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Adaptive Scheduling and Capacity of Multiuser
MIMO MAC System with Transmit Antenna
Correlation
by
Abhishek Kumar Gupta
Y5827020
DEPARTMENT OF ELECTRICAL ENGINEERING
INDIAN INSTITUTE OF TECHNOLOGY KANPUR
Adaptive Scheduling and Capacity of Multiuser
MIMO MAC System with Transmit Antenna
Correlation
A Thesis Submitted
in Partial Fulfillment of the Requirements
for the Degree of
Master of Technology
by
Abhishek Kumar Gupta
to the
DEPARTMENT OF ELECTRICAL ENGINEERING
INDIAN INSTITUTE OF TECHNOLOGY KANPUR
June 2010
CERTIFICATE
It is certified that the work contained in the thesis entitled "Adaptive Scheduling and
Capacity of Multiuser MIMO MAC System with Transmit Antenna Correlation' by Abhishek
Kumar Gupta has been carried out under our supervision and that this work has not been
submitted elsewhere for a degree.
11 June '2.0TD
June 2010
~,Dr. Ajit Kumar Chaturvedi
Professor
+t-~ ~~~Dr. Adrish Banerjee
Department of Electrical Engineering,
Indian Institute of Technology,
Assistant Professor,
Department of Electrical Engineering,
Kanpur-208016.
Indian Institute of Technology,
Kanpur-208016.
Dedicated
to
My Mentors
and
My Parents
Acknowledgements
First of all, I express my sincere gratitude to God for giving me positive energy for my
work and my life. I would like to thank my advisors, Profs. Ajit Kumar Chaturvedi and
Adrish Banerjee for their encouragement and inspirational guidance without which nothing
could have been sprouted. The time spent in the several meetings with my advisors has
been a great learning experience for me. I am grateful to my parents and family who have
supported me at every moment in my life. I would also like to thank my friends Ankesh
Garg, Balaji Shrinivasan Prabhu and Alok Singh for helping and supporting me throughout
my stay at IIT Kanpur. I am also thankful to Mr Vinosh Babu and Ganesh at CEWIT, IIT
Madras. I am grateful to the students in the WC3 Laboratory and Mobile communication
laboratory, IIT Kanpur for providing so wonderful environment for working.
Abstract
In present wireless communication systems, multiuser multiple-input multiple-output
(MIMO) system provides a promising solution to enhance the performance of communi-
cation. Although for analysis purposes independent and uncorrelated antennas are generally
assumed but in practice, antenna correlation always exists caused by limited physical sizes
or spacing of transmitters/receivers and is regarded as a negative factor since it may result
in reduced degrees of freedom. Inspite of the previous fact, in multiuser MIMO multiple ac-
cess channel (MAC) system with covariance feedback, antenna correlation at mobile stations
(MS) can be potentially beneficial to the sum capacity. Especially, below a certain signal to
noise ratio (SNR), antenna correlation can actually lead to a performance improvement.
In this work, effect of transmit antenna correlation in multiuser MIMO-MAC has been
evaluated analytically. Two extreme cases has been considered - Full correlation (FC) and
No correlation case (NC). It has been proved that full correlation can be better for low SNR
region especially at the cell edges which are far from base station (BS) and there exists a
crossover point where the channel capacity curves for these FC and NC modes intersects
each other. The approximate estimate of the crossover point is also calculated for the above
case. We have also described cases where the crossover does not exist and proved that full
correlation is always better in those cases.
These results also motivate us to design a scheduling scheme for users with adaptive
selection of their modes. We have proposed three schemes to select users mode and schedule
them to maximize the channel capacity.
In the first scheme named as ‘Distributive Scheme with Pre-Calculated Crossover In-
formation’, selection of mode is done at MS depending of Pre-Calculated crossover point
information. In second scheme named as ‘Centralized Scheme with Pre-Calculated Crossover
Information’, all users operate in the same mode decide by Base station while in the third
scheme named as ‘Centralized Scheme with No Crossover Information’, BS searches for the
best users and mode among all possible combinations of users and modes in the entire cell
and schedules them. It has been shown through simulations that all the schemes perform
better than the scenario when no adaptive mode selection schemes are used.
Contents
List of Figures xvi
List of Tables xviii
List of Acronyms xix
List of Symbols xx
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 System Model 8
2.1 MIMO Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Multi-user MIMO MAC System . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Correlation Matrixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.1 Full Correlation Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.2 No Correlation Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Channel Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.6 Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Single User Case 17
3.1 Channel Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
xiii
3.2 Full Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 No Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Crossing of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Sensitivity with Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.6 Capacity Expression for Full Correlation Scenario with N Transmit Antennas 30
3.7 Capacity Expression for No Correlation Scenario . . . . . . . . . . . . . . . . 31
3.8 Crossover Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Multi-user MIMO 36
4.1 Full Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.1.1 Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.1.2 Capacity Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.3 Sensitivity with Power . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 No Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2.1 Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2.2 Capacity Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2.3 Sensitivity with Power . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 Crossing of the Capacity Curves . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.4 Simulation Results: Crossover Capacity Curves . . . . . . . . . . . . . . . . . 47
4.4.1 Case-I: MN < Nr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.4.2 Case-II: M > Nr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4.3 Case-III: M < Nr,MN > Nr . . . . . . . . . . . . . . . . . . . . . . . 48
4.5 Crossover Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 Correlation Adaptive User Scheduling 55
5.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Distributive Scheme with Pre-Calculated Crossover Information . . . . . . . . 57
5.2.1 Adaptive Correlation Mode Selection at Mobile Station . . . . . . . . 57
5.2.2 User Scheduling at Base Station . . . . . . . . . . . . . . . . . . . . . 57
5.2.3 Simulation Results: DS-PCI . . . . . . . . . . . . . . . . . . . . . . . . 58
5.3 Centralized Scheme with Pre-Calculated Crossover Information . . . . . . . . 59
5.3.1 Adaptive Correlation Mode Selection at Base Station . . . . . . . . . 60
5.3.2 User Scheduling at Base Station . . . . . . . . . . . . . . . . . . . . . 60
5.3.3 Simulation Results: CS-PCI . . . . . . . . . . . . . . . . . . . . . . . . 61
5.4 Centralized Scheme with No Crossover Information . . . . . . . . . . . . . . . 62
5.4.1 Simulation Results: CS-NCI . . . . . . . . . . . . . . . . . . . . . . . . 64
5.5 Suboptimal CS-NCI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.5.1 Simulation Results: SCS-NCI . . . . . . . . . . . . . . . . . . . . . . . 70
5.6 Comparison of all Schemes: Scheme Gain . . . . . . . . . . . . . . . . . . . . 72
6 Conclusion and Future Work 73
7 Appendix 75
References 86
List of Figures
2.1 A common MIMO system with three transmit antennas and receive antennas 8
2.2 A common MIMO-MAC system showing both SU and MU modes . . . . . . 11
3.1 Capacity curves for single user for FC and NC cases with two transmit antennas
and 4 receiver antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Variation of λQ11opt with correlation ρ and SNR (below one), the upper one is
showing SNR constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Variation of λQ11opt with correlation ρ at various SNR values . . . . . . . . . . 28
3.4 Comparison of Exact expression with approximated Capacity . . . . . . . . . 34
3.5 Comparison of Exact difference with approximated difference between FC and
NC Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1 Capacity curves for case-I (M < Nr and MN < Nr): single user (M=1) . . . 47
4.2 Capacity curves for case-I (M < Nr and MN < Nr): two users (M=2) . . . . 48
4.3 Capacity curves for case-II (M > Nr and MN > Nr): four users (M=4) . . . 49
4.4 Capacity curves for case-III (M < Nr and MN > Nr): three users (M=3) . . 49
4.5 Variation of crossover point with Nr for various values of M and N . . . . . . 50
4.6 Variation of crossover point with N for various values of M and Nr . . . . . . 50
5.1 Hexagonal cellular system with BS at the center . . . . . . . . . . . . . . . . 55
5.2 Capacity curves in FC and NC cases for calculation of crossover information
(PCI) for DS-PCI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.3 Capacity curves for DS-PCI compared with no adaptive mode selection . . . 59
xvi
5.4 Capacity curves in FC and NC cases for calculation of crossover information
(PCI) for CS-PCI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.5 Ratio of capacity values in FC and NC case versus SNR (in dB) . . . . . . . 61
5.6 Capacity curves for CS-PCI compared with no adaptive mode selection . . . 63
5.7 Capacity curves for CS-NCI compared with no adaptive mode selection . . . 64
5.8 CS-NCI: selected mode distribution for different SNR values with x-axis show-
ing different modes (refer Table 5.1) . . . . . . . . . . . . . . . . . . . . . . . 65
5.9 CS-NCI: mode selection distribution at different SNR values versus distance
of MS from BS in a 10 radius cell (refer Table 5.1) . . . . . . . . . . . . . . . 66
5.10 CS-NCI: dominant modes at different SNR values over the hexagonal cell (refer
Table 5.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.11 Capacity curves for suboptimal CS-NCI compared with no adaptive mode
selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.12 SCS-NCI:selected mode distribution for different SNR values with x-axis show-
ing different modes (refer Table 5.1) . . . . . . . . . . . . . . . . . . . . . . . 69
5.13 SCS-NCI: mode selection distribution at different SNR values versus distance
of MS from BS in a 10 radius cell (refer Table 5.1) . . . . . . . . . . . . . . . 70
5.14 SCI-NCI: dominant modes at different SNR values over the hexagonal cell
(refer Table 5.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.15 Comparison of all proposed schemes: scheme gain . . . . . . . . . . . . . . . . 72
7.1 Plot of f with scaling parameter a . . . . . . . . . . . . . . . . . . . . . . . . 77
7.2 Plot of f with order N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.3 Plot of F with scaling parameter a . . . . . . . . . . . . . . . . . . . . . . . . 79
7.4 Plot of F with order N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
List of Tables
3.1 Comparison between PI,approx and PI for single user two transmit antenna cases 34
4.1 Comparison between PI,approx and PI for case-I (M < Nr and MN > Nr) . . 53
4.2 Comparison between PI,approx and PI for Case-III (M < Nr and MN > Nr) . 53
5.1 Indexes showing different mode combinations for two users . . . . . . . . . . 65
xviii
List of Acronyms
3GPP Third Generation Partnership Project
BS Base Station
CPI Crossover Point Information
CSI Channel State Information
CS-PCI Central Scheme with Pre-Calculated Crossover Information
CS-NCI Central Scheme with No Crossover Information
FC Full Correlation
KKT Karush-Kuhn-Tucker Conditions
LTE Long Term Evolution
MAC Multiple Access Channel
MRC Maximum Ratio Combining
MIMO Multiple input Multiple output
MS Mobile Stations
MU Mobile Units
MU-MIMO Multi-user MIMO
NC No Correlation
NCS-PCI Non-Central Scheme with Pre-Calculated Crossover Information
PCI Pre-Calculated Crossover Information
QoS Quality of Service
RB Resource Block
SNR Signal to Noise Ratio
SU-MIMO Single user MIMO
xix
List of Symbols
E [.] Expectation operator
f Small f-function
F Capital f-function
Hk Channel Matrix for user k
K Number of total users in a cell
M Number of users transmitting simultaneously
N Number of transmit antennas
Nr Number of receive antennas
Pk Power allocated to user k
PI Crossover Point
Qk Power Allocation Matrix for user k
Tk Transmit antenna correlation matrix for user k
λ Eigen value
ψ psi function
ρ Correlation coefficient between antennas
xx
Chapter 1
Introduction
In this chapter, we will first describe the motivation for our work followed by an overview
and organization of the thesis.
1.1 Motivation
In present wireless communication systems, multiple-input multiple-output (MIMO) provide
a promising solution to enhance the performance of communication [1, 2, 3, 4]. In a MIMO
system, multiple antennas are used at transmitter (Tx) and receiver (Rx) side. The trans-
mitter sends multiple data symbols through multiple antennas which are received by multiple
antennas at receiver. All these data symbols are attenuated (faded) by the channel between
transmitter and receiver antenna by a random number which is Rayleigh distributed. How-
ever since we are using multiple antennas, probability that channels corresponding to all the
links between Tx antennas and Rx antennas fall in deep fade (i.e. become poor) at any
instant simultaneously is very less.
So even if some channels become poor at any instant, we still have one or more good
channels from which we can receive good transmitted signal. This is known as spatial diversity
and number of maximum independent channels is called diversity order. Now the original
data symbols can be extracted using any standard diversity combining technique.
In general, any communication system’s performance is measured by its channel capacity
which is defined as the maximum information which can be transmitted reliably in that
1.1 Motivation 2
system. It is also given by the mutual information between input (transmitted symbols) and
output (received symbols) maximized over all possible input distribution [5], [6]. In simple
language, the capacity represents the maximum number of bits which can transmitted in a
system without any error in a unit frequency band.
When we use multiple antennas, the capacity of the system increases. The improve-
ment achieved in capacity when using diversity is known as diversity gain. But antennas
can increase the diversity order (hence capacity) only when they are uncorrelated otherwise
channels corresponding to them would not be independent of each other. So independent
and identically distributed (i.i.d.) fading at antennas is taken as common assumption in the
research work related to MIMO systems.
However, antenna correlation does exist in practice. In many cases, it can be caused by
limited physical sizes or spacing of transmitters/receivers or by the lack of sufficient scatters
in the transmission environments. Antenna correlation is commonly regarded as a negative
factor since it may result in reduced degrees of freedom [2, 4].
Antenna correlation forces channel realization at each antenna to be same and can result
in all the channels in deep fade simultaneously. At extreme cases, when correlation is highest,
system starts working as single antenna system and thus reduces the capacity significantly[2].
Due to above mentioned reasons, the existing work related to antenna correlation is mainly
focused on its negative impact which is relevant in single-user MIMO environments [7].
In present wireless technologies, transmitter tries to maximize the capacity of a system by
optimizing the data vector direction according to channel conditions (e.g. if a channel stream
is poor at any instant, transmitter will not send any (or send less) data symbols through this
stream and use other channel streams). These techniques are known as beamforming or more
specifically transmit beamforming. But for achieving optimized beamforming, the transmitter
need to know all the channels corresponding to antennas at each instant. This information
is termed as channel state information (CSI) and it is fedback to transmitter from receiver.
Also the effective beamforming depends on rank of the channel matrix formed by combining
all the channels. So if antennas are correlated, channels observed at different antennas are
related and the potential for beamforming increases. Also since they are correlated, the
feedback required at transmitter will be less thus reducing the feedback overhead. In such a
1.1 Motivation 3
scenario, antenna correlation improves channel capacity. It has been observed that whether
correlation is beneficial or not depends on the Signal-to-Noise-Ratio (SNR) region in which
system is operating. The capacity for correlated case is higher in low SNR region while the
capacity for uncorrelated case is higher in high SNR region.
In the work [8], Louie et al has considered multiple input multiple output systems with
maximum ratio combining (MIMO-MRC) with multiple terminals and spatial correlation
at either or both the transmitter and receiver ends. In this paper, authors have presented
theoretical capacity approximations by deriving expansions for the cumulative distribution
function (c.d.f.) of the maximum eigenvalue of uncorrelated, semi, and double-correlated
Wishart matrices. The results can be applied for both downlink (channel from base station
(BS) to mobile stations (MS) which is broadcast channel) and uplink (channel from MS to BS
which is multiple access channels (MAC) ) channels using duality principle of communication
[9]. They have also analyzed the effects of correlation on capacity. Authors have considered
a case with semi-correlation and have shown that capacity approximation increases with
correlation and a case with a large number of terminals (with a fixed number of antennas).
They have quantified the benefits of correlation for both the semi and double-correlated
scenarios. The results demonstrate that capacity scales logarithmically with the maximum
eigenvalue of the correlation matrix at either/both the transmitter or/and receiver end. This
generalizes the results of [10, 11] to arbitrary correlation models and multiple antennas.
We assume the partial channel information is available at the transmitter side that means
transmitter knows the statistics (i.e. mean and variance) of the channel and this information
is fedback to transmitter in the form of channel covariance information. Receiver is assumed
to have perfect channel state information. In such scenario, transmitter can optimize channel
capacity by choosing a optimal power allocation. Total power available at receiver is finally
distributed in different antennas in a way to utilize maximum eigen values of channel.
In the case of single user, this optimum allocation can be achieved using water filling
algorithm [12]. In this work, Soysal et al have considered both the single user and the mul-
tiuser power allocation problems in above mentioned MIMO systems. In a single-user MIMO
system, authors have considered an iterative algorithm that solves for the eigenvalues of the
optimum transmit covariance matrix that maximizes the rate. The algorithm is based on
1.1 Motivation 4
enforcing the Karush-Kuhn-Tucker (KKT) optimality conditions of the optimization prob-
lem at each iteration. The same principle has been extended to the multiuser case where
the eigenvalues of the optimum transmit covariance matrices of all users that maximize the
sum rate of the MIMO multiple access channel (MIMO-MAC) is calculated. An iterative
algorithm has been proposed that finds the unique optimum power allocation policies of all
users. At a given iteration, the multi-user algorithm updates the power allocation of one user,
given the power allocations of the rest of the users, and iterates over all users in a round-robin
fashion.
In another paper [13], a power allocation algorithm based on game theory is proposed.
A power allocation game for uplink multiple input multiple output access channels has been
considered. Consider competing users each equipped with several antennas at the transmitter
and common multiple antennas at the receiver (base station), a game theoretic framework
is constructed to analyze the optimum pre-coding matrix (power allocation and eigenvector
transmit structure) such that each user maximizes selfishly his own rate under power con-
straint (assuming single user decoding at the receiver). Interestingly, as the dimensions of
the system grow i.e. the numbers of transmitting and receiving antennas grow to infinity but
the ratio stays constant, a Nash equilibrium [14] has been shown to exist and is unique. The
results are based on random matrix theory and provide, in the asymptotic case, a closed-form
expression of the Nash equilibrium operating point. Each terminal can compute the power
allocation independently based only on the knowledge of the statistics of the channel (spatial
correlation structure at the transmitter and the receiver) and not its instantaneous realiza-
tions. This reduces dramatically the downlink overhead signaling protocol which becomes
substantial as the number of users grow. The asymptotic claims are then validated using
simulations with a finite number of transmit and receive antennas.
In the works [8, 15, 16], specific transmission strategies in time-division multiple-access
format has been presented. The advantage of antenna correlation reported in these results
comes purely from beamforming gain, while that reported in the work [17] results from
besides beamforming gain, the space diversity related to the locations of multiple Mobile
stations (users) MSs.
In current wireless systems, there are more than one users at a single time. In this multi-
1.2 Overview of the thesis 5
user scenario, the base station has to schedule more than one user at an instance. While
the previous work talks about the case when BS only selects one user at a time depending
on the channel, in the work [18], Wang et al investigated the impact of antenna correlation
at mobile stations (MSs) in multi-user MIMO environments. In contrast to the common
impression that antenna correlation is detrimental, antenna correlation at MSs is potentially
beneficial to the sum capacity of MIMO multiple access channels (MACs) with covariance
feedback. It was shown that below a certain crossover point, antenna correlation can actually
lead to a performance improvement. Furthermore, such a point occurs at a rate increasing
with the number of mobile stations (MSs) which is denoted by K below, so the range where
antenna correlation is beneficial increases with K. While analytic results were presented for
the asymptotic case of K → ∞, for finite number of MSs, the gain due to correlation was
examined numerically. The paper claims that the results can be extended to broadcast
channels straightforwardly using the duality principle [9].
In a similar work [17], Wang et al studied multi-user multiple input multiple output
(MIMO) systems with rate constraints. They showed that antenna correlation at MS is
actually beneficial from the capacity point of view. This understanding is useful in practice
as minimizing the physical size of MSs is highly desirable, but it may result in antenna
correlation.
In recent 3GPP advancements, multiuser scenarios [19] have been discussed and many
schemes have been proposed to assign the same resources to more than one user simulta-
neously until it starts degrading the capacity thus exploiting the channel capacity fully. In
this cases, one user is treated as primary user and other users are scheduled only when they
don’t make the capacity of first user below certain Quality of Service (QoS) requirement.
The BS has all the information about each user’s transmitting channel and based on these
information, it selects the users for transmission.
1.2 Overview of the thesis
In the current work, we analyzed the effect of transmit antenna correlation analytically.
We considered two extreme cases for our analysis. One is full correlation scenario when
1.2 Overview of the thesis 6
each antenna is fully correlated to another antennas of the same user and independent of
other user’s antennas. Second is the no correlation scenario where the antennas are fully
uncorrelated.
We have calculated the closed form finite length series expansion for the capacity ex-
pression. Then we have generalized the capacity expression for multiuser cases for MAC
channels.
We have proposed two standard function namely f{a;N} and F{a; N} with scaling pa-
rameter a and order N . We then showed that the channel capacity of a MIMO system can
be expressed as a finite sum of these two functions only.
We have first taken single user case with two transmits antennas and calculated the
expression for both two scenarios. We then proved analytically that the capacity curves
intersects each other at some point. We also calculated the upper bound of this intersecting
point. We have also shown an approximation for the intersecting point.
We extend the analysis for multi user case for N transmit antennas using the capacity
series expansion calculated above and have shown that the two curves will intersect in most
of the cases. We also found the cases where these curves will not cross and have shown
that for such cases, full correlation curve is always better. Finally, we give an approximate
expression for the crossover point and also compare it with the exact value of crossover
calculated through simulations.
The above discussion motivates us to exploit the advantage of crossover between these
scenarios for an adaptive correlation mode selection scheme and user scheduling algorithm.
Since in a cellular system environment, each user faces different effective SNR’s, different
mode can be beneficial at different position and instances.
We also propose three combined two level ‘selection and scheduling’ schemes to increase
the sum capacity of the channel and we have compared the results with no adaptive selection.
One of the proposed schemes is distributive where each mobile user has the information
about crossover point between the capacity curves while other two schemes are centralized
schemes where both mode selection and scheduling are done by base station.
1.3 Organization 7
1.3 Organization
The rest of the thesis is organized as following. Chapter 2 describes the MIMO MAC system
and basics of power allocation among multiple users. Chapter 3 deals with single user case
with two transmit antennas and proves that the capacity curves intersects each other at some
point.
Chapter 4 extends the above results to Multi-user MIMO system with N transmit antennas
at each user. Chapter 5 describes various scheduling schemes for correlation mode selection
and scheduling and also presents its performance through simulation. Finally in chapter 6,
we give some concluding remarks and scope of further research. We have added an appendix
that gives important proof of theorems and statements used throughout the thesis.
Chapter 2
System Model
The following chapter describes the basics of MIMO system and multi-user scenarios. We
also presents the system model used in our analysis with the power allocation algorithm.
2.1 MIMO Systems
1
2
3
1
2
3
Tx Rx
h11 h21
h31 h12
h22 h32 h13 h23
h33
Figure 2.1: A common MIMO system with three transmit antennas and receive antennas
In multiple-input and multiple-output or MIMO systems, we use multiple antennas at
both the transmitter and the receiver to improve communication system performance. The
transmitter sends multiple streams by multiple transmit antennas. The transmit streams go
2.1 MIMO Systems 9
through a channel which consists of all NNr paths between the N transmit antennas at the
transmitter and Nr receive antennas at the receiver. The receiver receives the signal vectors
using multiple receive antennas and decodes these signal vectors to get back the original
information.
A typical MIMO System is shown in figure 2.1 where hij represents the channel between
ith receiver and jth transmitter antenna.
If x1, x2 and x3 are transmitted values from three antennas at any instant, the transmit
vector x can be written as
x =[x1 x2 x3
]TLet h11, h12, h13 are fading coefficients seen by signals through channel between first re-
ceive antenna and three transmitter antennas respectively. The received value at first receiver
antenna can be given as
y1 = h11x1 + h12x2 + h13x3 + n1
y1 =[h11 h12 h13
]x + n1
where n1 is the additive white Gaussian noise.
Similarly received value at second and third antenna can be given as
y2 =[h21 h22 h23
]x + n2
y3 =[h31 h32 h33
]x + n3
The received vector is
y =[y1 y2 y3
]TWe can combine all above equations in the following expression,
y =
⎡⎢⎢⎢⎣
h11 h12 h13
h21 h22 h23
h31 h32 h33
⎤⎥⎥⎥⎦x + n
where noise vector n is
n =[n1 n2 n3
]T
2.2 Multi-user MIMO MAC System 10
We can also write the above expression as
y = Hx + n
where H = {hij} is known as channel matrix. where hij indicates the channel between ith
receiver and jth transmitter antenna.
The (ergodic) channel capacity of such a system is given as
C = E[log(det(INr + HQHH))
]where Q is equal to E
[xxH
]and its diagonal elements indicates the power allocated to
various streams (or transmit antennas here). The constraint on Q is
tr(Q) ≤ P
where P is the total power allocated to the transmitter.
2.2 Multi-user MIMO MAC System
MIMO MAC (Multiple Access Channel) refers to MIMO uplink case where multiple users
are transmitting and one access point (AP) is receiving.
In cellular systems, these users are known as Mobile Stations (MS) or Mobile Users (MU)
while the AP is known as Base Station (BS). Figure 2.2 shows a MIMO-MAC system with
M users.
In multi-user scenario, base station can select more than one user on same resource block
and ask them transmit simultaneously. A resource block is a set of some frequency carriers
and time slots and represents a time-frequency grid.
In 3GPP-LTE system, a resource block (RB) consists of 12 sub-carriers (spanning 180
kHz) and 0.5 ms time. Maximum of 2 users can be scheduled on the same resource block.
The scheduling is done by base station depending on channel conditions of users which
takes care of orthogonality of channels and interference scenarios. Base Station can also
change system mode from Multi User (MU) to Single User (SU) in case interference is high
or a minimum QoS requirement is not being met.
2.3 System Model 11
Figure 2.2: A common MIMO-MAC system showing both SU and MU modes
2.3 System Model
We consider a multiuser MAC System with M users. There are N antennas at each MS and
Nr antennas at BS. Let Hk is the channel matrix and xk is the transmitted vector for kth
MS. Since all the users are transmitting simultaneously, the received data y at BS is
y =M∑
k=1
Hkxk + n (2.1)
where n is a vector of complex additive white Gaussian noise samples with zero mean and
unit variance.
If the antennas were uncorrelated, channel matrix would consist of independent complex
Gaussian random values. But in case of correlation, channel matrix can be expressed as [12]
Hk = R1/2k ZkT
1/2k (2.2)
where Zk is matrix with independent complex circular Gaussian variables with zero mean and
2.4 Correlation Matrixes 12
unit variance and Rk and Tk are antenna correlation matrixes at receiver and transmitter
side. For our purpose, we assume that there is no correlation at receiver side i.e. Rk = IN
and the correlation comes only from transmitter (MS) side. Therefore Hk can be written as
Hk = ZkT1/2k (2.3)
Also, for normalization, we let trace of Tk equal to N i.e.
tr(Tk) = N
This normalization ensures that Tk does not alter the sum of the average channel gains
for all the antenna links between kth MS and the BS.
2.4 Correlation Matrixes
Correlation Matrix Tk indicates the correlation between different antennas at kth MS. We
assume that correlation between any two antennas for a user is ρ. So Tk can be written as
Tk =
⎡⎢⎢⎢⎢⎢⎢⎣
1 ρe−jθ ... ρe−j(N−1)θ
ρejθ 1 ... ρe−j(N−2)θ
... ... ... ...
ρej(N−1) ρej(N−2)θ ... 1
⎤⎥⎥⎥⎥⎥⎥⎦
where θ is random number from uniform distribution U [0, 2π].
Eigen values of the above correlation matrix is given by
λTk =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
1 − ρ
1 − ρ
...
1 − ρ
1 + (N − 1)ρ
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
2.5 Channel Capacity 13
2.4.1 Full Correlation Case
For full correlation case, the above T matrix takes the form of rank 1 matrix and can be
given by Tk|ρ=1
Tk|ρ=1 =
⎡⎢⎢⎢⎢⎢⎢⎣
1 e−jθ ... e−j(N−1)θ
ejθ 1 ... e−j(N−2)θ
... ... ... ...
ej(N−1) e−j(N−2)θ ... 1
⎤⎥⎥⎥⎥⎥⎥⎦
Out of N eigen values of Tk, one value is N and other N − 1 values are zero.
2.4.2 No Correlation Case
For no correlation case, the above T matrix takes the form of identity matrix and can be
given by Tk|ρ=0
Tk|ρ=0 =
⎡⎢⎢⎢⎢⎢⎢⎣
1 0 ... 0
0 1 ... 0
... .... . . ...
0 0 ... 1
⎤⎥⎥⎥⎥⎥⎥⎦
All of N eigen values of Tk are 1.
2.5 Channel Capacity
If user k transmits xk vector and all users are transmitting simultaneously, the channel sum
capacity is given by
C = E
[log
∣∣∣∣∣INr +M∑
k=1
HkQkHHk
∣∣∣∣∣]
(2.4)
where Qk = E[xkxHk ] is transmit covariance matrix and it signifies the power allocation
at different transmit antennas and expectation is over all realizations of Hk.
The power allocation to different antennas for a user k can be optimized to maximize the
capacity. So the capacity expression can be written as
2.6 Power Allocation 14
C = maxtr(Qk)≤Pk∀k
E
[log
∣∣∣∣∣INr +M∑
k=1
HkQkHHk
∣∣∣∣∣]
(2.5)
where Pk is the allocated power to kth user. We assume here that Pk is same for all users
and given by
Pk =Psys
M
where Psys is total power of system.
2.6 Power Allocation
We assume that Channel State Information (CSI) available is partial and MSs only know
the statistics of the channel i.e. mean and variance of it. So power allocation is done in
a manner to optimize the channel capacity averaged over all realization of channel. This
averaged channel capacity is given by (2.4).
Power allocation is represented by transmit covariance matrix Qk whose ith diagonal term
shows the average power in ith transmit antenna. So as stated earlier, the power allocation
problem is basically choosing Qk such that average capacity is maximized.
Let singular value decomposition (SVD) of Qk is given as Qk = UkΛkUHk . It has been
shown in [12] that equation (2.4) will be maximized if eigen vectors of Qk, Uk is equal to
eigen vectors of Tk . Let’s assume that λQki’s are eigen values of matrix Qk i.e. the diagonal
elements of Λk.
In that case, the problem of optimization over Qk reduces to optimization over λQki only
and (2.5) reduces to
C = max∑Ni=1 λki≤Pk∀k
E
[log
∣∣∣∣∣INr +M∑
k=1
N∑i=1
λTkiλ
QkizkizH
ki
∣∣∣∣∣]
(2.6)
where zki is the uncorrelated channel vector corresponding to ith transmitter antenna of
kth user or simply ith column of Zk matrix [12].
To solve this optimization problem, we take the iterative approach described in [12].
When differentiating the expressions, we get the following set of KKT conditions:
Eki
(λQ)
= E
[λT
kizHkiA
−1ki zki
1 + λQkiλ
Tkiz
HkiA
−1ki zki
]≤ μk
2.6 Power Allocation 15
over i and k where Aki is defined as following:
Aki = A − λTkiλ
QkizkizH
ki
where
A = IN +∑
k
∑j
λTkjλ
QkjzkjzH
kj
Here μk is the Lagrange multiplier.
The above inequality is strict inequality only when optimum λQki is zero otherwise it is
satisfied with equality.
So the above condition can also be written in the form of equality if we multiply both
sides with λQki giving,
λQkiEki
(λQ)
= λQkiμk
When we add all the conditions over i for a fixed k, we get the value of μk which can be
put back in the above condition to give
λQki =
λQkiEki∑
j EkjλQkj
Pk = gki
(λQ)
The above expression can be solved using iterative approach for which following algorithm
can be deduced easily [12],
1. Initialize λQki with some initial values, say Pk/N .
2. Calculate
Eki
(λQ)
= E
[λT
kizHkiA
−1ki zki
1 + λQkiλ
Tkiz
HkiA
−1ki zki
](2.7)
for all i and k. Here
A = IN +∑
k
∑j
λTkjλ
QkjzkjzH
kj
and
Aki = A − λTkiλ
QkizkizH
ki
2.6 Power Allocation 16
3. Update λ′s using following equations
λQki =
λQkiEki∑
j EkjλQkj
Pk (2.8)
4. Repeat step-2 until convergence condition is satisfied which is given as
maxk,i
∣∣∣λQki − λQ
ki(old)∣∣∣∣∣∣λQ
ki
∣∣∣ ≤ ε (2.9)
where ε is pre-specified tolerance (say 0.01).
Once λQki are known, transmitters can transmit with these optimum powers.
If the transmitter has perfect CSI, power allocation algorithm can be derived similarly.
Chapter 3
Single User Case
In this chapter, we consider a simple case of single user MIMO with two transmit antennas
only. We calculate the expression for capacity in both full correlation and no correlation
case. For calculation of capacity expression, in full correlation case we have taken N transmit
antennas and in no correlation, we have taken two transmit antennas for simplicity. We will
also prove that the antenna correlation can improve channel capacity in low Signal-to-Noise
Ratio (SNR) region.
3.1 Channel Capacity
In the single user case with two transmit antennas and Nr receive antennas, channel capacity
expression is given by
C = max∑i λ1i≤P1
E
[log
∣∣∣∣∣INr +2∑
i=1
λT1iλ
Q1iz1izH
1i
∣∣∣∣∣]
(3.1)
where z1i are channel vectors corresponding to ith transmit antenna. The above equation
can also be written in expanded form as following:
C = max∑i λ1i≤P1
E[log∣∣∣INr + λT
11λQ11z11zH
11 + λT12λ
Q12z12zH
12
∣∣∣] (3.2)
We know that if A and C are column vectors and B and D are row vectors, then we can
write [20]
3.1 Channel Capacity 18
det(I + AB + CD) = det
⎛⎝I +
[A C
]⎡⎣B
D
⎤⎦⎞⎠
det(I + AB + CD) = det
⎛⎝I2 +
⎡⎣B
D
⎤⎦[A C
]⎞⎠
det(I + AB + CD) = det
⎛⎝I2 +
⎡⎣BA BC
DA DC
⎤⎦⎞⎠ (3.3)
Using equation 3.3, we can write the channel capacity as,
C = max∑λ1i≤P1
E
⎡⎣log
∣∣∣∣∣∣INr +[λT
11λQ11z11 λT
12λQ12z12
]⎡⎣zH11
zH12
⎤⎦∣∣∣∣∣∣⎤⎦
C = max∑λ1i≤P1
E
⎡⎣log
∣∣∣∣∣∣I2 +
⎡⎣zH
11
zH12
⎤⎦[λT
11λQ11z11 λT
12λQ12z12
]∣∣∣∣∣∣⎤⎦
C = max∑λ1i≤P1
E
⎡⎣log
∣∣∣∣∣∣I2 +
⎡⎣zH
11λT11λ
Q11z11 zH
11λT12λ
Q12z12
zH12λ
T11λ
Q11z11 zH
12λT12λ
Q12z12
⎤⎦∣∣∣∣∣∣⎤⎦
C = max∑λ1i≤P1
E
⎡⎣log
∣∣∣∣∣∣I2 +
⎡⎣λT
11λQ11‖z11‖2 λT
12λQ12z
H11z12
λT11λ
Q11z
H12z11 λT
12λQ12‖z12‖2
⎤⎦∣∣∣∣∣∣⎤⎦
C = max∑λ1i≤P1
E
⎡⎣log
∣∣∣∣∣∣⎡⎣1 + λT
11λQ11‖z11‖2 λT
12λQ12z
H11z12
λT11λ
Q11z
H12z11 1 + λT
12λQ12‖z12‖2
⎤⎦∣∣∣∣∣∣⎤⎦
After solving the determinant in the above equation, we get
C = max∑λ1i≤P1
E[log(1 + λT
11λQ11‖z11‖2
)(1 + λT
12λQ12‖z12‖2
)− λT
12λQ12z
H11z12λ
T11λ
Q11z
H12z11
]
C = max∑λ1i≤P1
E
⎡⎢⎣log
⎛⎜⎝ 1 + λT
11λQ11‖z11‖2 + λT
12λQ12‖z12‖2
+ λT11λ
Q11λ
T12λ
Q12‖z11‖2‖z12‖2 − λT
12λQ12λ
T11λ
Q11z
H11z12zH
12z11
⎞⎟⎠⎤⎥⎦ (3.4)
Since the correlation coefficient is ρ, correlation matrix (Tk) for two transmit antennas
is given by
3.1 Channel Capacity 19
⎡⎣ 1 ρe−jθ
ρejθ 1
⎤⎦ (3.5)
where θ is random variable taken from uniform distribution U [0, 2π].
The eigen values of this Tk matrix are given by
λT11 = 1 − ρ
λT12 = 1 + ρ
Putting these eigen values back in the capacity expression, we get
C = max∑λQ1i≤P1
E
⎡⎢⎣log
⎛⎜⎝ 1+(1 − ρ)λQ
11‖z11‖2 + (1 + ρ)λQ12‖z12‖2
+ (1 − ρ2)λQ11λ
Q12
(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)⎞⎟⎠⎤⎥⎦ (3.6)
In (3.6), λQ1i are needed to be optimized to maximize the capacity under the following
constraint:
λ11 + λ12 ≤ P1
Now all four terms inside log in (3.6), are positive (last one being positive from Cauchy
Schwartz inequality) which makes overall term greater than 1 for all values of z1i. So the
expectation of the term will increase as we increase the values of last three terms. Hence we
can say that the expectation is monotonically increasing with values of λQ1i for all i. Now it
is obvious that the capacity will be maximized when λQ1i’s are equal to their maximum value
possible. Hence the above inequality constraint changes to the following equality condition,
λ11 + λ12 = P1
and the expression (3.6) can be written as
C = max∑λQ1i=P1
E
⎡⎢⎣log
⎛⎜⎝ 1+(1 − ρ)λQ
11‖z11‖2 + (1 + ρ)λQ12‖z12‖2
+ (1 − ρ2)λQ11λ
Q12
(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)⎞⎟⎠⎤⎥⎦
3.2 Full Correlation 20
The above two variable optimization can be changed to single variable optimization using
the above constraint and thus capacity is given as
C = max0<λQ
11≤P1
E
⎡⎢⎣log
⎛⎜⎝ 1+(1 − ρ)λQ
11‖z11‖2 + (1 + ρ)(P1 − λQ11)‖z12‖2
+ (1 − ρ2)λQ11(P1 − λQ
11)(‖z11‖2‖z12‖2 − ‖zH
11z12‖2)⎞⎟⎠⎤⎥⎦ (3.7)
Now we will consider the full correlation and no correlation case separately and derive
the capacity expressions for these cases using the above discussion.
3.2 Full Correlation
In full correlation (FC) case, correlation coefficient ρ is equal to 1 and one of the eigen values
of Tk becomes zero. Hence for this case, capacity expression reduces to
C = maxλQ11≤P1
E[log(
1+2(P1 − λQ11)‖z12‖2
)](3.8)
Equation 3.8 is monotonically decreasing with the eigen value λQ11. So the capacity (3.8)
will be maximized at minimum value of λQ11 which is λQ
11 = 0. Therefore the capacity for FC
is given by
CFC = E[log(1 + 2P1‖z12‖2
)](3.9)
3.3 No Correlation
Similarly in no correlation (NC) case, the correlation coefficient ρ is 0 and both of the eigen
values of Tk become 1. Hence for this case, capacity expression reduces to
CNC = maxλQ11≤P1
C ′ (3.10)
where
C ′ = E
⎡⎢⎣log
⎛⎜⎝ 1 + λQ
11‖z11‖2+(P1 − λQ11)‖z12‖2
+λQ11(P1 − λQ
11)(‖z11‖2‖z12‖2 − ‖zH
11z12‖2)⎞⎟⎠⎤⎥⎦ (3.11)
3.4 Crossing of Curves 21
To maximize the value of C ′, we differentiate it with respect to eigen value and we get
the following condition
∂C ′
∂λQ11
= E
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
‖z11‖2 − ‖z12‖2 + (P1 − 2λQ11)(‖z11‖2‖z12‖2 − ‖zH
11z12‖2)
1 + λQ11‖z11‖2+(P1 − λQ
11)‖z12‖2
+ λQ11(P1 − λQ
11)(‖z11‖2‖z12‖2 − ‖zH
11z12‖2)
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
(3.12)
For maximization of C ′, the condition is
∂C ′
∂λQ11
= 0
Putting 2P1 = λQ11 in the above expression, we see that
∂C ′
∂λQ11
= E
[‖z11‖2 − ‖z12‖2
1 + P12 ‖z11‖2 + P1
2 ‖z12‖2 + P 214
(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)]
(3.13)
∂C ′
∂λQ11
=E
[‖z11‖2
1 + P12 ‖z11‖2 + P1
2 ‖z12‖2 + P 214
(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)]
− E
[‖z12‖2
1 + P12 ‖z11‖2 + P1
2 ‖z12‖2 + P 214
(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)] (3.14)
The two terms in the above expressions are essentially equal which makes∂C ′
∂λQ11
zero.
Therefore we can say that C ′ is maximized at λQ11 = P1/2 giving the capacity CNC as
CNC = E[log(
1 +P1
2(‖z11‖2 + ‖z12‖2
)+
P 21
4(‖z11‖2‖z12‖2 − ‖zH
11z12‖2))]
(3.15)
3.4 Crossing of Curves
These two expressions (3.9) and (3.15) when plotted against power, intersect at a point.
Capacity in no correlation case dominates the capacity in full correlation case in high SNR
3.4 Crossing of Curves 22
region. As we decrease the power, the full correlation capacity starts dominating intersecting
the no correlation capacity. Figure 3.1 shows the above said behavior. We here present proof
of the crossing of these curves using standard tools from calculus.
10−2 10−1 100 101 102
10−1
100
101
Power
Cap
acity
(nat
s/s/
Hz)
Full CorrelationNo Correlation
Figure 3.1: Capacity curves for single user for FC and NC cases with two transmit antennas
and 4 receiver antennas
Theorem 3.4.1 There exists a point in the interval(0, 4Nre
−E[log( 14(‖z11‖2‖z12‖2−‖zH
11z12‖2))])
where CFC and CNC will intersect. Below this point, CFC is higher than CNC and vice-versa.
Proof
Let us define G(P1) as
G = CNC − CFC
3.4 Crossing of Curves 23
At P1 = 0
At zero power, both the expression becomes zero giving
CFC = CNC = 0
G = 0
At low P1 region
We will first analyze G for low SNR region where derivative of G can be given as
∂G
∂P1=
∂
∂P1
[E[log(
1 +P1
2(‖z11‖2 + ‖z12‖2
)+
P 21
4(‖z11‖2‖z12‖2 − ‖zH
11z12‖2))]]
− ∂
∂P1
[E[log(1 + 2P1‖z12‖2
)]]=
12E
[‖z11‖2 + ‖z12‖2 + P 2
1
(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)1 + P1
2 (‖z11‖2 + ‖z12‖2) + P 214
(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)]− E
[1‖z12‖2
1 + 2P‖z12‖2
](3.16)
At P = 0,∂G
∂P1|P=0 =
12[‖z11‖2 + ‖z12‖2
]− 2E[‖z12‖2
]
= E[‖z11‖2
]− 2E[‖z11‖2
]= −E
[‖z11‖2]
= −Nr (3.17)
Therefore, we can say that in the positive neighborhood (low SNR region) of 0,
∂G
∂P1< 0
Along with the fact that G = 0 at P1 = 0, it gives
G < 0 at low P1
CNC < CFC at low P1 (3.18)
3.4 Crossing of Curves 24
At High P1
Now capacity at no correlation
CNC = E[log(
1 +P1
2(‖z11‖2 + ‖z12‖2
)+
P 21
4(‖z11‖2‖z12‖2 − ‖zH
11z12‖2))]
> E[log(
P 21
4(‖z11‖2‖z12‖2 − ‖zH
11z12‖2))]
= E [2 log (P1)] + E[log(
14(‖z11‖2‖z12‖2 − ‖zH
11z12‖2))]
= 2 log P1 + E[log(
14(‖z11‖2‖z12‖2 − ‖zH
11z12‖2))]
From Jensen’s inequality,
E [log f(x)] ≤ log E [f(x)]
We can say that
CFC < log E[1 + 2P‖z12‖2
]= log (1 + 2P1Nr)
< log (4P1Nr)
= log P1 + log 4Nr
Since we are considering high P1 for crossover point, we have assumed that P >1
2Nrin
the third step of above calculation.
Now for P > 4Nre−E[log( 1
4(‖z11‖2+‖z12‖2−‖zH11z12‖2))], comparing the above expressions for
FC and NC case, we see that
CFC < CNC (3.19)
Combining (3.19) and (3.18), we can say that there exists a point in the interval
(0, 4Nre
−E[log( 14(‖z11‖2+‖z12‖2−‖zH
11z12‖2))])
where CFC and CNC will intersect.
3.5 Sensitivity with Correlation 25
3.5 Sensitivity with Correlation
In this section, we will discuss how the sensitivity of capacity behave against correlation.
From (3.1), we get
C = max∑λQ1i=P1
E
⎡⎢⎣log
⎛⎜⎝ 1 + (1 − ρ)λQ
11‖z11‖2+(1 + ρ)λQ12‖z12‖2
+ (1 − ρ2)λQ11λ
Q12
(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)⎞⎟⎠⎤⎥⎦
(3.20)
Let us assume λQ11opt and λQ
12opt are the optimum value for the above expression giving opti-
mum capacity as
Copt = E
⎡⎢⎣log
⎛⎜⎝ 1 + (1 − ρ)λQ
11opt‖z11‖2+(1 + ρ)λQ12opt‖z12‖2
+ (1 − ρ2)λQ11optλ
Q12opt
(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)⎞⎟⎠⎤⎥⎦
(3.21)
Sensitivity of the capacity with respect to correlation is given by
Cs =∂Copt
∂ρ=E
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
−λQ11opt‖z11‖2 + λQ
12opt‖z12‖2 − 2ρλQ11optλ
Q12opt
(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)⎛⎜⎝ 1 + (1 − ρ)λQ
11opt‖z11‖2+(1 + ρ)λQ12opt‖z12‖2
+ (1 − ρ2)λQ11optλ
Q12opt
(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)⎞⎟⎠
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
+E
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
(1 − ρ)∂λQ
11opt
∂ρ‖z11‖2 + (1 + ρ)
∂λQ12opt
∂ρ‖z12‖2
+(1 − ρ2){∂λQ11opt
∂ρλQ
12opt + λQ11opt
∂λQ12opt
∂ρ} (‖z11‖2‖z12‖2 − ‖zH
11z12‖2)
⎛⎜⎝ 1 + (1 − ρ)λQ
11opt‖z11‖2+(1 + ρ)λQ12opt‖z12‖2
+ (1 − ρ2)λQ11optλ
Q12opt
(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)⎞⎟⎠
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(3.22)
3.5 Sensitivity with Correlation 26
Cs =∂Copt
∂ρ= −E
[λQ
11opt‖z11‖2
(Cx)
]+ E
[λQ
12opt‖z12‖2
(Cx)
]
−E
[2ρλQ
11optλQ12opt
(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)(Cx)
]
+E
⎡⎢⎣(1 − ρ)
∂λQ11opt
∂ρ ‖z11‖2
(Cx)
⎤⎥⎦+ E
⎡⎢⎣+(1 + ρ)
∂λQ12opt
∂ρ ‖z12‖2
(Cx)
⎤⎥⎦
+E
⎡⎢⎣(1 − ρ2){∂λQ
11opt
∂ρ λQ12opt}
(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)(Cx)
⎤⎥⎦
+E
⎡⎢⎣(1 − ρ2){∂λQ
12opt
∂ρ λQ11opt}
(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)(Cx)
⎤⎥⎦
(3.23)
where
Cx = 1+(1−ρ)λQ11opt‖z11‖2+(1+ρ)λQ
12opt‖z12‖2+(1−ρ2)λQ11optλ
Q12opt
(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)Again solving further we get the following,
Cs =∂Copt
∂ρ
=(λQ
12opt − λQ11opt
)E[‖z11‖2
(Cx)
]− 2ρλQ
11optλQ12optE
[(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)(Cx)
]
+
((1 − ρ)
∂λQ11opt
∂ρ+ (1 + ρ)
∂λQ12opt
∂ρ
)E[‖z11‖2
(Cx)
]
+ (1 − ρ2)
(∂λQ
11opt
∂ρλQ
12opt + λQ11opt
∂λQ12opt
∂ρ
)E[‖z11‖2‖z12‖2 − ‖zH
11z12‖2
(Cx)
](3.24)
Considering the constraint λQ11opt + λQ
12opt = P1, we have
3.5 Sensitivity with Correlation 27
Cs =∂Copt
∂ρ
=(P1 − 2λQ
11opt
)E[‖z11‖2
(Cx)
]− 2ρλQ
11opt
(P1 − λQ
11opt
)E
[(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)(Cx)
]
+
(−2ρ
∂λQ11opt
∂ρ
)E[‖z11‖2
(Cx)
]
+ (1 − ρ2)∂λQ
11opt
∂ρ
(P1 − 2λQ
11opt
)E
[(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)(Cx)
]
=(P1 − 2λQ
11opt
)E[‖z11‖2
(Cx)
]− 2ρλQ
11opt
(P1 − λQ
11opt
)E
[(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)(Cx)
]
+ 2ρ
(−∂λQ
11opt
∂ρ
)E[‖z11‖2
(Cx)
]
− (1 − ρ2)
(−∂λQ
11opt
∂ρ
)(P1 − 2λQ
11opt
)E
[(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)(Cx)
]
(3.25)
Now all the expectation terms are positive as they are expectation of positive terms. It has
been observed that λQ11opt decreases with respect to correlation ρ from P1/2 to 0 (See figure
3.2 and 3.3),
So∂λQ
11opt
∂ρ≤ 0
Now let us represent the expectation terms by A(ρ, P1) and B(ρ, P1) as
A(ρ, P1) = E[‖z11‖2
(Cx)
]
B(ρ, P1) = E
[(‖z11‖2‖z12‖2 − ‖zH11z12‖2
)(Cx)
]
We have
Cs =(P1 − 2λQ
11opt
)A−2ρλQ
11opt
(P1 − λQ
11opt
)B+
(−∂λQ
11opt
∂ρ
)[2ρA − (1 − ρ2)
(P1 − 2λQ
11opt
)B]
For full correlation, ρ = 1
λQ11opt = 0
3.5 Sensitivity with Correlation 28
00.5
10
2040
0
5
10
15
20
25
30
35
ρSNR
λQ 11
Figure 3.2: Variation of λQ11opt with correlation ρ and SNR (below one), the upper one is
showing SNR constraint
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
16
ρ
λQ 11
P=1WP=3.16WP=10WP=17.78WP=25.11WP=31.62W
Figure 3.3: Variation of λQ11opt with correlation ρ at various SNR values
3.5 Sensitivity with Correlation 29
So sensitivity is given by
Cs,FC = P1A − 2 (P1) 0B +
(−∂λQ
11opt
∂ρ
)[2ρA − (1 − 1) (P1 − 2.0)B]
Cs,FC = P1A +
(−∂λQ
11opt
∂ρ
)2A
which is always positive irrespective of P1.
For no correlation, ρ = 0
λQ11opt = P1/2
So sensitivity is given by
Cs,NC = (P1 − P1)A − 2.0(P1/2)(P1/2)B +
(−∂λQ
11opt
∂ρ
)[2.0.A − (1 − 0) (0) B]
Cs,NC = 0
i.e. sensitivity is always zero irrespective of P1.
We will now see what is the behavior of sensitivity for a fixed correlation. At low Power
P1, let us say P1 → 0, higher order terms of P1 can be neglected in comparison to the lower
order terms, giving
Cs =(P1 − 2λQ
11opt
)A +
(−∂λQ
11opt
∂ρ
)[2ρA]
which is positive.
As we increase the power of user, higher order terms start becoming significant making
sensitivity negative with ρ,
Cs = −2ρλQ11opt
(P1 − λQ
11opt
)B +
(−∂λQ
11opt
∂ρ
)[−(1 − ρ2)
(P1 − 2λQ
11opt
)B]
which has been observed to be negative for lower ρ through simulations.
3.6 Capacity Expression for Full Correlation Scenario with N TransmitAntennas 30
3.6 Capacity Expression for Full Correlation Scenario with N
Transmit Antennas
In this section, we will give a closed form expression for capacity for single user case with N
transmit antennas in Full Correlation scenario.
Theorem 3.6.1 In Full Correlation scenario, the capacity with single user is given by
C =2−Nr
(Nr − 1)!NP1
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
Nr−2∑i=0
2i (Nr − 1)!(Nr − 1 − i)!
⎡⎢⎢⎢⎢⎣
Nr−1−i∑j=1
Γ(j)2j(−1)Nr−1−i−j
(NP1/2)(Nr−i−j)
+e1/(NP1)(−1)Nr−1−i
(NP1/2)Nr−iEi
[1
NP1
]⎤⎥⎥⎥⎥⎦
+2(Nr)(Nr − 1)!e1/(NP1)Ei
[1
NP1
]
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(3.26)
which can be approximated for high value of NP1 by
C ≈ log(NP1) + ψ(Nr)
Proof:
As from our previous description and from (3.9), for single user case we get
C = E[log(1 + NP1‖z12‖2)
]Now since entries of vector z11 are taken from complex Gaussian with unit variance, z12 will
follow distribution of scaled chi-squared random variable. Numerically,
‖z12‖2 =Nr∑n=1
|z12n|2 =Z
2
where
Z ∼ χ2 (2Nr)
3.7 Capacity Expression for No Correlation Scenario 31
which gives
C = E[log(
1 + NP1Z
2
)]
=∫ ∞
0log(1 + NP1
z
2
) 2−Nr
(Nr − 1)!zNr−1e−
z2 dz
=2−Nr
(Nr − 1)!
∫ ∞
0log(1 + NP1
z
2
)zNr−1e−
z2 dz
=2−Nr
(Nr − 1)!F{NP1/2;Nr − 1} (3.27)
where F{a;N} is given by
F{a; N} = 2aN−1∑i=0
2i n!(n − i)!
⎡⎣N−i∑
j=1
Γ(j)(−1)N−i−j
aN−i−j+1+
e1/(2a)(−1)N−i
aN−i+1Ei
[12a
]⎤⎦+2N+1N !e1/2aEi
[12a
] (3.28)
where Ei is the exponential integral function.
Note: Last step in (3.27) comes from Appendix A2.
For higher values of NP1,
C ≈ E[log(NP1‖z12‖2
]= log(NP1) + E
[log(‖z12‖2
]= log(NP1) + ψ(Nr)
where last step follows from [21].
ψ() function is defined as
ψ(N) = −γ +N−1∑i=1
1i
where γ is Euler-Mascheroni constant[22].
3.7 Capacity Expression for No Correlation Scenario
In this section we will give a closed form expression for capacity for single user case with two
transmit antennas in no correlation scenario.
3.7 Capacity Expression for No Correlation Scenario 32
Theorem 3.7.1 In no correlation scenario, the capacity for single user is given by
CNC =1
(Nr − 1)!2Nr+1
⎡⎢⎣ 4Nr(Nr − 1)F{P
4; Nr − 2}
− 2(Nr − 1)F{P
4; Nr − 1} + F{P
4;Nr}
⎤⎥⎦ (3.29)
Proof:
From the above discussion, we can write the expression of the capacity for 2 transmit antenna
single user in No correlation scenario as
CNC = E[log∣∣∣∣(I2 +
P1
2ZT Z)
∣∣∣∣]
(3.30)
where Z = [z11z12]
CNC = E
⎡⎣log
∏i=1,2
(1 +P1
2λi(ZT Z))
⎤⎦
where λi(.) denotes the ith eigen value of the argument matrix.
CNC =∑i=1,2
E[log(1 +
P1
2λi(ZT Z))
]
= 2E[log(1 +
P1
2λ1(ZT Z))
]
Now Z is a 2 × 2 matrix with complex Gaussian distributed elements of zero mean and unit
variance. Its eigen value’s distribution can be given by [1],
p(λ) =1
2(Nr − 1)![(Nr − 1) + λ2 + (Nr − 1)2 − 2λ(Nr − 1)
]λNr−2e−λ
3.8 Crossover Point 33
which gives
CNC = 2∫ ∞
0log(1 +
P1
2λ)
12(Nr − 1)!
[(Nr − 1) + λ2 + (Nr − 1)2 − 2λ(Nr − 1)
]λNr−2e−λdλ
= 2∫ ∞
0log(1 +
P1
2λ)
12(Nr − 1)!
[Nr(Nr − 1)λNr−2 + λNr − 2λNr−1(Nr − 1)
]e−λdλ
=1
(Nr − 1)!
∫ ∞
0log(1 +
P1
2λ)Nr(Nr − 1)λNr−2e−λdλ
+1
(Nr − 1)!
∫ ∞
0log(1 +
P1
2λ)λNre−λdλ
− 1(Nr − 1)!
∫ ∞
0log(1 +
P1
2λ)2λNr−1(Nr − 1)e−λdλ
Now using the property of F{a; N}∫ ∞
0log(1 + Px)xne−xdx =
12n+1
F{P/2;n}
we can write the above expression as
CNC =1
(Nr − 1)!2Nr+1
⎡⎢⎣ 4Nr(Nr − 1)F{P
4; Nr − 2}
− 4(Nr − 1)F{P
4; Nr − 1} + F{P
4;Nr}
⎤⎥⎦ (3.31)
3.8 Crossover Point
When we compare (3.27) and (3.31), we see that crossover point PI satisfies the following
expression:
4Nr(Nr − 1)F{PI
4; Nr − 2} − 4(Nr − 1)F{PI
4; Nr − 1} + F{PI
4; Nr} − 2F{PI ; Nr − 1} = 0
(3.32)
The above can be solved using any standard numerical techniques. However an approximated
solution is presented below.
Using the property of F for high P (See Appendix A4), we can write (3.32) as
log(PI,approx) − 3 log(2) + Nr (ψ(Nr − 1) + ψ(Nr + 1) − 2ψ(Nr)) + ψ(Nr) = 0
log(PI,approx) − 3 log(2) +1
Nr − 1+ ψ(Nr) = 0(3.33)
3.8 Crossover Point 34
100 101 102 103 104
100
101
Power
Cap
acity
(nat
s/s/
Hz)
Exact FCExact NCApprox. FCApprox NC
Figure 3.4: Comparison of Exact expression with approximated Capacity
Table 3.1: Comparison between PI,approx and PI for single user two transmit antenna cases
Nr PI PI,approx
2 4.38 14.24
3 2.09 5.24
4 1.3781 3.17
5 1.0257 2.2781
6 0.8167 1.7742
10 0.4497 0.9406
Solving (3.33), we get the crossover point as
PI,approx = 8e−(ψ(Nr)+1/(Nr−1))
The calculated approximate PI,approx serves as upper bound for the actual PI . The table
3.1 shows some values for different Nr. Figure 3.4 shows the used approximation of capacity
3.8 Crossover Point 35
10−1 100 101 102 103 104−10
−8
−6
−4
−2
0
2
4
P
CFC
−CN
C
ExactApproximate C
Figure 3.5: Comparison of Exact difference with approximated difference between FC and
NC Capacity
curves for Nr = 4. Figure 3.5 shows the difference of these capacity and its approximations for
Nr = 4 and shows exact crossover point and approximated crossover point as the intersection
of curves with x axis.
Chapter 4
Multi-user MIMO
In this chapter, we calculate the capacity of MU-MIMO System. We first calculate the
capacity expression for both full correlation and no correlation cases and then compare the
two expressions.
4.1 Full Correlation
We consider a MIMO MAC system with M users and N transmit antennas and Nr receiver
antennas. We will first calculate the optimal power allocation. Then we will calculate the
capacity expression based on this optimized power allocation.
4.1.1 Power Allocation
As described in chapter 2, for full correlation scenario, λTk1 will be N for all users k while
other eigen values would be zero. When this values are put in the expression (2.7), we get
that
Eki =
⎧⎪⎪⎨⎪⎪⎩
0 for i �= 1
E
[NzH
k1A−1k1 zk1
1 + λQk1NzH
k1A−1k1 zk1
]for i = 1
and (2.8) becomes
λQki =
λQkiEki
EQk1λ
Qk1
Pk
4.1 Full Correlation 37
which gives
λQki =
⎧⎪⎨⎪⎩
Pk for i = 1
0 for i �= 1
4.1.2 Capacity Expressions
Putting the above optimally calculated λQ values in the capacity expression to get
CFC = E
[log
∣∣∣∣∣INr +M∑
k=1
NPkzk1zHk1
∣∣∣∣∣]
(4.1)
Since the rank of the later matrix term depends on the system parameters, we will consider
separate cases.
For Case-I, let us assume that number of users are less than number of receive antennas
i.e. M < Nr
CFC = E
⎡⎢⎢⎢⎢⎢⎢⎣
log
∣∣∣∣∣∣∣∣∣∣∣∣INr + NPk
[z11 z21... zM1
]⎡⎢⎢⎢⎢⎢⎢⎣
zH11
zH21
...
zHM1
⎤⎥⎥⎥⎥⎥⎥⎦
∣∣∣∣∣∣∣∣∣∣∣∣
⎤⎥⎥⎥⎥⎥⎥⎦
= E
⎡⎢⎢⎢⎢⎢⎢⎣
log
∣∣∣∣∣∣∣∣∣∣∣∣IM + NPk
⎡⎢⎢⎢⎢⎢⎢⎣
zH11
zH21
...
zHM1
⎤⎥⎥⎥⎥⎥⎥⎦[z11 z21... zM1
]∣∣∣∣∣∣∣∣∣∣∣∣
⎤⎥⎥⎥⎥⎥⎥⎦
= E[log∣∣IM + NPkZHZ
∣∣]= E
[log
M∏i=1
Eig(IM + NPkZHZ
)]
=M∑i=1
E[log 1 + Eig
(NPkZHZ
)]= ME [log 1 + NPkλ]
where λ is an eigen value of(ZHZ
)
4.1 Full Correlation 38
The pdf of λ has been computed in Appendix A-4 and can be expressed as
p(λ) =1M
M−1∑k=0
k!(k + Nr − M)!
2k∑r=0
Ak,Nr−Mr λr+Nr−Me−λ
where
Ak,pr = (−1)r ((p + k)!)2
(2p + r)!(k!)2
t=min {r,k}∑t=max {0,r−k}
(2p + r
t + p
)(k
t
)(k
r − t
)
which gives
CFC = M
∫ ∞
0log(1 + NPkλ)p(λ)dλ
= M
∫ ∞
0log(1 + NPkλ)
1M
M−1∑k=0
k!(k + Nr − M)!
2k∑r=0
Ak,Nr−Mr λr+Nr−Me−λdλ
=M−1∑k=0
2k∑r=0
k!Ak,Nr−Mr
(k + Nr − M)!
∫ ∞
0log(1 + NPkλ)λr+Nr−Me−λdλ
=M−1∑k=0
2k∑r=0
k!Ak,Nr−Mr
(k + Nr − M)!1
2r+Nr−M+1F{NPk/2; r + Nr − M}
=1
2Nr−M+1
M−1∑k=0
2k∑r=0
k!Ak,Nr−Mr
(k + Nr − M)!12r
F{NPk/2; r + Nr − M}
=1
2Nr−M+1
2(M−1)∑r=0
⎡⎣ M−1∑
k=�r/2�
k!Ak,Nr−Mr
(k + Nr − M)!
⎤⎦ F{NPk/2; r + Nr − M}
2r
Similarly for M > NR, the capacity expression is given by
CFC =1
2M−Nr+1
2(Nr−1)∑r=0
⎡⎣ Nr−1∑
k=�r/2�
k!Ak,M−Nrr
(k + M − Nr)!
⎤⎦ F{NPk/2; r + Nr − M}
2r
4.1.3 Sensitivity with Power
Sensitivity of the capacity with power can be got by differentiating the above expression and
using derivative property of F{a;N} (See Appendix A4)
4.2 No Correlation 39
Case-I : M < NR
∂CFC
∂Pk=
N
21
2Nr−M+1
2(M−1)∑r=0
⎡⎣ M−1∑
k=�r/2�
k!Ak,Nr−Mr
(k + Nr − M)!
⎤⎦ f{NPk/2; r + Nr − M + 1}
2r
at Pk=0, the sensitivity can be further simplified to
∂CFC
∂Pk|Pk=0 =
N
21
2Nr−M+1
2(M−1)∑r=0
⎡⎣ M−1∑
k=�r/2�
k!Ak,Nr−Mr
(k + Nr − M)!
⎤⎦ 2r+Nr−M+2(r + Nr − M + 1)!
2r
=N
22Nr−M+2
2Nr−M+1
2(M−1)∑r=0
⎡⎣ M−1∑
k=�r/2�
k!Ak,Nr−Mr
(k + Nr − M)!
⎤⎦ (r + Nr − M + 1)!
= N
2(M−1)∑r=0
⎡⎣ M−1∑
k=�r/2�
k!Ak,Nr−Mr
(k + Nr − M)!
⎤⎦ (r + Nr − M + 1)!
Case-II: M > Nr
∂CFC
∂Pk|Pk=0 = N
2(Nr−1)∑r=0
⎡⎣ Nr−1∑
k=�r/2�
k!Ak,M−Nrr
(k + M − Nr)!
⎤⎦ (r + M − Nr + 1)!
4.2 No Correlation
4.2.1 Power Allocation
For no correlation scenario, λTki will be 1 for all users k. Since we are interested in a solution
of the algorithm which simultaneously satisfies both equations (2.7) and (2.8), we try to put
all λQki equal in (2.7) which gives all EQ
ki equal and equal to (Let’s say EQ).
When this value is put in (2.8), it gives
λQki =
EQ
NEQPk
4.2 No Correlation 40
which gives
λQki =
Pk
N∀i, k
Since power Pk is assumed to be equal for each user, all λQki are equal which was the earlier
assumption.
So λQki = Pk
N ∀i, k is the optimal power allocation in no correlation case
4.2.2 Capacity Expressions
Putting the above optimally calculated λQ values in the capacity expression to get
CNC = E
[log
∣∣∣∣∣INr +M∑
k=1
N∑i=1
Pk
NzkizH
ki
∣∣∣∣∣]
(4.2)
CNC = E
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
log
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
INr +Pk
N
[z11 .. zM1..z1N .. zMN
]
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
zH11
...
zHM1
...
zH1N
...
zHMN
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
Again the rank of the later matrix depends on various system parameters, We will consider
different cases separately.
4.2 No Correlation 41
Case-I: MN < Nr
CNC = E
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
log
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
IMN +Pk
N
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
zH11
...
zHM1
...
zH1N
...
zHMN
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
[z11 .. zM1..z1N .. zMN
]
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
= E[log∣∣∣∣IMN +
Pk
NZHZ
∣∣∣∣]
= E
[log
M∏i=1
Eig
(IMN +
Pk
NZHZ
)]
=MN∑i=1
E[log 1 + Eig
(Pk
NZHZ
)]
= MNE[log 1 +
Pk
Nλ
]
where λ is an Eigen value of(ZHZ
)The pdf of λ has been computed in Appendix A-4 and can be expressed as
p(λ) =1
MN
MN−1∑k=0
k!(k + Nr − MN)!
2k∑r=0
Ak,Nr−MNr λr+Nr−MNe−λ
where
Ak,pr = (−1)r ((p + k)!)2
(2p + r)!(k!)2
t=min {r,k}∑t=max {0,r−k}
(2p + r
t + p
)(k
t
)(k
r − t
)
4.2 No Correlation 42
Which gives
CNC = MN
∫ ∞
0log(1 +
Pk
Nλ)p(λ)dλ
= MN
∫ ∞
0log(1 +
Pk
Nλ)
1MN
MN−1∑k=0
k!(k + Nr − MN)!
2k∑r=0
Ak,Nr−MNr λr+Nr−MNe−λdλ
=MN−1∑
k=0
2k∑r=0
k!Ak,Nr−MNr
(k + Nr − MN)!
∫ ∞
0log(1 +
Pk
Nλ)λr+Nr−MNe−λdλ
=MN−1∑
k=0
2k∑r=0
k!Ak,Nr−MNr
(k + Nr − MN)!1
2r+Nr−MN+1F{Pk/(2N); r + Nr − MN}
=1
2Nr−MN+1
MN−1∑k=0
2k∑r=0
k!Ak,Nr−MNr
(k + Nr − MN)!12r
F{Pk/(2N); r + Nr − MN}
=1
2Nr−MN+1
2(MN−1)∑r=0
⎡⎣MN−1∑
k=�r/2�
k!Ak,Nr−MNr
(k + Nr − MN)!
⎤⎦ F{Pk/2N ; r + Nr − MN}
2r
Case-II: MN > Nr
CNC =1
2MN−Nr+1
2(Nr−1)∑r=0
⎡⎣ Nr−1∑
k=�r/2�
k!Ak,MN−Nrr
(k + MN − Nr)!
⎤⎦ F{Pk/2N ; r + MN − Nr}
2r
4.2.3 Sensitivity with Power
Sensitivity of the capacity with power can be obtained by differentiating the above expression
and using derivative property of F{a; N} (See Appendix A4)
Case-I: MN < Nr
∂CNC
∂Pk=
12N
12Nr−MN+1
2(MN−1)∑r=0
⎡⎣MN−1∑
k=�r/2�
k!Ak,Nr−MNr
(k + Nr − MN)!
⎤⎦ f{Pk/2N ; r + Nr − MN + 1}
2r
4.3 Crossing of the Capacity Curves 43
At Pk=0, the sensitivity can be further simplified to
∂CNC
∂Pk|Pk=0 =
12N
12Nr−MN+1
2(MN−1)∑r=0
⎡⎣MN−1∑
k=�r/2�
k!Ak,Nr−MNr
(k + Nr − MN)!
⎤⎦ 2r+Nr−MN+2(r + Nr − MN + 1)!
2r
=N
22Nr−MN+2
2Nr−MN+1
2(MN−1)∑r=0
⎡⎣MN−1∑
k=�r/2�
k!Ak,Nr−MNr
(k + Nr − MN)!
⎤⎦ (r + Nr − MN + 1)!
=1N
2(MN−1)∑r=0
⎡⎣MN−1∑
k=�r/2�
k!Ak,Nr−MNr
(k + Nr − MN)!
⎤⎦ (r + Nr − MN + 1)!
Case-II: MN > Nr
∂CNC
∂Pk|Pk=0 =
1N
2(Nr−1)∑r=0
⎡⎣ Nr−1∑
k=�r/2�
k!Ak,MN−Nrr
(k + MN − Nr)!
⎤⎦ (r + Nr − MN + 1)!
4.3 Crossing of the Capacity Curves
We will adopt the same strategy used in single user case to prove that the two curves cross each
other. Since the expression are complex, we have to take the help of numerical calculation
to show the results at some points. At Pk = 0,
F{NPk/2;n} = F{Pk/2N ; n} = 0∀n > 0
giving the both capacity expression to be zero,
CFC = CNC
Let G = CNC − CFC .
At low Pk region,
4.3 Crossing of the Capacity Curves 44
Case-I: M < Nr and MN < Nr
∂G
∂Pk=
12N
12Nr−MN+1
2(MN−1)∑r=0
⎡⎣MN−1∑
k=�r/2�
k!Ak,Nr−MNr
(k + Nr − MN)!
⎤⎦ f{Pk/2N ; r + Nr − MN + 1}
2r
−N
21
2Nr−M+1
2(M−1)∑r=0
⎡⎣ M−1∑
k=�r/2�
k!Ak,Nr−Mr
(k + Nr − M)!
⎤⎦ f{NPk/2; r + Nr − M + 1}
2r
At Pk = 0, the following expression becomes
G′(0) =1N
2(MN−1)∑r=0
⎡⎣ r∑
k=�r/2�
k!Ak,Nr−MNr
(k + Nr − MN)!
⎤⎦ (r + Nr − MN + 1)!
−N
2(M−1)∑r=0
⎡⎣ r∑
k=�r/2�
k!Ak,Nr−Mr
(k + Nr − M)!
⎤⎦ (r + Nr − M + 1)!
=1N
(MNNr) − N(MNr)
= −M(N − 1)Nr
at Pk = 0. which is non positive for all values of M , N and Nr. It becomes zero for N = 1
making both cases equal.
Case-II: M > Nr and MN > Nr
∂G
∂Pk=
12N
12MN−Nr+1
2(Nr−1)∑r=0
⎡⎣ Nr−1∑
k=�r/2�
k!Ak,MN−Nrr
(k + MN − Nr)!
⎤⎦ f{Pk/2N ; r + MN − Nr + 1}
2r
−N
21
2M−Nr+1
2(Nr−1)∑r=0
⎡⎣ Nr−1∑
k=�r/2�
k!Ak,M−Nrr
(k + M − Nr)!
⎤⎦ f{NPk/2; r + M − Nr + 1}
2r
4.3 Crossing of the Capacity Curves 45
At Pk = 0, the following expression becomes
G′(0) =1N
2(Nr−1)∑r=0
⎡⎣ r∑
k=�r/2�
k!Ak,MN−Nrr
(k + MN − Nr)!
⎤⎦ (r + MN − Nr + 1)!
−N
2(Nr−1)∑r=0
⎡⎣ r∑
k=�r/2�
k!Ak,M−Nrr
(k + M − Nr)!
⎤⎦ (r + M − Nr + 1)!
=1N
(MNNr) − N(MNr)
= −M(N − 1)Nr
at Pk = 0. which is non positive for all values of M , N and Nr. It becomes zero for N = 1
making both cases equal.
Case-III: M < Nr and MN > Nr
Similarly in this case,
G′(0) =1N
(MNNr) − N(MNr)
= −M(N − 1)Nr
at Pk = 0. which is non positive for all values of M , N and Nr. It becomes zero for N = 1
making both cases equal.
Now since G = 0 at Pk = 0 and G′ < 0 for Pk = 0, this gives that G < 0 near Pk = 0
giving
CFC > CNC
for low Pk
For high Pk, using the properties of F{a; N},we can write G′(Pk) as
4.3 Crossing of the Capacity Curves 46
Case-I: M < Nr and MN < Nr
G′(Pk → ∞) =1N
2N
Pk
2(MN−1)∑r=0
⎡⎣ r∑
k=�r/2�
k!Ak,Nr−MNr
(k + Nr − MN)!
⎤⎦ (r + Nr − MN)!
−N2
PkN
2(M−1)∑r=0
⎡⎣ r∑
k=�r/2�
k!Ak,Nr−Mr
(k + Nr − M)!
⎤⎦ (r + Nr − M)!
=1Pk
(MN − M)
=1Pk
M(N − 1)
Case-II: M > Nr and MN > Nr
G′(Pk → ∞) =1N
2N
Pk
2(Nr−1)∑r=0
⎡⎣ Nr−1∑
k=�r/2�
k!Ak,MN−Nrr
(k + MN − Nr)!
⎤⎦ (r + MN − Nr)!
−N2
PkN
2(Nr−1)∑r=0
⎡⎣ Nr−1∑
k=�r/2�
k!Ak,M−Nrr
(k + M − Nr)!
⎤⎦ (r + M − Nr)!
=1Pk
(Nr − Nr)
= 0
Case-III: M < Nr and MN > Nr
G′(Pk → ∞) =1N
2N
Pk
2(Nr−1)∑r=0
⎡⎣ Nr−1∑
k=�r/2�
k!Ak,MN−Nrr
(k + MN − Nr)!
⎤⎦ (r + MN − Nr)!
−N2
PkN
2(M−1)∑r=0
⎡⎣ r∑
k=�r/2�
k!Ak,Nr−Mr
(k + Nr − M)!
⎤⎦ (r + Nr − M)!
=1Pk
(Nr − M)
As shown above, the value of G′(Pk → ∞) is positive (because M < Nr ) in case I and
case III cases which shows that after a point Pk,I , CNC will start increasing with a rate
4.4 Simulation Results: Crossover Capacity Curves 47
faster than CFC . However it has been observed that Pk,I increases with M and for M → ∞,
CNC cannot cross CFC in simulation results. In case II, derivative remains always negative
and reaches zero as P tends to infinity which makes No correlation curve never cross full
correlation.
4.4 Simulation Results: Crossover Capacity Curves
Above three cases are simulated in MATLAB to validate our results. Range of M is taken
to be between 2 and 6, N and Nr is usually taken between 2 to 8.
4.4.1 Case-I: MN < Nr
Figure 4.1 shows the capacity curves for single user for case I. Figure 4.2 shows the capacity
curves for two users for case I. As obvious form the figure than the curves cross each other.
The value of crossover point increases as we increase number of users or number of transmitter
antennas.
100 1010
2
4
6
8
10
12
P
Cap
acity
(nat
s/s/
Hz)
M=1 N=4 Nr=4 FCM=1 N=4 Nr=4 NCM=1 N=3 Nr=4 FCM=1 N=3 Nr=4 NCM=1 N=2 Nr=4 FCM=1 N=2 Nr=4 NCM=1 N=1 Nr=4 FCM=1 N=1 Nr=4 NC
Figure 4.1: Capacity curves for case-I (M < Nr and MN < Nr): single user (M=1)
4.4 Simulation Results: Crossover Capacity Curves 48
10−1 100 1010
5
10
15
P
Cap
acity
(nat
s/s/
Hz)
M=2 N=2 Nr=6 NCM=2 N=3 Nr=6 NCM=2 N=3 Nr=6 FCM=2 N=1 Nr=6 FCM=2 N=1 Nr=6 NCM=2 N=2 Nr=6 FC
Figure 4.2: Capacity curves for case-I (M < Nr and MN < Nr): two users (M=2)
4.4.2 Case-II: M > Nr
Figure 4.3 shows the capacity curves for 4 users for case II and shows that there are no
crossover between curves.
4.4.3 Case-III: M < Nr,MN > Nr
Figure 4.4 shows the capacity curves for 3 users for case III. The value of crossover point
is higher in this case. Again the same behavior is observed here with number of users and
transmitter antennas.
Figures 4.5 and 4.6 shows the variation of crossover point with M , N , and Nr.
4.4 Simulation Results: Crossover Capacity Curves 49
10−1 100 101 1020
5
10
15
20
P
Cap
acity
(nat
s/s/
Hz)
M=4 N=1 Nr=3 NCM=4 N=1 Nr=3 FCM=4 N=2 Nr=3 NCM=4 N=2 Nr=3 FCM=4 N=3 Nr=3 NCM=4 N=3 Nr=3 FC
Figure 4.3: Capacity curves for case-II (M > Nr and MN > Nr): four users (M=4)
100 101 102 103 1040
10
20
30
40
50
P
Cap
acity
(nat
s/s/
Hz)
M=3 N=4 Nr=4 FCM=3 N=4 Nr=4NCM=3 N=2 Nr=4 FCM=3 N=2 Nr=4 NCM=3 N=6 Nr=4 FCM=3 N=6 Nr=4 NC
Figure 4.4: Capacity curves for case-III (M < Nr and MN > Nr): three users (M=3)
4.4 Simulation Results: Crossover Capacity Curves 50
2 3 4 5 6 7 8 9 1010−1
100
101
102
Nr
p I
M=2 N=2M=3 N=2M=4 N=2M=5 N=2M=2 N=3M=3 N=3M=4 N=3M=5 N=3
Figure 4.5: Variation of crossover point with Nr for various values of M and N
2 4 6 8 10 12
100
101
102
103
N
p I
M=3 Nr=2
M=3 Nr=4
M=3 Nr=6
M=3 Nr=8
M=3 Nr=10
M=2 Nr=2
M=2 Nr=4
M=2 Nr=6
M=2 Nr=8
M=2 Nr=10
Figure 4.6: Variation of crossover point with N for various values of M and Nr
4.5 Crossover Point 51
4.5 Crossover Point
Crossover point (if exists) can be given by P following satisfying the following equation
CNC − CFC = 0
Case-I: MN < Nr
Let’s assume the first case where M and MN both are less than Nr. For which case the
above equation becomes
12Nr−MN+1
2(MN−1)∑r=0
⎡⎣MN−1∑
k=�r/2�
k!Ak,Nr−MNr
(k + Nr − MN)!
⎤⎦ F{Pk/2N ; r + Nr − MN}
2r
− 12Nr−M+1
2(M−1)∑r=0
⎡⎣ M−1∑
k=�r/2�
k!Ak,Nr−Mr
(k + Nr − MN)!
⎤⎦ F{NPk/2; r + Nr − M}
2r= 0
The above can be solved using any standard numerical technique however an approximate
solution is also possible analytically. Using the asymptotic properties of F , we can write
expression as
12Nr−MN+1
2(MN−1)∑r=0
⎡⎣MN−1∑
k=�r/2�
k!Ak,Nr−MNr
(k + Nr − MN)!
⎤⎦
×(r + Nr − MN)!2r+Nr−MN+1 [log(Pk/N) + ψ(r + Nr − MN + 1)] 2−r
− 12Nr−M+1
2(M−1)∑r=0
⎡⎣ M−1∑
k=�r/2�
k!Ak,Nr−Mr
(k + Nr − MN)!
⎤⎦
×(r + Nr − M)!2r+Nr−M+1 [log(NPk) + ψ(r + Nr − M + 1)] 2−r = 0
2(MN−1)∑r=0
⎡⎣MN−1∑
k=�r/2�
k!Ak,Nr−MNr
(k + Nr − MN)!
⎤⎦
×(r + Nr − MN)! [log(Pk/N) + ψ(r + Nr − MN + 1)]
−2(M−1)∑
r=0
⎡⎣ M−1∑
k=�r/2�
k!Ak,Nr−Mr
(k + Nr − MN)!
⎤⎦
×(r + Nr − M)! [log(NPk) + ψ(r + Nr − M + 1)] = 0
4.5 Crossover Point 52
log(Pk)
⎡⎢⎢⎢⎢⎢⎢⎣
2(MN−1)∑r=0
⎡⎣MN−1∑
k=�r/2�
k!Ak,Nr−MNr
(k + Nr − MN)!
⎤⎦× (r + Nr − MN)!
−2(M−1)∑
r=0
⎡⎣ M−1∑
k=�r/2�
k!Ak,Nr−Mr
(k + Nr − MN)!
⎤⎦× (r + Nr − M)!
⎤⎥⎥⎥⎥⎥⎥⎦
− log(N)
⎡⎢⎢⎢⎢⎢⎢⎣
2(MN−1)∑r=0
⎡⎣MN−1∑
k=�r/2�
k!Ak,Nr−MNr
(k + Nr − MN)!
⎤⎦× (r + Nr − MN)!
+2(M−1)∑
r=0
⎡⎣ M−1∑
k=�r/2�
k!Ak,Nr−Mr
(k + Nr − MN)!
⎤⎦× (r + Nr − M)!
⎤⎥⎥⎥⎥⎥⎥⎦
+2(MN−1)∑
r=0
⎡⎣MN−1∑
k=�r/2�
k!Ak,Nr−MNr
(k + Nr − MN)!
⎤⎦
×(r + Nr − MN)!ψ(r + Nr − MN + 1)
−2(M−1)∑
r=0
⎡⎣ M−1∑
k=�r/2�
k!Ak,Nr−Mr
(k + Nr − MN)!
⎤⎦
×(r + Nr − M)!ψ(r + Nr − M + 1) = 0
log(Pk)(MN − M) − log(N)(MN + M)S1MN,Nr
− S1M,Nr
= 0
where
Sil,m =
2(l−1)∑r=0
⎡⎣ l−1∑
k=�r/2�
k!Ak,m−lr
(k + m − l)!
⎤⎦ (r + m − l)!ψ(r + m − l + i)
log(Pk)(MN − M) − log(N)(MN + M) + S1MN,Nr
− S1M,Nr
= 0
log(Pk) = log(N)N + 1N − 1
+S1
M,Nr− S1
MN,Nr
M(N − 1)
The calculated approximate PI,approx serves as upper bound for the actual PI . The table 4.1
shows some values for different Nr.
Similarly for other cases the approximated crossover point is given by as following
Case-II: M > Nr
log(Pk)(Nr − Nr) − log(N)(2Nr) + S1Nr,M − S1
Nr,MN = 0
4.5 Crossover Point 53
Table 4.1: Comparison between PI,approx and PI for case-I (M < Nr and MN > Nr)
M Nr N PI PI,approx
1 4 2 1.3784 3.1793
1 4 3 1.5318 4.5926
2 6 3 1.217 5.4255
2 8 3 0.76672 2.3383
3 9 3 0.81129 3.7601
3 12 3 0.51123 1.5646
2 12 3 0.43993 1.1354
Which shows that the difference in the capacity is independent of power and the curves will
never cross. In this case, full correlation is always better than the no correlation scenario
and the crossover point can not be computed. As obvious from the previous section’s graphs
that there exists no crossover point for this case.
Case-III: M < Nr and MN > Nr
Table 4.2: Comparison between PI,approx and PI for Case-III (M < Nr and MN > Nr)
M Nr N PI PI,approx
2 4 4 3.6026 5.463
2 8 4 0.83631 4.211
3 4 4 21.338 22.6224
4 6 4 6.1125 7.083
5 6 4 59.4581 59.9513
5 6 2 6.2277 6.997
6 7 2 6.5631 6.7955
7 8 2 6.8786 6.6553
log(Pk)(Nr − M) − log(N)(Nr + M) + S1Nr,MN − S1
M,Nr= 0
4.5 Crossover Point 54
log(Pk) = log(N)Nr + M
Nr − M+
S1M,Nr
− S1Nr,M
(Nr − M)
The calculated approximate PI,approx serves as estimate for the actual PI . The following
table 4.2 shows some values for different M , N and Nr.
The values in table 4.2 shows that the approximation is very good when number of user
increases.
Chapter 5
Correlation Adaptive User
Scheduling
5.1 System Description
BS MS
x1
H1
x2
H2
Figure 5.1: Hexagonal cellular system with BS at the center
5.1 System Description 56
We assume a single cell system where K users are distributed randomly and base station
is situated at center as described in figure 5.1. Each mobile station (user) has N antennas.
The base system is equipped with Nr antennas.
Since from the previous discussion, it has been observed that received power for any user
decides whether the full correlation is better or no correlation. We assume the inverse square
loss in power with respect to the distance. So the received signal at base station is given by
y = Hkxk + n
where
Hk =
√P
r2Hw,k
where Hw,k consists of complex Gaussian with unit variance, P is the power at unit distance
and r is the distance of BS to the user.
So the effective SNR can be expressed as
Pk =P
r2
At any instant, two level optimization is required one at BS and one at MS. At first level,
mode selection is done at MS. Then on second level, BS selects M users such that channel
capacity is maximized. We assume that the mobile users have only partial CSI information
but know their distance from the base station and thus the power at which their transmitted
data will be received at Base Station.
We propose three different scheduling schemes depending on the channel scenario. In the
remaining part of this chapter, we will discuss these three scheduling schemes each one of
which is based on two level optimization.
For simulation purposes, we have taken a hexagonal cell of radius 10 km to run our
simulations. Two different set of values of M, N, Nr has been simulated which are as follows:
M N Nr K r
2 2 4 8 10
3 4 4 8 10
5.2 Distributive Scheme with Pre-Calculated Crossover Information 57
5.2 Distributive Scheme with Pre-Calculated Crossover Infor-
mation
In distributive schemes, two level optimization is done distributively. Users selects their
modes independent of other users. We assume that user know the crossover point information.
We can describe this schemes in two parts:
5.2.1 Adaptive Correlation Mode Selection at Mobile Station
To make the scheme simple, we let MS assume that other (M − 1)users have same effective
SNR as it has. So mobile stations know the crossover point information Pk,I for the case
where there are only M users and no scheduling is done by BS which allows all M users to
communicate to BS.
Now user calculates the following decision variable
dk = Pk > Pk,I
Depending upon this decision variable, users put itself in the NC mode if dk is true and
in FC mode if it is false and informs BS about its mode and channel state information.
The above procedure is followed at each mobile stations.
Switching of the mode can either be done by smart antennas or layer mapping (i.e.
multiplication by a matrix with desired rank).
5.2.2 User Scheduling at Base Station
At base station, BS collects all the mode indexes and channel information from all users. The
following scheduling algorithm is used by BS to select M users from K.
1. Assign Selection set S to null and Leftover set L to contain all the indexes
2. BS station calculates the maximum achievable capacity by any user and select the user
with maximum achievable channel capacity and put it in set S and removes from L.
3. BS then calculates the following value for each remaining user
5.2 Distributive Scheme with Pre-Calculated Crossover Information 58
Cj = E
⎡⎣log
∣∣∣∣∣∣INr +∑
k∈S⋃{j}
Dk
∣∣∣∣∣∣⎤⎦ (5.1)
where Dk is given by
Dk =
⎧⎪⎨⎪⎩
NPkzk1zHk1 if mode=FC
PkN
∑i<Ni=0 zkizH
ki if mode=NC(5.2)
It selects the users having maximum Cj value and put it in set S and removes from L.
4. Previous step is repeated until maximum number of users have been scheduled in the
resource block.
5.2.3 Simulation Results: DS-PCI
0.5 1 1.5 2 2.5 34
4.5
5
5.5
6
6.5
7
7.5
8
P
Cap
acity
(nat
s/s/
Hz)
FCNC
Figure 5.2: Capacity curves in FC and NC cases for calculation of crossover information
(PCI) for DS-PCI
We first calculate the the crossover point assuming that all other M − 1 users are at the
same point. Figure 5.2 shows the crossover point which is equal to 1.1220 (in normalized
units). We assume that users have this PCI and they select their modes on this information.
5.3 Centralized Scheme with Pre-Calculated Crossover Information 59
−5 0 5 10 15 20
101
P in dB
Cap
acity
(nat
s/s/
Hz
Scheduling M users all in NCScheduling M users all in FCScheduling M users using DS−PCI
Figure 5.3: Capacity curves for DS-PCI compared with no adaptive mode selection
For comparison of capacity, we have taken three cases
• When all the users are transmitting in Full Correlation mode without any adaptive
selection.
• When all the users are transmitting in No Correlation mode without any adaptive
selection.
• When all the users select their mode using DS-PCI adaptive selection.
Figure 5.3 shows the capacity curves for these three cases and we can observe the perfor-
mance improves near crossover points.
5.3 Centralized Scheme with Pre-Calculated Crossover Infor-
mation
In centralized schemes, two level optimization is done at center. Users modes are jointly
selected by BS centrally. We assume that BS know the crossover point information. We can
describe this schemes in two parts:
5.3 Centralized Scheme with Pre-Calculated Crossover Information 60
5.3.1 Adaptive Correlation Mode Selection at Base Station
Since Base station knows the crossover point pI information for the case where there are K
users and the scheduling was done in the same scenario allowing only M users to communicate
to BS without any adaptive mode selection. we denote this crossover as PI .
Now BS calculates the following decision variable for each of the mobile user
d = P > PI
Depending upon this decision variable, BS put the all the user the NC mode if d is true
and in FC mode if it is false and informs users about their mode to adapt. .
5.3.2 User Scheduling at Base Station
At Base Station, BS has the information about all the mode indexes and channel information
from all users. The same scheduling algorithm is used by BS to select M users from K as
described in previous section.
1. Assign Selection set S to null and Leftover set L to contain all the indexes
2. BS station calculates the maximum achievable capacity by any user and select the user
with maximum achievable channel capacity and put it in set S and removes from L.
3. BS then calculates the following value for each remaining user
Cj = E
⎡⎣log
∣∣∣∣∣∣INr +∑
k∈S⋃{j}
Dk
∣∣∣∣∣∣⎤⎦ (5.3)
where Dk is given by
Dk =
⎧⎪⎨⎪⎩
NPkzk1zHk1 if mode=FC
PkN
∑i<Ni=0 zkizH
ki if mode=NC(5.4)
It selects the users having maximum Cj value and put it in set S and removes from L.
4. Previous step is repeated until maximum number of users have been scheduled in the
resource block.
5.3 Centralized Scheme with Pre-Calculated Crossover Information 61
5.3.3 Simulation Results: CS-PCI
−5 0 5 10 15 20
101
P in dB
Cap
acity
(nat
s/s/
Hz
Scheduling M users out of K in NCScheduling M users out of K in FC
Figure 5.4: Capacity curves in FC and NC cases for calculation of crossover information
(PCI) for CS-PCI
−5 0 5 10 15 200.8
0.9
1
1.1
1.2
1.3
1.4
1.5
P
CN
C/C
FC
Figure 5.5: Ratio of capacity values in FC and NC case versus SNR (in dB)
To compute the pre-calculated crossover information, we first calculate the capacity as-
5.4 Centralized Scheme with No Crossover Information 62
suming that all K users are at random positions in the cell and P is the power at which users
is transmitting and it is same for all users. Figure 5.4 shows the crossover point which is
equal to 2.2 (in normalized units). In figure 5.5, we have plotted ratio between FC and NC
curves showing the crossover point.
In this scheme base station collects all users’ allocated power (which is assumed to be
same here) and compares it with cross point (CP) sets all users in the same mode (either FC
and NC) depending on whether allocated power is lower or greater than CP.
For comparison of capacity, we have taken three cases
• When all the users are transmitting in Full Correlation mode without any adaptive
selection.
• When all the users are transmitting in No Correlation mode without any adaptive
selection.
• When users mode selection is done using CS-PCI adaptive selection by BS.
Figure 5.6 shows the capacity curves for these three cases and we can observe the perfor-
mance improves near crossover points.
5.4 Centralized Scheme with No Crossover Information
In this scheme, users modes are selected by BS jointly and at centralized level. In this
scenario, we assume that BS does not know the crossover point information. The mode
selection is done extensively examining each combination of modes and users. In this scheme
scheduling and mode selection of users are done jointly at same level.
The algorithm can be described in the following steps:
1. Make set S which contains all combinations of M users out of K users (there can beKCM total elements).
2. Select an element S ∈ S .
3. For all combinations of modes of users, we calculate capacity by
5.4 Centralized Scheme with No Crossover Information 63
−5 0 5 10 15 20
101
P in dB
Cap
acity
(nat
s/s/
Hz
Scheduling M users all in NCScheduling M users all in FCScheduling M users using CS−PCI
Figure 5.6: Capacity curves for CS-PCI compared with no adaptive mode selection
Cj = E
[log
∣∣∣∣∣INr +∑k∈S
Dk
∣∣∣∣∣]
(5.5)
where Dk is given by
Dk =
⎧⎪⎨⎪⎩
NPkzk1zHk1 if mode=FC
PkN
∑i<Ni=0 zkizH
ki if mode=NC(5.6)
4. BS calculates the maximum capacity C(S) from all selected modes 2M combinations
possible.
5. BS repeats step 2 for all possible combinations of M out of K users.
6. BS selects users set Ssel which maximizes capacity.
Ssel = arg maxS∈S
C(S)
5.4 Centralized Scheme with No Crossover Information 64
and schedules the users of the selected set Ssel.
5.4.1 Simulation Results: CS-NCI
In this scheme, BS station uses no pre calculated CI. It checks for each combination of modes
of M users out of K and schedules users.
Figure 5.7 shows the capacity curves for these three cases and we can observe the per-
formance improves not only near crossover points but also at each power values. Although
at low power and high power, the improvement is low and decreases as we move in to one
extreme, near crossover points the improvement is large.
−5 0 5 10 15 20
101
P in dB
Cap
acity
(nat
s/s/
Hz
Scheduling M users all in NCScheduling M users all in FCScheduling M users using CS−NCI
Figure 5.7: Capacity curves for CS-NCI compared with no adaptive mode selection
Figure 5.8 shows the distribution of mode selection at various values of SNR. The four
modes indicated by 1-4 indexes are given in Table 5.1.
where two values separated by a comma show the modes of two MSs. It can be observed
that at low power, mode index 1 is the preferable choice while the significant occurrence of
mode 2 and 3 accounts for the increase in the capacity in comparison with no adaptive mode
selection. At high power, mode index 4 is the preferable choice while at inbetween values of
5.4 Centralized Scheme with No Crossover Information 65
1 2 3 40
0.5
1P=−8 dB
1 2 3 40
0.5
1P=−4 dB
1 2 3 40
0.5
1P=0 dB
1 2 3 40
0.5
1P=4 dB
1 2 3 40
0.5
1P=8 dB
1 2 3 40
0.5
1P=12 dB
1 2 3 40
0.5
1P=16 dB
1 2 3 40
0.5
1P=20 dB
1 2 3 40
0.5
1P=24 dB
Figure 5.8: CS-NCI: selected mode distribution for different SNR values with x-axis showing
different modes (refer Table 5.1)
Table 5.1: Indexes showing different mode combinations for two users
Index Modes
1 FC,FC
2 FC,NC
3 NC,FC
4 NC,NC
power, all the modes are selected with same probability.
Figure 5.9 shows the distribution of modes at different distances from BS. The first (named
1) graph shows the probability that the user along with the second user are selected in FC
modes given the user at that r distance. The second (named 2) graph shows the probability
that the user is selected in FC while the second user is selected in NC mode given that user at
that r distance. The third (named 3) graph shows the probability that the user is selected in
NC while the second user is selected in FC mode given that user at that r distance. The last
5.4 Centralized Scheme with No Crossover Information 66
0 5 100
0.2
0.4
0.6
0.8
P=−8 dB
0 5 100
0.2
0.4
0.6
0.8
P=−4 dB
0 5 100
0.2
0.4
0.6
0.8P=0 dB
0 5 100
0.2
0.4
0.6
P=4 dB
0 5 100
0.2
0.4
0.6P=8 dB
0 5 100
0.2
0.4
0.6
0.8
P=12 dB
0 5 100
0.2
0.4
0.6
0.8
P=16 dB
0 5 100
0.2
0.4
0.6
0.8
P=20 dB
0 5 100
0.2
0.4
0.6
0.8
P=24 dB
1234
Figure 5.9: CS-NCI: mode selection distribution at different SNR values versus distance of
MS from BS in a 10 radius cell (refer Table 5.1)
(named 4) graph shows the probability that the user along with the second user are selected
in NC modes given a user at that r distance. It can be seen from graph 1 that as distance
increases, probability of selecting mode index 1 decreases.
Figure 5.10 shows the dominant mode at various distance in the cell at different SNR
values. The small region in the center shows the BS. It is obvious that as the SNR increases,
dominant mode changes from 1 to 4. But at the moderate SNR, mode index 2 becomes
5.4 Centralized Scheme with No Crossover Information 67
Figure 5.10: CS-NCI: dominant modes at different SNR values over the hexagonal cell (refer
Table 5.1)
dominant mainly near edges. Near edges, since the power becomes less, user prefers to put
itself into FC and let other user be selected in NC mode. Similar behavior is observed near
center, mode 3 becomes preferable, showing that user put itself in NC mode as power is high
near center and let other user be selected in FC Mode.
At high powers, all the users comes in high power region, thus mode index 4 (signifying
both users selected in NC modes) becomes dominant irrespective of the distance of BS from
MS users. Although mode index 2 shows an increasing behavior with distance from BS as
5.5 Suboptimal CS-NCI 68
predicted in earlier discussion.
5.5 Suboptimal CS-NCI
We also propose a suboptimal less-expensive version of CS-NCI scheduling scheme. In this
scheme, we reverse the order of selection and scheduling. BS first schedules the users and
then allocate modes to them. This scheme can be described as following:
1. Select all users in FC mode and put M best users in set S1.
2. Now select all users in NC mode and put M best users in set S2.
3. Make set S equal to {S1, S2}
4. Select an element S ∈ S .
−5 0 5 10 15 20
101
P in dB
Cap
acity
(nat
s/s/
Hz
Scheduling M users all in NCScheduling M users all in FCScheduling M users using Suboptimal CS−NCI
Figure 5.11: Capacity curves for suboptimal CS-NCI compared with no adaptive mode se-
lection
5. For all combinations of modes of users, we calculate capacity by
5.5 Suboptimal CS-NCI 69
1 2 3 40
0.5
1P=−8 dB
1 2 3 40
0.5
1P=−4 dB
1 2 3 40
0.5
1P=0 dB
1 2 3 40
0.5
1P=4 dB
1 2 3 40
0.5
1P=8 dB
1 2 3 40
0.5
1P=12 dB
1 2 3 40
0.5
1P=16 dB
1 2 3 40
0.5
1P=20 dB
1 2 3 40
0.5
1P=24 dB
Figure 5.12: SCS-NCI:selected mode distribution for different SNR values with x-axis showing
different modes (refer Table 5.1)
Cj = E
[log
∣∣∣∣∣INr +∑k∈S
Dk
∣∣∣∣∣]
(5.7)
where Dk is given by
Dk =
⎧⎪⎨⎪⎩
NPkzk1zHk1 if mode=FC
PkN
∑i<Ni=0 zkizH
ki if mode=NC(5.8)
6. BS calculates the maximum Capacity C(S) from all selected modes 2M combinations
possible.
7. BS repeats step 2 for all possible combinations of M out of K users.
8. BS selects users set Ssel which maximizes capacity.
Ssel = arg maxS∈S
C(S)
5.5 Suboptimal CS-NCI 70
and schedules the users of the selected set Ssel.
The complexity of this scheme is O(K + 2M ) as compared to O( KCM2M ) complexity of
the CS-NCI.
5.5.1 Simulation Results: SCS-NCI
5 10
P=−8 dB
0 5 100
0.2
0.4
0.6
P=−4 dB
0 5 100
0.2
0.4
0.6
P=0 dB
5 10
P=4 dB
0 5 100
0.2
0.4
0.6
P=8 dB
0 5 100
0.2
0.4
0.6
0.8P=12 dB
5 10
P=16 dB
0 5 100
0.2
0.4
0.6
0.8P=20 dB
0 5 100
0.2
0.4
0.6
0.8
P=24 dB
Figure 5.13: SCS-NCI: mode selection distribution at different SNR values versus distance of
MS from BS in a 10 radius cell (refer Table 5.1)
We have simulated SCS-NCI scheme with same parameters. Figure 5.11 shows the perfor-
mance of the suboptimal version of CS-NCI compared with scheduling plan with no adaptive
mode selection. We can seen that the performance improvement (sum capacity) is less than
the earlier CS-NCI but the complexity required in computing the best pattern is significantly
reduced and is almost equal to earlier two algorithms with PCI (DS-PCI and CS-PCI) and
performing better than them.
5.5 Suboptimal CS-NCI 71
Figure 5.14: SCI-NCI: dominant modes at different SNR values over the hexagonal cell (refer
Table 5.1)
Figure 5.12 shows how the mode selection patterns are distributed for various SNR values.
Figure 5.13 shows how modes selection patterns is distributed for different SNR values at
different distances from BS in the cell. Similarly figure 5.14 shows dominant modes in a
cell at various positions. As obvious from these figures, mode selection follows the similar
behavior as the optimal CS-NCI thus resulting in near performance of it.
5.6 Comparison of all Schemes: Scheme Gain 72
5.6 Comparison of all Schemes: Scheme Gain
−5 0 5 10 15 20−0.1
0
0.1
0.2
0.3
0.4
0.5
P in dB
SG
(nat
s/s/
Hz)
CS−PCIDS−PCICS−NCISCS−NCI
Figure 5.15: Comparison of all proposed schemes: scheme gain
For comparison of the schemes, we define scheme gain SG by subtracting maximum of
FC and NC mode capacity from the capacity when using that scheme and is expressed as
SG = Cscheme − max (CFC , CNC) (5.9)
Figure 5.15 shows scheme gain for all proposed schemes at different SNRs. We can observe
that suboptimal version of CS-NCI performs almost equal to the CS-NCI. It should be noted
that in all the schemes, the scheme gain is always positive (or zero) indicating that these
schemes atleast perform better than both FC and NC cases.
Chapter 6
Conclusion and Future Work
In this work, we have studied the effect of transmit antenna correlation in multiuser MIMO-
MAC. We have taken two extreme cases - corresponding to the case when antennas are fully
correlated (FC) and another where antennas are uncorrelated (NC). We have calculated the
expressions for capacity in both these cases. We have first taken the single user case with
two transmit antennas.
Using standard techniques of calculus, we have proved that full correlation is better than
no correlation for low SNR region. Then we have extended the same for multiuser MIMO
MAC system. Also there exist some cases depending on the number of users, number of
transmit antennas and receive antennas, where full correlation is better than no correlation
at every SNR. We have also described all such cases. We have also given an approximate
value of the intersecting point of the FC and NC capacity curves.
It has been observed that value of crossover point increases as we increase number of users
or number of transmit antennas and decreases when we increase number of receiver antennas.
Then we have described three proposed schemes jointly known as ‘adaptive correlation mode
selection based user scheduling algorithm’ to select users mode and schedule them to maximize
the channel capacity. These three schemes are named as
• Distributive Scheme with Pre-Calculated Crossover Information(DS-PCI)
• Centralized Scheme with Pre-Calculated Crossover Information(CS-PCI)
• Centralized Scheme with No Crossover Information(CS-NCI)
74
The last scheme CS-NCI uses the extensive search over the cell. It is clear from the
histogram of mode selection across cell that all users near to the BS are scheduled in full
correlation and users which are at cell edge are mostly scheduled in no correlation. Also at
high SNR, all users are scheduled in NC mode and at low SNR, all the users are scheduled
in FC mode. While at moderate SNR values, the most preferable selection is when one user
is selected in FC positioned near the cell center and one is selected in NC mode positioned
near cell edge.
We have also presented a simple suboptimal but less expensive version of CS-NCI which
shows almost similar behavior of mode selection.
In a future work, theoretical analysis for the variation of crossover point with different
system parameters can be done. Instead of a dual mode (FC and NC) selection, a tri or more
mode (FC, PC and NC) selection can also be used to improve the capacity more where PC
is partial correlation case (say ρ = 0.5). These schemes can also be simulated in 3GPP-LTE
scenario for performance improvement.
Chapter 7
Appendix
A.1: Definition of small f function
To Prove that
f{a; n} =∫ ∞
0
e−z/2zn
1 + azdz
=n∑
i=1
Γ(i)2i(−1)n−i
an−i+1+
e1/(2a)(−1)n
an+1Ei [1/(2a)] (7.1)
Proof:
f{a; n} =∫ ∞
0
e−z/2zn
1 + azdz
=∫ ∞
0
e−z/2(1 + az − 1)zn−1
a(1 + az)dz
=∫ ∞
0
e−z/2zn−1
adz − 1
a
∫ ∞
0
e−z/2zn−1
1 + azdz
=2n
aΓ(n) − 1
af{a;n − 1} (7.2)
Applying this recursion many times, we get
f{n} =1aΓ(n)2n − 1
a2Γ(n − 1)2n−1 +
1a3
Γ(n − 2)2n−3 − 1a3
f{a;n − 3}
=n∑
i=1
(−1)n−i
a2iΓ(i) +
(−1)n
anf{a; 0}
where
f{a; 0} =∫ ∞
0
e−z/2
1 + azdz (7.3)
76
Put 1 + az = 2at
f{a; 0} =∫ ∞
12a
e−(t− 12a
)
2at2dt
=e
12a
at
∫ ∞
12a
e−(t)
tdt
=e
12a
aEi
[12a
]
where Ei(x) is the exponential integral function defined as
Ei(x) =∫ ∞
x
e−t
tdt
putting the value in (7.3), we get the desired result.
A.2: Definition of capital F function
To Prove that
F{a; n} =∫ ∞
0log(1 + az)zne−
z2 dz
= 2an−1∑i=0
f(n − i)2i n!(n − i)!
+ 2n+1n!e1/(2a)Ei [1/(2a)]
Proof:
F{a; n} =∫ ∞
0log(1 + az)zne
z2 dz
=∫ ∞
0{log(1 + az)zn}e− z
2 dz
= {log(1 + az)zn}∫ ∞
0e−
z2 dz −
∫ ∞
0
∂
∂z(log(1 + az)zn)
∫ ∞
0e−
z2 dz dz
= (−2){log(1 + az)zn}e− z2 |∞0 −
∫ ∞
0
(a
zn
1 + az+ log(1 + az)nzn−1
)∫ ∞
0e−
z2 dz dz
= −∫ ∞
0
(a
zn
1 + az+ log(1 + az)nzn−1
)e−
z2 (−2) dz
= 2∫ ∞
0
(a
zn
1 + az+ log(1 + az)nzn−1
)e−
z2 dz
= 2a
∫ ∞
0
(zn
1 + az
)e−
z2 dz + 2n
∫ ∞
0
(log(1 + az)zn−1
)e−
z2 dz
= 2af (a; n) + 2nF{a; n − 1}
77
Applying this recursion expression many times we get,
F{a;n} = 2af (a; n) + 2n.2af (a; n − 1) + 22n(n − 1).2af (a; n − 2) + 23n(n − 1)(n − 2)F{a;n − 3}
= 2an−1∑i=0
f (a; n − i) 2i n!(n − i)!
+ 2nn!F{a; 0}
where
F{a; 0} =∫ ∞
0{log(1 + az)}e− z
2 dz
Put 1 + az = 2at in the above equation to get
F{a; 0} = 2e1/(2a)
∫ ∞
1/(2a)log(2at)e−tdt (7.4)
= 2e1/(2a)Ei
[12a
](7.5)
Putting this value into equation (7.4), we get the desired result.
A.3: Properties of Small f and Capital F functions
Properties of f{a;n}:
0 2 4 6 8 1010−1
100
101
102
103
104
→ a
→ f{
a;N
}
N=1N= 2N= 3N= 4N= 5
Figure 7.1: Plot of f with scaling parameter a
78
1 1.5 2 2.5 3 3.5 4 4.5 510−1
100
101
102
103
→ N
→ f{
a;N
}
a=1 a=3 a=7 a=10
Figure 7.2: Plot of f with order N
In this section some of the important properties of f{a; N} has been discussed.
1. We will first show the variation of f{a; N} with respect to the scaling parameter and
order. Figures 7.1 and 7.2 shows the change in f with respect to its parameters.
2. f{0;N} = 2N+1N !
3. lima→∞ f{a; n} = 1a2n(n − 1)!
Properties of F{a;n}:
In this section some of the important properties of F{a; N} has been discussed.
1. We will first show the variation of F{a; N} with respect to the scaling parameter and
order. Figures 7.3 and 7.4 shows the change in F with respect to its parameters.
2. F{0;N} = 0
79
1 2 3 4 5 6 7 8 9 10100
101
102
103
104
→ a
→ F
{a;N
}
N=1
N= 2
N= 3
N= 4
Figure 7.3: Plot of F with scaling parameter a
1 1.5 2 2.5 3 3.5 4
101
102
103
→ N
→ F
{a;N
}
a=1 a=3 a=7 a=10
Figure 7.4: Plot of F with order N
80
3. ∂F{a;N}∂a = f{a;N + 1}
4. lima→∞ F{a; n} = n!2n+1 (log(2a) + ψ(n + 1)) where ψ(n) = −γ +∑n−1
i=1 1/i
A.4: Distribution of eigen values of ZHZ matrix where Z is m × n matrix
with n > m
From [1], the pdf of eigen value can be written as
p(λ) =1m
m−1∑k=0
k!(k + n − m)!
(Ln−m
k (λ))2
λn−me−λ
where Lpk(λ) is the associated Laguerre polynomial of order k and is given by
Lpk(λ) =
k∑s=0
(−1)s (k + p)!(k − s)!(p + s)!s!
xs
The square of associated laguerre polynomial is given by
LSpk(x) =
[k∑
s=0
(−1)s (k + p)!(k − s)!(p + s)!s!
xs
][k∑
t=0
(−1)t (k + p)!(k − t)!(p + t)!t!
xt
]
LSpk(x) =
k∑s=0
k∑t=0
(−1)s+t (k + p)!(k + p)!(k − s)!(p + s)!s!(k − t)!(p + t)!t!
xs+t
using the transformation
r = s + t
t = t
we get
LSpk(x) =
2k∑r=0
t=min {r,k}∑t=max {0,r−k}
(−1)r (k + p)!(k + p)!(k − r + t)!(p + r − t)!(r − t)!(k − t)!(p + t)!t!
xr
Let us write the above expression as power series of x
LSpk(x) =
2k∑r=0
Ak,pr xr
where Ak,pr are squared associated laguerre polynomial coefficients and are given by
Ak,pr = (−1)r
t=min {r,k}∑t=max {0,r−k}
(k + p)!(k + p)!(k − r + t)!(p + r − t)!(r − t)!(k − t)!(p + t)!t!
81
Ak,pr = (−1)r (k + p)!(k + p)!
k!k!(2p + r)!
t=min {r,k}∑t=max {0,r−k}
2p+rCt+pkCt
kCr−t
using the definition of binomial coefficients of negative indexes, we can write as
Ak,pr = (−1)r (k + p)!(k + p)!
k!k!(2p + r)!
r∑t=0
2p+rCt+pkCt
kCr−t
A.5
To Prove
M∑t=0
(−1)t MCtp+tCs =
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
0 s ≤ M − 1
(−1)M s = M
(−1)Mp s = M + 1
(7.6)
We know from binomial expansion that
(1 + x)M =M∑t=0
MCtxM−t
(1 + x)−(s+1) =∞∑
t=0
s+kCsxk(−1)k
Multiplying the above two expression, we get
(1 + x)M =M∑t=0
MCtxM−t
(1 + x)M−(s+1) =M∑t=0
∞∑k=0
MCtxM−t s+kCsx
k(−1)k
(1 + x)M−(s+1) =M∑t=0
∞∑k=0
MCtxM−t+k s+kCs(−1)k
Collecting the coefficients of xm+p−s, we get
(−1)M+p−s s+1−M+M+p−s−1CM+p−s = (−1)p−sM∑t=0
(−1)t MCtp+tCs
(−1)M pCM+p−s =M∑t=0
(−1)t MCtp+tCs
82
Putting different values of s, we can get the desired results.
A.6
To Prove
M−1∑k=0
k∑s=0
k∑t=0
k!(−1)s+t(k + Nr − M)!(Nr − M + s + t + 1)!(k − s)!s!(Nr − M + s)!(k − t)!t!(Nr − M + t)!
= MNr (7.7)
Proof: Let us write
I(M, Nr) =M−1∑k=0
k∑s=0
k∑t=0
k!(−1)s+t(k + Nr − M)!(Nr − M + s + t + 1)!(k − s)!s!(Nr − M + s)!(k − t)!t!(Nr − M + t)!
(7.8)
We will prove the above identity using mathematical induction over M and Nr,
For M = 1, we have
I(1, Nr) =0!(−1)0(Nr − 1)!(Nr)!
(0)!0!(Nr − 1)!(0)!0!(Nr − 1)!= Nr
The identity is true for M = 1.
Now let’s assume that this is true for all m ≤ M and nr ≤ Nr, i.e.
I(m,nr) = m.nr∀m ≤ M&nr ≤ Nr
I(M + 1, Nr) =M∑
k=0
k∑s=0
k∑t=0
k!(−1)s+t(k + Nr − M − 1)!(Nr − M − 1 + s + t + 1)!(k − s)!s!(Nr − M − 1 + s)!(k − t)!t!(Nr − M − 1 + t)!
= I(M, Nr − 1) +M∑
s=0
M∑t=0
M !(−1)s+t(Nr − 1)!(Nr − M − 1 + s + t + 1)!(M − s)!s!(Nr − M − 1 + s)!(M − t)!t!(Nr − M − 1 + t)!
= I(M, Nr − 1) +M∑
s=0
(−1)s(Nr − 1)!(s + 1)!(M − s)!s!(Nr − M − 1 + s)!
M∑t=0
(−1)t MCt(Nr−M+s+t)Cs+1
= I(M, Nr − 1) +M∑
s=0
(−1)s(Nr − 1)!(s + 1)!(M − s)!s!(Nr − M − 1 + s)!
(−1)M Nr−M+sCM+Nr−M+s−s−1
83
I(M + 1, Nr) = I(M, Nr − 1) +M∑
s=0
(−1)s(Nr − 1)!(s + 1)!(M − s)!s!(Nr − M − 1 + s)!
(−1)M Nr−M+sCNr−1
= I(M, Nr − 1) +(−1)M−1(Nr − 1)!(M)!
(M − M + 1)!(M − 1)!(Nr − M − 1 + M − 1)!(−1)M Nr−1CNr−1
+(−1)M (Nr − 1)!(M + 1)!
(M − M)!s!(Nr − M − 1 + M)!(−1)M NrCNr
= I(M, Nr − 1) − (Nr − 1)!(M)!(1)!(M − 1)!(Nr − 2)!
+(Nr − 1)!(M + 1)!
M !(Nr − 1)!Nr
= M(Nr − 1) − (Nr − 1)(M) + (M + 1)Nr
= (M + 1)Nr
Similarly,
I(M, Nr + 1) =M−1∑k=0
k∑s=0
k∑t=0
k!(−1)s+t(k + Nr + 1 − M)!(Nr − M + 1 + s + t + 1)!(k − s)!s!(Nr + 1 − M + s)!(k − t)!t!(Nr + 1 − M + t)!
= I(M − 1, Nr)
+M−1∑s=0
M−1∑t=0
(M − 1)!(−1)s+t(Nr)!(Nr − M + s + t + 2)!(M − 1 − s)!s!(Nr − M + 1 + s)!(M − 1 − t)!t!(Nr − M + 1 + t)!
= I(M − 1, Nr)
+M−1∑s=0
(−1)s(Nr)!(s + 1)!(M − 1 − s)!s!(Nr − M + 1 + s)!
M−1∑t=0
(−1)t M−1Ct(Nr−M+s+t+2)Cs+1
= I(M − 1, Nr)
I(M, Nr + 1) +M−1∑s=0
(−1)s(Nr)!(s + 1)!(M − 1 − s)!s!(Nr − M + 1 + s)!
(−1)M−1 Nr−M+s+2CM−1+Nr−M+s+2−s−1
= I(M − 1, Nr)
+M−1∑s=0
(−1)s(Nr)!(s + 1)!(M − 1 − s)!s!(Nr − M + 1 + s)!
(−1)M−1 Nr−M+s+2CNr
= I(M − 1, Nr) +(−1)M−2(Nr)!(M − 1)!
(M − 1 − M + 2)!(M − 2)!(Nr − M + 1 + M − 2)!(−1)M−1 NrCNr
+(−1)M−1(Nr)!(M)!
(M − 1 − M + 1)!(M − 1)!(Nr − M + 1 + M − 1)!(−1)M−1 Nr+1CNr
84
I(M, Nr + 1) = I(M − 1, Nr) − (Nr)!(M − 1)!(1)!(M − 2)!(Nr − 1)!
+(Nr)!(M)!
(M − 1)!(Nr)!(Nr + 1)
= (M − 1)Nr − (Nr)(M − 1) + (M)(Nr + 1)
= M(Nr + 1)
which completes the proof.
A.7
To Prove
M−1∑k=0
k∑s=0
k∑t=0
k!(−1)s+t(k + Nr − M)!(Nr − M + s + t)!(k − s)!s!(Nr − M + s)!(k − t)!t!(Nr − M + t)!
= M (7.9)
Proof: We will follow the same steps as in A6 to prove this. Let us write
I(M,Nr) =M−1∑k=0
k∑s=0
k∑t=0
k!(−1)s+t(k + Nr − M)!(Nr − M + s + t)!(k − s)!s!(Nr − M + s)!(k − t)!t!(Nr − M + t)!
(7.10)
We will prove the above identity using mathematical induction over M and Nr,
For M = 1, we have
I(1, Nr) =0!(−1)0(Nr − 1)!(Nr − 1)!
(0)!0!(Nr − 1)!(0)!0!(Nr − 1)!= 1
The identity is true for M = 1.
Now let’s assume that this is true for all m ≤ M and nr ≤ Nr, i.e.
I(m,nr) = m∀m ≤ M&nr ≤ Nr
85
I(M + 1, Nr) =M∑
k=0
k∑s=0
k∑t=0
k!(−1)s+t(k + Nr − M − 1)!(Nr − M − 1 + s + t)!(k − s)!s!(Nr − M − 1 + s)!(k − t)!t!(Nr − M − 1 + t)!
= I(M, Nr − 1) +M∑
s=0
M∑t=0
M !(−1)s+t(Nr − 1)!(Nr − M − 1 + s + t)!(M − s)!s!(Nr − M − 1 + s)!(M − t)!t!(Nr − M − 1 + t)!
= I(M, Nr − 1) +M∑
s=0
(−1)s(Nr − 1)!(s)!(M − s)!s!(Nr − M − 1 + s)!
M∑t=0
(−1)t MCt(Nr−M+s+t−1)Cs
= I(M, Nr − 1) +M∑
s=0
(−1)s(Nr − 1)!(s)!(M − s)!s!(Nr − M − 1 + s)!
(−1)M Nr−M+s−1CM+Nr−M+s−s−1
I(M + 1, Nr) = I(M,Nr − 1) +M∑
s=0
(−1)s(Nr − 1)!(s)!(M − s)!s!(Nr − M − 1 + s)!
(−1)M Nr−M+s−1CNr−1
= I(M,Nr − 1) +(−1)M (Nr − 1)!(M)!
(M − M)!M !(Nr − M − 1 + M)!(−1)M Nr−1CNr−1
= I(M,Nr − 1) +(Nr − 1)!(M)!M !(Nr − 1)!
= M + 1
= (M + 1)
Similarly,
I(M, Nr + 1) = I(M − 1, Nr) + 1
= M
which completes the proof.
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