energy efficiency optimization of 5g chain · pdf filearxiv:1604.02665v1 [cs.ni] 10 apr 2016...

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ENERGY EFFICIENCY OPTIMIZATION OF 5G

RADIO FREQUENCY CHAIN SYSTEMS

Ran Zi1, Xiaohu Ge1, John Thompson2, Cheng-Xiang Wang3, Haichao Wang1, Tao Han1

1Department of Electronics and Information EngineeringHuazhong University of Science and Technology, Wuhan 430074, Hubei, P. R. China.

Email: xhge,[email protected], [email protected] for Digital Communications,

University of Edinburgh, Edinburgh, EH9 3JL, UK.Email: [email protected]

3Joint Research Institute for Signal and Image Processing,School of Engineering & Physical Sciences,

Heriot-Watt University, Edinburgh, EH14 4AS, UK.Email: [email protected]

Abstract—With the massive multi-input multi-output (MIMO)antennas technology adopted for the fifth generation (5G) wirelesscommunication systems, a large number of radio frequency(RF) chains have to be employed for RF circuits. However,a large number of RF chains not only increase the cost ofRF circuits but also consume additional energy in 5G wirelesscommunication systems. In this paper we investigate energyand cost efficiency optimization solutions for 5G wireless com-munication systems with a large number of antennas and RFchains. An energy efficiency optimization problem is formulatedfor 5G wireless communication systems using massive MIMOantennas and millimeter wave technology. Considering the non-concave feature of the objective function, a suboptimal iterativealgorithm, i.e., the energy efficient hybrid precoding (EEHP)algorithm is developed for maximizing the energy efficiencyof5G wireless communication systems. To reduce the cost of RFcircuits, the energy efficient hybrid precoding with the minimumnumber of RF chains (EEHP-MRFC) algorithm is also proposed.

Submitted to IEEE Journal on Selected Areas in Communications Special Issue onEnergy-Efficient Techniques for 5G Wireless CommunicationSystems.

Correspondence author: Dr. Xiaohu Ge, Tel: +86 27 8755 7942,Fax: +86 27 87557943, Email: [email protected]

The authors would like to acknowledge the support from the International Scienceand Technology Cooperation Program of China under grants 2015DFG12580 and2014DFA11640, the National Natural Science Foundation of China (NSFC) under thegrants 61471180, NFSC Major International Joint Research Project under the grant61210002, the Fundamental Research Funds for the Central Universities under thegrant 2015XJGH011. This research is partially supported bythe EU FP7-PEOPLE-IRSES, project acronym S2EuNet (grant no. 247083), projectacronym WiNDOW (grantno. 318992) and project acronym CROWN (grant no. 610524), the EU H2020 5GWireless project (Grant no. 641985), EU FP7 QUICK project (Grant no. PIRSES-GA-2013-612652), National 863 project in 5G by Ministry of Science and Technology inChina (Grant no. 2014AA01A701), National international Scientific and TechnologicalCooperation Base of Green Communications and Networks (No.2015B01008) andHubei International Scientific and Technological Cooperation Base of Green BroadbandWireless Communications.

Moreover, the critical number of antennas searching (CNAS)anduser equipment number optimization (UENO) algorithms arefurther developed to optimize the energy efficiency of 5G wirelesscommunication systems by the number of transmit antennasand UEs. Compared with the maximum energy efficiency ofconventional zero-forcing (ZF) precoding algorithm, numericalresults indicate that the maximum energy efficiency of theproposed EEHP and EEHP-MRFC algorithms are improved by220% and 171%, respectively.

Index Terms—Energy efficiency, cost efficiency, radio fre-quency chains, baseband processing, 5G wireless communicationsystems I. I NTRODUCTION

The massive multi-input multi-output (MIMO) antennas andthe millimeter wave communication technologies have beenwidely known as two key technologies for the fifth generation(5G) wireless communication systems [1]–[6]. Compared withconventional MIMO antenna technology, massive MIMO canimprove more than 10 times spectrum efficiency in wirelesscommunication systems [7]. Moreover, the beamforming gainbased on the massive MIMO antenna technology helps toovercome the path loss fading in millimeter wave channels.For MIMO communication systems with traditional radiofrequency (RF) chains and baseband processing, one antennacorresponds to one RF chain [8], [9]. In this case, a largenumber of RF chains has to be employed for massive MIMOcommunication systems. These RF chains not only consumea large amount of energy in wireless transmission systems butalso increase the cost of wireless communication systems [10].Therefore, it is an important problem to find energy efficientsolutions for 5G wireless communication systems with a largenumber of antennas and RF chains.

To improve the performance of multiple antenna transmis-sion systems, hybrid precoding technology combining digital

http://arxiv.org/abs/1604.02665v1

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. XX, NO. Y, MONTH 2015 2

baseband precoding with analog RF precoding was investi-gated in [9], [11]–[17]. Based on the joint design of RFchains and baseband processing, a soft antenna subset selectionscheme was proposed for multiple antenna channels [9]. Whena single data stream is transmitted, the soft antenna subsetselection scheme was able to achieve the same signal-to-noiseratio (SNR) gain as a full-complexity scheme involving allantennas in MIMO wireless communication systems. By for-mulating the problem of millimeter wave precoder design as asparsity-constrained signal recovery problem, algorithms weredeveloped to approximate optimal unconstrained precodersand combiners in millimeter wave communication systemswith large antenna arrays [11]. To maximize the sum rateof MIMO communication systems, a hybrid beamforming ap-proach was designed indirectly by considering a weighted summean square error minimization problem incorporating thesolution of digital beamforming systems [12]. Based on a low-complexity channel estimation algorithm, a hybrid precodingalgorithm was proposed to achieve a near-optimal performancerelative to the unconstrained digital solutions in the single usermillimeter wave communication system [13]. To reduce thefeedback overhead in millimeter wave MIMO communicationsystems, a low complexity hybrid analog/digital precodingscheme was developed for the downlink of the multi-user com-munication systems [14]. To maximize the minimum averagedata rate of users subject to a limited RF chain constraint anda phase-only constraint, a two stage precoding scheme wasproposed to exploit the large spatial degree of freedom gainin massive MIMO systems with reduced channel state infor-mation (CSI) signaling overhead [15]. Based on phase-onlyconstraints in the RF domain and a low-dimensional basebandzero-forcing (ZF) precoding, a low complexity hybrid precod-ing scheme was presented to approach the performance ofthe traditional baseband ZF precoding scheme [16]. Comparedwith the digital beamforming scheme, the hybrid beamformingscheme was shown to achieve the same performance with theminimum RF chains and phase shifters in multi-user massiveMIMO communication systems [17]. In the above studies,hybrid precoding schemes have rarely been investigated tooptimize the energy efficiency of massive MIMO communica-tion systems. While in cellular communication systems, energyefficiency has been considered as a critical performance metric[18].

To improve the energy efficiency of MIMO communicationsystems, some digital precoding schemes have been studiedin [19]–[21]. Based on static and fast-fading MIMO channels,an energy efficient precoding scheme was investigated whenthe terminals are equipped with multiple antennas [19]. Jointlyconsidering the transmit power, power allocation among datestreams and beamforming matrices, a power control andbeamforming algorithm was developed for MIMO interferencechannels to maximize the energy efficiency of communicationsystems [20]. Transforming the energy efficiency of MIMObroadcast channel into a concave fractional program, an opti-mization approach with transmit covariance optimization andactive transmit antenna selection was proposed to improve theenergy efficiency of MIMO systems over broadcast channels[21]. With the massive MIMO concept emerging as a key

technology in 5G communication systems, the energy effi-ciency of massive MIMO has been studied in several papers[10], [22]–[26]. Based on a new power consumption model formulti-user massive MIMO, closed-form expressions involvingthe number of antennas, number of active users and grossrate were derived for maximizing the energy efficiency ofmassive MIMO systems with ZF processing [10]. When linearprecoding schemes are adopted at the BS, it is proved thatmassive MIMO can improve the energy efficiency by threeorders of magnitude [22]. Considering the transmit powerand the circuit power in massive MIMO systems, a powerconsumption model has been proposed to help optimize theenergy efficiency of multi-cell mobile communication systemsby selecting the optimal number of active antennas [23].When the sum spectrum efficiency was fixed, the impactof transceiver power consumption on the energy efficiencyof ZF detector was investigated for the uplinks of massiveMIMO systems [24]. Considering the power cost by RFgeneration, baseband computing, and the circuits associatedwith each antenna, simulation results in [25] illustrated thatmassive MIMO macro cells outperforms LTE macro cells inboth spectrum and energy efficiency. To maximize the energyefficiency of the massive MIMO OFDMA systems, an energyefficient iterative algorithm was proposed by optimizing thepower allocation, data rate, antenna number, and subcarrierallocation in [26].

However, in all the aforementioned studies, only the trans-mission rate and the hardware complexity of hybrid precod-ing systems were analyzed for MIMO or massive MIMOcommunication systems. Moreover, energy efficient solutionsfor the RF chains and the baseband processing of 5G wire-less communication systems is surprisingly rare in the openliterature. On the other hand, the energy and cost increasethrough using a large number of RF chains is an inevitableproblem for 5G wireless communication systems. Motivatedby the above gaps, in this paper we propose energy and costefficient optimization solutions for 5G wireless communicationsystems with a large number of antennas and RF chains. Thecontributions and novelties of this paper are summarized asfollows.

1) The BS energy efficiency including the energy consump-tion of RF chains and baseband processing is formulatedas an optimization function for 5G wireless communi-cation systems adopting massive MIMO antennas andmillimeter wave technologies.

2) Considering the non-concave feature of the optimizationobjective function, a suboptimal solution is proposedto maximize the BS energy efficiency using the energyefficient hybrid precoding (EEHP) algorithm.

3) To reduce the cost of RF circuits, the energy efficienthybrid precoding with the minimum number of RFchains (EEHP-MRFC) algorithm is developed to tradeoffthe energy and cost efficiency for 5G wireless commu-nication systems.

4) To utilize the user scheduling and resource manage-ment schemes, the critical number of antennas search-ing (CNAS) and user equipment number optimization


RFN

BBB

TxNK

RFB

1UE

2UE

UEk

UEK

Fig. 1 system model.

(UENO) algorithms are developed to maximize the en-ergy efficiency of 5G wireless communication systems.

The remainder of this paper is outlined as follows. SectionII describes the system model of 5G wirelss communicationsystems and the energy efficiency optimization problem is for-mulated. In Section III, an iteration algorithm, i.e., the EEHPalgorithm is developed to maximize the energy efficiency of5G wireless communication systems. To save the cost of RFcircuits, the EEHP-MRFC algorithm is developed to tradeoffthe energy and cost efficiency in Section IV. Moreover, theCNAS and UENO algorithms are developed to optimize theenergy efficiency of 5G wireless communication systems con-sidering the number of transmit antennas and user equipments(UEs). Detailed numerical simulations are presented in SectionV. Finally, conclusions are drawn in Section VI.

II. SYSTEM MODEL

Considering the impact of massive MIMO antennas on theRF chains and the baseband processing, the energy efficiencyof 5G wireless communication systems has to be rethought.In the following, we describe the system configuration of 5Gwireless communication systems, including massive MIMOantennas, RF chains and the baseband processing. Moreover,the energy efficiency optimization problem of 5G wirelesscommunication systems is formulated.

A. System Configuration

Without loss of generality, a single cell scenario is illustratedin Fig. 1, where a BS andK UEs are located in the 5Gwireless communication system. The BS is assumed to beequipped withNTx antennas. And based on the massive MIMOconfiguration, we assumeNTx > 100 in our system [7], [22].There areK active UEs, each with a single antenna, that areassociated with the BS. Moreover, the transmission systemof BS is equipped withNRF RF chains. One baseband datastream is assumed to be associated with one UE in 5G wirelesscommunication systems. In this paper, our studies focuses onthe downlinks of 5G wireless communication systems.

The received signal at thekth UE is expressed as

yk = hHk BRFBBBx+ wk, (1)

wherex = [x1, ..., xk, ..., xK ]H is the signal vector transmit-ted from the BS toK UEs, where thexk, k = 1, ...,K, areassumed to be independently and identically distributed (i.i.d.)Gaussian random variables with zero mean and variance of 1;BBB ∈ CNRF×K is the baseband precoding matrix, where thekth column ofBBB is denoted asbBB,k which is the basebandprecoding vector associated with thekth UE;BRF ∈ CNTx×NRF

is the RF precoding matrix which is performed byNRF RFchains;wk is the noise received by thekth UE. Moreover, allnoises received by UEs are denoted asi.i.d. Gaussian randomvariables with zero mean and variance of 1. The vectorh

Hk is

the downlink channel vector between the BS and thekth UE.The downlink channel matrix between the BS andK UEs isdenoted asHH = [h1, ...,hk, ...,hK ]

H . The power consumedto transmit signals for thekth UE is expressed as

Pk = ‖BRFbBB,kxk‖2 = ‖BRFbBB,k‖2. (2)

Note that the consumed power for transmitting signals in(2) is expended for a given bandwidthW , which is set to be20MHz in this paper [10]. The millimeter wave communica-tion technology is adopted for 5G wireless communicationsystems. Considering the propagation characteristic of mil-limeter waves in wireless communications, a geometry-basedstochastic modeling (GBSM) is used to express the millimeterwave channel as follows [11], [12], [27], [28]

hk =

√

NTxβkNray

Nray∑

i=1

ρkiu (ψi, ϑi), (3)

whereNray is the number of the multipath between the BSandK UEs. βk = ζ/lγk is the large scale fading coefficientover the wireless link between the BS and thekth UE. ζ isthe lognormal random variable with the zero mean and thevariance of 9.2 dB.lk is the distance between the BS andthe kth UE. γ is the path loss exponent.ρki is the complexgain of theith multipath over thekth UE link, which denotesthe small-scale fading in wireless channels and is governedby a complex Gaussian distribution. Moreover,ρki is i.i.d. fordifferent values ofk(k = 1, ...,K) and i(i = 1, ..., Nray).ψi and ϑi are the azimuth and the elevation angle of theith multipath at the BS antenna array, respectively.u (ψi, ϑi)is the response vector of BS antenna array with the azimuthψi and the elevation angleϑi. Without loss of generality, theBS antenna array is assumed as the uniform planar antennaarray in this paper. Therefore, the response vector of BSantenna array with the azimuthψi and the elevation angleϑi is expressed as [29]

u (ψi, ϑi) =1√NTx

[1, . .., ej2πλd(m sin(ψi)sin(ϑi)+ncos(ϑi)),

..., ej(NTx−1) 2πλd((M−1) sin(ψi)sin(ϑi)+(N−1)cos(ϑi))

]T,

(4)whered is the distance between adjacent antennas,λ is thecarrier wave length,M andN are the row and column numberof the BS antenna array, respectively.m is denoted as themth


antenna in the row of the BS antenna array,1 6 m < M ; n isdenoted as thenth antenna in the column of the BS antennaarray,1 6 n < N .

B. Problem Formulation

Based on the system model in Fig. 1, the link spectrumefficiency of thekth UE is expressed as

Rk = log2

1 +hHk BRFbBB,kb

HBB,kB

HRFhk

K∑

i=1,i6=khHk BRFbBB,ib

HBB,iB

HRFhk + σ2

n

.

(5)

Note that from (5) we get the instantaneous spectrumefficiency. And in this paper, we assume that the BS transmitterhas perfect CSI, i.e. channel vectorshk, k = 1, ...,K areknown at the BS. This assumption is widely adopted for the in-vestigation of precoding problems in massive MIMO systemsand millimeter wave transmission systems [11], [12], [22],[23]. In practical wireless communication systems, the CSIcanbe obtained through uplink channel estimation then appliedto downlink precoding based on the channel reciprocity inthe time division duplex (TDD) mode [7], [30]. Moreover,the millimeter wave multipath channel estimation utilizingcompressed channel sensing was investigated in [13] and[31]. Furthermore, considering all the UEs, the sum spectrumefficiency is expressed by

Rsum=

K∑

k=1

Rk. (6)

The total BS power is expressed as

Ptotal =1

α

K∑

k=1

‖BRFbBB,k‖2 +NRFPRF + PC, (7)

whereα is the efficiency of the power amplifier, and the termK∑

k=1

‖BRFbBB,k‖2 is the power consumed to transmit signals

for K UEs over the given bandwidthW , PRF is the powerconsumed at every RF chain which is comprised by converters,mixers, filters, phase shifters, etc. Considering the number ofantennas is fixed in a wireless communication system, thenumber of phase shifters is also fixed since each phase shifteris associated with one antenna.PC is the power consumed forsite-cooling, baseband processing and synchronization intheBS. To simplify the derivation,PC is fixed as a constant.

In this paper, we focus on how to maximize the BSenergy efficiency (bits per Joule) by optimizing the basebandprecoding matrixBBB, the RF precoding matrixBRF andthe number of RF chainsNRF. This optimization problem is

formed by

(

NoptRF ,B

optRF,B

optBB

)

= arg maxNRF,BRF, BBB

η =WRsum

Ptotal

s.t.∣

∣

∣[BRF]i,j

∣

∣

∣

2

=1

NTx

Rk > Γk, k = 1, ...,KK∑

k=1

‖BRFbBB,k‖2 6 Pmax

, (8)

where W is the transmission bandwidth,Γk is the mini-mum spectrum efficiency required by UEk, andPmax is themaximum transmit power required for wireless downlinks. Ingeneral, the RF precoding is performed by phase shifters,which can change the signal phase but can not change thesignal amplitude. Therefore, the amplitude of RF precodingmatrix is fixed as a constant, which is added as a constraint

for (8), i.e.,∣

∣

∣[BRF]i,j

∣

∣

∣

2

= 1NTx

. Furthermore, there exists aminimum data rate required by each UE in practical appli-cations. The minimum data rate is obtained by multiplyingthe corresponding minimum spectrum efficiency by the band-width. Since the bandwidth is fixed in this paper, an equivalentminimum spectrum efficiency constraint is used to satisfythe minimum data rate requirement, which is expressed as

Rk > Γk, k = 1, ...,K in (8). AndK∑

k=1

‖BRFbBB,k‖2 6 Pmax

is the maximum transmit power constraint.

III. E NERGY EFFICIENT HYBRID PRECODING DESIGN

To maximize the energy efficiency in (8), an EEHP algo-rithm is developed to jointly optimize the baseband precodingmatrix, the RF precoding matrix and the number of RF chainsin the following.

A. Upper Bound of Energy Efficiency

Based on definitions of the baseband precoding matrix andthe RF precoding matrix, the size ofBRF ∈ CNTx×NRF andBBB ∈ CNRF×K is related with the number of RF chainsNRF. To simplify the derivation, we first fix the number ofRF chains. Based on the optimization objective function inthe Section II, (8) is a non-concave function with regard toBRF andBBB. In general, there does not exist an analyticalsolution for such non-concave functions. To tackle this prob-lem,B = BRFBBB is configured as a digital precoding matrixB ∈ CNTx×K , B = [b1, ...,bk, ...,bK ] whose size does notdepend on the number of RF chainsNRF. Furthermore, the

amplitude ofB is free from the constraint∣

∣

∣[BRF]i,j

∣

∣

∣

2

= 1NTx

.Substituting the digital precoding matrixB into (8) and

neglecting the constraint∣

∣

∣[BRF]i,j

∣

∣

∣

2

= 1NTx

, the optimization


Algorithm 1 Energy Efficient Hybrid Precoding-A (EEHP-A) al gorithm .

Begin:

1) AssumingB(0) to be the initial digital precoding matrix,Ω(0)k andΞ(0)

k are calculated by (10) for the UE UEk, k =

1, ...,K;

2) At the nth, n = 1, 2, ..., iteration step, the iterative step lengthµ(n)k is searched within[0, 1] for all K UEs

µ(n)k = arg max

µ(n)k

∈[0,1]

η

[

INTx + µ(n)k

(

[

Ξ(n−1)k

]−1

Ω(n−1)k − INTx

)]

b(n−1)k

s.t. Rk

([

INTx + µ(n)k

(

[

Ξ(n−1)k

]−1

Ω(n−1)k − INTx

)]

b(n−1)k

)

> Γk

K∑

k=1

∥

∥

∥

∥

[

INTx + µ(n)k

(

[

Ξ(n−1)k

]−1

Ω(n−1)k − INTx

)]

b(n−1)k

∥

∥

∥

∥

2

6 Pmax

; (12)

3) Based onµ(n)k , the digital precoding matrixb(n)

k is calculated at thenth n = 1, 2, ... iteration step

b(n)k =

[

INTx + µ(n)k

(

[

Ξ(n−1)k

]−1

Ω(n−1)k − INTx

)]

b(n−1)k ; (13)

4) Based onb(n)k and (10),Ω(n)

k andΞ(n)k are updated;

5) Return to step 2 and keep iterating tillb(n)k converges.

end Begin

problem in the Section II is transformed as

Bopt = arg max

B

η =

WK∑

k=1

Rk

1α

K∑

k=1

‖bk‖2 +NRFPRF + PC

s.t. Rk > Γk, k = 1, ...,KK∑

k=1

‖bk‖2 6 Pmax

, (9a)

where

Rk = log2

1 +hHk bkb

Hk hk

K∑

i=1,i6=khHk bib

Hi hk + σ2

n

. (9b)

Based on (9), the energy efficient is maximized by op-timizing the digital precoding matrixB, where B is onlyconstrained by the maximum transmit powerPmax and theminimum spectrum efficiencyΓk. Assume that the maximumenergy efficiency in (8) is denoted asηmax. Similarly, assumethat the maximum energy efficiency in (9) is denoted asηmax.Compared (8) with (9), two optimization problems have thesame objective function. But (9) has less constraints than thatof (8). Therefore, the solution of (9), i.e., the maximum energyefficiency of (9) is larger than or equal to the solution of(8), i.e., the maximum energy efficiency of (8). Therefore,the maximum energy efficiencyηmax is upper bounded by themaximum energy efficiencyηmax, i.e., ηmax 6 ηmax.

B. Energy Efficiency Local Optimization

Considering (9) is a non-concave function, it is difficultto find a global optimization solution for this optimization

problem. A local optimization solutionηopt with the optimaldigital precoding matrixBopt =

[

bopt1 , · · · ,bopt

k , · · · ,boptK

]

isfirst derived for the energy efficiency optimization in (9).

Denoting the energy efficiency and spectrum efficiency in(9) as functions ofbk, i.e. η (bk) and Rk (bk). The gradientof η (bk) with respect tobk is derived by

∂η (bk)

∂bk=

2

P 2[Ωk −Ξk]bk, (10a)

with

Ωk =Phkh

Hk

K∑

j=1

hHk bjbHj hk + σ2

n

, (10b)

Ξk =

K∑

i=1

Ri

α ln 2INTx+P

K∑

i=1,i6=k

(

hHi bib

Hi hi

(δi)2 + δihHi bib

Hi hi

hihHi

)

,

(10c)

P =1

W

(

1

α

K∑

k=1

‖bk‖2 +NRFPRF + PC

)

, (10d)

δi =

K∑

j=1,j 6=ihHi bjb

Hj hi + σ2

n, (10e)

whereδi is the sum of the received interference power andthe noise power for the UE UEi, i = 1, ...,K. Ri is obtainedfrom (9b) by replacingk with i.

When the zero-gradient condition∂η(bk)∂bk

= 0 is applied,the local optimization solution for the UE UEk, k = 1, ...,Kis derived as

Ξkbk = Ωkbk. (11)


To obtain the optimal digital precoding matrixBopt of thelocal optimization solution, an iterative algorithm is developedand called the energy efficiency hybrid precoding-A (EEHP-A) algorithm on the top of the previous page.

Based on the EEHP-A algorithm, theboptk is obtained by the

convergedb(n)k . After obtainingbopt

k , k = 1, · · · ,K for all KUEs, the local optimization solutionηopt is achieved. To ensurethe convergence of EEHP-A algorithm, the correspondingproof is given as follows.

Proof : AssumingX =[

Ξ(n−1)k

]−1

Ω(n−1)k b

(n−1)k , (12)

and (13) are rewritten as

µ(n)k = arg max

µ(n)k

∈[0,1]

η

µ(n)k X+

(

1− µ(n)k

)

b(n−1)k

s.t. Rk(

b(n)k

)

> Γk

K∑

k=1

∥

∥

∥b(n)k

∥

∥

∥

2

6 Pmax

, (14)

b(n)k = µ

(n)k

[

Ξ(n−1)k

]−1

Ω(n−1)k b

(n−1)k +

(

1− µ(n)k

)

b(n−1)k

= µ(n)k X+

(

1− µ(n)k

)

b(n−1)k

.

(15):

For the UE UEk, k = 1, ...,K, Ξ(n−1)k based on (10c) is a

Hermitian symmetric positive matrix. Hence,Ξ(n−1)k can be

denoted asΞ(n−1)k = ZZ

H , whereZ is a symmetric positivedefinite matrix. Furthermore, the following result is derived as(16):

∂η(

b(n−1)k

)

∂b(n−1)k

H

(

X− b(n−1)k

)

=2

P 2

[

b(n−1)k

]H (

Ω(n−1)k −Ξ

(n−1)k

)

×(

[

Ξ(n−1)k

]−1

Ω(n−1)k − 1

)

b(n−1)k

=2

P 2

[

b(n−1)k

]H(

Ω(n−1)k

[

Ξ(n−1)k

]−1

Ω(n−1)k −

2Ω(n−1)k +Ξ

(n−1)k

)

b(n−1)k

=2

P 2

[

b(n−1)k

]H(

Z−1

Ω(n−1)k −Ω

(n−1)k

)H

×(

Z−1

Ω(n−1)k −Ω

(n−1)k

)

b(n−1)k ≻

−

0

. (16)

Based on (16) and the proposition in [33],η(

b(0)k

)

6

η(

b(1)k

)

6 · · · 6 η(

b(n)k

)

are non-decreasing sequences

for the UE UEk, k = 1, ...,K. Moreover, η (bk) is upper-bounded according to the proposition in [20]. It is easilyknown that the upper-bounded non-decreasing sequence ap-proaches to a convergence value. Therefore,η (bk) in EEHP-Ais proved to converge.

C. Hybrid Precoding Matrices Optimization

Based on (9) and the EEHP-A algorithm, the local opti-mization solutionηopt is obtained. When the hybrid precodingmatrices BRFBBB approach the optimal digital precoding

matrix Bopt with the constraint

∣

∣

∣[BRF]i,j

∣

∣

∣

2

= 1NTx

, the energyefficiency η will approaches the local optimization solutionηopt. Therefore, the optimal hybrid precoding matricesB

optRF

andBoptBB can be solved by minimizing the Euclidean distance

betweenBRFBBB andBopt [11], [12], [34]

(

BoptRF,B

optBB

)

= arg minBRF, BBB

∥

∥Bopt −BRFBBB

∥

∥

F

s.t.∣

∣

∣[BRF]i,j

∣

∣

∣

2

=1

NTx

. (17)

Considering the non-convex constraint∣

∣

∣[BRF]i,j

∣

∣

∣

2

= 1NTx

[32], it is not tractable to analytically solve the optimizationproblem in (17). Based on the millimeter wave channel in(3), the entries of BS antenna array steering matrixU =[

u (ψ1, ϑ1) , ...,u (ψi, ϑi) , ...,u(

ψNray, ϑNray

)]

∈ CNTx×Nray

are constant-amplitude which can be implemented by phaseshifters in BS RF circuits. Meanwhile, as pointed out in [11],the columns vectors of steering matrixU are independentfrom each other in millimeter wave channels. Moreover,U ∈ CNTx×Nray and BRF ∈ CNTx×NRF have the same rownumbers. To simplify the engineering application,NRF columnvectors are selected fromU to form the column vectors ofBRF. The detailed vector selection method is described inthe EEHP-B algorithm. Furthermore, the baseband precodingmatrixBBB is optimized by approachingBRFBBB toB

opt. As aconsequence, the optimization problem in (17) is transformedas follows

B

opt

BB = arg minBBB

∥

∥

∥

∥

Bopt −U

BBB

∥

∥

∥

∥

F

s.t.

∥

∥

∥

∥

∥

diag

(

BBB

B

H

BB

)∥

∥

∥

∥

∥

0

= NRF

∥

∥

∥

∥

U

BBB

∥

∥

∥

∥

2

=∥

∥Bopt∥

∥

2

, (18)

where

BBB ∈ CNray×K is a digital precoding matrix con-

strained by

∥

∥

∥

∥

∥

diag

(

BBB

B

H

BB

)∥

∥

∥

∥

∥

0

= NRF, and

BBB hasNRF

non-zero rows;

∥

∥

∥

∥

U

BBB

∥

∥

∥

∥

2

= ‖Bopt‖2 is the transmit power

constraint. As a result, the energy efficient hybrid precoding-B(EEHP-B) algorithm is developed for the optimization problemin (18) as follow.


Algorithm 2 Energy Efficient Hybrid Precoding-B (EEHP-

B) algorithm .

Begin:

1) PresetBRF as anNTx × NRF empty matrix, and set

Btemp = Bopt;

2) For i = 1 : 1 : NRF

∆ = UHBtemp;

v = arg maxv=1,...,Nray

[

∆∆H]

v,v;

BRF =[

BRF

∣

∣

∣ [U] : , v

]

;

BBB,temp=(

BHRFBRF

)−1BHRFB

opt;

Btemp =B

opt−BRFBBB,temp

‖Bopt−BRFBBB,temp‖F

;

End for

3) BoptRF is solved by the step 2. The optimal base-

band precoding matrix is calculated byBoptBB =

‖Bopt‖FBBB,temp

‖BoptRFBBB,temp‖

F

.

end Begin

Since the EEHP-B algorithm has the specified cycle number,i.e.,NRF, the EEHP-B algorithm is guaranteed to converge.

When the optimal hybrid precoding matricesBoptRF andBopt

BBare submitted into (8), an optimal energy efficiencyηopt can beobtained. Based on the EEHP-B algorithm, the value ofηopt

approaches to the value ofηopt, i.e., ηopt 6 ηopt. Consideringηopt is a locally optimal solution for the energy efficiencyoptimization in (9),ηopt is also a locally optimal solution forthe energy efficiency optimization in (8).

D. number of RF chains Optimization

Based on EEHP-A and EEHP-B algorithms, a local opti-mization solutionηopt with optimized hybrid precoding ma-trices B

optRF and B

optBB is available for the energy efficiency

optimization in (8). However, the number of RF chains is fixedfor the solutionηopt. To maximize the energy efficiency, thenumber of RF chains is further optimized based on the locallyoptimal solutionηopt.

By analyzing (9) and (17), the number of RF chainsNRF isrelated not only to the size of the optimized hybrid precodingmatricesBopt

RF and BoptBB but also to the entries ofBopt

RF andB

optBB. Therefore, it is difficulty to derive an analytical solution

for the optimal number of RF chains. However, the numberof RF chainsNRF is an positive integer and is limited in thespecific range[K,NTx]. In this case, we can utilize the ergodicsearching method to find the optimal number of RF chainsmaximizing the energy efficiency in (8). Therefore, the EEHPalgorithm is developed to achieve the global optimizationsolution for the energy efficiency optimization in (8).

Algorithm 3 Energy Efficient Hybrid Precoding (EEHP)

algorithm .

Begin:

1) For NRF = K : 1 : NTx (search all the possible values

of NRF from K to NTx)

For a certain value ofNRF, calculateBopt (NRF)

according to the EEHP-A algorithm;

Based on Bopt (NRF) and NRF, calculate

BoptRF (NRF) and B

optBB (NRF) according to the

EEHP-B algorithm;

Calculateηopt (NRF) with BoptRF (NRF) andBopt

BB (NRF);

End for2) Find the optimal number of RF chainsNopt

RF maximiz-

ing the energy efficiency;

3) Configure the global optimal hybrid precoding matri-

ces asBoptRF

(

NoptRF

)

andBoptBB

(

NoptRF

)

.

end Begin

Based on the EEHP algorithm, the global maximum energyefficiency ηopt

global is achieved by configuring the number ofRF chainsNopt

RF , the RF chain precoding matrixBoptRF

(

NoptRF

)

and the baseband precoding matrixBoptBB

(

NoptRF

)

. Consideringη

optglobal 6 ηopt 6 ηmax, an suboptimal energy efficiency

optimization solution is found by the EEHP algorithm in thissection.

Furthermore, according to the computational complexityof matrix calculation and iterative algorithms in [35]and [36], the computational complexity of the proposedalgorithm is explained as follows: The complexity ofAlgorithm 1 is calculated asO (NTxK) + O

(

N3Tx

)

floatingpoint operations (flops); the complexity of Algorithm 2is calculated asO

(

N2TxK +N3

RF +N2RFNTx +NTxNRFK

)

flops; combining Algorithm 1 and 2, the complexityof the EEHP algorithm, i.e. Algorithm 3 iscalculated as O

[

(NTx −K)(

N2TxK +N3

RF +N2RFNTx

+NTxNRFK +N3Tx

)]

flops.

IV. ENERGY EFFICIENT OPTIMIZATION WITH THE

M INIMUM NUMBER OF RF CHAINS

In general, the cost of RF chain is very high in wirelesscommunication systems. To reduce the cost of massive MIMOsystems, an energy efficient solution with the minimum num-ber of RF chains is investigated in the following.

A. Energy Efficiency Hybrid Precoding with the Minimum

number of RF chains

Based on the function of RF chains and the hybrid precodingscheme, the number of RF chains is larger than or equalto the number of baseband data streams in massive MIMOcommunication systems [9], [39]. In this paper the number ofbaseband data streams is assumed to be equal to the number ofactive UEs. Without loss of generality, the minimum number


of RF chains is configured asNminRF = K, whereK is the

number of active UEs in Fig. 1.When the minimum number of RF chains is configured, the

RF precoding matrixBRF has a size ofNTx × K, which isexactly the size of the conjugate transpose of the downlinkchannel matrix. Therefore, the entry of RF precoding matrixBRF is directly configured as [16]

[BRF]i,j =1√NTx

ejθi,j , (19)

where[BRF]i,j denotes the(i, j) th entry of the RF precodingmatrix BRF, andθi,j is the phase of the(i, j) th entry of theconjugate transpose of the downlink channel matrix.

When the downlink channel matrixHH and the RF chainprecoding matrix are combined together, an equivalent down-link channel matrix for allK UEs is given by

HHeq = H

HBRF =

[

BHRFh1, ...,B

HRFhk, ...,B

HRFhK

]H

= [h1,eq, ...,hk,eq , ...,hK,eq]H ∈ C

K×NRF. (20)

Substitute (20) into (8), the energy efficiency optimizationproblem is transformed as

BoptBB = arg max

BBB

η =

WK∑

k=1

Rk,eq

1α

K∑

k=1

‖BRFbBB,k‖2 +KPRF + PC

s.t. Rk,eq > Γk, k = 1, ...,KK∑

k=1

‖BRFbBB,k‖2 =

K∑

k=1

Pk 6 Pmax

,

(21a)where

Rk,eq = log2

1 +hHk,eqbBB,kb

HBB,khk,eq

K∑

i=1,i6=khHk,eqbBB,ib

HBB,ihk,eq + σ2

n

.

(21b)Similar to the optimization problem in (8), the energy

efficiency η in (20) is maximized by the optimal basebandprecoding matrixBopt

BB. Considering the objective functionin (21) is non-concave, it is intractable to solve for theglobal optimum ofη. Therefore, a local optimum solution isdeveloped for (21) as follow.

Given thatη is a function of the baseband precoding vectorof the kth UE, i.e. η (bBB,k), the gradient ofη (bBB,k) isderived by

∂η (bBB,k)

∂bBB,k=

2

P 2

[

Ωk − Ξk

]

bBB,k, (22a)

with [22b – 22e].

Ωk =Phk,eqh

Hk,eq

K∑

j=1

hHk,eqbBB,jbHBB,jhk,eq + σ2

n

, (22b)

Fig. 2 Energy efficiency of the EEHP-MRFC algorithm.

Ξk =

K∑

i=1

Ri,eq

α ln 2BHRFBRF+

P

K∑

i=1,i6=k

hHi,eqbBB,ib

HBB,ihi,eq

(

δi

)2

+ δihHi,eqbBB,ibHBB,ihi,eq

hi,eqhHi,eq

,

(22c)

P =1

W

(

1

α

K∑

i=1

‖BRFBBB,i‖2 +KPRF + PC

)

, (22d)

δi =K∑

j=1,j 6=ihHi,eqbBB,jb

HBB,jhi,eq + σ2

n, (22e)

where δi is the sum of the received interference and noisepower for the ith UE, Rk,eq is obtained by replacingkwith i in (21b). Replacing the results of (10) by the resultsof (22), the baseband precoding matrixBopt

BB is solved bythe EEHP-A algorithm. Substituting the baseband precodingmatrix B

optBB into the EEHP algorithm, a local optimum of

η is solved. To differentiate this approach from the EEHPalgorithm, this algorithm is denoted as the energy efficienthybrid precoding with the minimum number of RF chains(EEHP-MRFC) algorithm.

To analyze the performance of EEHP-MRFC algorithm, nu-merical simulation results are illustrated in Fig. 2. In generally,the optimization result of EEHP-MRFC algorithm depends onthe initial values of the baseband precoding matrix, i.e.B

(0)BB .

Without loss of generality,B(0)BB can be configured to have

equal entries as [20]

B(0)BB =

√

Pmax

K1K×K , (23)

where1K×K denotes theK × K matrix whose entries are

equal to 1, the coefficient√

Pmax

Kis due to the maximum


TABLE I Simulation parameters of EEHP algorithm [10], [37]–[39]

Parameter ValueMaximum transmit powerPmax 33 dBmMinimum spectrum efficiency for each UEΓk 3 bit/s/HzPower consumed by each RF chainPRF 48 mWPower consumed by other parts of the BSPC 20WThe number of BS antennasNTx 200Power amplifier efficiencyα 0.38Noise power spectral density -174 dBm/HzCarrier frequency 28 GHzBandwidth 20 MHzCell radius 200 mMinimum distance between the UE and the BS10 mPath loss exponentγ 4.6The number of multipathsNray 30Azimuth ψi and elevation angleϑi Uniformly distributed within[0, 2π]

transmit power constraint in the BS. The detailed simulationparameters are list in Table I at the top of this page. Based onthe results in Fig. 2, the EEHP-MRFC algorithm convergesafter a limited iteration number. Besides, the convergencerate decreases the number of UEs increases. Moreover, theconverged value of the energy efficiency also decreases withthe increase of the number of UEs. This result indicates thatthe energy efficiency of massive MIMO system is inverselyproportional to the number of active UEs.

B. Energy Efficiency Optimization Considering the Numbers

of Antennas and UEs

In practical engineering applications, the optimal numberof transmit antennas and the optimal number of UEs can beperformed by the resource management and the user scheduleschemes for maximizing the energy efficiency of 5G wirelesscommunication systems. However, the EEHP algorithm cannot directly derive the analytical number of transmit antennasand UEs for maximizing the energy efficiency of 5G wirelesscommunication systems. Therefore, we try to derive the op-timal number of transmit antennas and UEs for maximizingthe energy efficiency of 5G massive MIMO communicationsystems. To simplify the derivation, a specified scenario withrich scattering and multipaths propagation in a millimeterwave wireless channels is considered for the ergodic capacitycalculation in this paper. Based on measurement results ofmillimeter wave channels in [28], [40], [41], the Rayleighfading model can be adopted to describe the consideredmillimeter wave wireless channels.

Assume that the transmitter, i.e., the BS has the perfect CSI.When the minimum number of RF chains is configured andthe ZF precoding is adopted in the system model [8], basedon (19) and (20), the baseband precoding matrix is given by

BBB = Heq

(

HHeqHeq

)−1D, (24)

whereD is aK×K diagonal matrix, which aims to normalizeBBB. Assume thatPout is the total BS downlink transmit powerconsumed byK active UEs. To simplify the derivation, theequal power allocation scheme is assumed to be adopted for allK active UEs. Substituting (24) into (5), the normalized link

capacity, i.e. the link spectral efficiency of UEk is expressedas [22]

Rk,ZF = log2

1 +

Pout

K[

(

HeqHHeq

)−1]

k,k

. (25)

Based on Jensen inequality, the upper-bound of the ergodiclink capacity of UEk is derived by

E (Rk,ZF) 6 Rk,ZF = log2

1 + E

Pout

K[

(

HeqHHeq

)−1]

k,k

,

(26)where E () is the expectation operation taken over theRayleigh fading channelH within Heq = B

HRFH. Diagonal

and off-diagonal entries ofHeq are given by

[Heq]k,k = hHk bRF,k =

1√NTx

NTx∑

i=1

∣

∣

∣[H]i,k

∣

∣

∣, (27a)

[Heq]j,k = hHj bRF,k =

1√NTx

NTx∑

i=1

[H]i,jejθi,k , (27b)

wherebRF,k is thekth column ofBRF. With Rayleigh fadingchannel considered and based on results in [16], the diagonalentry of Heq is governed by a normal distribution, i.e.,

[Heq]k,k ∼ N(

π√NTx2 , 1− π

4

)

and the off-diagonal entryof Heq is governed by a standard normal distribution, i.e.,[Heq]j,k ∼ CN (0, 1). Considering massive MIMO antennasare equipped at the BS, without loss of generality, the numberof BS antennas is assumed to be larger than or equal to 100, i.e.NTx > 100. In this case the expected value of diagonal entriesis much larger than the expected value of off-diagonal entriesin Heq. Therefore, the expected value of the off-diagonalentries can be set to zero inHeq, andHeq is approximatedas a diagonal matrix [16]. Based on (26), the upper-bound ofthe ergodic link capacity of UEk is approximated as

Rk,ZF ≈ log2

[

1 + E

(

Pk [Heq]2k,k

)]

= log2

[

1 + E

(

Pk

(

NTxπ2 − π + 4

4

))], (28)


When the ergodic link capacity is replaced by the upper-bound of the ergodic link capacity, the upper-bound of the BSenergy efficiency is derived by

ηZF =Klog2

(

1 + PoutK

(

NTxπ2−π+44

))

1αPout +K (PRF + PBB) + P ′

C, (29)

wherePBB is the power consumed by the baseband processingfor the baseband data stream,P ′

C is the fixed BS powerconsumption without the power consumed for the downlinktransmit, the RF chains and the baseband processing.

For the upper-bound of the BS energy efficiency in (29),the following proposition is given.

Proposition: Considering the impact of the number of UEsand transmit antennas on the BS energy efficiency, a functionG (K,NTx) is formed as (30)

G (K,NTx) =

[

Pout (PRF + PBB)(

NTxπ2 − π + 4

)

(

4αPout + 4P ′

C)

ln2+

Pout(

NTxπ2 − π + 4

)

4K ln 2

]

-

[

1 +Pout

(

NTxπ2 − π + 4

)

4K

]

×

log2

[

1 +Pout

(

NTxπ2 − π + 4

)

4K

]

.

(30)

When G (1, 100) ≥ 0, there exists a critical number ofantennasNCri

Tx . The critical number of antennasNCriTx is the

smallest integer which is larger than or equal to the root ofthe equationG (1, NTx) = 0. WhenNTx ≥ NCri

Tx , there existsan optimal number of UEsKopt ≥ 1 which maximizes the BSenergy efficiency asηMax

ZF . Kopt is solved by the integer closestto the root of the equationG (K,NTx) = 0. WhenNTx < NCri

Tx ,the maximum BS energy efficiencyηMax

ZF is achieved with thenumber of UEsK = 1. Moreover, the BS energy efficiencyηZF decreases with the increase of the number of UEsK.

When G (1, 100) < 0, there exists an optimal numberof UEs Kopt which maximizes the BS energy efficiency asηMax

ZF . When the number of antennasNTx is given, the optimalnumber of UEsKopt is solved by the integer closest to theroot of the equationG (K,NTx) = 0.

Proof: The proposition is proved in Appendix.

Note that in the above proposition and the correspondingappendix, the minimum number of antennas is set as 100considering the massive MIMO scenario. Based on the resultsin [16], the approximation in (28) is guaranteed to hold whenthe number of antennas is larger than or equal to 100. Based onthe Appendix,G (1, NTx) is a monotonous decreasing functionwith respect toNTx. Utilizing the bisection method, the criticalnumber of antennas searching (CNAS) algorithm is developedto solve the critical number of antennasNCri

Tx .

Algorithm 4 Critical Number of Antennas Searching

(CNAS) algorithm.

Begin:

1) Preset the initial lower bound of searching range

NLowTx = 100 and the initial upper bound of searching

rangeNHighTx = 1000;

2) Adjust the upper bound of the range until

G

(

1, NHighTx

)

is less than or equal to 0.

While G

(

1, NHighTx

)

> 0

Do NHighTx = N

HighTx × 2;

End

3) Preset the median value asNTempTx =

(

NLowTx +NHigh

Tx

)

/2 .

4) Keep searching until the median value satisfies

G

(

1, NTempTx

)

= 0.

While G

(

1, NTempTx

)

! = 0

If G

(

1, NTempTx

)

> 0

NLowTx = NTemp

Tx ;

ElseNHigh

Tx = NTempTx ;

EndNTemp

Tx =(

NLowTx +NHigh

Tx

)

/2 ;

End

5) Configure the critical number of antennas as the small-

est integer not less than the final median value, i.e.,

NCriTx =

⌈

NTempTx

⌉

.

end Begin

When the critical number of antennasNCriTx is obtained

by the CNAS algorithm, the optimal number of UEsKopt

is solved by G (K,NTx) = 0 which maximizes the BSenergy efficiencyηMax

ZF . ConsideringG (K,NTx) monotonouslydecreases with the increase of the number of UEsK, the UEnumber optimization (UENO) algorithm is developed to solvethe optimal number of UEsKopt by the bisection method.

Since the above CNAS and UENO algorithm are based onthe bisection method, the computational complexity of theCNAS and UENO algorithms are calculated asO [log2

(

NHighTx

−NLowTx

)]

andO [log2 (KHigh −KLow)], respectively [42].


Algorithm 5 The UE Number Optimization (UENO) algo-

rithm .Begin:

1) Preset the initial lower bound of the searching range

KLow = 1 and the initial upper bound of the searching

rangeKHigh = 40;

2) Adjust the upper bound of the range until

G (KHigh, NTx) is less than or equal to 0;

While G (KHigh, NTx) > 0

Do KHigh = KHigh × 2;

End

3) Preset the median value asKTemp =

(KLow +KHigh)/2;

4) Keep searching until the median value satisfies

G (KTemp, NTx) = 0;

While G (KTemp, NTx) ! = 0

If G (KTemp, NTx) > 0

KLow = KTemp;

ElseKHigh = KTemp;

EndKTemp= (KLow +KHigh)/2;

End5) Configure the optimal number of UEs as the inte-

ger closest to the final median value, i.e.,Kopt =

⌈KTemp− 1/2⌉.end Begin

V. SIMULATION RESULTS

Energy efficiency optimization solutions with respect tothe number of RF chains, transmit antennas and active UEsare simulated in the following. Without loss of generality,the number of active UEs is configured as 10. Other defaultparameters are listed in Table I.

The EEHP algorithm is proposed in Section III to maximizethe BS energy efficiency. To tradeoff the energy and costefficiency of the BS RF circuits, the EEHP-MRFC algorithmis developed in Section IV. To analyze the proposed EEHPand EEHP-MRFC algorithms, the energy efficient digitalprecoding (EEDP) algorithm and sparse precoding algorithmare simulated for performance comparisons. In the EEDPalgorithm, the energy efficiency of 5G wireless communicationsystems is obtained by substituting the optimal digital pre-coding vectorsboptk derived from the EEHP-A algorithm into(9). The sparse precoding algorithm in [11] is also simulated,which implements a spectrum efficiency maximization hybridprecoding scheme.

The two proposed algorithms along with the EEDP algo-rithm, the sparse precoding algorithm and the traditional ZFprecoding algorithm are compared in Fig. 3. In order to bettercompare the energy efficiency performance of the algorithms,

Fig. 3 Energy efficiency with respect to the total transmit power.

Fig. 4 Energy efficiency with respect to the number of transmitantennas considering different numbers of multipaths.

the total transmit power for all the UEs is used as thecomparison standard [19], [43]. For each algorithm, the energyefficiency first increases with the increasing total transmitpower. When the total transmit power exceeds a given thresh-old, the energy efficiency starts to decrease. The variationtendencies of the curves coincide with the previously publishedresults [19], [43]. When the total transmit power is fixed,the EEDP algorithm has the highest energy efficiency whichis consistent with the analysis result in Section III. The ZFprecoding algorithm has the lowest energy efficiency becauseit uses the same number of RF chains as antennas, which leadsto the highest power consumption for the RF chains. Besides,the sparse precoding algorithm performs more poorly than theEEHP algorithm but outperforms the EEHP-MRFC algorithmin terms of energy efficiency for 5G wireless communicationsystems. Although the energy efficiency of the EEHP-MRFCalgorithm is not the best result, this algorithm is valuablefor


Fig. 5 Energy efficiency with respect to the number of RF chainsconsidering different numbers of multipaths.

reducing the transmitter cost and the design complexity byminimizing the number of RF chains. In practical applications,the selection of the EEHP algorithm or the EEHP-MRFCalgorithm depends on the desired tradeoff between the energyand cost efficiency for 5G wireless communication systems.

Fig. 4 illustrates the BS energy efficiency with respect tothe number of transmit antennas considering different numbersof multipath components. The sparse precoding and EEDPalgorithms are also simulated to compare with the two pro-posed EEHP and EEHP-MRFC algorithms. It can be seen thatthe BS energy efficiency increases with increasing numbers ofthe transmit antennas. Meanwhile, increasing the number ofmultipath components yields the highest BS energy efficiencywhen the number of transmit antennas is fixed. Decreasingenergy efficiency results are yielded by the algorithms in theorder EEDP, EEHP, sparse precoding and finally the EEHP-MRFC algorithm, respectively.

The BS energy efficiency with respect to the number ofRF chains is shown in Fig. 5. As for the EEDP and EEHPalgorithms, the BS energy efficiency first increases then de-creases with increasing numbers of the RF chains. For thesparse precoding and EEHP-MRFC algorithms, the BS energyefficiency always decreases when increasing the number of RFchains. When the number of multipaths is fixed as 30 and thenumber of RF chains is less than or equal to 26, the energyefficiency of the EEHP-MRFC algorithm is larger than thatof the sparse precoding algorithm. Further, when the numberof RF chains is larger than 26, the energy efficiency of theEEHP-MRFC algorithm is always less than that of the sparseprecoding algorithm.

In Fig. 6, the spectrum efficiency as a function of thenumber of RF chains and the number of multipaths is il-lustrated. It can be seen that sparse precoding algorithm hasthe highest spectrum efficiency. As for EEDP and EEHPalgorithms, the spectrum efficiency increases with increasingthe numbers of the RF chains. Considering that the number

Fig. 6 Spectral efficiency with respect to the number of RF chainsconsidering different numbers of multipaths.

Fig. 7 Energy efficiency with respect to the number of UEsconsidering ZF baseband precoding and the minimum number of

RF chains.

of RF chains always equals the number of UEs in the EEHP-MRFC algorithm, the spectrum efficiency of the EEHP-MRFCalgorithm decreases with the increase of the number of RFchains. When the number of the RF chains is fixed, decreasingspectrum efficiency results are yielded by the algorithms intheorder sparse precoding, EEDP, EEHP, and finally the EEHP-MRFC algorithm, respectively.

The impact of the number of transmit antennas and thenumber of UEs on the BS energy efficiency is investigatedin Section IV.B. Based on the CNAS and UENO algorithms,numerical simulations are shown in Fig. 7. Without loss ofgenerality, the number of transmit antennas are configured as100, 150 and 200, respectively. The power consumed by otherparts of the BS is configured asPC = 20W in Fig. 7. The BSenergy efficiency first increases then decreases with increasing


of the number of UEs. The maximums of the BS energyefficiency correspond to the optimal numbers of UEs 35, 50and 55, respectively, which is consistent with the optimalnumber of UEs obtained from the proposition.

VI. CONCLUSION

In this paper, the BS energy efficiency considering theenergy consumption of RF chains and baseband processingis formulated as an optimization problem for 5G wirelesscommunication systems. Considering the non-concave featureof the objective function, an available suboptimal solutionis proposed by the EEHP algorithm. To tradeoff the energyand cost efficiency in RF chain circuits, the EEHP-MRFCalgorithm is developed for 5G wireless communication sys-tems. Based on the CNAS and UENO algorithms, the energyefficiency of 5G wireless communication systems can bemaximized by optimizing the number of UEs and BS antennas,which is easily employed in the user scheduling and resourcemanagement schemes. Compared with the maximum energyefficiency of conventional ZF precoding algorithm, numericalresults indicate that the maximum energy efficiency of theproposed EEHP and EEHP-MRFC algorithms are improvedby 220% and 171%, respectively. Moreover, the differencebetween the EEHP algorithm and the EEHP-MRFC algorithmis illustrated by numerical simulation results. Furthermore, ourresults provide some available suboptimal energy efficiencysolutions and insights into the energy and cost efficiency ofRFchain circuits for 5G wireless communication systems. For thefuture study, we will try to investigate the energy and spectralefficiency optimization of 5G radio frequency chain systemsin the multi-cell scenario.

APPENDIX

Proof to Proposition:To simplify the derivation, some variables are defined as

follows: z = 1K

, z ∈ (0, 1] , a = Pout

(

NTxπ2−π+44

)

, b =1αPout + P ′

C and c= (PRF + PBB). The derivative ofηZF withrespect to z is given by

dηZF

dz=

d(

log2(1+az)c+zb

)

dz

=

a(c+zb)(1+az) ln 2 − blog2 (1 + az)

(c+ zb)2

=

(

acb ln 2 + a

ln 2z)

− (1 + az) log2 (1 + az)1b(c+ zb)

2(1 + az)

. (31)

When the numerator of (31) is defined by

f (z, a) =( ac

b ln 2+

a

ln 2z)

− (1 + az) log2 (1 + az) , (32)

(31) is simply rewritten asdηZFdz = f(z,a)

1b(c+zb)2(1+az)

.

Since ∂f(z,a)∂z

= aln 2 −

(

a log (1 + az) + aln 2

)

< 0, f (z, a)monotonously decreases with increase ofz ∈ (0, 1] . More-over, the limitation off (z, a) is derived by

limz→0+

f (z, a) =ac

b ln 2> 0. (33)

If f (1, a) > 0, thenf (z, a) > 0, z ∈ (0, 1] . Furthermore,we have the resultdηZF

dz > 0. Therefore,ηZF monotonouslyincreases with increase ofz ∈ (0, 1] . In other words,ηZF

monotonously decreases with the increase ofK. In this case,the BS energy efficiencyηZF is maximized byK = 1. Iff (1, a) < 0, there must exist an optimal valuezopt, zopt ∈(0, 1] , which ensuresf (zopt, a) = 0. The integer closest to1/zopt is the optimal number of UEsKopt which maximizesthe BS energy efficiencyηZF.

The differential off (1, a) with respect toa is derived by

∂f (1, a)

∂a=

c

b ln 2− log2 (1 + a) . (34)

(34) implies that∂f(1,a)∂a

decreases with increase ofa. Whenthe number of transmit antenna is configured asNTx = 1, i.e.a is minimized asamin, the corresponding differential resultis ∂f(1,a)

∂a

∣

∣

∣

a=amin

< 0 considering the practical value range of

Pout, P ′C, PRF and PBB [10], [25], [44], [45]. Therefore, we

have the result∂f(1,a)∂a

< 0 for all available values ofa. Asa consequence,f (1, a) monotonously decreases with increaseof a ∈ [amin,∞).

Substitutez = 1K

, z ∈ (0, 1] , a = Pout

(

NTxπ2−π+44

)

,

b = 1αPout + P ′

C and c= (PRF + PBB) into (32), the functionG (K,NTx) is transformed byf (z, a). Whenf (1, amin) < 0,i.e., G (1, 100) < 0, there exists an optimal valuezopt whichensuresf (zopt, a) = 0 and maximizes the BS energy effi-ciency ηZF. The integer closest to1/zopt is the correspondingoptimal number of UEsKopt.

When f (1, amin) > 0, i.e. G (1, 100) > 0, thereexist a critical number of antennasNCri

Tx which ensures

f (1, a)∣

∣

∣NTx=NCriTx

= 0. WhenNTx > NCriTx , there exists an op-

timal valuezopt which ensuresf (zopt, a) = 0 and maximizesthe BS energy efficiencyηZF. As a consequence, the integerclosest to1/zopt is the corresponding optimal number of UEsKopt. WhenNTx < NCri

Tx , ηZF monotonously decreases withthe increase ofK. In this case, the BS energy efficiencyηZF

is maximized byK = 1.

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Ran Zi (S’14) received the B.E. degree in Commu-nication Engineering and M.S. degree in Electronicsand Communication Engineering from HuazhongUniversity of Science and Technology (HUST),Wuhan, China in 2011 and 2013, respectively. He iscurrently working toward the Ph.D. degree in HUST.His research interests include MIMO systems, mil-limeter wave communications and multiple accesstechnologies.

Xiaohu Ge (M’09-SM’11) is currently a full Pro-fessor with the School of Electronic Information andCommunications at Huazhong University of Scienceand Technology (HUST), China. He is an adjunctprofessor with with the Faculty of Engineering andInformation Technology at University of TechnologySydney (UTS), Australia. He received his PhD de-gree in Communication and Information Engineer-ing from HUST in 2003. He has worked at HUST

since Nov. 2005. Prior to that, he worked as a researcher at Ajou University(Korea) and Politecnico Di Torino (Italy) from Jan. 2004 to Oct. 2005. He wasa visiting researcher at Heriot-Watt University, Edinburgh, UK from June toAugust 2010. His research interests are in the area of mobilecommunications,traffic modeling in wireless networks, green communications, and interferencemodeling in wireless communications. He has published about 100 papers inrefereed journals and conference proceedings and has been granted about 15patents in China. He received the Best Paper Awards from IEEEGlobecom2010. He is leading several projects funded by NSFC, China MOST, andindustries. He is taking part in several international joint projects, such as theEU FP7-PEOPLE-IRSES: project acronym WiNDOW (grant no. 318992) andproject acronym CROWN (grant no. 610524).

Dr. Ge is a Senior Member of the China Institute of Communicationsand a member of the National Natural Science Foundation of China and theChinese Ministry of Science and Technology Peer Review College. He hasbeen actively involved in organizing more the ten international conferencessince 2005. He served as the general Chair for the 2015 IEEE InternationalConference on Green Computing and Communications (IEEE GreenCom). Heserves as an Associate Editor for theIEEE ACCESS, Wireless Communicationsand Mobile Computing Journal (Wiley)and the International Journal ofCommunication Systems (Wiley), etc. Moreover, he served as the guesteditor for IEEE Communications MagazineSpecial Issue on 5G WirelessCommunication Systems.

John Thompson (M’03) Prof. John S. Thompsonis currently a Professor in Signal Processing andCommunications at the School of Engineering in theUniversity of Edinburgh. He specializes in antennaarray processing, cooperative communications sys-tems and energy efficient wireless communications.He has published in excess of three hundred paperson these topics, including one hundred journal paperpublications. He is currently the project coordinator

for the EU Marie Curie International Training Network project ADVANTAGE,which studies how communications and power engineering canprovide future“smart grid” systems). He was an elected Member-at-Large for the Board ofGovernors of the IEEE Communications Society from 2012-2014, the secondlargest IEEE Society. He is also a distinguished lecturer onthe topic ofenergy efficient communications and smart grid for the IEEE CommunicationsSociety during 2014-2015. He is an editor for the Green Communications andComputing Series that appears regularly in IEEE Communications Magazine.

Cheng-Xiang Wang (S’01-M’05-SM’08) receivedthe BSc and MEng degrees in Communicationand Information Systems from Shandong University,China, in 1997 and 2000, respectively, and the PhDdegree in Wireless Communications from AalborgUniversity, Denmark, in 2004.

He has been with Heriot-Watt University, Edin-burgh, U.K., since 2005, and was promoted to aProfessor in wireless communications in 2011. He

was a Research Fellow at the University of Agder, Grimstad, Norway, from2001-2005, a Visiting Researcher at Siemens AG-Mobile Phones, Munich,Germany, in 2004, and a Research Assistant at Technical University ofHamburg-Harburg, Hamburg, Germany, from 2000-2001. His current researchinterests focus on wireless channel modelling and 5G wireless communicationnetworks. He has edited 1 book and published over 230 papers in refereedjournals and conference proceedings.

Prof. Wang served or is currently serving as an editor for 9 internationaljournals, including IEEE Transactions on Vehicular Technology (since 2011),IEEE Transactions on Communications (since 2015), and IEEETransactionson Wireless Communications (2007-2009). He was the leadingGuest Editorfor IEEE Journal on Selected Areas in Communications, Special Issue onVehicular Communications and Networks. He served or is serving as a TPCmember, TPC Chair, and General Chair for over 80 international conferences.He received the Best Paper Awards from IEEE Globecom 2010, IEEE ICCT2011, ITST 2012, IEEE VTC 2013-Spring, and IWCMC 2015. He is aFellowof the IET, a Fellow of the HEA, and a member of EPSRC Peer ReviewCollege.

Haichao Wang received the bachelor degree inelectronic science and technology from Wuhan Uni-versity of Technology, Wuhan, China, in 2013,Now he is working toward the master degree inHuazhong University of Science and Technology,Wuhan, China. His research interests include themutual coupling effect in antenna arrays and op-timization of the number of RF chains in antennaarrays.


Tao Han (M’13) received the Ph.D. degree incommunication and information engineering fromHuazhong University of Science and Technology(HUST), Wuhan, China in December, 2001. He iscurrently an Associate Professor with the School ofElectronic Information and Communications, HUST.From August, 2010 to August, 2011, he was a Visit-ing Scholar with University of Florida, Gainesville,FL, USA, as a Courtesy Associate Professor. His

research interests include wireless communications, multimedia communica-tions, and computer networks. Dr. Han is currently serving as an Area Editorfor the EAI Endorsed Transactions on Cognitive Communications.

energy efficiency optimization of 5g chain · pdf filearxiv:1604.02665v1 [cs.ni] 10 apr 2016...

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