Interference Alignment and DOF Analysis of Interference

Interference Alignment and DOF Analysis ofInterference and Interference Broadcast

Channels

by

Jhanak Parajuli

A thesis submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

in Electrical Engineering

Approved Dissertation Committee:

Prof. Dr. Eng. Giuseppe Abreu(Jacobs University Bremen)

Prof. Dr. Ing. Werner Henkel(Jacobs University Bremen)

Dr. Antti Tolli(CWC, University of Oulu)

Date of Defense: September 22, 2016.

Computer Science & Electrical Engineering, Jacobs University Bremen

ii

Dedication

“Behind every child who believes in himself/herself is a parent who believed first.”

Matthew Jacobson

I dedicate this thesis to my parents:

Jayaram Parajuli

and

Chuna Devi Parajuli.

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iv

Acknowledgements

I would like to express my deep thanks and sincere gratitude to all who have helped me in

one way or other during my complete Ph.D program. I am very very grateful to my Ph.D.

supervisor Prof. Dr. Giuseppe Abreu, whose constant motivation, strong support, quality

guidance, valuable discussions and strict requirements could make my work a success. I am

very thankful for the convenient and friendly academic environment that he provided during

my Ph.D and masters studies at Jacobs University Bremen. I could really grow academically

during this time. Thanks to the university for such a wonderful environment . I am also very

grateful to Prof. Dr. Ing. Werner Henkel and Dr. Antti Tolli, who have agreed to be on my

dissertation committee.

I am also thankful to the European Union FP7 project BUTLER (uBiquitous, secUre

inTernet-of-things with Location and contEx-awaReness) and H2020 project HIGHTS (High-

precision positioning for cooperative Intelligent Transport systems) for providing me economic

assistance, without which I could not imagine my Ph.D degree. My special thanks to Dr.

Stefano Severi for his constant efforts to work on new projects which made it really easy

for me and I could focus on my Ph.D work. I am very thankful to my previous and present

work-group and colleagues Samip, Satya, Tayo, Simona, Yohannes, Remun, Ali, Andrei,

Cristian, Jonathan and all other who are involved in our work-group in one way or the other.

Fruitful discussions during our group meeting always helped me a lot.

My sincere thanks to my Nepali friends in Jacobs University, Samip, Asman, Anup, Riwaj,

Shailesh, Dileep, Chhabi, Ujjwal, Dev and my friends outside the university Shankar, Ashish,

Sudha, Bishal, Gyanendra, Aabhushan, Jwalanta , Gunjan and Binod who supported me in

every possible ways they could and helped me feel homely every time. A very very sincere

thanks to my very nice and lovely host family in Bremen (Gerlind, Axel and Marius) who

helped me a lot during my initial days in Bremen. I always appreciate their help.

My special thanks to the MIT open courseware, edX and coursera that allows a framework

to take a number of online courses and make myself clear on the topics which I had not

understood before. It was really very helpful many times. Also, a very special thanks to the

badminton club members in the university and members of my house in Cigarrenmanufaktur,

especially to Shailesh, Viri, Satya and Harsha.

Last but not the least, I am very thankful to my parents, who always devoted their life for

v

giving me better education and helping me in every possible ways they could; and to all my

family members, my brothers, sisters, brother-in-laws and sister-in-laws for providing me a

strong mental and economic support when needed. Thanks to all my well-wishers and to all

those who helped me and I forgot to mention here.

vi

Statutory Declaration

I, Jhanak Parajuli, hereby declare, under penalty of perjury, that I am aware of the

consequences of a deliberately or negligently wrongly submitted affidavit, in particular the

punitive provisions of § 156 and § 161 of the Criminal Code (up to 1 year imprisonment or a

fine at delivering a negligent or 3 years or a fine at a knowingly false affidavit).

Furthermore I declare that I have written this PhD thesis independently, unless where

clearly stated otherwise. I have used only the sources, the data and the support that I have

clearly mentioned.

This PhD thesis has not been submitted for the conferral of a degree elsewhere.

Bremen, October 10, 2016.

Signature

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viii

Abstract

Multi-user wireless communication system is interference limited and determining the capacity

region for such network is a challenging problem. Information theoretically, even the capacity

region of a two-user Gaussian interference channel (IC) remains an open problem since

1970s. However, recently a new cooperative interference management technique, called the

interference alignment (IA), has been proposed and shown to achieve the optimum degrees of

freedom (DOF) in interference networks such as multiple access channel (MAC), broadcast

channel (BC), IC and the X-channel (XC). Since DOF is the first order approximation of the

capacity at high signal to noise ratios (SNRs), IA has been a topic of tremendous interests to

the wireless communication engineers and information theorists.

Interference alignment is a technique in which all the interfering signals observed from

different transmitters are aligned onto the same direction or onto a common subspace while

maintaining independence with the desired signal in a particular receiver. In a K-user single

input single output (SISO) interference network with K > 2, IA allows all K − 1 interference

signals to be aligned over only 1 dimension as opposed to K−1 dimensions in other spectrum

sharing techniques like frequency division multiple access (FDMA) and time division multiple

access (TDMA). Thus, each user can achieve maximum of 12

DOF, which is much higher than1K

in traditional FDMA and TDMA systems.

Aligning K − 1 independent interference signals observed from different sources onto a

single direction becomes mathematically challenging as K increases. Initial ideas in [1]

considered infinite uses of the channel and proved that 12

DOF is mathematically achievable

for any SISO-IC. Later, it is observed that for the multiple input multiple output (MIMO)

settings, the spatial dimensions can be exploited and infinite channel uses are not required

for the alignment, but the interference can be aligned by designing proper precoding matrices

at all the transmitters [2].

However, one of the main drawbacks of these approaches are the requirement of instanta-

neous channel state information at the transmitter (CSIT) and the explosion of parameters

as K increases. In fact, a robust generalized IA algorithm for any interference network still

remains an open problem.

In this thesis, we aim to address some of these existing challenges in the interference

management of multi-user networks by providing following three fold contributions:

ix

• Improvement of the existing IA algorithms: Using cooperation among the receivers, it

is shown that a proper precoding and zero-forcing matrix can be designed iteratively by

decreasing the interference power and increasing the desired signal power simultaneously

for a K-user MIMO-IC. Such precoding matrices improves the system capacity measured

in bits per second per channel use than previously existing algorithms. Simulation

results are provided.

• Relaxation of the requirement of CSIT: A space-time transmission scheme is designed

which uses only the delayed CSIT instead of instantaneous CSIT and such scheme is

shown to achieve greater than one DOF in any multi-cell multiple input single output

(MISO) BC. Such interference broadcast channel (IBC) requires both the inter cell

interference (ICI) and the inter user interference (IUI) be canceled properly. It is shown

that by proper transmission strategy, the achievable DOF converges to 85

when the

number of users per cell becomes very large.

• Proposition of a new alignment technique: A new grouping based IA technique is

proposed where the receivers within a group cooperatively align the interference from a

common source onto an overlapping space. By doing so, the receivers can take benefit

from the unused spatial dimensions of each other. Such concept of alignment is termed

as vertical alignment. Proper precoding matrices are designed to align the ICI onto a

common overlap space in two cell MIMO BC and optimal overlapping dimensions are

determined for ring and star topologically arranged three user MIMO IC.

DOF analysis of three user MIMO IC with any arbitrary number of transmit and receive

antennas is a challenging problem and very few works are done in this regard. By

determining the optimal overlapping between any two adjacent receivers, a relatively

new and simple chain based DOF analysis technique is proposed, where the DOF can

be expressed in terms of the length of such chain. The scheme is termed as receiver

chain alignment (RCA).

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Contents

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Interference Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Basic Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Precoding Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Asymptotic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Ergodic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.3 IA Based on Real Numbers . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.4 Lattice-Based Approach . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.5 Topology Based Approach . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 MIMO Interference Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 From Global to Local CSIT . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.1 Minimization of Leakage Interference . . . . . . . . . . . . . . . . . . 23

2.4.2 Maximization of Signal to Interference plus Noise Ratio . . . . . . . . 26

2.4.3 Alternating Minimization of Interference . . . . . . . . . . . . . . . . 27

2.5 Feasibility of Interference Alignment . . . . . . . . . . . . . . . . . . . . . . 29

3 Interference Alignment with Receiver Cooperation and Greedy Transmission . . . 33

3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Coalition Game Theory and Full Cooperation . . . . . . . . . . . . . . . . . 39

3.4 Simulation Results and Feasibility . . . . . . . . . . . . . . . . . . . . . . . . 40

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Space-Time Transmission with Delayed CSIT . . . . . . . . . . . . . . . . . . . . 45

4.1 Single Cell Two-user MISO BC with Delayed CSIT . . . . . . . . . . . . . . 47

4.2 K-user MISO BC with Delayed CSIT . . . . . . . . . . . . . . . . . . . . . . 51

4.3 Two cell MISO IBC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

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4.4 Space-Time Transmission Scheme . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4.1 Case I ( M=1, K=1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4.2 Case II (M=1, K=2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4.3 Case III (M=2, K=1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4.4 Case IV (M=2, K=2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4.5 Case V (M=3, K=1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4.6 Case VI (M=3, K=3) . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4.7 Case VII (Generalized Scheme for M = K) . . . . . . . . . . . . . . . 76

4.5 Improved Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.6 More Than Two-Cell MISO-IBC . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5 Interference Alignment and Optimal Overlapping in MIMO IBC and IC . . . . . 85

5.1 MIMO Interference Broadcast Channels . . . . . . . . . . . . . . . . . . . . 86

5.1.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.1.2 Designing Post-processing Matrices . . . . . . . . . . . . . . . . . . . 88

5.1.3 Designing the Precoding Matrices . . . . . . . . . . . . . . . . . . . . 89

5.2 MIMO Interference Channels . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2.2 Determining the Rank of N (Φ) . . . . . . . . . . . . . . . . . . . . . 98

5.2.3 Ring Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2.4 Star Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6 DOF Analysis of Three User MIMO IC via Receiver Chain Alignment . . . . . . . 111

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.2 Receiver Chain Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.3 DoF Analysis and Achievability . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.3.1 Case 1 : 0 ≤ N−MM≤ 1

2. . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.3.2 Case 2 : 12< N−M

M≤ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.3.3 Case 3 : 1 < N−MM≤ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.4 Information theoretic outer bound for DOF . . . . . . . . . . . . . . . . . . 119

6.5 Achievability of the DOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7 Conclusions and Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

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7.2 Discussion and Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

A Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

B Proof of Theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

C Proof of Theorem 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

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xiv

List of Figures

2.1 A K-user single input single output interference channel, where the solid lines

represent a direct or the desired links and the dashed line represent the cross

or interference links. All the receivers receive one desired link and K − 1

interference links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 A geometric explanation of IA with four interference aligned in two dimensions

and one desired signal in the orthogonal direction. Green is the desired signal

while blue, red, black and magenta are the interference signal. . . . . . . . . 10

2.3 A partially connected 5 user SISO IC. A solid line represents the desired

channel while the dashed lined represents the interference channel. . . . . . . 19

2.4 Alignment conflict graph for the network in Figure 2.3. Solid line represents

the alignment graph and the dashed line represents the conflict graph. . . . . 19

2.5 Interference alignment in three user MIMO IC. The solid line represents a

desired channel matrix and the dashed and dotted lines represent the interfer-

ence channel matrices. For ease of representation, the alignment of multiple

spatial dimensions is represented by an arrow. For example, dashed green

arrow represents a d dimensional subspace observed at RX1 from TX3. . . . 21

2.6 A K-user MIMO IC with M transmit and N receive antennas. Vjs are the

precoding matrices and Uis are the zero-forcing matrices, ∀ i, j = {1, 2, · · · , K}.Solid lines represent desired channels and dotted and dashed lines represent

interference channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7 For K = 3 user MIMO IC with M = 7 transmit antennas and N = 9 receive

antennas, only d = 4 DOF is achievable for a proper system. . . . . . . . . . 32

3.1 Sum rate achieved measured in bits/sec/Hz as a function of SNR measured

in dB for different transmit antennas, receive antennas and feasible DOF

that achieves IA for such case. The results are compared for the proposed

cooperative algorithm, minimization of leakage interference (MLI) algorithm

and maximization of signal to interference plus noise ratio (max-SINR) algorithm. 42

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3.2 Feasibility of interference alignment for the proposed cooperative and MLI

algorithms. The feasibility is measured as the percentage of the leakage

interference after each iteration measured for the given total data streams.

The proposed cooperative algorithm has less leakage interference than the MLI

for M = N = 3 and M = N = 4 with different number of total transmit streams. 43

4.1 A two user MISO BC with two antennas at the transmitter. The two indepen-

dent data streams are transmitted as a vector x(t) at any time instance t and

the channel vector is represented by hi1(t) and hi2(t), ∀i = {1, 2}. . . . . . . 48

4.2 A two-cell MISO interference broadcast channel with M transmit antennas in

each base station and K single antenna users per cell. The solid line represents

the signal received from the same cell and the dashed line represents the inter

cell interference received from the adjacent cell. The dotted line represents

that the delayed channel state information (CSI) feedback is provided from

any receiver to the transmitter. The two cells are connected via back-haul

connection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 DOF of two-cell MISO IBC with delayed and instantaneous CSIT for odd and

even number of users. For odd number of users, the DOF converges to 43

and

for the even number of users achievable DOF is always 43. . . . . . . . . . . . 79

4.4 Achievable DOF for two cell MISO IBC with delayed and instantaneous CSIT.

The improved transmission scheme achieves better DOF which converges

to 85

unlike the earlier approach that converges to 43. The DOF with the

instantaneous CSIT converges to 2. . . . . . . . . . . . . . . . . . . . . . . 82

5.1 Two cell MIMO Broadcast Channel with two users per cell and arbitrary

M antennas at each base station and N antennas per user. The solid line

represents the desired signal plus IUI, while the dashed line represents the ICI

observed by each user. The channels from base station j to user k in cell `

is represented as H(`)kj , while Vj are the precoding matrices and Uk` are the

interference suppressing matrices or the post processors at user k in cell `. . 87

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5.2 Plot of the total interference dimensions and the overlap dimensions for given

transmit streams d varying with the number of overlapping rows. The upper

part plots the total number of free dimensions after each user receives d dimen-

sional interference signal varying with the overlapping rows of the channels

between two users. The lower part plots the dimensions of the common region

where all interferences are aligned by two users. . . . . . . . . . . . . . . . . 90

5.3 Plot of the rank(N (Φ)

)with the number of overlapping rows per user when

different number of data streams are transmitted. The optimal number of

overlapping rows are the number of overlapping rows when the rank(N (Φ)

)=

d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.4 MAC (left side) and BC (right side) duality for the three-user MIMO IC where

only the interference signals are shown. U1 is determined with the knowledge

of V2 and V3 and V1 is determined with the knowledge of U2 and U3. . . . 93

5.5 Relationship between Φ and Z. . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.6 Interference dimension for different M . . . . . . . . . . . . . . . . . . . . . 100

5.7 An example of ring topology for antennas overlapping. Each antenna is indexed

with the alphabets. ‘a,b,c,d’ are the antennas in RX1, ‘e,f,g,h’ are the antennas

in RX2, ‘i,j,k,l’ are the antennas in RX3. All receivers share different antennas

with the adjacent receivers. The antenna elements in the dashed box are the

antennas shared between two receivers. . . . . . . . . . . . . . . . . . . . . . 101

5.8 Rank of N (Φj) for different M , when different antennas overlap between

the adjacent receivers. The null space is decreasing as the overlapping rows

increase. The graph shows two regions and the point where the graph changes

as shown by the dashed line gives the optimal number of overlapping rows. 104

5.9 An example of star topology for antennas overlapping in three user MIMO IC.

Each antenna is indexed with the alphabets. Same antennas overlap for all

the receivers. For example, the antennas in the dashed box ‘a’,’e’,’i’ overlap in

all three receivers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.10 Rank of N (Φ) for different transmit antennas varying with the number of

overlapping antennas in the ring topological overlap structure. The graph

shows two regions and the point where the region changes shows the optimal

overlapping rows as shown by the dashed line. . . . . . . . . . . . . . . . . . 108

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6.1 RCA for M = N = 4 and M = N = p (general case) in 3-user IC with optimal

overlapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2 RCA for M = 4, N = 5 with optimal overlapping. . . . . . . . . . . . . . . . 114

6.3 RCA for M = 4, N = 7 with optimal overlapping. . . . . . . . . . . . . . . . 116

6.4 RCA for M=6 and N=10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.5 RCA for M = 2, N = 5 in 3-user IC for optimal overlapping. . . . . . . . . 118

6.6 DOF plot for different transmit and receive antennas under different DOF

achievable schemes. The ‘*’ line represents the DOF achieved using the

proposed RCA scheme, the ‘o’ line represents the DOF achieved with the

scheme proposed by Jafar et. al [3] and the ‘square’ line represents the DOF

achieved with subspace alignment chain (SAC) scheme proposed by Wang et.

al [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.7 Overlap interference and transmission in K = 3 user IC with M = 2 and N = 3.122

6.8 An example analysis of alignment due to antenna sharing. . . . . . . . . . . 125

7.1 Alignment error for different total overlap dimensions due to two receivers.

The error is decreasing as the overlap is increasing. . . . . . . . . . . . . . . 134

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List of Tables

4.1 Two desired equations are observed by each user in both the cells after 9

channel uses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2 The signal received by all the users in both the cells during time instances

t = 1, 2, 3. The red color signal represents the signal to be swapped between

the first and second user in each cell and the blue colored signal represents the

signal to be swapped between the first and the third users in each cell. . . . 69

4.3 Three desired equations are observed by each user in both the cells after 15

time instances. All users can solve three independent desired data streams

using these three equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

xix

xx

Acronyms

AMI alternating minimization of interference.

AWGN additive white Gaussian noise.

BC broadcast channel.

CDI channel direction information.

CFF characteristic function form.

CQI channel quality information.

CSI channel state information.

CSIR channel state information at the receiver.

CSIT channel state information at the transmitter.

DOF degrees of freedom.

DPC dirty paper coding.

FDMA frequency division multiple access.

GDOF generalized degrees of freedom.

i.i.d independent and identically distributed.

IA interference alignment.

IA-AM interference alignment using alignment matrix.

IBC interference broadcast channel.

IC interference channel.

ICI inter cell interference.

xxi

IMAC interference multiple access channel.

IUI inter user interference.

MAC multiple access channel.

max-SINR maximization of signal to interference plus noise ratio.

MIMO multiple input multiple output.

MISO multiple input single output.

MLI minimization of leakage interference.

PFF partition function form.

RCA receiver chain alignment.

SAC subspace alignment chain.

SINR signal to interference plus noise ratio.

SISO single input single output.

SNR signal to noise ratio.

SVD singular value decomposition.

TDD time division duplexing.

TDMA time division multiple access.

XC X-channel.

xxii

Chapter 1

Introduction

Modern wireless communication system users demand higher data rate and better quality

of service, unconcerned with the fact that the transmit power, bandwidth and the design

complexity are expensive and limited resources and these demands continue to increase in

the following years. Wireless communication engineers and researchers are developing new

ideas and tools to meet these demands. Inherently, any wireless communication comes up

with three major challenges-noise, multi-path fading and interference. While nothing much

can be done about the noise, a tremendous amount of research from the last few decades are

concerned about mitigating multi-path fading and interference.

One of the ground-breaking ideas that utilized the inherent multi-path property of wireless

channels to mitigate fading is the introduction of multiple antennas at the transmitter and/or

at the receiver. By transmitting or receiving the signal that carry same information multiple

times through different independent channel paths, a multiple version of the same signal with

different fades are observed at the receiver that provides more freedom to choose the best

signal for detection. Such spatial diversity of the signal increases the reliability or quality of

service of the wireless system [5].

Further, by using multiple antennas both at the transmitter and at the receiver, not only

diversity gain is achieved but multiple data streams can be transmitted and decoded at the

same time, thus, increasing the degrees of freedom (DOF) for communication [5–7]. This

multi-antenna scheme called multiple input multiple output (MIMO) exploits the spatial

domain to improve the capacity and spectral efficiency significantly and several data streams

are spatially multiplexed onto the MIMO channel [5–7]. The multiplexing gain of the MIMO

channel is proportional to the minimum of the number of transmit and receive antennas for a

given total transmit power, under the assumptions of independent fading channels and the

noise, i.e, the capacity of a point to point MIMO channel with N transmit and N receive

antennas increases by N folds than that of the point to point single input single output

(SISO) channel with the same transmit power [8]. Here the independent fading channel means

that an independent transmission path exists between each transmit and receive antenna due

1

Introduction

to rich scattering environment such that the channel matrix between all transmit and receive

antennas is a full rank matrix.

In the real world communication, multiple users transmit and receive simultaneously. One of

the ways to achieve multi-user communication is the sharing of the available resources between

the users, for example, time division multiple access (TDMA) uses only a fraction of time per

user and frequency division multiple access (FDMA) divides the available frequency among

the users. This, however, reduces the DOF by a fraction of 1K

if there are K users transmitting

and receiving simultaneously. On the other hand, when all the transmitters transmit at the

same time in the same frequency, the interference from the adjacent transmitters become

unavoidable due to the superposition nature of the wireless channel and the desired signal

can not be decoded properly at the desired receiver. Such channels are called interfering

channels. Interfering wireless channels such as two way communication channel [9], broadcast

channel (BC) [10], multiple access channel (MAC) [11] and interference channel (IC) [12,13]

are well-known among the researchers since 1970s.

However, till date a very little is known about the information theoretic capacity of such

channels and a general solution for most of the problems are still open. Only the capacity

region of a degraded BC and general MAC is well-known and given as the closure of convex

hull of the rate vectors that satisfy user-by-user power constraint for the given product input

distribution [14]. Quite a few results are known about the capacity of MIMO MAC and

MIMO BC for constant and fading channels with different degree of channel state information

at the transmitter (CSIT) and channel state information at the receiver (CSIR) as in in [15].

The results are obtained using Costa’s dirty paper coding (DPC) [16] and the concept of

MAC-BC duality [17].

The capacity region of even a two-user Gaussian IC is not solved in general, except that

it is known for a very strong interference regime [12, 18]. Also, lately using simple Han

Kobayashi type scheme [18], Etkin and Tse has shown that the capacity of two user Gaussian

IC is approximated to within one bit per second per Hertz of the bound for all channel

parameters [19]. This finding highlighted the fact that approximation of the capacity region

is one of the ways to solve the existing open problems regarding Gaussian IC and the most

preliminary form of capacity characterization is its linear approximation, which is the DOF

characterization as can be observed directly from the Shannon’s capacity expression [20] for

a very high signal to noise ratio (SNR) as

C = DOF log2(1 + SNR) = DOF log2(SNR) + DOF log2(1 +1

SNR). (1.1)

2

Introduction

When SNR→∞, DOF log2(1 + 1/SNR)→ 0 and log2(SNR)→ k, some constant, then

C ≈ kDOF⇒ C ∝ DOF. (1.2)

One of the important aspects of this observation is that time, frequency and space all

offer DOF in terms of the orthogonal dimensions and it is easier to analyze multi-user single

antenna and multi-antenna networks in this regard. Hence, a majority of the research works

regarding the interference network is focused on the DOF characterization instead of capacity

characterization though the DOF characterization is not a trivial problem in itself. The

initial work in DoF characterization conjectured that the K-user single antenna IC has only

1 DOF while the achievable best outer bound was found to be K2

[21]. Interference alignment

(IA) was introduced to reduce the gap between this inner and outer bound and to answer the

question ‘what is the optimal achievable DOF of a K-user IC?’ [1].

By aligning the interference from all the unwanted transmitters in a single direction, all

K − 1 unwanted signal occupy only 1 dimension instead of K − 1 dimensions as thought

previously. Hence, it behaves as if there is only one interference signal and one desired signal

per transmitter-receiver pair, thus achieving 12

DOF per pair and K2

total DOF. The alignment

is proved to be achievable for K-user time-varying IC and shown that every user almost surely

achieves reliable communication at rates approaching one-half of the achievable capacity at

very high SNRs by using the channels infinite times in [1]. This result is interesting because

whatever is the number of users each user can use half of the resources all the time. In other

words, everyone gets half of the cake irrespective of the number of people and the total DOF

increase from previously thought is

DOFinc = (K

2− 1)× 100%. (1.3)

This shows that the DOF increase is tremendous as the number of transmit-receive pairs

(sometimes referred to as ‘users’ in this article) increase, for example, if K = 3, DoFinc = 50%,

if K = 10, DoFinc = 400% and if K = 50, DoFinc = 2400%. The result is a breakthrough

achievement in the field of information theory and communication systems and it clearly

contributes an important message that the “K-user IC is not interference limited”.

Although it is mathematically possible that every user gets half of the cake, this is a huge

practical challenge in itself because it is achievable only under idealistic conditions such as

deterministic channels or infinite channel uses or global channel information as in [1, 22,23].

These challenges provide plethora of opportunities for the researchers to work in this topic

and a huge number of research articles have been published in designing IA algorithms and

obtaining the feasibility conditions of IA in a very few years time under different network

3

Thesis Outline

scenarios and channel conditions such as [2, 3, 24–45].

More antennas at the transmitter and the receiver provide higher spatial DOF. Hence, on

one hand, it is easier to achieve IA in a MIMO channel because it does not require infinite

channel uses but on the other hand the achievable DOF is limited by M and N , where M

represents the number of transmit antennas and N represent the number of receive antennas.

Also, the complexity increases as the number of users grow. This makes K-user MIMO-IA

an interesting problem to the researchers. The main focus of this Ph.D. thesis is to obtain

the in-depth understanding of IA techniques and algorithms with regard to MIMO ICs, to

design a new and better alignment algorithm for the K-user MIMO-IC under suitable channel

assumptions and to analyze the achievable DOF for such networks. As such a complete DOF

expression for a general K-user MIMO-IC is still an open problem. Hence, most of our works

are based on three user MIMO-IC. Perfect CSIT is a very optimistic assumption for wireless

channel. In this regard, we also design a space time transmission scheme to obtain the DOF

of multiple input single output (MISO) and MIMO interference broadcast channel (IBC)

using delayed CSIT. Following section outlines the structure of the thesis.

1.1 Thesis Outline

After the introduction in chapter 1 where we outlined the motivation behind the invention

of interference alignment algorithms for multi-user interfering networks, the rest of the

dissertation is structured as follows:

Chapter 2 is a brief overview of the well-known interference alignment algorithms and

detailed description of the idea of interference alignment. In this chapter, we describe the basic

concept of IA with examples, we consider K-user SISO IC and explain in detail various well

known precoding techniques to achieve the alignment of interference such as the asymptotic

approach as proposed by Cadambe and jafar [1], the ergodic approach as proposed by Bobak

and Nazar [26], IA using the concept of Diophantine approximation which deals with the

approximation of real numbers by rational numbers as proposed by Motahari et al. [30,46],

the lattice based IA approach which deals with the structured coding and further introduces

the concept of generalized degrees of freedom (GDOF), a more generic way to describe the

DOF as a function of some parameter α and the topology based alignment approach, where

the knowledge of network topology is helpful in a partially connected network to achieve

better DOF using the concept of IA. Furthermore, chapter 2 also deals with the concept of

IA in multi-user MIMO IC, we discuss the algorithms that require global CSIT and some

4

Thesis Outline

of the well-known iterative algorithms that require only the distributed CSIT to achieve

the alignment. Last but not the least, this chapter describes the feasibility conditions for

achieving IA. Only those networks, where IA is feasible are the networks of our interest.

Chapter 3 describes a proposed cooperative IA algorithm, which outperforms some of the

well-known algorithms such as minimization of leakage interference (MLI) and maximization

of signal to interference plus noise ratio (max-SINR) [2]. The main idea of this algorithm is

to use the hybrid (cooperative + greedy ) optimization approach. Cooperation at the receiver

side helps to design the zero-forcing matrix and the greedy approach at the transmitter side

helps to design the precoding matrix. The other benefit of this approach is that the receivers

can estimate the precoding matrices themselves. Using coalition game theoretic approach,

we show that the full cooperation is always optimal.

Chapter 4 deals with the DOF analysis of single cell MISO BC and multi-cell MISO BC

with the help of delayed CSIT. Delayed CSIT is the CSIT of all previous time instances till

t − 1 observed at any time instance t. With the help of the CSIT from the previous time

instances, the interference signal could be aligned by using proper transmission technique

and hence the achievable DOF is improved. The DOF expression for the single cell K user

MISO BC is derived and the concept is extended to two-cell MISO BC. With two cells,

there are not only inter user interference (IUI) but also inter cell interference (ICI), which

is a great challenge to mitigate. However, we show that by proper space-time transmission

scheme ICI can be mitigated and gain in DOF can be achieved. We consider a number of

example transmission schemes and generalize the result with a conjecture. We also provide

an improved transmission strategy to obtain better results.

In chapter 5, we define and introduce the concept IA, where a group of receivers create

a common overlap subspace to align the interference observed from any transmitter. We

termed such an alignment approach as a vertical alignment because the interfered receivers

align the interference observed from a single transmitter unlike a single receiver aligning the

interference observed from multiple transmitters in traditional IA schemes. We show that it

is a very useful scheme in MIMO IBC and we also show that it is a useful idea to study the

optimal overlapping dimensions when the receivers have the common channel information

and independent channel information. Optimal overlapping dimensions are determined with

the help of a so called alignment matrix as defined in non linear dimensionality reduction [47].

In chapter 6, we characterize the achievable DOF of any three-user MIMO IC for any

arbitrary number of transmit and receive antennas when they are not equal by creating

an optimal overlap space between the receivers and forming a simple chain as long as the

5

Notation

optimal overlap is possible, called the receiver chain alignment. We provide an information

theoretic proof and also show that it is possible to design the precoder as long as the optimal

overlapping space is created.

Chapter 7 presents the conclusions and state some of the future works that can be done

with the help of this thesis.

1.2 Notation

(1) C1 Field of complex scalars.

(2) R1 Field of real scalars.

(3) Cd×1 Field of complex vectors of dimension d.

(4) CN×M Field of complex matrices with dimensions N ×M .

(5) RN×M Field of real matrices with dimensions N ×M .

(6) X A matrix X.

(7) x A vector x.

(8) x A scalar x.

(9) X† Moore Penrose pseudo inverse of a matrix X.

(10) Ir Identity matrix of dimensions r × r.(11) XT Transpose of a matrix X.

(12) XH Conjugate transpose of a matrix X.

(13) Tr(X) Trace of a matrix X.

(14) ||X||2 Euclidean norm of a matrix X.

(15) ||X||F Frobenius norm of a matrix X.

(16) span(X) Range or linear combination of column vectors of a matrix X.

(17) N (X) Null space of a matrix X.

(18) X = {·, ·, ·} A set X with a number of elements {·, ·, ·}.(19) X ∪Y Union of two sets X and Y.

(20) X ∩Y Intersection of two sets X and Y.

(21) E(X) Expected value of a random variable X.

(22) min(x, y) Minimum of the two scalars x and y.

(23) max(x, y) Maximum of the two scalars x and y.

(24)(nk

)Binomial coefficient.

6

Chapter 2

Interference Alignment

2.1 Basic Concept

Consider a K-user SISO IC as shown in Figure 2.1, where the signal from transmitter

j to receiver i is represented by hij ∈ C1, a complex scalar, ∀i = {1, 2, · · · , K} and

∀j = {1, 2, · · · , K}. The desired channels are represented by the solid line and the interfering

channels are represented by the dashed lines. Let xj ∈ C1 be the transmitted complex symbol

from any transmitter j, then the received signal at each receiver is the superposition of the

signals transmitted from each transmitter, affected by the fading channel and the noise, as

given by :

yi = hiixi +K∑

j=1,j 6=i

hijxj + ni, (2.1)

where, hiixi is the desired part and the rest K−1 parts inside summation are the interfering

parts while ni is the noise part, which means that there are K−1 unwanted symbols and only

1 desired symbol per receiver. In elementary linear algebraic sense, there are K unknowns to

be solved per receiver with only one desired unknown. Hence, in order to solve one desired

unknown, we need at least K linear equations per receiver with the channel coefficients

drawn from the continuous distribution. In other words, we need K independent signaling

dimensions, e.g., we need to use independent channels K times, to decode 1 desired symbol,

thus limiting the DOF to 1K

. But, we can obviously do better than this. Let us consider that

each receiver uses the channels T times (T < K) such that the system is under-determined

with T equations and K unknowns. For any receiver i, this system of linear equations can be

written as

yi(1) = hi1(1)x1 + hi2(1)x2 + · · ·+ hiK(1)xK + ni(1),

yi(2) = hi1(2)x1 + hi2(2)x2 + · · ·+ hiK(2)xK + ni(2),

......

...

yi(T ) = hi1(T )x1 + hi2(T )x2 + · · ·+ hiK(T )xK + ni(T ), (2.2)

7

Basic Concept

TX1

TX2

TXK

RX1

RX2

RXK

h11

h22

hKK

h1K

h12hK1

h21

h2KhK2

Figure 2.1: A K-user single input single output interference channel, where the solid linesrepresent a direct or the desired links and the dashed line represent the cross or interferencelinks. All the receivers receive one desired link and K − 1 interference links.

8

Basic Concept

which can be further expressed in the vector form as

yi = hi1x1 + hi2x2 + · · ·+ hiKxK + ni, (2.3)

where the vectors yi =[yi(1) yi(2) · · · yi(T )

]T, hij =

[hij(1) hij(2) · · · hij(T )

]Tand ni =

[ni(1) ni(2) · · · ni(T )

]T.

From equation (2.3), we observe that the received signal space is the linear combination of

T dimensional channel vectors. Since only one data stream, lets say x1, is to be decoded at

RX1, we want the vector hi1 to be independent of all the other vectors hij,∀j 6= 1, i.e., we

should not be able to express hi1 as the linear combination of hij,∀j 6= 1 or mathematically,

hi1 /∈ span({hij,∀j 6= 1}), (2.4)

where span(A) represents the range or linear combination of column vectors of A.

If span({hij,∀j 6= 1}) spans all the available dimensions, i.e., min(T,K − 1), then x1 is not

still decoded because there are no free dimensions available to resolve x1. Thus, we want to

consolidate the span({hij,∀j 6= 1}) in as much smaller dimension as possible so that there is

enough free dimension to resolve x1. This approach of consolidating the interference subspace

onto the lower dimensional space is achieved by aligning or overlapping the interference

vectors as much as possible, which is called interference alignment.

Consider a system with K = 5 and T = 3, also assume that the channel states are real and

the signal observed at any receiver i in all the time instances, which desires to decode x1 are

given by

yi(1) = −3x1 + 2x2 + 3x3 + x4 + 5x5, (2.5)

yi(2) = −2x1 + 4x2 + x3 − 3x4 + 5x5, (2.6)

yi(3) = −4x1 + 3x2 + 5x3 + 2x4 + 8x5. (2.7)

Although, this is an under-determined system, x1 is solvable because the coefficients of

the interfering signals x5 is the linear combination of coefficients of interfering signal x2 and

x3 and also the coefficients of x2 are the linear combination of the coefficients of x3 and x4.

This dependence allows all the four interference signal to be aligned over two dimensions as

is clearly observed in the Figure 2.2, where all the interference (shown by blue, red, magenta

and black colored vectors) span only two dimensions while allowing an independent dimension

for the desired signal x1 ( shown by green colored vector). Thus, the whole signal space can

be projected on a plane that zero-forces the interference to obtain the desired data.

At the same time, the other receivers j 6= i,∀j = {1, 2 · · ·K} also receive a linear combina-

tion of all the transmit signals from all transmitters. But interestingly, the linear combination

9

Basic Concept

−2y

2

0

4

0

−2

−4

−2 x

24

0

−4

z 2

4

6

8

Figure 2.2: A geometric explanation of IA with four interference aligned in two dimensionsand one desired signal in the orthogonal direction. Green is the desired signal while blue, red,black and magenta are the interference signal.

of the interference signal observed by all these receivers are different because of the fact that

each one of them desire a data stream different from the other. Hence, the requirement of

one receiver for alignment do not conflict with the requirement of the other receivers. This

interesting feature, called relativity of alignment, is an essential premise to ensure IA [1,48].

For the given example with deterministic and well-defined channel states, the IA principle

looks very simple and the alignment is easily achieved. But in real world scenario, the channel

states are completely random and can not be controlled by the designer. Hence, it is a great

challenge to make the coefficients of all the interfering signals as much dependent as possible

to consolidate them in the lower dimensional space. One of the ideas is to precode all the

transmitted signal before transmitting such that the interfering signal align with each other

as much as possible in each receiver. This, however, requires that all the transmitters need

to know all the channel states (global CSIT) perfectly beforehand and the channel need to

remain constant within the transmission period. This is a very optimistic assumption in any

wireless communication scenario and is a major drawback of the IA scheme.

Also, in the example scheme, only 53

DOF is achieved while the outer bound on the DOF of

any 5-user SISO-IC is 52

[1,21] and the obvious question is “ How close can we go to the outer

bound ?” This depends on how effectively all the receivers align the available interference. If

all the receivers can align the available interference in 2T time instances in order to decode

10

Precoding Techniques

T desired streams by each of them, then the outer bound is achieved. In other words, if each

receiver aligns all the interference over one-half of the available space leaving other half for

the desired streams, then the outer bound is achieved. In [1], Cadambe and Jafar showed

that this condition is not strictly achievable but it can be achieved with some relaxation in

the number of channel uses. For 3-user SISO-IC, [1] shows that 3n+ 1 streams of data can

be decoded in 2n+ 1 channel uses for any positive integer n, thus allowing 3n+12n+1

DOF, which

achieves the outer bound of 32

as n→∞.

2.2 Precoding Techniques

In this section, we briefly discuss about some of the well-known precoding techniques that

have been proposed in the literature to achieve IA. This provides us a better understanding

in this topic and acts as a framework for further discussions in the future chapters.

2.2.1 Asymptotic Approach

In K-user SISO-IC, one of the approaches to achieve alignment is by beamforming over

multiple symbol extensions of the time-varying wireless channel as proposed in [1]. By

extending the channel over (n+ 1)N + nN symbols, (n+ 1)N + (K − 1)nN independent data

streams can be decoded, where N = (K − 1)(K − 2)− 1 and n is any natural number, thus

providing K2

DOF asymptotically as

limn→∞

(n+ 1)N + (K − 1)nN

(n+ 1)N + nN= lim

n→∞

(n+ 1)N − nN(n+ 1)N + nN

+KnN

(n+ 1)N + nN(2.8)

= limn→∞

(1 + 1n)N − 1

(1 + 1n)N + 1

+K

(1 + 1n)N + 1

(2.9)

=K

2. (2.10)

For K = 3, N = 1 and (n+1)+2n = 3n+1 data streams are decoded for (n+1)+n = 2n+1

symbol extensions of the channel, which means that one of the transmitters transmits (n+ 1)

data streams and the rest transmit n data streams free of error in 2n+ 1 channel uses. This

is achieved by the proper alignment of interference signal at the receivers. Hence, the aim of

this approach is to design the precoders at each transmitter that achieves certain alignment

conditions.

Assume that each transmitted message is precoded by the same precoding vector v and

Hij is the diagonal channel matrix obtained from the symbol extensions of time-varying

11


channel observed at receiver i corresponding to transmitter j. The signal observed at any

receiver i without considering the noise is then given by

yi = Hiivsi +K∑

j=1,j 6=i

Hijvsj, (2.11)

where vsj represents the precoded symbol at transmitter j. The first part is the desired

signal and the second part in the summation is the interference part.

For simplicity, the channels are normalized over Hii and the received signal is

yi = vsi +K∑

j=1,j 6=i

Tijvsj, (2.12)

where Tij is the diagonal channel normalized with respect to Hii.

Let V be the space spanned by the desired part (corresponding to the precoding vector v)

and assume that same space is set aside by any receiver for the interference part then the

total space observed by the receiver i and the space set aside for interference is given by

N =⋃

i=1,i 6=j

(V ∪TijV). (2.13)

This is true for all receivers k 6= i. The total interference space is the union of the

interference space observed at all the receivers and the space set aside by each receiver.

There are total of N = K(K − 1) observed interference space and a space V set aside by

each receiver. If we represent all the normalized channels as T1,T2, · · · ,TN , then the total

interference space is

NT = (V ∪T1V ∪T2V ∪ · · · ∪TNV). (2.14)

The aim of IA is to consolidate all the observed interference space to the space set aside by

each receiver. This is possible only if

V ≈ T1V ≈ T2V ≈ · · · ≈ TNV. (2.15)

In order to achieve (2.15), any random V is initially chosen and updated iteratively such

that the next V contains all the interference space previously observed. Since V corresponds

to the precoding vector v, we obtain v by updating V. As in [36], choose V1 = 1, the vector

of all 1s then

N1 = 1 ∪T11 ∪T21 ∪ · · · ∪TN1. (2.16)

Clearly, the interference space is N + 1 dimensional, while the desired space V1 is 1

dimensional and the ratio dim(V1)dim(N1)

= 1N+1

, which is much less than 1. Now set the new desired

12


space V2 = N1, which is N + 1 dimensional, then the new interference space is

N2 = 1 ∪T11 ∪T21 ∪ · · · ∪TN1 ∪T1(1 ∪T11 ∪T21 ∪ · · · ∪TN1) ∪T2(1∪T11 ∪T21 ∪ · · · ∪TN1) ∪ · · · ∪TN(1 ∪T11 ∪T21 ∪ · · · ∪TN1), (2.17)

= 1 ∪T11 ∪T21 ∪ · · · ∪TN1 ∪T11 ∪T211 ∪T1T21 ∪ · · · ∪T1TN1 ∪T21

T2T11 ∪T221 ∪ · · · ∪T2TN1 ∪ · · · ∪TN1 ∪TNT11 ∪TNT21 ∪ · · · ∪T2

N1. (2.18)

Further , assuming that the diagonal interference channels are commutative, i.e., TiTj =

TjTi,∀i 6= j and i, j = {1, 2, · · · , N}, we have

N2 = 1 ∪T11 ∪T21 ∪ · · · ∪TN1 ∪T211 ∪T1T21 ∪ · · · ∪T1TN1∪

T221 ∪ · · · ∪T2TN1 ∪ · · · ∪T2

N1, (2.19)

= 1 ∪ · · · ∪Ti1 ∪ · · ·TiTj1 ∪ · · · ∪T2i1,∀i 6= j. (2.20)

The dimensions of N2 is (N + 1) +N(N − 1)/2 +N = (N + 1)(N + 2)/2 and the ratio

dim(V2)

dim(N2)=

N + 1

(N + 1)(N + 2)/2=

2

N + 2, (2.21)

which is still much less than 1 and we continue the next iteration for which V3 = N2 and

obtain N3 similarly as before whose dimensions is (N + 1)(N + 2)(N + 3)/6 and the ratiodim(V3)dim(N3)

= 3N+3

, which is still much less than one but better than previous ratios. Hence,

continuing similarly, it is observed that after n iterations the ratiodim(Vn)

dim(Nn)=

n

N + n, (2.22)

which approaches 1 as n→∞.

2.2.2 Ergodic Approach

The ergodic IA, initially presented by Bobak Nazar et al. in [26], considers that the channel

coefficients are independent, time-varying and are drawn from the distribution with uniform

phase. In a K-user SISO IC, all the transmitters transmit at time t and wait for the next

transmission until the complementary channel matrix occurs at time tc. The complementary

channel matrix for the given channel matrix H(t) =

h11 h12 · · · h1K

h21 h22 · · · h2K

......

. . ....

hK1 hK2 · · · hKK

, observed at all

the K receivers corresponding to all the K transmitters at any time t, is

13


Hc(tc) =

h11 −h12 · · · −h1K

−h21 h22 · · · −h2K

......

. . ....

−hK1 −hK2 · · · hKK

, (2.23)

such that H(t) + Hc(tc) is a diagonal matrix, which means all the interferences cancel

automatically if same data streams are transmitted at time t over channel H and at time tc

over channel Hc. Thus, at any receiver i,

yi(t) + yi(tc) = 2hii + ni(t) + ni(tc), (2.24)

where yi(t) and yi(tc) are the signal received at time t and tc respectively and ni(t) and

ni(tc) are the noise at time t and tc respectively.

However, the assumption that the exact complementary channel occurs after certain time

is very unrealistic and will happen with zero probability for any channel states distributed

continuously. Thus, the channel coefficients are quantized and matched up as closely as

possible with the available complementary channel. The error probability decreases for finer

quantization and the targeted rate is achieved in the limit as described in detail in [26].

2.2.3 IA Based on Real Numbers

The idea of real IA is to transform the single antenna system into pseudo multiple antenna

systems with infinitely many pseudo antennas, and simultaneously align interference at all

receivers. Hence, unlike ergodic approach channel need not be time-varying. In [30], it is

shown that K2

DOF is achievable for K-user Gaussian IC by selecting an appropriate transmit

directions at all the transmitters.

Consider a 3-user SISO Gaussian IC, the received signal at each receiver is given by

y1 = h11x1 + h12x2 + h13x3 + n1, (2.25)

y2 = h21x1 + h22x2 + h23x3 + n2, (2.26)

y3 = h31x1 + h32x2 + h33x3 + n3, (2.27)

where xi are the input symbols, hij are the channels from transmitter j to receiver i and

ni are the additive white Gaussian noise (AWGN) at receiver i.

Since the linear operations at the transmitter and the receiver do not affect the capacity

region of the channel, the following transmission strategy is adopted in [30], to generate an

equivalent standard channel with respect to DOF:

14


(i) Transmitter 1 transmits x1 = h23h12h21

x1,

(ii) Transmitter 2 transmits x2 = h13x2 and

(iii) Transmitter 3 transmits x3 = h12x3,

then the modified received signal is represented as

y1 =h11h23h12

h21

x1 + h12h13x2 + h13h12x3 + n1,

⇒ y1 =h11h23h12

h21h12h13

x1 + x2 + x3 + n1, (2.28)

y2 =h21h23h12

h21

x1 + h22h13x2 + h23h12x3 + n2,

⇒ y2 = x1 +h22h13

h23h12

x2 + x3 + n2, (2.29)

y3 =h31h23h12

h21

x1 + h32h13x2 + h33h12x3 + n3,

⇒ y3 = x1 +h21h32h13

h31h23h12

x2 +h33h12h21

h31h23h12

x3 + n3. (2.30)

Further, consider

G0 =h13h21h32

h12h23h31

=(h13

h12

)(h21

h23

)(h32

h31

), (2.31)

G1 =h11h12h23

h12h21h13

=(h11

h13

)(h23

h21

), (2.32)

G2 =h22h13

h12h23

=(h13

h12

)(h22

h23

), (2.33)

G3 =h33h12h21

h12h23h31

=(h21

h23

)(h33

h31

). (2.34)

Here, G0 is important because it is the product of the ratios of all the cross channels in

all receivers and in [30, 46], it is proved that for 3-user IC, if G0 is rational, then 32

DOF

is achievable. Hence, only G0 is considered as a generator function to create the transmit

directions. Since each transmit direction is independent of the other, the transmit directions

for d pseudo antennas which carry d streams are the basis set {1, G0, G20, · · · , Gd−1

0 } over the

rational numbers and the transmitted data is represented as the linear combination given by

xj = A(sj1 +d−1∑i=1

Gi0sj(i+1)), (2.35)

where A is any real number and sji are independent data streams transmitted from

15


transmitter j. Thus, for 3 user IC, we have

x1 = As11 + A(G0s12 +G20s13 + · · ·+Gd−1

0 s1d), (2.36)

x2 = As21 + A(G0s22 +G20s23 + · · ·+Gd−1

0 s2d), (2.37)

x3 = As31 + A(G0s32 +G20s33 + · · ·+Gd−1

0 s3d). (2.38)

Substituting the values of x1, x2 and x3 from (2.36), (2.37) and (2.38) in (2.28), we obtain

y1 =AG1(s11 +G0s12 +G20s13 + · · ·+Gd−1

0 s1d) + A(s21 + s31) + AG0(s22 + s32)+

AG20(s23 + s33) + · · ·+ AGd−1

0 (s2d + s3d) + n1, (2.39)

where all the interference from transmitter 2 and transmitter 3 are aligned over d inde-

pendent directions. Similarly substituting the values of x1, x2 and x3 from (2.36), (2.37)

and (2.38) in (2.29), it is easily possible to align all the interferences from transmitter 1

and transmitter 3 over d independent dimensions as before. But substituting the values of

x1, x2 and x3 from (2.36), (2.37) and (2.38) in (2.30), the alignment is observed over d+ 1

dimensions as given by

y3 =As11 + A(G0s12 +G20s13 + · · ·+Gd−1

0 s1d) +G0(As21 + A(G0s22 +G20s23 + · · ·

+Gd−10 s2d)) +G3(As31 + A(G0s32 +G2

0s33 + · · ·+Gd−10 s3d)) + n3, (2.40)

=As11 + AG0(s12 + s21) + AG20(s13 + s22) + · · ·+ AGd−1

0 (s1d + s2(d−1))+

AGd0s2d + AG3(s31 +G0s32 +G2

0s33 + · · ·+Gd−10 s3d) + n3. (2.41)

But since there are only d basis vectors, one of the dimensions is dependent which is evident

from the fact that Gd0 can be represented as a linear combination of {1, G0, G

20, · · · , Gd−1

0 }with rational coefficients. Hence, in all three receivers the interference signal is aligned over d

dimensions and the desired signal occupy other d dimensions independent of the interference

signal. In [46], it is proved that if such conditions exist, the achievable DOF for almost all

realization of the system is given by

dtot =total interference dimensions

maximum received dimensions=d+ d+ d

2d=

3

2. (2.42)

On the other hand, if G0 is not rational but a transcendental number, it is again proved

in [46] that the total achievable DOF is given by

dtot =3n+ 1

2n+ 1, (2.43)

as also given by [1] for any integer n.

16


2.2.4 Lattice-Based Approach

The structured codes are used to achieve IA in some K-user IC, called the lattice based IA

approach. Since the sum of the lattice points, which are the codewords is also a lattice point,

a new codeword, it is possible to decode the sum even if the individual codewords are not

decodable [49]. When the interference is strong and decoding the interference helps to decode

the desired signal, lattice codes are useful. In [49], it is shown that the capacity of many

to one IC is achieved within a constant bits per second per Hertz regardless of the channel

parameters using the lattice alignment.

When some signals are significantly stronger or weaker than the other, the DOF is not a

proper metric as it forces all the channels to be equally strong. In such case, a new metric

called GDOF is used which is a function of some parameter α that captures the signal

strength of particular channel [19,50–52]. Like DOF, GDOF is mathematically defined as :

GDOF(α) = limSNR→∞

C(SNR, α)

log(SNR), (2.44)

where C(SNR, α) is the channel capacity as a function of SNR and α, the ratio of interference

channel (in dB) to the desired channel (in dB).

Since layered code structure can be used when signal levels are different, lattice alignment

is used to optimize the GDOF of such IC. However, the wireless channel is random and

uncertain in practice. Hence the random codes are preferred than the structured codes most

of the time for wireless channel. Robust lattice alignment for wireless channels is still an

open problem and only few works are available in this regard such as [53].

2.2.5 Topology Based Approach

Requirement of complete channel knowledge is one of the major drawbacks of IA, because

wireless channels are uncertain and difficult to predict. It is relatively easier to predict the

channel states at the receiver side but more difficult to predict the channel states at the

transmitter side. Hence, a number of research works on IA are focused on relaxing the

requirement of complete perfect CSIT, such as [27,28,39–41,54–56].

The concept of blind IA is introduced in [27], where the alignment is observed only based

on the knowledge of autocorrelation of the channels seen by different receivers, whereas IA

is achieved with imperfect CSIT in [54]. Other approaches such as in [28,39–41] uses the

knowledge of CSIT from the previous time instances to achieve the alignment of interference.

One of the relatively new approaches that relaxes the explicit requirement of CSIT is

17

MIMO Interference Alignment

introduced by Jafar [45], which uses the knowledge of the network topology to align the

interference at the receiver using the concept of index coding [57,58].

For a partially connected network, alignment graph and the conflict graph is created from

which alignment set and number of internal conflicts are determined. Alignment graph is the

graph of all the interference messages which need to be aligned and conflict graph is the graph

of all the messages that need to be kept separate. For better understanding, consider a 5-user

partially connected SISO IC as shown in Figure 2.3, where wj represents the message from

any transmitter j and wi is the message desired by any receiver i, the solid line represents the

desired channel and the dashed line represents the interference channel and the corresponding

alignment conflict graph for the network is shown in Figure 2.4, where the solid line represents

the alignment graph and the dashed line represents the conflict graph.

As seen from the figure w1 and w5 need to be aligned at receiver 3 and w3 and w4 need to

be aligned at receivers 1 and 2. Hence, w1, w5 and w3, w4 are connected by solid lines. Thus

there are total of three alignment sets {w1, w5}, {w3, w4} and {w2}. If two messages in the

same alignment graph have conflict, it is called internal conflict, e.g., if w3 and w4 are also

connected by dashed line in Figure 2.4 or if receiver 4 receives the signal from transmitter 3

in Figure 2.3 then the internal conflict exists between w3 and w4. However, there exists no

internal conflict for the given network.

When there exists no internal conflict in the network, a symmetric DOF of 12

per user is

always achievable without the knowledge of CSIT [45]. For all the other networks, where 12

DOF is not achievable, internal conflict exists and the symmetric DOF per user is expressed

in terms of minimum internal conflict distance (∆) as given by:

DOF ≤ ∆

2∆ + 1, (2.45)

where the minimum conflict distance ∆ is defined as the minimum number of alignment

graph edges traversed by the messages nodes with internal conflict to go from one message

node to the other.

2.3 MIMO Interference Alignment

Multiple antennas at the transmitter and the receiver offer DOF in terms spatial dimensions.

Spatial dimensions are interesting because of their distributed nature and can be as large as

the number of transmit and receive antennas. Also, spatial dimensions allow operations such

as beam-forming to direct (beam) the interference signal onto the null space while keeping

18


1

2

3

4

5

1

2

3

4

5

w1

w2

w3

w4

w5

w1

w2

w3

w4

w5

~

~

~

~

~

Figure 2.3: A partially connected 5 user SISO IC. A solid line represents the desired channelwhile the dashed lined represents the interference channel.

w1

w2

w3 w4

w5

Figure 2.4: Alignment conflict graph for the network in Figure 2.3. Solid line represents thealignment graph and the dashed line represents the conflict graph.

19


the desired signal independent of interference signal. This nulling of interference is called

spatial zero-forcing [59]. Spatial dimensions provide spatial DOF as high as min(M,N) in a

single user MIMO channel, where M and N are the number of transmit and receive antennas

respectively [5].

Mathematically, beam-forming is achieved by sending the coded information to the receiver

with prior knowledge of all the channel states. This operation of coding before transmission is

called precoding. Multiple streams transmitted via spatial dimensions are coded by multiple

precoding vectors individually. All these precoding vectors form columns of the precoding

matrix. Since precoding is performed for beaming the signal streams to particular direction,

the signal amplitude is preserved while changing the phase of the signal. Hence, a precoding

matrix is assumed to be a unitary matrix. A number of literature such as [5,8,14] provide an

extensive detail on precoding in point to point MIMO channels.

Like a multi-user SISO channel, the major limitation of a multi-user MIMO channel is the

interference. More spatial dimensions allow more independent interference streams to be

observed at any receiver. If the total number of spatial dimensions available at any receiver

is less than the total number of interference observed by that receiver, it is unable to decode

the desired signal and no reliable communication is possible. Thus, MIMO IA is a technique

by which all the observed interference are allocated as minimum dimensions as possible

thus allowing the possible maximum dimensions for the desired streams, which increases

the spatial DOF of the system. This is achieved by designing proper precoding matrix at

each transmitter which aligns all the interference signal received by a receiver onto the same

subspace, meaning that the spatial dimensions allocated for the interference signal from all

interferer is the same as the spatial dimensions allocated for a single interferer [2, 3, 32].

Consider a simple example as depicted in Figure 2.5, where alignment of interference

is observed in a three user MIMO IC. Here, Hij ∈ CN×M is a channel matrix from any

transmitter j to receiver i and Vj is a precoding matrix whose dimensions depend on the

number of transmit streams from transmitter j. The precoding matrices Vj,∀j = {1, 2, 3}are unknown and they need to be designed such that alignment of interference is achieved in

all three receivers. When M = N , each channel can be separated into M parallel channels

and the whole network acts as KM user SISO IC, the achievable DOF for which is given by

DOF =KM

2. (2.46)

.

But for the cases when M 6= N , it is not trivial to determine the achievable DOF. In fact,

20


TX 1

TX 2

TX 3

RX 1

RX 2

RX 3

V1

V2

V3

H11

H22

H33

H21

H31

H32

H12H

13 H23

H13V3

H12V2

H11V1

H22V2

H32V2

H21V1

H23V3

H33V3

H31V1

Figure 2.5: Interference alignment in three user MIMO IC. The solid line represents a desiredchannel matrix and the dashed and dotted lines represent the interference channel matrices.For ease of representation, the alignment of multiple spatial dimensions is represented by anarrow. For example, dashed green arrow represents a d dimensional subspace observed atRX1 from TX3.

21


the alignment conditions increase exponentially with the increasing number of users and

designing a proper precoding matrix becomes a great challenge. Even for a simple three user

MIMO IC, the alignment is achieved when the following conditions are satisfied:

(I) At RX1,

span(H12V2) = span(H13V3), (2.47)

(II) At RX2,

span(H21V1) = span(H23V3), (2.48)

(III) At RX3,

span(H31V1) = span(H32V2), (2.49)

where span(A) represents the linear combination of the column vectors of A. From a very

simple observation, the linear combination of column vectors of two matrices are same when

two matrices are equal. Hence

(I) At RX1,

H12V2 = H13V3, (2.50)

⇒V2 = (H12)†H13V3, (2.51)

(II) At RX2,

H21V1 = H23V3, (2.52)

⇒V1 = (H21)†H23V3, (2.53)

(III) At RX3,

H31V1 = H32V2, (2.54)

Substituting the values of V2 and V1 from (2.51) and (2.53) in (2.54), we obtain

H31(H21)†H23V3 = H32(H12)†H13V3, (2.55)

⇒V3 = (H23)†H21(H31)†H32(H12)†H13V3, (2.56)

⇒[IM − (H23)†H21(H31)†H32(H12)†H13

]V3 = 0, (2.57)

where IM represents the M × M identity matrix and V3 is given by the eigenvectors

corresponding to the zero eigenvalues of A =[IM − (H23)†H21(H31)†H32(H12)†H13

].

Clearly, the solution to V3 exists only if the null space of A exists and the dimensions of V3

is the dimensions of the null space. This means the complete knowledge of A is required in

22

From Global to Local CSIT

order to determine V3. Since A is a function of all the interfering channel matrices to all the

receivers, the transmitters need to know all channels globally to precode the transmit streams

in order to achieve IA. Since wireless channels are time-varying, global channel knowledge at

the transmitter is unlikely almost surely.

In the next section, we present some of the IA algorithms to design the precoding matrices

that do not require the CSIT globally but locally. Local CSIT, here refers to the CSIT from one

transmitter to all the receivers, i.e., TX1 knows only Hi1 and not Hi2 and Hi3; ∀i = {1, 2, 3}.

2.4 From Global to Local CSIT

Local CSIT is easier to obtain because of the reciprocity of wireless channels. Reciprocity of

wireless channels means that the channels observed at any receiver i from the transmitter j

are the same as the channels observed in the reciprocal network where the receiver i acts as

a transmitter and the transmitter j acts as a receiver. Transmitter j can obtain the channel

states from all the receivers i = {1, 2, · · · , K}, by allowing the receivers to transmit in the

reciprocal direction in different synchronized time slot. Thus, local CSIT is available naturally

in time division duplexing (TDD) systems. A number of iterative algorithms such as MLI,

max-SINR [2], alternating minimization of interference (AMI) [32] and interference alignment

using alignment matrix (IA-AM) [60] are proposed to design the optimum precoder matrices

for MIMO IC in the literature. In this section, we discuss MLI, max-SINR and AMI, while

IA-AM is discussed in detail in chapter 5.

2.4.1 Minimization of Leakage Interference

The MLI algorithm iteratively minimizes the leakage interference at each receiver by designing

zero-forcing matrix at the receiver and precoding matrix at the transmitter. Leakage

interference is the amount of interference power still observed after zero-forcing the available

interference. The precoding matrices in all transmitters are randomly chosen initially, based

on which zero-forcing matrices are determined in all the receivers. Assuming that the zero-

forcing matrices acts as precoding matrices in the reciprocal network, the actual precoding

matrices are iteratively updated by minimizing the amount of leakage interference [2].

Consider a K-user MIMO IC with M transmit and N receive antennas as shown in

Figure 2.6. The channels from transmitter j to receiver i is represented by a matrix

Hij ∈ CN×M . Each element hk`,∀k = {1, 2, · · · , N}, ` = {1, 2, · · · ,M} of Hij are independent

23


and identically distributed (i.i.d) with zero mean and |hk`|2 variance. The precoding matrices

are represented by Vj ∈ CM×d, ∀j = {1, 2, · · · , K} and the zero-forcing matrices are

represented by Ui ∈ CN×d, ∀i = {1, 2, · · · , K}, where d is the total independent data

streams transmitted from any transmitter j. Such system where the number of all transmit

antennas are equal, the number of all receive antennas are equal and also the number of

independent data streams transmitted from all transmitters are equal is called a symmetric

system.

The total leakage interference power at any receiver i is the sum of the leakage interference

power due to all interfering transmitters j 6= i, which can be mathematically expressed as:

Fi = Tr[UHi Q

(f)i Ui

], (2.58)

where

Q(f)i =

K∑i 6=j

P

dHijVjV

Hj HH

ij , (2.59)

is the interference covariance matrix in the forward channel, P is the transmit power and

Tr[A] represents the trace of matrix A.

In order to satisfy the power constraint, the precoding matrix Vj is assumed to be an

orthonormal matrix, i.e., VHj Vj = Id and for any randomly initialized Vj, ∀j = {1, 2, · · · , K},

the optimum Ui is obtained by solving the following optimization problem:

minUi

Fi(Ui,Q(f)i ), (2.60)

s.t. VHj Vj = Id, (2.61)

UHi Ui = Id. (2.62)

The interference covariance Q(f)i ∈ CN×N is a positive semi-definite matrix and can be

eigen-decomposed. Hence, the optimum Ui is given by the eigenvectors corresponding to the

d smallest eigenvalues of Qi [2],

Ui = eigvd(Q(f)i ). (2.63)

The precoding matrix is obtained in all the transmitters by observing the dual network

where each receiver acts as a transmitter and each transmitter acts as a receiver. In that

case, previously determined zero-forcing matrix Ui plays the role of a new precoding matrix

and new zero-forcing matrix is obtained by minimizing the leakage interference in the reverse

direction. The reverse leakage interference is given by:

Rj = Tr[VHj Q

(r)j Vj

], (2.64)

24


TX1

TX2

TXK

RX1

RX2

RXK

V1

V2

VK

H11

H22

HKK

H21HK1

HK2

H 12

H1K

H 2K

U1

U2

UK

Figure 2.6: A K-user MIMO IC with M transmit and N receive antennas. Vjs are theprecoding matrices and Uis are the zero-forcing matrices, ∀ i, j = {1, 2, · · · , K}. Solid linesrepresent desired channels and dotted and dashed lines represent interference channels.

25


where

Q(r)j =

K∑j 6=i

P

dHjiUiU

Hi HH

ji , (2.65)

is the interference covariance matrix observed in the reverse direction.

The optimum Vj is thus obtained by solving the following optimization problem:

minVj

Rj(Vj,Q(r)j ), (2.66)

s.t. VHj Vj = Id, (2.67)

UHi Ui = Id, (2.68)

the solution to which is given by the eigenvectors corresponding to the d minimum

eigenvalues of Q(r)j . This Vj acts as a new precoding matrix to determine new Ui and the

operation is repeated iteratively till the leakage interference converges to a very small value

and the convergence is guaranteed for such optimization as proved in [2].

Here, we note that the alignment is possible by choosing d streams of data both in the

forward and the backward directions. This requires that feasible number of independent data

streams to be transmitted (the feasible DOF) be known beforehand and the alignment is not

observed if chosen d is greater than the feasible d.

2.4.2 Maximization of Signal to Interference plus Noise Ratio

In the MLI algorithm, only the interference signal observed in each receiver is minimized and

the desired signal is not considered at all. However in the max-SINR algorithm signal to

interference plus noise power ratio is maximized to design the precoding and zero-forcing

matrices at the transmitters and at the receivers respectively [2].

One of the main benefits of this scheme is that it is even valid for small SNR and the

precoding vectors need not be orthogonal to each other as before. In fact, orthogonal precoding

vectors are sub-optimal [2]. Hence, the precoding vector per data stream is determined. This

also requires the knowledge of feasible data streams beforehand. The signal to interference

plus noise ratio (SINR) of the tth stream of any receiver i is given by :

SINRit =(u

(i)t )

HHiiv

(t)i (v

(t)i )

HHHii u

(t)i

(u(t)i )

HBitu

(t)i

, (2.69)

where u(t)i is the tth column vector of any zero-forcing matrix Ui obtained at receiver i, v

(t)i

is the tth column vector of any precoding matrix Vi obtained at any transmitter i and the

matrix Bit is the interference covariance observed by receiver i due to tth column of precoding

26


matrices Vj, ∀j = {1, 2, · · · , K} as given by :

Bit =K∑j=1

d∑`=1

Hijv(`)j (v

(`)j )

HHHij −Hiiv

(t)i (v

(t)i )

HHHii + IN , (2.70)

where the dimensions of all the channels are same as defined in the previous algorithm.

We can then solve for the tth column of the zero-forcing matrix Ui as the unit vector u(t)i

which maximizes (2.69) as given by :

u(t)i =

(Bit)−1Hiiv

(t)i

||(Bit)−1Hiiv(t)i ||

. (2.71)

The dual network with reciprocal channel states is considered as in the MLI algorithm to

determine the tth column of any precoding matrix Vi, which is obtained by considering u(t)i

as the new precoding vector and the operation is repeated iteratively till convergence. The

convergence for this algorithm is guaranteed though global optimum is not achieved due to

non-convex nature of the objective function as proved in [2].

2.4.3 Alternating Minimization of Interference

Unlike the previous algorithms, this alternating minimization of interference approach pro-

posed by Peters and Heath in [32] uses the concept of projection to determine the optimal

subspace where all the interference is aligned. The zero-forcing matrix or sometimes called

interference suppression matrix Ui at any receiver i is the basis of the optimal subspace

where all the interference observed by receiver i is aligned.

The precoding matrices designed at each transmitter helps to align all the interference.

Since the precoding matrices and the zero-forcing matrices are not known initially, they

are alternately optimized assuming one is known at an instance. Consider that all the

precoding matrices are orthonormal and are randomly initialized, then the receiver i observes

d dimensional interference HijVj, ∀j 6= i from all transmitters j 6= i. Here, we assume that

all the symbols and dimensions are same as defined in previous algorithms and the system

is symmetric, i.e, all transmitters have M antennas, all receivers have N antennas and all

transmitters transmit d independent streams of data.

At any receiver i, Ui ∈ CN×d is the basis of the subspace where all the interference is

aligned. Since there are total of (K − 1)d interference streams, all interference projected onto

d dimensional subspace is aligned. Thus, the aligned interference due to any receiver j is

UiUHi HijVj, ∀j 6= i and the optimum Ui is obtained by minimizing the total error between

the initial and the projected interference streams as given by,

27


minUi

K∑j=1,j 6=i

||HijVj −UiUHi HijVj||2F , (2.72)

s.t. VHj Vj = Id, (2.73)

UHi Ui = Id, (2.74)

where ||A||F represents the Frobenius norm of matrix A.

Using the relation between Frobenius norm and the trace of a matrix, we can further

express (2.72) as

minUi

K∑j=1,j 6=i

Tr[HijVj −UiU

Hi HijVj)(HijVj −UiU

Hi HijVj)

H], (2.75)

which is further simplified to

minUi

K∑j=1,j 6=i

Tr[HijVj(HijVj)

H −UiUHi HijVj(HijVj)

H], (2.76)

= maxUi

K∑j=1,j 6=i

Tr[UiU

Hi HijVj(HijVj)

H], (2.77)

= maxUi

Tr[UHi

( K∑j=1,j 6=i

HijVjVHj HH

ij

)Ui

], (2.78)

whereK∑

j=1,j 6=i

HijVjVHj HH

ij is the positive semi-definite interference covariance matrix and

the d dimensional optimum basis Ui that maximizes the trace is given by the eigenvectors

corresponding to the d dominant eigenvalues ofK∑

j=1,j 6=i

HijVjVHj HH

ij .

Further, the optimum precoding matrix is obtained by minimizing ||HijVj−UiUHi HijVj||2F

over Vj with the distributed channel knowledge and with the knowledge of Uis obtained

previously as given by

minVj

K∑i=1,i 6=j

||HijVj −UiUHi HijVj||2F , (2.79)

s.t. VHj Vj = Id, (2.80)

UHi Ui = Id, (2.81)

28

Feasibility of Interference Alignment

which is further simplified to

minVj

K∑i=1,i 6=j

||(IN −UiUHi )HijVj||2F , (2.82)

= minVj

K∑i=1,i 6=j

Tr[(IN −UiU

Hi )HijVj((IN −UiU

Hi )HijVj)

H], (2.83)

= minVj

Tr[VHj

K∑i=1,i 6=j

HHij (IN −UiU

Hi )HijVj

], (2.84)

under the same constraints.

SinceK∑

i=1,i 6=j

HHij (IN −UiU

Hi )HijVj is a positive semi-definite matrix, the d dimensional

optimum Vj is given by the eigenvectors corresponding to the d minimum eigenvalues ofK∑

i=1,i 6=j

HHij (IN −UiU

Hi )HijVj . Using this Vj , (2.72) is optimized to get new Ui which is used

to solve (2.82) to obtain new Vj, and the process is repeated iteratively till the objective

converges to a small value. However, the global optimum is not guaranteed [32].

In all these distributed algorithms, the achievable DOF or the number of independent data

streams to be transmitted, are defined beforehand and when this is not known precisely, then

the algorithms do not converge to the desired value. So one of the challenges in designing IA

algorithms is to determine the achievable DOF for the given system or to determine if the

given system is feasible for achieving IA. In the next section, we discuss some of the criteria

and conditions for the feasibility of IA.

2.5 Feasibility of Interference Alignment

Interference alignment achieved by designing precoding matrices at the transmitters and

zero-forcing matrices at the receivers if needed is a linear operation and it is practically a

feasible assumption. Such linear IA is not feasible in all the systems, with any number of

users ( transmitters and receivers ) and transmit and receive antennas. In all the infeasible

systems, we can not achieve the benefits of IA. Hence the feasibility of IA is investigated in a

number of literature such as [2, 32, 34,61,62] to state a few.

In the distributed algorithms like MLI,max-SINR and AMI [2,32] as discussed before, IA

is obtained only if the following conditions are feasible at any receiver i and these are the

29


necessary and sufficient conditions to achieve IA:

UHi HijVj = 0, ∀j = {1, 2, · · · , K} (2.85)

rank(UHi HiiVi) = d, (2.86)

where we again consider the symmetric system with M transmit and N receive antennas

and d independent data streams transmitted from each transmitter, Ui ∈ CN×d,VM×dj

and Hij ∈ CN×M represent the zero-forcing matrix at receiver i, the precoding matrix at

transmitter j and the channel matrix corresponding to receiver i and transmitter j respectively.

Hence, the feasibility of IA is determined by the non-trivial solutions of system of polynomial

equations obtained from conditions (2.85) and (2.86). In fact, condition (2.86) is easily

achieved when the desired channels are sufficiently independent of the interference channels

by choosing d dimensional Ui and Vi. Thus, the non-trivial solutions of system of polynomial

equations obtained from condition (2.85) is enough to determine the feasibility of linear IA

and the multi-variable polynomial equations are solvable in general only if the number of

equations do not exceed the number of variables ( Bezout’s theorem ). IA problem where the

number of equations exceed the number of variables are termed as improper and the rest as

proper in [34].

The total number of equations obtained from condition (2.85) observed over any receiver i

due to a signal from transmitter j is d× d = d2, due to d rows of Ui and d columns of Vj.

Since there are K − 1 such transmissions observed by receiver i, total number of equations

per user for this symmetric system is given by

Ne = (K − 1)d2. (2.87)

On the other hand, the total number of variables observed at any receiver also depends on

the number of transmit and receive antennas in order to incorporate the available aligned

variables. Then the total number of variables per user observed in the symmetric system is

given by the following equation [34],

Nv = d(M +N − d). (2.88)

Thus for a proper system, Ne ≤ Nv which implies that

(K − 1)d2 ≤ d(M +N − d), (2.89)

⇒ d ≤ M +N

K + 1, (2.90)

is the feasible condition of linear interference alignment for symmetric network . However,

the feasibility condition for any type of generic interference network is still an open problem

and in fact is proved to be an NP-hard problem by Razaviyayn et al. in [63].

30


The distributed algorithms such as MLI and AMI achieve IA only for the proper system.

Based on the proposed idea of precoding matrix and zero-forcing matrix design, we observe

that interference is iteratively aligned and zero-forced. In that case, the information about the

interference signal is contained in the eigenvalues of sum of interference covariance matrices.

In each iteration, the amount of interference observed in the projection space (AMI algorithm)

or leakage interference (MLI algorithm) is measured in terms of sum of d minimum eigenvalues

of interference covariance (MLI algorithm) or the projection matrices (AMI algorithm), which

decreases and approaches zero for a proper system. When the system is improper, the

zero-forcing of interference is not possible.

This fact is depicted in depicted in Figure 2.7, which plots the sum of d minimum

eigenvalues of iteratively zero-forced and aligned interference covariance matrix with the

number of iterations. For M = 7 and N = 9 and K = 3, the achievable d is 7+93+1

= 4.

Hence, interference is completely zero-forced only when d = 4 and for d = 5 and d = 6, there

still exists a large amount of interference power. As long as interference is not completely

zero-forced it is not possible to decode the d dimensional transmitted data. Hence, we achieve

d = 4 DOF at maximum with M = 7, N = 9 and K = 3.

31


0 20 40 60 80 100 120 140 160 180 2000

5

10

15

20

25

30

35

40

Feasibility of Interference Alignment.

(M = 7, N = 9, K = 3)

No. of iterations

Sum

ofdminim

um

eigenvalues

d = 4d = 5d = 6

Figure 2.7: For K = 3 user MIMO IC with M = 7 transmit antennas and N = 9 receiveantennas, only d = 4 DOF is achievable for a proper system.

32

Chapter 3

Interference Alignment with Receiver

Cooperation and Greedy Transmission

In this chapter, we propose an improved interference alignment (IA) algorithm with hybrid

optimization, in which both leakage interference is minimized and desired signal power is

maximized simultaneously, unlike minimization of leakage interference (MLI) which minimizes

only leakage interference and maximization of signal to interference plus noise ratio (max-

SINR) which maximizes only signal to interference plus noise ratio (SINR) per transmit

stream of data. Through receiver cooperation the precoding matrix computed at transmitters

need not be known at each receiver, but rather are implicitly estimated cooperatively. The

cooperative structure of the algorithm benefits from the fact that only knowledge of the

sum of interference covariance matrices is enough to characterize the alignment problem

statistically. The choice of full cooperation amongst all the receivers is shown to be optimum

via a cooperative game theoretic argument. Simulation results show that the proposed

algorithm with receiver cooperation and hybrid optimization outperforms the well known

distributed IA schemes (MLI and max-SINR) proposed by Gomadam et al. [2].

All the distributed IA algorithms we discussed in section 2.4, i.e., MLI, max-SINR and

alternating minimization of interference (AMI) algorithms assume that the receivers have

perfect knowledge of the latest form of the precoding matrices at each iteration. Nevertheless,

these solutions do have the property that each receiver can act independently by aligning and

suppressing their own interference iteratively. This interesting feature of these algorithms

motivate us to think further about cooperation amongst receivers and the benefits of such

cooperation in K-user multiple input multiple output (MIMO) interference channel (IC).

In fact, it has been shown in [64] that cooperation amongst receivers can help mitigate

interference in the two-user IC.

We consider that the receivers cooperate by sharing amongst themselves their backward

channel state information (CSI), which enables each receiver to estimate the precoding matrix

at each iteration of IA optimization problem independently. This cooperation at at receivers is

33

System Model

aimed to minimize the leakage interference after zero-forcing the received signal by designing

proper zero-forcing matrices. However we consider a greedy approach at the transmitter

side. All transmitters aim to aim to maximize their own desired signal power and do not

care about the interference they cause to the other receivers. Such cooperative optimization

at receiver and greedy (non-cooperative) optimization at the transmitter side is termed as

“hybrid optimization”.

Such hybrid optimization arises a number of questions for further analysis. One of the

important questions is concerned with the amount of cooperation and the other question is

concerned with the convergence of such problems. Using coalition game theory [65,66], we

show that the spectrum efficiency of such scheme is maximized only if all the receivers are

the part of the cooperation and considering this iterative scheme as a class of evolutionary

process and relating to similar mathematical problem such as population migration problem as

described in [67], we show that such optimization converges after certain number of iterations.

The interesting observation is that such hybrid optimization gives better results in terms of

the data rate of the system (bits/sec/Hertz) and the feasibility, than the MLI and max-SINR

algorithms [2].

3.1 System Model

Consider a linear IA scheme over the K user MIMO IC with Mj transmit and Ni receive

antennas, ∀ i, j = {1, 2, · · · , K} in each transmitter-receiver pair (j, i). Let dj denotes the

independent data streams transmitted from transmitter j, i.e,degrees of freedom (DOF)

corresponding to the j-th transmitter, which is outer bounded by dj ≤ min(Mj, Nj). Thus for

any transmitter-receiver pair (j, i), there are dj desired links ∀j = i and (K − 1)dk interfering

links, such that each dk ≤ min(Mk, Nk), ∀k 6= i.

The received signal at any receiver i is the sum of a desired signal and K − 1 interfering

signals from all interfering transmitters j 6= i plus noise, as given by

yi = HiiVisi +∑j 6=i

HijVjsj + zi, (3.1)

where yi ∈ CNi×1 is the received signal vector at receiver i, Hij ∈ CNi×Mj is the MIMO

channel matrix corresponding to transmitter j and receiver i, Vj ∈ CMj×dj is the precoding

matrix at any transmitter j, sj ∈ Cdj×1 is the transmit symbol vector at any transmitter j,

and zi ∈ CNi×1 is the additive white Gaussian noise (AWGN) vector at receiver i, with zero

mean and unit variance. Each element of Hij is independent and identically distributed (i.i.d)

34

System Model

with zero mean and |hkl|2 variance, where ∀k = {1, 2, · · · , Ni} and ∀` = {1, 2, · · · ,Mj}.The transmit power at transmitter j is constrained by the available power Pj as :

E(sjsHj ) ≤ Pj. (3.2)

Also, the power constraint does not change after precoding the transmit signal, thus

E[||Vjsj||2

]= Tr

[E(Vjsj)(Vjsj)

H]

= Tr[VjE(sjs

Hj )VH

j

]≤ Pj (3.3)

The condition (3.3) is satisfied only if the precoding matrix is a unitary matrix, hence

VHj Vj = Idj . (3.4)

As discussed in section 2.4, linear IA schemes are mechanisms to design the precodeing

matrices Vj at any transmitter j that helps to beamform the transmitted signal in a particular

direction such that it aligns with the other interference signal in the undesired receivers

and the zero-forcing matrix or the interference suppressing matrix Ui at any receiver i that

zero-forces the aligned interference signal observed at any receiver i.

Since Vj and Ui behave exactly the same way one in the forward direction and another in

the backward direction assuming the reciprocity of the channels as observed in [2], Ui is also

orthonormal with the same number of columns as Vj. Hence,

UHi Ui = Idi . (3.5)

Our aim is to design Ui and Vj, ∀i, j = {1, 2, · · · , K} that satisfy the feasible IA conditions

as discussed in section 2.5. We employ the similar approach of Gomadam et al. [2] to use the

forward and backward channel and propose a hybrid iterative algorithm with cooperation

amongst the receiver and greedy transmission from the transmitters to achieve the following

linear feasible IA conditions:

UHi HijVj = 0, ∀ i 6= j, (3.6)

rank(UHi HiiVi

)= di, (3.7)

under the constraints

UHi Ui = Idi , (3.8)

VHi Vi = Idi , ∀i = 1, 2, · · · , K. (3.9)

Since MIMO channels are assumed non-degenerate, i.e., they are sampled from i.i.d

complex Gaussian random variables with zero mean and unit variance, they are full rank

with probability one and such assumption is sufficient to state that if equation (3.6) is

achievable, then equation (3.7) is also achievable with probability one. In other words, when

the interference subspace is nulled, the total desired subspace is contained in the signal space.

35

Proposed Algorithm

3.2 Proposed Algorithm

The main drawbacks of distributed algorithms such as MLI and max-SINR as discussed in

section 2.4 is that they require both full knowledge of forward channel and precoding matrices

Vj ’s in order to design the receive supressors Ui’s. Worse, both full knowledge and reciprocal

CSI of the backward channel, as well as the latest form of Ui’s are needed in order to design

the next precoding matrix Vj’s. In other words, the price payed for the distributiveness of

the schemes is considerable overhead and a rather strong limitation to channels with long

coherence time.

In addition to the aforementioned limitations, these distributed algorithms as proposed

in [2] are also in a sense incomplete, as each are designed to address only part of the

parameters of interest, namely, the interference and the received signal power. To clarify, the

MLI algorithm minimizes the interference that still remains after zero-forcing in both the

forward and backward channel and does not consider the desired signal power. Similarly, the

max-SINR algorithm maximizes SINR per data streams in both forward and reverse channel.

Thus, we propose a hybrid IA algorithm that contributes to mitigate all the aforementioned

limitations by simultaneously minimizing the leakage interference interference in the forward

step and maximizing the desired signal power in the backward step. The cooperation employed

amongst the receivers ensure that the exact knowledge of precoding matrices is not required

to be known at all the receivers and the zero-forcing matrices at all the transmitters [68].

Forward Step: Cooperative Interference Minimization

Let xj , Vjsj be the transmitted signal at the j-th transmitter, then the signal received at

the i-th receiver as given by (3.1) can be re-written as

yi = Hiixi +∑j 6=i

Hijxj + zi. (3.10)

Assume that each receiver has knowledge of the CSI between itself and corresponding

transmitters, and consider that the receivers cooperate with each other by sharing such CSI

amongst them, which means each receiver has the perfect knowledge of the channel matrix

H ∈ C(∑Ki=1Ni×

∑Kj=1Mj) as given by

H =

H11 H12 · · · H1K

H21 H22 · · · H2K

......

. . ....

HK1 HK2 · · · HKK

. (3.11)

36

Proposed Algorithm

Also, assume that the receivers cooperatively share the received signal amongst themselves

such that every receiver has the information of the received signal given by the vector

y ∈ C(∑Ki=1Ni)×1 expressed as

y =[yT1 yT2 · · · yTK

]T, (3.12)

where each yi is given by (3.10), ∀i = {1, 2, · · · , K}.With knowledge of channel matrix H and received signal vector y, the receivers cooperatively

estimate the transmitted signal x ∈ C∑Kj=1Mj×1 from all the transmitters using the least

square estimate [67] as given by

x = (HHH)−1HHy, (3.13)

which contains the estimates of the transmitted signal from each transmitter as given by

x =[xT1 xT2 · · · xTK

]T. (3.14)

Now, the receiver i designs a Ni × di dimensional zero-forcing matrix Ui based on the esti-

mated knowledge of xj,∀j = {1, 2, · · · , K} that minimizes the estimated leakage interference

observed at the receiver i as given by

Ii(UHi , {Qij}) =

∑j 6=i

Pjdj

Tr(UHi HijXjX

Hj HH

ijUi),

=∑j 6=i

Tr(UHi

Pjdj

HijXjXHj HH

ij︸︷︷︸Qij

Ui

),

=∑j 6=i

Tr(UHi QijUi), (3.15)

where Pj is the average transmit power of the j-th transmitter and the implicitly defined

quantities Qij are the interference covariance matrices of the i-th receiver, corresponding to

the interference estimate vectors xj expressed in terms of the corresponding diagonal matrices

Xj = diag(xj).

Notice that unlike [2], Ii(UHi , {Qij}) is an estimated leakage interference, which is computed

at each receiver without exact knowledge of all Vj’s but only the cooperative knowledge of

all the channel states and the received signal vector.

The i-th receiver then proceeds to solve the following optimization problem:

minUi

Ii(UHi , {Qij})

s.t. UHi Ui = Idi . (3.16)

Since Qij ∈ CNi×Ni is the positive semi-definite matrix, the solution to this trace min-

imization problem is well known as in [2, 32] and is obtained as follows. Consider the

37

Proposed Algorithm

eigendecomposition of the sum∑

j 6=i Qij, namely∑l 6=k

Qij = W ·ΛUi·WH , (3.17)

where the Ni eigenvectors of the matrix W are in descending order, i.e., W = [w1, · · · ,wNi ].

Then the optimum interference suppressing matrix at receiver i, U∗i is given by

U∗i =[wNi−di+1

,wNi−di+2

, · · · ,wNi

]. (3.18)

At this point we may remark that since the solution of the leakage interference minimization

problem actually depends only on the sum∑

l 6=k Qkl, the cooperative scheme here discussed

can be efficiently implemented with the help of a coordinator which collects all Hkl’s and

yk’s, calculates the sums∑

l 6=k Qkl for each k, and retransmits that information to the

corresponding receiver.

Backward Step: Power Maximization

While in each of the schemes in [2], the same optimization problem is solved on both forward

and backward stages of the iterative procedure, we remark here that this does not need

to be necessarily so. Relying on the fact that the receivers are capable of zero-forcing the

interference, a greedy power maximization approach can be employed at the backward stage

of the algorithm, such that the j-th transmitter performs

maxVj

Tr(VHj HH

ijUjUHj HijVj),

s.t. VHj Vj = Idj . (3.19)

Again, it is known that dj dimensional precoding matrix that solves this problem is

built from the dj largest eigenvectors of the desired covariance matrix. That is, given the

eigendecomposition

HHijUjU

Hj Hij = P ·ΛVj

·PH , (3.20)

with eigenvectors P = [p1, · · · ,pMj] in increasing order, the optimum precoding matrix is

obtained as

V∗j =[p1,p2, · · · ,pdj

]. (3.21)

This V∗j is now used to precode d independent data streams in the next iteration. The

receivers estimate the received signal and design optimum U∗i , with that knowledge the

transmitter again maximizes its desired power and design new V∗j and two steps iterate until

they converge. The convergence is guaranteed as long as the number of transmitted data

38

Coalition Game Theory and Full Cooperation

streams are feasible for IA as given by the feasibility condition in [34]

d ≤∑K

j=1Mj +∑K

i=1 Ni

K(K + 1), (3.22)

when all the transmitters transmit the same number of independent data streams d.

The exact feasibility condition for different number of transmitted independent data streams

is still an open problem. Hence, we consider a symmetric system for the simulation. The

algorithm can be stopped after the norm of the difference of consecutive precoding/zero-forcing

matrix is inferior to a pre-determined tolerance value.

3.3 Coalition Game Theory and Full Cooperation

Before we continue to show results on the performance of the cooperative interference

alignment described above, let us take the time to address the optimality of the full cooperation

assumed in section 3.2. This question can be approached from a game theoretical perspective,

specifically under the prism of Coalition Game Theory [65,66]. Let us be clear on some of

the terminologies that we have used in the later discussion.

Terminlogy: In a coalition game, several players form a coalition and act jointly to

improve the individual payoff. If there are no restrictions in the distribution of the payoff

among the members forming the coalition, the payoff is called transferable. When all the

available players form a single coalition, the coalition is called a grand coalition. Finally,

a coalition game is said to have a characteristic function form (CFF) if the value of the

coalition does not depend on outside coalitions. Otherwise, the coalition game is said to have

a partition function form (PFF).

Let R = {r1, r2, · · · , rK} be the set of all receivers and S ⊆ R consist of a coalition

(subset) of |S| players, where |.| represents the cardinality of the set. Suppose the function v

associates to each non-empty subset S the real value v(S), which is the total payoff available

for partition among the members of S. Then the coalition game with transferrable payoff

is denoted by 〈R, v〉 in CFF [66]. A coalition game 〈R, v〉 with transferable payoff v is

superadditive if for any two disjoint coalitions S1, S2 ⊆ R,

v(S1 ∪ S2

)≥ v(S1

)+ v(S2

). (3.23)

The core of a coalition game is a solution concept in CFF, where the set of players do

not break the coalition but act jointly to make the coalition better. When the core exists

and is non-empty, it indicates that the coalition formed is stable and can be considered as a

solution.

39

Simulation Results and Feasibility

Assuming no cost is incurred in forming a coalition amongst receivers, the payoff assigned

can be considered as the sum rate achieved by any receiver ri, which is given by

v({ri}) = I(xj,yi

), (3.24)

where I(xj,yi

)= H

(xj)− H

(xj|yi

)is the maximum mutual information between the

transmitter j and receiver i with the maximum taken over all possible input distributions,

H(xj)

is the input uncertainty, and H(x|y)

is the conditional uncertainty.

Assume that the receivers form a coalition S whose value is not influenced by the action of

players outside the coalition. Such a game is called coalition game with no externalities. The

payoff of a coalition S achieved by all the members is

v(S) =∑i∈S

v({ri}) = I(xS ,yS

), (3.25)

where I(xS ,yS

)is the maximum mutual information between transmitters and receivers

that are members of S.

Under the given system model and for the game with no externalities, the payoff of a

coalition v(S)

is superadditive, i.e., for any two disjoint coalitions S1 and S2, the following

condition is satisfied [65]

I(xS1∪S2 ,yS1∪S2

)≥ I

(xS1 ,yS1

)+ I

(xS2 ,yS2

). (3.26)

This is achieved by expanding I(xS1∪S2 ,yS1∪S2

)as follows using the chain rule of mutual

information [14,65] :

I(xS1∪S2 ,yS1∪S2

)=I(xS1 ,yS1

)+I(xS1 ,yS2|yS1

)+ I

(xS2 ,yS2|xS1

)+ I

(xS2 ,yS1 |yS2 ,xS1

),

(3.27)

and further expressing the mutual information in terms of entropy [14] as

I(xS2 ,yS1|yS2 ,xS1

)= H

(xS2)−H

(xS2|yS2 ,xS1

), (3.28)

≥ I(xS2 ,yS2

), (3.29)

and comparing the obtained results.

In [65], it is also shown that in such games the core always exists, which means that grand

coalition always gives improved and stable results, which finally implies the optimality of the

choice of full receiver cooperation in the proposed algorithm.

3.4 Simulation Results and Feasibility

We consider a three-user symmetric MIMO ICwith : M = N = 2, d = 1 and M = N = 4, d =

2. The independent streams of data transmitted from each transmitter d is determined before

40


transmission so that IA is feasible for the given system. In each case, we plot the achieved

sum rate measured in bits/sec/Hertz against the signal to noise ratio (SNR) measured in

deciBel (dB) as shown in Figure 3.1 and compare the results for our proposed algorithm with

MLI and max-SINR algorithms proposed in [2].

Interestingly, we observe that the proposed algorithm performs better than MLI for both

the cases and this algorithm achieves the same sum rate as max-SINR algorithm for the

case M = N = 2, d = 1; and achieves better sum rate for the case M = N = 4 and d = 2.

This shows that the estimation error is not high when all the receivers share the channel

information with each other and maximizing the direct desired power can improve the sum

rate. Other interesting observation here is that when there are more than one data streams,

the max-SINR algorithm provides better rate at low SNR but the MLI algorithm provides

better rate at high SNR, while the proposed cooperative performs better for all SNRs. This

result is justified because, max-SINR maximizes SINR per data stream [2], unlike MLI and

proposed cooperative approach. Thus, when there is only one data stream max-SINR provides

the same result as the proposed cooperative algorithm.

Since the leakage interference power is minimized in each iteration of the IA algorithms,

the feasibility of IA can be measured by determining the percentage of interference in the

desired signal space, which is mathematically measured as

ρi =

∑Λ∗Ui∑

j 6=iTr(Qij)

, (3.30)

where Λ∗Uiis the diagonal matrix of di minimum eigenvalues as given by equation (3.17).

When the percentage of interference calculated as in equation (3.30) is zero, perfect

alignment is achieved. The maximum total data streams for which the percentage of leakage

interference is zero is the total achievable DOF for the given system.

In Figure 3.2, the percentage of leakage interference is plotted against the achievable total

DOF in K = 3 user MIMO IC for the proposed cooperative and well-known MLI algorithm

in [2] at 0 dB transmit power for the symmetric cases with M = N = 3 and M = N = 4. We

observe here that though both algorithms perform better for feasible total number of data

streams, the proposed cooperative algorithm is slightly better than the MLI algorithm.

It can also be observed that the achieved sum DOF nearly meet the theoretical limit of

achievable sum DOF for the particular case of linear IA with all transmitters and all receivers

having the same number of antennas, as given by [34]

d = |s| ≤ M +N

K + 1. (3.31)

41


0 5 10 15 20 25 30 35 40 45 500

10

20

30

40

50

60

70

80

90

Sum Rate against Received SNR

(Number of Users: K=3)

Sum

Rate

Received SNR in dB

Proposed CooperativeMinimum LeakageMaximum SINR

DoF=2;M=4;N=4

DoF=1;M=2;N=2

Figure 3.1: Sum rate achieved measured in bits/sec/Hz as a function of SNR measured in dBfor different transmit antennas, receive antennas and feasible DOF that achieves IA for suchcase. The results are compared for the proposed cooperative algorithm, MLI algorithm andmax-SINR algorithm.

42


3 4 5 6 7 8

0

1

2

3

4

5

6

7

Sum DoF against percentage of leakage interference

(Number of Users: K=3)

Percentage

ofinterference

indesired

sign

alspace

Total number of transmit streams in the network

Proposed CooperativeMinimum Leakage

M=3;N=3

M=4;N=4

Figure 3.2: Feasibility of interference alignment for the proposed cooperative and MLIalgorithms. The feasibility is measured as the percentage of the leakage interference aftereach iteration measured for the given total data streams. The proposed cooperative algorithmhas less leakage interference than the MLI for M = N = 3 and M = N = 4 with differentnumber of total transmit streams.

43

Conclusions

Specifically, for M = 4 and K = 4, the outer bound on the achievable sum DoF, as given

by equation (3.31) yield DoF = 4+43+1× 3 = 6, which is achieved accurately by the proposed

algorithm. In turn, for M = 3 and K = 3, the achievable outer bound on the sum DoF

is given by DoF = 3+33+1× 3 = 4.5, which is nearly achieved, in the sense that only a small

amount of residual interference remains at that point.

3.5 Conclusions

We proposed a cooperative and hybrid distributed interference alignment algorithm based on

the MLI and max-SINR schemes presented in [2]. The hybrid aspect of our algorithm arises

in the fact that both leakage interference is minimized (in the forward step) and desired signal

power at there receiver is maximized (at the backward step) simultaneously. Employing

receiver cooperation, the requirement that precoding vectors computed at transmitters be

communicated to receivers at each step is mitigated.

The choice of full cooperation amongst all the receivers is shown to be optimum via a

cooperative game theoretic argument. Simulation results show that the proposed algorithm

with receiver cooperation and hybrid optimization outperforms the well known distributed

interference alignment scheme proposed by Gomadam et al. [2].

An interesting outcome of the approach is that, since only knowledge of the sum of

interference covariance matrices is enough to characterize the alignment problem statistically,

a robust (stochastic) version of the distributed alignment concept can be envisioned, motivating

future work.

44

Chapter 4

Space-Time Transmission with

Delayed CSIT

One of the major drawbacks of most of the interference alignment (IA) algorithms described

in the previous chapters are the requirement of perfect channel state information at the

transmitter (CSIT). Proper precoding and zero-forcing can achieve the optimal degrees of

freedom (DOF) only if the instantaneous CSIT is known perfectly. However, perfect CSIT is

a very optimistic assumption when the channels are varying rapidly with time and it is no

more possible to achieve the benefits of IA.

Previously, it was believed that CSIT from the previous time instances could not improve

the achievable DOF of the given multi-user multi-antenna channels. But recently, Maddah Ali

and Tse published an article [41, 69] with a curious result which states that the knowledge of

delayed or the outdated CSIT improves the DOF of the multiple input single output (MISO)

broadcast channel (BC). They showed that the per-user DOF of 23

is easily achievable in

two-user MISO BC only with the knowledge of delayed CSIT using the idea of retrospective

IA, which is the improvement from previously thought 12

per-user DOF [41,69].

Knowledge of delayed CSIT means that at time t, the transmitters know all the channel

states till time t − 1 but does not know the channel state at time instance t. This is a

valid model when the channel changes from symbol to symbol with each channel state being

independent and identically distributed (i.i.d).

In this chapter, we discuss the transmission scheme that improves the previously thought

achievable DOF in any K-user MISO BC as proposed in [41, 69] with the help of delayed

CSIT. We will further consider another scheme that improves the achievable DOF by not only

considering the delayed CSIT but also assuming that some portion of imperfect instantaneous

CSIT is available at any time instance t as proposed in [43, 70]. Following the same basis,

we propose our own space-time based transmission scheme to show that better DOF is

achievable in K user multi-cell MISO BC or simply a K-user interference broadcast channel

(IBC) with the knowledge of perfect delayed CSIT. The proposed space-time transmission

45

Space-Time Transmission with Delayed CSIT

scheme is based on retrospective interference alignment for two-cell interfering MISO BC

with M transmit antennas and K single antenna users in each cell. We suggest that using

the improved transmission scheme, the total DOF achieved converges to 85

in the case when

M = K and delayed CSIT is known perfectly. This is an interesting observation because it

shows that better than 1 DOF is achievable even if the CSIT is not known instantaneously.

Even the DOF of the MISO IBC with perfect instantaneous CSIT is not well explored

in literature. Only few works such as [71, 72] study the achievable DOF in any B cell M

antennas transmitter and K single antenna users MISO BC using the concept of IA. The

authors in [71]design the precoders using IA and zero-forcing to align the inter-cell and

intra-cell interference and achieve the optimum DOF which is outer bounded by BK2

, when

M = K. As such they show that for any β = max(M,K)min(M,K)

, the DOF of any B MISO IBC with

M transmit antennas and K single antenna users per cell is given by :

d =

BK if B ≤ β & M ≥ K,

BK1+ 1

β

if B ≥ β + 1 & M ≥ K,

BM if B ≤ β & M < K,

BM1+ 1

β

ifB ≥ β + 1 & M < K.

(4.1)

This satisfies the DOF achieved per user in a single input single output (SISO)interference

channel (IC), i.e., 12

as in [1].

Also, the authors in [72] have developed the transmission scheme based on subspace IA

to align interference over multiple dimensions and analyze the achievable DOF per cell for

B-cells interfering system with K users in each cell using multiple sub-carriers. The authors

have proved that B cells BC is the dual of B cells interference multiple access channel (IMAC)

and show that the following DOF is achievable for both channels with K users per cell:

d =K(

B−1√K + 1

)B−1, (4.2)

which approaches to unity as K approaches to infinity.

Some other works like [73,74] have been proposed to analyze the achievable DOF of multiple

input multiple output (MIMO) IBC based on IA and linear beam-forming schemes. The

authors in [74] have proved that the achievable DOF per user for a B cell MIMO IBC with

M transmit antennas per base-station and N receive antennas per user with K users per cell

is outer bounded by :

d ≤ max( M

Kη + 1,

Nη

Kη + 1

), (4.3)

46

Single Cell Two-user MISO BC with Delayed CSIT

where η ∈{pq

: p ∈ {1, 2, · · · , B − 1}, q = {1, 2, · · · , (B − p)K}}

.

For B = 2, p = {1} and q = {1, 2, · · · , K} and η ∈ {1, 12, · · · , 1

K} and d is outer bounded

by MK+1

for η = 1 and M2

for η = 1K

.

The main challenges in the analysis of any multi-cell MISO or MIMO BC or multiple access

channel (MAC) are the following:

• All the base stations transmit without co-operation. So, for the cell edge users, the

system acts as an interference channel.

• Both inter cell interference (ICI) and inter user interference (IUI) are to be managed

when there are multiple users in each cell.

Due to these challenges even in the presence of the instantaneous CSIT, the DOF analysis

of the multi-cell system in any generic channel is an open problem and is one of the interesting

topics for information theorists and communication engineers.

Next, we consider a transmission scheme to analyze the DOF for a simple network scenario

with single cell MISO BC with two transmit antennas and only two single antenna users per

cell using delayed CSIT.

4.1 Single Cell Two-user MISO BC with Delayed CSIT

Consider a MISO BC with two transmit antennas and two single antenna users as shown

in Figure 4.1, where the channel is independently varying at each time instance. The

channel from the transmitter to user i, ∀i = {1, 2} at time instance t is represented as

hi(t) =[hi1(t) hi2(t)

]and the precoded transmit streams from the transmitter at any time

instance t is x(t), which contains two independently encoded streams of data, u(i)1 (t) and

u(i)2 (t), intended to user i, ∀i = {1, 2}.Using the following transmission scheme, the delayed CSIT is useful to transmit and decode

four independent streams of data in three channel uses, thus gaining 43

total DOF.

• Time instance t = 1 : During this time instance which requires one time slot, the

transmitter transmits two independent data streams both intended to user 1 from two

antennas, i.e.,

x(1) =[u

(1)1 u

(1)2

]T. (4.4)

47


1

2

x(t)

[h 11(t)

h 12(t)]

[h21 (t) h22 (t)]

Figure 4.1: A two user MISO BC with two antennas at the transmitter. The two independentdata streams are transmitted as a vector x(t) at any time instance t and the channel vectoris represented by hi1(t) and hi2(t), ∀i = {1, 2}.

48


Thus, the signal observed at both the users at t = 1 is expressed as:

y1(1) =[h11(1) h12(1)

]x(1) + z1(1), (4.5)

= h11(1)u(1)1 + h12(1)u

(1)2 + z1(1), (4.6)

y2(1) =[h21(1) h22(1)

]x(1) + z2(1), (4.7)

= h21(1)u(1)1 + h22(1)u

(1)2 + z2(1), (4.8)

where zi(1) is the additive white Gaussian noise (AWGN) at time instance t = 1.

• Time instance t = 2: During this time instance which requires one time slot, the

transmitter transmits two independent data streams both intended to user 2 from two

antennas, i.e.,

x(2) =[u

(2)1 u

(2)2

]T. (4.9)

The signal received by both the receivers at time instance t = 2 is thus given by

y1(2) =[h11(2) h12(2)

]x(2) + z1(2), (4.10)

= h11(2)u(2)1 + h12(2)u

(2)2 + z1(2), (4.11)

y2(2) =[h21(2) h22(2)

]x(2) + z2(2), (4.12)

= h21(1)u(2)1 + h22(1)u

(2)2 + z2(2), (4.13)

• Time instance t = 3: From the previous instances, we observe that y2(1) and y1(2)

are the interference signal for user 2 and user 1 respectively. However, y2(1) and y1(2)

contain the desired data streams for user 1 and user 2. During this time instance which

uses one time slot, the transmitter transmits the linear combination of y2(1) and y1(2)

minus noise, because the transmitter has the knowledge of all the channel states till

time t = 2, as given by:

x(3) =[h11(2)u

(2)1 + h12(2)u

(2)2 + h21(1)u

(1)1 + h22(1)u

(1)2 0

]T. (4.14)

The signal received by both the receivers at time instance t = 3 is then given by

y1(3) =[h11(3) h12(3)

]x(3) + z1(3), (4.15)

= h11(3)(h11(2)u

(2)1 + h12(2)u

(2)2 + h21(1)u

(1)1 + h22(1)u

(1)2

)+ z1(3), (4.16)

49


and

y2(3) =[h21(3) h22(3)

]x(3) + z2(3), (4.17)

= h21(3)(h11(2)u

(2)1 + h12(2)u

(2)2 + h21(1)u

(1)1 + h22(1)u

(1)2

)+ z2(3), (4.18)

where zi(3) is the AWGN at time instance t = 3.

Further, we can express the signal received by any user i, ∀i = {1, 2} over all the time

instances in the form of matrix equation as,y1(1)

y1(2)

y1(3)

=

h11(1) h12(1)

0 0

h11(3)h21(1) h11(3)h22(1)

︸︷︷︸

rank= 2

x(1) +

0 0

h11(2) h12(2)

h11(3)h11(2) h11(3)h12(2)

︸︷︷︸

rank= 1

x(2) +

z1(1)

z1(2)

z1(3)

,(4.19)y2(1)

y2(2)

y2(3)

=

h21(1) h22(1)

0 0

h21(3)h21(1) h21(3)h22(1)

︸︷︷︸

rank= 1

x(1) +

0 0

h21(2) h22(2)

h21(3)h11(2) h21(3)h12(2)

︸︷︷︸

rank= 2

x(2) +

z2(1)

z2(2)

z2(3)

,(4.20)

Clearly, we observe here that the two data streams desired by user 1, i.e., x(1) =[u

(1)1 u

(1)2

]Tis spanned by rank 2 matrix over two independent dimensions, while the

undesired data streams are spanned by a rank 1 matrix over a single dimension and similarly,

the two data streams desired by user 2, i.e., x(2) =[u

(2)1 u

(2)2

]Tis spanned by rank 2 matrix

over two independent dimensions while the undesired data streams are spanned by rank 1

matrix over a single dimension. Hence both receivers can decode total of 4 data streams in

three channel uses.

The achievable DOF is measured as:

d =total decodable data streams

total number of time slots required, (4.21)

=4

3, (4.22)

which is greater than 1 as previously believed. This result from [69] is further extended for

K user MISO BC as discussed in the section below:

50

K-user MISO BC with Delayed CSIT

4.2 K-user MISO BC with Delayed CSIT

Consider a MISO BC with K transmit antennas and K single antenna users. As in the case

for two-user MISO BC, linear combination of K streams are transmitted intended to each

user for the first K channel uses. Thus, there are total of K2 data streams to be decoded

and the DOF is obtained by determining the total number of channels required in K phases.

The nth phase determines the number of channels required to decode degree n messages.

As shown in [69], the phase j has(Kj

)sub-phases each with one time slot which takes

(K − j + 1)(Kj

)symbols with degree j and generate j

(Kj+1

)symbols with degree j + 1. Thus,

the DOF corresponding to any degree j for K user MISO BC with K transmit antennas is

given by:

dj(K,K) =total number of degree j symbols

total channel uses. (4.23)

The total number of degree j symbols is given by (K − j + 1)(Kj

)while the total number of

channel uses is the total number of time slots required, i.e,(Kj

)time slots for

(Kj

)sub-phases

and the extra time slots to decode j(Kj+1

)symbols with degree j + 1, hence

dj(K,K) =(K − j + 1)

(Kj

)(Kj

)+

j( Kj+1)

dj+1(K,K)

, (4.24)

=K − j + 1

1 + j (K−j)j+1

1dj+1(K,K)

, (4.25)

where dj+1(K,K) is the DOF of symbols with degree j + 1. Further simplifying, we can

express (4.25) as:

K − (j − 1)

j

1

dj(K,K)=

1

j+K − jj + 1

1

dj+1(K,K), (4.26)

=1

j+

1

j + 1+K − (j + 1)

j + 2

1

dj+2(K,K), (4.27)

The maximum possible degree of the symbol is K and there are at maximum one data

symbols of degree K, thus it is easy to verify that dK(K,K) = 1 and continuing the similar

pattern as (4.27) up until j + 1 = K, we can further simplify (4.27) as:

K − (j − 1)

j

1

dj(K,K)=

1

j+

1

j + 1+

1

j + 2+ · · ·+ 1

K. (4.28)

51

Two cell MISO IBC

We require the DOF for degree one symbols, hence j = 1 and

K1

d(K,K)= 1 +

1

2+

1

3+ · · ·+ 1

K, (4.29)

⇒d(K,K) =K

1 + 12

+ 13

+ · · ·+ 1K

, (4.30)

is the achievable DOF for K-user MISO BC with delayed CSIT.

This idea of DOF analysis of single cell MISO BC is extended to the case of two-cell MISO

IBC with delayed CSIT in [75], where we show that delayed CSIT is useful to mitigate ICI

and IUI with proper space-time transmission technique and more than unity DOF is easily

achievable in that case. We consider all the cases where the number of transmit antennas is

not necessarily equal to the number of users per cell. The space-time transmission scheme

and DOF analysis of two-cell MISO IBC is discussed in the next section.

4.3 Two cell MISO IBC

Consider a multi-cell MISO IBC with B = 2 cells, where a base station per cell with M

transmitting antennas transmit data to K single antenna users in each cell. Since there are

two base stations transmitting at the same time, the cell-edge users in both the cells receive

inter cell interference from the neighboring cell. Also, all the base-stations are broadcasting

the message to all the users at the same time, every user in the cell receive the signal

transmitted to K− 1 other users as an interference, called inter user interference. The system

model for this two-cell MISO IBC system with M transmit antennas per base station and K

users per cell is depicted in Figure 4.2.

We assume that each user has the perfect CSI for both the serving and the interfering

link. Since the transmitters have no instantaneous CSIT, the system is designed such that

each user feedbacks the available CSI to the serving base-station. In doing so, the user sends

the channel direction information (CDI) and the channel quality information (CQI) of both

the serving and the interfering links to the serving base-station. The base stations, thus,

exchange only the CDI and CQI for the interfering links by using the error and delay free

back-haul channel. Such information exchange technique is also discussed in detail in [76].

Let us denote the channel between the base-station b and user k in the `th cell at any

time-instant t by the vector h(`)kb (t) ∈ CM×1, the transmitted signal from the base-station b

intended to the user k in the same cell at time t by a vector s(b)k (t) ∈ CM×1, the received

signal at the user k in the `th cell at any time instant t by a scalar y(`)k (t) and the AWGN at

52

Two cell MISO IBC

Cell 1

M antennas

K users CSI feedback 1

CSI feedback 2

Cell 2 M antennas

K users

Back-haul connection

Inter-cell interference

Figure 4.2: A two-cell MISO interference broadcast channel with M transmit antennas ineach base station and K single antenna users per cell. The solid line represents the signalreceived from the same cell and the dashed line represents the inter cell interference receivedfrom the adjacent cell. The dotted line represents that the delayed channel state information(CSI) feedback is provided from any receiver to the transmitter. The two cells are connectedvia back-haul connection.

53

Space-Time Transmission Scheme

user k in the cell ` at time instant t as z(`)k (t). Each channel element is assumed to be i.i.d

with zero mean. The received signal at any user k at time instant t can be expressed as:

y(`)k (t) = [h

(`)k` (t)]

Hs

(`)k (t) +

K∑m 6=k,m=1

[h(`)k` (t)]

Hs(l)m (t) +

B∑b 6=`

K∑m=1

[h(`)kb (t)]

Hs(b)m (t) + z

(`)k (t). (4.31)

Following the similar information theoretic definition of DOF as in [1,14], the DOF per

cell is defined as:

d =1

Blim

SNR→∞

CT (SNR)

log SNR, (4.32)

where CT (SNR) is the total capacity of the system as a function of signal to noise ratio

(SNR).

This implies that this capacity pre-log factor can be obtained as the total number of

independent data streams or signals at the receivers per channel use. Since the instantaneous

CSIT is not available, proper precoding vectors at the transmitter side and zero-forcing vectors

at the receiver side are difficult to design. In that case, the spatio-temporal transmission

scheme to transmit the data streams from transmitter to the receiver is utilized to analyze

the DOF of the system. The delayed CSIT is useful in this regard and the DOF is expressed

in terms of the ratio of total number of independent data streams transmitted to the total

number of channel uses as also described in previous section. Hence, for MISO IBC with

delayed CSIT, the DOF per cell is measured as:

d =total independent streams decoded per cell

total number of channel uses per cell. (4.33)

In the following section, we present the space time transmission technique to analyze the

DOF of two-cell MISO IBC with the help of delayed CSIT for different values of transmit

antennas M and the number of users per cell K.

4.4 Space-Time Transmission Scheme

We consider various cases for different values of M and K before we generalize the result for

achievable DOF. Some of them are discussed here:

54


4.4.1 Case I ( M=1, K=1)

The simplest possible case is the one where each base-station with single antenna transmits

to a single antenna user in each cell. We observe that such system acts as a two-user SISO IC

and most of the fact about the DOF of such system is well known. It is easily observed that

even in the absence of any CSIT, the DOF of 12

is achievable and the outdated CSIT does

not help much to improve this DOF because it is the outer bound on the achievable DOF

for two-user SISO IC as we can not improve much with the help of IA. In other words, we

require at least two time instances to decode two transmitted data streams, thus, providing12

DOF per cell.

• At time instant t = 1, the signal received by each user is given by

y(1)1 (1) = h

(1)11 (1)s

(1)11 + h

(1)12 (1)s

(2)12 + z

(1)1 (1), (4.34)

y(2)1 (1) = h

(2)11 (1)s

(1)11 + h

(2)12 (1)s

(2)12 + z

(2)1 (1). (4.35)

• At time instant t = 2, each user feedbacks the CSI to the corresponding transmitter,

which share the interfering channel information with each other. Since each of them

have all the CSIT from previous time instances, the base-station 1 transmits the

side information observed by the user in cell 2, which is h(2)11 (1)s

(1)11 and similarly the

base-station 2 transmits the side information observed by the user in cell 1, which is

h(1)12 (1)s

(2)12 . Thus the received signal is

y(1)1 (2) = h

(1)11 (2)h

(2)11 (1)s

(1)11 + h

(1)12 (2)h

(1)12 (1)s

(2)12 + z

(1)1 (2), (4.36)

y(2)1 (2) = h

(2)11 (2)h

(2)11 (1)s

(1)11 + h

(2)12 (2)h

(1)12 (1)s

(2)12 + z

(2)1 (2). (4.37)

• Since we perform the DOF analysis at high SNR, we neglect the noise terms, which is

reasonable assumption. Thus, decoding is easily possible with the available information

in this case by canceling the ICI as given by:

s(1)11 =

y(1)1 (2)− h(1)

12 (2)y(1)1 (1)

h(1)11 (2)h

(2)11 (1)− h(1)

12 (2)h(1)11 (1)

, (4.38)

s(2)12 =

y(2)1 (2)− h(2)

11 (2)y(2)1 (1)

h(2)12 (2)h

(1)12 (1)− h(2)

11 (2)h(2)12 (1)

. (4.39)

4.4.2 Case II (M=1, K=2)

This is an example of a scheme where K > M and we observe here that the scheme can’t

achieve better than one DOF, which is not an improvement. The key reason is the lack of

55


independent channel coefficients to cancel out the ICI. The main idea of the proposed space-

time transmission scheme is the repetitive alignment of interferences by repetitive transmission

of the similar signal in different time instances. This is valid as long as independent channel

coefficients can be created to decode all the streams.

From now onwards, we remove the noise terms in each analysis for simplicity as the noise

plays no role in high SNR analysis and we consider following transmission scheme:

• At time instant t = 1, base-station 1 transmits the sum of the data streams intended to

both the users in cell 1, which is s(1)11 + s

(1)21 and the base-station 2 transmits the sum of

data streams intended to both users in cell 2, which is s(2)12 + s

(2)22 . The signals received

by all the users at t = 1 is given by:

y(1)1 (1) = h

(1)11 (1)

(s

(1)11 + s

(1)21

)+ h

(1)12 (1)

(s

(2)12 + s

(2)22

), (4.40)

y(1)2 (1) = h

(1)21 (1)

(s

(1)11 + s

(1)21

)+ h

(1)22 (1)

(s

(2)12 + s

(2)22

), (4.41)

y(2)1 (1) = h

(2)11 (1)

(s

(1)11 + s

(1)21

)+ h

(2)12 (1)

(s

(2)12 + s

(2)22

), (4.42)

y(2)2 (1) = h

(2)21 (1)

(s

(1)11 + s

(1)21

)+ h

(2)22 (1)

(s

(2)12 + s

(2)22

). (4.43)

• At time instance t = 2, both the base-stations use the delayed CSIT and transmit the

linear combination of the ICI observed by each cell to align those interference. We

consider only the signal received by the first user in each cell because all the other users

behave symmetrically and use extra channel. The signal received by the first user in

each cell after t = 2 is

y(1)1 (2) = h

(1)11 (2)h

(2)11 (1)

(s

(1)11 + s

(1)21

)+ h

(1)12 (2)h

(1)12 (1)

(s

(2)12 + s

(2)22

), (4.44)

y(2)1 (2) = h

(2)11 (2)h

(2)11 (1)

(s

(1)11 + s

(1)21

)+ h

(2)12 (2)h

(1)12 (1)

(s

(2)12 + s

(2)22

). (4.45)

This implies the following decoding condition

s(1)11 + s

(1)21 =

y(1)1 (2)− h(1)

12 (2)y(1)1 (1)

h(1)11 (2)h

(2)11 (1)− h(1)

12 (2)h(1)11 (1)

, (4.46)

s(2)12 + s

(2)22 =

y(2)1 (2)− h(2)

11 (2)y(2)1 (1)

h(2)12 (2)h

(1)12 (1)− h(2)

11 (2)h(2)12 (1)

. (4.47)

• Since it is not decoded perfectly, we repeat the transmission in the next time instant

56


t = 3, which gives the similar decoding conditions given by:

s(1)11 + s

(1)21 =

y(1)1 (3)− h(1)

12 (3)y(1)1 (1)

h(1)11 (3)h

(2)11 (1)− h(1)

12 (3)h(1)11 (1)

, (4.48)

s(2)12 + s

(2)22 =

y(2)1 (3)− h(2)

11 (3)y(2)1 (1)

h(2)12 (3)h

(1)12 (1)− h(2)

11 (3)h(2)12 (1)

. (4.49)

It is clearly visible that from (4.46), (4.47), (4.48) and (4.49), the decoding of all the four

streams is not possible with these four channel conditions. This shows that when there are

more streams to be transmitted than the number of transmit antennas, the decoding is not

possible with this proposed scheme. So, we consider only the cases when M ≥ K.

4.4.3 Case III (M=2, K=1)

When the number of transmit antennas is greater than the number of users in each cell, it is

possible to have enough independent channel coefficients and the decoding is easily possible.

For the case with M = 2 and K = 1, we show that using the space time transmission scheme

4 data streams can be transmitted and decoded using only 3 channel instances, thus providing

total DOF of 43

or per cell DOF of 23.

• At time instance t = 1, each base-station transmits two data streams from the available

two antennas and the signal received by the users in each cell is given by

y(1)1 (1) = h

(1)11 (1)

Hs

(1)11 + h

(1)12 (1)

Hs

(2)12 , (4.50)

y(2)1 (1) = h

(2)11 (1)

Hs

(1)11 + h

(2)12 (1)

Hs

(2)12 , (4.51)

where h(`)kb (t) ∈ CM×1 is the channel vector from base station b to user k at cell ` during

time instance t and s(`)kb ∈ CM×1 is the transmit signal vector intended to user k in cell

` transmitted from base station b.

• At time instance t = 2, each base-station has the knowledge of previous channel states.

Thus, both the base stations transmit in such a way that the ICI observed by the users

in each cell is aligned. Each base-station uses only one antenna for this transmission as

given by:

y(1)1 (2) = h

(1)11 (2)

(h

(2)11 (1)

Hs

(1)11

)+ h

(1)12 (2)

(h

(1)12 (1)

Hs

(2)12

), (4.52)

y(2)1 (2) = h

(2)11 (2)

(h

(2)11 (1)

Hs

(1)11

)+ h

(2)12 (2)

(h

(1)12 (1)

Hs

(2)12

). (4.53)

57


• At time instance t = 3, we use the concept of repetition of the transmission assuming

all the channel elements are i.i.d and the signal received at each user is, thus, given by:

y(1)1 (3) = h

(1)11 (3)

(h

(2)11 (1)

Hs

(1)11

)+ h

(1)12 (3)

(h

(1)12 (1)

Hs

(2)12

), (4.54)

y(2)1 (3) = h

(2)11 (3)

(h

(2)11 (1)

Hs

(1)11

)+ h

(2)12 (3)

(h

(1)12 (1)

Hs

(2)12

). (4.55)

• The desired signals can be easily decoded by each user with the available information

by simply observing the following conditions:[y

(1)1 (2)− h(1)

12 (2)y(1)1 (1)

y(1)1 (3)− h(1)

12 (3)y(1)1 (1)

]=

[h

(1)11 (2)h

(2)11 (1)

H − h(1)12 (2)h

(1)11 (1)

H

h(1)11 (3)h

(2)11 (1)

H − h(1)12 (3)h

(1)11 (1)

H

]s

(1)11 , (4.56)

and

[y

(2)1 (2)− h(2)

11 (2)y(2)1 (1)

y(2)1 (3)− h(2)

11 (3)y(2)1 (1)

]=

[h

(2)12 (2)h

(1)12 (1)

H − h(2)11 (2)h

(2)12 (1)

H

h(2)12 (3)h

(1)12 (1)

H − h(2)11 (3)h

(2)12 (1)

H

]s

(2)12 . (4.57)

4.4.4 Case IV (M=2, K=2)

The simplest example with more than one users per cell and more than one transmit antennas

per base station is the symmetric MISO IBC with M = 2 in all base-stations and K = 2 in

all cells. Using the similar space-time transmission scheme, we can easily verify in this case

that the total of 8 data independent streams can be transmitted over 6 channel uses, thus

achieving the total DOF of 86

= 43

and per cell DOF of 23.

• At time instant t = 1, both the base stations transmit all the data streams intended to

the first user in each cell. Hence base station 1 transmits two data streams intended to

user 1 at cell 1 using two independent antennas and base station 2 transmits two data

streams intended to user 1 at cell 2 using two independent antennas. Thus, the signal

received by all the users in both the cells at t = 1 is given by

y(1)1 (1) = [h

(1)11 (1)]

Hs

(1)11 + [h

(1)12 (1)]

Hs

(2)12 , (4.58)

y(1)2 (1) = [h

(1)21 (1)]

Hs

(1)11 + [h

(1)22 (1)]

Hs

(2)12 , (4.59)

y(2)1 (1) = [h

(2)11 (1)]

Hs

(1)11 + [h

(2)12 (1)]

Hs

(2)12 , (4.60)

y(2)2 (1) = [h

(2)21 (1)]

Hs

(1)11 + [h

(2)22 (1)]

Hs

(2)12 , (4.61)

where y(`)k (t) is a scalar signal received by a single antenna user k at cell ` during

time instance t, h`kb(t) ∈ C2×1 is the channel vector from base station b to user k in

58


cell ` during time instance t and s(`)kb ∈ C2×1 is the independent transmit data vector

transmitted from base station b intended to user k at cell `.

• At time instance t = 2, both the base stations transmit all the data streams intended

to the second user in each cell, which means base station 1 transmits two independent

data streams intended to user 2 in cell 1 and base station 2 transmits two independent

data streams intended to user 2 in cell 2. Hence, the signal received by all the users at

time instance t = 2 is given by:

y(1)1 (2) = [h

(1)11 (2)]

Hs

(1)21 + [h

(1)12 (2)]

Hs

(2)22 , (4.62)

y(1)2 (2) = [h

(1)21 (2)]

Hs

(1)21 + [h

(1)22 (2)]

Hs

(2)22 , (4.63)

y(2)1 (2) = [h

(2)11 (2)]

Hs

(1)21 + [h

(2)12 (2)]

Hs

(2)22 , (4.64)

y(2)2 (2) = [h

(2)21 (2)]

Hs

(1)21 + [h

(2)22 (2)]

Hs

(2)22 . (4.65)

• During t = 1, user 2 in both cells observe IUI plus ICI while user 1 in both cells observes

only ICI. Also during t = 2, user 1 in both cells observe IUI plus ICI while user 2 in

both cells observe only ICI.

At time instance t = 3, base-station 1 and base-station 2 have the knowledge of all the

previous channel states. Each user in the corresponding cell feedback the CSI to the

relevant base-station and the base stations then share all the information through a

back-haul connection. Hence base station 1 transmits the linear combination of all the

ICI observed by user 1 in cell 2 during time instance t = 1 using only one antenna and

base station 2 transmits the linear combination of all the ICI observed by user 1 in cell

1 during time instance t = 1 using only one antenna. The received signal at all the

users at time instance t = 3 is given by:

y(1)1 (3) = h

(1)11 (3)[h

(2)11 (1)]

Hs

(1)11 + h

(1)12 (3)[h

(1)12 (1)]

Hs

(2)12 , (4.66)

y(1)2 (3) = h

(1)21 (3)[h

(2)11 (1)]

Hs

(1)11 + h

(1)22 (3)[h

(1)12 (1)]

Hs

(2)12 , (4.67)

y(2)1 (3) = h

(2)11 (3)[h

(2)11 (1)]

Hs

(1)11 + h

(2)12 (3)[h

(1)12 (1)]

Hs

(2)12 , (4.68)

y(2)2 (3) = h

(2)21 (3)[h

(2)11 (1)]

Hs

(1)11 + h

(2)22 (3)[h

(1)12 (1)]

Hs

(2)12 . (4.69)

Please note here the scalar channels used during time instance t = 3 to indicate that

the linear combination of the data streams is transmitted from only one antenna.

• Similarly at time instance t = 4, the base-station 1 transmits the linear combination of

all the ICI observed by user 2 in cell 2 during time instance t = 2 and base station 2

59


transmits the linear combination of all the ICI observed by user 2 in cell 1 during time

instance t = 2. In this case also, both the base stations use only one antenna. The

received signal by all the users during time instance t = 4 is given by:

y(1)1 (4) = h

(1)11 (4)[h

(2)21 (2)]

Hs

(1)21 + h

(1)12 (4)[h

(1)22 (2)]

Hs

(2)22 , (4.70)

y(1)2 (4) = h

(1)21 (4)[h

(2)21 (2)]

Hs

(1)21 + h

(1)22 (4)[h

(1)22 (2)]

Hs

(2)22 , (4.71)

y(2)1 (4) = h

(2)11 (4)[h

(2)21 (2)]

Hs

(1)21 + h

(2)12 (4)[h

(1)22 (2)]

Hs

(2)22 , (4.72)

y(2)2 (4) = h

(2)21 (4)[h

(2)21 (2)]

Hs

(1)21 + h

(2)22 (4)[h

(1)22 (2)]

Hs

(2)22 . (4.73)

Also note the use of scalar channels during time instance t = 4, this is because of only

one antenna being used and the other being muted by the base-station.

• We employ the repetition of transmission technique to obtain other set of equations

to decode the desired streams. Hence at time instance t = 5, base station 1 and base

station 2 transmits the same linear combination as in time instance t = 3 through single

antenna and the signal observed by all the users in this case is given by :

y(1)1 (5) = h

(1)11 (5)[h

(2)11 (1)]

Hs

(1)11 + h

(1)12 (5)[h

(1)12 (1)]

Hs

(2)12 , (4.74)

y(1)2 (5) = h

(1)21 (5)[h

(2)11 (1)]

Hs

(1)11 + h

(1)22 (5)[h

(1)12 (1)]

Hs

(2)12 , (4.75)

y(2)1 (5) = h

(2)11 (5)[h

(2)11 (1)]

Hs

(1)11 + h

(2)12 (5)[h

(1)12 (1)]

Hs

(2)12 , (4.76)

y(2)2 (5) = h

(2)21 (5)[h

(2)11 (1)]

Hs

(1)11 + h

(2)22 (5)[h

(1)12 (1)]

Hs

(2)12 . (4.77)

• Similarly, at time instance t = 6, base station 1 and base station 2 transmit the same

linear combination of data streams as in time instance t = 4, which again helps to

mitigate the ICI by obtaining other independent set of equations in order to decode the

desired streams by all the users. The signal received by all the users at time instance

t = 6 is given by:

y(1)1 (6) = h

(1)11 (6)[h

(2)21 (2)]

Hs

(1)21 + h

(1)12 (6)[h

(1)22 (2)]

Hs

(2)22 , (4.78)

y(1)2 (6) = h

(1)21 (6)[h

(2)21 (2)]

Hs

(1)21 + h

(1)22 (6)[h

(1)22 (2)]

Hs

(2)22 , (4.79)

y(2)1 (6) = h

(2)11 (6)[h

(2)21 (2)]

Hs

(1)21 + h

(2)12 (6)[h

(1)22 (2)]

Hs

(2)22 , (4.80)

y(2)2 (6) = h

(2)21 (6)[h

(2)21 (2)]

Hs

(1)21 + h

(2)22 (6)[h

(1)22 (2)]

Hs

(2)22 . (4.81)

• The decoding of independent data streams is possible from the signals observed during

all the time instances by performing simple manipulations. Let us consider the decoding

for the user 1 in cell 1. By subtracting the signal received at time instance t = 1

60


multiplied by CSI at t = 3 from the signal received at time instance t = 3, we obtain

an equation which contains only the data streams intended to user 1 in cell 1. Further

by subtracting the signal received at time instance t = 1 multiplied by CSI at t = 5

from the signal received at time instance t = 5, we get another equation which also

contains only the data streams intended to user 1 in cell 1. Since there are two data

streams transmitted from base-station 1 intended to user 1 in cell 1, two independent

equations are enough to decode these two streams as explained mathematically by

following equation:

[y

(1)1 (3)− h(1)

12 (3)y(1)1 (1)

y(1)1 (5)− h(1)

12 (5)y(1)1 (1)

]=

[h

(1)11 (3)[h

(2)11 (1)]

H − h(1)12 (3)[h

(1)11 (1)]

H

h(1)11 (5)[h

(2)11 (1)]

H − h(1)12 (5)[h

(1)11 (1)]

H

]︸︷︷︸

A

s(1)11 . (4.82)

Since the channel coefficients are i.i.d and non-degenerate, A is a full rank matrix

almost surely with rank=2, and hence the two dimensional data vector s(1)11 is easily

decoded.

• Similarly, we can decode the data streams intended to user 2 in cell 1 by observing the

signals received at time instance t = 2, time instance t = 4 and time instance t = 6 as

given by:[y

(1)2 (4)− h(1)

22 (4)y(1)2 (2)

y(1)2 (6)− h(1)

22 (6)y(1)2 (2)

]=

[h

(1)21 (4)[h

(2)21 (2)]

H − h(1)22 (4)[h

(1)21 (2)]

H

h(1)21 (6)[h

(2)21 (2)]

H − h(1)22 (6)[h

(1)21 (2)]

H

]︸︷︷︸

B

s(1)21 . (4.83)

Also, B is full rank matrix for all i.i.d channel coefficients and hence we can easily

decode s(1)21 .

With the similar calculations, we can easily decode all the data streams intended to

user 1 and user 2 in cell 2. Hence, overall 8 data streams are decoded in just 6 channel

uses with the help of past channel states and DOF of 43

is achieved in two-cell MISO

IBC with M = K = 2.

The repetition scheme works fine for M = 2 and K = 2 and we achieved all the required

equations in all the users just by canceling the ICI and we not much analysis regarding the

cancellation and alignment of IUI is needed. But the repetition scheme is not suitable when

M > 2 because this scheme provides only two sets of independent equations. For M > 2,

the matrices constructed, i.e., A and B as observed in (4.82) and (4.83) are not full rank

61


matrices anymore. We verify this by considering the case M = 3 and K = 1 in the next

subsection.

4.4.5 Case V (M=3, K=1)

The transmission scheme for this case is observed as:

• Since there are three transmit antennas and only one user per cell, both the transmitters

transmit 3 independent streams of data at time instance t = 1 and the observed signal

is given by:

y(1)1 (1) =

[h

(1)11 (1)

]Hs

(1)11 +

[h

(1)12 (1)

]Hs

(2)12 , (4.84)

y(2)1 (1) =

[h

(2)11 (1)

]Hs

(1)11 +

[h

(2)22 (1)

]Hs

(2)12 , (4.85)

where y(`)k (t) is the scalar signal observed by user k at cell ` during time instance t,

h(`)kb (t) ∈ C3×1 is the channel vector from base station b to user k in cell ` during time

instance t and s(`)kb ∈ C3×1 is the independent transmit data vector from base station b

intended to user k in cell `.

• In order to cancel ICI, the base station in cell 1 transmits the linear combination of the

signal which is the ICI as observed by the user in cell 2 and the base station in cell 2

transmits the linear combination of the signal which is the ICI as observed by the user

in cell 1 using only one antenna in the time instance t = 2 and the signal received by

each user is given by:

y(1)1 (2) = h

(1)11 (2)

[h

(2)11 (1)

]Hs

(1)11 + h

(1)12 (2)

[h

(1)12 (1)

]Hs

(2)12 , (4.86)

y(2)1 (2) = h

(2)11 (2)

[h

(2)11 (1)

]Hs

(1)11 + h

(2)22 (2)

[h

(1)12 (1)

]Hs

(2)12 . (4.87)

• Similarly, the same operation is repeated in the next time instance t = 3 and the

independent time varying channels ensure that the signal received by the users is

different than that of the previous time instance and is given by:

y(1)1 (3) = h

(1)11 (3)

[h

(2)11 (1)

]Hs

(1)11 + h

(1)12 (3)

[h

(1)12 (1)

]Hs

(2)12 , (4.88)

y(2)1 (3) = h

(2)11 (3)

[h

(2)11 (1)

]Hs

(1)11 + h

(2)22 (3)

[h

(1)12 (1)

]Hs

(2)12 . (4.89)

• We still need another equation since 3 independent streams are transmitted. Repetition

based space time transmission scheme suggests that, we cancel ICI by transmitting the

62


linear combination of the interference as before over independent channels. The signal

received after t = 4 is thus given by:

y(1)1 (4) = h

(1)11 (4)

[h

(2)11 (1)

]Hs

(1)11 + h

(1)12 (4)

[h

(1)12 (1)

]Hs

(2)12 , (4.90)

y(2)1 (4) = h

(2)11 (4)

[h

(2)11 (1)

]Hs

(1)11 + h

(2)22 (4)

[h

(1)12 (1)

]Hs

(2)12 . (4.91)

• Decoding: Decoding is done by canceling the ICI observed by each user. The following

equations show the decoding in the case of user 1 in cell 1 as:y(1)1 (2)− h(1)

12 (2)y(1)1 (1)

y(1)1 (3)− h(1)

12 (3)y(1)1 (1)

y(1)1 (4)− h(1)

12 (4)y(1)1 (1)

=

h

(1)11 (2)

[(h

(2)11 (1)

]H− h(1)

12 (2)[h

(1)11 (1))

]Hh

(1)11 (3)

[(h

(2)11 (1)

]H− h(1)

12 (3)[h

(1)11 (1))

]Hh

(1)11 (4)

[(h

(2)11 (1)

]H− h(1)

12 (4)[h

(1)11 (1))

]H

︸︷︷︸C

s(1)11 , (4.92)

=

h(1)11 (2) −h(1)

12 (2)

h(1)11 (3) −h(1)

12 (3)

h(1)11 (4) −h(1)

12 (4)

︸︷︷︸

3× 2

[(h(2)11 (1)

]H[(h

(1)11 (1)

]H

︸︷︷︸2× 3

s(1)11 . (4.93)

Since s(1)11 is a three dimensional vector, we are unable to decode the data streams because

the rank of the matrix C is only 2 since it is the product of a 3 × 2 and a 2 × 3 matrices

as expressed by (4.93). Hence, for such case, it requires 3 time instances to determine two

equations and another 3 time instances to determine another equation, thus providing no

DOF gain.

However, we observe that for the cases with M = K, we can still achieve greater than

one DOF using the repetition transmission scheme and swapping of the equations required

by each user from other users where they act as an interference signal as explained with an

example case with M = K = 3 in the subsection below.

4.4.6 Case VI (M=3, K=3)

The transmission scheme for M = 3 and K = 3 is divided into the following three phases,

where each phase requires certain number of time instances.

63


Phase I: Transmission of M = K independent data streams intended to each user.

In this phase, each base station transmits independent data streams intended to each user

using one time instance from M antennas. Thus K = 3 users require 3 time instances. The

signal received by each user in the given time instance are as follows:

• At time instance t = 1, both the transmitters transmit all the three independent data

streams intended to the first user in each cell similar to the case of M = 2, K = 2 and

the signal received by all three users in both cell is given by:

y(1)1 (1) = [h

(1)11 (1)]

Hs

(1)11 + [h

(1)12 (1)]

Hs

(2)12 , (4.94)

y(1)2 (1) = [h

(1)21 (1)]

Hs

(1)11 + [h

(1)22 (1)]

Hs

(2)12 , (4.95)

y(1)3 (1) = [h

(1)31 (1)]

Hs

(1)11 + [h

(1)32 (1)]

Hs

(2)12 , (4.96)

y(2)1 (1) = [h

(2)11 (1)]

Hs

(1)11 + [h

(2)12 (1)]

Hs

(2)12 , (4.97)

y(2)2 (1) = [h

(2)21 (1)]

Hs

(1)11 + [h

(2)22 (1)]

Hs

(2)12 , (4.98)

y(2)3 (1) = [h

(2)31 (1)]

Hs

(1)11 + [h

(2)32 (1)]

Hs

(2)12 , (4.99)

where y(`)k (t) is a scalar signal received by a single antenna user k at cell ` during

time instance t, h`kb(t) ∈ C3×1 is the channel vector from base station b to user k in

cell ` during time instance t and s`kb ∈ C3×1 is the independent transmit data vector

transmitted from base station b intended to user k at cell `.

• At time instance t = 2, both the transmitter transmit three independent data streams

intended to the second user in each cell and using the similar notational pattern as

before, the signal received by all the users in both the cells is expressed as:

y(1)1 (2) = [h

(1)11 (2)]

Hs

(1)21 + [h

(1)12 (2)]

Hs

(2)22 , (4.100)

y(1)2 (2) = [h

(1)21 (2)]

Hs

(1)21 + [h

(1)22 (2)]

Hs

(2)22 , (4.101)

y(1)3 (2) = [h

(1)31 (2)]

Hs

(1)21 + [h

(1)32 (2)]

Hs

(2)22 , (4.102)

y(2)1 (2) = [h

(2)11 (2)]

Hs

(1)21 + [h

(2)12 (2)]

Hs

(2)22 , (4.103)

y(2)2 (2) = [h

(2)21 (2)]

Hs

(1)21 + [h

(2)22 (2)]

Hs

(2)22 , (4.104)

y(2)3 (2) = [h

(2)31 (2)]

Hs

(1)21 + [h

(2)32 (2)]

Hs

(2)22 . (4.105)

64


• Similarly, at time instance t = 3, both the transmitters transmit three independent

data streams intended to the third user and the signal observed by each user is given

by:

y(1)1 (3) = [h

(1)11 (3)]

Hs

(1)31 + [h

(1)12 (3)]

Hs

(2)32 , (4.106)

y(1)2 (3) = [h

(1)21 (3)]

Hs

(1)31 + [h

(1)22 (3)]

Hs

(2)32 , (4.107)

y(1)3 (3) = [h

(1)31 (3)]

Hs

(1)31 + [h

(1)32 (3)]

Hs

(2)32 , (4.108)

y(2)1 (3) = [h

(2)11 (3)]

Hs

(1)31 + [h

(2)12 (3)]

Hs

(2)32 , (4.109)

y(2)2 (3) = [h

(2)21 (3)]

Hs

(1)31 + [h

(2)22 (3)]

Hs

(2)32 , (4.110)

y(2)3 (3) = [h

(2)31 (3)]

Hs

(1)31 + [h

(2)32 (3)]

Hs

(2)32 . (4.111)

Phase II: Use of repetition transmission scheme

Repetition transmission scheme as discussed for the case M = K = 2 is used to cancel

ICI observed by the k−th user in cell 1 from the k−th user in cell 2 and obtain B desired

equations per user, where B is the number of cells. This phase requires BK = 6 time

instances. The transmission scheme and the signal received by each user in the given time

instance during this phase are listed below:

• Let L1 = [h(2)11 (1)]Hs

(1)11 be the linear combination of the ICI observed by the first user

in cell 2 due to the transmission from the base station in cell 1 in time instance t = 1

and L2 = [h(1)12 (1)]Hs

(2)12 be the linear combination of the ICI observed by the first user

in cell 1 due to the transmission from the base station in cell 2 at time instance t = 1.

During the time instance t = 4, the base station from cell 1 transmits L1 using a single

antenna and the base station from the cell 2 transmits L2 also using a single antenna.

65


The signal received by all the users is then given by:

y(1)1 (4) = h

(1)11 (4)L1 + h

(1)12 (4)L2, (4.112)

y(1)2 (4) = h

(1)21 (4)L1 + h

(1)22 (4)L2, (4.113)

y(1)3 (4) = h

(1)31 (4)L1 + h

(1)32 (4)L2, (4.114)

y(2)1 (4) = h

(2)11 (4)L1 + h

(2)12 (4)L2, (4.115)

y(2)2 (4) = h

(2)21 (4)L1 + h

(2)22 (4)L2, (4.116)

y(2)3 (4) = h

(2)31 (4)L1 + h

(2)32 (4)L2. (4.117)

• Similarly, let L3 = [h(2)21 (2)]Hs

(1)21 be the linear combination of ICI observed by the second

user in cell 2 due to transmission from the base station in cell 1 during time instance

t = 2 and L4 = [h(1)22 (2)]Hs

(2)22 be the linear combination of ICI observed by the second

user in cell 1 due to transmission fro the base station in cell 2 during time instance

t = 2. During t = 5 base station from cell 1 transmits L3 and the base station from cell

2 transmits L4 using a single antenna and the signal received by all the users is given

by:

y(1)1 (5) = h

(1)11 (5)L3 + h

(1)12 (5)L4, (4.118)

y(1)2 (5) = h

(1)21 (5)L3 + h

(1)22 (4)L4, (4.119)

y(1)3 (5) = h

(1)31 (5)L3 + h

(1)32 (4)L4, (4.120)

y(2)1 (5) = h

(2)11 (5)L3 + h

(2)12 (4)L4, (4.121)

y(2)2 (5) = h

(2)21 (5)L3 + h

(2)22 (4)L4, (4.122)

y(2)3 (5) = h

(2)31 (5)L3 + h

(2)32 (4)L4. (4.123)

• Also let L5 = [h(2)31 (3)]Hs

(1)31 and L6 = h

(1)32 (3)]Hs

(2)32 be the linear combination of ICI

observed by the third user in cell 2 due to transmission from base station in cell 1 and

the linear combination of ICI observed by the third user in cell 1 due to transmission

from base station in cell 2 during t = 3 respectively. During t = 6 the base station from

the cell 1 transmits L5 and the base station from the cell 2 transmits L6 using a single

66


antenna and the signal received by all the users is given by :

y(1)1 (6) = h

(1)11 (6)L5 + h

(1)12 (6)L6, (4.124)

y(1)2 (6) = h

(1)21 (6)L5 + h

(1)22 (4)L6, (4.125)

y(1)3 (6) = h

(1)31 (6)L5 + h

(1)32 (4)L6, (4.126)

y(2)1 (6) = h

(2)11 (6)L5 + h

(2)12 (4)L6, (4.127)

y(2)2 (6) = h

(2)21 (6)L5 + h

(2)22 (4)L6, (4.128)

y(2)3 (6) = h

(2)31 (6)L5 + h

(2)32 (4)L6. (4.129)

• During the next three time instances t = 7, 8, 9, the repetition transmission takes

place and L1, L2 , L3, L4, and L5, L6 are transmitted respectively over the independent

channels and each transmitter use only one antenna. The signal received by all the

users is obtained similarly as in previous three instances, only changing the channel

parameters. For example, the signal received by the first user in cell 1 and cell 2 during

t = 7 is given by:

y(1)1 (7) = h

(1)11 (7)L1 + h

(1)12 (7)L2, (4.130)

y(2)1 (7) = h

(2)11 (7)L1 + h

(2)12 (7)L2. (4.131)

This allows us to solve L1 and L2 in the first user of cell 1 and cell 2 independently,

using the signals received during time instances t = 4 and t = 7 as given by the following

for the first user of cell 1:

L1 =h

(1)12 (7)y1(4)− h(1)

12 (4)y1(7)

h(1)11 (4) + h

(1)11 (7)

, (4.132)

L2 =h

(1)11 (7)y1(4)− h(1)

11 (4)y1(7)

h(1)12 (4) + h

(1)12 (7)

. (4.133)

Similarly, all the users can solve for L1 and L2 using the two available equations. Also,

using the signal observed during the time instance t = 5 and t = 8, all the users can

solve for L3 and L4. Consider the signal received by the second user in each cell during

t = 8 as given by:

y(1)1 (8) = h

(1)11 (8)L3 + h

(1)12 (8)L4, (4.134)

y(2)1 (8) = h

(2)11 (8)L3 + h

(2)12 (8)L4, (4.135)

and L3 and L4 is obtained by the second user in cell 1 by solving the signals received

67


Cell 1 Cell 2

User 1 L1 = [h(2)11 (1)]Hs

(1)11 , L2 = [h

(1)12 (1)]Hs

(2)12 ,

y(1)1 (1)− L2 = [h

(1)11 (1)]Hs

(1)11 y

(2)1 (1)− L1 = [h

(2)12 (1)]

Hs

(2)12

User 2 L3 = [h(2)21 (2)]Hs

(1)21 , L4 = [h

(1)22 (2)]Hs

(2)22 ,

y(1)2 (2)− L4 = [h

(1)21 (2)]Hs

(1)21 y

(2)2 (2)− L3 = [h

(2)22 (2)]

Hs

(2)22

User 3 L5 = [h(2)31 (3)]Hs

(1)31 , L6 = h

(1)32 (3)]Hs

(2)32 ,

y(1)3 (3)− L6 = [h

(1)31 (3)]Hs

(1)31 y

(2)3 (3)− L5 = [h

(2)32 (3)]

Hs

(2)32

Table 4.1: Two desired equations are observed by each user in both the cells after 9 channeluses.

during time instance t = 5 and t = 8 as given by:

L3 =h

(1)12 (8)y1(5)− h(1)

12 (5)y1(8)

h(1)11 (5) + h

(1)11 (8)

, (4.136)

L4 =h

(1)11 (8)y1(5)− h(1)

11 (5)y1(8)

h(1)12 (5) + h

(1)12 (8)

. (4.137)

Using the similar approach all the users in both the cells can solve for L5 and L6. Hence

at the end of 9 channel uses, all users have the knowledge of two desired channels as

shown in table 4.1

Phase III: Swapping of desired signal from the adjacent users and further can-

celing the ICI.

Since two equations per user are not enough to solve three independent streams of data,

we need one more equation per user. The aim of the next phase is to obtain the required

equation in all the users. This is achieved by swapping the required information from the

adjacent users and again canceling the extra ICI observed due to the transmission in both

the cells. This phase requires 6 time instances. The signal received during each time instance

in this phase and the transmission strategy is listed below:

• By swapping the signal observed by the first user during the time instance t = 2 with

the signal observed by the second user during the time instance t = 1 in both the cells,

both the first and the second user obtains the required equations. However, the ICI is

68


Cell 1 Cell 2

User 1 y(1)1 (1) = [h

(1)11 (1)]

Hs

(1)11 + [h

(1)12 (1)]

Hs

(2)12 , y

(2)1 (1) = [h

(2)11 (1)]

Hs

(1)11 + [h

(2)12 (1)]Hs

(2)12 ,

y(1)1 (2) = [h

(1)11 (2)]Hs

(1)21 + [h

(1)12 (2)]Hs

(2)22 , y

(2)1 (2) = [h

(2)11 (2)]

Hs

(1)21 + [h

(2)12 (2)]Hs

(2)22 ,

y(1)1 (3) = [h

(1)11 (3)]Hs

(1)31 + [h

(1)12 (3)]

Hs

(2)32 y

(2)1 (3) = [h

(2)11 (3)]

Hs

(1)31 + [h

(2)12 (3)]Hs

(2)32

User 2 y(1)2 (1) = [h

(1)21 (1)]Hs

(1)11 + [h

(1)22 (1)]

Hs

(2)12 , y

(2)2 (1) = [h

(2)21 (1)]

Hs

(1)11 + [h

(2)22 (1)]Hs

(2)12 ,

y(1)2 (2) = [h

(1)21 (2)]

Hs

(1)21 + [h

(1)22 (2)]

Hs

(2)22 , y

(2)2 (2) = [h

(2)21 (2)]

Hs

(1)21 + [h

(2)22 (2)]

Hs

(2)22 ,

y(1)2 (3) = [h

(1)21 (3)]

Hs

(1)31 + [h

(1)22 (3)]

Hs

(2)32 , y

(2)2 (3) = [h

(2)21 (3)]

Hs

(1)31 + [h

(2)22 (3)]

Hs

(2)32 ,

User 3 y(1)3 (1) = [h

(1)31 (1)]Hs

(1)11 + [h

(1)32 (1)]

Hs

(2)12 , y

(2)3 (1) = [h

(2)31 (1)]

Hs

(1)11 + [h

(2)32 (1)]Hs

(2)12 ,

y(1)3 (2) = [h

(1)31 (2)]

Hs

(1)21 + [h

(1)32 (2)]

Hs

(2)22 , y

(2)3 (2) = [h

(2)31 (2)]

Hs

(1)21 + [h

(2)32 (2)]

Hs

(2)22

y(1)3 (3) = [h

(1)31 (3)]

Hs

(1)31 + [h

(1)32 (3)]

Hs

(2)32 y

(2)3 (3) = [h

(2)31 (3)]

Hs

(1)31 + [h

(2)32 (3)]

Hs

(2)32 .

Table 4.2: The signal received by all the users in both the cells during time instances t = 1, 2, 3.The red color signal represents the signal to be swapped between the first and second user ineach cell and the blue colored signal represents the signal to be swapped between the firstand the third users in each cell.

not completely canceled.

In the next time instance t = 10, the base station from cell 1 transmits [h(1)11 (2)]Hs

(1)21

and[h(1)21 (1)]Hs

(1)11 using two antennas; and also the base station from cell 2 transmits

[h(2)12 (2)]Hs

(2)22 and [h

(2)22 (1)]Hs

(2)12 using two antennas. These signals are shown by red

color in table 4.2. Thus the signal received during time instance t = 10 is given by:

y(1)1 (10) = h

(1)11 (10)[h

(1)11 (2)]Hs

(1)21 + g

(1)11 (10)[h

(1)21 (1)]Hs

(1)11

+ h(1)12 (10)[h

(2)12 (2)]Hs

(2)22 + g

(1)12 (10)[h

(2)22 (1)]Hs

(2)12 , (4.138)

y(1)2 (10) = h

(1)21 (10)[h

(1)11 (2)]Hs

(1)21 + g

(1)21 (10)[h

(1)21 (1)]Hs

(1)11

+ h(1)22 (10)[h

(2)12 (2)]Hs

(2)22 + g

(1)22 (10)[h

(2)22 (1)]Hs

(2)12 , (4.139)

y(1)3 (10) = h

(1)31 (10)[h

(1)11 (2)]Hs

(1)21 + g

(1)31 (10)[h

(1)21 (1)]Hs

(1)11

+ h(1)32 (10)[h

(2)12 (2)]Hs

(2)22 + g

(1)32 (10)[h

(2)22 (1)]Hs

(2)12 , (4.140)

69


y(2)1 (10) = h

(2)11 (10)[h

(1)11 (2)]Hs

(1)21 + g

(2)11 (10)[h

(1)21 (1)]Hs

(1)11

+ h(2)12 (10)[h

(2)12 (2)]Hs

(2)22 + g

(2)12 (10)[h

(2)22 (1)]Hs

(2)12 , (4.141)

y(2)2 (10) = h

(2)21 (10)[h

(1)11 (2)]Hs

(1)21 + g

(2)21 (10)[h

(1)21 (1)]Hs

(1)11

+ h(2)22 (10)[h

(2)12 (2)]Hs

(2)22 + g

(2)22 (10)[h

(2)22 (1)]Hs

(2)12 , (4.142)

y(2)3 (10) = h

(2)31 (10)[h

(1)11 (2)]Hs

(1)21 + g

(2)31 (10)[h

(1)21 (1)]Hs

(1)11

+ h(2)32 (10)[h

(2)12 (2)]Hs

(2)22 + g

(2)32 (10)[h

(2)22 (1)]Hs

(2)12 , (4.143)

where h(1)ij and h

(2)ij are the scalar channels from cell j to user i corresponding to antenna

1 (single antenna) observed in cell 1 and cell 2 respectively and g(1)ij and g

(2)ij are the scalar

channels from cell j to user i corresponding to antenna 2 (single antenna) observed in

cell 1 and cell 2 respectively.

Now, the first and the second user swaps the relevant signal as follows:

(I) First user in cell 1 performs the following operation,

y(1)1 (10)− h(1)

11 (10)y(1)1 (2) = g

(1)11 (10)[h

(1)21 (1)]Hs

(1)11︸︷︷︸

desired

+ g(1)12 (10)[h

(2)22 (1)]Hs

(2)12︸︷︷︸

ICI from the first user

+ h(1)12 (10)[h

(2)12 (2)]Hs

(2)22 − h(1)

11 (10)[h(1)12 (2)]Hs

(2)22︸︷︷︸

ICI from the second user

, (4.144)

which provides a desired signal but ICI from the first and second user in cell 2

exists. We can break [h(2)22 (1)]Hs

(2)12 as:

[h(2)22 (1)]Hs

(2)12 = y

(2)2 (1)− [h

(2)21 (1)]Hs

(1)11 , (4.145)

and substitute in (4.144) to obtain

y(1)1 (10)− h(1)

11 (10)y(1)1 (2) = g

(1)11 (10)[h

(1)21 (1)]Hs

(1)11 − g(1)

12 (10)[h(2)21 (1)]Hs

(1)11︸︷︷︸

desired

+

h(1)12 (10)[h

(2)12 (2)]Hs

(2)22 − h(1)

11 (10)[h(1)12 (2)]Hs

(2)22 + g

(1)12 (10)y

(2)2 (1)︸︷︷︸


.

(4.146)

(II) Second user in cell 1 performs the following operation

y(1)2 (10)− g(1)

21 (10)y(1)2 (1) = h

(1)21 (10)[h

(1)11 (2)]Hs

(1)21︸︷︷︸

desired

+h(1)22 (10)[h

(2)12 (2)]Hs

(2)22︸︷︷︸


+ g(1)22 (10)[h

(2)22 (1)]Hs

(2)12 − g(1)

21 (10)[h(1)22 (1)]Hs

(2)12︸︷︷︸


, (4.147)

70


which can be further simplified by substituting [h(2)12 (2)]Hs

(2)22 as:

[h(2)12 (2)]Hs

(2)22 = y

(2)1 (2)− [h

(2)11 (2)]Hs

(1)21 , (4.148)

to obtain

y(1)2 (10)− g(1)

21 (10)y(1)2 (1) = h

(1)21 (10)[h

(1)11 (2)]Hs

(1)21 − h(1)

22 (10)[h(2)11 (2)]Hs

(1)21︸︷︷︸

desired

+

g(1)22 (10)[h

(2)22 (1)]Hs

(2)12 − g(1)

21 (10)[h(1)22 (1)]Hs

(2)12 + h

(1)22 (10)y

(2)1 (2)︸︷︷︸


.

(4.149)

(III) First user in cell 2 performs the following operation

y(2)1 (10)− h(2)

12 (10)y(2)1 (2) = g

(2)12 (10)[h

(2)22 (1)]Hs

(2)12︸︷︷︸

desired

+ g(2)11 (10)[h

(1)21 (1)]Hs

(1)11︸︷︷︸


+ h(2)11 (10)[h

(1)11 (2)]Hs

(1)21 − h(2)

12 (10)[h(2)11 (2)]Hs

(1)21︸︷︷︸


, (4.150)


(1)11 as:

[h(1)21 (1)]Hs

(1)11 = y

(1)2 (1)− [h

(1)22 (1)]Hs

(2)12 , (4.151)

to obtain

y(2)1 (10)− h(2)

12 (10)y(2)1 (2) = g

(2)12 (10)[h

(2)22 (1)]Hs

(2)12 − g(2)

11 (10)[h(1)22 (1)]Hs

(2)12︸︷︷︸

desired

+

h(2)11 (10)[h

(1)11 (2)]Hs

(1)21 − h(2)

21 (10)[h(2)11 (2)]Hs

(1)21 + g

(2)11 (10)y

(1)2 (1)︸︷︷︸


.

(4.152)

(IV) Similarly, the second user in cell 2 performs the following operation:

y(2)2 (10)− g(2)

22 (10)y(2)2 (1) = h

(2)22 (10)[h

(2)12 (2)]Hs

(2)22︸︷︷︸

desired

+h(2)21 (10)[h

(1)11 (2)]Hs

(1)21︸︷︷︸


+ g(2)21 (10)[h

(1)21 (1)]Hs

(1)11 − g(2)

22 (10)[h(2)21 (1)]Hs

(1)11︸︷︷︸


, (4.153)


(1)21 as:

[h(1)11 (2)]Hs

(1)21 = y

(1)1 (2)− [h

(1)12 (2)]Hs

(2)22 , (4.154)

71


to obtain

y(2)2 (10)− g(2)

22 (10)y(2)2 (1) = h

(2)22 (10)[h

(2)12 (2)]Hs

(2)22 − h(2)

21 (10)[h(1)12 (2)]Hs

(2)22︸︷︷︸

desired

+

g(2)21 (10)[h

(1)21 (1)]Hs

(1)11 − g(2)

22 (10)[h(2)21 (1)]Hs

(1)11 + h

(2)21 (10)y

(1)1 (2)︸︷︷︸


.

(4.155)

• In the next time instance t = 11, transmit the signal

U11 = g(2)21 (10)[h

(1)21 (1)]Hs

(1)11 − g(2)

22 (10)[h(2)21 (1)]Hs

(1)11 + h

(2)21 (10)y

(1)1 (2)

from the base station in cell 1 using a single antenna and transmit the signal

U22 = h(1)12 (10)[h

(2)12 (2)]Hs

(2)22 − h(1)

11 (10)[h(1)12 (2)]Hs

(2)22 + g

(1)12 (10)y

(2)2 (1)

from the base station in cell 2 using single antenna. This cancels all the ICI observed

by the first user in cell 1 and the the second user in the cell 2. This requires some extra

previous signal received to be known at the receiver which can be easily obtained like

the CSIT. The signal received by the first user in cell 1 and the second user in cell 2

during t = 11 is given by:

y(1)1 (11) = h

(1)11 (11)U11 + h

(1)12 (11)U22, (4.156)

y(2)2 (11) = h

(2)21 (11)U11 + h

(2)22 (11)U22. (4.157)

Next, the first user in cell 1 and the second user in cell 2 can cancel the observed ICI

as follows:

y(1)1 (11)− h(1)

12 (11)(y

(1)1 (10)− h(1)

11 (10)y(1)1 (2)

)=h

(1)11 (11)U11 − h(1)

12 (11)(g

(1)11 (10)[h

(1)21 (1)]Hs

(1)11 − g(1)

12 (10)[h(2)21 (1)]Hs

(1)11

)=h

(1)11 (11)

[g

(2)21 (10)[h

(1)21 (1)]H − g(2)

22 (10)[h(2)21 (1)]H

]s

(1)11 − h(1)

12 (11)[g

(1)11 (10)[h

(1)21 (1)]H

− g(1)12 (10)[h

(2)21 (1)]H

]s

(1)11 + h

(1)11 (11)h

(2)21 (10)y

(1)1 (2). (4.158)

72


Thus,

y(1)1 (11)− h(1)

12 (11)(y

(1)1 (10)− h(1)

11 (10)y(1)1 (2)

)− h(1)

11 (11)h(2)21 (10)y

(1)1 (2)

=h(1)11 (11)

[g

(2)21 (10)[h

(1)21 (1)]H − g(2)

22 (10)[h(2)21 (1)]H

]s

(1)11 − h(1)

12 (11)[g

(1)11 (10)[h

(1)21 (1)]H

− g(1)12 (10)[h

(2)11 (1)]H

]s

(1)11

=[(h

(1)11 (11)g

(2)21 (10)− h(1)

12 (11)g(1)11 (10)

)[h

(1)21 (1)]H +

(h

(1)12 (11)g

(1)12 (10) −

h(1)11 (11)g

(2)22 (10)

)[h

(2)21 (1)]H

]s

(1)11 (4.159)

is the third equation required by the first user in cell 1.

Similarly,

y(2)2 (11)− h(2)

21 (11)(y

(2)2 (10)− g(2)

22 (10)y(2)2 (1)

)=h

(2)22 (11)U22 − h(2)

21 (11)(h

(2)22 (10)[h

(2)12 (2)]Hs

(2)22 − h(2)

21 (10)[h(1)12 (2)]Hs

(2)22

)=h

(2)22 (11)

[h

(1)12 (10)[h

(2)12 (2)]H − h(1)

11 (10)[h(1)12 (2)]H

]s

(2)22 − h(2)

21 (11)[h

(2)22 (10)[h

(2)12 (2)]H

− h(2)21 (10)[h

(1)12 (2)]H

]s

(2)22 + h

(2)22 (11)g

(1)12 (10)y

(2)2 (1)), (4.160)

and

y(2)2 (11)− h(2)

21 (11)(y

(2)2 (10)− g(2)

22 (10)y(2)2 (1)

)− h(2)

22 (11)g(1)12 (10)y

(2)2 (1))

=h(2)22 (11)

[h

(1)12 (10)[h

(2)12 (2)]H − h(1)

11 (10)[h(1)12 (2)]H

]s

(2)22 − h(2)

21 (11)[h

(2)22 (10)[h

(2)12 (2)]H

− h(2)21 (10)[h

(1)12 (2)]H

]s

(2)22 ,

=[(h

(2)22 (11)h

(1)12 (10)− h(2)

21 (11)h(2)22 (10)

)[h

(2)12 (2)]H +

(h

(2)21 (11)h

(2)21 (10)−

h(2)22 (11)h

(1)11 (10)

)[h

(1)12 (2)]H

]s

(2)22 , (4.161)

is the third equation required by the second user in cell 2.

• In the next time instance t = 12, the second user in cell 1 and the first user in cell 2

cancel the ICI observed by each of them by transmitting U21 from the base station in

cell 1 and U12 from the base station in cell 2 using a single antenna where

U21 = h(2)11 (10)[h

(1)11 (2)]Hs

(1)21 − h(2)

21 (10)[h(2)11 (2)]Hs

(1)21 + g

(2)11 (10)y

(1)2 (1),

U12 = g(1)22 (10)[h

(2)22 (1)]Hs

(2)12 − g(1)

21 (10)[h(1)22 (1)]Hs

(2)12 + h

(1)22 (10)y

(2)1 (2),

and the signal received by the second user in cell 1 and the first user in cell 2 during

73


t = 12 is given by:

y(1)2 (12) = h

(1)21 (12)U21 + h

(1)22 (12)U12, (4.162)

y(2)1 (12) = h

(2)11 (12)U21 + h

(2)12 (12)U12. (4.163)

Hence, the second user in cell 1 and the first user in cell 2 determines the required third

equation by canceling the ICI as follows:

y(1)2 (12)− h(1)

22 (12)(y

(1)2 (10)− g(1)

21 (10)y(1)2 (1)

)=h

(1)21 (12)U21 − h(1)

22 (12)(h

(1)21 (10)[h

(1)11 (2)]Hs

(1)21 − h(1)

22 (10)[h(2)11 (2)]Hs

(1)21

)=h

(1)21 (12)

[h

(2)11 (10)[h

(1)11 (2)]H − h(2)

21 (10)[h(2)11 (2)]H

]s

(1)21 − h(1)

22 (12)[h

(1)21 (10)[h

(1)11 (2)]H

− h(1)22 (10)[h

(2)11 (2)]H

]s

(1)21 + h

(1)21 (12)g

(2)11 (10)y

(1)2 (1), (4.164)

which is further simplified as:

y(1)2 (12)− h(1)

22 (12)(y

(1)2 (10)− g(1)

21 (10)y(1)2 (1)

)− h(1)

21 (12)g(2)11 (10)y

(1)2 (1)

=h(1)21 (12)

[h

(2)11 (10)[h

(1)11 (2)]H − h(2)

21 (10)[h(2)11 (2)]H

]s

(1)21 − h(1)

22 (12)[h

(1)21 (10)[h

(1)11 (2)]H

− h(1)22 (10)[h

(2)11 (2)]H

]s

(1)21

=[(h

(1)21 (12)h

(2)11 (10)− h(1)

22 (12)h(1)21 (10)

)[h

(1)11 (2)]H +

(h

(1)22 (12)h

(1)22 (10)−

h(1)21 (12)h

(2)21 (12)

)[h

(2)11 (2)]H

]s

(1)21 . (4.165)

Also,

y(2)1 (12)− h(2)

11 (12)(y

(2)1 (10)− h(2)

12 (10)y(2)1 (2)

)=h

(2)12 (12)U12 − h(2)

11 (12)(g

(2)12 (10)[h

(2)22 (1)]Hs

(2)12 − g(2)

11 (10)[h(1)22 (1)]Hs

(2)12

)=h

(2)12 (12)

[g

(1)22 (10)[h

(2)22 (1)]H − g(1)

21 (10)[h(1)22 (1)]H

]s

(2)12 − h(2)

11 (12)[g

(2)12 (10)[h

(2)22 (1)]H

− g(2)11 (10)[h

(1)22 (1)]H

]s

(2)12 + h

(2)12 (12)h

(1)22 (10)y

(2)1 (2), (4.166)

74


which is further simplified as:

y(2)1 (12)− h(2)

11 (12)(y

(2)1 (10)− h(2)

12 (10)y(2)1 (2)

)− h(2)

12 (12)h(1)22 (10)y

(2)1 (2)

=h(2)12 (12)

[g

(1)22 (10)[h

(2)22 (1)]H − g(1)

21 (10)[h(1)22 (1)]H

]s

(2)12 − h(2)

11 (12)[g

(2)12 (10)[h

(2)22 (1)]H

− g(2)11 (10)[h

(1)22 (1)]H

]s

(2)12

=[(h

(2)12 (12)g

(1)22 (10)− h(2)

11 (12)g(2)12 (10)

)[h

(2)22 (1)]H +

(h

(2)11 (12)g

(2)11 (10)−

h(2)12 (12)g

(1)22 (10)

)[h

(1)22 (1)]H

]s

(2)12 . (4.167)

Thus, after 12 channel uses all the four users have the required three equations to solve three

desired data streams. In order to obtain the third equation for the third user in both the cells,

same approach of swapping of desired signal and canceling of ICI is used. Let us assume that

the swapping takes place between the first user and the third user in both the cells. Note,

here that the swapping can also take place between the second and the third users in both

the cells.

Following all the approaches as we considered before for the swapping between the first

and second users in both the cells, we require next three time instances to obtain the third

equation for the third user in both cells. The first time instance t = 13 transmits the linear

combination of IUI from the first user and the third user in both cells. Next time instance

t = 14 is used to cancel ICI and obtain the third equation for the third user in cell 2 and

similarly other time instance t = 15 is used to obtain the third equation for the third user in

cell 1. The following are the third equations observed by the third user in both cells.

y(2)3 (14)− h(2)

31 (14)(y

(2)3 (13)− g(2)

33 (13)y(2)3 (1)

)− h(2)

33 (14)g(1)13 (10)y

(2)3 (1))

=[(h

(2)33 (14)h

(1)13 (13)− h(2)

31 (14)h(2)33 (13)

)[h

(2)13 (2)]H +

(h

(2)31 (14)h

(2)31 (13)−

h(2)32 (14)h

(1)11 (13)

)[h

(1)13 (2)]H

]s

(2)32 , (4.168)

and

y(1)3 (15)− h(1)

33 (15)(y

(1)3 (13)− g(1)

31 (13)y(1)3 (1)

)− h(1)

31 (15)g(2)11 (13)y

(1)3 (1)

=[(h

(1)31 (15)h

(2)11 (13)− h(1)

33 (15)h(1)31 (13)

)[h

(1)11 (2)]H +

(h

(1)33 (15)h

(1)33 (13)−

h(1)31 (15)h

(2)31 (15)

)[h

(2)11 (2)]H

]s

(1)31 . (4.169)

Thus, we show that in a two-cell MISO BC with three users per cell, we can decode total

of 18 streams of data in just 15 time instances only using the delayed CSIT and very few

75


delayed signal from previous time instances. The achievable DOF is 1815

= 65, which is greater

than 1 even though there are enough ICI and IUI. All three desired equations for each user

in both the cells are tabulated in the table 4.3.

4.4.7 Case VII (Generalized Scheme for M = K)

The three phase proposed scheme is valid for all the general number of users K such that

M = K as in the case of K = 3. In the general scheme, the first phase requires K time

instances to transmit 2K2 independent data streams from two cells such that K independent

data streams are intended to each user. Similarly, the second phase or the repetition phase

requires another 2K time instances to obtain 2 independent desired equations per user.

Next, each user needs to obtain K − 2 desired equations by swapping the desired streams

from the adjacent users. As observed for K = 3 case, we can only swap between two adjacent

users in the same cell at a time and each swapping provides one independent desired equation.

But there occurs the ICI from the adjacent cell and require two more time instances to cancel

all the ICI observed by both the users in both the cells. Thus, 4 users require three time

instances to obtain one desired equation. In other words, four desired equations are obtained

in three time instances.

Since there are 2K users and K − 2 desired equation per user, the total of 2K(K − 2)

desired equations are obtained in 34× 2K(K − 2) = 3× K(K−2)

2. However, we observed that

if K(K − 2) is not divisible by 2, it requires another extra three time instances as in the case

of K = 3. Hence, we can express the total time required in the third phase as :

t3 = 3×⌈K(K − 2)

2

⌉.

Based on this observation, we make a following conjecture on the achievable DOF of a

two-cell MISO IBC with delayed CSIT:

Conjecture 4.1. The total achievable DOF of a two-cell MISO-IBC with M transmit anten-

nas and K users in each cell, such that M = K, with the proposed space-time transmission

scheme is given by

dtot =2K2

K + 2K + 3⌈K(K−2)

2

⌉ . (4.170)

Thus, we observe here that if K is even (divisible by 2), the achievable DOF is

dtot =2K2

3K(1 + K−22

)=

4

3, (4.171)

independent of the number of users.

76


Cell 1 Cell 2

User 1 1.) L1 = [h(2)11 (1)]

Hs(1)11 , 1.) L2 = [h

(1)12 (1)]

Hs(2)12 ,

2.) y(1)1 (1)− L2 = [h

(1)11 (1)]

Hs(1)11 , 2.) y

(2)1 (1)− L1 = [h

(2)12 (1)]

Hs(2)12 ,

3.) y(1)1 (11)− h

(1)12 (11)

(y(1)1 (10)− h

(1)11 (10)y

(1)1 (2)

)3.) y

(2)1 (12)− h

(2)11 (12)

(y(2)1 (10)− h

(2)12 (10)y

(2)1 (2)

)−h(1)

11 (11)h(2)21 (10)y

(1)1 (2) =

[(h(1)11 (11)g

(2)21 (10) −h(2)

12 (12)h(1)22 (10)y

(2)1 (2)

)=[(

h(2)12 (12)g

(1)22 (10)

−h(1)12 (11)g

(1)11 (10)

)[h

(1)21 (1)]

H +(h(1)12 (11)g

(1)12 (10) −h(2)

11 (12)g(2)12 (10)

)[h

(2)22 (1)]

H +(h(2)11 (12)g

(2)11 (10)

−h(1)11 (11)g

(2)22 (10)

)[h

(2)21 (1)]

H]s(1)11 −h(2)

12 (12)g(1)22 (10)

)[h

(1)22 (1)]

H]s(2)12

User 2 1.) L3 = [h(2)21 (2)]

Hs(1)21 , 1.) L4 = [h

(1)22 (2)]

Hs(2)22 ,

2.) y(1)2 (2)− L4 = [h

(1)21 (2)]

Hs(1)21 , 2.) y

(2)2 (2)− L3 = [h

(2)22 (2)]

Hs(2)22 ,

3.) y(1)2 (12)− h

(1)22 (12)

(y(1)2 (10)− g

(1)21 (10)y

(1)2 (1)

)3.) y

(2)2 (11)− h

(2)21 (11)

(y(2)2 (10)− g

(2)22 (10)y

(2)2 (1)

)−h(1)

21 (12)g(2)11 (10)y

(1)2 (1) =

[(h(1)21 (12)h

(2)11 (10) −h(2)

22 (11)g(1)12 (10)y

(2)2 (1)) =

[(h(2)22 (11)h

(1)12 (10)

−h(1)22 (12)h

(1)21 (10)

)[h

(1)11 (2)]

H +(h(1)22 (12)h

(1)22 (10) −h(2)

21 (11)h(2)22 (10)

)[h

(2)12 (2)]

H +(h(2)21 (11)h

(2)21 (10)

−h(1)21 (12)h

(2)21 (12)

)[h

(2)11 (2)]

H]s(1)21 −h(2)

22 (11)h(1)11 (10)

)[h

(1)12 (2)]

H]s(2)22

User 3 1.) L5 = [h(2)31 (3)]

Hs(1)31 , 1.) L6 = h

(1)32 (3)]

Hs(2)32 ,

2.) y(1)3 (3)− L6 = [h

(1)31 (3)]

Hs(1)31 , 2.) y

(2)3 (3)− L5 = [h

(2)32 (3)]

Hs(2)32 ,

3.) y(1)3 (15)− h

(1)33 (15)

(y(1)3 (13)− g

(1)31 (13)y

(1)3 (1)

)3.) y

(2)3 (14)− h

(2)31 (14)

(y(2)3 (13)− g

(2)33 (13)y

(2)3 (1)

)−h(1)

31 (15)g(2)11 (13)y

(1)3 (1) =

[(h(1)31 (15)h

(2)11 (13) −h(2)

33 (14)g(1)13 (10)y

(2)3 (1)) =

[(h(2)33 (14)h

(1)13 (13)

−h(1)33 (15)h

(1)31 (13)

)[h

(1)11 (2)]

H +(h(1)33 (15)h

(1)33 (13) −h(2)

31 (14)h(2)33 (13)

)[h

(2)13 (2)]

H +(h(2)31 (14)h

(2)31 (13)

−h(1)31 (15)h

(2)31 (15)

)[h

(2)11 (2)]

H]s(1)31 h

(2)32 (14)h

(1)11 (13)

)[h

(1)13 (2)]

H]s(2)32

Table 4.3: Three desired equations are observed by each user in both the cells after 15 timeinstances. All users can solve three independent desired data streams using these threeequations.

77

Improved Scheme

If K is odd (not divisible by 2), K − 2 is also odd and K(K−2)2

is not divisible by 2. Thus⌈K(K−2)2

⌉can be simplified as⌈K(K − 2)

2

⌉=K(K − 2)− 1

2+ 1 =

K2 − 2K + 1

2,

and

dtot =2K2

3K + 3K2−2K+1

2

=4K2

3K2 + 3,

=4

3(1 + 1K2 )→ 4

3as K →∞. (4.172)

However, in all the cases, the achievable DOF is always greater than 1 and this fact is

depicted in Figure 4.3, where the achievable DOF is plotted with varying number of users

when the number of users are odd and even. We plot the achievable DOF with instantaneous

CSIT as observed in [72] and the achievable DOF using the proposed scheme with delayed

CSIT.

4.5 Improved Scheme

Further, we observe that the lower DOF achieved for the odd values of K is due to extra

time instances required during the third phase (swapping and ICI cancellation phase) and by

improving the transmission scheme, we can also achieve 43

and better DOF when K ≥ 3.

Let us again consider the case of K = 3 where we observe that during the swapping and

ICI cancellation phase the first user in each cell requires only one desired equation but obtains

two desired equation because it does the swapping with the the second user and also with the

third user in each cell. By using these extra equations, we can improve the achievable DOF.

By the improved transmission strategy, we perform following improvements in the respective

phases:

Phase I:

We transmit another three independent data streams intended to each user using extra three

time instances. Another independent data streams intended to user 1, user 2 and user 3 are

transmitted during the time instances t = 4, t = 5, and t = 6. Hence, the first phase now

requires 6 time instances.

78

Improved Scheme

0 5 10 15 20 25 30 35 40 45 50

0.8

1

1.2

1.4

1.6

1.8

2

Achie

vable

DO

F

0 5 10 15 20 25 30 35 40 45 501.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Achie

vable

DO

F

DoF of two cell MISO IBC.(M = K, K is odd)

DoF of two cell MISO IBC.(M = K, K is even)

Number of Users (K)

Number of Users (K)

Delayed CSIT

Delayed CSIT

Instantaneous CSIT

Instantaneous CSIT

Figure 4.3: DOF of two-cell MISO IBC with delayed and instantaneous CSIT for odd andeven number of users. For odd number of users, the DOF converges to 4

3and for the even

number of users achievable DOF is always 43.

79

Improved Scheme

Phase II:

The second phase is the repetition scheme and also requires 6 time instances. Like in the

previous example, the linear combination of ICI observed by the first user in cell 2 due to

base station in cell 1 during t = 1 and the linear combination of all the ICI observed by

the first user in cell 1 due to base station in cell 2 during t = 1 is transmitted during t = 7.

Similarly, the linear combination of all the ICI observed by the second user in cell 2 due to

base station in cell 1 and the linear combination of all the ICI observed by the second user

in cell 1 due to the base station in cell 2 during t = 2 is transmitted in the time instance

t = 8 and the linear combination of all the ICI observed by the third user in both the cells

during t = 3 is transmitted in the time instance t = 9. Same transmission is repeated for

another three time instance. This allows us to obtain two equations observed by each user in

both the cells as described before. The only difference is that each equation has six variables

instead of three.

Phase III:

In this phase, we need to obtain four more equations instead of only one as in the previous

case. The swapping and cancellation of ICI between any two users within the cell in both

the cells require three time instances as observed previously. For example, the swapping and

ICI cancellation between the first and the second user in both the cells require three time

instances and provide one more equation to the first and second user in both cells. Similarly

the swapping and ICI cancellation between the first and the third user require another three

time instance and so does the swapping between the second and the third user in both the

cells. Thus, after 9 more time instances, all the users in both the cells have 2 more desired

equations.

Phase IV:

After the third phase, each user still requires two more equations. However, all the users

already have the linear combination of all the required equations as observed in the third

phase. Hence, we easily observe that by transmitting a linear combination of the desired

streams per user, we do not need more than another 6 time instances to obtain another 2

equations per user. Thus, we require total of 27 time instances to decode all 36 data streams,

which allows us to achieve 3627

= 43

total DOF.

80

Improved Scheme

For the general case with K users, following are the time instances required during each

phase:

(I) Phase I: It requires total of 2K time instances to transmit 4K2 data streams from two

cells.

(II) Phase II: This repetition phase also requires 2K time instances as discussed in the

previous example and two desired equations are obtained per user.

(III) Phase III: In this swapping and ICI cancellation phase, all the desired equations are

swapped between the adjacent users. Since there are K users and each time swapping

occurs between any two users, the total number of possible swapping is

ns =

(K

2

)=

K!

(K − 2)!2!=K(K − 1)

2,

As observed previously, each swapping and ICI cancellation requires 3 time instances,

ns swapping requires 3 × ns time instances and each user obtains all K − 1 desired

equations from the adjacent users.

(IV) Phase IV: Each user has already obtained 2+(K−1) = K+1 equations from the second

and third phase and at the same time these users also have the linear combination of

all the desired equations during the third phase. Since 2K streams were transmitted

per user, 2K − (K + 1) = K − 1 desired equations per user are to be obtained in the

fourth phase. We observe that all the K users require K(K − 1) time instances to do

so.

Hence, total time instances required during all the phases are

ttot = 2K + 2K +3K(K − 1)

2+K(K − 1)

= K(4 +

5

2(K − 1)

)=K

2(3 + 5K). (4.173)

Now, we can express the total achievable DOF as:

dtot =4K2

K2

(3 + 5K)

=8K

3 + 5K→ 8

5as K →∞. (4.174)

This improved achievable DOF for K ≥ 3 is depicted in Figure 4.4, which shows that the

DOF converges to 85

as K → ∞. The achievable DOF with instantaneous CSIT as given

by [72] is also plotted which converges to 2 as K →∞.

81

Improved Scheme

0 5 10 15 20 25 30 35 40 45 501.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Achie

va

ble

DO

F

DoF of two cell MISO IBC.(M = K,K ≥ 3)

Number of Users (K)

Delayed CSITInstantaneous CSITDelayed CSIT (Improved)

Figure 4.4: Achievable DOF for two cell MISO IBC with delayed and instantaneous CSIT.The improved transmission scheme achieves better DOF which converges to 8

5unlike the

earlier approach that converges to 43. The DOF with the instantaneous CSIT converges to 2.

82

Conclusions

4.6 More Than Two-Cell MISO-IBC

As the number of cells increase, the ICI increase considerably and it becomes difficult to align

the interference in the absence of instantaneous CSIT. Even in the presence of instantaneous

CSIT, the multi-cell MISO is difficult to analyze because of the ICI and IUI. The DOF

analysis with delayed CSIT for B ≥ 3, where B is the number of cells, is an open problem

and an effective transmission scheme that provides a DOF gain greater than 1 is not known.

Hence, we leave the DOF analysis for multi-cell MISO IBC for B ≥ 3 with delayed analysis

as a future work.

Other important topic which has been less discovered is the multi-cell MIMO BC or MIMO-

IBC. Only a few works such as [73, 76–78] are well-known on MIMO IBC with instantaneous

CSIT and lately a few works such as [79,80] has been done with delayed CSIT. In the next

section, we present an interference alignment scheme in MIMO-IBC, where the users within

a cell or within a cluster collectively align the observed ICI onto a common subspace.

4.7 Conclusions

The space time transmission scheme to obtain the DOF of two-cell MISO IBC with delayed

CSIT is presented, mainly for the cases when the number of transmit antennas M is equal

to the number of user K in each cell. A detailed scheme is presented for M = K = 2 and

M = K = 3 and the achievable DOF measured in terms of ratio of number of independent

data streams to the number of time instances required is observed to be greater than one.

We presented an improved transmission scheme which shows that the DOF of two cell MISO

IBC converges to 85

as the number of users K increases. We also observed that it becomes

much harder to define a transmission scheme due to the introduction of ICI and occurs large

amount of delay to decode all the desired streams.

83

Conclusions

84

Chapter 5

Interference Alignment and Optimal

Overlapping in MIMO IBC and IC

In this chapter, we present an interference alignment (IA) scheme where a group of users

within a cell or within a cluster collectively align all the unwanted signal or interference onto

a common subspace. We assume that such group exists within a cell, where all the users

collectively assign a common overlap subspace to the observed inter cell interference (ICI)

such as in multiple input multiple output (MIMO) interference broadcast channel (IBC) or

the number of users in MIMO interference channel (IC) form different topological structures

to overlap the unwanted signal onto a common subspace.

We term such IA technique of aligning the interference from a same source onto a common

subspace by different users, as vertical alignment, which is unlike the conventional horizontal

alignment where the same user align all the interference coming from different sources in a

common subspace. This approach of vertical IA is not well studied in the literature except

few slightly different version of such scheme presented in [77] and [78].

One of the major challenges in the vertical IA is the determination of the common

overlapping subspace and the optimal dimensions of this overlapping subspace. In this

work, we present a scheme to determine the overlap subspace and also the dimensions of

optimal overlapping required between any two or more users to create such an overlap. We

observe that the common subspace (overlap region) exists between the users only if they

share common spatial dimensions.

The next generation communication in a high speed train and local buses or in a crowded

stadium can be a combination of wired plus wireless communication. All users can plug in

certain device, which allows all the users get connected and also provide all of them with

the common spatial dimensions. However, all the users observe independent channels from

the wireless transmitters at the same time. The concept of vertical alignment and the idea

of optimal overlapping plays a pivotal role in such scenarios where each user observes the

common channel and the private channels.

85

MIMO Interference Broadcast Channels

Some of the antennas in each users are dedicated to common channels and other to the

private channels. We assume that the knowledge of which antennas are common and which

are private are given by the topological matrices Si, ∀i = {1, 2, · · · , K}, where K is the

number of users. Si also determines how the antennas are topologically shared among the

users. In this regard, we also propose a ‘ring topological sharing scheme’ and ‘star topological

sharing scheme’ for the case of three user MIMO IC.

Next, we describe the idea of vertical alignment and optimal overlapping in two-cell MIMO

channels with two users per cell, where each user receive the ICI from the adjacent cell.

5.1 MIMO Interference Broadcast Channels

5.1.1 System Model

Consider the case of two-cell MIMO-IBC as depicted in the Figure 5.1, where two cells each

having a base station with M antennas serve two users with N antennas in each cell. Each

user observes both the inter user interference (IUI) and the ICI, which requires the precoders

Vj =[V1j V2j

]and post-processors Ukj,∀k = {1, 2} be properly designed at any cell j so

as to align both IUI and ICI as much as possible.

Here, all the users within a cell (two) are assumed to form a group and they can collectively

align the ICI observed from the unwanted transmitter onto a common subspace, denoted by

Gj for any cell j, ∀j = {1, 2}. In order to get a clear picture of the idea, let us consider there

are M = 6 transmit antennas in each base station and N = 4 receive antennas per user and

each transmitter is transmitting d = 3 independent data streams intended to both the users.

We assume the data streams are intended to both users just for simplicity. However different

users require different data streams from the transmitter.

This causes ICI of d = 3 dimensions to be observed by each user. Since there are N = 4

antennas per user, each user can resolve only N − d = 1 data streams. However, if two users

align all the ICI in a common d = 3 dimensional space, even with N = 4 antennas per user

total of 5 data streams can be resolved and this is a considerable gain from total of 2 data

streams in the absence of overlapping.

86


BS1

BS2

1

2

1

2

cell 1

cell 2

H11(1)

H21 (1)

H11 (2)H

21 (2)H 12

(1)

H 22(1)

H12(2)

H22 (2)

U11

U12

U21

U22

V1 = [V11 V21]

V2= [ V12 V22]

Figure 5.1: Two cell MIMO Broadcast Channel with two users per cell and arbitrary Mantennas at each base station and N antennas per user. The solid line represents the desiredsignal plus IUI, while the dashed line represents the ICI observed by each user. The channelsfrom base station j to user k in cell ` is represented as H

(`)kj , while Vj are the precoding

matrices and Uk` are the interference suppressing matrices or the post processors at user kin cell `.

87


5.1.2 Designing Post-processing Matrices

As discussed earlier, the challenge is again to determine the overlapped subspace and the

overlapped dimensions. Let Ukj,∀k = {1, 2} at any cell j be the orthonormal basis for the

projected signal space by user k, then both the users align the ICI onto the common subspace

which are spanned over the same dimensions. Hence, for the given two-cell system, the users

in cell 1 align the ICI observed from cell 2 onto a common subspace G1 such that

span(UH

11H(1)12 V2

)= span

(UH

21H(1)22 V2

)= G1. (5.1)

The common subspace region only exists when there is enough overlap between the signals

observed by the two users. Otherwise, the users obtain interference individually and can

not align them collectively. One of the ideas to create an overlap space is to assume that

the interference channel observed by both the users is correlated. We assume that there is a

strong correlation and consider that the two channels H(1)12 ∈ CN×M and H

(1)22 ∈ CN×M are

the sub-matrices of another bigger channel matrix H(1)2 ∈ C2N×M with enough overlapping

such that the common overlap space G1 exists between the first and second user in the cell 1.

Thus, the interference observed by both the users is a part of H(1)2 V2 as

H(1)12 V2 = S1H

(1)2 V2, (5.2)

H(1)22 V2 = S2H

(1)2 V2, (5.3)

where S1 ∈ R(N+No/2)×2N and S2 ∈ R(N+No/2)×M are considered as the 0 − 1 selection

matrices and No is the total number of overlapping rows.

If G1 exists then we can design U11 and U21 from (5.1), (5.2) and (5.3) and using the

property span(A) = span(AAH) [81] as

span(UH

11S1PSH1 U11

)= span

(UH

21S2PSH2 U21

), (5.4)

where

P = H(1)2 V2V

H2 H

(1)2

H. (5.5)

We can further express (5.4) as

UH11S1PSH1 U11 = UH

21S2PSH2 U21, (5.6)

⇒UH11S1PSH1 U⊥11 −UH

21S2PSH2 U⊥21 = 0, (5.7)

⇒[UH

11 UH21

] [S1PSH1 0

0 −S2PSH2

][U11

U21

]= 0, (5.8)

⇒UH1 ∆U1 = 0, (5.9)

88


where

U1 =

[U11

U21

],

∆ =

[S1PSH1 0

0 −S2PSH2

],

and we can determine U⊥1 by determining the null space of ∆, i.e., U1 ∈ N (∆).

For the given P and Sk,∀k = {1, 2}, U1 exists only if N (∆) exists. Since the dimensions of

P depends on the dimensions of V2, which is the number of interference streams transmitted

to both the users in cell 1 from cell 2, this dimension can vary up to min(M, 2N). The

N (∆) does not exist when ∆ is a full column rank matrix, but the overlapping ensures that

N (∆) always exists. Thus, we can determine U1 and also the basis of overlap subspace.

For different dimensions of V2, the rank of N (∆) and the available overlap dimensions are

plotted in the upper part and the lower part of the Figure 5.2 respectively.

The plot on the upper part of Figure 5.2 clearly shows that when the dimensions (d) of

V2 is less than N , N (∆) still exists even with no overlapping rows. This is because there

are N − d free dimensions available at each user independent of the d dimensions. The plot

on the lower part of Figure 5.2 shows that the common region or the overlap space exists

only when there are overlapping rows between the channels. We also observe that the N (∆)

contains the overlapped region for all d ≥ N .

5.1.3 Designing the Precoding Matrices

Similarly, the users in cell 2 design the respective post-processing or interference suppressing

matrices U12 and U22. Based on the knowledge of all the post-processing matrices Ukj, the

precoding matrix Vj is updated such that Vj aligns and zero forces all ICI. If we consider

V2 that zero forces all ICI, such V2 is contained in the null space of Φ, where

Φ = SH1 U11UH11S1 + SH2 U21U

H21S2, (5.10)

is a modified version of the so called alignment matrix as defined in [47,82–84]. The null

space of Φ, hereafter denoted N (Φ), is interesting because it allows all the interference

streams to be aligned on the common subspace. If the dimension of N (Φ) is greater than the

dimension of the allocated streams d, the overlap space affects the available free dimensions.

Hence the optimum dimension of the overlap space is that number of overlapping rows when

rank(N (Φ)

)= d. (5.11)

89


0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

10

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

Dim. of N (∆) increasing with Overlapping Rows

(M = 6, N = 4, K = 2)

Dim. of Overlap space increasing with Overlapping Rows

(M = 6, N = 4, K = 2)

Number of overlapping rows


Ran

kof

N(∆

)Ran

kof

Overlap

Space

d = 3

d = 3

d = 4

d = 4

d = 5

d = 5

d = 6

d = 6

Figure 5.2: Plot of the total interference dimensions and the overlap dimensions for giventransmit streams d varying with the number of overlapping rows. The upper part plots thetotal number of free dimensions after each user receives d dimensional interference signalvarying with the overlapping rows of the channels between two users. The lower part plotsthe dimensions of the common region where all interferences are aligned by two users.

90

MIMO Interference Channels

In the Figure 5.3, we plot the rank(N (Φ)

)for different number of transmit streams d and

we observe that the optimum number of overlapping rows from each user is always equal to

dd2e. Intuitively, this analysis makes sense since the overlapping occurs between two users and

we assume that there are equal number of antennas overlapping from each user. After total

overlap dimensions of d is achieved, increasing the overlapping rows does not affect since all

interfering dimensions are aligned on the overlap space.

For example, the optimal number of overlapping rows per user is 2, when d = 3 and

d = 4. This is true because the overlap dimension of odd numbers is not created due to equal

antennas overlapping between two users.

5.2 MIMO Interference Channels

In this section, we present the idea of vertical alignment and overlapping subspace in the

case of three user MIMO IC. We achieve MIMO-IA by estimating the projection matrices at

each receiver, called the local projectors, embedding the orthogonal complements of the local

projectors onto some common dimensions and adding them to obtain the so called alignment

matrix, the null space of which contains the optimal precoders. Such concept of the alignment

matrix, hereafter denoted Φ is also introduced in dimensionality reduction for non-linear

manifold learning [47,82–84].

The embedding of the orthogonal complements of the local projectors is achieved by the

antenna selection matrices, which also provides flexibility in selecting different structural

topology for the receivers as well as introduces the notion of partial antennas sharing among

the receivers such as the ‘ring topological sharing’ and the ‘star topological sharing’ and

discussed before. Before that let us redefine and reformulate the problem of MIMO IA in

K-user MIMO IC in the next section .

5.2.1 Preliminaries

For K-user MIMO-IC, IA is achieved by designing the optimal post-processor matrices at

each receiver i, here denoted by U∗i and the optimal precoding matrices at each transmitter j,

here denoted by V∗j . The aim of IA is to align all K−1 interfering signals, each of dimensions

d in a common subspace and at the same time allocate d dimensions for the desired signal

independent of interference signal. Consider that U∗i is the optimal basis for the common

subspace, where all the interference is aligned.

91


0 0.5 1 1.5 2 2.5 3 3.5 43

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8

Optimal Overlapping Dimensions

(M = 6, N = 4, K = 2)


Overlap

Dim

ension

d = 3d = 4d = 5d = 6

Figure 5.3: Plot of the rank(N (Φ)

)with the number of overlapping rows per user when

different number of data streams are transmitted. The optimal number of overlapping rowsare the number of overlapping rows when the rank

(N (Φ)

)= d.

92


Tx 2

Tx 3

Rx 1 Tx 1

Rx 2

Rx 3

V2

V3

U1 V1

U2

U3

Figure 5.4: Multiple access channel (MAC) (left side) and broadcast channel (BC) (rightside) duality for the three-user MIMO IC where only the interference signals are shown. U1

is determined with the knowledge of V2 and V3 and V1 is determined with the knowledge ofU2 and U3.

In order to design U∗i , the IC is treated as a MAC (left side of Figure 5.4), where the

receiver i observes all the interference signal from the transmitters j 6= i and the desired

signal from the transmitter i. Since U∗i spans d dimensional subspace, d needs to be properly

defined such that the alignment of interference is achievable. For a K user MIMO-IC with M

transmit and N receive antennas, the d that achieves IA is outer bounded by the following

expression [34]

d ≤ M +N

K + 1. (5.12)

Similarly, in order to design V∗j , the IC is treated as a BC (right side of Figure 5.4) and the

signal from any transmitter j is precoded in such a way that these signals align themselves

as much as possible in the subspace spanned by U∗i , ∀i 6= j. Since the alignment is possible

for only d independent streams, the precoding matrix consists of d column vectors.

Thus, the optimal precoders and the optimal post-processor matrices are obtained iteratively

by using the property of MAC/BC duality or the uplink/downlink duality. Let Vj ∈ CM×dand Ui ∈ CN×d be the randomly initialized precoder and post-processor for any transmitter

j and any receiver i respectively, then in order to maintain the power constraints Vj and Ui

93


are orthonormal such that VHj Vj = Id and UH

i Ui = Id. Since d ≤ min(M,N), UiUHi 6= Id.

Let Pi = UiUHi be the local projection matrix observed at each receiver i and P⊥i = IN −Pi

be a projection matrix onto the orthogonal complement of the subspace spanned by Ui [81].

Also, the random channels from any transmitter j to any receiver i is represented by a

matrix Hij ∈ CN×M , where each element hnm is the independent and identically distributed

(i.i.d) Gaussian channel state corresponding to the n-th antenna element of receiver i and

m-th antenna element of transmitter j with zero mean and |hnm|2 variance. For any random

Vj and Hij, ∀i 6= j, HijVj is the span of d orthogonal transmitted interference signal.

Optimization of Ui

Ui is the basis of subspace where all these interference signal are projected. Hence these

interference signal are minimized as much as possible on the orthogonal complement to the

subspace spanned by Ui. Thus, Ui is obtained by minimizing the following:

minUi

K∑j=1,j 6=i

||P⊥i HijVj||2F , (5.13)

minUi

K∑j=1,j 6=i

trace[P⊥i HijVjV

Hj HH

ij

], (5.14)

minUi

K∑j=1,j 6=i

trace[HijVjV

Hj HH

ij −UiUHi HijVjV

Hj HH

ij

], (5.15)


UHi Ui = Id, VH

j Vj = Id, (5.16)

where (5.14) follows from (5.13) by norm-trace relationship and observing the fact that trace

is invariant under cyclic permutations, that is , trace(ABC) = trace(BCA) = trace(CAB)

for some matrices A,B and C; the property of projection matrix P2 = P for some projection

matrix P. Equation (5.15) follows immediately from (5.14). We can now equivalently

express (5.15) as

maxUi

K∑j=1,j 6=i

trace[UHi HijVjV

Hj HH

ijUi

], (5.17)

under the same constraints.

Here we observe that (5.13) and (5.17) are two equivalently different ways to obtain

the optimum Ui. Since Hij is known at the receiver perfectly, (5.17) is a well-known

trace optimization problem with orthonormal constraints and the solution is given by the

94


eigenvectors corresponding to the d maximum eigenvalues of∑K

j=1,j 6=i HijVjVHj HH

ij . Thus,

the local projectors are obtained as Pi = UiUHi . However, the independence of the interference

signal from the desired signal is still not guaranteed.

In the downlink, each transmitter then updates the corresponding precoding matrices in

such a way that the interference power observed in the unwanted receivers is minimized as

much as possible. This guarantees that the aligned interfering signals are not contained in

the subspace orthogonal to the one spanned by Ui.

Optimization of Vj

The d dimensional independent signals from any transmitter j to the unwanted receivers

i, ∀i 6= j is HijVj . Please note here that the channel Hij in the uplink is from all transmitters

j 6= i to the receiver i while the channel Hij in the downlink is from the transmitter j to all

the receivers i 6= j and Vj is obtained by minimizing the following :

minVj

K∑i=1,i 6=j

||P⊥i HijVj||2F (5.18)

minVj

K∑i=1,i 6=j

trace[P⊥j HijVjV

Hj HH

ij

], (5.19)

minVj

trace[ K∑i=1,i 6=j

VHj (HH

ijP⊥i Hij)Vj

], (5.20)


UHi Ui = Id, VH

j Vj = Id. (5.21)

Equation (5.19) and (5.20) follow from (5.18) by the trace norm relationship and the

cyclical invariant properties of the trace. From (5.20), it is clear that Vj is determined if

we know P⊥i and the channel states Hij exactly. At that time, it is again a well-known

trace optimization problem and the optimum Vj is given by the eigenvectors corresponding

to the d minimum eigenvalues of∑

i=1,i 6=j HHijP

⊥i Hij, which is the basis of most of the IA

algorithms such as [2, 32]. However, channel state information at the transmitter (CSIT) is

always difficult to determine in the time-varying wireless channels and also the knowledge of

optimum value of d is not available.

95


Grouping of Receivers and Reformulation of Optimization of Vj

As discussed in the previous section, we would like to form a group among the receivers and

collectively align the interference onto a common subspace. Hence, we imagine a system

where each receiver has some common channel parts and other independent channel parts

and transmitter knows the common channel and independent channels from each receiver

locally. Such system model can be used in future wired plus wireless communication systems,

such as in trains and buses as we discussed before.

The topological information about the common and independent channels are presented

in the 0-1 selection matrices Si ∈ RN×NT , ∀i = {1, 2, · · · , K}. If Hj ∈ CNT×M is the

channel matrix available at the transmitter j, the elements of which are the channel states

corresponding to the available spatial dimensions (antenna elements ) in all the interfering

receivers, then the channel matrix at each receiver i corresponding to transmitter j can be

written as:

Hij = SiHj.

The optimization problem in (5.20) is then expressed as:

minVj

trace[VHj HH

j

( K∑i=1,i 6=j

SHi P⊥i Si)

HjVj

], (5.22)

under the same constraints, where the term Φ =∑K

i=1,i 6=j SHi P⊥i Si is the alignment matrix

as defined in non-linear manifold learning [47,82–84].

Expression (5.22) is particularly significant because we can express HjVj as a single

variable, we can play around with another parameter Si (that provides flexibility in topology)

and we can easily analyze the error if P⊥i is not known perfectly. Mathematically, HjVj is

the d dimensional precoded interference signal from the transmitter j and we can clearly

observe that the trace in (5.22) is minimum when HjVj ∈ N (Φ), where N (Φ) represents

the null space of Φ. This fact motivates us to ask the following questions: a) Does N (Φ)

exist? b) If it does exist, what is the dimension of N (Φ) ?

We observe that the N (Φ) exists as long as there is overlapping channels or physically the

overlapping spatial dimensions between any two receivers. But such overlapping is not always

optimal. Our aim is to determine the optimal overlapping between them. Next, we discuss

the concept of optimal overlapping between any two sub-spaces created by the projection of

any two sub-matrices.

96


Z1

Z2

S1

S2

ZΦf(S1,S2,Z1,Z2)

ΦZ = 0 for any S1 and S2.

Figure 5.5: Relationship between Φ and Z.

Optimal Overlapping

Consider Zk and Z`, ∀k 6= ` are two sub-matrices of Z and the corresponding projection

matrices PZk and PZ` are the projection onto any two subspaces M and N , then the two

subspaces are said to be optimally overlapped if one of the sub-matrix has full column rank

and

min[

rank(Zk), rank(Z`)]

= rank(Zk`), (5.23)

where Zk` = Zk ∩ Z` represents the matrix of the common rows between Zk and Z`.

When they are optimally overlapped, it is no longer possible to find two independent

vectors wk and w` such that Zk`wk = Zk`w`. Thus, a unique solution exists that satisfies

this condition. The study of the uniqueness of the solutions of a merging or the realization

problem is called rigidity [85]. In other words, the subspaces are optimally overlapped when

the rigid solution exists and in that case the total space is exactly equal to the overlapped

subspace, which is exactly equal to the null space of the alignment matrix obtained from two

sub-matrices as given by

R(Z) = R(Zk) ∩R(Z`) = N (Φ). (5.24)

The relationship between Z and Φ is clearly depicted in Figure 5.5 for any two sub-matrices

of Z named as Z1 and Z2. N (Φ) and Z are exactly same when optimal overlap exists between

Z1 and Z2.

How do we obtain the optimal overlapping subspace?

One idea is to determine the optimally overlapped sub-matrices of the given matrix Z. In

order to do so, we define the selection matrices Si such that Zi = SiZ. Now, we can tune

97


Sis optimally to determine the optimal Φ. The optimal tuning of Sis help to optimally

overlap the rows of Zis, which allows more flexibility in the system in terms of the topological

arrangement of the receivers. In fact, each rows of Zis represent the number of antenna

elements in the receiver i and hence the overlapping of the rows can be mathematically

viewed as the concept of virtual antennas overlapping or the antennas sharing. We call this

virtual sharing because the sharing is only mathematically observed and not physically.

If rank(Zk) = rank(Z`) = d, then when Zk and Z` are optimally overlapped, rank(Zk`) =

dim(N (Φ)) = d is satisfied. So, we can choose any arbitrary d and determine the optimal

Φ such that dim(N (Φ)) = d by properly tuning the selection matrices. In this regard, we

require a scheme to determine the rank of N (Φ) for different Sis.

5.2.2 Determining the Rank of N (Φ)

Theorem 5.1. Consider that Z1 and Z2 are two sub-matrices of Z ∈ CNT×M with some

overlapping rows, expressed as

Z1 = S1Z =

[Z11

Z12

], Z2 = S2Z =

[Z21

Z22

], (5.25)

with Z11 ∈ CN11×M and Z22 ∈ CN22×M are the non overlapped parts and Z12 = Z21 ∈ CN12×M

are the overlapped parts, and Φ = SH1 (IN11+N12−Z1Z†1)S1 +SH2 (IN22+N12−Z2Z

†2)S2 as defined

previously then

rank(Φ) = N11 +N12 +N22 − (r1 + r2 + r3), (5.26)

rank(N (Φ)) = r1 + r2 + r3, (5.27)

where

r1 = rank{Z12}, (5.28)

r2 = rank{(Z11R1)(:,r1+1:M)}, (5.29)

r3 = rank{(Z22R1)(:,r1+1:M)}. (5.30)

R1 ∈ CM×M is the right singular vector obtained from singular value decomposition (SVD) of

Z12.

Proof. The proof is provided in the appendix A.

The rank ofN (Φ) for different values of M and the same value of NT for varying overlapping

dimensions is plotted in Figure 5.6. One of the interesting things we observe here is the case

when there is no overlapping. In that case, r1 + r2 + r3 = 2M but we expected all r1, r2

98


and r3 to be zero since they depend on R1, the right singular vector of an empty matrix.

However, we observe that though Z12 is an empty matrix, the right singular vector R1 is an

identity matrix IM . Thus, r1 = 0, r2 = M and r3 = M . Intuitively if we observe the rank

of matrix Φ, this makes sense because Φ is the embedding of the orthogonal projections of

dimensions N11 −M and N22 −M in this case and hence the rank of N (Φ) is 2M .

As the overlapping increases, r2 and r3 depends on r1 and the rank of N (Φ) is r1 + 2(M −r1) = 2M − r1. r2 and r3, in fact, represent the number of independent dimensions at the two

receivers and r1 represents the overlapping dimensions between them. For all r1 > M , the

rank is always M . The N (Φ) does not exist when 2M − r1 ≥ 2N because the complement of

the projection matrix becomes close to zero in that case. In the simulation shown in 5.6, the

rank of N (Φ) is not defined for the cases when (M = 10, r1 = 0) and (M = 12, r1 = 0, 2, 4).

The optimal overlapping is observed when r1= number of transmit antennas=M . Since we

assume that each receiver observes equal overlapping all the time, the number of overlapping

rows is always considered to be even in the simulations.

The rank of N (Φ) is observed for two overlapping receivers that share common antennas

(spatial dimensions) between them. As we discussed before, the Sis provide flexibility to

define different topology of antennas sharing and in the next section, we discuss the approach

to determine the rank of N (Φ) for the ring topological sharing and the star topological

sharing in the case of three user MIMO-IC.

5.2.3 Ring Topology

System Model

If the overlapping of certain number of antennas is observed between the adjacent receivers,

we call this scheme the ‘ring topology’. Symmetric antennas overlapping for ring topology of

three user MIMO-IC is depicted in Figure 5.7, where the antennas in each receiver are indexed

with alphabets. Different antennas overlap between the adjacent receivers. For example, the

antennas indexed ‘d’ and ‘e’ overlap between RX1 and RX2, the antennas indexed ‘h’ and ‘i’

overlap between RX2 and RX3 and the antennas indexed ‘l’ and ‘a’ overlap between RX1 and

RX3 as shown in Figure 5.7. In that case, RX1 observes all the information in antennas ‘l, a,

b ,c ,d, e’, RX2 observes all the information in antennas ‘d, e, f, g, h, i’ and RX3 observes all

the information in antennas ‘h, i, j, k, l, a’.

Thus, for the ring topology each receiver observes three parts of channel information- the

part corresponding to the common channel with the previous receiver, the part corresponding

99


0 2 4 6 8 10 12 14 16 18 202

4

6

8

10

12

14

16

18

20

Rank of N (Φ)

(N = 10, K = 2)

Ran

k(r

1+r2+r3)

Overlapping Dimensions(r1 = M1)

M = 2M = 4M = 6M = 8M = 10M = 12

Figure 5.6: Interference dimension for different M

100


Rx1

Rx3

Rx2

a b c d

d e

e f g h

i j k l l a h i

Figure 5.7: An example of ring topology for antennas overlapping. Each antenna is indexedwith the alphabets. ‘a,b,c,d’ are the antennas in RX1, ‘e,f,g,h’ are the antennas in RX2, ‘i,j,k,l’are the antennas in RX3. All receivers share different antennas with the adjacent receivers.The antenna elements in the dashed box are the antennas shared between two receivers.

101


to the independent channel and the part corresponding to the common channel with the next

receiver. The signal received by any receiver i, transmitted from all the transmitters j, that

observes overlap of antennas with the adjacent receivers i+ 1 and i− 1 is given by

yi =

H(i)i(i−1)

H(i)ii

H(i)i(i+1)

Visi +K∑

j=1,j 6=i

H(j)i(i−1)

H(j)ii

H(j)i(i+1)

Vjsj + ni, (5.31)

where H(j)i(i−1) ∈ C2Nc×M is the common channel between the receivers i and (i−1) observed

at receiver i corresponding to the transmitter j, H(j)ii ∈ C(N−Nc)×M is the independent channel

observed at receiver i corresponding to transmitter j, and H(j)i(i+1) ∈ C2Nc×M is the common

channel between the receivers i and (i + 1) observed at receiver i corresponding to the

transmitter j; and ni ∈ CNk×1 is the additive white Gaussian noise (AWGN) with zero mean

and unit variance. Nc is the number of overlapped antennas per user between the adjacent

receivers and Nk = N + 2Nc; sj ∈ Cd×1 are the d dimensional data streams transmitted from

any transmitter j and Vj ∈ CM×d are the precoding matrix corresponding to transmitter j.

Now, the channel from any transmitter j to all the receivers is Hj ∈ CKN×M and each

channel is selected from Hj by the selection matrices Si ∈ R(N+2Nc)×M corresponding to

receiver i. Thus SiHjVj is the sub-matrix of HjVj. We use the same MAC-BC duality

approach as described before to determine all Uis and the corresponding Φ, only differing

in the dimensions. The aim is again to obtain optimum Vj which is contained in N (Φ)

from (5.22). Hence, we obtain the optimal dimensions of N (Φ) for ring topology next.

Optimal Dimensions of N (Φ) for Ring Topology

Theorem 5.2. For K = 3 user IC with ring topological antennas overlap, consider any matrix

Z = HjVj ∈ CKN×M due to any transmitter j, the selection matrices Si ∈ R(N+2Nc)×KN

observed at any receiver i and the sub-matrices Zi = SiHjVj ∈ C(N+2Nc)×M , where N is the

number of receive antennas, Nc is the number of overlap antennas per user between adjacent

receivers and M is the number of transmit antennas; then the alignment matrix formed by

the receivers due to any transmitter j Φj is defined as Φj = SHj PZjSj +∑K

i=1,i 6=j SHi (P⊥Zi)Si

and the rank of null space of Φj is given by

rank(N (Φj)) =

N +M − (K + 1)Nc, 0 ≤ Nc ≤ M

(K−1)

N − (K − 1)Nc,M

(K−1)≤ Nc ≤ N

(K−1)

0, Nc ≥ N(K−1)

. (5.32)

102


Proof. The proof is provided in appendix B.

Intuitively, we observe that the N (Φj) is the possible space created by the interfered

receivers where the observed interference is aligned, plus the available free dimensions in

the intended receiver. If d = M streams of data is transmitted from any transmitter Tx1,

then the interference occupies M dimensions in Rx2 and M dimensions in Rx3 and there are

N −M free dimensions in Rx1. Hence, the dimension of N (Φj) is N −M +M +M = N +M .

When the antennas overlap, the overlap dimension belongs to the null space. If there is

overlap of 1 antenna per user, then two interference streams from each user belong to that

space, thus reducing the dimension of null space by (K − 1)1 = 2. Also, due to ring topology

2 ∗Nc = 2× 1 = 2 free dimensions at the desired receiver overlap with other receivers and

available free dimensions is N −M − 2Nc. Hence, for Nc overlap, the dimension of the null

space is reduced by (K + 1)Nc = 4Nc for the case when Nc ≤ M2

as given by theorem 5.2.

For M2≤ Nc ≤ N

2, the two interfered receivers completely overlap all the interference

signal, thus contributing only M dimensions. The overlap, however varies the free dimensions

available in the desired receiver as N −M − 2Nc. Hence the rank of null space is M +N −M − 2Nc = N − 2Nc

The number of optimal overlapping rows is achieved when all the interference observed by

the interfered receivers are aligned, i.e, when only M dimensions are occupied by interference

in two receivers. In that case, the rank(N (Φj)) = N −M + M − 2Nc = N − 2Nc, where

N −M − 2Nc is due to the free dimensions in the first receiver and M is due to other two

receivers. Hence, the optimal number of overlapping rows Nc is achieved when

N − 2Nc = N +M − (K + 1)Nc ⇒ Nc =M

K − 1. (5.33)

The rank of the N (Φj) for different number of transmit antennas and N = 10 receive

antennas with varying overlapping dimensions is plotted in Figure 5.8 for three users that

observe common channels over different antennas (ring topology). These results are plotted

for the Φj obtained from the designed post processing matrices Uis. We also consider that

each transmitter transmits d = M streams of data and the number of antennas in the receiver

is always greater than the number of transmit antennas.

The plot in Figure 5.8 clearly shows the two region, the optimal overlapping rows is given

by the number of overlapping rows corresponding to that point where the two regions are

separated. This is also shown in the graph by drawing a dotted line from the the point where

the two regions are separated in the graph. For example, the number of optimal overlapping

rows when M = 3 is 1, which is bM2c as obtained in (5.33). This makes sense because it

103


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

10

12

14

16

18

Rank of N (Φ) vs Overlap for Ring topology

(N = 10, K = 3)

Overlapping Rows

Ran

kof

N(Φ

)

M = d = 3M = d = 4M = d = 5M = d = 6M = d = 7M = d = 8

Figure 5.8: Rank of N (Φj) for different M , when different antennas overlap between theadjacent receivers. The null space is decreasing as the overlapping rows increase. The graphshows two regions and the point where the graph changes as shown by the dashed line givesthe optimal number of overlapping rows.

104


requires at least one overlapping rows between the users that receive interference in order to

create the optimal interference overlap space for three independent interference streams. If

two rows per user overlap, it uses four channels to align three independent streams, which

does not improve more than using one rows per user to overlap.

5.2.4 Star Topology

System Model

When the overlapping of the group of same antennas is observed in all the receivers, we call

such overlapping the ‘star topology’. An example scenario for the star topology is shown in

Figure 5.9, where all the antennas in each receiver are indexed with certain alphabets and all

the receivers have the same overlapping antennas ‘a, e, i’. Thus, RX1 observes ‘a, b, c, d, e,

i’, RX2 observes ‘a, e, f, g, h, i’ and RX3 observes ‘a, e, i, j, k, l’ antennas.

The concept is similar to the ‘ring topology’ except that the overlapping antennas are

same in all receivers. This approach of antennas overlapping may have a lot of practical

applications in future networks, where both wired and wireless system serve the same user

at the same time. All the common channel from the wired server includes all the common

control information and all other private channels from the wireless server includes the private

data, which allows better system performance.

The signal received by any receiver i from all the transmitters j = {1, 2, · · · , K} for the

star topology is expressed as :

yi =

[H

(p)ii

H(c)i

]Visi +

K∑j=1,j 6=i

[H

(p)ij

H(c)j

]Vjsj + ni, (5.34)

where H(p)ij ∈ C(N−Nc/K)×M is the private channel states observed at receiver i from the

transmitter j and H(c)j is the common channel states at all receivers due to transmitter j.

Nc/K is the common rows per receiver, assuming that the overlapping is symmetric.

Optimal Dimensions of N (Φ) for Star Topology

For this channel setup with star topology antennas overlapping, the selection matrix Sis

are defined accordingly and then the alignment matrix Φ is obtained by estimating suitable

Uis as described before and then the optimal dimensions for N (Φ) is determined. The

rank(N (Φ)) for star topological antennas overlapping is given by the following theorem:

105


Rx1

a b c d

Rx2

e f g h

Rx3

i j k l

a e i

a e i a e i

Figure 5.9: An example of star topology for antennas overlapping in three user MIMO IC.Each antenna is indexed with the alphabets. Same antennas overlap for all the receivers. Forexample, the antennas in the dashed box ‘a’,’e’,’i’ overlap in all three receivers.

106


Theorem 5.3. For K = 3, consider any matrix Z = HjVj ∈ CKN×M with star topo-

logical setup having Nc overlapping antennas per user, such that selection matrices Si ∈C(N+(K−1)Nc)×KN and the sub-matrices Zi = SiHjVj ∈ C(N+(K−1)Nc)×M for which the align-

ment matrix due to any transmitter j is defined as Φj = SHj PZjSj +∑K

i=1,i 6=j SHi (P⊥Zi)Si,

then the rank of the null space of Φj is given by:

rank(N (Φj)) =

N + (K − 2)M − (K − 1)2Nc, 0 ≤ Nc ≤ M

K

N −Nc,MK≤ Nc ≤ N

0 Nc ≥ N

, (5.35)

Proof. The proof is provided in appendix C.

Similar to the ring topology, the null space of Φj in the star topology is also the available

possible interference space observed by the receivers which aim to align interference and

the space of free dimensions in the receiver which does not receive any interference. When

there is no overlap, the star topology and the ring topology behaves similarly and hence the

rank(N (Φj)) = N +M .

When the overlap is 0 ≤ Nc ≤ MK

, the total interference dimensions in the interfered users

is (K − 1)(M −KNc

)+KNc because of the K overlap unlike K − 1 overlap in ring topology,

while the total free dimensions is N −M −Nc. Hence,

rank(N (Φj)) = (K − 1)(M −KNc

)+KNc +N −M −Nc,

= (K − 2)M −K2Nc + 2KNc −Nc +N,

= (K − 2)M − (K2 − 2K + 1)Nc +N,

= N + (K − 2)M − (K − 1)2Nc.

When Nc ≥ MK

, all interfered receivers only contribute M dimensions and total free

dimensions in desired receiver is N −M −Nc. So

rank(N (Φj)) = N −M −Nc +M = N −Nc.

The rank of rank(N (Φj)) for star topology with 10 receive antennas and varying transmit

antennas such that each transmitter transmits d = M streams of data is plotted in the

Figure 5.10. The simulation is obtained by designing the suitable selection matrices Sis and

hence alignment matrix Φj. In the graph, we clearly see the two regions as given by the

two conditions and the number of optimal overlapping rows is given by the overlapping rows

corresponding to the point where the two regions are separated as indicated by the dashed

lines in the graph in Figure 5.10. We observe that the number of optimal overlapping rows is

107


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

10

12

14

16

18

Rank of N (Φ) vs Overlap for Star topology

(N = 10, K = 3)

Overlapping Rows

Ran

kof

N(Φ

)

M = d = 3M = d = 4M = d = 5M = d = 6M = d = 7M = d = 8

Figure 5.10: Rank of N (Φ) for different transmit antennas varying with the number ofoverlapping antennas in the ring topological overlap structure. The graph shows two regionsand the point where the region changes shows the optimal overlapping rows as shown by thedashed line.

108

Conclusions

given by

N∗c = bMKc.

This makes sense because the overlap is formed by the K users unlike K − 1 users in ring

topology. We also observe that the graph does not go to zero at just Nc = 5 overlap unlike

the ring topology. This is because in ring topology there occurs 5 overlap from two users and

no free dimensions are available at the desired receiver. However, in the star topology there

are still 10− 5 = 5 free dimensions.

5.3 Conclusions

In this chapter, we introduced the concept of vertical alignment by forming a group amongst

the receivers, which can collectively align the interference onto a common overlap space.

These receivers can only form a group when there is enough correlation or overlap between

them. Thus, we assume that these receivers have common channels and independent channels.

This is a reasonable assumption for future communication networks in crowded trains, buses

and stadiums where we can obtain common channels from wired media and the independent

channels from the wireless media. We introduced the alignment matrices for different

topological antennas overlapping and determine the optimal overlapping condition for each of

the topology. This new idea of collective interference management can provide new directions

in the future research.

109

Conclusions

110

Chapter 6

DOF Analysis of Three User MIMO

IC via Receiver Chain Alignment

6.1 Introduction

Demand for higher data rates and the better quality of service in wireless systems continue to

increase, unconcerned with the fact that transmit power, bandwidth and design complexity

are limited and expensive resources. Multi-user communications in a single input single

output (SISO) setting and/or in a MIMO setting is believed to be an effective solution to

these exponentially increasing demands. However, multi-user communications are interference

limited due to multiple users transmitting and receiving at the same time with the same

frequency and majority of today’s research is focused on designing the effective interference

management and mitigation techniques, both academically [1, 38,64,86,87] and industrially

[88–90].

One of such interference management techniques is the IA, which received significant

interests amongst the researchers, because of the fact that this is proved to be the degrees

of freedom (DOF) optimal [1, 22]. DOF is an important metric to study the interference

problem as it is viewed as a first order capacity approximation at high signal to noise ratio

(SNR) [91,92].

The DOF of a variety of interfering networks with different channel assumptions are studied

under the prism of IA in a number of recent articles such as [2, 3, 27–29, 68, 69, 75] but an

effective and generalized IA algorithm for all networks and for all channel conditions still fail

and is an open problem.

The DOF of the three user MIMO-IC is of special interest to the researchers because it is

the smallest MIMO network where the effect of IA is distinctly visible in terms of spatial

extensions, which is achieved by proper precoding in the absence of infinite channel uses in

time or frequency [2]. Further, due to less number of unknown parameters this channel acts

as a gateway towards the generalization to large number of users.

111

Receiver Chain Alignment

The DOF of interference channels are well studied for almost all channel realizations

when the transmitters and receivers are equipped with same number of antennas, i.e, for

a symmetric system with M = N , where M is the number of transmit antennas and N

is the number of receive antennas. However, when the transmitters and the receivers are

equipped with different number of transmit and receive antennas, the generalized scheme for

the characterization of DOF for more than two users IC is not still well studied. Some of the

recent works by Wang et al. [4] proposed that the DOF of three user IC can be characterized

by the notion of subspace alignment chains (SACs) which identifies the extra dimensions

required for the optimal IA schemes.

From the last chapter, we learned that the optimal overlapping is always possible when

there is certain collective cooperative operation between any two receivers. Motivated by

this fact, in this chapter, we describe a new notion of cooperative receiver chain alignment

scheme to characterize the DOF of three user IC with different number of transmit and

receive antennas.

In this regard, we explain a simple method in terms of overlapping spaces observed by the

adjacent receivers as a chain diagram. Such overlapping chain diagrams among the receivers is

termed as receiver chain alignment (RCA) [93]. Note here that the DOF analysis of three-user

MIMO-IC with arbitrary number of transmit and receive antennas is still unsolved and by

considering the symmetry, we reduce the number of parameters for ease of analysis. Unlike

the SAC described in [4], RCA is achieved by the knowledge of optimal overlap dimensions

as obtained from the null space of Φ observed by all the receivers as defined in the previous

chapter. In the next section, we describe the RCA scheme in detail.

6.2 Receiver Chain Alignment

Receiver chain alignment is a scheme at the receiver side, where the adjacent receivers form

an optimal overlap space to align the observed interference. The optimal space is allocated

as long as the spatial dimensions available in each receiver are enough.

In order to better understand the RCA, consider an example with M = N = 4 and K = 3,

where the adjacent receivers RX2, RX3; RX3, RX1 and RX1, RX2 optimally allocate an

overlapping space for the interference represented in the form of chain as shown in Figure 6.1.

Here, each overlap space is of dimension M , with the contribution of M2

dimensions from

each receiver. We observed in chapter 5 that this is the optimal number of overlapping rows

when two receivers collectively align the interference. The total overlap dimensions between

112

Receiver Chain Alignment

any two adjacent receivers are denoted by the dotted lines and the remaining free dimensions

are denoted by the solid lines in the Venn-diagram like structure as shown in Figure 6.1.

Let us assume that each optimal overlap is of length 1. Since there are no more free

dimensions after three optimal overlaps for M = N = 4, the total length of the chain (L) is

3. We also observe that for any M = N = p(even), L is always K (3 here) as shown in the

lower part of the Figure 6.1.

M = N = 4

M = N = p (even)

RX2 RX3 RX3 RX1 RX1 RX2


p2

p2

p2

p pp

−Free dimensions −Overlapped dimensions

Figure 6.1: RCA for M = N = 4 and M = N = p (general case) in 3-user IC with optimaloverlapping.

When N > M , we observe that there are still N −M free dimensions per user after the

chain length K. Hence, the total chain length is

L = K +KN −MM

. (6.1)

The fraction N−MM

comes from the fact that the overlap space of dimension M is assumed

to be of length 1, which means the N −M dimensions has length N−MM

. The RCA for

N = 5,M = 4 is depicted in Figure 6.2 and this is same for all M and N with N −M = 1

and K ≤M .

113

DoF Analysis and Achievability

M = 4, N = 5


RX2RX3RX1


Chain I -

Chain II-

Figure 6.2: RCA for M = 4, N = 5 with optimal overlapping.

6.3 DoF Analysis and Achievability

Theorem 6.1. For symmetric K = 3 user IC with M transmit and N receive antennas

(N ≥M , M is even), the achievable DOF per user when optimal (M) overlapping rows exist

between the adjacent receivers, is given by

d =

2M3, ∀ 1

2< N−M

M≤ 1

N1+(1+min(N−M

M,1)), otherwise,

(6.2)

where 1 + N−MM

is the length of RCA normalized per user.

Proof. Clearly, we observe that the achievable DOF is dependent on N−MM

, which constitutes

the length of RCA. Thus, we prove the theorem by constructing RCA for the following cases,

6.3.1 Case 1 : 0 ≤ N−MM ≤ 1

2

Assume that d0 is the spatially normalized achievable DOF, i.e., the DOF per transmit

antenna, then the total desired streams transmitted by K users with M antennas per user is

K × (Md0). Since the chain length is L, there are also L overlapped interference space, each

of dimension M that contribute L× (Md0) interference streams. In order to decode all the

desired streams independent of the interference streams, the total (desired + interference)

114


streams can not exceed the total number of antennas in all the receivers. Hence,

(K + L)Md0 ≤ KN ⇒Md0 ≤KN

K + L. (6.3)

The RCA for N = M was discussed in the previous section and depicted in Figure 6.1,

where L = K. The achievable DOF per user in that case is

d = Md0 ≤KN

2K=N

2. (6.4)

The RCA for N−MM

= 14

is depicted in Figure 6.2. The total length of the chain in that

case is given by (6.1) and also holds true for all values of M and N , when N−MM≤ 1

2. Hence

the achievable DOF per user is

d = Md0 ≤KN

2K +K N−MM

=NM

N +M, (6.5)

In fact (6.4) is obtained from (6.5) with M = N . We observe that similar result is observed

in [3] and [4] for this case.

6.3.2 Case 2 : 12 <

N−MM ≤ 1

The RCA for one of the cases that satisfies 12< N−M

M≤ 1 is depicted in Figure 6.3, where

M = 4 and N = 7. It is observed that after the first K chain lengths, each receiver has

N −M free dimensions and one more overlap of M dimensions is still possible. This means

that K + 1 complete overlaps are possible. But, after K + 1 overlaps, K − 2 receivers (here

RX1) has N −M free dimensions, while K − 1 receivers (here RX2 and RX3) have N − 3M2

free dimension each.

Since there are enough available free dimensions, the next chain (RX3, RX1) is still possible.

However, in order to create the optimum overlap space, RX3 and RX1 both require equal free

dimensions (M2

each). Since RX1 has (N −M) dimensions, and RX3 is in short of 2M −Ndimensions, RX3 borrows the required dimensions from the RX1 to form the optimum chain

(RX3, RX1). This borrowing phenomenon acts like adding one virtual antenna at the RX3

and the total antennas at all receivers is KN + (2M −N) = (K − 1)N + 2M instead of KN .

The first K + 1 chain length always exist and still there are (K − 1)(N − 3M2

) + (K −2)(N −M) = 3N − 4M free dimensions for K = 3. After borrowing of required dimensions,

another chain of dimension M is possible, leaving 3N − 4M −M = 3N − 5M free dimensions.

Hence, total chain length is

L1 = K + 2 +3N − 5M

M. (6.6)

Let us consider another example with M = 6 and N = 10 as shown in Figure 6.4. Here,

115


M = 4, N = 7


RX2RX3RX3RX1RX2


Chain I -

Chain II-

Figure 6.3: RCA for M = 4, N = 7 with optimal overlapping.

RX3 is in short of two more dimensions in order to create the optimum overlap space of

dimension M and to complete the chain (RX3, RX1). But, RX3 can borrow one extra

dimension from RX1 and the other one from RX2, equivalently adding two virtual antennas.

In both the examples, 2M − N virtual antennas are added and chain length is given

by (6.6). Hence,

(K + L1)Md0 ≤ (K − 1)N + 2M, (6.7)

d = Md0 ≤(K − 1)N + 2M

K +K + 2 + 3N−5MM

=2M

3,∀K = 3. (6.8)

6.3.3 Case 3 : 1 < N−MM ≤ 2

The RCA for one of the example scenarios that satisfies 1 < N−MM

< 2 is shown in Figure 6.5,

for M = 2 and N = 5. In this case, we clearly observe that there are 2K complete overlaps

and still (N −M)−M = N − 2M free dimensions per user. Hence, the total length of the

chain is

L2 = 2K +K(N − 2M)

M, (6.9)

116


M = 6, N = 10


RX2RX3RX3RX1RX2


Chain I -

Chain II-

Figure 6.4: RCA for M=6 and N=10

and the total data streams (desired + interference) is given by

Ts = KMd0 + (2K +K(N − 2M)

M)Md0. (6.10)

However, for the K user IC, each receiver only observes the maximum of (K − 1)Md0

interference streams and Md0 desired streams. For K = 3, the maximum possible total

(desired + interference) streams is 9Md0 and not Ts as given by (6.10). Hence,

9Md0 ≤ 3N ⇒ d = Md0 ≤N

3. (6.11)

Hence for case 1 and case 3, d is expressed in general as :

d ≤ N

1 + (1 + min(N−MM

, 1)). (6.12)

One of the interesting features of RCA is the pictorial representation of the ratio R = N−MM

.

When R = 0, the chain length (L) is K and the RCA terminates in chain I, while for

0 < R ≤ 1, we have K < L ≤ 2K and the RCA terminates anywhere in the chain II.

Similarly, for 1 < R ≤ 2, we have 2K < L ≤ 3K as shown in Figure 6.5. Also, each

intermediate value of L, e.g., L = K + α, ∀α ≤ K represent all the intermediate values of R,

e.g., β : 0 < β ≤ 1.

For all the cases when N−MM≥ 2, no IA is required and the achievable DOF is always M

117


M = 2, N = 5


RX2RX3RX3RX1RX1RX2

RX2 RX3 RX1


Chain I -

Chain II -

Chain III-

Figure 6.5: RCA for M = 2, N = 5 in 3-user IC for optimal overlapping.

118

Information theoretic outer bound for DOF

(N > M). No RCA is created.

Although the RCA evaluates the DOF of IC with N ≥M and M is even, the achievable

DOF is completely symmetric over the ICs and the DOF evaluated for the case N > M holds

true also when N and M are interchanged. The DOF is also the linear function of the ratio

of M and N , and hence the DOF for odd values of M are easily determined by evaluating

the DOF for 2M and 2N and dividing the obtained value by 2. For example, the achievable

DOF for M = 3, N = 4 is obtained by evaluating the achievable DOF for M = 6, N = 8 and

dividing the result by 2. In this regard, RCA is useful to evaluate the DOF of three user

MIMO IC for any values of M and N .

The variation of DOF per user for different transmit antennas (M) with respect to different

receive antennas (N) antennas is plotted in the Figure 6.6, where the variation of DOF clearly

shows specific patterns in certain region, mainly four; the first linear region is observed for

N ≤ M/3, the second non-linear region exists for M/3 < N ≤ 2M , the third linear region

exists for 2M < N ≤ 3M and the fourth constant region for N ≥ 3M . The second non-linear

is especially important because it is the region where MIMO processing and IA is both

possible. This is the region where improvement is still possible and the problem is still open.

6.4 Information theoretic outer bound for DOF

Let Wk be the independent transmitted messages from TXk to RXk and Rk(ρ) be the

achievable rate as a function of SNR, denoted by ρ, corresponding to user k, ∀k = {1, 2, 3}.The rate is achieved by choosing appropriately large n, where n can be considered as the

number of channel uses [1, 14]. If dk is the DOF corresponding to user k then

dk = limρ→∞

Rk(ρ)

log(ρ). (6.13)

Let Rsum =∑3

k=1Rk and Yk = (YkD, YkI) is the signal received at receiver k, where YkD is

the desired signal and YkI is the interference signal. Also, let h(X) be the differential entropy,

I(X, Y ) be the differential mutual information, h(X|Y ) be the differential conditional entropy,

I(X;Y |Z) represents the differential conditional mutual information and W = {W1,W2,W3}

119


0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10

DoF of three user (M ×N) MIMO IC.

Receive Antennas(N)

Degrees

ofFreedom

DoF RCA

DoF JafarDoF Wang

M = 4

M = 6

M = 10

M = 15

Figure 6.6: DOF plot for different transmit and receive antennas under different DOFachievable schemes. The ‘*’ line represents the DOF achieved using the proposed RCAscheme, the ‘o’ line represents the DOF achieved with the scheme proposed by Jafar et. al [3]and the ‘square’ line represents the DOF achieved with SAC scheme proposed by Wang et.al [4].

120


[14] then from definitions, we have

nRsum = h(W ) = I(W ;Y nk ) + h(W |Y n

k ), (6.14)

≤ I(W ;Y nk ) + nεn, (6.15)

= I(W ;Y nkD, Y

nkI) + nεn, (6.16)

= I(W ;Y nkD) + I(W ;Y n

kI |Y nkD) + nεn, (6.17)

≤ nN log ρ+ n o(log ρ) + I(W ;Y nkI |Y n

kD) + nεn. (6.18)

Equation (6.15) follows from the Fano’s inequality [14], while (6.17) follows from the chain

rule of mutual information [14] and (6.18) follows from the fact that the DOF can not be

more than the number of antennas at the receiver (antenna bound) and o(log ρ) approaches

zero as ρ approaches infinity. From definitions,

I(W ;Y nkI |Y n

kD) = h(Y nkI |Y n

kD)− h(Y nkI |Y n

kD,W ),

≤ h(Y nkI |Y n

kD) + o(n). (6.19)

Equation (6.19) follows from the fact that given the input messages and the received signal

at particular receiver, the interference signal is decoded with small noise distortion [ [4]

Lemma 3 ]. Hence from (6.18) and (6.19),

nRsum ≤ nN log ρ+ h(Y nkI |Y n

kD) + n o(log ρ) + o(n). (6.20)

Consider the example scheme where M = 2, N = 3, with optimum overlap as shown in the

Figure 6.7, where the interference and desired signals observed at the receiver k after the

first overlap are: YkI = (Y(k+1)a, Y(k−1)c), YkD = (Yka, Ykc), ∀k = {1, 2, 3} and k is cyclic.

Hence, we can further express (6.20) as

nRsum ≤ nN log ρ+ h(Y n(k+1)a, Y

n(k−1)c|Y n

kc) + n o(log ρ) + o(n), (6.21)

and also as

nRsum ≤ nN log ρ+ h(Y n(k+1)a, Y

n(k−1)c|Y n

ka) + n o(log ρ) + o(n). (6.22)

Using the chain rule for conditional differential entropy and expressing Yka and Ykc as

a function of X(k+1)a and X(k−1)c respectively as shown in Figure 6.7, where Xka, Xkb,

X(k+1)a, X(k+1)b, X(k−1)a, X(k−1)b are the signals transmitted from all the transmitters and (6.21)

is further simplified to,

nRsum ≤ nN log ρ+ h(Xn(k−1)a|Xn

(k−1)b) + h(Xn(k+1)b|Xn

(k−1)b, Xn(k−1)a) + n o(log ρ) + o(n),

= nN log ρ+ nR(k−1) − h(Xn(k−1)b) + n o(log ρ) + o(n), (6.23)

where (6.23) follows from the fact that the achievable rate can be expressed in terms of

the joint entropy nRk = h(Xka, Xkb) and the corresponding chain rule of joint differential

121


.

.

.

.

.

.

.

.

.

.

.

.

X1a

X1b

X2a

X2b

X3a

X3b

TX1

TX2

TX3

RX1

RX2

RX3

Y1a = f(X2a)

Y1b

Y1c = f(X3b)

Y2a = f(X3a)

Y2b

Y2c = f(X1b)

Y3a = f(X1a)

Y3b

Y3c = f(X2b)

Figure 6.7: Overlap interference and transmission in K = 3 user IC with M = 2 and N = 3.

122


entropy. Similarly, (6.22) is simplified to

nRsum ≤ nN log ρ+ nR(k+1) − h(Xn(k+1)a) + n o(log ρ) + o(n). (6.24)

Adding (6.23) and (6.24) and summing over all k = {1, 2, 3},3∑

k=1

2nRsum ≤3∑

k=1

(2nN log ρ+ nR(k−1) + nR(k+1) − h(Xn(k−1)b)− h(Xn

(k+1)a) + n o(log ρ) + o(n)).

(6.25)

Further simplifying, we obtain

4nR ≤ 2nN log ρ− 1

3

[h(Xn

a ) + h(Xnb )]

+ n o(log ρ) + o(n), (6.26)

where h(Xna ) =

∑3k=1 h(Xn

ka), h(Xnb ) =

∑3k=1 h(Xn

kb) and R = Rsum

3, the rate per user.

Similarly, in the next overlap, (chain II in RCA), there is only one free antenna per receiver,

which satisfies

nRk ≤ h(Xnka) + h(Xn

kb). (6.27)

Summing over all k = {1, 2, 3}, (6.27) is further expressed as

nRsum ≤ h(Xna ) + h(Xn

b )⇒ nR ≤ 1

3

[h(Xn

a ) + h(Xnb )]. (6.28)

Now, adding (6.26) and (6.28),

5nR ≤ 2nN log ρ+ n o(log ρ) + o(n), (6.29)

⇒ 5R

log ρ≤ 2N +

o(log ρ)

log ρ+ o(n). (6.30)

As n→∞ and ρ→∞, we getR

log ρ≤ 2N

5=

6

5, (6.31)

which is the per user achievable DOF for M = 2, N = 3 and K = 3.

For general case with M transmit antennas and N receive antennas, with 0 < N−MM≤ 1

2, we

follow the similar approach but there are M inputs instead of two and total of 3M conditions

in the first overlap (chain I in RCA). Adding all the conditions and simplifying, we obtain:

2MnR ≤MNn log ρ− 1

3

[h(Xn

a ) + h(Xnb ) + · · ·+ h(Xn

m)]

+ n o(log ρ) + o(n). (6.32)

Also, for next overlap (chain II in RCA) there are N −M free antennas. Thus,

n(N −M)R ≤ 1

3

[h(Xn

a ) + h(Xnb ) + · · ·+ h(Xn

m)]. (6.33)

Adding (6.32) and (6.33), we get:

(M +N)nR ≤MNn log ρ+ n o(log ρ) + o(n), (6.34)

⇒ (M +N)R

log ρ≤MN +

o(log ρ)

log ρ+ o(n). (6.35)

123


As n→∞ and ρ→∞,R

log ρ≤ MN

M +N= d. (6.36)

Thus, we show that the DOF is information theoretically achievable for the the case when

0 ≤ N−MM≤ 1

2.

When N ≥ 3M , no alignment is required and M DOF is easily achievable. Let us consider

the case when 2M < N ≤ 3M . As given by RCA, the DOF per user in this case is outer

bounded by N3

. We validate this result using information theoretic approach.

Consider an example scenario with M = 2 and N = 5, that satisfies 2M < N ≤ 3M . Our

approach is based on optimum overlapping between the receivers as depicted in the figure 6.8,

where we observe that two layers of alignment are possible unlike the previous case.

By using the same information theoretic approaches and same notational definitions

explained for the previous case, we obtain the expression similar to (6.26) for the case of

M = 2 and N = 3 corresponding to the chain I of RCA. This is true because each transmitter

transmits M = 2 streams of data, which is same as in the previous case. Thus,

4nR ≤ 2nN log ρ− 1

3

[h(Xn

a ) + h(Xnb )]

+ n o(log ρ) + o(n). (6.37)

Each receiver still has two antennas for optimal overlapping the interfering signals, hence

they perform second layer of overlapping as shown in Figure 6.8. The second layer contains

two antennas more than the first layer, but still behaves as if there are only N2

antennas

due to complete overlapping of the first layer over second layer. The signals observed at

interfering antennas of RX1 is Y1I = (Y3d, Y3e, Y2a, Y2b), where Y3e is overlapped with Y3d

and Y2a is overlapped with Y2b; similarly, the signals observed at desired antennas of RX1

is Y1D = (Y1a, Y1b, Y1d, Y1e), where Y1a is overlapped with Y1b and Y1e is overlapped with Y1d.

Similar analysis is true for the signals observed at all the other receivers.

The conditional differential entropy corresponding to the signal at interfering antennas

given the signals at desired antennas for n channel uses are expressed as

h(Y n1I |Y n

1D) = h(Y n3d, Y

n2b|Y n

1b) ≤ h(Xn2a, X

n2b, X

n3a, X

n3b|Xn

2a, Xn2b),

= h(Xn3a, X

n3b|Xn

2a, Xn2b). (6.38)

Since (Xn3a, X

n3b) and (Xn

2a, Xn2b) are independent, we can further express (6.38) as

h(Y n1I |Y n

1D) ≤ h(Xn3a, X

n3b) = nR3. (6.39)

Similar analysis for the conditional entropy is valid for the other received signals at desired

antennas, which gives h(Y n1I |Y n

1D) ≤ nR2. Also, we perform similar analyses on the signals

received at the interfering and desired antennas corresponding to the RX2 and RX3, for which

124


.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

X1a

X1b

X2a

X2b

X3a

X3b

TX1

TX2

TX3

RX1

RX2

RX3

Y1a = f(X2a)

Y1b = f(X2b)

Y1c

Y1d = f(X3a)

Y1e = f(X3b)

Y2a = f(X3a)

Y2b = f(X3b)

Y2c

Y2d = f(X1a)

Y2e = f(X1b)

Y3a = f(X1a)

Y3b = f(X1b)

Y3c

Y3d = f(X2a)

Y3e = f(X2b)

Figure 6.8: An example analysis of alignment due to antenna sharing.

125


we obtain the conditional differential entropies h(Y n2I |Y n

2D) ≤ nR1, h(Y n2I |Y n

2D) ≤ nR3 and

h(Y n3I |Y n

3D) ≤ nR2, h(Y n3I |Y n

3D) ≤ nR1 respectively. For all the conditional entropies, we can

obtain the corresponding expression for Rsum as given by expressions (6.18) and (6.19) and

sum them all to obtain:

6nRsum ≤ 6nN

2log ρ+ 2nRsum + n o(log ρ) + o(n),

4nRsum ≤ 3nN log ρ+ n o(log ρ) + o(n),

4nR ≤ nN log ρ+ n o(log ρ) + o(n). (6.40)

imilarly, for the third chain, there is still one antenna remaining in each receiver, and

similar to the case of previous example, we obtain the expression

nR ≤ 1

3

[h(Xn

a ) + h(Xnb )]. (6.41)

Adding up (6.37), (6.40) and (6.41),

9nR ≤ 3nN log ρ+ n o(log ρ) + o(n), (6.42)

9R

log ρ≤ 3N +

o(log ρ)

log ρ+ o(n). (6.43)


log ρ≤ 3N

9=N

3=

5

3= D0, (6.44)

which is the per user DoF when M = 2 and N = 3 as obtained from the RCA.

For the general case with M transmit and 2M < N ≤ 3M receive antennas, the general

expression for chain I is given by (6.32). Since there are M more transmit antennas in chain

II, which is effectively halved due to overlapping,

3MnRsum ≤3MN

2log ρ+MnRsum + n o(log ρ) + o(n),

2MnR ≤ MN

2log ρ+ n o(log ρ) + o(n). (6.45)

For the chain III, with M transmit antennas, there are M2

antennas remaining at each

receiver which is exactly enough for optimal overlap and the general expression is given by

Mn

2Rsum ≤ h(Xn

a ) + h(Xnb ) + · · ·+ h(Xn

m),

Mn

2R ≤ 1

3[h(Xn

a ) + h(Xnb ) + · · ·+ h(Xn

m)], (6.46)

Adding (6.32), (6.45) and (6.46), we obtain

(4Mn+Mn

2)R ≤ 3MNn

2+ n o(log ρ) + o(n). (6.47)

126

Achievability of the DOF


log ρ≤ 3MN

2(9M2

)=N

3= D0. (6.48)

6.5 Achievability of the DOF

The precoding matrices are designed which align all the interferences in the common subspace

between any two interfering receivers. For the random post processing matrices designed at

the receiver, there are always two parts, one due to the common channel states and the other

due to the independent channel states. Thus at any receiver k, the post-processing matrix

Uk is defined as:

Uk =

[Uk1

Uk2

]∈ C(mk1+mk2)×d, (6.49)

where for the symmetric case with optimal overlapping,

mk2 = M and mk1 = N − M

2.

Hence, we consider M is even to ensure that N − M2

is an integer.

We know that for optimal overlapping, M is the number of overlapping antennas between

any two adjacent receivers where the interfering signals are aligned. Consider the case when

the interference signal from TX1 as observed by RX2 and Rx3 is aligned in the common

subspace formed by RX2 and Rx3, which is achieved by designing proper post-processing

matrix. In fact, the post processing matrix Uk at any receiver k has two parts, the first

part zero-forces the interference observed and the second part aligns the interference in the

common subspace formed by two adjacent receivers.

The alignment conditions for RX2 and RX3 in order to align the interference observed from

TX1 are then given by

(I) Aligning the observed interference over the subspace created from common antenna

elements:

UH22H

(c)21 V1 = UH

31H(c)21 V1, (6.50a)

⇒(UH22 −UH

31)H(c)21 V1 = 0, (6.50b)

⇒V1 ∈ N((UH

22 −UH31)H

(c)21

), (6.50c)

(II) Zero forcing the interference observed over the subspace created from independent

127


antenna elements in RX2:

UH21H

(i)21V1 = 0, (6.51a)

⇒V1 ∈ N(UH

21H(i)21

), (6.51b)

(III) Zero forcing the interference observed over the subspace created from independent

antenna elements in RX3:

UH32H

(i)31V1 = 0, (6.52a)

⇒V1 ∈ N(UH

32H(i)31

). (6.52b)

From these three conditions, we can state that

V1 ∈ {N((UH

22 −UH31)H

(c)21

)∩N

(UH

21H(i)21

)∩N

(UH

32H(i)31

)}. (6.53)

Furthermore, utilizing the property of null space that N (A) = N (AHA) for any given

matrix A [81], we can express (6.53) as:

V1 ∈ {N(CH

1 C1

)∩N

(CH

2 C2

)∩N

(CH

3 C3

)}, (6.54)

where C1 = (UH22 −UH

31)H(c)21 , C2 = UH

21H(i)21 and C3 = UH

32H(i)31 . Then,

CH1 C1 = H

(c)21

H(U22U

H22 + U31U

H31 −U22U

H31 −U31U

H22

)H

(c)21 ,

= H(c)21

H(U22U

H22 + U31U

H31 −U22U

H31 − (U22U

H31)H

)H

(c)21 , (6.55a)

CH2 C2 = H

(i)21

HU21U

H21H

(i)21 , (6.55b)

CH3 C3 = H

(i)31

HU32U

H32H

(i)31 . (6.55c)

Now, realizing the fact that N (A + B) = N (A) ∩ N (B for any positive semidefinite

matrices A and B [81, 84], we can express (6.54) as:

V1 ∈ {N(CH

1 C1 + CH2 C2 + CH

3 C3

)}. (6.56)

Since N (AH) = R(A), we have N (AH) ∩N (A) = {φ}, an empty set. Hence,

N (CH1 C1) = N

(H

(c)21

HU22U

H22H

(c)21

)∩N

(H

(c)21

HU31U

H31H

(c)21

),

= N(H

(c)21

H(U22U

H22 + U31U

H31)H

(c)21

). (6.57)

Hence from equations (6.55b), (6.55c), (6.56) and (6.57), the alignment is achieved when

V1 ∈ N(H

(c)21

H(U22U

H22 + U31U

H31

)H

(c)21 + H

(i)21

HU21U

H21H

(i)21 + H

(i)31

HU32U

H32H

(i)31

). (6.58)

Since H(i)21 , H

(c)21 , H

(i)31 and H

(c)31 are obtained from the channel matrices H21 and H31, let us

128


assume that,

H(i)21 = T11H21, H

(c)21 = T12H21 (6.59)

H(i)31 = T22H31, H

(c)31 = T21H31, (6.60)

where T11,T12,T21 and T22 are the 0− 1 selection matrices of relevant dimensions, that

depends on the dimensions of H21,H31 and H(i)21 , H

(c)21 , H

(i)31 , H

(c)31 .

Further, let us consider the summation terms in (6.58) be denoted by Σ such that

Σ =H(c)21

HU22U

H22H

(c)21 + H

(c)21

HU31U

H31H

(c)21 + H

(i)21

HU21U

H21H

(i)21 + H

(i)31

HU32U

H32H

(i)31 , (6.61)

=(T12H21)HU22UH22T12H21 + (T21H31)HU31U

H31T21H31 + (T11H21)HU21U

H21T11H21

+ (T22H31)HU32UH32T22H31, (6.62)

=HH21(TH

12U22UH22T12 + TH11U21U

H21T11)H21 + HH

31(TH21U31U

H31T21 + TH

22U32UH32T22)H31,

(6.63)

=HH21

{[0 0

0 U22UH22

]+

[U21U

H21 0

0 0

]}H21 + HH

31

{[U31UH31 0

0 0

]+

[0 0

0 U32UH32

]}H31,

(6.64)

=HH21

{[U21UH21 0

0 U22UH22

]}H21 + HH

31

{[U31UH31 0

0 U32UH32

]}H31. (6.65)

Now realize that H21 and H31 are obtained as sub-matrices of some matrix H1 such that

H21 = S1H1 and H31 = S2H1, then Σ is expressed as:

Σ = HH1

(SH1

[U21U

H21 0

0 U22UH22

]S1 + SH2

[U31U

H31 0

0 U32UH32

]S2

)H1. (6.66)

Hence, for any V1 which has the same dimensions as the dimensions of the null space of Σ,

V1 ∈ N (Σ)⇒ ΣV1 = 0⇒ VH1 ΣV1 = 0. (6.67)

However, the N (Σ) may not exist and V1 can have only trivial solution. But when we

substitute the value of Σ, we can express (6.67) as:

VH1 HH

1 ΦH1V1 = 0⇒ H1V1 ∈ N (Φ), (6.68)

where

Φ = SH1

[U21U

H21 0

0 U22UH22

]S1 + SH2

[U31U

H31 0

0 U32UH32

]S2. (6.69)

As we discussed previously, Φ is the higher dimensional embedding of the lower dimensional

subspace and the N (Φ) always exist and the achievable DOF with optimal overlap is the

dimensions of the N (Φ).

129

Conclusions

Since

H1V1 ∈ N(SH1 U2U

H2 S1 + SH2 U3U

H3 S2

), (6.70)

we can determine V1 for M dimensional overlapping Rx2 and Rx3. Similarly, we can

determine V2 and V3 for the M dimensional overlapping Rx3, Rx1 and Rx1, Rx2. This shows

that the precoder always exist as long as the optimal overlap is possible. Hence the chain

stops when the optimal overlap is not possible. This proofs the achievability of the scheme.

6.6 Conclusions

The DOF characterization of three user MIMO IC with arbitrary M transit and N receive

antennas is achieved via a scheme based on creating an overlapping space by the adjacent

receivers, called receiver chain alignment. The DOF is expressed in terms of the length of

the alignment chain. We provide a proof based on information theory and also presented

an achievable scheme in order to create an overlap space and designing a precoder which

achieves the overlap.

130

Chapter 7

Conclusions and Future Works

7.1 Concluding Remarks

Interference management is one of the challenging issues in the future multi-user wireless

communication systems and interference alignment (IA) is the promising scheme to improve

the degrees of freedom (DOF) in multi-user systems. In this thesis, we incorporated the

IA technique in the multi-user interference channels (ICs) and the interference broadcast

channels (IBCs).

A novel IA achieving algorithm is presented in the first part to design the precoder and

the zero-forcing matrix for any K-user IC. We showed that by using the hybrid scheme

where the receivers cooperate and the transmitters transmit greedily, we can design precoders

and zero-forcing matrices that achieve better results than the existing ones in the literature.

This scheme minimizes the interference power cooperatively and maximizes the signal power

greedily while most of the existing literatures such as in [2] only minimizes interference power

or maximizes signal power.

Instantaneous channel state information at the transmitter (CSIT) is difficult to obtain in

a time varying wireless channels and this limits the performance of IA. Existing literature

suggests that no benefits of IA can be achieved in the absence of instantaneous CSIT. However,

we presented schemes that achieve DOF gain with the help of delayed CSIT. Our main

contribution was to present a space time based transmission scheme to achieve better than

1 total DOF in a two-cell multiple input single output (MISO) IBC. In fact, we showed

that the total achievable DOF converges to 85

as the number of users approaches infinity.

Interference management in multi-cell MISO IBC is not well studied in literature and our

results which showed that with proper transmission scheme and proper knowledge of past

channel states, inter cell interference (ICI) and inter user interference (IUI) can be mitigated

wisely to improve the DOF is a strong contribution.

We showed that receivers can collectively align the received interference from any transmitter

onto a common subspace when there is enough overlapping between received signal dimensions.

131

Discussion and Future Works

In this regard, we considered multiple input multiple output (MIMO) IBC and three user

MIMO IC to show that optimal overlapping can be achieved. We showed that the interference

signal is contained in the null space of the so called alignment matrix, denoted by Φ and

derived the expression to determine the dimension of these null spaces which is the dimensions

of the overlap space.

The receiver chain alignment (RCA) scheme is presented to determine the DOF of three

user MIMO IC with arbitrary number of transmit and receive antennas by forming the chain

of optimal overlapping space between any two receivers and we showed that the DOF can

be expressed in terms of the length of such alignment chain. We provided an information

theoretic outer bound for such schemes and also showed that such schemes are achievable.

Thus, overall this thesis is a contribution in the interference management and DOF analysis

of the multi-user interference channels and interference broadcast channels and we present a

number of directions for the future research.

7.2 Discussion and Future Works

Most of the interference management works in multi-user communication are open. Especially,

the works regarding the DOF in multi-user (more than two) in different network scenarios

are not well studied. Hence, we list here some of the works where we can further work in this

thesis:

More than two cell IBC

We observed that even with the two cells, it is a challenging task to manage both IUI and

ICI. However, we also observed that the users can cooperatively align the received ICI onto a

common subspace. With more than two cells, precoders can be properly designed so as to

align the ICI observed from all the cells and the users in each cell can cooperatively create a

subspace to place these interference. However, the problem is still challenging due to more

variables.

Rank Deficient MIMO Channels

The concept of alignment matrix can be useful to solve number of MIMO channels and one

of them could be a rank-deficient MIMO IC [94]. We can easily create an alignment matrix

in such case and determine the overlap space by obtaining the null space of such matrix.

132


Alignment Matrix With Error

Most of the analysis we performed in chapter 5 are by assuming that the alignment matrix

were known perfectly but this is not always true. So we can further study on the alignment

matrix with error and design better practical approaches to create the overlap space.

By some abuse of notation, let us represent that Φ by Φ and the actual Φ obtained

trace[(HjVj)

H(Φ−Φ∗

)(HjVj)

], (7.1)

= trace[(HjVj)

H( K∑i=1,i 6=j

SHi (P⊥i −P∗i⊥)Si

)(HjVj)

], (7.2)

Since P∗i⊥ is not known, we would like to obtain the difference between the observed and

the exact value, which is measured as an error given by

||E||2 =K∑

i=1,i 6=j

||SHi(P⊥i −P∗i

⊥)Si||2, (7.3)

≤K∑

i=1,i 6=j

||P⊥i −P∗i⊥||2, (7.4)

≤K∑

i=1,i 6=j

||P⊥i P∗i ||2, (7.5)

=K∑

i=1,i 6=j

||(IN −UiUHi )SiHjVj(SiHjVj)

†||2, (7.6)

≤K∑

i=1,i 6=j

||(IN −UiUHi )SiHjVj||2||(SiHjVj)

†||2, (7.7)

=K∑

i=1,i 6=j

||Ei||2||SiHjVj||2

κ(SiHjVj) ≤K∑

i=1,i 6=j

||Ei||2, (7.8)

where κ(SiHjVj) = ||SiHjVj|| ||(SiHjVj)†|| is the condition number for matrix SiHjVj,

||Ei||2 = ||(IN −UiUHi )SiHjVj||2 is the local projection error observed at any receiver i. The

inequality (7.5) follows from the difference and product relationship of any two projection

matrices given by Krein-Krasnoselskii-Milman equality [95] and stated as

||P−Q||2 = max{||P(I−Q)||2, ||Q(I−P)||2}, (7.9)

for any two projection matrices P and Q and (7.7) follows directly from (7.6) by the property

of 2-norm.

By designing Ui and Vj iteratively, we aim to minimize the alignment error as much as

possible. We observed that the alignment error is decreasing iteratively and it is minimum

133


when the overlap is more as shown in Figure 7.1.

0 10 20 30 40 50 60 70 80 90 100

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Alignment Error for Varying Overlap

(N = 10,M = 8)

No. of iterations

Align

mentError

Overlap=8Overlap=6Overlap=4Overlap=2Overlap=0

Figure 7.1: Alignment error for different total overlap dimensions due to two receivers. Theerror is decreasing as the overlap is increasing.

134

Appendices

135

Appendix A

Proof of Theorem 5.1

We consider all the symbols and terms as defined in theorem 5.1. Using the singular value

decomposition (SVD), we decompose Z12 as:

Z12 = Z21 = Q1

[Σ1 0

0 0

]RH

1 , (A.1)

and further Z11 and Z22 as

Z11 = Q2

[Γ1 Σ2 0

Γ1 0 0

][I 0

0 RH2

]RH

1 , (A.2)

Z22 = Q3

[Γ2 Σ3 0

Γ2 0 0

][I 0

0 RH3

]RH

1 . (A.3)

where Q1 ∈ Cm12×m12 ,Q2 ∈ Cm11×m11 ,Q3 ∈ Cm22×m22 , R1 ∈ CM×M ,R2 ∈ CM−r1×M−r1 and

R3 ∈ CM−r2×M−r2 are the the unitary matrices; Σ1 ∈ Cr1×r1 ,Σ2 ∈ Cr2×r2 and Σ3 ∈ Cr3×r3are the diagonal matrices. Also, Γ1 and Γ1 are the top r2 rows and bottom m11 − r2 rows of

QH2 (Z11R1)(:,1:r1) respectively and Γ2 and Γ2 are the top r3 rows and bottom m22 − r3 rows

of QH3 (Z22R1)(:,1:r1) respectively, and

r1 = rank{Z12}, (A.4)

r2 = rank{(Z11R1)(:,r1+1:M)}, (A.5)

r3 = rank{(Z22R1)(:,r1+1:M)}. (A.6)

For the further simplification, consider Z1 = QH21Z1, for

Q21 =

[Q2 0

0 Q1

]∈ C(m11+m12)×(m11+m12) (A.7)

is the unitary matrix and based on the decompositions (A.1), (A.2) and (A.3), we obtain

span{Z1

}= span

{QH

21Z1

}= span

{Y1

}, where

Y1 =

0 I

W1 0

I 0

0 0

[R1 0

0 I

]∈ C(m11+m12)×(r1+r2), (A.8)

137


W1 = Γ1Σ−11 and R1 = (I + WH

1 W1)−12 . We are interested in the orthogonal complement of

span{Y1

}, the basis of which is given by

Y⊥1 =

0 0

I 0

−WH1 0

0 I

[D1 0

0 I

]∈ C(m11+m12)×(m11−r2+m12−r1), (A.9)

for D1 = (I + W1WH1 )−

12 .

Similarly, considering Z2 = QH13Z2 for the unitary matrix

Q13 =

[Q1 0

0 Q3

]∈ C(m12+m22)×(m12+m22). (A.10)

Also, based on the decompositions (A.1), (A.2) and (A.3) span{Z2

}= span

{QH

13Z2

}=

span{Y2

}, where

Y2 =

I 0

0 0

0 I

W2 0

[R2 0

0 I

]∈ C(m12+m22)×(r1+r3), (A.11)

W2 = Γ2Σ−11 and R2 = (I + WH

2 W2)−12 . The orthogonal complement of span

{Y2

}is given

by

Y⊥2 =

−WH

2 0

0 I

0 0

I 0

[D2 0

0 I

]∈ C(m12+m22)×(m22−r3+m12−r1), (A.12)

for D2 = (I + W2WH2 )−

12 .

By embedding the projection matrix due to Y⊥1 and the projection matrix due to Y⊥2 onto

m11 +m12 +m22 dimensional space and adding them , we obtain the alignment matrix Φ as

given by

Φ = ST1 P⊥Y1S1 + ST2 P⊥Y2

S2, (A.13)

for the given selection matrices S1 and S2. Since Y⊥1 and Y⊥2 are known, Φ is determined by

obtaining the row-wise embedding E1 and E2

E1 =

[Y⊥1

0

], E2 =

[0

Y⊥2

], (A.14)

138


and expressing as Φ = E1EH1 + E2E

H2 = EEH , where

E =[E1 E2

]=

0 0 0 0

D1 0 0 0

−WH1 D1 0 −WH

2 D2 0

0 I 0 I

0 0 0 0

0 0 D2 0

. (A.15)

Since E spans the same space as EEH , dim(Φ) = dim(E) = m11 +m12 +m22−(r1 +r2 +r3),

which is clear from the fact that m12 − r1 columns of E are the same.

The alignment matrix due to Z1 and Z2, Φ is related to the alignment matrix due to Z1

and Z2, Φ by the expression

Φ = QHΦQ, (A.16)

for Unitary matrix Q =

Q2 0 0

0 Q1 0

0 0 Q3

.

Hence rank(Φ) = rank(Φ) = m11 +m12 +m22−(r1 +r2 +r3) and rank(N (Φ)) = r1 +r2 +r3.

For the fully overlapped case, m11 + m12 + m22 = 2N, r1 = M, r2 = 0, r3 = 0, hence

rank(Φ) = 2N −M and rank(N (Φ)) = M = r1.

139


140

Appendix B


For the given ring topological setup with M transmit and N receive antennas, where M < N ,

we define the alignment matrix collectively formed by all the receivers due to any transmitter

j, i.e, Φj as:

Φj = SHj PZjSj +K∑

i=1,i 6=j

SHi P⊥ZiSi, (B.1)

as defined in theorem 5.2, where PZj = UjUHj are the projection observed due to the basis

matrix Uj at any receiver j and P⊥Zi = Ir −UiUHi are the projection observed due to the

orthogonal complement of the basis matrix Ui.

Now we can express N (Φj) as:

N (Φj) = N(SHj PZjSj +

K∑i=1,i 6=j

SHi P⊥ZiSi

), (B.2)

Using the property of the null space, i.e, the null space of the sum of any two positive

definite matrices A and B is the intersection of their null spaces as [81]

N (A + B) = N (A) ∩N (B), (B.3)

we can express

N (Φj) = N(SHj PZjSj

)⋂N( K∑i=1,i 6=j

SHi P⊥ZiSi

). (B.4)

Let us look at N(SHj PZjSj

)and N

(∑Ki=1,i 6=j SHi P⊥ZiSi

)separately. We have assumed

that M data streams are transmitted from any transmitter. Hence N(SHj PZjSj

)contains

all the space in KN dimensional region due to K receivers each with N antennas except the

M dimensional space because PZj is M dimensional.

Similarly N(∑K

i=1,i 6=j SHi P⊥ZiSi

)is the intersection of the null spaces of each K − 1 terms.

If K = 3, there are two terms N(SH2 P⊥Z2

S2

)and N

(SH3 P⊥Z3

S3

). Due to the projection onto

the orthogonal complement, N(SH2 P⊥Z2

S2

)contains only M dimensional space corresponding

to Rx2 and all other spaces corresponding to Rx1 and Rx3. Also, N(SH3 P⊥Z3

S3

)contains

141


only M dimensional space corresponding to Rx3 and all other spaces corresponding to Rx1

and Rx2. Hence when there is no overlap, the intersection between these two null spaces

is M dimensional spaces corresponding to each of the receiver and N dimensional space

corresponding to Rx1.

The total intersection space observed by all the receivers is thus N −M dimensions in Rx1,

M dimensions in Rx2 and M dimensions in Rx3. Since they are independent, the rank of the

null space when there is no overlapping is N −M +M +M = N +M .

When there areNc overlap between the adjacent receivers, then the null spaceN(SH1 PZ1S1

)still contains all the available space except M dimensional space in Rx1. However the

N(SH2 P⊥Z2

S2

)contains Nc less dimensions from Rx1 and another Nc less from Rx3. Similarly,

N(SH3 P⊥Z3

S3

)contains Nc dimensions less from Rx1 and Nc dimensions less from Rx2. Hence

the intersection space decreases by 4Nc and the rank of the intersection space is N +M −4Nc

as stated in the theorem.

After Nc = M2

overlapping, the total intersection space depends only on the space observed

at Rx1 because then Rx2 and Rx3 completely overlap all the received interference signal and

increasing Nc does not affect the intersection space at Rx2 and Rx3. However increasing Nc

decreases the intersection space observed by Rx1 by two times due to the overlap of Rx1 with

Rx2 and Rx3. Hence the dimension of the N (Φj) varies as N − 2Nc.

142

Appendix C


The proof follows similarly as in the ring topology by designing the suitable alignment matrix

due to any transmitter j collectively observed by all the receivers and given by:

Φj = SHj PZjSj +K∑

i=1,i 6=j

SHi P⊥ZiSi. (C.1)

Similarly, we can express N (Φj) as

N (Φj) = N(SHj PZjSj

)⋂N( K∑i=1,i 6=j

SHi P⊥ZiSi

). (C.2)

When there is no overlap, the rank of N (Φj) is same as ring topology and is given by

N +M . When there is overlap of Nc rows per user, then total of KNc common dimensions

always decrease and Nc dimensions decrease from Rx1 due to being in the overlap. Hence

the total intersection dimension is again N +M − (K + 1)Nc. This is valid as long as the

number of overlapping rows is MK

. When overlapping rows per user is greater than MK

, the

total number of overlaps is greater than M and the dimensions of N (Φj) no longer varies

with M but only varies linearly with N −Nc.

143

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