be solved by using astandard SOCP solver [26, 256]. Let us denote by δ?, m?

s its solution. The solution ofproblem (127) is given by β? = δ?2, v?

s = m?s/

∥∥m?s

∥∥2, p?

s =∥∥m?

s

∥∥2

2, s = 1, . . . , S.

4.3.2 Generalization to other system performance criteria

A close investigation of Algorithm 4.3.1 described in Section 4.3.1 reveals the followinginteresting property: the particular expression of the objective function in problem (126)is used only at the second step of the algorithm. Therefore, Algorithm 4.3.1 can be eas-ily extended to other system performance criteria, described as a general nondecreasingfunction of SINR values of the data streams f(γs, . . . , γS).

Similarly to (126), the general linear transceiver design problem can be formulatedas

maximize f(γs, . . . , γS)

subject to γs ≤ps

∣∣uHs Hksvs

∣∣2

1 +∑S

k=1,k 6=s pk

∣∣uHs Hksvk

∣∣2 , s = 1, . . . , S

∑Ss=1 ps

∥∥v[n]s

∥∥2

2≤ Pn, n = 1, . . . , N

‖us‖2 = 1, ‖vs‖2 = 1, ps ≥ 0, s = 1, . . . , S

(132)

where the variables are vs ∈CT , us ∈CRks , ps ∈ IR, γs ∈ IR.The problem is solved by using a modified version of Algorithm 4.3.1, where at the

second step problem (126) is replaced by problem (132). Recall that this step computesthe optimum power allocation by considering the transmit and receive beamformersfixed. Consequently, this optimization problem is identical to the problem of computingthe power allocation vector at Step 3 of Algorithm 4.2.1, except that the sum powerconstraint is replaced by a set of N power constraints (one for each antenna groupAn). Notice that all power constraints are linear, and, thus, the algorithms described inSection 4.2.3 can be used to solve the resulting power allocation problem [214–218].

4.3.3 Numerical examples

For the numerical results, the same simulation setup as in Section 4.2.5 has been con-sidered, i.e., a MIMO DL channel with T = 4 antennas at the BS and Rk = 2 antennas

127

at each user terminal operating in a frequency-flat fading channel.Figure 21 shows the sum-rate achieved after different numbers of iterations of Al-

gorithm 4.3.1. In order to facilitate the comparison to the sum-capacity, a sum powerconstraint Pmax for all transmit antennas has been considered first. All S = 4 datastreams were equally weighted, i.e., w = [1, 1, 1, 1]T. Figure 21(a) shows the sum-rateversus SNR for K = 20 users, and Figure 21(b) shows the sum-rate versus the numberof users for SNR = 10dB. As a reference, the sum-rate achieved when only opportunis-tic beamforming is used and the sum-capacity of the MIMO DL channel (where onlythe scheduled users are considered) are also presented. As the results show, the sum-rate provided by the proposed optimization method is within 0.5 - 1.5 bits/second/Hz ofthe sum-capacity. The algorithm converges relatively fast, i.e., depending on the SNRpoint and K, it requires about 30-70 iterations to arrive at a sum-rate stationary point.

Figure 22 compares the sum-rates achieved by the proposed algorithm under threedifferent power constraints: a single sum power constraint Pmax for all four transmitantennas, two sum power constraints Pmax/2 for the antenna groups A1 = {1, 2} andA2 = {3, 4}, and an individual power constraint Pmax/4 for each antenna. Clearly, thelast power constraint is the most restrictive. The sum-rate loss is about 5% comparedto the case of sum power constraint for all transmit antennas. Figure 22 shows also thesum-rate provided by the ZF design [202] with a sum power constraint. The resultsshow that the proposed sum-rate maximization method provides a large gain, even inthe case of per antenna power constraint.

128

2 4 6 8 10 12 14 16 18 204

6

8

10

12

14

16

18

20

22

24

26

SNR

Sum

−rat

e [b

its/s

ec/H

z]

1,2,4,8,16,32,64,100iterations

sum−capacity of the scheduled users (DPC)sum−rate with opportunistic beamformingsum−rate with proposed rate maximization alg.

(a)

101

102

6

7

8

9

10

11

12

13

14

15

16

Number of users

Sum

−rat

e [b

its/s

ec/H

z]

1,2,4,8,16,32,100iterations

sum−capacity of the scheduled users (DPC)sum−rate with opportunistic beamformingsum−rate with proposed rate maximization alg.

(b)

Fig 21. Sum-rate performance of Algorithm 4.3.1 after different numbers of iter-ations for T = 4T = 4T = 4, Rk = 2Rk = 2Rk = 2 antennas and w = [1, 1, 1, 1]= [1, 1, 1, 1]= [1, 1, 1, 1] T; (a) K = 20K = 20K = 20 users, SNR =2÷ 202÷ 202÷ 20 dB, (b) SNR = 10 dB, K = 2÷ 256K = 2÷ 256K = 2÷ 256 users.

129

2 4 6 8 10 12 14 16 18 204

6

8

10

12

14

16

18

20

22

24

26

SNR

Sum

−rat

e [b

its/s

ec/H

z]

sum−capacity of the scheduled users (DPC)per all antenans sum power constraintper groups of 2 antennas sum power constraintper each antenna power constraintzero forcingopportunistic beamforming only

(a)

101

102

6

7

8

9

10

11

12

13

14

15

16

Number of users

Sum

−rat

e [b

its/s

ec/H

z]

sum−capacity of the scheduled users (DPC)per all antenans sum power constraintper groups of 2 antennas sum power constraintper each antenna power constraintzero forcingopportunistic beamforming only

(b)

Fig 22. Sum-rate performance of Algorithm 4.3.1 under different power constraintsfor T = 4T = 4T = 4, Rk = 2Rk = 2Rk = 2 antennas and w = [1, 1, 1, 1]= [1, 1, 1, 1]= [1, 1, 1, 1] T; (a) K = 20K = 20K = 20 users, SNR = 2÷ 202÷ 202÷ 20 dB,(b) SNR = 10 dB, K = 2÷ 256K = 2÷ 256K = 2÷ 256 users.

130

4.4 Summary and discussion

The problem of linear transceiver design for MIMO downlink channels was consideredin this chapter. For the case in which the channel of the scheduled users is availableat the BS, a general framework for a joint transmit and receive beamformer design ac-cording to different optimality criteria was provided. The optimality criteria were fairlygeneral and include sum power minimization subject to minimum SINR constraints perdata stream, balancing of SINR values among data streams, minimum SINR maximiza-tion, weighted sum-rate maximization, and sum MSE minimization.

The proposed algorithms can handle multiple antennas at the BS and at the mobileterminals with an arbitrary number of data streams per scheduled user. Furthermore,they can be easily modified to accommodate certain supplementary constraints imposedby a specific application (e.g., minimum QoS requirements for a subset of users and/ordata streams), and the feasibility of the resulting problems can be verified. Besides thetraditional sum power constraint on the transmit beamformers, solutions for a more gen-eral power constraint model were provided. The model considered allows for multiplepower constraints on arbitrary subsets of transmit antennas. Such power constraintsare required, e.g., in the case of distributed MIMO antenna architectures, where theBS performs cooperative antenna heads transmission subject to individual sum powerconstraints for each antenna head. Furthermore, certain applications may require a perantenna power constraint due to issues related to the linearity of the power amplifiers.

The optimization problems encountered in the beamformer design are not convex ingeneral. Therefore, in some cases, the problem of finding the global optimum is intrin-sically non-tractable and suboptimal methods must be used. The original optimizationproblems were decomposed as a series of remarkably simpler optimization problemswhich can be efficiently solved by using standard convex optimization methods. Eventhough each subproblem is optimally solved, and the objective function is monotoni-cally improved at each iteration, there is no guarantee that the global optimum has beenfound due to the nonconvexity of the original problems. However, the simulations haveshown that the proposed algorithms converge rapidly to a solution, which can be a localoptimum, but is still efficient. As an example, the sum MSE lower bound developedin Section 4.2.4 was asymptotically attained by the proposed algorithm for all channelrealizations for which its attainability can be guaranteed. The simulations showed alsothat even few iterations in the proposed algorithm are enough to outperform the existingZF based solutions.

131

The weighted sum-rate maximization algorithm can also be a useful tool in findingthe gap between the achievable rate region with linear processing only and the rateregion achievable with DPC. The weighted sum MSE minimization algorithm can beused to compute the achievable MSE region, and the SINR balancing algorithm enablesone to study the impact of different fairness requirements on the total system throughput.Finally, the sum power minimization subject to minimum SINR constraints allows oneto evaluate the feasibility of a set of QoS requirements and enables an effective useradmission control mechanism.

132

5 Conclusions and future work

Advanced MIMO transceiver structures capable of utilizing the knowledge of CSI atthe TX side to optimize certain system performance criteria under different constraintswere considered in this thesis. The first chapter included a literature review related tothe topic under consideration. Adaptive transmission schemes for point-to-point MIMOsystems were addressed in Chapter 2. A robust link adaptation method for TDD systemsemploying MIMO-OFDM channel eigenmode based transmission was developed. Thescheme maintains a constant FER by controlling the instantaneous transmitted powerin such a way that the average SINR at output of the linear receivers is kept constant.A bit and power loading algorithm which copes with reduced signaling overhead to in-form the receiver about the transmission parameters was proposed. Despite its reducedcomputational complexity, the numerical results have shown that the throughput degra-dation compared to the optimal Hughes-Hartogs algorithm is negligible. The effect ofspatial correlation on the achievable spectral efficiency was studied by numerical sim-ulations. It is shown that by using a simple scalar channel encoder in an unbalancedantenna MIMO system, the achieved spectral efficiency at low and medium SNR val-ues is significantly higher than the outage MIMO capacity with unknown CSI at theTX.

Numerical algorithms for computing the sum-capacity of the MIMO downlink chan-nel were considered in Chapter 3. By using the uplink-downlink duality, two algorithmswere derived. The first one, denoted as IWF-RUP, is an extension of the original IWFalgorithm used to optimize the transmit covariance matrices of the MIMO multiple-access channel with individual per-user power constraints. In contrast to the previouslyproposed SPC-IWF algorithms, it has lower complexity and requires no additional pre-cautions to ensure the convergence. At each step, the IWF-RUP algorithm increases thesum-rate by updating just two randomly selected covariance matrices while preservingtheir sum power. It was proved analytically that the IWF-RUP algorithm is conver-gent with probability one. The computer simulations demonstrated that it convergessignificantly faster than the previously proposed SPC-IWF algorithms and the differ-ence is more accentuated as the number of users, transmit antennas and/or subcarriersincreases.

The second algorithm studied is based on the dual decomposition method and con-

133

sists of two nested loops. The inner loop optimizes the transmit covariance matrices fora fixed waterfilling level, and the outer loop finds the optimum waterfilling level by us-ing a subgradient method. The problem of tracking the instantaneous error of the innerloop was considered, and an upper bound was derived by using the Lagrange dualitytheory. The upper bound was used to construct a stopping criterion for the inner loop.The simulations demonstrated that the proposed algorithm based on the derived upperbound performs very close to a hypothetical implementation where the true error of theinner loop could be computed. It was further shown that the rate of convergence canbe improved substantially by introducing a covariance normalization step between twosuccessive outer iterations. When the proposed algorithms are compared to each other,the numerical results showed that the IWF-RUP algorithm provides the best tradeoffbetween complexity and speed of convergence for a moderate number of users. Nev-ertheless, as the number of users increases, the dual decomposition based algorithmconverges substantially faster.

The problem of linear transceiver design for MIMO downlink channels was studiedin Chapter 4. A general framework for joint transmit and receive beamformer design ac-cording to different optimality criteria was provided. The optimality criteria were fairlygeneral and include sum power minimization subject to minimum SINR constraintsper data stream, balancing of SINR values among data streams, minimum SINR maxi-mization, weighted sum-rate maximization, and sum MSE minimization. The proposedalgorithms can handle multiple antennas at the BS and at the mobile terminals with anarbitrary number of data streams per scheduled user. Besides the traditional sum powerconstraint on the transmit beamformers, solutions for a more general power constraintmodel were provided. The considered model allows for multiple power constraints onarbitrary subsets of transmit antennas. Such power constraints are required, e.g., in thecase of distributed MIMO antenna architectures, where the BS performs cooperative an-tenna heads transmission subject to individual sum power constraints for each antennahead. Furthermore, certain applications may require a per antenna power constraintdue to issues related to the linearity of the power amplifiers. The computer simulationsshowed that the sum-rate degradation due to such supplementary constraints is minor.

The weighted sum-rate maximization algorithm is a useful tool for studying thegap between the achievable rate region with linear transceiver architectures and the rateregion achievable with optimal nonlinear pre-coders using DPC. As the linear archi-tectures are computationally less demanding, this comparison reveals more insight onthe performance complexity tradeoff. The weighted sum MSE minimization algorithm

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can be used to compute the achievable MSE region, and the SINR balancing algorithmenables one to study the impact of different fairness requirements on the total systemthroughput. Finally, the sum power minimization subject to minimum SINR constraintsallows one to evaluate the feasibility of a set of QoS requirements and enables an ef-fective user admission control mechanism. The transceiver design solutions based onconvex optimization techniques can be easily extended to accommodate various sup-plementary constraints, such as upper and/or lower bounds for the SINR values andguaranteed QoS for different subsets of users. The ability to handle transceiver op-timization problems where a network-centric objective (e.g., aggregate throughput ortransmitted power) is optimized subject to user-centric constraints (e.g., minimum QoSrequirements) is an important feature which must be supported by future broadbandcommunication systems.

When mobile users are equipped with multiple antennas, the linear transceiver de-sign leads to nonconvex optimization problems. In such cases, the problem of findingthe global optimum is intrinsically non-tractable and suboptimal methods must be used.The proposed iterative algorithms decompose the original transceiver design problemsas a series of remarkably simpler optimization subproblems which can be efficientlysolved by using standard convex optimization techniques. Even though the subprob-lems are optimally solved, and the objective function is monotonically improved at eachiteration, there is no guarantee that the global optimum is found due to the nonconvex-ity of the original problem. However, the simulations showed that algorithms convergerapidly to a solution, which can be a local optimum, but is still efficient. As an example,the sum MSE lower bound was asymptotically attained for all channel realizations forwhich its attainability can be guaranteed. The iterative solutions, where most of theimprovement is achieved during the first few iterations, make the proposed methods at-tractive for practical implementation in future broadband communication systems. Thesimulations demonstrate that even few iterations in the proposed algorithm are enoughto outperform the existing ZF based solutions. A simple heuristic SINR vector maxi-mization algorithm with lower complexity than ZF solutions was also derived. Eventhough it was not specifically designed to optimize a certain objective function, it wasshown to perform remarkably well according to different system performance criteria.

While issues related to the point-to-point MIMO system design have been analyzedthoroughly during the last two decades, there are still many open problems in the areaof transceiver design for MIMO multiuser systems. Since the capacity region of theMIMO downlink channel was an open problem until recently, only the sum-capacity

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maximization has been considered in this thesis. Numerical algorithms for computingthe entire capacity region more efficiently than by using general interior-point methodsdeserve future investigations. The linear transceiver optimization algorithms for down-link must be evaluated with more realistic channel models and system parameters forinvestigating the system level benefits. The single-cell environment used in the stud-ies is useful to obtain only preliminary qualitative results. However, accurate modelsfor data traffic and inter-cell interference should be also taken into consideration in fur-ther studies. The MIMO multiuser transceiver optimization with imperfect CSI at thetransmitter side is still an open problem of great interest. Such an optimization prob-lem is probably intractable, but recent results from the robust optimization theory couldprovide possible ways to approach this problem.

As opportunistic communication systems are gaining more interest, one could alsostudy the different dimensions of opportunism with MIMO techniques. These include,e.g., spectrum, location, user cooperation, channel knowledge and behavior, traffic mod-els as well as topology awareness. Finding a practical combination of these paves theroad for future cognitive radio systems.

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152

Appendix 1 KKT optimality conditions

Since problem (59) is convex, the following KKT optimality conditions are necessaryand sufficient for the variables W?, {S?

k}Kk=1, {p?

k}Kk=1, Γ?, {Ψ?

k}Kk=1, {ν?}K

k=1 to beprimal and dual optimal.

W? ¹ ∑Kk=1 HkS?

kHHk + I

S?k º 0,

tr{S?k} ≤ p?

k,

k = 1, . . . , K

k = 1, . . . , K

Γ? º 0Ψ?

k º 0,

ν?k ≥ 0,

k = 1, . . . , K

k = 1, . . . , K

Γ? = W?−1

HHkΓ?Hk = ν?

kI−Ψ?k,

ν?k = λ,

k = 1, . . . , K

k = 1, . . . , K

tr[Γ?

(W? −∑K

k=1 HkS?kH

Hk − I

)]= 0

tr(Ψ?kS

?k) = 0,

ν?k [tr(S?

k)− p?k] = 0,

k = 1, . . . , K

k = 1, . . . , K

(133)

From ν?k = λ > 0, k = 1, . . . ,K, it follows that the corresponding constraints are

tight, i.e., tr(S?k) = p?

k. Furthermore, Γ? = W?−1 Â 0, implies that the correspondingconstraint is tight, i.e., W? =

∑Kk=1 HkS?

kHHk + I. This confirms the equivalence of

problems (56) and (59), and implies that

Γ? =

(K∑

k=1

HkS?kH

Hk + I

)−1

(134)

and

HHk

(K∑

k=1

HkS?kH

Hk + I

)−1

Hk = λI−Ψ?k, k = 1, . . . ,K . (135)

The last equation is very similar to the single-user waterfilling condition. The onlydifference is that λ (i.e., the inverse of the waterfilling level) is not obtained from a sumpower constraint. It is a fixed value that itself determines the sum power. Since Ψ?

k º 0,from (135) it follows that λmax

({S?k}K

k=1

)defined in (69) satisfies

λ({S?

k}Kk=1

)=

{λ0 < λ if Ψ?

k  0 for all k = 1, . . . , K

λ otherwise. (136)

153

From the complementary slackness, i.e., tr(Ψ?kS

?k) = 0, it follows that Ψ?

k  0 forall k = 1, . . . ,K implies that S?

k = 0 for all k = 1, . . . , K. By letting S?k = 0 in (135),

it is easy to observe that Ψ?k  0 if and only if λ > λmax

(HH

kHk

)for all k = 1, . . . , K.

Physically, this situation corresponds to a very small water level in (135), which doesnot inundate any of the channel eigenmodes. Clearly, such λ values can not be optimalfor the master problem (57). Thus, the search domain of the master problem (57) can berestricted to λ < maxk=1,...,K λmax

(HH

kHk

). In such conditions, λ

({S?k}K

k=1

)= λ,

and, therefore, the dual variable defined in (68) become an optimal dual variable, i.e.,

Γ({S?

k}Kk=1

)=

(K∑

k=1

HkS?kH

Hk + I

)−1

= Γ? . (137)

154

Appendix 2 Uplink-downlink SINR duality viaLagrange duality

Let us denote by γdl∆= [γdl

1 , . . . , γdlS ]T and γul

∆= [γul1 , . . . , γul

S ]T the data streams’SINRs (82) and (84), and let γ

∆= [γ1, . . . , γS ]T, γ Â 0 be a set of minimum SINRsrequirements. From (82) and (84) follows that

γdl º γ iff Gp º 1 and γul º γ iff GTq º 1 (138)

where the matrix G ∈ IRS×S is given by

[G]s,k∆=

{−gs,k if s 6= k

gs,s/γs if s = kwhere gs,k = |uH

s Hksvk|2 . (139)

The downlink sum power minimization problem under γdl º γ constraint can beexpressed as

minimize 1Tp

subject to Gp º 1

p º 0

(140)

and has the Lagrange dual problem [26, Sect. 5.2]

maximize 1Tλ1

subject to −GTλ1 − λ2 + 1 = 0λ1 º 0 ; λ2 º 0

(141)

where the dual variables are λ1, λ2 ∈ IRS . By eliminating λ2 between the last twoconstraints the following equivalent formulation is obtained

maximize 1Tλ1

subject to GTλ1 ¹ 1

λ1 º 0

. (142)

The physical meaning of the dual problem (142) can be interpreted as follows. Sincethe dual variable λ1 º 0, it can be interpreted as the uplink power allocation, i.e.,q = λ1. The first constraint is equivalent to γul ¹ γ (see (138)). Therefore, thedual problem can be interpreted as the maximization of the uplink sum power under themaximum SINR constraints γul ¹ γ.

155

First, the case in which problems (140) and (142) are feasible is considered. Thesolutions of problems (140) and (142) are denoted by p? and q?, respectively. Thefollowing conclusions can be made:

– The sum powers of the downlink and the dual uplink channel are equal, i.e., 1Tp? =1Tq?. This follows from the zero duality gap of a feasible linear program [26].

– The SINR values of the downlink and the dual uplink channel are equal, and theyare equal to γ, i.e., γ?

dl(p?) = γ?

ul(q?) = γ. This follows from the fact that at the

optimal point all the SINR constraints have to be active, i.e., they hold with equality.

The last fact can be easily proved by contradiction as follows. Suppose, for example,that there is a positive margin (or slack) in the kth constraint of problem (140), i.e.,γdl

k > γk. In this case, p?k can be decreased until the constraint is satisfied with equality

without violating any other constraint. This strictly decreases the objective function andcontradicts the assumption that p? is optimal. A similar reasoning can be used to provethat all SINR constraints are active in problem (142).

In the following, the feasibility conditions for problems (140) and (142) are estab-lished. Since they are dual linear programs and the objective of (140) is bounded frombelow, they can be either both feasible or both infeasible [26]. Therefore, only problem(140) will be analyzed in the sequel. From (139) it follows that all off-diagonal entriesof the matrix G are smaller or equal to zero. Note that since γ  0, (82) implies thatp  0, i.e., if (140) is feasible, the solution must be strictly positive. Therefore, [101,Theorem 2.5.3.12] implies that the problem (140) is feasible if and only if the matrix G

is an M-matrix, that is, all eigenvalues of G have positive real part.Note that by expressing the matrix G as G = D−1(I −DC) with D, C given by

(97), the classical feasibility condition18 obtained in [19, 90, 171] under the assumptionthat the matrix A = DC is primitive can be also obtained. In fact, this type of decom-position is a standard trick to connect the theory of M-matrices to the Perron-Frobeniustheory of nonnegative matrices (see also [101, Lemma 2.5.2.1]). Note that, in contrastto the classical proofs [19, 90, 171], here it was not assumed that the matrix A is primi-tive for proving the converse. This extends the generality of the result, e.g., to the caseof Costa precoder, where the matrix A is upper triangular and, therefore, not primitive.

Since at the optimal point the SINR constraints are active, i.e., Gp? = 1 and

18Problem (140) is feasible if and only if the Perron-Frobenius eigenvalue of A is strictly less than 1.

156

GTq? = 1, the solutions are unique and given by{

p? = G−11 = (I−DC)−1D1

q? = (GT)−11 = D(I−CTD)−11. (143)

In sum, it has been established that a set of minimum SINR requirements, γ Â 0,are achievable in the downlink channel if and only if they are achievable in the dualuplink channel. Furthermore, when they are achievable, the sum power required inuplink is smaller or equal to the sum power required in downlink. By replacing G

by GT in (140) the converse is obtained, i.e., the sum power required in downlink issmaller or equal to the sum power required in uplink. Therefore, when the SINR valuesγ are achievable, they require the same sum power in both the original downlink andthe dual uplink channels.

157

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