Adaptive Scheduling and Capacity of Multiuser MIMO MAC …home.iitk.ac.in/~adrish/abhithesis.pdf · 2010. 6. 23. · CERTIFICATE It is certified that the work contained in the thesis

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


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


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]


det(I + AB + CD) = det

⎛⎝I +

[A C

]⎡⎣B

D

⎤⎦⎞⎠


⎛⎝I2 +

⎡⎣B

D

⎤⎦[A C

]⎞⎠


⎛⎝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


⎡⎣ 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


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


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)


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)


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


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


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


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)


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


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


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

)


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


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.


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


(2p + r

t + p

)(k

t

)(k

r − t

)


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,


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


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



Case-III: M < Nr and MN > Nr

Similarly in this case,

G′(0) =1N

(MNNr) − N(MNr)

= −M(N − 1)Nr



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


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


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)


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.


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)


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

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


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


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.


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


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:


5.3.1 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)


Dk =

⎧⎪⎨⎪⎩


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.


5.3.3 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 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)


Dk =

⎧⎪⎨⎪⎩


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)


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


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


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


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


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)


Dk =

⎧⎪⎨⎪⎩


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)


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.


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


)e−

z2 (−2) dz

= 2∫ ∞

0

(a

zn


)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


(−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


(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)!


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.

References

[1] E. Telatar. Capacity of the multiple-antenna Gaussian channel. Euro.Trans. Telecom-

mun., vol. 10, pp. 585-595, 1999.

[2] D. S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn. Fading correlation and its

effect on the capacity of multielement antenna systems. IEEE Trans. Comm., vol. 48,

pp. 502-513, Mar. 2000.

[3] W. Yu, W. Rhee, S. Boyd, and J. Cioffi. Iterative water-filling for Gaussian vector

multiple access channels. IEEE Trans. Inform. Theory,vol. 50, no. 1, pp.145-151, 2004.

[4] C. N. Chuah, D. Tse, J. M. Kahn, and R. A. Valenzuela. Capacity scaling in MIMO

wireless systems under correlated fading. IEEE Trans.Inform. Theory, vol. 48, no. 3,

pp. 637-650, Mar. 2002.

[5] T. M. Cover and J. A. Thomas. Elements of Information Theory. John Wiley and Sons,

New York, 2006.

[6] C. E. Shannon. A mathematical theory of communication. Bell System Technical Jour-

nal, Vol. 27, pp. 379423, 623656, 1948.

[7] A. Tulino, A. Lozano, and S. Verdu. Impact of antenna correlation on the capacity of

multiantenna channels. IEEE Trans. Inf. Theory,51, pp. 637-650, 2005.

[8] R. Louie, M. McKay, and I. Collings. Impact of correlation on the capacity of multiple

access and broadcast channels with MIMO-MRC. IEEE Trans. Wireless Commun., vol.

7, no. 6, pp. 2397-2407, June 2008.

REFERENCES 87

[9] S. Vishwanath, N. Jindal, and A. Goldsmith. Duality, achievable rates and sum-rate

capacity of Gaussian MIMO broadcast channels. IEEE Trans. Inform. Theory, vol. 49,

pp. 2658-2668, Oct. 2003.

[10] Q. Zhou and H. Dai. Asymptotic analysis on spatial diversity versus multiuser diversity

in wireless networks. Proc. IEEE Int. Conf. on Commun. (ICC), Istanbul, Turkey, June

2006.

[11] D. Park and S. Y. Park. Effect of transmit antenna correlation on multiuser diversity.

Proc. IEEE Int. Symp. Inform. Theory (ISIT), Adelaide, Australia, Sept. 2005.

[12] A. Soysal and S. Ulukus. Optimum power allocation for single user MIMO and multiuser

MIMO-MAC with partial CSI. IEEE Journal on Selected Areas in Communication, Vol.

25, No. 7, pp 1402-1411, Sept 2007.

[13] S. Lasaulce, A. Suarez, and M. Debbah. Power allocation game for fading MIMO multiple

access channels with antenna correlation. france.

[14] J. Nash. Equilibrium points in n-person games. Proceedings of the National Academy of

Sciences, 36(1):48-49, 1950.

[15] P. Viswanath, D. Tse, and R. Laroia. Opportunistic beamforming using dumb antennas.

IEEE Trans. Inform. Theory, vol. 48, no. 6, pp. 1277-1294, June 2002.

[16] N. Zhang and B. Vojcic. Multiuser diversity scheduling in MIMO systems with correlated

fading. IEEE GlobeCom, St. Louis, Nov. 2005.

[17] H. Wang, P. Wang, and L. Ping. On the impact of antenna correlation in multi-user

MIMO systems with rate constraints. IEEE Communication Letters, Vol. 13, No. 12,pp

935-937, Dec 2009.

[18] H. Wang, P. Wang, L. Ping, and X. Lin. Beneficial impact of antenna correlation at

mobile units in MIMO MACs with covariance feedback. Electronics letters Vol. 45 No.

20, 2009.

REFERENCES 88

[19] C. B. Ribero, K. Hugl, M. Lampinen, and M. Kuusela. Performance of linear multi-user

MIMO precoding in LTE system. International Symposium on Wireless and Pervasive

Computin (ISWPC), pp. 410-415, 2008.

[20] D. A. Harville. Matrix Algebra from a statistician’s perspective. Springer, 1997.

[21] A. Lapidoth and S. M. Moser. Capacity bounds via duality with appilcation to multiple

antenna system on flat fading channels. IEEE Transaction on Information theory, vol

49, no 10, pp 2426-2467, oct 2003.

[22] L. Mascheroni. Adnotationes ad calculum integralem Euleri, Vol. 1 and 2. Ticino, Italy,

1790 and 1792.

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