Interference Alignment inCellular Networks: Theory ... · Interference Alignment in Cellular Networks: Theory, Algorithms, and System Design Gokul Sridharan Doctor of Philosophy Graduate

Interference Alignment in Cellular Networks: Theory, Algorithms,and System Design

by

Gokul Sridharan

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Electrical and Computer EngineeringUniversity of Toronto

Copyright © 2015 by Gokul Sridharan

Abstract

Interference Alignment in Cellular Networks: Theory, Algorithms, and System Design

Gokul Sridharan

Doctor of Philosophy

Graduate Department of Electrical and Computer Engineering

University of Toronto

2015

Degrees-of-freedom (DoF) is a useful yet tractable metric in characterizing the capacity of wireless multi-

antenna cellular networks. DoF provides a first-order approximation (pre-log factor) to capacity and

reflects capacity scaling at high signal-to-noise ratios. Interference alignment is a concept that is used to

design linear beamformers that achieve the optimal DoF of multi-antenna wireless networks. This thesis

focuses on the role of interference alignment in establishing the maximal DoF of multi-antenna cellular

networks and its relevance in practical cellular network optimization. The primary focus is on studying

the symmetric DoF of cellular networks that consist of G mutually interfering cells with K users/cell,

having N antennas at each base station and M antennas at each user.

The first part of this thesis investigates the achievable DoF using two different strategies for inter-

ference alignment. The first is an asymptotic alignment scheme that is typically used in single-antenna

networks and relies on symbol extensions in time, and the second is a linear beamforming strategy that

exploits overlap of transmission subspaces in multi-antenna receivers. It is shown that there are distinct

regimes of G, K, M , and N , where one technique outperforms the other. Using a set of outer bounds

on the sum-DoF, conditions for optimality of the DoF achieved using the two techniques is established.

Studying the optimal DoF highlights the importance of developing algorithmic techniques for in-

terference alignment. The second part of this thesis focuses on the design of linear beamformers for

interference alignment. Conditions for interference alignment are formulated as a rank minimization

problem and insights from compressive sensing are used to develop algorithms that can efficiently solve

this problem. The proposed algorithms are shown to be more effective in designing aligned beamformers

for interference alignment than other existing algorithms.

The final part of this thesis explores the value of interference alignment in practical cellular network

optimization. Using a two-stage optimization framework, it is established that interference alignment is

valuable in dense cellular networks where interference alignment provides an altered interference land-

scape that subsequent network utility maximization algorithms can take advantage of, but is otherwise

difficult to reach.

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iii

Acknowledgements

This thesis would not have been possible without the help, advice, and guidance I received from

several people. The first and foremost on this list is my advisor, Prof. Wei Yu. Prof. Yu has been

instrumental in shaping my research outlook and I have learnt a great deal from my interactions with

him. I am grateful to him for giving me considerable latitude in choosing my area of research, while

constantly shaping the overall objectives through our weekly meetings. In particular, his ability to ask

the right questions have helped me to consider things from a new perspective and have often led to new

avenues of research. I would also like to thank him for his time and effort in editing and improving

my academic papers. I always looked forward to our weekly meetings and it is something that I will

certainly miss.

I would like to thank my thesis committee members, Prof. Syed A. Jafar, Prof. Frank R. Kschischang,

Prof. Ashish Khisti and Prof. Costas D. Sarris, for spending considerable time reading my thesis

and providing valuable feedback. In particular, I would like to thank Prof. Frank R. Kschischang

for scrutinizing and helping improve the language and notation in several parts of this thesis. Their

comments and suggestions have helped bring better clarity to this document.

I am particularly indebted to my friend and colleague Siyu Liu for his help in proving certain im-

portant results in the last chapter of this thesis. Without his expertise and generosity with his time,

it would not have been possible to establish these results. I also thank Kianoush Hosseini for several

insightful discussions, at all odd hours, on various aspects of this thesis.

I also extend my thanks to Mary Stathopoulos of the Communications Group, Jayne Leake at the

Undergraduate Office, and Judith Levene and Darlene Gorzo at the Graduate Office, for all their help

in handling several academic and administrative matters. Their role in allowing graduate students focus

on their research is highly understated.

My stay at the University of Toronto would not have been as enjoyable without the company of

my colleagues in BA7114 and BA4162. In particular, I thank Binbin Dai, Soroush Tabatabaei, Yuhan

Zhou, Yicheng Lin, Siyu Liu, Kianoush Hosseini, Pratik Patil, Louis Tan, and Kaveh Mahdaviani for

providing a congenial atmosphere to work in. I am grateful for their friendship and will cherish the long

conversations we have had on various topics. I would be remiss not to mention my flatmates, Shreyas

Potnis and Vijay Shankar Venkataraman, whose invaluable company made my grad-school journey all

that more memorable.

Finally, I would like to thank my family for their unrelenting support and encouragement all through

my meandering academic endeavours. My heartfelt gratitude also goes to my wife for her patience and

understanding over all these years.

I am grateful to the Graduate Student Fellowship program at the University of Toronto and the

Natural Science and Engineering Research Council (NSERC) of Canada for providing financial assistance

during my graduate studies.

iv

Contents

1 Introduction 1

1.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Degrees of Freedom and Interference Alignment . . . . . . . . . . . . . . . . . . . . 5

1.2.2 Algorithms For Designing Aligned Beamformers . . . . . . . . . . . . . . . . . . . 8

1.3 Role of Interference Alignment in System Design . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Degrees of Freedom of MIMO Cellular Networks 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Decomposition Based Schemes: Achievable DoF and Conditions for Optimality . . . . . . 16

2.3.1 Achievable DoF using decomposition based schemes . . . . . . . . . . . . . . . . . 16

2.3.2 Outer Bounds on the DoF of MIMO Cellular Networks . . . . . . . . . . . . . . . 19

2.3.3 Optimality of the DoF Achieved Using Decomposition . . . . . . . . . . . . . . . . 20

2.3.4 Insights on the Optimal DoF of MIMO Cellular Networks . . . . . . . . . . . . . . 22

2.4 Linear Beamforming: Structured Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.2 Packing Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.3 Extending packing ratios to larger networks . . . . . . . . . . . . . . . . . . . . . . 32

2.5 Linear Beamforming Design: Unstructured Design . . . . . . . . . . . . . . . . . . . . . . 32

2.5.1 The Unstructured Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5.2 Numerical Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5.3 Unstructured Approach for MIMO Interference Channel . . . . . . . . . . . . . . . 37

2.5.4 USAP-uplink for MIMO Cellular Networks . . . . . . . . . . . . . . . . . . . . . . 40

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Interference Alignment Via Rank Minimization 44

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.1 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

v

3.1.2 Proposed Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4 Interference Alignment as Rank Minimization . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4.1 RFNM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.2 RNNM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.5 Interference Alignment Using AM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5.1 AM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5.2 Convergence and Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.6 Comparison to Existing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.7.1 The (4, 1, 2× 3) and (4, 1, 3× 3) networks . . . . . . . . . . . . . . . . . . . . . . . 61

3.7.2 Larger Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 Role of Interference Alignment in Cellular Network Optimization 69

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3 Feasibility of Partial Interference Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 Optimization Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4.1 Stage I: Partial Interference Alignment . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.4.2 Stage II: Utility Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5 Conclusion 80

Appendices 83

A DoF Outer Bound for the Two-Cell Three-Users/Cell Network 83

B Achievability of the Optimal sDoF 89

C Proof of Theorem 4.3.1 94

C.1 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

C.1.1 Transcendental Field Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

C.1.2 Zariski Topology and a Theorem of Chevalley . . . . . . . . . . . . . . . . . . . . . 95

C.2 Proof of Theorem 4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Bibliography 100

vi

List of Tables

2.1 Packing ratios for the two-cell two-user/cell network . . . . . . . . . . . . . . . . . . . . . 31

2.2 Packing ratios for the two-cell three-user/cell network . . . . . . . . . . . . . . . . . . . . 31

3.1 Systems used in simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

vii

List of Figures

1.1 Multi-cell network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 A three-user interference channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Interference Alignment in a three-user interference channel . . . . . . . . . . . . . . . . . . 7

2.1 2-D Wyner model of a cellular network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Insights on DoF of MIMO Cellular Networks . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 sDoF of the 2-cell, 2-user/cell MIMO cellular network . . . . . . . . . . . . . . . . . . . . 26

2.4 sDoF/user of the 2-cell, 3-user/cell MIMO cellular network . . . . . . . . . . . . . . . . . 27

2.5 Achieving 3 DoF/user in a (2, 3, 9× 12) network . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 Inner and outer bounds on the DoF of the G-user interference channel . . . . . . . . . . . 36

2.7 Results of the numerical experiment for the three-user interference channel . . . . . . . . 37

2.8 Results of the numerical experiment in region I of the the four-user interference channel . 38

2.9 Results of the numerical experiment in region II of the four-user interference channel . . . 39

2.10 Inner and outer bounds on the DoF of the G-cell, K-user/cell network . . . . . . . . . . . 40

2.11 Results of the numerical experiment for the two-cell, four-user/cell network . . . . . . . . 41

2.12 Results of the numerical experiment in region II of the three-cell, two-user/cell network . 42

3.1 Interference alignment in the (4, 1, 2× 3) network . . . . . . . . . . . . . . . . . . . . . . 59

3.2 Interference alignment in the (4, 1, 3× 3) network . . . . . . . . . . . . . . . . . . . . . . . 60

3.3 Interference alignment in the (4, 1, 2× 3) and (4, 1, 3× 3) networks using RNNM algorithm 62

3.4 Interference-free dimensions in the (4, 1, 2×3) and (4, 1, 3×3) networks using the WRCRM,

RCRM and RNNM algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5 Network-wide alignment using the RFNM, AM and ILM algorithms. . . . . . . . . . . . . 64

3.6 Network-wide alignment using the RFNM, AM and ILM algorithms. . . . . . . . . . . . . 65

3.7 Interference-free dimensions in the (3, 2, 3×4) and (3, 2, 4×4) networks using the WRCRM,

RCRM and RNNM algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.8 Network-wide alignment using RNNM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1 Illustration of the sufficient condition for feasibility of PIA in Corollary 4.3.2. . . . . . . 73

4.2 The proposed optimization framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3 Network topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.4 Per-cell throughput in a (3,K, 3× 4) network . . . . . . . . . . . . . . . . . . . . . . . . . 77


viii


A.1 The signal structure obtained after linear transformation for the case when γ ≤ 2/3. Note

that the figure does not include signals from the same cell. . . . . . . . . . . . . . . . . . . 84

A.2 The signal structure obtained after linear transformation when γ ≥ 2/3. The figure does

not include signals from the same cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

C.1 Structure of the Jacobian matrix. Shaded regions depict non-zero partial derivatives. . . 99

ix

Chapter 1

Introduction

With ever-increasing demand for data on wireless networks, networks are getting denser while also max-

imally reusing the available bandwidth. When multiple transmitter-receiver pairs communicate over a

shared wireless medium, interference is inevitable. Such networks are fundamentally limited by interfer-

ence and effective interference management is critical to their efficient operation. Addressing interference

in such networks calls for network enhancements by either adding infrastructure for cooperation among

the transmitters or by adding more physical resources, such as antennas. These enhancements can then

be exploited to devise effective strategies for interference mitigation.

In multi-antenna cellular networks, additional infrastructure for cooperation among the base stations

(BSs) can be used for sharing control signals, user data, and channel-state information (CSI). Sharing

information about the network permits a centralized design of the transmission parameters. When both

data and CSI are shared, the resulting network architecture is commonly referred to as coordinated

multi-point transmission (CoMP) or network-MIMO (multiple-input and multiple-output). When only

CSI or other control parameters are shared, the resulting architecture is known as inter-cell interference

coordination (ICIC). The addition of a large number of antennas, results in an architecture called

massive-MIMO where the BSs are equipped with a large antenna array and transmit in an uncoordinated

manner. Such an architecture is founded upon the premise that a large number of spatial dimensions

leads to a natural separation between interference and signal, and requires no further coordination to

mitigate interference.

A common thread connecting all three architectures is the use of spatial resources (antennas) to

manage interference and to serve more users. Unlike bandwidth, which is a scarce and expensive resource,

adding antennas is a relatively low-cost solution to increase the capacity of a multi-user network and

is particularly useful in addressing interference. Through the use of transmit and receive beamforming

vectors, transmitters can steer the signal beams to minimize interference at the unintended receivers.

The receivers can in turn choose to receive the transmitted signal in a direction that is best suited to

reject interference from concurrent transmissions while preserving signal strength. In essence, spatial

resources are useful in three ways: (a) they can be used to serve multiple users simultaneously; (b)

they can be used to enhance signal strength (matched filtering); and (c) they can be used to cancel

interference (zero-forcing beamformers). Understanding this multi-faceted use of spatial resources is one

1

Chapter 1. Introduction 2

Figure 1.1: A multi-cell network having three mutually interfering cells with four users per cell.

of the goals of this thesis.

From an information-theoretic standpoint, characterizing the capacity of multi-antenna cellular net-

works remains a challenging open problem. In recent years, while the capacity of a single-cell multi-

antenna network (Gaussian vector broadcast channel) has been established [1], generalizations to the

multi-cell case appear non-trivial. With exact capacity characterization proving to be very challeng-

ing, an alternate metric called degrees-of-freedom (DoF) has emerged as a useful yet tractable metric

to understand the fundamental limits of such networks [2–4]. DoF reflects capacity scaling at high

signal-to-noise ratios (SNRs) and can be thought of as the number of interference-free dimensions that

can be created in a network. While studies on finer characterizations of capacity tend to get caught in

the peculiarities of the network under consideration, DoF appears to strike the right balance between

the granularity of capacity characterization and the ability to provide broad insights on a wide class

of cellular networks. With the intent of providing a broad perspective on the fundamental limits of

multi-antenna cellular networks, DoF is the primary metric of interest in this thesis.

In particular, this thesis studies the DoF of a cellular network consisting of G mutually interfering

cells with K users/cell having M antennas at each user and N antennas at each BS—denoted as a

(G,K,M ×N) network. Fig. 1.1 shows one such network with three interfering cells having four users

each. Of specific interest is the symmetric DoF of such networks, i.e., the maximum DoF per user that

can be simultaneously supported. Due to the symmetry of the networks under consideration this is

equivalent to studying the sum-DoF of these networks. The focus is on symmetric networks because it

allows us to establish insights that are broadly applicable without getting drawn into the peculiarities

that arise from asymmetric network configurations. The wireless channels between the BSs and the users

are assumed to be perfectly known at a centralized location, but user data is not assumed to be shared,

i.e., the focus is on an ICIC architecture.

There are three parts to this thesis. The first part is a theoretical study on the symmetric DoF of

a (G,K,M ×N) network. The primary focus is on the DoF achieved using two contrasting techniques

that are both developed based on a concept called interference alignment (IA). In simple terms, IA is the

notion of minimizing the dimensions spanned by interference at a multi-antenna receiver where the total

number of dimensions at a receiver is equivalent to the total number of antennas at that receiver. In

addition to spatial dimensions, extensions in time or frequency can also contribute to the total number

of dimensions at a receiver. IA is pivotal to all DoF studies so far, and plays an important role in this

thesis.

Motivated by insights from the DoF studies of the first part, the second part of this thesis focuses on

developing algorithms for IA. These algorithms are particularly useful in networks where non-algorithmic


ways of achieving IA are not yet known.

The last part of this thesis focuses attention on the role of interference alignment in cellular net-

work optimization. Cellular network optimization focuses on designing transmission parameters such as

beamformers and transmit powers to maximize a network-wide utility function [5]. While there exist

several novel techniques to solve these problems, due to the non-convex nature of such an optimization,

these algorithms only succeed in finding a locally optimal solution. The suboptimal nature of these

techniques provides scope for improvement through theoretical insights on the capacity of multi-antenna

cellular networks. This part of the thesis investigates whether insights from the DoF analysis are useful

in practical cellular network optimization.

A minor point worth mentioning here is that although the term “cellular network” is widely used

throughout this thesis, several results presented here are applicable to any wireless network with a

similar architecture. In general, these results can be applied to any wireless network where a transmitter

simultaneously serves multiple receivers in the presence of several other similar transmissions (interfering

broadcast channels). These results apply to such networks irrespective of their scale, geometry, or

channel models. For example, a cluster of WiFi routers, a distributed antenna system with dedicated

backhaul networks or heterogeneous networks with transmitters of different capabilities are all possible

applications.

Before proceeding further, the signal and transmission model used in this thesis is presented next.

1.1 System Model

Consider a (G,K,M × N) network. Let the complex channel gains in the uplink from the kth user

in the gth cell to the ith BS be denoted as the N × M matrix H(gk,i). Let H(i,gk) denote the cor-

responding downlink channel. Uplink-downlink channel reciprocity is assumed throughout, so that

H(gk,i) = HH(i,gk). Since most wireless channels are frequency selective, such a channel model assumes an

underlying orthogonal-frequency-division-multiplexing (OFDM) implementation and the matrix H(i,gk)

represents the wireless channel gains over a single sub-carrier.

The channels gains are assumed to be randomly drawn according to a cumulative distribution function

in CG2KMN that is absolutely continuous. Further all the conditional cumulative distribution functions

of this distribution are also assumed to be absolutely continuous. The first condition allows us to assume

that the probability of a measure zero set of channel gains occurring is equal to zero and the second

condition precludes scenarios where a channel gain becomes deterministic once a subset of the channel

gains is known. Two closely related notions that are repeatedly used in this thesis are the notions of

generic properties and generic channels. A generic property, as defined in Appendix C, is any property

that is true over a non-empty Zariski open set of the channel matrices. Informally, Zariski closed sets

are defined to be roots of a system of polynomial equations and are known to have measure zero (w.r.t.

CG2KMN in the present context). Since Zariski closed sets have measure zero, any generic property is

almost surely true for any instance of channel matrices drawn from a distribution satisfying the first two

conditions. Any channel satisfying a particular generic property is said to be a generic channel of that

property. Thus, if generic channels of a generic property are proven to satisfy a given condition, it can

then be concluded that this condition is almost surely true under the assumed distribution of channel


matrices. Often in this thesis, generic channels are assumed without explicitly stating the generic

property that they satisfy. The property can however be easily inferred from the context. Examples of

generic properties include non-collinearity of three points on a plane, linear independence of the columns

of a channel matrix, etc.

The channels are also assumed to be time varying when extensions in time are considered.

In the uplink, assuming the kth user in the gth cell transmits the M×1 signal vector xgk, the received

signal at the ith BS is given by

yi =

G∑

g=1

K∑

k=1

H(gk,i)xgk + ni, (1.1)

where yi is an N × 1 vector and ni is the N × 1 vector representing circularly-symmetric additive white

Gaussian noise ∼ CN (0, σ2I). The received signal is defined similarly for the downlink.

If the transmit signal vector xgk is formed through an M × d transmit beamforming matrix Vgk,

then the received signal can be written as

yi =

G∑

g=1

K∑

k=1

H(gk,i)Vgksgk + ni, (1.2)

where sgk is the d× 1 symbol vector transmitted by the kth user in the gth cell. Note that each user is

simultaneously served by d data steams and the matrices Vgk consist of d column vectors. To recover

the signal transmitted by jth user in the ith cell, the signal received by the ith base station is processed

using an N × d receive beamformer Uij and the received signal after being projected by Uij is written

as

UHijyi =

G∑

g=1

K∑

k=1

UHijH(gk,i)Vgksgk +UH

ijni. (1.3)

In the uplink, users are constrained by a sum-power constraint that requires the transmit beamformers

to satisfy ‖Vgk‖2F ≤ Pmax, where Pmax is maximum transmitted power and ‖ · ‖F denotes the Frobenius

norm. When the beamformers Ugk are used for downlink transmission, they are required to satisfy

K∑

k=1

‖Ugk‖2F ≤ Pmax, ∀g. (1.4)

Mathematically, the total uplink DoF (referred to as the optimal DoF) of a network is defined as

lim supρ→∞

[

sup{Rgk(ρ)}∈C(ρ)

(R11(ρ) +R12(ρ) + . . .+RGK(ρ)

)

log(ρ)

]

(1.5)

where ρ is the transmit SNR (Pmax/σ2), {Rgk(ρ)} is an achievable rate tuple for a given SNR where Rij

denotes the rate to the jth user in the ith cell, and C(ρ) is the capacity region for a given SNR. DoF in

the downlink is defined similarly.

Although such a description gives the impression that the capacity region is to be known first, this is

however not necessary. Suppose there exists an appropriate choice of transmit and receive beamformers

such that the interfering signals are completely nulled at each receiver, this results in a network consisting


of multiple point-to-point links each with a received signal of the form y = Hx + n, where H is the

effective channel matrix. Once the network is reduced to a set of point-to-point links, the total DoF

that can be achieved is simply the sum of the DoFs that can be achieved over the individual links. The

number of DoF that can be achieved over one such point-to-point link is determined by the rank of the

effective channel matrix. The focus of this thesis is on reducing a given network to a set of parallel

point-to-point links through a careful design of the transmit and receive beamformers while ensuring the

effective channel matrix has full rank. Such an approach has proved effective in achieving the optimal

DoF of several wireless networks.

Throughout this thesis, the direct and cross channel gains are assumed to be perfectly known and

available at a centralized location. However, in practice, channel gains can only be estimated to a finite

precision and even this entails significant overhead. CSI is typically obtained using either uplink or

downlink pilots [6]. When CSI is estimated in the uplink, the users transmit orthogonal pilot sequences

that are received at various BSs in the network and used to estimate the direct and cross channel gains.

The quality of the estimate is influenced by factors such as pilot contamination (due to pilot sequence

reuse), pilot transmission powers etc. Further, due to processing delays, there is always a delay between

when a CSI estimation is made and when it is available. Thus, in practice, it is reasonable to assume

that only delayed CSI, estimated to a certain precision, is available. The estimated CSI is only valuable

for the duration during which the channel remains unchanged. This is determined by the coherence time

of the channel and affects the rate at which pilots must be transmitted.

Perfect knowledge of CSI is a significant assumption made in this thesis and is vital to establish

several theoretical results. It must be acknowledged that measuring CSI is rarely straightforward and

significant thought must be exercised before even deciding to measure these channels. Mobility and the

time varying nature of wireless channels play a crucial role in such decisions (refer to [7] and [8] for a

detailed analysis on these issues). It is nevertheless important to explore the possibilities when such

information is available. Even when the results are not directly applicable, some key insights that such

a study provides may still be applicable when only coarse or partial CSI is available. Further, there

could be applications in future where the theoretical insights and algorithmic techniques developed here

are readily applicable. In particular, the design of wireless backhaul networks where channel coherence

times are much higher, and mobility is no longer an issue hold significant promise for such techniques.

1.2 Thesis Overview

This section elaborates further on the three key aspects of this thesis and highlights some of the important

contributions.

1.2.1 Degrees of Freedom and Interference Alignment

As mentioned earlier, recent years have witnessed a shift to understanding cellular networks from a

DoF perspective. Starting with the work on the DoF of a 2-user MIMO interference channel, where

two transmitters communicate with their corresponding receivers over a wireless medium [2], subsequent

papers have studied the DoF of various other networks. A key breakthrough that accelerated these studies


Figure 1.2: A three-user interference channel.

emerged from an investigation of the DoF of the 2-user X network where two transmitters communicate

with two receivers, with each transmitter having an independent message for each receiver [4, 9]. This

work introduced a novel concept called interference alignment to establish the optimal DoF.

To illustrate the basic idea behind IA, consider a 3-user MIMO interference channel as shown in Fig.

1.2. Assume that each node has two antennas. It is known that in such a channel 1 DoF/user can be

achieved simultaneously, i.e., one interference free data stream can be transmitted concurrently between

the three transmitter-receiver pairs [3]. Let v11, v21 and v31 be the 2× 1 linear transmit beamforming

vectors at each of the three transmitters and u11, u21 and u31 be the corresponding receive beamformers.

IA requires these beamformers to be designed such that interference is completely nulled at all three

receivers so that an interference-free direction for transmission is available at each of the receivers.

Eliminating interference allows the transmission rate between each transmitter-receiver pair to grow

linearly in log(SNR) thus proving that 1 DoF/receiver is achievable in this network.

In mathematical terms, the beamformers are required to satisfy

uHi H(g1,i)vg1 = 0, ∀g, i ∈ {1, 2, 3}, g 6= i,

while ensuring that uHi H(i1,i)vi 6= 0 ∀i ∈ {1, 2, 3}. Beamformers that satisfy such a condition are called

aligned beamformers and it can be shown that beamformers satisfying such constraints exist for such a

network.

The term alignment refers to the fact that although each receiver sees three transmissions, the

two interfering transmissions are aligned into a single dimension at each of the receivers, leaving one

dimension for receiving the signal of interest. This is illustrated in Fig. 1.3 where overlapping arrows

represent interfering vectors that are aligned and lie along the same direction. Note that different sets of

vectors need to align at different receivers. Further note that it suffices for the aligned interfering vectors

and the received signal vectors to be linearly independent and not orthogonal for them to be separable.

Establishing such a result for the 3-user single-antenna interference channel is significantly more

challenging, and relies upon an asymptotic scheme for IA that was developed in a landmark paper by

Cadambe and Jafar [3]. This scheme assumes a time varying channel and considers beamforming across

multiple time slots. Over N time slots, this is equivalent to a N ×N MIMO interference channel where

the channel matrices are diagonal. Relying on the commutativity of such channels (diagonal matrices

commute), an ingenious scheme for designing aligned beamformers is proposed and this scheme is shown

to achieve 1 DoF/receiver in the limit as N → ∞.

The two schemes for IA, namely, the asymptotic scheme over time-extended channels where channel

matrices are diagonal and the linear beamforming scheme over MIMO channels where channel matrices


Figure 1.3: IA in a three-user interference channel. The three transmissions are represented in three

different colours. At each receiver the interfering vectors align while the desired signal vector is separable

from aligned interference.

are full (i.e., matrices have no structure), form the core components of our analysis of the symmetric DoF

of a (G,K,M ×N) network. Note that while both of the above two schemes design linear beamformers,

they differ over the kind of channel matrices on which they are applied—one relies on diagonal channels

and exploits their commutativity while the other assumes full channel matrices and exploits the co-

location of multiple antennas at a node. For convenience, the two schemes are informally referred to as

the asymptotic scheme and the linear beamforming scheme in the rest of the thesis.

Chapter 2 of this thesis investigates the achievable DoF using the above two schemes in a (G,K,M×N) network and shows that there are distinct regimes where one outperforms the other. Since the asymp-

totic scheme relies on commutativity, applying this technique requires decomposing the multi-antenna

nodes of a MIMO network into multiple single-antenna nodes. Using decomposition of multi-antenna

nodes, the DoF achieved by the asymptotic scheme in a given (G,K,M ×N) network is characterized.

On the linear beamforming front, two contrasting approaches for designing aligned beamformers are pro-

posed. A structured design takes a methodical approach to designing aligned beamformers by carefully

accounting for alignment of interfering vectors and the dimensions occupied by interference and signal

at each multi-antenna receiver. While such a approach is shown to be useful in small networks (i.e.,

small G and K), an alternate approach that uses an unstructured design is shown to be more versatile

in designing aligned beamformers for networks of any size. This versatility comes at the expense of

requiring a one-time numerical check to prove the feasibility of designing aligned beamformers using this

approach.

Through information-theoretic outer bounds, the conditions for optimality of the DoF achieved using

the proposed methods is established. Important optimality results include:

1. The optimal symmetric DoF of any G-cell network with single-antenna users, i.e., any (G,K, 1×N)

network, is characterized.

2. The structured approach to design aligned beamformers is shown to achieve the optimal sym-

metric DoF of the 2-cell 2-user/cell network and the 2-cell 3-user/cell network for any antenna

configuration.

3. When linear beamforming schemes are expected to outperform the asymptotic scheme, numerical

tests suggest that the unstructured approach is capable of designing the DoF-optimal aligned

beamformers for networks of any size.


1.2.2 Algorithms For Designing Aligned Beamformers

From the results of the analysis in Chapter 1, it is observed that there exist scenarios where design-

ing aligned beamformers to achieve a certain DoF in a given network is not possible using either the

structured or the unstructured approaches. Such scenarios motivate the need for developing numerical

techniques to design aligned beamformers.

Going back to the earlier example on the 3-user interference channel, IA requires the transmit and

receive beamformers to satisfy

uHi1H(g1,i)vg1 =0, g, i ∈ {1, 2, 3}, g 6= i,

uHi1H(i1,i)vi1 6=0, ∀i ∈ {1, 2, 3}.

(1.6)

Ignoring the second set of conditions, observe that the aligned beamformers are solutions to a polyno-

mial system of equations. Assuming this system of equations to be feasible, solving this system and

subsequently verifying the second set of conditions provides one way of designing aligned beamformers.

In general, the question of feasibility is resolved through techniques based on algebraic-geometry without

explicitly constructing aligned beamformers. Relying on such results for feasibility, several algorithms

aim to solve a polynomial system of equations, similar to that in (1.6), that emerge under more general

settings.

Chapter 3 takes a different approach to finding aligned beamformers. The key insight comes from

recognizing that IA imposes a set of rank constraints on the set of interfering vectors at each receiver.

For example, in the 3-user interference channel, IA requires the two interfering vectors at each receiver

to occupy a single dimension. This translates to the following set of equivalent conditions for IA:

rank([H(g1,i)vg1 H(j1,i)vj1]) ≤1, ∀i, j, g ∈ {1, 2, 3}, i 6= j 6= g,

uHi1H(i1,i)vi1 6=0, ∀i ∈ {1, 2, 3}.

In general, this perspective frames IA as designing a set of low-rank interference matrices and natu-

rally lends itself to algorithmic techniques first developed in the context of compressive sensing. The

interference matrix is the collection of all interfering vectors at a receiver. For example, in the 3-user

MIMO interference channel considered above, the interference matrices for the three receivers are given

by [H(21,1)v21 H(31,1)v31], [H(31,2)v31 H(11,2)v11] and [H(11,3)v11 H(21,3)v21].

By reformulating the conditions for IA as a set of rank constraints, Chapter 3 develops two sets of

algorithms for IA. The first part of this chapter develops rank minimization algorithms that design aligned

transmit beamformers by iteratively minimizing a weighted matrix norm of the interference matrix. Prior

knowledge of the expected rank of the interference matrices plays a crucial role in algorithms designed

to minimize the matrix norm. The second part of this chapter utilizes the prior rank knowledge in a

different way. It devises an algorithm where the rank-deficient interference matrices are expressed as a

product of two lower-dimensional matrices. The two components are alternately optimized while keeping

the other fixed. Simulation results suggest that all three algorithms outperform existing algorithms. In

particular the choice of nuclear norm as the matrix norm leads to an algorithm that is well suited for

designing aligned beamformers in a wide range of networks.


1.3 Role of Interference Alignment in System Design

The last part of the thesis explores the value of insights from the DoF analysis and the relevance of IA

in practical cellular network optimization.

In cellular networks with an ICIC architecture, transmission parameters such as beamformers and

transmit powers are optimized to maximize a network utility function. Joint optimization of these pa-

rameters is an active area of research and typically involves solving an optimization problem to maximize

the utility function.

In spite of several novel techniques to solve this problem, due to its non-convex nature, globally

optimal solutions are still elusive to find and any means of enhancing the performance of the algorithms

proposed for network utility maximization (NUM) is of significant interest. The optimality of aligned

beamformers from a DoF perspective motivates an investigation into their value in NUM.

Note that NUM and IA share a similar overhead since both require global CSI and a centralized

design, and it is therefore reasonable to assess the value of IA in relation to NUM. However, due to the

limited focus of IA on interference suppression while neglecting signal strength, IA cannot be viewed as

a substitute for NUM, and must instead be considered as a potential augmentation to the optimization

process. Unlike NUM algorithms that strive to strike the right balance between signal strength and

interference at each receiver in the network, IA focuses exclusively on nulling interference. While this

is clearly not optimal from an NUM perspective, IA might in fact be better suited to navigate the

interference landscape in dense cellular networks.

Chapter 4 proposes a two-stage optimization framework for NUM. The first stage of the proposed

framework focuses exclusively on nulling interference from a set of dominant interferers using IA, while

the second stage optimizes the transmit and receive beamformers iteratively to maximize a network-wide

utility using the IA solution as the initial condition. Such a framework counters the myopic nature of

straightforward NUM algorithms by leveraging IA’s ability to comprehensively address interference from

the dominant interferers while subsequently relying on NUM algorithms to account for signal strength

and maximize the network utility. Specifically, the proposed framework is used to maximize the minimum

rate to the scheduled users (max-min fairness) subject to per-BS power constraints.

Through simulations on specific topologies of isolated clusters of BSs with realistic channel models,

it is observed that the two-stage optimization framework outperforms straightforward optimization on

these topologies. The results further indicate that, as the distance between adjacent BSs decreases, the

number of scheduled users must be decreased so that more dominant interferers can be nulled in the

first stage.

Chapter 5 summarizes the main contributions in the thesis and presents some concluding remarks.

1.4 Notation

The following notation is used all through this thesis. Column vectors are represented in bold lower-case

letters and matrices in bold upper-case letters. The conjugate transpose and Euclidean norm of a vector

v are denoted as vH and ‖v‖ respectively. The diagonal matrix formed using the vector v is denoted

as diag(x). The notation v(1:k) is used to represent the length-k vector formed using the first k entries


of v.

If M is a matrix, the span of the columns is denoted as span(M); tr(M) represents the trace of the

matrix and diag(M) represents the diagonal matrix formed using the diagonal entries of M; and the

notation M(1 : i, 1 : j) is used to refer to the submatrix of M consisting of the first i rows and the first

j columns of M. The simplified notation {Mij} to represent a set of matrices Mij where the indices i

and j are varied within a range that is clear from the context. Similarly, the notation {Mij}jmax

j=1 is used

when the index i is held constant and j is varied from 1 to jmax. The Frobenius norm of a matrix M

is defined as√∑

i,j |M(i, j)|2 and is denoted as ‖M‖F . The Frobenius norm is also equal to the square

root of the sum of squares of the singular values. The nuclear norm, defined as the sum of singular

values, is denoted as ‖M‖∗.The all-ones vector and the identity matrix are denoted as 1 and I respectively. Calligraphic letters

(e.g., Q) are used to denote sets. The notation CN (µ, σ2I) is used to denote the multi-variate circularly-

symmetric complex Gaussian density function with mean µ and variance σ2 in each dimension. The

notation for random variables and the expectation operator is established where necessary. The mutual

information and differential entropy are denoted as I(·; ·) and h(·) respectively. The notation Z+ is used

to represent the set of positive integers.

Chapter 2

Degrees of Freedom of MIMO

Cellular Networks

2.1 Introduction

The study of DoF started with the work on the two-user MIMO interference channel [2]. In [4, 9],

the authors investigated the DoF of the 2 × 2 X network1 for which linear beamforming based on

interference alignment was used to establish the optimal DoF. This was followed by the landmark paper

of [3], where it is shown that the K-user single-input single-output (SISO) interference channel has a

total of K/2 DoF. The crucial contribution of [3] is an asymptotic scheme for interference alignment over

multiple symbol extensions in time or frequency for establishing the optimal DoF. This scheme requires

channels to be time/frequency varying and crucially relies on the commutativity of diagonal channel

matrices obtained from symbol extensions in time or frequency. Subsequently, the asymptotic scheme

was extended to MIMO cellular networks [10] and MIMO X networks [11]. We note that instead of

relying on infinite symbol extensions over time or frequency varying channels, a signal space alignment

scheme based on rational dimensions developed in [12] achieves the same DoF as the scheme in [3],

but over constant channels. Since these early results, the asymptotic schemes of [3, 12] and the linear

beamforming schemes have emerged as the leading techniques for establishing the optimal DoF of various

networks.

In this work, we study the DoF achieved using the asymptotic scheme of [3] and the linear beam-

forming schemes along with conditions for their optimality in the context of MIMO cellular networks.

Optimizing either scheme for general MIMO cellular networks is not straightforward. While the asymp-

totic schemes require the multi-antenna nodes in a MIMO network to be decomposed into independent

single-antenna nodes, linear beamforming schemes require significant customization for each MIMO cel-

lular network. This work draws upon the observations in [13], where it is shown that for the K-user

MIMO interference channel the two techniques have distinct regimes where one outperforms the other

and that both play a critical role in establishing the optimal DoF. We observe that the same insight also

1An X-network is a network where every transmitter has an independent message for every receiver.

11

Chapter 2. Degrees of Freedom of MIMO Cellular Networks 12

applies to MIMO cellular networks, but the characterization of the optimal DoF is more complicated

because of the presence of multiple users per cell. We make progress on this front by studying the

optimality of decomposition based schemes for a general (G,K,M ×N) network, and by developing two

contrasting approaches to linear beamforming that emerge from two different perspectives on interfer-

ence alignment. In a parallel and independent investigation, Liu and Yang [14] develop a new set of outer

bounds on the DoF of MIMO cellular networks and a structured approach to characterize the optimal

DoF under linear beamforming. While some of the results of this chapter overlap with those of [14], the

approach taken in this chapter for establishing these results is considerably different from that of [14].

2.1.1 Literature Review

Decomposition Based Schemes

The asymptotic scheme developed in [3] for the SISO K-user interference channel can be extended

to other MIMO networks, including the X-network [11, 15], and cellular networks [10, 16] having the

same number of antennas at each node. Since the original scheme in [3] relies on the commutativity of

channel matrices, applying this scheme to MIMO networks requires decomposing multi-antenna nodes

into multiple single-antenna nodes. Two-sided decomposition involves decomposing both transmitters

and receivers into single-antenna nodes, while one-sided decomposition involves decomposing either the

transmitters or the receivers. Formally, the three forms of decomposition can be defined as follows.

Definition For a given (G,K,M × N) network, one-sided decomposition on the user side splits the

multi-antenna users into independent single-antenna users, thus resulting in a (G,KM, 1×N) network

where each cell has KM single-antenna users.

Definition For a given (G,K,M × N) network, one-sided decomposition on the BS side splits the

multi-antenna BSs into independent single-antenna BSs, thus resulting in a network with a total of GN

single-antenna BSs and GK multi-antenna users, with the K users in each cell being served by the set

of N single-antenna BSs that originally constituted the N -antenna BS serving that cell.

Definition For a given (G,K,M ×N) network, two-sided decomposition splits the multi-antenna BSs

into independent single-antenna BSs while also splitting the multi-antenna users into independent single-

antenna users. Such a decomposition results in a in a network with a total of GN single-antenna BSs

and GKM single-antenna users, with each of the KM single-antenna users in a cell being served by the

set of N single-antenna BSs that originally constituted the N -antenna BS serving that cell.

Note that except for one-sided decomposition on the user side, the other two forms of decomposition

do not result in a typical cellular network where a user is served by a single BS. The resulting networks

in the other two cases are variations of the X-network where certain messages are absent. Nevertheless,

once a network has been decomposed, the scheme in [3] can be applied to get an inner bound on the

DoF of the original network.

Two-sided decomposition is first used to prove that the K-user interference channel with M antennas

at each node has KM/2 DoF [3]. This shows that the network is two-side decomposable, i.e., no DoF

are lost by decomposing multi-antenna nodes into single antenna nodes. Two-sided decomposition is


also known to achieve the optimal DoF of MIMO cellular networks with the same number of antennas

at each node [10]. In particular, it is shown that a (G,K,N × N) network has N/(K + 1) DoF/user.

However, for X-networks with A transmitters and B receivers having N antennas at each node, two-

sided decomposition is shown to be suboptimal and that one-sided decomposition achieves the optimal

DoF of ABN/(A+B− 1) [15]. In [17,18], the DoF of the K-user interference channel with M antennas

at the transmitters and N antennas at the receivers is studied and the optimal DoF is established for

some M and N (e.g., when M and N are such that max(M,N)min(M,N) is an integer) using the rational dimensions

framework developed in [12]. In [13], it is shown that decomposition based schemes achieve the optimal

DoF of the K-user interference channel whenever K−2K2−3K+1 ≤ M

N ≤ 1 for K ≥ 4.

Linear Beamforming

Linear beamforming techniques that do not require decomposition of multi-antenna nodes play a crucial

role in establishing the optimal DoF of MIMO networks with different number of antennas at the

transmitters and receivers. In particular, the work of Wang et al. [19] highlights the importance of linear

beamforming techniques in achieving the optimal DoF of the MIMO three-user interference channel.

In [19], the achievability of the optimal DoF is established through a linear beamforming technique

based on a notion called subspace alignment chains. A more detailed characterization of the DoF of

the MIMO K-user interference channel is provided in [13] where antenna configuration (values of M

and N) is shown to play an important role in determining whether the asymptotic schemes or linear

beamforming schemes achieve the optimal DoF.

The study of the design and feasibility of linear beamforming for interference alignment without

symbol extensions has received significant attention [20–25]. Designing transmit and receive beamformers

for linear interference alignment is equivalent to solving a system of bilinear equations. Interference

alignment is said to be feasible if for all generic channels, there exists a set of transmit and receive

beamformers with linearly independent columns that satisfy the system of bilinear equations. A widely

used necessary condition to check for the feasibility of linear interference alignment is to verify if the

total number of variables exceeds the total number of constraints in the system of equations. If a system

has a greater number of variables than constraints, it is called a proper system. Otherwise, it is called

an improper system [20]. In particular, when d DoF/user are desired in a (G,K,M ×N) network, the

system is said to be proper if M +N ≥ (GK +1)d and improper otherwise [25]. While it is known that

almost all improper systems are infeasible [21, 22], feasibility of proper systems is still an area of active

research. In [21–23] a set of sufficient conditions for feasibility are established, while numerical tests to

check for feasibility are provided in [24].

While the optimality of linear beamforming for the K-user MIMO interference channel has been well

studied, the role of linear beamforming in MIMO cellular networks having different number of antennas

at the transmitters and receivers has not received significant attention. Partial characterizations of the

optimal DoF achieved using linear beamforming for two-cell networks are available in [26–29], while [30]

establishes a set of outer bounds on the DoF for the general (G,K,M,N) network. Linear beamforming

techniques to satisfy the conditions for interference alignment without symbol extensions are presented

in [29–32].


Characterizing linear beamforming strategies that achieve the optimal DoF for larger networks is

challenging primarily because multiple subspaces can interact and overlap in complicated ways. Thus

far in the literature, identifying the underlying structure of interference alignment for each given network

(e.g. subspace alignment chains for the three-user MIMO interference channel) has been a prerequisite

for (a) developing counting arguments that expose the limitations of linear beamforming strategies, and

(b) developing DoF optimal linear beamforming strategies. Concurrent to this work, significant recent

progress has been made in [14] on characterizing the DoF of MIMO cellular networks. By identifying a

genie chain structure, the optimality of linear beamforming is established for certain regimes of antenna

configuration. In contrast to [14], the current work on one hand establishes a simpler structure called

packing ratios for smaller networks, yet on the other hand, through numerical observation, establishes

that even an unstructured approach can achieve the optimal DoF for a wide range of MIMO cellular

networks, thus significantly alleviating the challenge in identifying structures in DoF-optimal beamformer

design for larger networks.

2.1.2 Main Contributions

This chapter aims to understand the DoF of MIMO cellular networks using both decomposition based

schemes and linear beamforming. On the use of decomposition, we first note that both the asymp-

totic scheme of [17] for the MIMO interference channel and the asymptotic scheme of [11] for the

X-network can be applied to MIMO cellular networks. Extending the scheme in [17] to MIMO cellu-

lar networks requires one-sided decomposition on the user side (multi-antenna users are decomposed

to multiple single antenna users), while extending the scheme in [11] requires two-sided decomposition.

More importantly, both approaches achieve the same degrees of freedom. We develop a set of outer

bounds on the DoF of MIMO cellular networks and use these bounds to establish conditions under

which decomposition based approaches are optimal. The outer bounds that we develop are based on

an outer bound for MIMO X-networks established in [11]. In particular, we establish that for any

(G,K,M ×N) network, max(

MKη+1 ,

NηKη+1

)is an outer bound on the achievable DoF per user, where

η ∈{

pq : p ∈ {1, 2, . . . , G− 1}, q ∈ {1, 2, . . . , (G− p)K}

}

.

In order to study linear beamforming strategies for MIMO cellular networks, similar in spirit to [19],

we allow for spatial extensions of a given network and study the spatially-normalized DoF (sDoF)

(defined in the next section). Spatial extensions are analogous to time/frequency extensions where spatial

dimensions are added to the system through addition of antennas at the transmitters and receivers.

Unlike time or frequency extensions where the resulting channels are block diagonal, spatial extensions

assume generic channels with no additional structure—making them significantly easier to study without

the peculiarities associated with additional structure. Using the notion of sDoF, we first develop a

structured approach to linear beamforming that is particularly useful in two-cell MIMO cellular networks.

We then focus on an unstructured approach to linear beamforming that can be applied to a broad class

of MIMO cellular networks.

Structured approach to linear beamforming: This chapter develops linear beamforming strategies

that achieve the optimal sDoF of two-cell MIMO cellular networks with two or three users per cell. We

characterize the optimal sDoF/user for all values of M and N and show that the optimal sDoF is a


piecewise-linear function of the ratio M/N , with either M or N being the bottleneck. We introduce

the notion of packing ratio that describes the interference footprint or shadow cast by a set of uplink

transmit beamformers and exposes the underlying structure of interference alignment. Specifically, the

packing ratio of a given set of beamformers is the ratio between the number of beamformers in the set

and the number of dimensions these beamformers occupy at an interfering base-station (BS).

Packing ratios are useful in determining the extent to which interference can be aligned at an inter-

fering BS. For example, for the two-cell, three-user/cell MIMO cellular network, when M/N ≤ 2/3, the

best possible packing ratio is 2 :1, i.e., a set of two beamformers corresponding to two users aligns onto

a single dimension at the interfering BS. This suggests that if we have sufficiently many such sets of

beamformers, no more than 2N/3 DoF/cell are possible. This in fact turns out to be a tight upper bound

whenever 59 ≤ M

N ≤ 23 . Through the notion of packing ratios, it is easier to visualize the achievability

of the optimal sDoF using linear beamforming and the exact cause for the alternating behaviour of the

optimal sDoF where either M or N is the bottleneck becomes apparent. In particular, we establish the

sDoF of two-cell networks with two or three users/cell.

Unstructured approach to linear beamforming: In order to circumvent the bottleneck of identifying

the underlying structure of interference alignment and to establish results for a broad set of networks,

we propose a structure-agnostic approach to designing linear beamformers for interference alignment.

In such an approach, depending on the DoF demand placed on a given MIMO cellular network, we first

identify the total number of dimensions that are available for interference at each BS. We then design

transmit beamformers in the uplink by first constructing a requisite number of random linear vector

equations that the interfering data streams at each BS are required to satisfy so as to not exceed the

limit on the total number of dimensions occupied by interference. This system of linear equations is

then solved to obtain a set of aligned transmit beamformers.

The crucial element in such an approach is the fact that we construct linear vector equations with

random coefficients. This is a significant departure from typical approaches to constructing aligned

beamformers where the linear equations that identify the alignment conditions emerge from notions

such as subspace alignment chains or packing ratios and are predefined with deterministic coefficients.

The flexibility in choosing random coefficients allows us to use this technique for interference alignment

in networks of any size, without having to explicitly infer the underlying structure.

Such an approach is also discussed in a limited context in [31] where it is used to design aligned

transmit beamformers when only 1 DoF/user is desired. We significantly expand the scope of such

an approach by proposing the use of a polynomial identity test to resolve certain linear independence

conditions that need to be satisfied when more than 1 DoF/user are desired. In our work we outline

the key steps to designing aligned transmit beamformers using this approach and take a closer look at

the DoFs that can be achieved. We then proceed to numerically examine the optimality of the DoF

achieved through such a scheme. Numerical evidence suggests that for any given (G,K,M×N) network,

the unstructured approach to linear beamforming achieves the optimal sDoF whenever M and N are

such that the decomposition inner bound(

MNKM+N

)lies below the proper-improper boundary

(M+NGK+1

).

Remarkably, the result of the polynomial identity test also appears to identify the optimal sDoF in this

regime.

The first part of this chapter, presented in Section 2.3, discusses the achievable DoF using decompo-


sition based approaches, establishes outer bounds on the DoF of MIMO cellular networks, and identifies

the conditions under which such an approach is DoF optimal. In the second part, we present a structured

and an unstructured approach to linear beamforming design for MIMO cellular networks. In particular,

in Section 2.4, we establish the optimal sDoF of the two-cell MIMO network with two or three users per

cell through a linear beamforming strategy based on packing ratios. Section 2.5 introduces the unstruc-

tured approach to interference alignment and explores the scope and limitations of such a technique in

achieving the optimal sDoF of any (G,K,M ×N) network.

2.2 System Model

The system model is exactly the same as that presented in the previous chapter. The focus is on achieving

d DoF/user in a (G,K,M ×N) network. The M × d beamforming matrix on the user side is denoted

as Vgk and the N × d beamforming matrix on the BS side is denoted as Ugk. All channels are assumed

to generic, or equivalently, drawn from a continuous distribution. Assuming sgk to be the d× 1 symbol

vector transmitted by the kth user in the gth cell, the effective received signal after being projected by

Uij at the ith BS is given by

UHijyi =

G∑

g=1

K∑

k=1


ijni. (2.1)

where yi is the received signal at the ith BS, ni is the N × 1 vector representing circular symmetric

additive white Gaussian noise ∼ CN (0, σ2I). The signal model is defined similarly for the downlink.

We denote the space occupied by interference at the ith BS as the column span of a matrix Ri

formed using the column vectors from the set {H(gk,i)vgkj : g ∈ {1, 2, . . . , G}, k ∈ {1, 2, . . . ,K}, j ∈{1, 2, . . . , d}, g 6= i}, where we use the notation vgkj to denote the jth beamformer associated with user

(g, k) as given by the jth column of Vgk.

2.3 Decomposition Based Schemes: Achievable DoF and Con-

ditions for Optimality

In this section we discuss the DoF/user that can be achieved in a MIMO cellular network using the

asymptotic scheme presented in [3] and establish the conditions under which such an approach is DoF

optimal.

2.3.1 Achievable DoF using decomposition based schemes

Applying the asymptotic scheme in [3] to a MIMO network requires us to decompose either the trans-

mitters or the receivers, or both, into independent single-antenna nodes. When using the asymptotic

scheme on the decomposed network, the DoF achieved per user in the original network is simply the

sum of the DoFs achieved over the individual single-antenna nodes.


One-sided decomposition of a (G,K,M ×N) cellular network on the user side reduces the network

to a G-cell cellular network with KM single antenna users per cell. Since user-side decomposition

of both the MIMO interference channel and the MIMO cellular network results in a MISO cellular

network, the results of [17, 18] naturally extend to MIMO cellular networks. Two-sided decomposition

of a (G,K,M × N) cellular network results in GN single-antenna BSs and KM single-antenna users,

which form a GN×GKM X-network with a slightly different message requirement than in a traditional

X-network since each single-antenna user is interested in a message from only N of the GN single-

antenna BSs. The asymptotic alignment scheme developed in [11] for X-networks can also be applied to

this GN × GKM X-network. It turns out that one-sided decomposition and two-sided decomposition

achieve the same DoF in a (G,K,M × N) network. Using the results in [11, 17, 18], the achievable

DoF for general MIMO cellular networks using decomposition based schemes is stated in the following

theorem.

Theorem 2.3.1. For the (G,K,M × N) cellular network, using one-sided decomposition on the user

side or two-sided decomposition, KMNKM+N DoF/cell are achievable when (G− 1)KM ≥ N .

Proof. In the following, we present a short proof of the above theorem when user-side decomposition is

used. This proof is based on the techniques discussed in [11,17,18] and is presented here for completeness.

Since user-side decomposition is considered, the proof is presented for the case when M = 1.

Consider a r symbol extension of the channel so that each transmitter (user) has r dimensions and

each receiver (BS) has rN dimensions. In the uplink, the received signal at the ith BS for this extended

channel can be written as

yi =

G∑

g=1

K∑

k=1

H(gk,i)xgk + ni. (2.2)

=

G∑

g=1

K∑

k=1

[

H(gk,i1)H(gk,i2) . . . H(gk,iN)

]T

xgk + ni (2.3)

where xgk is the r × 1 transmitted signal vector from the kth user in the gth cell, H(gk,ic) is the r × r

diagonal channel matrix from the kth user in the gth cell to the cth antenna of the ith BS.

Assuming each user transmits d data streams, each user selects the same r × d beamforming matrix

V so that xgk = Vsgk where sgk is the d × 1 symbol vector to be transmitted. We let |V| denotethe number of columns of V. Note that since each user can access only r dimensions of the total rN

dimensions available at any BS, interference alignment is not possible using any set of N users. In order

to align interference, we first define a (N |V|)-dimensional interference space at each of the BSs and then

try to align interference from all the out-of-cell users within this N |V| dimensional space. We set this

N |V| dimensional space to be the space spanned by the columns of the rN ×N |V| matrix

Q =

V 0 · · · 0

0 V · · · 0

...... · · · 0

0 0 · · · V

. (2.4)


In other words, the received signal vector at the ith BS from every interfering user (g, k) must satisfy

span(

H(gk,i)V)

⊆ span (Q) ∀ g 6= i (2.5)

⇒ span

H(gk,i1)V

H(gk,i2)V

...

H(gk,iN)V

⊆ span (Q) ∀ g 6= i (2.6)

.

Equivalently, we require

span(

H(gk,ic)V)

⊆ span(V) ∀ i, g ∈ {1, 2, . . . , G}, i 6= g, k ∈ {1, 2, . . . ,K}, c ∈ {1, 2, . . . , N}. (2.7)

Noting that all matrices H(gk,ic) are diagonal and hence commute, we adopt the same iterative procedure

as outlined in [15] to design the beamforming matrix V. We begin by first setting V = 1r×1. In each

subsequent iteration we update V to be the set of interfering vectors received at all the BSs, i.e., if after

s iterations, V = Vs, then Vs+1 is given by

Vs+1 ={H(gk,ic)Vs | i, g ∈ {1, 2, . . . , G}, i 6= g, k ∈ {1, 2, . . . ,K}, c ∈ {1, 2, . . . , N}

}. (2.8)

A more useful, non-recursive manner of describing Vs+1 is to list all the column vectors contained

in Vs+1. This is given by the set

∏

i,g ∈{1:G}, i6=gk=1:K, c=1:N

(H(gk,ic))β(gk,ic)

1 s.t.

∑

i,g ∈{1:G}, i6=gk=1:K, c=1:N

β(gk,ic) ≤ s+ 1, β(gk,ic) ∈ Z+, β(gk,ic) ≥ 0

.

(2.9)

Note that in all there are L = (G− 1)GN constraints of the form shown in (2.7). Since the matrices

Hgk,ic commute, the number of columns in V after s iterations is given by(s+Ls

). Thus we see that as

the number of iterations s → ∞,

|Vs+1||Vs|

=s+ L+ 1

s+ 1→ 1. (2.10)

This suggests that asymptotically, the column space of the matrix Vs is invariant to linear transfor-

mations by the set of matrices H(gk,ic), and hence satisfies the span constraints in (2.7). Through this

procedure we ensure that at each BS, interference from all the out-of-cell users is contained within a

sufficiently small number of dimensions (≈ N |Vs| for large s).

We now need to show that (a) the columns of Vs are linearly independent; and (b) the received signal

from the K users in a cell is separable from the interfering signals at the corresponding BS. We first set

the number of channel extensions to be such that r =⌈N |Vs|+K|Vs|

N

⌉

for some large s. The columns of

Vs can be shown to be linearly independent using Lemma 1 and 2 in [11]. The set of received signals at

the ith BS can also be shown to be linearly independent by considering the received signal matrix

[

H(i1,i)V H(i2,i)V · · · H(iK,i)V

]

rN×K|Vs|(2.11)


and again invoking Lemma 1 and 2 in [11]. We now need to show that the received signal H(ij,i)V is

linearly independent of the columns in Q. We prove this using a contradiction. Suppose this is not

true, then there exists a non-zero vector w such that [(H(ij,i)V) Q]w = 0 ⇒ [(H(ij,i1)V) V]w1:r = 0.

Now, since direct channels play no role in the design of V and because all channels are assumed to be

generic, (H(ij,i1)V) is a generic linear transformation of the subspace spanned by V and the existence of

a non-trivial solution to [(H(ij,i1)V) V]w1:r = 0 requires the matrix [(H(ij,i1)V) V] to be rank deficient.

This is however not possible under the assumption of generic channels, thus presenting a contradiction

to our initial assumption. Thus, at all the BSs the received signal from the K users is separable from

interference. This shows that |Vs|r = |Vs|

⌈(N |Vs|+K|Vs|)/N⌉ ≈ NN+K DoF/user are achievable as s → ∞.

Note that when (G− 1)KM < N , there is no scope for interference alignment and random transmit

beamforming in the uplink turns out to be the DoF optimal strategy. This theorem generalizes the

result established in [10], where it is shown that SISO cellular networks with K-users/cell have K/K+1

DoF/cell. By duality of linear interference alignment, this result applies to both uplink and downlink.

While we consider decomposing multi-antenna users into single-antenna users for one-sided decompo-

sition here, we can alternately consider decomposing the multi-antenna BSs. It can however be shown

that the achievable DoF remains unchanged. Designing the achievable scheme is similar to [15], where

separation between signal and interference is no longer implicitly assured.

2.3.2 Outer Bounds on the DoF of MIMO Cellular Networks

We derive a new set of outer bounds on the DoF of MIMO cellular networks that are based on a result

in [11], where MIMO X-networks with A transmitters and B receivers are considered. By focusing on

the set of messages originating from or intended for a transmitter-receiver pair and splitting the total

messages in the network into AB sets, [11] derives a bound on the total DoF of this set of messages. Let

di,j represent the DoF between the ith transmitter and the jth receiver. The following lemma presents

the outer bound obtained in this manner.

Lemma 2.3.2 ( [11] ). In a wireless X-network with A transmitters and B receivers, the DoF of all

messages originating at the ath transmitter and the DoF of all the messages intended for the bth receiver

are bounded by

B∑

i=1

da,i +

A∑

j=1

dj,b − da,b ≤ max(M,N), (2.12)

where M is the number of antennas at the ath transmitter and N is the number of antennas at the bth

receiver. By symmetry, this bound also holds when the direction of communication is reversed.

Before we proceed to establish outer bounds on the DoF of a MIMO cellular network, we define the

set Q as

Q =

{p

q: p ∈ {1, 2, . . . , G− 1}, q ∈ {1, 2, . . . , (G− p)K}

}

. (2.13)

The following theorem presents an outer bound on the DoF.


Theorem 2.3.3. If a (G,K,M×N) network satisfies M/N ≤ p/q, for some p/q ∈ Q, then Np/(Kp+q)

is an outer bound on the DoF/user of that network. Further, if M/N ≥ p/q, for some p/q ∈ Q, then

Mq/(Kp+ q) is an outer bound on the DoF/user of that network.

Proof. To prove this theorem, we first note that a cellular network can be regarded as an X-network

with some messages set to zero. Further, Lemma 2.3.2 is applicable even when some messages are set

to zero. Now, suppose MN ≤ p

q for some pq ∈ Q, then consider a set of p cells and allow the set of BSs in

these p cells to cooperate fully. Let B denote the set of indices corresponding to the p chosen cells. From

the remaining G− p cells, we pick q users and denote the set of indices corresponding to these users as

UB and allow them to cooperate fully.

Applying Lemma 2.3.2 to the set of BSs B and the set of users UB, we get

∑

i∈B

K∑

j=1

dij,i +∑

(g,h)∈UB

dgh,g ≤ max(pN, qM). (2.14)

By summing over similar bounds for all the(Gp

)sets of p BSs and the corresponding

((G−p)K

q

)sets

of q users for each set of p BSs, we obtain

[K

q+

1

p

] G∑

i=1

K∑

j=1

dij,i ≤GK

pqmax(pN, qM)

⇒G∑

i=1

K∑

j=1

dij,i ≤GK

Kp+ qmax(pN, qM) = pN. (2.15)

Thus, the total DoF in the network is bounded by GKNpKp+q . Hence, DoF/user ≤ Np

Kp+q whenever

p/q ∈ Q. The outer bound is established in a similar manner when MN ≥ p

q . Note that wheneverMN = p

q ,Np

Kp+q = MqKp+q = MN

KM+N .

In [30], outer bounds on the DoF for MIMO cellular network are derived which are also based on the

idea of creating multiple message sets [11]. The DoF/user of a (G,K,M × N) network is shown to be

bounded by

DoF/user ≤ min(

M, NK , max[KM,(G−1)N ]

K+G−1 , max[N,(G−1)M ]K+G−1

)

. (2.16)

While it is difficult to compare this set of bounds and the bounds in Theorem 2.3.3 over all parameter

values, we can show that under certain settings the bounds obtained in Theorem 2.3.3 are tighter. For

example, since p/q ∈ Q, let us fix p/q = 1/K, then set M/N = p/q = 1/K. Further, let us assume

that (G− 1) < K. Under such conditions, (2.16) bounds the DoF/user by MKK+G−1 while Theorem 2.3.3

states that DoF/user ≤ M2 . Since we have assumed K > G − 1, it is easy to see that the latter bound

is tighter.

2.3.3 Optimality of the DoF Achieved Using Decomposition

Using the results in previous two sections, we establish conditions for the optimality of one-sided and

two-sided decomposition of MIMO cellular networks in the following theorem.


Figure 2.1: 2-D Wyner model of a cellular network. Two cells are connected to each other if they

mutually interfere. Cells that are not directly connected to each other are assumed to see no interference

from each other. Note that each user in a given cell sees interference from the four adjacent BSs.

Theorem 2.3.4. The optimal DoF for any (G,K,M ×N) network with MN ∈ Q is MN

KM+N DoF/user.

The optimal DoF is achieved by either one-sided or two-sided decomposition with asymptotic interference

alignment.

This result follows immediately from Theorems 2.3.1 and 2.3.3. We observe that this result is

analogous to the results in [17, 18] where it is shown that the G-user interference channel has MNM+N

DoF/user whenever η = max(M,N)min(M,N) is an integer and G > η. It is easy to see that the results of [17, 18]

can be easily recovered from the above theorem by setting K = 1 and letting G represent the number

of users in the interference channel.

The result in Theorem 2.3.4 has important consequences for cellular networks with single-antenna

users. The following corollary describes the optimal DoF/user of any cellular network with single antenna

users that satisfies (G− 1)K ≥ N .

Corollary 2.3.5. The optimal DoF of a (G,K, 1×N) network with (G− 1)K ≥ N , is NK+N DoF/user.

For example, this corollary states that a three-cell network having four single-antenna users per cell

and four antennas at each BS has 1/2 DoF/user. Using this corollary and the DoF achieved using zero-

forcing beamforming, the optimal DoF of cellular networks with single-antenna users can be completely

characterized and is stated in the following theorem.

Theorem 2.3.6. The DoF of a G-cell cellular network with K single-antenna users per cell and N


antennas at each BS is given by

DoF/user =

NN+K N < (G− 1)K

NGK (G− 1)K ≤ N < GK

1 N ≥ GK

. (2.17)

The optimal DoF is achieved through zero-forcing beamforming when N ≥ (G−1)K and through asymp-

totic interference alignment when N < (G− 1)K.

Another interesting consequence of Theorem 2.3.4 for two-cell cellular networks is stated in the

following corollary.

Corollary 2.3.7. For a (G = 2,K,M ×N) cellular network with K = NM , time sharing across cells is

optimal and the optimal DoF/user is N2K .

Proof. Using Theorem 2.3.4, the optimal DoF/user of this network is N2K . Since the K-user MAC/BC

with MN = 1

K has NK DoF/user, accounting for time sharing between the two cells gives us the required

result.

This result recovers and generalizes a similar result obtained in [26] for two-cell MISO cellular net-

works, which shows that in dense cellular networks where K = N/M , when two closely located cells

cause significant interference to each other, simply time sharing between the two mutually interfering

BSs is a DoF-optimal way to manage interference in the network. This result can be further extended

to the 2-D Wyner model for MIMO cellular networks and is stated in the following corollary.

Corollary 2.3.8. Consider a two-dimensional square grid of BSs with K users/cell, M antennas/user,

and N antennas/BS, such that each BS interferes only with the four neighbouring BSs as shown in Fig.

2.1. When KM = N , time sharing between adjacent cells so as to completely avoid interference is a

DoF optimal strategy and achieves N/2K DoF/user.

2.3.4 Insights on the Optimal DoF of MIMO Cellular Networks

When the achievable DoF using decomposition, the outer bounds on the DoF, and the proper-improper

boundary are viewed together, an insightful (albeit incomplete) picture of the optimal DoF of MIMO

cellular networks emerges. Fig. 2.2 plots the normalized DoF/user (DoF/user/N) achieved by the de-

composition based approach as a function of the ratio M/N (γ) along with the outer bounds derived

in Theorem 2.3.3 for a set of two-cell networks with different number of users/cell. We also plot the

proper-improper boundary (M +N ≶ (GK +1)d) that acts as an upper bound on the DoF that can be

achieved using linear beamforming (A more detailed discussion on this is presented in the next section.).

Although Fig. 2.2 only considers two-cell networks, several important insights on general MIMO cellular

networks can be inferred and are listed below.

(a) Two distinct regimes: Depending on the network parameters G, K, M and N , there are two distinct

regimes where decomposition based schemes outperform linear beamforming and vice versa.

(b) Optimality of decomposition based schemes for large networks: For large networks, the decomposition


M/N

0.1

0.2

0.3

0.4

0.5

0.5 1 2 3

DoF/user/N

2 cells and 2 users/cell

(a)

M/N

0.1

0.2

0.3

1 2 3

DoF/user/N


(b)

M/N

0.1

0.2

1 2 3

DoF/user/N


(c)

M/N

0.1

0.2

1 2 3

DoF/user/N


(d)

M/N

0.05

0.1

1 2 3

DoF/user/N


(e)

Figure 2.2: The proper-improper boundary (red), decomposition inner bound (blue), and the DoF outer

bounds (green) for a set of two-cell networks with different number of users per cell. Note the increasing

dominance of the decomposition based inner bound as the network size increases.

based approach is capable of achieving higher DoF than linear beamforming and the range of γ over

which the decomposition based approach dominates over linear beamforming increases with network size.

The outer bounds on the DoF suggest that when the decomposition based inner bound lies above the

proper-improper boundary, the inner bound could well be optimal. Fig. 2.2(e) is particularly illustrative

of this observation.

(c) Importance of linear beamforming for small networks: For small networks (e.g. two-cell, two-

users/cell; two-cell, three-users/cell), the decomposition based inner bound lies below the proper-improper

boundary, suggesting that linear beamforming schemes can outperform decomposition based schemes.

In the next section, we study the DoF of the two smallest cellular networks and design a linear beam-

forming strategy that achieves the optimal DoF of these two networks. In the subsequent section a

general technique to design linear beamformers for any cellular network is presented.

(d) Inadequacy of existing outer bounds: The outer bounds listed in Theorem 2.3.3 are not exhaustive,

i.e., in some cases, tighter bounds are necessary to establish the optimal DoF. This observation is drawn

from Fig. 2.2(b), where it is seen that some part of the outer bound lies above both the proper-improper

boundary and the decomposition based inner bound suggesting that tighter outer bounds may be pos-

sible. In the next section, we indeed derive a tighter outer bound for specific two-cell three-users/cell

networks.

Motivated by the above observations, we now turn to linear beamforming schemes for MIMO cellular

networks.


2.4 Linear Beamforming: Structured Design

Consider a (G,K,M ×N) network with the goal of serving each user with d data streams. Using (2.1),

when no symbol extensions are allowed, the linear beamformers Vij and Uij need to satisfy the following

two conditions for linear interference alignment [20]:

UHijH(gk,i)Vgk = 0 ∀ (i, j) 6= (g, k) (2.18)

rank(UHijH(ij,i)Vij) = d ∀ (i, j). (2.19)

For a given system, it is not always possible to satisfy the conditions in (2.18) and (2.19) and a preliminary

check on feasibility is to make sure that the given system is proper [20, 25]. As mentioned earlier, a

(G,K,M × N) network with d DoF/user is said to be proper if M + N ≥ (GK + 1)d and improper

otherwise [25]. While not all proper systems are feasible, improper systems have been shown to be

almost surely infeasible [21, 22]. For proper-feasible systems, solving the system of bilinear equations

(2.18) typically requires the use of iterative algorithms such as those developed in [33–36]. In certain

cases where max(M,N) ≥ GKd, it is possible to solve the system of bilinear equations by randomly

choosing either the receive beamformers {Uij} or the transmit beamformers {Vij} and then solving the

resulting linear system of equations.

Assuming the channels to be generic allows us to restate the conditions in (2.18) and (2.19) in a

manner that is more useful in developing DoF-optimal linear beamforming schemes. Condition (2.18)

captures both intra-cell interference nulling and inter-cell interference nulling. For now, ignoring the

conditions for intra-cell interference nulling, note that the direct channels do not otherwise play a role

in (2.18). Hence any set of beamformers that satisfy the conditions for inter-cell interference nulling

automatically satisfy the conditions in (2.19) whenever Uij and Vij have rank d and whenever the

channels are generic [20]. As a further consequence of channels being generic, satisfying the intra-cell

interference conditions in (2.18) is equivalent to the condition that the set of uplink transmit beam-

formers {Vij} is such that there are at least d interference-free dimensions at each receiver before any

linear processing. In essence, generic channels ensure that at each BS, the intersection between useful

signal subspace (span([H(i1,i)Vi1,H(i2,i)Vi2, . . . ,H(iK,i)ViK ]) and interference subspace (span(Ri)) is

almost surely zero dimensional, provided that the rank(Ri) ≤ (N −Kd) ∀i. Thus the requirements for

interference alignment can be alternately stated as

rank(Ri) ≤ N −Kd ∀ i, (2.20)

rank(Vjl) = d ∀ j, l. (2.21)

The rank constraint in (2.20) essentially requires the (G − 1)Kd column vectors of Ri to satisfy L =

GKd−N distinct linear vector equations. Given a set of transmit precoders {Vjl} that satisfy the above

conditions, designing the receive filters is then straightforward.

This alternate perspective on interference alignment lends itself to counting arguments that account

for the number of dimensions at each BS occupied by signal or interference. These counting arguments

in turn lead to the development of DoF-optimal linear beamforming strategies such as the subspace

alignment chains for the 3-user interference channel [19].


In this section we take a structured approach to constructing the L distinct linear vector equations

that need to satisfy (2.20) and (2.21). Such an approach is DoF-optimal for small networks such as the

two-cell two-user/cell and the two-cell, three-user/cell networks.

2.4.1 Main Results

We consider two of the simplest cellular networks, namely the two-cell two-user/cell and the two-cell,

three-user/cell networks, and establish a linear beamforming strategy that achieves the optimal sym-

metric DoF. In particular, we establish the spatially-normalized DoF of these two networks for all values

of the ratio γ = M/N . The spatially-normalized DoF of a network is defined as follows [19].

Definition Denoting the DoF/user of a (G,K,M × N) cellular network as DoF(M,N), the spatially-

normalized DoF/user is defined as

sDoF(M,N) = supq∈Z+

DoF(qM, qN)

q. (2.22)

Analogous to frequency and time domain symbol extensions, the definition above allows us to permit

extensions in space, i.e., adding antennas at the transmitters and receivers while maintaining the ratio

M/N to be a constant. Unlike time or frequency extensions where the resulting channels are block

diagonal, spatial extensions assume generic channels with no additional structure. The lack of any

structure in the channel obtained through space extensions makes it significantly easier to analyze

the network. Although mathematically sDoF appears to be a weaker notion than DoF, it has been

conjectured that the two are equivalent. This is known as spatial scale invariance, where the optimal

DoF for the (G,K, qM × qN) network is expected to be q times the optimal DoF of the (G,K,M ×N)

network [15].

We now present the main results concerning the sDoF of the two cellular networks under considera-

tion.

Let the function f(ω,K)(·) be defined as

f(ω,K)(M,N) = max

(Nω

Kω + 1,

M

Kω + 1

)

, (2.23)

where ω ≥ 0 and K ∈ Z+. Further, define the function D(2,2)(·) to be

D(2,2)(M,N) =min(N,KM, f( 1

2 ,2)(M,N), f(1,2)(M,N)

), (2.24)

and the function D(2,3)(·) to be

D(2,3)(M,N) =min(N,KM, f( 1

3 ,3)(M,N), f( 1

2 ,3)(M,N),

f( 23 ,3)

(M,N), f(1,3)(M,N)). (2.25)

The following theorem characterizes an outer bound on the DoF/user of the two-cell two-user/cell

network and the two-cell three-user/cell network.

Theorem 2.4.1. The DoF/user of a two-cell, K-user/cell MIMO cellular network with K ∈ {2, 3},having M antennas per user and N antennas per BS is bounded above by D(2,K)(M,N), i.e.,

DoF/user ≤ D(2,K)(M,N). (2.26)


0

1/4

1/3

1/2

0 14

12

23 1 3

2

N2

M3

N3

M2

N4 M

γ+15

γ2γ+1

γ (M/N)

Norm

alizedsD

oF/user

Proper-improper boundary

Optimal sDoF

Decomposition based inner bound

Figure 2.3: The sDoF/user (normalized by N) of a 2-cell, 3-user/cell MIMO cellular network as a

function of γ.

Note that since this outer bound is piecewise-linear in either M or N , this bound is invariant to

spatial normalization and hence is also a bound on sDoF and not just DoF. The outer bounds for

the two-cell, two-user/cell case follows directly from either the bounds established in Section 2.3.2 (for

1/4 ≤ γ ≤ 3/2) or through DoF bounds on the multiple-access/broadcast channel (MAC/BC) obtained

by letting the two cells cooperate (for γ ≤ 1/4) and γ ≥ 3/2). In the case of the two-cell, three-user/cell

network, the bounds when γ ≤ 1/6 or γ ≥ 4/3 follow from DoF bounds on the MAC/BC obtained by

letting the two cells cooperate, while the bounds when 1/6 ≤ γ ≤ 5/9 and 3/4 ≤ γ ≤ 4/3 follow from

the bounds established in Section 2.3.2. When 5/9 ≤ γ ≤ 3/4, we derive a new set of genie-aided outer

bounds on the DoF. Our approach to deriving these new bounds is similar to the approach taken in [19]

and the exact details of this derivation are presented in Appendix A.

The outer bound presented in the previous theorem turns out to be tight. The main theorem of

this section is a characterization of the sDoF/user of the two-cell, two-or-three-user/cell MIMO cellular

network. The proof of achievability is deferred to the next section.

Theorem 2.4.2. The spatially-normalized DoF of a 2-cell, K-user/cell cellular network with K ∈ {2, 3},having M antennas per user and N antennas per BS is given by

sDoF/user = D2,K(M,N). (2.27)

This result states that when spatial-extensions are allowed, the outer bound presented in Theorem

2.4.1 is tight. The achievability part of the result in Theorem 2.4.2 is based on a linear beamforming


0

1/5

2/9

1/4

1/6

1/3

0 113

12

16

59

25

23

34

43

M N6

M2

N5

2M5

2N9

M3

N4

M4

N3

γ+17

γ3γ+1

γ (M/N)

Norm

alizedsD

oF/user


Optimal sDoF

Decomp. based inner bound

Figure 2.4: The sDoF/user (normalized by N) of a 2-cell, 3-user/cell MIMO cellular network as a

function of γ.

strategy developed using the notion of packing ratios. We elaborate further on this scheme in the next

subsection.

Figs. 2.3 and 2.4 capture the main results presented in the above theorems and plot sDoF/user

normalized by N as a function of γ. It can be seen in both the figures that, just as in the 3-user

interference channel [19], there is an alternating behaviour in the sDoF with either M or N being the

bottleneck for a given γ.

The figures also plot the boundary separating proper systems from improper systems. It is seen from

the two figures that not all proper systems are feasible. For example, for the two-cell three-users/cell

case, networks with γ ∈ {1/6, 2/5, 5/9, 3/4, 4/3} are the only ones on the proper-improper boundary

that are feasible.

For the two-cell two-users/cell network, we can see from Fig. 2.3 that when γ ∈ {1/4, 2/3, 3/2},neither M nor N has any redundant dimensions, and decreasing either of them affects the sDoF. On the

other hand, when M/N ∈ {1/2, 1}, both M and N have redundant dimensions, and some dimensions

from either M or N can be sacrificed without losing any sDoF. For all other cases, only one of M or N

is a bottleneck. Similar observations can also be made for the 2-cell 3-users/cell network from Fig. 2.4.

Figs. 2.3 and 2.4 also plot the achievable DoF using the decomposition based approach. Interestingly,

the only cases where the decomposition based inner bound achieves the optimal sDoF is when both M

and N have redundant dimensions i.e., γ ∈ {1/2, 1} in the case of the two-cell, two-user/cell network

and when γ ∈ {1/3, 1/2, 2/3, 1} in the case of the two-cell, three-user/cell network.


2.4.2 Packing Ratios

We now present the linear transmit beamforming strategy that achieves the optimal sDoF of the two

networks under consideration. We consider achievability only in the uplink as duality of interference

alignment through linear beamforming ensures achievability in the downlink as well. We start by intro-

ducing a new notion called the packing ratio to describe a collection of transmit beamforming vectors.

Definition Consider the uplink of a two-cell network and let S be a collection of transmit beamformers

used by users belonging to the same cell. If the number of dimensions occupied by the signals transmitted

using this set of beamformers at the interfering BS is denoted by c, then the packing ratio η of this set

of beamformers is given by |S| :c.

As an example, consider a two-cell, two-users/cell cellular network with 2 antennas at each user and

3 antennas at each BS. Suppose we design two beamformers v and w for two different users in the

same cell so that H(11,2)v = H(12,2)w, then the set of vectors S = {v,w} is said to have a packing

ratio of 2 : 1. Another way to picture this is to think of H(11,2) and H(12,2) as providing access to a

two-dimensional subspace of the three-dimensional received signal space at the second BS. Since two

two-dimensional subspaces in a three-dimensional space must have at least one dimension in common,

these two subspaces are said to overlap in one dimension. Exploiting this one-dimensional overlap leads

to a packing ratio of 2 : 1. As another example, for the same network, consider the case when M > N .

Since users can now zero-force all antennas at the interfering BS, we can have a set S of beamformers

with packing ratio |S| : 0.When designing beamformers for the two-cell network, it is clear that choosing sets of beamformers

having a high packing ratio is desirable as this reduces the number of dimensions occupied by interference

at the interfering BS. The existence of beamformers satisfying a certain packing ratio is closely related

to the ratio γ (M/N). For example, it is easily seen that when γ < 23 , it is not possible to construct

beamformers having a packing ratio of 3 :1. Further even when beamformers satisfying a certain packing

ratio exist, there may not be sufficient sets of them to completely use all the available dimensions at a

BS. In such a scenario, we need to consider designing beamformers with the next best packing ratio.

Using the notion of packing ratios, we now describe the achievability of the optimal sDoF of the

two-cell three-users/cell cellular network. We first define the set P23 = {1 : 0, 3 : 1, 2 : 1, 3 : 2, 1 : 1}to be the set of fundamental packing ratios for the two-cell, three-users/cell cellular network. For any

given γ, our strategy is to first construct the sets of beamformers that have the highest possible packing

ratio from the set P23. If such beamformers do not completely utilize all the available dimensions at

the two BSs, we further construct beamformers having the next best packing ratio in P23 until all

the dimensions at the two BSs are either occupied by signal or interference. Our proposed strategy is

essentially a greedy strategy to minimize the dimensions occupied by interference. Greedy strategies

for aligning interference, including the notion of subspace alignment chains developed in [19] where an

alignment chain is terminated until no more interference can be aligned, are observed to be capable of

achieving the optimal sDoF. The strategy we develop is illustrated in the following example.

Consider the case when 2/3 < γ < 3/4. Since M < N , no transmit zero-forcing is possible. Further,

each user can access only M of the N dimensions at the interfering BS. Since we assume all channels to


be generic, and 2M > N , the subspaces accessible to any two users overlap in 2M −N dimensions. This

2M − N dimensional space overlaps with the M dimensions accessible to the third user in 3M − 2N

dimensions. Note that such a space exists as we have assumed 2/3 < γ. Thus, we can construct 3M−2N

sets of three beamformers (one for each user) that occupy just one dimension at the interfering BS and

thus have a packing ratio of 3 :1. Assuming that the same strategy is adopted for users in both cells, at

any BS, signal vectors occupy a total of 3(3M − 2N) dimensions while interference occupies 3M − 2N

dimensions. Thus a total of 4(3M − 2N) dimensions are occupied by signal and interference. Since

4(3M − 2N) < N whenever 4M < 3N , we see that such vectors do not completely utilize all the N

dimensions at a BS.

In order to utilize the remaining 9N − 12M dimensions, we additionally construct beamformers with

the next highest packing ratio (2 : 1). Let M ′ = M − (3M − 2N) = 2N − 2M denote the unused

dimensions at each user. At the interfering BS, each pair of users has 2M ′ − N dimensions that can

be accessed by both users. Note that since 2M ′ − N = 2(2N − 2M) − N = 3N − 4M > 0, such an

overlap exists almost surely. For a fixed pair of users in each cell, we choose (3N − 4M) sets of two

beamformers (one for each user in the pair) whose interference aligns onto a single dimension, so that

each set has a packing ratio of 2 : 1. After choosing beamformers in this manner, we see that signal

and interference span all N dimensions at each of the two BSs. Through this process, each BS receives

3(3M − 2N) + 2(3N − 4M) signalling vectors while interfering signals occupy (3M − 2N) + (3N − 4M)

dimensions. We have thus shown that 3(3M−2N)+2(3N−4M) = M DoF/cell are achievable. To ensure

that M/3 DoF/user are achieved, we can either cycle through different pairs of users when designing the

second set of beamformers, or we can simply pick (3N − 4M)/3 sets of beamformers for every possible

pair of users in a cell. If (3N − 4M)/3 is not an integer, we simply scale N and M by a factor of 3 to

make it an integer. We can afford the flexibility to scale M and N because we are only characterizing

the sDoF of the network.

As another example, consider the two-cell, three-users/cell network with 3/4 ≤ γ ≤ 1. When

3/4 ≤ γ ≤ 1, all three users of a cell can access a 3M − 2N dimensional space at the interfering BS,

thus 3M − 2N sets of three beamformers having a packing ratio of 3 : 1 are possible. Note that 3 : 1 is

still the highest possible packing ratio. If users in both cells were to use such beamformers, signal and

interference from such beamformers can occupy at most 4(3M − 2N) > N dimensions at any BS. Thus,

when 3/4 ≤ γ < 1, we have sufficient sets of beamformers with packing ratio 3 : 1 to use all available

dimensions at the BSs. Choosing N/4 such sets provides us with 3N/4 DoF/cell. Fig. 2.5 illustrates

this idea for a 2-cell 3-user/cell network with 12 antennas at each BS and 9 antennas at each user. Note

that a packing ratio of 3:1 is feasible and there exist 3 (3M −2N) such sets of beamformers. Using these

three sets of beamformers, 3 DoF/user are achieved.

Such an approach to designing the linear beamformers provides insight on why the optimal sDoF

alternates between M and N . When γ is such that there are sufficient sets of beamformers having the

highest possible packing ratio, it is the number of dimensions at the BSs that proves to be a bottleneck

and the DoF bound becomes dependent on N . On the other hand, when there are not enough sets of

beamformers having the highest possible packing ratio, we are forced to design beamformers with a lower

packing ratio so as to use all available dimensions at the two BSs. Since for a fixed N , the number of

sets of beamformers having the highest packing ratio is a function of M , the bottleneck now shifts to M .


999999

1212

(a)

999999

1212

(b)

999999

1212

(c)

999999

1212

(d)

Figure 2.5: In a (2, 3, 9 × 12) network, three sets of beamformers have a packing ratio of 3 : 1. Using

these three sets lets us achieve 3 DoF/user. Of the 12 dimensions at each BS, 9 are used to recover the

desired signals while the remaining three are lost to aligned interference. Signalling dimensions are show

in red and aligned interference is shown in blue. Note that the same strategy is used in both the cells.

We thus see that for a large but fixed N , as we gradually increase M , we cycle through two stages—the

first stage where beamformers with a higher packing ratio become feasible but are limited to a small

number, then gradually, the second stage where there are sufficiently many such beamformers. As M is

increased even further, we go back to the scenario where the next higher packing ratio becomes feasible

however with only limited set of beamformers, and so on.

The design strategy described for the case 2/3 < γ ≤ 1 is also applicable to other intervals of γ, as well

as the two-cell two-users/cell network. For the two-cell three-user/cell network, when 1/3 < γ ≤ 1/2, we

design as many sets of beamformers having packing ratio 3 : 2 as possible, then use beamformers having

a packing ratio of 1 : 1 (random beamforming) to fill any unused dimensions at the two BSs. When

1/2 < γ ≤ 2/3 we first design as many sets of beamformers having packing ratio 2 : 1 as possible and

then use beamformers having a packing ratio of 3 : 2. When γ ≤ 1/3, it is easy to see that interference

alignment is not feasible and that a random beamforming strategy suffices. Finally, when γ ≥ 1, we

first design beamformers that zero-force the interfering BS (packing ratio 1 : 0), then use beamformers

having a packing ratio of 3 : 1 to fill any remaining dimensions at each BS.

For the two-cell two-user/cell network we define the set P22 = {1 : 0, 2 : 1, 2 : 1, 1 : 1} to be the set

of fundamental packing ratios. When γ > 1, we first design beamformers that zero-force the interfering

BS (packing ratio 1 : 0), then if necessary, use beamformers having a packing ratio of 2 : 1 to fill any

remaining dimensions at each BS. When 1/2 < γ ≤ 1, the highest possible packing ratio is 2 : 1, hence

we first design beamformers having packing ratio 2 : 1 to occupy as many dimensions as possible at the

two BSs, then if there are unused dimensions at the two BSs, we use random beamformers (packing

ratio 1 : 1) to occupy the remaining dimensions. When γ ≤ 1/2, interference alignment is not feasible

and simple random beamforming achieves the optimal DoF.


Table 2.1: The sets of beamformers and their corresponding packing ratios used to prove achievability

of the optimal sDoF of the two-cell two-user/cell network for different values of γ.

γ (M/N)Set of beamformers DoF/cell (No. of

signal-vectors per cell)Packing ratio No. of sets Packing ratio No. of sets

0 < γ < 14 1:1 2M – – 2M

14 ≤ γ ≤ 1

2 1:1 N2 – – N

2

12 < γ < 2

3 2:1 2M −N 1:1 4N−6M2 M

23 ≤ γ ≤ 1 2:1 2M −N – – 2N

3

1 < γ < 32 1:0 2(M −N) 2 :1 3N−2M

32M3

32 ≤ γ 1:0 N – – N

Table 2.2: The sets of beamformers and their corresponding packing ratios used to prove achievability

of the optimal sDoF of the two-cell three-user/cell network for different values of γ.

γSet of beamformers DoF/cell (No. of

signal-vectors per cell)Packing ratio No. of sets Packing ratio No. of sets

0 < γ < 16 1:1 3M – – 3M

16 ≤ γ ≤ 1

3 1:1 N2 – – N

2

13 < γ < 2

5 3:2 3M −N 1:1 6N−15M2

3M2

25 ≤ γ ≤ 1

2 3:2 N5 – – 3N

5

12 < γ < 5

9 2:1 3(2M −N) 3 :2 10N−18M5

6M5

59 ≤ γ ≤ 2

3 2:1 N3 – – 2N

3

23 < γ < 3

4 3:1 3M − 2N 2:1 3N − 4M M

34 ≤ γ ≤ 1 3:1 N

4 – – 3N4

1 < γ < 43 1:0 3(M −N) 3 :1 N − 3M

43M4

43 ≤ γ 1:0 N – – N

In Tables 2.1 and 2.2, we summarize the strategies used for different intervals of γ, and list the

number of sets of beamformers of a certain packing ratio required to achieve the optimal DoF along with

the DoF achieved per cell. Note that fractional number of sets can always be made into integers as we

allow for spatial extensions. We discuss finer details on constructing beamformers using packing ratios

in Appendix B.


2.4.3 Extending packing ratios to larger networks

It is possible to extend the notion of packing ratios to certain larger networks. For example, the following

theorem establishes the optimal sDoF of two-cell networks with more than three users per cell for certain

values of γ.

Theorem 2.4.3. The optimal sDoF/user of a two-cell, K-user/cell MIMO cellular network with M

antennas per user and N antennas per BS when γ = MN ∈ (0, 1

K−1 ] is given by

DoF/user ≤ min(M,max

(N2K , M2

), N2K−1

),

and when γ = MN ≥ K

K+1 , the optimal sDoF/user are given by

DoF/user ≤ min(max

(N

K+1 ,M

K+1

), NK

).

The proof of this theorem follows directly from the outer bounds established in Section 2.3.2 and

designing beamformers using the notion of packing ratios. The optimal sDoF in the interval (0, 1K−1 ]

consists of four piecewise-linear regions and a combination of random beamforming in the uplink and

beamformers having a packing ratio of K : (K − 1) achieves the optimal sDoF. When γ ≥ KK+1 ,

the optimal DoF consists of three piecewise-linear regions achieved using a combination of zero-forcing

beamformers and beamformers having packing ratio K : 1.

Extending the notion of packing ratios to any general cellular network and for all values of γ requires

us to first identify the set of fundamental packing ratios that play a crucial role in identifying the best

set of beamformers that can be designed for any given system. Identifying these fundamental packing

ratios requires an understanding of how multiple subspaces in a large network interact. In the absence of

a coherent theory characterizing such interactions, this is a major bottleneck in extending packing ratios

to general cellular networks. Different from the approach taken here, the notion of subspace alignment

chains of [19] proves useful in establishing the optimal-DoF of the three-user interference channel, while

[14] proposes a notion called irresolvable subspace chains to construct DoF-optimal beamformers for

general cellular networks.

2.5 Linear Beamforming Design: Unstructured Design

In contrast to the structured approach presented previously, we develop an alternative approach to

designing linear beamformers by relying on random linear vector equations to satisfy (2.20). Since this

approach does not require us to explicitly infer the underlying structure of interference alignment, it

bypasses the need for counting arguments and is applicable to a wide class of cellular networks. We call

this the unstructured approach (USAP) to designing linear beamformers for interference alignment and

discuss the scope and limitations of such an approach.

Our main observation is the following. For any (G,K,M × N) network, in the regime where the

proper-improper boundary lies above the decomposition based inner bound, i.e.,(

MNKM+N < M+N

GK+1

), an

unstructured approach appears to be able to achieve the optimal sDoF. The sDoF obtained numerically


04×3 04×3 α1211,1H(21,1) α1

221,1H(22,1) α1311,1H(31,1) α1

321,1H(32,1)

04×3 04×3 α2211,1H(21,1) α2

221,1H(22,1) α2311,1H(31,1) α2

321,1H(32,1)

α1111,2H(11,2) α1

121,2H(12,2) 04×3 04×3 α1311,2H(31,2) α1

321,2H(32,2)

α2111,2H(11,2) α2

121,2H(12,2) 04×3 04×3 α2311,2H(31,2) α2

321,2H(32,2)

α1111,3H(11,3) α1

121,3H(12,3) α1211,3H(21,3) α1

221,3H(22,3) 04×3 04×3

α2111,3H(11,3) α2

121,3H(12,3) α2211,3H(21,3) α2

221,3H(22,3) 04×3 04×3

v111

v121

v211

v221

v311

v321

= 0.

(2.29)

from this unstructured approach matches the optimal sDoF characterized in a parallel and independent

work [14] using a structured approach. The key advantage of the unstructured approach advocated in

this section is that it is conceptually much simpler. Further, it also achieves a significant portion of the

DoF in the regime where decomposition based inner bound lies above the proper-improper boundary.

The broad applicability of the unstructured approach with minimal dependence on network parameters

provides a single unified technique for linear beamforming design in MIMO cellular networks. This

approach along with the asymptotic scheme of [3] form the two main techniques needed to establish

the optimal DoF of MIMO cellular networks. The remainder of this section describes the unstructured

approach and presents the results of numerical experiments that identify the scope and limitations of

this approach.

2.5.1 The Unstructured Approach

Consider a (G,K,M ×N) cellular network with the goal of achieving d DoF/user without any symbol

extensions. In the uplink, note that each BS observes GKd streams of transmission of which (G− 1)Kd

streams constitute interference. Setting aside Kd dimensions at each BS for the received signals from

the in-cell users, to satisfy (2.20) the (G − 1)Kd interfering data streams must occupy no more than

N −Kd dimensions at each BS. Assuming (G− 1)Kd > N −Kd (no interference alignment is necessary

otherwise), we require the (G−1)Kd transmit beamformers of the interfering signals to satisfy GKd−N

(= L) distinct linear equations. In other words, for the ith BS, we require

G∑

l=1,l 6=i

K∑

m=1

d∑

n=1

αplmn,iH(lm,i)vlmn = 0, (2.28)

where αplmn,i refers to the coefficient associated with the interfering transmit beamformer vlmn in the pth

linear equation corresponding to the ith BS. Thus, we have GL linear vector equations, each involving

a set of (G − 1)Kd transmit beamforming vectors. Concatenating the transmit beamforming vectors

vlmn into a single vector v = [v111,v112, . . . ,v11d, . . . ,vGKd] and by appropriately defining the matrix

M, the GL linear vector equations can be expressed as the matrix equation Mv = 0. Note that M is a

GLN ×GKMd matrix.

As an example, for the (3, 2, 3× 4) network with d = 1, the linear matrix equation Mv = 0 is given


by (2.29). It is known that for the above example, interference alignment is feasible. In other words, it

is known that there exists a set of coefficients {αplmn,i} such that the system of equations in (2.29) has

a non-trivial solution. Note that the matrix M in this case is a 24× 18 matrix (system of 24 equations

with 18 unknowns), and that a random choice of coefficients {αplmn,i} results in a matrix M having full

column rank, rendering the system of equations infeasible. Determining the right set of coefficients is

non-trivial and highlights a particular difficulty in finding aligned beamformers using the set of equations

characterized by Mv = 0.2

Now, suppose we append an additional antenna to each BS, thereby creating a (3, 2, 3× 5) network

and then consider designing transmit beamformers to achieve 1 DoF/user, it can be shown that the

transmit beamformers now need to satisfy a system of equations of the form Mv = 0, where M is a

12×18 matrix. It is easy to see that even a random choice of coefficients permits non-trivial solutions to

this system of equations. The ability to choose a random set of coefficients is quite significant as instead

of solving a set of bilinear polynomial equations for interference alignment, we now only need to solve a

set of linear equations. We thus have two networks, namely, the (3, 2, 3× 4) network and the (3, 2, 3× 5)

network that significantly differ in how aligned beamformers can be computed. This points to a much

broader divide among MIMO cellular networks.

While aligned beamformers satisfy the system of equations Mv = 0 for a set of coefficients, not all

solutions to Mv = 0 with a fixed set of coefficients form aligned beamformers. A vector v satisfying

Mv = 0, can be considered to constitute a set of aligned beamformers provided (a) the set of beamformers

corresponding to a user are linearly independent, i.e., Vij is full rank ∀i, j; (b) the signal received from

a user at the intended BS is full rank i.e., H(ij,i)Vij is full rank; and (c) signal and interference are

separable at each BS. Since we assume generic channel coefficients and since direct channels are not

used in forming the matrix M, (c) is satisfied almost surely, while (b) is true under the assumption of

generic channel coefficients provided (a) is true. While the idea of satisfying conditions for interference

alignment through random linear equations is also discussed in [31], the presentation in [31] is limited

to achieving 1 DoF/user, thereby avoiding the necessity to check for linear independence of the transmit

beamformers.

Since M is a GLN×GKMd matrix, whenever LN < KMd the system of equations Mv = 0 permits

a non-trivial solution for any random choice of coefficients. When LN < KMd, a solution to the equation

Mv = 0 can be expressed as v = det(MMH)(I −MH(MMH)−1M)r, where r is a GKMd× 1 vector

with randomly chosen entries. For v to qualify as a solution for interference alignment, we need to

ensure that condition (a) is satisfied, i.e., we need to ensure that the set of transmit beamformers vij1,

vij2 . . . vijd obtained from v are linearly independent for any i ∈ {1, 2, . . . , G}, j ∈ {1, 2, . . . ,K}. LettingVij be the M × d matrix formed using vij1, vij2 . . . vijd, checking for linear independence is equivalent

to checking if the determinant of the matrix [Vij Rij ], where Rij is a (M − d) × d matrix of random

entries, is non-zero or not.

Since the determinant of [Vij Rij ] is a polynomial in the variables Rij , r, the coefficients {αplmn,i},

2A classic example in this context is the three-user interference channel with two antennas at each node, where it is

known that 1 DoF per receiver is achievable [3]. The matrix M in this case is a 6× 6 matrix with no non-trivial solutions

to Mv = 0 unless the coefficients are chosen carefully. The set of aligned transmit beamformers in this case are the eigen

vectors of an effective channel matrix, with the coefficients being related to the eigen values of this effective channel matrix.


and the channel matrices {H(lm,i)}, checking for linear independence of the transmit beamformers is

equivalent to checking if this polynomial is the zero-polynomial or not. This problem is known as poly-

nomial identity testing (PIT) and is well studied in complexity theory [37]. While a general deterministic

algorithm to solve this problem is not known, a randomized algorithm based on the Schwartz-Zippel

lemma [38, 39] is available and it involves evaluating this polynomial at a random instance of Rij , r,

{αplmn,i}, and {H(lm,i)}. If the value of the polynomial at this point is non-zero, then this polynomial is

determined to be not identical to the zero-polynomial. Further, it can be concluded that this polynomial

evaluates to a non-zero value for almost all values ofRij , r, {αplmn,i}, and {H(lm,i)}. If on the other hand,

the polynomial evaluates to the zero, the polynomial is declared to be identical to the zero-polynomial

and this statement is true with a very high probability as a consequence of the Schwartz-Zippel lemma.

Thus, whenever LN < KMd, we propose a two step approach to designing aligned beamformers.

We first pick a set of random coefficients, form the linear equations to be satisfied by the transmit

beamformers and compute a set of transmit beamformers by solving the system of linear equations. We

then perform the numerical test outlined above to ensure that the transmit beamformers are indeed

linearly independent. If the transmit beamformers pass the numerical test then they can be considered

to be a set of aligned transmit beamformers. Further, if such a procedure works for a (G,K,M × N)

network with d DoF/user for a particular generic channel realization, then it works almost surely for

all generic channel realizations of this network. This observation allows us to construct a numerical

experiment to verify the limits of using such an approach.

2.5.2 Numerical Experiment

The numerical experiment we perform is outlined as follows. We consider a network with G cells and

K users/cell. For this network, we consider all possible pairs of M and N such that M ≤ Mmax and

N ≤ Nmax, where Mmax and Nmax are some fixed positive integers. For a fixed M and N , we then

consider the feasibility of constructing aligned beamformers using the method described above in order

to achieve d DoF/user where d is such that L > 03, LN < KMd, d ≤ M , Kd ≤ N , M < GKd4, gcd(M,N, d) = 15 and (G,K,M × N, d) form a proper system. For such a set of M , N , and d,

we generate an instance of generic channel matrices and proceed to carry out the two step procedure

outlined earlier. Such a procedure is said to be successful if the polynomial test returns a non-zero value

and unsuccessful otherwise. If successful, we conclude that such a procedure can be reliably used to

design transmit beamformers for almost all channel instances of the (G,K,M × N, d) network under

consideration. When unsuccessful, we conclude that with a very high probability such a procedure does

not yield a set of aligned transmit beamformers for almost all channel instances.

While we considered designing transmit beamformers in the uplink (USAP-uplink) using random

linear vector equations, we can alternately consider designing transmit beamformers in the downlink

(USAP-downlink) using the same process. For the (G,K,M × N, d) network, it can be shown that

3When L ≤ 0, random transmit beamforming in the uplink achieves the necessary DoF.4When M ≥ GKd, random transmit beamforming in the downlink achieves the necessary DoF.5Spatial scale invariance states that if d DoF/user are feasible for a (G,K,M × N) network, then sd DoF/user are

feasible in a (G,K, sM × sN) network where s ∈ Z+ denotes the scale factor. While no proof of such a statement is

available, no contradictions to this statement exist to the best of our knowledge.


0

1G

γl

γl+1

12

1G

10 γl

γγ+1

γ+1G+1

γ

1G−γ

γ2

Gγ−1

I

II

γ (M/N)

Norm

alizedDoF/user

USAP-uplink applicable

MAC/BC DoF bound

USAP-uplink necessary condition


Random beamforming in uplink


USAP-downlink necessary condition

Piecewise-linear optimal sDoF

Figure 2.6: Inner and outer bounds on the DoF of the G-user interference channel. The optimal DoF

consists of infinitely many piecewise-linear components when γ < γl, while the decomposition based

approach determines the optimal DoF when γ ≥ γl.

GK(GKd − M)M < GKdN is a necessary condition for the linear system of equations obtained in

USAP-downlink to have a non-trivial solution. While there are no significant differences between USAP-

uplink and USAP-downlink for the interference channel (K = 1), a major difference emerges for cellular

networks where K > 1. For cellular networks, when designing transmit beamformers in the downlink,

direct channels get involved in the linear system of equations and as a result, a solution to the linear

system is no longer guaranteed to satisfy conditions (b) and (c) even when channel coefficients are

generic. In this respect, USAP-uplink has a significant advantage over USAP-downlink for cellular

networks. In addition, for cellular networks, the necessary condition GK(GKd−M)M < GKdN places

further restrictions on the applicability of USAP-downlink in the context of achieving the optimal DoF.

We discuss the scope and limitations of USAP-uplink and USAP-downlink in the next section. For

clarity, we present our observations for the interference channel (K = 1) and the cellular network

separately (K > 1).


310

13

25

37

49

12

15

13

12

35

57

12

45

911

56

23

34

1

γ (M/N)

Norm

alizedDoF/user

USAP-uplink successfulUSAP-uplink unsuccessful

USAP-uplink necessary cond.


Rand. beamforming in uplink

Decomp. based inner bound

Figure 2.7: Results of the numerical experiment for the three-user interference channel. Observe that a

clear piecewise-linear boundary emerges between the successful and unsuccessful trials of the proposed

method. The observed boundary matches with the characterization of the optimal DoF in [19].

2.5.3 Unstructured Approach for MIMO Interference Channel

In Fig. 2.6 we sketch some well known bounds on the normalized sDoF/user (sDoF/user/N) as a function

of γ ∈ (0, 1] for the G-user (G > 3) interference channel. By symmetry, it suffices to only consider

γ ≤ 1. Except for the three-user interference channel, the proper-improper boundary and decomposition

based inner bound intersect at a point γl < 1 and this point splits the optimal sDoF characterization

into a piecewise-linear region and a smooth region characterized by the decomposition based inner

bound [13, 14]. A simple DoF bound obtained by letting all the BSs or users6 cooperate (denoted as

MAC/BC DoF bound) is also plotted along with the maximum achievable sDoF using random transmit

beamforming in the uplink. We also plot the curves characterizing the necessary conditions for USAP-

uplink and USAP-downlink to be applicable. It can be shown that these two conditions, the proper-

improper boundary and decomposition inner bound all intersect at γl =(G−1)−

√(G−1)2−4

2 .

We first narrow our focus to region I (shaded blue) in Fig. 2.6, where the optimal sDoF exhibits a

piecewise-linear behaviour. For the 3-user interference channel, the point of intersection γl is equal 1, and

a complete characterization of this piecewise-linear behaviour for all γ ∈ (0, 1] is provided in [19]. Since

region I lies below the necessary condition for USAP-uplink/USAP-downlink, USAP-uplink/USAP-

downlink is applicable for any (M,N, d) such that (M/N, d/N) falls in this region. Since the optimal

sDoF of the three-user interference channel are known for all γ, we test the scope of USAP-uplink for

this channel.

We carry out the numerical experiment described earlier for the three-user interference channel with

6To be consistent with the previous sections, we refer to nodes with N antennas as BSs and nodes with M antennas

as users and use the usual notions of uplink and downlink.


1/4

3/11

8/293−

√5

5−√5· · · · ·

14

13

411

38

1129

...

3−√5

2

23

γ (M/N)

Norm

alizedDoF/user

USAP-uplink successful

USAP-uplink unsuccessful


USAP-downlink necessary condition




MAC/BC DoF bound

Figure 2.8: Results of the numerical experiment in region I of the four-user interference channel. Observe

that a clear piecewise-linear boundary emerges between the successful and unsuccessful trials of the

proposed method. The observed boundary matches with the optimal DoF as characterized in [14].

values of M , N , and d such that (M/N, d/N) falls in region I, with Nmax = Nmax = 75. The results of

this experiment are shown in Fig. 2.7, where we observe that a clear piecewise-linear boundary emerges

between the successful and unsuccessful trials on the polynomial identity test. This boundary exactly

matches with the piecewise-linear optimal sDoF as detailed in [19], suggesting that such an approach is

capable of achieving the optimal sDoF of the three-user interference channel. We also observe that the

boundary characterizing the necessary conditions for USAP-uplink has no particular significance and

the success or failure of the proposed method is completely determined by the polynomial identity test.

A similar piecewise linear boundary also emerges in the case of the four-user interference channel as

seen in Fig 2.8 for γ ∈ (0, γl). These results are in-line with the results on the optimal sDoF of this

network as established in [14].

These observations lead us to conjecture that for any G-user interference channel, whenever γ ∈(0, γl), the optimal sDoF exhibits a piecewise-linear behavior and the optimal sDoF in this regime can

be achieved by constructing linear beamformers using the proposed method.

Shifting focus to region II (shaded yellow) in Fig. 2.6, note that this region lies entirely below

the decomposition based inner bound and does not impact the characterization of the optimal sDoF.

Also note that this region lies below the proper-improper boundary and the necessary condition for

USAP-uplink, thus making USAP-uplink applicable in this region. This region is bounded below by the

maximum DoF that can be trivially achieved using random transmit beamforming in the uplink. In

order to verify the applicability of USAP-uplink in this region, we carry out the numerical experiment


3−√5

5−√5

14

14

3−√5

20 1

3

320

25

1

γ2

4γ−1

γγ+1

γ+15

14−γ

γ

γ (M/N)

Norm

alizedDoF/user


USAP-downlink necessary cond.




MAC/BC DoF bound

Figure 2.9: Results of the numerical experiment in region II of the four-user interference channel. Observe

that the necessary condition for USAP-uplink completely determines the success of failure of the proposed

approach, making the polynomial identity test redundant.

outlined earlier on the four-user interference channel for values of (M,N, d) such that the (M/N, d/N)

falls in region II, with Nmax = Nmax = 75. The results are presented in Fig. 2.9, where it is seen that the

necessary condition for USAP-uplink, LN < KMd, completely determines the success of the proposed

method, with the subsequent numerical test proving to be redundant. It is also significant to note that

these results bring to light a computational boundary that divides systems for which computing transmit

beamformers for interference alignment is easy (requires solving a system of linear equations; no worse

than O((GKMd)3)) in complexity) and systems that require techniques of higher complexity such as

iterative algorithms [33–36] to design such transmit beamformers.

So far, except for networks where the underlying structure for interference alignment is known (the

three-user interference channel etc.), solving for aligned beamformers of a given network meant solving a

system of bilinear equations through computationally intensive iterative algorithms that can sometimes

take several thousand iterations to converge [40]. Our observations suggest that except when the DoF

demand d placed on a (G, 1,M ×N) network is such that γ > γl and (γ, d/N) is sandwiched between

the necessary condition for USAP-uplink and the proper-improper boundary, iterative algorithms are

not necessary and that the aligned beamformers can be computed by simply solving a system of linear

equations.

It can be shown that USAP-downlink also exhibits a similar piecewise linear behavior whenever γ <

γl. When γ ≥ γl, since the necessary condition for USAP-uplink lies above the necessary condition for

USAP-downlink, the set of systems that can take advantage of the proposed method remains unchanged.


0

1GK

γl

Kγl+1 · · · ·

γr

Kγr+1

1K

0 1GK

γl 1 γr (G− 1) + 1K GGK−1

K

γKγ+1

γ+1GK+1

γGK

1K(G−γ)

γ2

GKγ−1

I

II

III

γ (M/N)

Norm

alizedDoF/user

USAP-uplink applicable

MAC/BC DoF bound

USAP-uplink necessary condtion


Rand. beamf. in uplink/downlinkDecomposition based inner bound


Piecewise-linear optimal sDoF

Figure 2.10: Inner and outer bounds on the DoF of the G-cell, K-user/cell network. The optimal DoF

consists of infinitely many piecewise-linear components for γ < γl and γ > γr, while the decomposition

based approach determines the optimal DoF when γl ≤ γ ≤ γr.

2.5.4 USAP-uplink for MIMO Cellular Networks

Fig. 2.10 is a sketch analogous to Fig. 2.6 and applies to any MIMO cellular network, with the

exception of the two-cell, two-user/cell and the two-cell, three-user/cell networks. Note that γ is no

longer restricted to (0, 1]. While the necessary condition for USAP-uplink, the proper-improper boundary

and the decomposition based inner bound all intersect at the same two points γl and γr, the same is

not true for the necessary condition of USAP-downlink. The points of intersection γl and γr can be

computed to be the pointsK(G−1)±

√K2(G−1)2−4K

2K . The optimal sDoF of a general cellular network is

recently investigated in [14]. The optimal sDoF as characterized in [14] has a piecewise-linear behaviour

in regions I (γ < γl) and III (γ > γr) (see Fig. 2.10). Based on the results in [13] for the MIMO

interference channel, the decomposition based inner bound is likely to characterize the optimal DoF

whenever γl ≤ γ ≤ γr.

Focusing on regions I and III, we note that USAP-uplink is applicable to all points in these two

regions. To gain insight on the scope of this technique for cellular networks, we perform the numerical

experiment outlined earlier for the 2-cell 4-user/cell network. For this network, the proper-improper

boundary and the decomposition based inner bound touch each other at γ = 1/2, i.e., γl = γr = 1/2,

with the decomposition based inner bound lying entirely below the proper-improper boundary. The

results of the numerical experiment are plotted in Fig. 2.11 and it is easy to see that a clear piecewise

linear boundary emerges between the successful and unsuccessful trials, with the successful or failure of

the proposed method completely determined by the polynomial identity test.

Remarkably, the boundary of the achievable sDoF determined by our unstructured approach matches


1/8

1/7

3/202/13

1/6

2/11

3/16

1/5

1/4

18

14

27

13

720

38

12

711

231116

34

45

54

1γ (M/N)

Norm

alizedDoF/user

USAP-uplink successfulUSAP-uplink unsuccessful

USAP-uplink necessary cond.


Rand. beamf. in uplinkDecomp. based inner bound

Figure 2.11: Results of the numerical experiment for the two-cell, four-user/cell network. Note the clear

piecewise-linear boundary that emerges between the successful and unsuccessful trials of the proposed

method. The observed boundary matches with the result in [14].

with the optimal sDoF claimed in [14]. This leads us to conjecture that for any G-cell K-user/cell cellular

network with (G,K) /∈ {(2, 2), (2, 3)}, when γ ∈ (0, γl) ∪ (γr∞) the optimal sDoF can be achieved by

constructing linear beamformers using the proposed method. Further, the optimal sDoF in this regime

exhibits a piecewise linear behaviour as also observed in [14], where a structured approach to linear

beamforming based on irresolvable subspace chains is used to establish these results, unlike the approach

discussed here.

Observations on the applicability of USAP-uplink in region II7 are similar to observations made in

the context of the interference channel. By running the numerical experiment on the 3-cell, two-user/cell

network for (M,N, d) such that (M/N, d/N) lies in region II, we note from Fig. 2.12 that the necessary

condition LN < KMd also ensures the success of the polynomial identity test. It is thus seen that even

in the regime where γl ≤ γ ≤ γr, a significant portion of the achievable sDoF can be achieved using the

unstructured approach.

A major difference between interference channels and cellular networks arises with respect to the

scope and limitations of USAP-downlink. It is clear from Fig. 2.10 that due to the nature of the

necessary condition associated with USAP-downlink, USAP-downlink cannot be used to establish the

7Note that for cellular networks with G > 4, the inner bound obtained through random transmit beamforming in the

downlink (GKd ≤ M) and the USAP-uplink’s necessary condition (LN < KMd) intersect at two points, thereby splitting

region II into two separate parts. This does not alter any of the observations made in this section.


12

2√2+1

7√2

· ·

2√2−1

7√2

· ·

00

16

√2−1√2

√2+1√2

16

52

31

γ2

6γ−1γ+17

12(3−γ)

γ2γ+1

γ6

γ (M/N)

Norm

alizedDoF/user

MAC/BC DoF bound



Rand. beamf. in uplink/downlinkDecomposition based inner bound


USAP-uplink successful

Figure 2.12: Results of the numerical experiment outlined in region II of the three-cell, two-user/cell

network. Observe that the necessary condition for USAP-uplink completely determines the success of

failure of the proposed approach.

same piecewise linear behaviour in regions I and III, as observed with USAP-uplink. Further, as stated

earlier, since direct channels get involved in the linear system generated by USAP-downlink, verifying

that a solution to the linear system also satisfies conditions for interference alignment involves further

checks such as ensuring the separability of signal and interference. Due to these reasons, the utility of

USAP-downlink for cellular networks is quite limited and offers no particular advantages over USAP-

uplink.

2.6 Summary

This chapter investigates the DoF of MIMO cellular networks. In particular we establish the achievable

DoF using the asymptotic scheme and the linear beamforming scheme. Through a new set of outer

bounds, we establish conditions for optimality of the decomposition based approach. Through these

outer bounds it is apparent that the optimal DoF of a general G-cell, K-users/cell network exhibits two

distinct regimes, one where decomposition based approach dominates over linear beamforming and vice

versa. With regard to linear beamforming, we develop a structured approach to linear beamforming

that is DoF-optimal in small networks such as the two-cell two-users/cell network and the two-cell three-

users/cell network. We also develop an unstructured approach to linear beamforming that is applicable

to general MIMO cellular networks, and through numerical experiments, show that such an approach is

capable of achieving the optimal-sDoF for a wide class of MIMO cellular networks.

Although the structured design of linear beamformers takes a disciplined approach to constructing


beamformers, the wide applicability of the unstructured approach and its apparent ability to achieve the

optimal sDoF in regimes where the sDoF curve exhibits a piecewise-linear behaviour renders it highly

attractive. The remarkable effectiveness of the unstructured approach warrants a deeper investigation

on the role of randomization and that of the polynomial identity test in designing aligned beamformers.

Chapter 3

Interference Alignment Via Rank

Minimization

This chapter proposes a new framework to the design of transmit and receive beamformers for IA

without symbol extensions in MIMO cellular networks. The proposed framework uses an alternate set

of conditions for IA. The alternate set of conditions express the conditions for IA as two sets of rank

constraints, one governing the rank of interference matrices, consisting of all the interfering vectors

at a BS, and the other on the transmit beamformers in the uplink. Using this set of conditions, this

chapter develops two sets of algorithms for IA. The first part of this chapter develops rank minimization

algorithms that design aligned transmit beamformers in the uplink by iteratively minimizing a weighted

matrix norm of the interference matrix. The choice between Frobenius norm and nuclear norm leads to

reweighted nuclear norm minimization (RNNM) or reweighted Frobenius norm minimization (RFNM)

with significantly different per-iteration complexities. We exploit the prior knowledge of the required

rank of interference matrices and propose a novel weight update rule that navigates the algorithm towards

aligned beamformers. As an alternative to solving rank minimization problems, the second part of this

chapter utilizes the prior rank knowledge in a different way and devises an alternating minimization

(AM) algorithm where the rank-deficient interference matrices are expressed as a product of two lower-

dimensional matrices. The two components are alternately optimized while keeping the other fixed.

Simulation results indicate that RNNM, which has a per-iteration complexity of a semidefinite program,

is effective in designing aligned beamformers for proper-feasible systems with or without redundant

antennas, while RFNM and AM, which have a per-iteration complexity of a quadratic program, are

better suited for systems with redundant antennas. All three algorithms are shown to outperform

previous state-of-the-art.

3.1 Introduction

Our interest in designing algorithms for interference alignment is twofold. First, in cooperative cellu-

lar networks that operate in an interference-limited regime, these beamformers identify regions in the

optimization landscape where interference is significantly mitigated. Second, while there exist several

44

Chapter 3. Interference Alignment Via Rank Minimization 45

algebraic-geometry-based techniques that establish feasibility of interference alignment and there exist

constructive approaches to design aligned beamformers for certain networks (e.g., subspace alignment

chains [19] for the 3-user interference channel), iterative algorithms [25,33–35,41–45] are still necessary to

design aligned beamformers for general (G,K,M×N) networks (for e.g., in Fig. 2.10, region sandwiched

between the proper-improper boundary and the USAP-uplink necessary condition).

In this chapter we develop two novel sets of iterative algorithms to design aligned beamformers. A

crucial insight of this chapter is that exploiting the prior knowledge of the rank of interference matrices

can significantly benefit the numerical convergence of the algorithms. In the first set of algorithms, we

use the prior rank knowledge to develop an effective algorithm to solve a rank minimization formulation.

In an alternate approach, we restrict the rank of interference matrices by expressing them in a product

form and develop a computationally efficient algorithm to find aligned beamformers.

3.1.1 Literature Survey

Several iterative algorithms are available to design beamformers for interference alignment [25,33–35,41–

47]. In [33], an iterative algorithm called iterative leakage minimization (ILM) is proposed for the MIMO

interference channel. It is based on minimizing the sum of interference powers at all the receivers. This

algorithm is extended to MIMO cellular networks in [25]. While algorithms of [33] and [25] are known

to converge, they typically need several hundred iterations. However, their per-iteration complexity is

low as they only require computing a small number of eigenvalue decompositions per iteration. More

recently, [42, 43] also propose algorithms that minimize the total interference power across all receivers.

In [35], a rank-constrained rank minimization (RCRM) framework for finding linear beamformers for

interference alignment is proposed. Recognizing that the algorithms proposed in [25,33] are equivalent to

minimizing the Frobenius norm (matrix analogue of the ℓ2-norm for vectors) of an interference matrix,

the authors instead propose to minimize its nuclear norm. The interference matrix is the projection

of the received interference vectors at a BS on the useful signal space as determined by the receive

beamforming vectors at each BS. Nuclear normminimization (matrix analogue of the ℓ1-norm for vectors)

is known to induce sparsity and is well suited for generating aligned solutions. This framework is also

capable of designing aligned beamformers when time or frequency extensions are allowed. Using the

same framework, [44, 45] further propose a reweighted version of nuclear norm minimization for finding

aligned beamformers. Due to the nuclear norm approximation, the algorithms of [35, 44, 45] involve

solving a series of semidefinite programs (SDPs). These algorithms are computationally more intensive,

on a per-iteration basis, than the algorithms of [25, 33], but they require fewer iterations. However, in

spite of using approximations which are sparsity inducing, these algorithms still lack explicit mechanisms

to generate the desired level of sparsity for interference alignment and often fall short of generating the

requisite number of interference free dimensions in practice—a shortcoming this work seeks to address.

Different from the above approaches, [46] and [47] develop algorithms to create sufficient number of

interference-free dimensions for signal-vectors by minimizing a subset of eigenvalues of the interference

matrix subject to a unitary constraint on the uplink transmit beamformers. The unitary constraints

require the optimization to be carried out on the complex Steifel manifold.


3.1.2 Proposed Framework

In this work, we design linear interference alignment for generic channels without symbol extensions using

the alternate set of conditions for interference alignment that impose two sets of rank constraints, the first

on the interference matrix of each BS and the second on the transmit beamforming matrices. As before,

the interference matrix we consider is the collection of interfering vectors at a given BS before being

projected by the receive beamforming matrices. The algorithms in [46, 47], developed originally for the

MIMO interference channel, represent one particular approach to designing algorithms for interference

alignment using such a framework.

Using this framework we develop three distinct and novel algorithms that aim to design uplink trans-

mit beamformers for interference alignment. The first two algorithms, namely, the reweighted Frobenius

norm minimization (RFNM) and reweighted nuclear norm minimization (RNNM) are developed by re-

casting the rank constraints as a rank minimization problem subject to linear constraints, while the third

algorithm, called alternating minimization (AM), is developed by expressing rank-deficient matrices that

satisfy the rank constraints as a product of two lower-dimensional matrices, then solving a simple convex

quadratic program. In each of these cases, the key observation is that exploiting prior knowledge of the

rank of interference matrices is crucial.

Reweighted Matrix Norm Minimization

The rank minimization formulation aims to minimize the total number of dimensions occupied by in-

terference at each BS by minimizing the rank of the interference matrix. Minimizing the dimensions

occupied by interference opens up more dimensions for signal vectors. While the formulation is similar

in spirit to [35, 46, 47], there are also significant differences. First, unlike [35], we only optimize over

uplink transmit beamformers and eliminate the need to alternately optimize over transmit and receive

beamformers. Second, instead of imposing rank constraints or unitary constraints on the uplink trans-

mit beamformers, we impose a simple set of affine constraints that are straightforward to satisfy. These

changes allow us to pose the problem of designing beamformers for interference alignment as a rank

minimization problem subject to affine constraints only.

Rank minimization subject to affine constraints is well studied in the context of compressive sensing

and there exist several effective algorithms to solve such problems [48–54]. Drawing inspiration from

these algorithms, we solve the rank minimization problem for interference alignment by approximating

rank using a series of weighted Frobenius or nuclear norms. The iterative reweighting of the matrix

norms provides improved approximation to the rank of the interference matrix after each iteration.

A crucial observation made in this chapter is that utilizing prior rank knowledge of the interference

matrix can significantly improve the convergence of the algorithms. Towards this end, we propose

a novel reweighting rule that couples the weights in the weighting matrices by taking advantage of

the prior knowledge of the expected rank of interference matrices, which is known once the degree of

freedom required per user is fixed. This coupling of weights acts as a control mechanism to indicate to

the algorithm the desired level of sparsity i.e., the desired ranks of the interference matrices.

The choice between reweighting the Frobenius and the nuclear norm results in two computationally

different algorithms. While RFNM, which minimizes a Frobenius norm, requires solving an unconstrained


convex quadratic program at every iteration, RNNM, which minimizes a nuclear norm, requires solving

a semidefinite program. Although the complexity-per-iteration of RNNM is significantly higher than

that of RFNM, RNNM requires much fewer iterations to converge. This raises an interesting trade-off

between complexity-per-iteration and the total number of iterations required.

Alternating Minimization

Recognizing rank deficiency of the interference matrices as the crucial bottleneck in designing beamform-

ers for interference alignment, this chapter further devises an alternative to rank minimization algorithms

by explicitly imposing the rank constraint on the interference matrices. Since the desired ranks of the

interference matrices are known a priori, we can express these rank-deficient matrices as a product of

two lower-dimensional matrices. Once expressed in this manner, we can alternately optimize one of the

two component matrices while holding the other constant. Such an approach, called alternating mini-

mization (AM), has been previously proposed in the context of robust rank minimization in the presence

of noisy observations [55, 56]. The optimization procedure can be interpreted as iterative minimization

of the difference between the actual interference matrix as determined by the set of uplink transmit

beamformers and a nominal interference matrix that identifies the subspace reserved for interference.

Key Insights

Key insights on system design from simulation results include: (a) Having redundant antennas in the

network significantly reduces the complexity of computing aligned beamformers. Algorithms with low

per-iteration complexity such as RFNM, AM and ILM are very effective for networks with redundant

antennas; (b) Systems with no redundant antennas are the more challenging systems to design aligned

beamformers for. The only effective algorithm for designing aligned beamformers for such systems is

RNNM, where per iteration complexity is that of a SDP.

3.2 System Model

The system model is the same as that presented in the earlier chapters and is restated briefly. Consider

a (G,K,M × N) network, assuming transmit and receive beamforming, the received signal at the ith

BS, after receiver processing is given by

UHijyi =

G∑

g=1

K∑

k=1


ijni, (3.1)

where all notations are the same as before. The space occupied by interference at the ith BS is denoted

as the column span of a matrix Ri formed using the column vectors from the set {H(gk,i)vgkl : g ∈{1, 2, . . . , G}, k ∈ {1, 2, . . . ,K}, l ∈ {1, 2, . . . , d}, g 6= i}, where we use the notation vgkl to denote the

lth beamformer associated with the kth user in the gth cell.


3.3 Problem Formulation

As shown in Chapter 2, the original conditions for interference alignment,

UHijH(gk,i)Vgk = 0 ∀ (i, j) 6= (g, k) (3.2)

rank(UHijH(ij,i)Vij) = d ∀ (i, j). (3.3)

can be alternately stated as

rank(Ri) ≤ N −Kd ∀ i, (3.4)

rank(Vij) = d ∀ i, j. (3.5)

In the rest of this chapter, we collectively refer to the set {Vij} as V .Since we need to design transmit beamformers that satisfy conditions (3.4) and (3.5), it is natural

to pose the problem of finding these beamformers as a feasibility problem, as given below:

minimizeV,{Ri}

1

subject to rank(Ri) ≤ N −Kd ∀ i,

rank(Vij) = d ∀ (i, j),

Li(V) = Ri ∀i.

(3.6)

where Li(V) = Ri implicitly captures the linear relationship between V and Ri1. While the rank

constraint on Vij is easily handled by restricting it to be in column-reduced echelon form, i.e., Vij(1 :

d, 1 : d) = I ∀ i, j, handling the rank constraint on Ri is not straightforward. However, the prior

knowledge of the rank of Ri allows us to re-frame the rank constraint into a more amenable form.

Towards this end, we propose two contrasting formulations. Substituting the rank constraints on Vij

with equivalent affine constraints leaves us with only the other two constraints to satisfy. In the first

formulation we transform the feasibility problem to a rank minimization problem by moving the rank

constraints on Ri to the objective function. Such a transformation leads to an optimization problem of

the formmin

V,{Ri}max

i∈{1,2,...,G}rank(Ri)

subject to Vij(1 : d, 1 : d) = I ∀ (i, j)

Li(V) = Ri ∀ i.

(3.7)

This is a minimax optimization problem where we minimize the maximum rank of the matrices R1,

R2,. . . , RG. Since we assume the given system to be proper and feasible, the global optimum of

this optimization problem is no more than N − Kd. Further, any set of beamformers that achieves

the objective rank of N − Kd constitutes a set of aligned solutions. As shown in the next section,

after a few more changes, this formulation allows us to use the prior knowledge of rank to devise an

effective algorithm to find aligned beamformers. This prior knowledge is rarely available in generic rank

minimization problems but is readily available in the context of interference alignment as the degrees of

1For example, in a (2, 2, 4 × 6) network when 2 DoF/user are desired, L1(V) = (R1) captures the linear relationship

given by R1 = [H(21,1)V21 H(22,1)V22 H(31,1)V31 H(32,1)V32].


freedom required for each user is fixed beforehand based on sufficient conditions that ensure feasibility

of interference alignment [21, 22]. Such prior knowledge proves vital in reducing just the right number

of singular values of the interference matrices to zero.

In the second formulation we relax the affine constraint governing V and Ri and move this constraint

to the objective function by imposing a quadratic penalty on their difference (Li(V)−Ri). This allows

us to treat Ri as a free variable that is defined independent of V and lets us handle the rank constraint

on Ri by expressing it as a product of two matrices Pi and Qi where Pi is a N × (N − Kd) matrix

and Qi is a (N − Kd) × (G − 1)Kd matrix, so that the rank of Ri is no more than N − Kd. This

reformulation leads to an optimization problem of the form

minV,{Pi},{Qi}

maxi∈{1,2,...,G}

‖Li(V)−Ri‖2F

subject to Ri = PiQi ∀ i,

Vij(1 : d, 1 : d) = I ∀ (i, j).

(3.8)

Since all matrices permit a singular value decomposition (SVD), there is no loss of generality in expressing

Ri as the product PiQi. Since we assume the given system to be proper and feasible, the global optimum

of this optimization problem occurs when the objective is reduced to zero. The set of beamformers

obtained at the global optimum satisfy the conditions for interference alignment.

Note that the two formulations differ in the constraint set over which the matrix Ri is optimized.

While the second formulation restricts the optimization to a set of rank deficient matrices, no such

condition is imposed in the first. Since rank deficient matrices constitute a measure-zero set over the

space of all matrices of a given dimension, restricting the search space to a set of rank deficient matrices

significantly prunes the search space. However, this gain comes at the cost having to define Ri as a

product of two matrices, thereby losing linearity.

Due to the non-smooth, non-convex nature of the rank function (3.7), the non-linearity in (3.8),

and the difficulty in dealing with minimax formulation in both cases, further approximations to these

formulations are necessary before developing algorithms to find aligned beamformers. We treat these

two formulations in further detail in the next two sections.

3.4 Interference Alignment as Rank Minimization

This section focuses on the first formulation where interference alignment is cast as a rank minimization

problem. We first propose to replace the minimax formulation (3.7) by a min-sum formulation given by

minimizeV

G∑

i=1

rank(Ri)


Li(V) = Ri ∀i, .

(3.9)

Although techniques such as the subgradient algorithm could have been used to solve the minimax formu-

lation directly, such techniques are known to converge very slowly. Replacing the minimax optimization

problem with a min-sum optimization problem significantly alters the problem landscape. However,


by taking advantage of the prior knowledge of the rank of optimal Ri, this min-sum formulation can

be effectively used to design aligned beamformers. This addresses a key difficulty in dealing with the

original minimax formulation (3.7). A similar approach can be adopted for (3.8).

Next, in order to apply standard optimization techniques for rank minimization, rank needs to be

approximated using a surrogate function. Well-known surrogate functions for rank include nuclear norm

(convex envelope of rank), Schatten-p function [54], log(det(·)) and −tr(inv(·)) [52, 53]. The choice of

surrogate function determines the per-iteration complexity of the resulting algorithms. Solving (3.9)

using the log(det(·)) approximation leads to a sequence of SDPs each minimizing a weighted Nuclear

norm [48], while using the Schatten-p function requires solving a series of unconstrained convex quadratic

programs each minimizing a weighted Frobenius norm. Either approximation leads to solving a series of

convex optimization problems of the form

minimizeV

G∑

i=1

f si (Ri)


Li(V) = Ri ∀i.

(3.10)

where f si (Ri) is a convex function of Ri that is used to approximate its rank in the sth iteration. The

crucial element of such an approach is the iterative update of fki (·). While the iterative updates suggested

in [48, 54] are effective for rank minimization problems, they do not always minimize the rank of Ri to

the required extent for interference alignment in the min-sum formulation. The main idea of this section

is to develop a new update rule that ensures the rank of Ri is minimized to the desired extent while

factoring in the minimax nature of the original formulation. This new approach is discussed for weighted

Frobenius norm and for weighted nuclear norm minimization separately below.

3.4.1 RFNM

We begin by discussing the standard approach for matrix rank minimization via reweighted Frobe-

nius norm minimization. Solving affine-constrained rank minimization by iteratively solving a series of

quadratic programs that minimize a weighted Frobenius norm is first discussed in [57], where the rank

of a matrix X is approximated using the Schatten-p function given by tr(XHX + γI)p/2 for 0 < p ≤ 1.

Noting that the derivative of the Schatten-p function is given by pX(XHX+ γI)p/2−1, it is shown that

the KKT conditions of the resulting affine-constrained optimization problem can be satisfied by itera-

tively solving a set of a weighted-least-squares problems. Mathematically, the affine constrained rank

minimization problem:

minimize rank(X)

subject to L(X) = b,(3.11)

is solved by iteratively solving the following optimization problem:

minimize tr(WsXHX) = ‖X(Ws)

1/2‖2Fsubject to L(X) = b,

(3.12)


where the weights Ws are positive semidefinite matrices and are updated using the rule Ws+1 =

((Xs)H(Xs) + γs+1I)

p2−1 where the optimal X obtained after the sth iteration is denoted as Xs and

γs+1 is the regularization parameter used in updating the (s+ 1)th weight. Further, the same iterative

procedure can also be applied when p = 0, where the weight update rule is justified by showing that

it solves a fixed point equation emerging from the KKT conditions that result when rank of X is

approximated as log(det(XHX+ γI)).

When weights are updated using the update rule given above, the weighting matrices can be inter-

preted to weight the singular values of the matrixXHs Xs. To see this, let the singular value decomposition

of Xs be given by PsΣsQHs , then W(s+1) = Qs(Σ

(p−2)s + γ(s+1)I)

−1QHs . Thus, the weighting matrix

Ws+1 imposes a penalty that is inversely proportional to the square of the magnitude of each non-zero

singular value of Xs. Since small, non-zero singular values are heavily penalized, the iterative procedure

is incentivized to reduce them to zero, thus reducing the rank of X.

Using a modified weight update rule, we use an adaptation of the reweighted Frobenius norm min-

imization approach outlined above to solve (3.9) while setting p = 0. This approach requires us to

iteratively solve an optimization problem of the form

minimizeV

G∑

i=1

‖Ri(W1/2i(s))

H‖2F


Li(V) = Ri ∀ i.

(3.13)

where Ri is weighted using Wi(s) in the sth iteration. Note that we implicitly assume that Ri has

more rows than columns, if not we simply replace Ri with RHi in the above formulation and all the

subsequent steps. This ensures that the number of singular values of Wi(s) and Ri are the same. It

turns out that a direct application of the weighting procedure outlined above to (3.13), while inducing

a low-rank Ri, almost never generates the total requisite number of interference-free dimensions. For

example, the formulation in (3.13) may lead to scenarios where we have more than the necessary number

of interference free dimensions at one BS with insufficient interference-free dimensions at other BSs. The

main idea of this section is that prior knowledge of the expected rank of Ri can be used to design better

weight update rules to address these issues.

Proposed Reweighting Technique

Since we are looking for transmit beamformers that ensure rank(Ri) ≤ N − Kd ∀ i, we require z =

min(Kd, (GKd −N)) singular values of Ri to be zero. To avoid local minima where rank of Ri is not

sufficiently minimized, we couple the penalties (weights) associated with each of the z smallest singular

values of R1,R2, . . . ,RG. Let {σir : r = 1, 2, . . . ,min ((G− 1)Kd,N)} be the set of singular values of

Ri obtained after the sth iteration, ordered in the descending order i.e., σir ≥ σi(r+1). Further, let

σ2min = min

i,rσ2ir, (3.14)


Algorithm 1 Reweighted Frobenius Norm Minimization

1: Initialize Wi(1) = I ∀i, set γ = γ1, s = 1.

2: for s=1 to smax do

3: Solve (3.13) using weights Wi(s) and denote the optimal interference matrices as Riopt.

4: Compute the reduced SVD of Riopt, and denote it as Pi(s)Σi(s)(Qi(s))H .

5: Set σ2min = mini,r σ

2ir , where σir are the singular values of Ri(s) arranged in descending order.

6: Set Di(s) = diag([σ2i1, . . . , σ

2i(N−Kd), σ

2min, . . . , σ

2min

︸︷︷︸

z times

]).

7: Set γ to max(σ2min, 10

−8).

8: Update Wi(s+1) = (Qi(s)Di(s)(Qi(s))H + γI)−1.

9: Return to Step (2a) if s < itermax.

10: end for

and define the diagonal matrix Di(s) as

Di(s) = diag([σ2i1, σ

2i2, . . . ,σ

2i(N−Kd),

σ2min, σ

2min, . . . , σ

2min

︸︷︷︸

z times

]). (3.15)

We set the weights for the (s+ 1)th iteration to be

Wi(s+1) = (Qi(s)Di(s)(Qi(s))H + γs+1I)

−1. (3.16)

Such an update equally penalizes each of the z smallest singular values of Ri, thereby encouraging the

algorithm to seek aligned beamformers where all z smallest singular values can be simultaneously set to

zero. The proposed iterative procedure is summarized as Algorithm 1.

The parameter γ acts as a regularization constant that makes sure the weighting matrices are positive

definite. It limits the penalty imposed on small non-zero singular values and is typically reduced with

each iteration to prevent the algorithm from prematurely converging to local minima.

Once the transmit beamformers in the uplink are designed, the receiver beamformers at the ith BS

to recover the data streams of the jth user can be chosen to be the left-singular vectors corresponding to

the d smallest singular values of the matrix Ri augmented with the interfering vectors from other users

in the same cell.

Convergence and Complexity

When weights are updated according to the original update rule given in [54], the surrogate function

monotonically decreases and since the function is bounded below, the value of the surrogate function

converges. A proof of convergence of the iterates when p = 0 is not known to the best of our knowledge.

For the proposed weight update rule convergence of the iterates and the surrogate function is not

guaranteed. However, since decoupling the weights ensures convergence, we can decouple the weights

after a fixed number of iterations to let the algorithm converge. In our simulations we simply run the

algorithm for a fixed number of iterations (itermax) with coupled weights.

To analyze the complexity of the algorithm, note that each iteration of the proposed heuristic requires

solving the quadratic program in (3.13) and computing SVDs of G matrices of size N × (G− 1)Kd. The


two linear constraints in (3.13) are easy to eliminate as Ri is just an auxiliary variable and the conditions

on Vij are easy to satisfy. Thus, (3.13) is an unconstrained quadratic program in GK(M − d)d (denote

this as ν) variables and requires solving a linear system in the same number of variables which can be

accomplished in O(ν3) time. Each SVD can be computed in O(max(N, (G− 1)Kd)3) time.

3.4.2 RNNM

We now present an analogous reweighting strategy for solving the interference alignment problem by

rank minimization where the rank is approximated by nuclear norm. Rank minimization is the matrix

analogue of ℓ0-norm minimization for vectors, and techniques developed for ℓ0-norm minimization can

be extended to rank minimization. In the compressive sensing literature, ℓ0-norm minimization is often

approximated by reweighted ℓ1-norm minimization [49]. The matrix counterpart of reweighted ℓ1-norm

minimization proposed in [49] is reweighted nuclear norm minimization [48]. This procedure reformulates

the rank minimization in (3.9) as

minimize

G∑

i=1

‖WiLRiWiR‖∗


Li(V) = (Ri) ∀i.

(3.17)

where WiL and WiR are two positive definite matrices that are interpreted to reweight the nuclear

norm of Ri. The iterative procedure involves solving an instance of (3.17) for fixed WiL and WiR, then

updating the weights for the next iteration. Every iteration requires solving a semidefinite program.

The choice of the weight update rule affects the overall performance and needs to be chosen carefully.

In [48], the authors derive a weight update rule by exploiting an equivalence relation between the rank of

Ri and a positive semidefinite matrix Z of the form

A Ri

RHi B

, and approximating the rank of Z using

the surrogate function log(det(·)). Such a weight update rule is interpreted to minimize this concave

surrogate function through a majorization-minimization procedure [48,49,52]. The resulting weights can

be interpreted as the reweighting of the singular values of the matrix Z.

Proposed Reweighting Technique

While we adopt the general framework of reweighting as proposed in [48] for solving the interference

alignment problem, we update weights using an update rule similar to that proposed in the previous

section. Note that we require z = min(Kd, (GKd−N)) of the y = min(N, (G− 1)Kd) singular values of

Ri to be zero. Let PisΣis(Qis)H be the full singular value decomposition of the optimum Ri obtained

after the sth iteration and let σmin = mini,j σij , where σij are the singular values of Σis. We once

again couple the penalties associated with those singular values that need to be set to zero and update

the weights for the (s+ 1)th iteration as follows

WiL(s+1) =Pis(DiL(s+1) + γI)−12 (Pis)

H (3.18)

WiR(s+1) =Qis(DiR(s+1) + γI)−12 (Qis)

H (3.19)


Algorithm 2 Reweighted Nuclear Norm Minimization

1: Initialize WiL1 = I, W1iR1 = I ∀i.


3: Solve (3.17) using weights WiLs and WiRs; Denote the optimal interference matrices as Ris.

4: Compute the full SVD of Ris, and denote it as PisΣis(Qis)H .

5: Set σmin = mini,j σij , where σij are the singular values of Ri arranged in descending order.

6: Set DiLs and DiRs as given in (3.20) and (3.21).

7: Compute WiL(s+1) and WiR(s+1) using (3.18) and (3.19).

8: Return to Step (2a) if s < itermax.

9: end for

where DiL(s+1) and DiR(s+1) are defined as

DiL(s+1) =diag([σi1, σi2, . . . , σi(y−z),

σmin, σmin, . . . , σmin︸︷︷︸

N−y+z times

])

(3.20)

DiR(s+1) =diag([σi1, σi2, . . . , σi(y−z),

σmin, σmin, . . . , σmin︸︷︷︸

(G−1)Kd−y+z times

])

(3.21)

where the singular values are assumed to be ordered in the descending order. Again, the idea is to

equally penalize the singular values that are to be reduced to zero so that Ri is of the requisite rank.

The proposed iterative procedure is summarized as Algorithm 2.

Convergence and Complexity

A key difference between RFNM and RNNM is that every iteration of RNNM requires solving an SDP.

SDPs are typically solved using the primal-dual interior point method [58], as in popular solvers such as

SDPT3 [59, 60]. The SDP in (3.17) requires minimizing the rank of a block diagonal matrix consisting

of G blocks each of size N × (G − 1)Kd. This problem can be posed in the standard SDP form using

Lemma 1 of [61]. Excluding auxiliary variables such as Ri, when posed in standard form there are a

total of νsdp free variables, where

νsdp = GKd(M − d) +N2/2 + ((G− 1)Kd)2/2. (3.22)

The per-iteration complexity of the primal-dual interior point method used in generic solvers when

applied to RNNM grows at least as fast as O(ν3sdp) [62]. Custom SDP solvers that exploit structure in

the SDP can reduce the per-iteration complexity to O(νsdp−custom) [62], where νsdp−custom is given by

νsdp−custom = N × (G− 1)Kd× (GK(M − d)d)2. (3.23)

In general the number of iterations of the primal-dual interior point method required to solve one

instance of (3.17) is between 10-50 iterations and is only weakly dependent on the network parameters.


After solving the SDP, further computations are necessary to compute G SVDs to reweight the nuclear

norm. Clearly, the complexity of solving one instance of (3.17) is considerably higher than that of (3.13).

However, RNNM only requires solving about 20-40 instances of (3.17) as opposed to several hundred

instances of (3.13) in the case of RFNM. It is hence difficult to compare the overall complexity of RFNM

and RNNM.

Although a proof of convergence for RNNM using the proposed weight update rule is not yet available,

simulation results indicate that the algorithm converges within tens of iterations. In our simulations we

run the algorithm for a fixed number of iterations (itermax).

3.5 Interference Alignment Using AM

3.5.1 AM Algorithm

We now turn to the second formulation (3.8) for the interference alignment problem. As seen in (3.8),

relaxing the linear constraint between the interference matrix Ri and the transmit beamformers V by

imposing a quadratic penalty on the difference allows us to treat Ri as a free variable and to express it as

the product PiQi. This new approach to satisfying rank constraints on Ri is inspired by the alternating

minimization algorithm proposed in the context of robust rank minimization [55,56]. Such an approach

proposes iteratively optimizing either Pi or Qi while holding the other constant. With computational

complexity in mind, we again simplify (3.8) to a min-sum formulation as given below:

minimizeV,{Pi},{Qi}

G∑

i=1

||Li(V)−Ri||2F

subject to Ri = PiQi ∀ i,

Vij(1 : d, 1 : d) = I ∀ (i, j).

(3.24)

Note that the global optimum is still zero and that all sets of aligned beamformers achieve the global

optimum. When either Pi or Qi is fixed, the above formulation is a convex quadratic program. We solve

(3.24) by alternately solving for Pi and Qi while keeping the other fixed. Intuitively, Pi and Qi jointly

define the space occupied by interference at the ith BS, and the beamformers are designed to make sure

that the residual interference that spills beyond the space identified by Pi and Qi is minimized at every

iteration. The proposed algorithm is summarized in Algorithm 3.

3.5.2 Convergence and Complexity

Since every step in alternating minimization decreases the quadratic objective, the value of the objective

converges as it is bounded below by zero. Although the iterates (Pi, Qi and Vij) themselves cannot be

guaranteed to converge, this has no significant impact on the algorithm.

To analyze the complexity of the proposed algorithm, note that the equality constraints in (3.25) and

(3.26) are trivial to satisfy. Solving (3.25) and (3.26) requires finding the minimum of an unconstrained

convex quadratic function involving GKd(M − d) + GN(N − Kd) (denote this as νp) complex scalar

variables in the case of (3.25) and GKd(M − d) + G(G − 1)Kd(N −Kd) (denote this as νq) complex

scalar variables in the case of (3.26). This requires solving a set of linear equations involving νp and


Algorithm 3 Alternating Minimization

1: Initialize Qi for i ∈ {1, 2, . . . , G} with random entries drawn from a continuous distribution.


3: Solve the following optimization problem using Qi obtained from Step 2b (for the first iteration

use Qi initialized to random entries):

minimizeV,{Pi}

G∑

i=1

||Li(V)−PiQi||2F

subject to Vij(1 : d, 1 : d) = I ∀ (i, j).

(3.25)

4: Solve the following optimization problem using Pi obtained from Step 2a.

minimizeV,{Qi}

G∑

i=1

||Li(V)−PiQi||2F


(3.26)

5: end for

νq variables respectively and the worst case complexity of solving this set of linear equations is given

by O(ν3p ) and O(ν3q ) respectively. This algorithm is comparable in complexity to RFNM as it requires

solving a simple quadratic program. In addition, it does not require computing any SVDs.

3.6 Comparison to Existing Algorithms

We provide a brief discussion on some of the existing algorithms for the purpose of performance compari-

son. Two well-known algorithms for interference alignment include ILM [25,33,34], and RCRM [35]. ILM

aims to minimize the overall interference power across the network by solving the following optimization

problem:

minimizeVij ,Uij

G∑

i=1

‖[Ui1, . . . ,UiK ]HRi‖2F

subject to rank(Vij) = d ∀ (i, j)

rank([Ui1, . . . ,UiK ]) = Kd ∀ i

(3.27)

Note that only inter-cell interference is taken into account as intra-cell interference can be nulled sub-

sequently by an appropriate linear transformation of the receive beamformers ({Uij}). The algorithm

alternately optimizes the transmit and receive beamformers and involves computing G(K+1) eigenvalue

decompositions in each iteration.

Interpreting interference alignment as reducing the rank of the projected interference matrix, RCRM

[35] replaces Frobenius norm with nuclear norm and considers solving the following optimization problem

minimizeVij ,Uij

G∑

i=1

‖[Ui1, . . . ,UiK ]HRi‖∗

subject to λmin(UHijH(ij,i)Vij) ≥ ǫ ∀ (i, j)

(3.28)


Table 3.1: Systems used in simulations.

System

(G,K,M ×N, d)Redundancy

Number of

data streams

Interference-free dim. to

be created per BS

1(4, 1, 2× 3, 1) No 4 1

(4, 1, 3× 3, 1) Yes 4 1

2(3, 2, 3× 4, 1) No 6 2

(3, 2, 4× 4, 1) Yes 6 2

3(3, 3, 4× 6, 1) No 9 3

(3, 3, 4× 7, 1) Yes 9 2

where ǫ > 0. The constraint on the minimum eigenvalue of the received signal space ensures that: (a) all

transmit and receive beamformer matrices have rank d, and (b) the received signal space UHijH(ij,i)Vij is

not rank deficient. For any choice of ǫ > 0, the constraint on the smallest eigenvalue places a restriction

on the received signal space and the optimization is not a pure pursuit to align interference. While

this is a necessary constraint when channels are not generic, this constraint can be dropped for generic

channels provided {Uij} and {Vij} are all full rank.

The optimization problem (3.28) is convex in either {Uij} or {Vij} when the other is held fixed and

is solved by alternately optimizing {Uij} and {Vij}. Each iteration requires solving two SDPs, one each

to update {Uij} and {Vij}.In [45], instead of approximating rank using nuclear norm in the RCRM formulation, rank is approx-

imated using the log(det(·)) surrogate function. A majorization-minimization routine that minimizes a

series of weighted nuclear norms is proposed. The resulting algorithm consists of two loops— the outer

loop updates the weights, while the inner loop alternately optimizes transmit and receive beamformers

for a fixed set of weights. The core optimization problem that is solved at each step is similar to (3.28)

and is given by

minimizeVij ,Uij

G∑

i=1

‖Wi[Ui1, . . . ,UiK ]HRi‖∗

subject to λmin((UHijH(ij,i)Vij) ≥ ǫ ∀ (i, j).

(3.29)

where {Wi} are the weighting matrices. In the outer loop, the weighting matrices are updated as

Pi(Σi + γI)−1PHi , where γ is a regularizing constant and PiΣiQ

Hi is the SVD of the optimal [Ui1, . . .,

UiK ]HRi matrix obtained from the inner loop. Note that unlike the reweighting rule in Section 3.4.2,

the weight update rule in [45] does not couple the weights. Further, since all the singular values of the

interference matrices are to be reduced to zero, the idea of coupling weights does not provide any benefit

as the formulation then becomes equivalent to RCRM.

The ILM framework introduced in [33] serves as an important starting point for the subsequent al-

gorithms discussed above. The subsequent RCRM formulations are developed with the goal of inducing

sparsity to reduce the dimensions occupied by interference (after being projected by receive beamform-

ers). When interference alignment is feasible, the singular values of all the matrices in the objective of


(3.28) and (3.29) are to be reduced to zero. However, this places an extreme sparsity requirement on the

optimization problems as the rank of a GKd×G(G− 1)Kd block diagonal matrix needs to be reduced

to zero. Typical rank minimization algorithms are not designed to generate this level of sparsity and

quite often do not succeed in completely eliminating interference.

As compared to ILM and RCRM, the new algorithms proposed in this chapter have several key

advantages. First, by focusing on the received interference vectors before being projected by the receive

beamformers, the algorithms developed in this chapter do not require alternately optimizing transmit and

receive beamformers. Further, we incorporate prior knowledge of the requisite number of interference-

free dimensions in designing our algorithms. Such a design prevents the algorithms from converging to

local minima and helps reduce the rank of the interference matrices to the required extent.

3.7 Simulation Results

To evaluate the algorithms developed in this chapter, we consider systems of varying degrees of com-

plexity as listed in Table 3.1. Note that in addition to increasing number of data streams, the number of

interference-free dimensions that need to be created at each BS also increase. Also note that for every

system that is on the proper-improper boundary and has no redundant dimensions, we also consider an

identical system with redundant antennas on either the user or the BS side.

Further, all systems listed in Table 3.1 are chosen so as to lie on or below the proper-improper

boundary (d ≤ (M+N)/(GK+1)) and also satisfy d ≥ N2/(K(GN−M)) — the boundary below which

aligned beamformers can be designed in a straightforward manner using the unstructured approach.

Since these algorithms are run on finite precision computers, a numerical threshold on interference

suppression is required to declare interference alignment. We adopt signal-to-interference (SIR) ratio as

a metric to evaluate interference suppression and declare interference to be aligned if SIR for every data

stream in the network exceeds a certain threshold. It is important to consider interference suppression

across all data streams since there may exist several local minima that eliminate interference only in a

subset of the streams. While a high SIR threshold is an accurate indicator of interference alignment, a

lower threshold is of practical interest since most applications only require interference to be suppressed

up to the noise floor. We consider SIR thresholds of 20dB, 40dB and 60dB. We opt to show the

number of interference-free dimensions and instances of network-wide alignment rather than system-level

performance metrics such as sum-rate because the former is more directly related to the optimization

objective considered in this chapter.

The algorithms are tested over at least 400 channel realizations with channel coefficients drawn

independently from a complex circularly-symmetric Gaussian distribution with unit variance in each

dimension. We compare the algorithms developed in this chapter with ILM proposed in [33], the RCRM

formulation proposed in [35] and the reweighted nuclear norm minimization approach proposed to solve

RCRM (WRCRM) in [45]. The convex optimization problems involved in these algorithms are solved

using CVX, a package for specifying and solving convex programs [63, 64]. All algorithms are run for a

fixed number of iterations.

One update each of the transmit and receive beamformers is considered to be one iteration of an

algorithm. In the case of ILM, each iteration requires computing G(K + 1) eigenvalue decompositions,


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Instancesofnetwork-w

idealignment

Number of iterations

RFNM SIR>60dBRFNM SIR>40dBRFNM SIR>20dB

AM SIR>60dBAM SIR>40dBAM SIR>20dB

ILM SIR>60dBILM SIR>40dBILM SIR>20dB

0 120 240 360 480 600

(a)

0

1

2

3

4

Number

ofiterference-freedim

ensions





0 120 240 360 480 600

(b)

Figure 3.1: Network-wide alignment (a) and interference-free dimensions (b) in the (4, 1, 2× 3) network

using the RFNM, AM and ILM algorithms.


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%


idealignment





0 12 24 36 48 60

(a)

0

1

2

3

4

Number


ensions





0 12 24 36 48 60

(b)

Figure 3.2: Network-wide alignment (a) and interference-free dimensions (b) in the (4, 1, 3× 3) network

using the RFNM, AM and ILM algorithms.


while RCRM and WRCRM require solving two SDPs per iteration. Each iteration of AM and RFNM

requires solving one unconstrained quadratic program while RNNM requires solving one SDP. Due to the

significant differences in the complexities, the results corresponding to ILM, AM and RFNM algorithms

are presented separately from the results for RNNM, RCRM and WRCRM.

We first present results pertaining to the simplest two networks in Table 3.1 — the (4, 1, 2× 3) and

the (4, 1, 3× 3) networks.

3.7.1 The (4, 1, 2× 3) and (4, 1, 3× 3) networks

The performance of the ILM, AM and RFNM algorithms on the (4, 1, 2×3) network is plotted in Fig. 3.1.

Note that this system has no redundant antennas and requires one interference-free dimension at each

BS. Fig. 3.1(a) plots the number of instances (channel realizations) where network-wide alignment occurs

as a function of the number of iterations. Network-wide alignment requires the SIR in each of the four

data streams to exceed a certain threshold. Fig. 3.1(b) plots the average number of interference-free

data streams created as a function of the number of iterations. In this case, we do not look for network-

wide alignment and only focus on the SIR of each individual data stream. Scenarios where partial

interference alignment occurs (interference eliminated in only some of the data streams) contribute

towards interference-free dimensions (Fig. 3.1(b)) but not towards network-wide alignment (Fig. 3.1(a)).

It is seen from Fig. 3.1(a) that both AM and RFNM show better performance than ILM. For example,

after 120 iterations of each algorithm, RFNM achieves an SIR exceeding 60dB in all data streams for

65% of the channel realizations, but ILM only achieves an SIR exceeding 60dB in 40% of the channel

realizations. It is observed that the performance difference between the different algorithms is minimal

if the algorithms are each run for more than 600 iterations. Thus, the number of iterations after which

these algorithms are terminated plays an important role in determining their relative performance. A

similar set of observations can be made from Fig. 3.1(b).

The (4, 1, 3× 3) network has a redundant antenna at each user and its impact is clearly seen in Fig.

3.2. ILM, AM and RFNM all converge to aligned solutions within 50-60 iterations, with AM and RFNM

significantly outperforming ILM. The single additional antenna on the user-side reduces the number of

iterations required by almost a factor of 10, thus highlighting the enormous value of redundant antennas

when designing aligned beamformers.

Fig. 3.3 plots the performance of RNNM for the (4, 1, 3× 3) and (4, 1, 2× 3) networks. It is seen that

aligned beamformers are designed within 10-20 iterations in both the cases. The performance comparison

of RCRM, WRCRM and RNNM is plotted in Fig. 3.4. In Fig. 3.4, we set the SIR threshold to 20dB and

plot the number of interference-free dimensions generated. It is seen that both RCRM and WRCRM do

not generate the requisite number of interference-free dimensions and hence do not achieve network-wide

interference alignment. It appears that within the first few iterations, these algorithms get trapped in

local minima that do not reduce a sufficient number of singular values to zero. Although redundant

antennas help in finding more interference-free dimensions, the RCRM and WRCRM algorithms still

fall short of achieving network-wide interference alignment. This highlights the importance of designing

algorithms that are able to exploit the prior knowledge of the required rank.


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%


idealignment


(4, 1, 3× 2) RNNM SIR>60dB

(4, 1, 3× 2) RNNM SIR>40dB

(4, 1, 3× 2) RNNM SIR>20dB

(4, 1, 3× 3) RNNM SIR>60dB

(4, 1, 3× 3) RNNM SIR>40dB

(4, 1, 3× 3) RNNM SIR>20dB

0 4 8 12 16 20 24 28 32 36 40

(a)

0

1

2

3

4

Number


ensions


(4, 1, 3× 2) RNNM SIR>60dB

(4, 1, 3× 2) RNNM SIR>40dB

(4, 1, 3× 2) RNNM SIR>20dB

(4, 1, 3× 3) RNNM SIR>60dB

(4, 1, 3× 3) RNNM SIR>40dB

(4, 1, 3× 3) RNNM SIR>20dB

0 4 8 12 16 20 24 28 32 36 40

(b)

Figure 3.3: Network-wide alignment (a) and interference-free dimensions (b) in the (4, 1, 2 × 3) and

(4, 1, 3× 3) networks using the RNNM algorithm.


0

1

2

3

4

Number


ensions


(4, 1, 3× 3) RNNM

(4, 1, 3× 3) RCRM

(4, 1, 3× 3) WRCRM

(4, 1, 2× 3) RNNM

(4, 1, 2× 3) RCRM

(4, 1, 2× 3) WRCRM

0 3 6 9 12 15 18 21 24 27 30

Figure 3.4: Interference-free dimensions in the (4, 1, 2× 3) and (4, 1, 3× 3) networks using the WRCRM,

RCRM and RNNM algorithms. The SIR threshold is set to 20 dB.

3.7.2 Larger Networks

To further test the observations made in the previous section we now consider larger networks where

more interference-free dimensions are required at each BS. Fig. 3.5 plots the performance of ILM, AM

and RFNM on the second and third set of networks listed in Table. 3.1 that do not have any redundant

antennas. A broad trend that can be observed from Fig. 3.5 is that as the network size increases, all three

algorithms need several thousand iterations to design aligned beamformers. Further, the performance

gap between AM, RFNM and ILM grows smaller as the network size increases.

Fig. 3.6 considers the same two networks but now appended with one redundant antenna at either

the BS or the user side. It is seen from Fig. 3.6 that the redundant antenna significantly reduces the the

number of iterations required to design aligned beamformers. In particular, AM and RFNM perform

significantly better than ILM and require up to 25% fewer iterations for a similar level of performance.

As before, we see in Fig. 3.7 that the RCRM and WRCRM algorithms do not achieve the necessary

number of interference-free dimensions.

Fig. 3.8 plots the performance of RNNM for the same two networks with and without redundant

antennas. It is seen that unlike the previous three algorithms, RNNM is able to generate aligned

beamformers, even for high SIR thresholds within about 20-40 iterations, irrespective of whether there

are redundant antennas or not. There is only a marginal increase in the number of iterations required

as the network size grows.

A conclusion drawn from these simulations is that systems with redundant antennas permit the use

of algorithms that are not necessarily well suited for sparsity but have the advantage of having low

per-iteration complexity. When this additional flexibility is absent, algorithms such as RNNM that are


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%


idealignment


RFNM SIR>60dB

RFNM SIR>40dB

RFNM SIR>20dB

AM SIR>60dB

AM SIR>40dB

AM SIR>20dB

ILM SIR>60dB

ILM SIR>40dB

ILM SIR>20dB

0 150 300 450 600 750 900 1050 1200 1350 1500

(a) (3, 2, 3× 4) network

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%


idealignment


SIR>20dB

SIR>40dB

SIR>60dB

ILM

AM

RFNM

0 240 480 720 960 1200 1440 1680 1920 2160 2400

(b) (3, 3, 4× 6) network

Figure 3.5: Network-wide alignment using the RFNM, AM and ILM algorithms.


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%


idealignment





0 20 40 60 80 100

(a) (3, 2, 4× 4) network

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%


idealignment





0 30 60 90 120 150 180 210 240 270 300

(b) (3, 3, 4× 7) network

Figure 3.6: Network-wide alignment using the RFNM, AM and ILM algorithms.


0

1

2

3

4

5

6

Number


ensions


(3, 2, 4× 4) RNNM

(3, 2, 4× 4) RCRM

(3, 2, 4× 4) WRCRM

(3, 2, 3× 4) RNNM

(3, 2, 3× 4) RCRM

(3, 2, 3× 4) WRCRM

0 3 6 9 12 15 18 21 24 27 30

Figure 3.7: Interference-free dimensions in the (3, 2, 3× 4) and (3, 2, 4× 4) networks using the WRCRM,

RCRM and RNNM algorithms. The SIR threshold is set to 20 dB.

designed specifically to generate a desired level of sparsity are necessary to design aligned beamformers.

Hence, this trade-off between redundant antennas and algorithmic complexity must be factored into

design considerations for multi-antenna wireless networks.

3.8 Summary

In this chapter we propose several new approaches to designing aligned beamformers in a cellular network

based on a reformulation of the conditions for interference alignment. Using these alternate conditions,

we formulate a rank minimization problem to design aligned transmit beamformers in the uplink and

solve the rank minimization using reweighted matrix norm minimization leading to RNNM and RFNM

algorithms. A crucial aspect of these algorithms is a novel reweighting rule that exploits prior knowledge

of the required rank of interference matrices.

This chapter also develops the AM algorithm for interference alignment where the prior knowledge

of the rank of the interference matrices is used to allow the rank-deficient matrices to be expressed as

a product of two matrices and the alternate optimization of these matrices for minimizing a quadratic

objective. Each iteration of this algorithm only requires solving an unconstrained convex quadratic

program. In terms of complexity, while RNNM requires solving an SDP in each iteration, RFNM and

AM only require solving an unconstrained quadratic program.

Simulation results indicate that RNNM is very effective in designing aligned beamformers irrespec-

tive of the presence of redundant antennas. It significantly outperforms previously proposed RCRM

and WRCRM algorithms (with similar per-iteration complexity) in terms of maximizing the number of


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%


idealignment


(3, 2, 4× 4) RNNM SIR>60dB

(3, 2, 4× 4) RNNM SIR>40dB

(3, 2, 4× 4) RNNM SIR>20dB

(3, 2, 3× 4) RNNM SIR>60dB

(3, 2, 3× 4) RNNM SIR>40dB

(3, 2, 3× 4) RNNM SIR>20dB

0 4 8 12 16 20 24 28 32 36 40

(a) (3, 2, 3× 4) and (3, 2, 4× 4) network

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%


idealignment


(3, 3, 4× 7) RNNM SIR>60dB

(3, 3, 4× 7) RNNM SIR>40dB

(3, 3, 4× 7) RNNM SIR>20dB

(3, 3, 4× 6) RNNM SIR>60dB

(3, 3, 4× 6) RNNM SIR>40dB

(3, 3, 4× 6) RNNM SIR>20dB

0 10 20 30 40 50

(b) (3, 3, 4× 6) and (3, 3, 4× 7) networks

Figure 3.8: Network-wide alignment using RNNM.


interference-free dimensions. In contrast, RFNM and AM show significantly faster convergence in sys-

tems with redundant antennas when compared to ILM (which has a similar per-iteration complexity).

Chapter 4

Role of Interference Alignment in

Cellular Network Optimization

This chapter explores the role of IA in joint optimization of transmission parameters for utility maximiza-

tion in cellular networks with an ICIC architecture. In this context, this chapter proposes a two-stage

optimization framework for interference coordination and utility maximization. The first stage of the

proposed framework focuses exclusively on nulling interference from a set of dominant interferers using

IA, while the second stage optimizes the transmit and receive beamformers iteratively to maximize a

network-wide utility using the IA solution as the initial condition. This chapter focuses on maximiz-

ing the minimum rate achieved in the network. Through simulations on specific topologies of isolated

clusters of BSs with realistic channel models, it is observed that the two-stage optimization framework

outperforms straightforward optimization on these topologies. The results further indicate that as the

distance between adjacent BSs decreases, the number of scheduled users must be decreased so that more

dominant interferers can be nulled in the first stage.

4.1 Introduction

Joint optimization in coordinated cellular networks is an area of active research [5,65–75]. Typically, the

optimization problem involves the maximization of a network-wide utility function (weighted sum-rate,

max-min-fairness rate, etc) over transmission parameters such as beamformers and transmit powers.

Several novel techniques that exploit equivalence relations between various problem formulations (e.g.,

weighted sum-rate maximization and weighted mean-squared-error minimization [67,72]) or use concepts

like uplink-downlink duality [68, 71] have been proposed in the context of NUM. However, irrespective

of the problem formulation and the proposed solution, due to the non-convex nature of these problems,

an efficient method for finding the globally optimal solution is still elusive and remains a worthwhile

pursuit.

In parallel to these developments, IA has significantly helped in establishing the DoF of multi-antenna

cellular networks [14, 15, 21, 22, 24]. However, due to the asymptotic nature of these results, the value

of IA in the context of NUM under realistic channel conditions that include pathloss, shadowing and

69

Chapter 4. Role of Interference Alignment in Cellular Network Optimization 70

fading is not yet known.

Since both IA (without symbol extensions) and NUM algorithms assume global channel-state infor-

mation (CSI) and a centralized design1 (thus having a similar overhead), it is pertinent to assess the

value of IA in relation to NUM. However, due to the limited focus of IA on interference suppression while

neglecting signal strength, IA cannot be viewed as a substitute for NUM, and must instead be considered

as a potential augmentation to the optimization process. It is worth mentioning that the assumption of

perfect CSI for IA has come under significant scrutiny [7,8]. However, given that the value of IA has not

been fully established in the NUM context even with perfect CSI, this chapter makes the perfect CSI

assumption for now and focuses solely on the role of IA in wireless cellular network optimization with

perfect CSI.

This chapter proposes a two-stage optimization framework for NUM in cellular networks with an ICIC

architecture. The first stage of this framework exclusively focuses on mitigating interference from the

dominant interferers using IA. The second stage uses this altered interference landscape to optimize the

network parameters to maximize a given utility function. Such a framework counters the myopic nature of

straightforward NUM algorithms by leveraging IA’s ability to comprehensively address interference from

the dominant interferers while subsequently relying on numerical optimization algorithms to account for

signal strength and to maximize the network utility.

Recent work on multilevel topological interference management also advocates a similar approach

to manage interference in wireless networks [76]. Such an approach, proposed to achieve a certain

generalized degrees of freedom (GDoF), requires decomposing the network into two components, one

consisting of links that correspond to interference that needs to be avoided or nulled and the other

consisting of links where interference is sufficiently weak and is handled through power control. Our

effort can be thought of in similar terms but in a more practical setting with the overall objective of

maximizing a utility function.

In this chapter, we use the two-stage framework to maximize the minimum rate to the scheduled

users (max-min fairness) subject to per-BS power constraints. We first establish theoretical results on

the number of dominant interfering BSs that can be nulled per user in a (G,K,M × N) network.

We then identify the requisite number of dominant interfering BSs to be nulled in the first stage.

After aligning interference from the dominant BSs, we alternately optimize the transmit and receive

beamformers to maximize the minimum rate. Simulations on specific topologies of isolated cluster of BSs

under realistic channel conditions indicate that (a) aligned beamformers do not naturally emerge from

straightforward NUM algorithms even at high signal-to-noise ratios; (b) aligned beamformers provide a

significant advantage as initial condition to NUM, especially when BSs are closely spaced; and (c) IA

provides insights on the optimal number of users to schedule per cell. In particular, fewer number of

users should be scheduled per cell as the BS-to-BS distance decreases until the number of users per cell

reaches K = ⌊M+N−1G ⌋.

1Feasibility of over-the-air implementation of the proposed decentralized algorithms is limited by their iterative nature.


4.2 System Model

Consider the downlink of a (G,K,M ×N) network. Assuming each user is served with one data stream,

the transmitted signal corresponding to the kth user in the gth cell is given by ugksgk where ugk is

a N × 1 linear transmit beamforming vector and sgk is the symbol to be transmitted. This signal is

received at the intended user using a M ×1 receive beamforming vector vgk and the received signal after

being processed by the receive beamforming vector can be written as

vHgkygk =

G∑

i=1

K∑

j=1

vHgkH(i,gk)uijsij + vH

gkngk, (4.1)

where ngk is the M × 1 vector representing circular symmetric additive white Gaussian noise.

In the next section, we present a set of results on the feasibility of IA when interference from only a

subset of BSs is cancelled at a user. Since interference from only a subset of the interferers is aligned,

we call this partial interference alignment (PIA). It is important to establish these results as complete

IA may not be feasible in a given network and sometimes, even unnecessary. This chapter aims to show

that beamformers designed for PIA are eventually useful for NUM.

4.3 Feasibility of Partial Interference Alignment

In the G-cell network described above, assume that we are given a list I of user-BS pairs where each

pair indicates the need to cancel interference from a specific BS to a specific user. Let the double index

gk denote the kth user in the gth cell and the single index l denote the lth BS. If the pair (12, 3) ∈ I,this implies that the interference from the 3rd BS is to be completely nulled at the 2nd user in the 1st

cell. Satisfying this condition requires solving the following K equations:

vH12H(3,12)u3j = 0, ∀j ∈ {1, 2, . . . ,K}. (4.2)

In addition to these conditions, we also require the set of transmit beamformers at any BS to be linearly

independent, i.e, rank([ug1,ug2, . . . ,ugK ]) = K.

Cancelling interference from only a subset of the interfering BSs is analogous to complete IA in

partially connected cellular networks where certain cross links are assumed to be completely absent [77].

When the set I consists of all the (G − 1)GK possible pairs (denoted as Iall), we get the familiar

set of conditions for IA [25]. Each of the K equations in (4.2) is quadratic and collectively form a

polynomial system of equations. Feasibility of the system of polynomial equations when I = Iall is wellstudied using tools from algebraic geometry. Several necessary and sufficient conditions for feasibility

are known [21, 22]. The same set of tools can also be used to establish conditions for feasibility for any

given I. The following theorem establishes one such result.

Theorem 4.3.1. Consider a (G,K,M × N) network where each user is served with one data stream.

Let ugk and vgk denote the transmit and receive beamformer corresponding to the (g, k)th user where the

set of beamformers {ug1,ug2, . . . ,ugK} is linearly independent for every g. Further, let I ⊆ {(gk, i) :g 6= i, 1 ≤ g, i ≤ G, 1 ≤ k ≤ K} be a set of user-BS pairs such that for each (gk, i) ∈ I the interference


caused by the ith BS at the (g, k)th user is completely nulled, i.e.,

vHgkH(i,gk)uij = 0, ∀j ∈ {1, 2, . . . ,K}. (4.3)

A set of transmit and receive beamformers {ugk} and {vgk} satisfying the polynomial system defined by

I exist if and only if

M ≥ 1, N ≥ K, (4.4)

and

|Jusers|(M − 1) + |JBS |(N −K)K ≥ |J |K (4.5)

where J is any subset of I and Jusers and JBS are the set of user and BS indices that appear in J .

The proof of this theorem uses the same technique as [21, 22, 78] and is presented in Appendix C.

Note that intra-cell interference can be subsequently eliminated as the transmit beamformers in each

cell are linearly independent. A useful corollary that emerges from this theorem is stated below.

Corollary 4.3.2. Suppose the set I is such that each user in a (G,K,M×N) network requires interfer-

ence from no more than q BSs to be cancelled, where 1 ≤ q ≤ G− 1, and each BS has no more than Kq

users that require this BS’s transmission to be nulled at these users, then a set of sufficient conditions

for the feasibility of IA is given by

M ≥ 1, N ≥ K, (4.6)

and

M +N ≥ K(q + 1) + 1. (4.7)

Note that when q = G− 1, we recover the well known proper-improper condition for MIMO cellular

networks [25]. Fig. 4.1 illustrates the conditions of Corollary 4.3.2 imposed on a (4, 2, 3× 4) network for

the feasibility of PIA with q = 2. Each entry in Fig. 4.1 represents a user-BS pair as identified by its

row and column indices. If a certain user-BS pair is in I, the corresponding entry is marked with a ‘×’.

Corollary 4.3.2 requires I to be such that each row has no more than q chosen entries and each column

has no more than Kq chosen entries, where q = ⌊M+N−1K ⌋ − 1.

This corollary provides us with a simpler set of guidelines on choosing the set of user-BS pairs (I) forPIA than Theorem 4.3.1. Unlike partially connected networks where the set I is predetermined, practical

cellular networks are fully connected although the channel strength can vary significantly. Thus, cellular

networks may require interference nulling from only the dominant interferers, necessitating a careful

design of the set I while ensuring feasibility of PIA. The conditions in this corollary play an important

role in the optimization framework developed in the next section. Note, however, that designing Iaccording to this corollary comes at the cost of simplifying restrictions on I that may otherwise be

unnecessary.


BS1 BS2 BS3 BS4

U11 × × ≤ q

U12 × × ≤ q

U21 × × ≤ q

U22 × × ≤ q

U31 × × ≤ q

U32 × × ≤ q

U41 × × ≤ q

U42 × × ≤ q

︸︷︷︸

≤Kq︸︷︷︸

≤Kq︸︷︷︸

≤Kq︸︷︷︸

≤Kq

Figure 4.1: Illustration of the sufficient condition for feasibility of PIA in Corollary 4.3.2.

4.4 Optimization Framework

The optimization framework developed in this section aims to leverage the strength of IA in nulling

interference to overcome the limitations imposed by the non-convexity of the NUM problem. In a

wireless cellular network, spatial resources can be used in one of three ways: (a) they can be used to

serve more users i.e, spatial multiplexing; (b) they can used to enhance the signal strength (e.g. matched

filtering); or (c) they can be used to null interference (zero-forcing/IA). NUM algorithms strive to strike

the right balance between these three competing objectives to maximize a certain utility. In dense

cellular networks, due to the conflicting nature of these objectives, NUM algorithms may not be able

to comprehensively navigate the entire optimization landscape. The main point of this chapter is that

in certain practical networks, it is better to introduce a pre-optimization step to exclusively focus on

interference nulling and subsequently use the NUM algorithm to re-balance these priorities to maximize

the utility function.

Given a (G,K,M×N) network, we propose a two-stage optimization framework where the first stage

focuses on nulling interference from the dominant interferers using IA followed by a joint optimization

of beamformers and transmit powers to maximize a network utility using the IA solution as the initial

condition. Specifically, the optimization objective is to maximize the minimum rate achieved in the

network2.

Such a framework is well suited for investigating the benefits of IA in the context of NUM. The

difference in performance with and without the first stage of interference cancellation sheds light on the

value of IA in enhancing the performance of NUM algorithms. For a given network topology, a significant

difference in performance reflects that: (a) IA solutions are valuable from a NUM perspective; and (b) IA

2For this particular utility, spatial multiplexing is handled by a external scheduler thus simplifying the NUM algorithm’s

task to simply balance signal strength and interference across the network. For utilities such as weighted sum-rate, power

control acts as a proxy for controlling the number of users served.


Optimized power, beamformers

Schedule K users/cell

iterate

Identify q dominantinterferers for each user;Ensure feasibility of IA

Compute aligned beamformers

Fix receive beamformers.Optimize transmit beamformers,

find optimal max-min rateusing bisection search.

Fix transmit beamformers. Com-pute MMSE receive beamformers.

Figure 4.2: The proposed optimization framework.

solutions (or close-to-IA solutions) do not organically emerge from NUM algorithms due to the conflicting

uses for spatial resources. IA solutions are expected to hold significant value in dense cellular networks

where interference nulling is critical. Details of the proposed optimization framework follow.

4.4.1 Stage I: Partial Interference Alignment

In the first stage, each user identifies q dominant interferers from whom we attempt to null interference

using IA. Note from Corollary 4.3.2 that for a given (G,K,M × N) network, the choice of q is closely

dependent on the number of scheduled users; in fact, it is necessary that q ≤ ⌊M+N−1K ⌋−1. This suggests

that higher the number of scheduled users, fewer the number of interferers that can be nulled and vice

versa. Thus, the number of scheduled users, K, emerges as a crucial parameter governing the usefulness

of IA and is discussed further in the next section.

For a fixed K, set q = ⌊M+N−1K ⌋ − 1. The q dominant interferers are identified by their interference

strength with the transmit and receive beamformers set to certain predetermined values. The dominant

interferers can also be identified using the strength of the pilot signals received from the surrounding

BSs during the user-BS association phase. In our simulations we set all beamformers to be equal to the

all-ones vector.

Once the dominant interferers are identified, we then ensure that the chosen set of user-BS pairs,

denoted as I, conforms to the condition for feasibility of PIA as stated in Corollary 4.3.2. Constructing

a matrix analogous to that shown in Fig. 4.1, it is easy to see that while the rows of this matrix have

no more than q chosen entries by construction, the columns may have more than Kq chosen entries. To


eliminate such cases, if any column has more than Kq chosen cells, we sort the chosen cells of this column

in the descending order of their interference strengths and prune this sorted list, from the bottom, until

no more than Kq cells are left. The set of user-BS pairs that result at the end of this process (denoted

as I), satisfies the conditions imposed by Corollary 4.3.2, ensuring the feasibility of PIA. As a result of

the pruning, not all users have interference from all their q dominant interferers nulled. This is however

unavoidable to ensure feasibility of PIA. Note that for the case q = G− 1, no such pruning is necessary.

Once I is obtained, aligned beamformers satisfying the conditions for PIA can be designed using any

algorithm developed for IA such as interference leakage minimization [25, 33, 77], iterative matrix norm

minimization (as discussed in the previous chapter), etc.

4.4.2 Stage II: Utility Maximization

This stage focuses on maximizing a given network utility function using the aligned beamformers obtained

in the previous stage as the initialization. As stated before, this chapter focuses on maximizing the

minimum rate for the scheduled users subject to per-BS power constraints. In mathematical terms, we

would like to solve the following optimization problem:

maximizeugk, vgk

t

subject to|vH

gkH(g,gk)ugk|2σ2 +

∑

(i,j) 6=(g,k)

|vHgkH(i,gk)uij |2

≥ t, ∀(g, k),

K∑

k=1

|ugk|2 ≤ Pmax, ∀g,

|vgk|2 = 1, ∀(g, k), (4.8)

where ugk, vgk are the variables for optimization and Pmax is the maximum transmit power permitted

at any BS. This problem is non-convex in its current form and no convex reformulation is known except

when the users have a single antenna. Several techniques for finding a local optimum of this problem have

been proposed [70–72]. We solve (4.8) by alternately optimizing the transmit and receive beamformers,

leveraging the convex reformulation that emerges when users have a single antenna [79, 80]. Fixing the

receive beamformers to be the aligned beamformers obtained from the first stage, we use a bisection

search over t to find the optimal max-min rate as proposed in [79]. Specifically, setting tmin and tmax to

be the limits of t, we fix t = t0, where t0 = (tmin + tmax)/2 and proceed to solve the following feasibility

problem:

maximizevgk

1

subject to|uH

gkH(g,gk)vgk|2σ2 +

∑

(i,j) 6=(g,k)

|uHgkH(i,gk)vij |2

≥ t0, ∀(g, k),

G∑

g=1

|vgk|2 ≤ Pmax, ∀g.

(4.9)

This problem is known to be equivalent to a second-order cone program and can be solved efficiently.

See [79, 80] for further details. If the problem is feasible for a particular t0, set tmin = t0, else set


Figure 4.3: Network topologies: a three-sector cluster, a 5-cell ring topology and a 7-cell hexagonal

layout.

Table 4.1: Simulation Parameters

Network

3-Sector (3,K, 3× 4)

Ring Topology (5,K, 5× 6)

Hexagonal Layout (7,K, 4× 4)

BS-to-BS distance 600m to 1800m

Transmit power PSD -35dBm/Hz

Thermal noise PSD -169dBm/Hz

Antenna gain 10dBi

SINR gap 6dB

Distance dependent pathloss 128.1 +37log10(d)

Shadowing Log-normal, 8dB SD

Fading Rayleigh

tmax = t0 and continue with the bisection search until the optimal t is resolved up to a desired precision.

For a fixed set of transmit beamformers, the optimal set of receive beamformers is given by the MMSE

beamformers, i.e, when the transmit beamformers are fixed, we can update the receive beamformers as

vgk =

∑

(i,j)

H(i,gk)vijuHijH

H(i,gk) + σ2I

−1

Hg,gkvgk. (4.10)

Once the receive beamformers are updated, we proceed to re-optimize the transmit beamformers and

this procedure is repeated for a fixed number of iterations. The flowchart in Figure 4.2 lists each step of

the proposed optimization framework.

4.5 Simulation Results

The value of IA is best illustrated in a dense cluster of BSs where interference mitigation plays an

increasingly important role as the distance between BSs decreases. Towards this end, we consider three

network topologies with increasing cluster sizes to test the proposed framework. The first network is a


4

6

8

10

12

14

16

18

600 900 1200 1500 1800

distance between BSs (in meters)

Averagecellthroughput(bits/s/Hz)

K = 2, no IA

K = 3, no IA

K = 4, no IA

K = 2, with IA, q = 2

K = 3, with IA, q = 1

K = 4, with IA, q = 0

Figure 4.4: Per-cell throughput in a (3,K, 3× 4) network forming a 3-sector cluster, with K varied from

2 to 4.

3-sector cluster, the second consists of 5 BSs spread out on a ring and the third is a 7-cell hexagonal

cluster. No out-of-cluster interference is taken into account in all three cases. The same pathloss,

shadowing and fading assumptions are made for all three networks. Users are assumed to be uniformly

distributed in each cell, and are served by one data stream each. Fig. 4.3 shows the three topologies

under consideration. Table 4.1 lists the antenna configuration for each of the networks, along with other

parameter settings.

For each network, the number of scheduled users per cell, K, is varied from⌊M+N−1

G

⌋to N . Note

that as K increases, the number of dominant BSs that can be cancelled in the first stage decreases.

Thus when K > M+N−12 , no dominant interferers can be nulled and the beamformers are chosen to only

cancel intra-cell interference.

For a given set of scheduled users, the proposed optimization framework is used to maximize the

minimum rate achieved in the network. For each user, interference from at most q = ⌊M+N−1K ⌋ − 1

interferers is nulled using the interference leakage minimization algorithm [33]. Using these aligned

beamformers as initialization, the convex optimization problem arising from (4.8) for a fixed set of

receive beamformers is solved using CVX, a package for specifying and solving convex programs [63,64].

The transmit and receive beamformers are alternately optimized over 5 iterations. The performance of

the proposed framework is compared to the setup where the first stage is omitted, i.e., the dominant

interferers are not nulled using IA (marked as ‘no IA’). The average cell throughput (i.e., K multiplied

by the the max-min rate) expressed in bit/s/Hz is obtained by averaging over 100 user locations.

Figs. 4.4, 4.5 and 4.6 plot the average cell throughput for each of the three networks as a function of

BS-to-BS distance and of the number of scheduled users. It is seen that IA solutions provide an altered

interference landscape that is otherwise non-trivial to find, and this altered landscape enhances the


7

8

9

10

11

12

13

14

15

16

17

600 900 1200 1500 1800



K = 2, no IAK = 3, no IAK = 4, no IAK = 5, no IAK = 6, no IAK = 2, with IA, q = 4K = 3, with IA, q = 2K = 4, with IA, q = 1K = 5, with IA, q = 1K = 6, with IA, q = 0

Figure 4.5: Per-cell throughput in a (5,K, 5 × 6) network having a 5-BS ring topology, with K varied

from 2 to 6.

performance of subsequent NUM algorithms. Focusing on Fig. 4.4, it is clear that IA has a significant

impact on optimization, especially when BSs are closely spaced. The gain of IA depends on the number

of users scheduled. In particular, when 2 users/cell are scheduled, it is possible to achieve 1 DoF/user

as interference can be completely nulled in the network (q = G − 1 = 2). In this case, IA provides

4-6 bit/s/Hz improvement at small BS-to-BS distances. When 3 users/cell are scheduled, IA can cancel

interference from up to one interferer for each user. Such IA solutions are seen to enhance the average cell

throughput by about 1 bit/s/Hz. However, when 4 users/cell are scheduled, only intra-cell interference

can be nulled, and IA has no impact on the optimization. Note also that because it is possible to

completely null inter-cell interference only when K = 2 (or equivalently, q = 2), this is the only scenario

where throughput does not saturate as the BS-to-BS distance decreases. Finally, we comment that for

a broad range of BS-to-BS distances, scheduling 2 users/cell appears to be optimal.

A similar set of observations can also be made in Fig. 4.5. In particular, IA provides about 1 bit/s/Hz

gain when K ≤ 5 and over good range of BS-to-BS distances. However, unlike the 3-sector network,

nulling interference from all interferers (i.e., q = 4, K = 2) is not necessarily the best strategy, except

at very small BS-to-BS distances. At larger distances it appears that nulling interference from the two

dominant interferers suffices (q = 2, K = 3).

Finally, Fig. 4.6 considers the 7-cell network—the only network, among the three considered here,

where not all cells are equivalent, and pruning the list of dominant interferers plays an important role

in ensuring feasibility of PIA. As expected, it can be seen that with increasing cluster size, scheduling

K = ⌊M+N−1G ⌋ users (in this case, K = 1, q = 6), is a good strategy only at much smaller BS-to-BS

distances. In fact IA does not provide consistent rate gain across all the cases. But the simulation does


6

7

8

9

10

600 900 1200 1500 1800



K = 1, no IAK = 2, no IAK = 3, no IAK = 4, no IAK = 1, with IA, q = 6K = 2, with IA, q = 2K = 3, with IA, q = 1K = 4, with IA, q = 0

Figure 4.6: Per-cell throughput in a (7,K, 4 × 4) network forming a 7-cell hexagonal topology, with K

varied from 1 to 4.

provide insight on the optimal number of users to schedule. It appears that the number of scheduled

users should be such that nulling interference from one or two of the dominant interferers for each user

is feasible.

Surprisingly, in all three networks, scheduling as many users as there are antennas does not appear to

be the right choice even at large BS-to-BS distances. Aggressive spatial multiplexing seems to severely

limit the use of spatial resources to enhance signal strength or to null interference.

4.6 Summary

This chapter investigates the role of IA in NUM. In order to leverage the strengths of IA and to overcome

the shortcoming of conventional NUM algorithms, a two-stage optimization framework is proposed.

This framework is used to evaluate the value of IA in practical cellular network optimization. Through

simulations on different network topologies for maximizing the minimum rate achieved in a given network,

it is established that IA is valuable in dense cellular networks where IA solutions provide an altered

interference landscape that the subsequent NUM algorithm can take advantage of, but is otherwise

difficult to reach. System-level simulation results reveal that, indeed while IA cannot be a substitute

for optimization algorithms due to its exclusive focus on nulling interference, it can be used to augment

their performance. IA also provides us with some important insights on the right number of users to

schedule in such networks. It is seen that fewer users should be scheduled in each cell as the BS-to-BS

distance decreases.

Chapter 5

Conclusion

This thesis studies the role of interference alignment in multi-antenna cellular networks. In particular,

this thesis focuses on the theoretical DoF results that can be established using interference alignment,

the algorithmic techniques for interference alignment and the relevance of IA in practical cellular network

optimization.

From a theoretical perspective, the achievable DoF using asymptotic IA and the conditions for

its optimality are established. Using these results, the symmetric DoF of MISO cellular networks is

completely characterized. Two contrasting linear beamforming strategies for interference alignment in

MIMO cellular networks are also developed. A structured approach based on a notion called packing

ratios is used to establish the symmetric DoF of the two smallest cellular networks—the 2-cell, 2-user/cell

network and the 2-cell, 3-user/cell network. An unstructured approach for designing linear beamformers

for interference alignment that is versatile and easily applied to networks of any size is proposed. Based

on numerical experiments, this approach appears to achieve the symmetric DoF of any (G,K,M ×N)

network whenever the decomposition based inner bound lies below the proper-improper boundary.

The second part of this thesis focuses on developing algorithmic techniques for IA. Using an alter-

nate set of conditions for IA, the problem of finding aligned beamformers for IA is posed as a rank

minimization problem. The rank minimization problem is cast as a reweighted matrix norm minimiza-

tion by approximating rank using a smooth function such as the Frobenius norm or the nuclear norm.

Using the prior knowledge of the expected rank of interference matrices, a novel weight update rule is

developed to iteratively solve the reweighted matrix norm minimization. As an alternate approach, the

prior rank knowledge is used to approximate the rank deficient interference matrices as a product of two

low-rank matrices. These low-rank matrices are alternately optimized to minimize a quadratic objec-

tive. Simulation results show that reweighted nuclear norm minimization is effective in designing aligned

beamformers irrespective of the presence of redundant antennas. It significantly outperforms previously

proposed algorithms, of similar complexity, in terms of maximizing the number of interference-free di-

mensions. Reweighted Frobenius norm minimization and alternating minimization are seen to be more

suitable when the system has some redundant antennas.

Finally, the thesis discusses the implications of IA on cellular network optimization. Using a two-

stage optimization framework to test the value of IA it is seen that IA can enhance the performance of

80

Chapter 5. Conclusion 81

NUM algorithms in dense cellular networks where inter-cell interference is significant. It is further seen

that the number of users scheduled in dense cellular networks must be reduced as the BS-to-BS distance

decreases.

The thesis provides a comprehensive treatment of IA in the context of multi-antenna cellular networks.

It is seen that IA is useful in establishing several theoretical DoF results that translate to useful insights

on practical cellular network operation.

Appendices

82

Appendix A

DoF Outer Bound for the Two-Cell

Three-Users/Cell Network

In this section we show that for the two-cell three-users/cell MIMO cellular network whenever 59 ≤ γ ≤

34 , no more than max

(2N9 , M

3

)DoF/user are possible. Since there is no duality associated with the

information theoretic proof presented here, we need to establish this result separately for uplink and

downlink. Similar to [19], we first perform an invertible linear transformation at the users and the base-

stations. The linear transformation involves multiplication by a full rank matrix at each user and BS. Let

the M ×M transformation matrix at user (i, j) be denoted as Tij and the N ×N transformation matrix

at BS i be denoted as Ri. Using these transformations the effective channel between user (i, j) and

BS i is given by RiH(ij,i)Tij . Subsequent to this transformation, we first consider the uplink scenario

and identify genie signals that enable the BSs to decode all the messages in the network and set up a

bound on the sum-rate of the network. Using the same transformation, we then identify genie signals to

establish the bound in the downlink. We start by considering the case when 5/9 ≤ γ ≤ 2/3.

Throughout this section we use the relative indices i and i when referring to the two cells and use

the notation ij to denote the jth user in ith cell. The vector random variables corresponding to the

transmit signal x, received signal y and additive noise z are denoted as X, Y and Z, respectively. W

denotes a uniform discrete random variable associated with the transmitted message at a transmitter.

DoF Outer Bound When 5/9 ≤ γ ≤ 2/3

We divide the set of N antennas at BS i into three groups and denote them as ia, ib and ic. The sets ia

and ic contain the first and last N −M antennas each while set ib has the remaining 2M −N antennas.

Let the M antennas at user ij be denoted as ijk where k ∈ {1, 2, · · · ,M}. Using a similar notation for

BS antennas, let H(ij,ip:iq) represent the channel from user ij to the subset of BS antennas from the pth

antenna to the qth antenna.

We first focus on the N ×M channel from user i1 to BS i. We set the first N − M rows of Ri to

be orthogonal to the columns of Hij,i. Since H(ij,i) spans only M of the N dimensions at BS i, it is

possible to choose such a set of vectors. Similarly, the next 2M −N and N−M rows of Ri are chosen to

83

Appendix A. DoF Outer Bound for the Two-Cell Three-Users/Cell Network 84

ia

ib

ic

N −M

2M −N

N −M

i1

i2

i3

M

M

M

gi1(xib,xic)

gi2(xia,xic)

gi3(xib,xic)

gia(xi2,xi31:i3(N−M))

gib(xi11:i1(2M−N) ,xi3(N−M+1):i3M )

gic(xi1(2M−N+1):i1M ,xi2)

Users in Cell i Base-station i

Figure A.1: The signal structure obtained after linear transformation for the case when γ ≤ 2/3. Note

that the figure does not include signals from the same cell.

be orthogonal to user i2 and user i3 respectively. Since all channels are assumed to be generic, matrix

Ri is guaranteed to be full rank almost surely.

On the user side, user i1 inverts the channel to the last M antennas of BS i, i.e., Ti1 =

(H(i1,iN−M+1:iN))−1, while user i3 inverts the channel to the first M antennas of BS i, i.e., Ti3 =

(H(i1,i1:iM))−1. We let Ti2 = I. The signal structure resulting from such a transformation is shown in

Fig. A.1.

DoF Bound in the Uplink Let wij be the message from user ij to BS i. This message is mapped

to a Mn × 1 codeword xnij , where n is the length of the code. We use the notation xn

ijp to denote the

transmitted signal on the kth antenna over the n time slots and the notation xijp:ijq to denote the signal

transmitted by user ij using antennas p, p + 1, . . . , q. We denote the rate to user ij as Rij , the total

sum-rate of the network as Rsum and the collection of all messages in the network as {wij}.Now, consider providing the set of signals S1 = {xn

i2, xni11:i1(2M−N)} to BS i. We use xn to denote

xn+zn where zn is circular symmetric Gaussian noise that is artificially added to the transmitted signal

xn. Since we seek to establish a converse, we assume that BS i can decode all the messages from its

users. After decoding and subtracting these signals from the received signal, the resulting signals at

the three antenna sets are given in Fig. A.1 where gi∗(·) represents a noisy linear combination of its

arguments. Given S1, we can subtract xni2 from gic(xi1(2M−N+1):i1M , xi2) and along with xn

i11:i1(2M−N)

from S1, we can decode wi1 subject to noise distortion. After decoding wi1, and subtracting xni1 and xn

i2

from the received signal, wi3 can also be decoded subject to noise distortion. Since BS i can recover all

the messages in the network given yniand S1 subject to noise distortion, we have

nRsum

a≤ I

({Wij};Yn

i ,S1

)+ no(log ρ) + o(n)

b≤ Nn log ρ+ h(Xn

i2, Xni11:i1(2M−N)|Yn

i ) + no(log ρ) + o(n)

c≤ Nn log ρ+ nRi2 + h(Xn

i11:i1(2M−N)) + no(log ρ) + o(n) (A.1)

where (a) follows from Fano’s inequality, (b) follows from Lemma 3 in [19] and (c) follows from the fact

that conditioning reduces entropy.

Next, consider providing the set of signals S2 = {xni3, x

ni1(2M−N+1):i1M} to BS i. After subtracting

xni3 from the received signal, the BS can recover wi2 from observations at antenna sets ia and ic subject

to noise distortion. Subsequently, BS i can also recover wi1 subject to noise distortion. Since BS i can

recover all messages when provided with the genie signal S2, using similar steps as before, we obtain


nRsum ≤ I({Wij};Yn

i ,S2


≤ Nn log ρ+ h(Xni3, X

ni1(2M−N+1):i1M |Yn


≤ Nn log ρ+ nRi3 + h(Xni1(2M−N+1):i1M |Xn

i11:i1(2M−N)) + no(log ρ) + o(n)

≤ Nn log ρ+ nRi3 + nRi1 − h(Xni11:i1(2M−N)) + no(log ρ) + o(n), (A.2)

where Xni1 denotes Xn

i1 corrupted by channel noise.

Adding (A.1) and (A.2) we get,

2nRsum ≤2nN log ρ+

3∑

j=1

nRij + no(log ρ) + o(n) (A.3)

Using a similar inequality for BS i, we can write

3nRsum ≤4nN log ρ+ no(log ρ) + o(n) (A.4)

Letting n → ∞ and ρ → ∞, we see that DoF/user ≤ 2N9 .

DoF Outer Bound in the Downlink Using same notation as before, consider providing user i1

with the genie signal S1 = (wi2, wi3,xnia). Since we are interested in establishing an outer bound, we

assume all the users in the network can decode their own messages. Since user i1 can decode wi1, using

S1, user i1 can reconstruct xnia, x

nib and xn

ic, and subtract them from the received signal yni1. Using the

signal obtained after subtracting xnia, x

nib and xn

ic from yni1 and using xia) from S1, user i1 can now

decode messages wi1, wi1 and wi1 subject to noise distortion. Since user i1 can decode all messages in

the network given yni1 and S1, we have

nRsum ≤ I ({Wij};Yni1,S1) + no(log ρ) + o(n)

≤ nM log ρ+ nRi2 + nRi3 + h(Xnia|Yn

i1,Wi2,Wi3) + no(log ρ) + o(n)

≤ nM log ρ+ nRi2 + nRi3 + h(Xnia|Xn

ib, Xnic) + no(log ρ) + o(n). (A.5)

Next, consider providing user i3 with the genie signal S3 = (wi1, wi2,xnic). Following the exact same

steps as before, we get

nRsum ≤ nM log ρ+ nRi2 + nRi3 + h(Xnia|Xn

ib, Xnic) + no(log ρ) + o(n). (A.6)

Now consider providing user i2 with the genie signal S2 = (wi1, wi3, xnib, xn

i(M+1):i(2N−2M)). Note

that xni(M+1):i(2N−2M)

forms a part of the signal xnic. After subtracting the transmitted signals from BS

i, user i2 has 2N − 2M noisy linear combinations of the signals xnia and xn

ic, which along with xnib

from

S2 can be used to decode all the messages from BS i subject to noise distortion. As before, we can write



≤ nM log ρ+ nRi1 + nRi3 + h(Xnib, X

ni(M+1):i(2N−2M)|Yn

i1,Wi1,Wi3) + no(log ρ) + o(n)

≤ nM log ρ+ nRi1 + nRi3 + h(Xni(M+1):i(2N−2M)) + h(Xn

ib|Xnia, X

nic) + no(log ρ) + o(n)

≤ nM log ρ+ nRi1 + nRi3 + n(2N − 3M) log ρ+ h(Xnib|Xn

ia, Xnic) + no(log ρ) + o(n)

≤ n(2N − 2M) log ρ+ nRi1 + nRi3 + h(Xnib|Xn

ia, Xnic) + no(log ρ) + o(n) (A.7)

Adding (A.5), (A.9) and (A.7), we get

n3Rsum ≤ n2N log ρ+

3∑

j=1

n2Rij + h(Xnia|Xn

ib, Xnic) + h(Xn

ib|Xnic, X

nia) + h(Xn

ic|Xnib, X

nia) + no(log ρ) + o(n)

≤ n2N log ρ+3∑

j=1

n2Rij + h(Xnia) + h(Xn

ib|Xnia) + h(Xn

ic|Xnib, X

nia) + no(log ρ) + o(n)

≤ n2N log ρ+

3∑

j=1

n2Rij +

3∑

j=1

nRij + no(log ρ) + o(n) (A.8)

Using a similar inequality for users in cell i, we can write

n6Rsum ≤ n4N log ρ+ n3Rsum + no(log ρ) + o(n)

(A.9)

Letting n → ∞ and ρ → ∞, we see that DoF/user ≤ 2N9 .

DoF Outer Bound when 2/3 ≤ γ ≤ 3/4

In this case, we again group the antennas at BS i into three groups exactly as before. The M antennas

at each user are also grouped into three sets as shown in Fig. A.2. The linear transformation at BS i is

also same as before, i.e., each group of antennas zero-forces one of three users.

On the user side, Ti1 for user i1 is chosen such that i1a zero-forces ib while i1b and i1c both zero-force

ic. Similarly, Ti3 is chosen so that i3c zero-forces ib, while i3b and i3c both zero-force ia and finally Ti2

is chosen such that i2a zero-forces ia, while i2b and i2c both zero-force ic. The resulting signal structure

at BS i after removing signals from Cell i is given in Fig. A.2.

DoF Outer Bound in the Uplink Consider providing the set of signals S1 = {xni1, x

ni2b, x

ni2c} to

BS i. After decoding the messages from users in Cell i, we see that using S3, we can first decode wi2

followed by wi3, subject to noise distortion. Since BS i can recover all the messages in the network given

yniand S1, subject to noise distortion, we have

nRsum ≤ I({Wij};Yn

i ,S1


≤ Nn log ρ+ h(Xni1, X

ni2b, X

ni2c|Yn


≤ Nn log ρ+ nRi1 + h(Xni2b, X

ni2c|Xn

i2a) + no(log ρ) + o(n)

≤ Nn log ρ+ nRi1 + nRi2 − h(Xni2a) + no(log ρ) + o(n), (A.10)


ia

ib

ic

N −M

2M −N

N −M

i1a

i1b

i1c

i2a

i2b

i2c

i3a

i3b

i3c

N −M

3M − 2N

N −M

gi1a(xic)

gi1b(xib)

gi1c(xib)

N −M

3M − 2N

N −M

gi2a(xic)

gi2b(xia)

gi2c(xia)

N −M

3M − 2N

N −M

gi3a(xib)

gi3b(xib)

gi3c(xia)

gia(xi2b,xi2c,xi3c)

gib(xi1b,xi1c,xi3a,xi3b)

gic(xi1a,xi2a)

Users in Cell i Base-station i

Figure A.2: The signal structure obtained after linear transformation when γ ≥ 2/3. The figure does

not include signals from the same cell.

where Xni2a denotes Xn

i2a corrupted by channel noise.

Next, we consider the genie signal S2 = {xni3, x

ni2a, x

ni2b}. It can once again be shown that BS i can

recover all the messages in the network given yniand S2. Going through similar steps as before, it can

be shown that

nRsum ≤ (3M −N)n log ρ+ nRi3 + h(Xni2a) + no(log ρ) + o(n). (A.11)

Adding (A.10) and (A.11), we get

2nRsum ≤3Mn logρ+

3∑

j=1

nRij + no(log ρ) + o(n). (A.12)

By symmetry we must also have an analogous inequality involving the rates Rij , and adding these two

inequalities, we get

3nRsum ≤6Mn log ρ+ no(log ρ) + o(n) (A.13)

Letting n → ∞ and ρ → ∞, we see that DoF/user ≤ M3 .

DoF Outer Bound in the Downlink Consider providing the genie signal S1 = {wi2, wi3, xia} to

user i1. It can be shown that user i1 can decode all the messages in the network using the received signal

and the genie signal subject to noise distortion. Hence, using similar steps as before, we can write


≤ nM log ρ+Ri2 +Ri3 + h(Xia|Xia, Xia) + no(log ρ) + o(n)

(A.14)

Using identical genie signals S2 = {wi1, wi3, xib} and S3 = {wi1, wi2, xic} for users i2 and i3

respectively, we obtain the following two inequalities:


nRsum ≤ nM log ρ+Ri1 +Ri3 + h(Xib|Xia, Xic) + no(log ρ) + o(n), (A.15)

nRsum ≤ nM log ρ+Ri1 +Ri2 + h(Xic|Xia, Xib) + no(log ρ) + o(n). (A.16)

Adding the inequalities in (A.14), (A.15) and (A.16), we get

3nRsum ≤3nM log ρ+

3∑

j=1

2nRij +

3∑

j=1

nRij + no(log ρ) + o(n). (A.17)

Using a similar set of genie signals for users in cell i, we can establish a corresponding inequality on

the sum-rate. Adding these two inequalities gives us

6nRsum ≤6Mn log ρ+ 3nRsum + no(log ρ) + o(n). (A.18)

Letting n → ∞ and ρ → ∞, we see that DoF/user ≤ M3 .

Appendix B

Achievability of the Optimal sDoF

In this section we provide further details on the linear beamforming strategy used to achieve the optimal

sDoF for the two-cell two-users or three-users per cell MIMO cellular networks. We consider designing

transmit beamformers in the uplink. By duality of linear interference alignment, the same strategy also

holds in downlink.

Linear Beamforming Strategy for the Two-Cell, Two-Users/Cell Network

We divide the discussion in this section into six cases, each corresponding to one of the six distinct

piece-wise linear regions in Fig. 2.3. Since we assume generic channel coefficients, we do not need to

explicitly check to make sure that (a) interference and signal are separable at each BS and (b) signal

received from a user at the intended BS occupies sufficient dimensions to ensure all data streams from

that user are separable (i.e., H(ij,i)Vij is full rank for all i and j). We however need to ensure that the

set of beamformers designed for a user are linearly independent.

Case i: 0 < γ ≤ 1/4: Each user here requires M DoF. It is easy to observe that since N ≥ 4M ,

random uplink transmit beamforming in the uplink suffices. The BSs have enough antennas to resolve

signal from interference. Note that no spatial extensions are required here.

Case ii: 1/4 ≤ γ ≤ 1/2: The goal here is to achieve N/4 DoF/user. If N/4 is not an integer,

we consider a space-extension factor of four, in which case we have 4M antennas at the users and 4N

antennas at the transmitter. Since we need N DoF/user and the BSs now have 4N antennas, we once

again see that random uplink transmit beamforming suffices.

Case iii: 1/2 < γ < 2/3: Since each user requires M/2 DoF/user, we consider a space-extension

factor of two so that there are 2M antennas at each user and 2N antennas at each BS. The two users

in the second cell each have access to a 2M dimensional subspace at the first BS. These two subspaces

overlap in 4M − 2N dimensions. Note that since γ > 1/2, 4M > 2N , such an overlap almost surely

exists. The two users in cell 2 pick 4M − 2N linear transmit beamformers so as to span this space and

align their interference. Specifically, the transmit beamformers v21j and v22j for j = 1, . . . , (4M − 2N)

89

Appendix B. Achievability of the Optimal sDoF 90

are chosen such that

H(21,1)v21j = H(22,1)v22j

⇒[

H(21,1) −H(22,1)

]

v21j

v22j

= 0. (B.1)

The 4M − 2N sets of solutions to (B.1) can be generated using the expression (I − AH(AAH)−1A)r

where A =[

H(21,1) −H(22,1)

]

and r is a random vector. Adopting the same strategy for cell 1 users,

we see that at both BSs interference occupies 4M − 2N dimensions while signal occupies 8M − 4N

dimensions, with 8N − 12M unused dimensions. Note that since γ ≤ 2/3, 8N − 12M ≥ 0. Letting each

user pick 2N−3M random beamformers, the remaining 8N−12M dimensions are equally split between

interference and signal at each of the BSs. We have thus designed M transmit beamformers for each user

while ensuring that at each BS, interference occupies no more than (4M−2N)+2(2N−3M) = 2N−2M

dimensions, resulting in M/2 sDoF/user.

Case iv: 2/3 ≤ γ ≤ 1: We need to achieve N/3 DoF/user. We consider a space-extension factor

of three, so that each user has 3M antennas and each BS has 3N antennas; and we need to design N

transmit beamformers per user. The two users in the second cell each have access to a 3M dimensional

subspace at the first BS. These two subspaces overlap in 6M − 3N dimensions. Since γ > 2/3, we note

that 6M − 3N > N , allowing us to pick a set of N transmit beamformers such that interference is

aligned at BS 1. Using the same strategy for users in cell 1, interference and signal together span 3N

dimensions. The transmit beamformers can be computed by solving the same set of equations as given

in (B.1).

Case v: 1 < γ < 3/2: In order to achieve M/3 DoF/user, we consider a space-extension factor

of three and design M beamformers per user. Since we now have more transmit antennas than receive

antennas, transmit zero-forcing becomes possible. Each user in cell 2 picks 3M−3N linearly independent

transmit beamformers so as to zero-force BS 1, i.e., the beamformers are chosen from the null space of

the channel H(2i,1) and satisfy

H(2i,1)v2ij = 0 ∀ i ∈ {1, 2}, j ∈ {1, 2, . . . (3M − 3N)}. (B.2)

We let users in cell 1 use the same strategy. Now, in order to achieve M DoF/user, we still need to

design 3N − 2M transmit beamformers per user. So far, both BSs do not see any interference and have

6M − 6N dimensions occupied by signals from their own users. The remaining 9N − 6M dimensions

at each BS need to be split in a 2 : 1 ratio between signal and interference to achieve M DoF/user.

To meet this goal, we choose the remaining 3N − 2M transmit beamformers for users in cell 2 such

that the interference from these users aligns at BS 1. This is accomplished by solving for the transmit

beamformers using (B.1) for users in cell 2, and using a similar strategy for users in cell 1, resulting in

(3M − 3N) + (3N − 2M) = M DoF/user over a space-extension factor of three.

Case vi: 3/2 ≤ γ: Assuming a space-extension factor of two, each user needs N transmit beamform-

ers. The null space of the channel from a user in cell 2 to BS 1 spans 2M − 2N dimensions and since

γ > 3/2, 2M − 2N > N . Choosing N transmit beamformers from such a null space and using the same

strategy for users in cell 1, we see that each BS sees no interference and hence is able to completely

recover signals from both of its users.


Linear Beamforming Strategy for the Two-cell, Three-Users/Cell Network

We divide the discussion in this section into ten cases, each corresponding to one of the ten distinct

piecewise-linear regions in Fig. 2.3. The cases γ < 1/6 and 1/6 ≤ γ ≤ 1/3 and γ ≥ 4/3 are identical to

cases (i), (ii) and (vi) in the previous section, where either random transmit beamforming or zero-forcing

achieve the optimal DoF. We omit the discussion of these three cases here.

Case iii: 13 < γ < 2

5 : We consider a space extension factor of two and prove that M DoF/user are

achievable. Since 4M < 2N , a many-to-one type of alignment between multiple interfering vectors is not

possible. However, since 6M > 2N , it is possible to design a set of three beamformers, one for each user

in a cell, such that the beamformers occupy only two dimensions at the interfering BS. In particular, to

design beamformers for the three users in cell 2, we solve the following system of equations

[H(21,1) H(22,1) H(23,1)

]

v21j

v22j

v23j

= 0. (B.3)

Note that this is a system of 2N equations in 6M unknowns, and there can be at most 6M − 2N

linearly independent solutions. These solutions yield 6M − 2N sets of three beamformers, with each set

having a packing ratio of 3 : 2. While the 6M − 2N solutions to the system of equations are linearly

independent, we need to prove that the 6M − 2N beamformers designed for each user are also linearly

independent. In other words, linear independence of the set of solutions {[vT21j vT

22j vT23j ]}6M−2N

j=1 does

not immediately imply the linear independence of the set {v2ij}6M−2Nj=1 for all i ∈ {1, 2, 3}. We prove

through a contradiction that this is indeed true. Suppose that the set {[vT21j v

T22j v

T23j ]}6M−2N

j=1 is linearly

independent, but the set {v2ij}6M−2Nj=1 is not, for some i. Without loss of generality, let i = 1. Then,

there exist a set of coefficients {βj} such that

6M−2N∑

j=1

βj v21j = 0. (B.4)

Let w denote the vector∑6M−2N

j=1 βj [vT21j vT

22j vT23j ]

T . Then,

[H(21,1) H(22,1) H(23,1)

]w =0, (B.5)

⇒[H(22,1) H(23,1)

]w(M + 1 : 3M) =0. (B.6)

Equation (B.6) is a system of N equations and 2M unknowns, and since 2M < N , (B.6) is satisfied only

if w(M + 1 : 3M) = 0 ⇒ w = 0 ⇒ the set {[vT21j vT

22j vT23j ]}6M−2N

j=1 is linearly dependent, which is a

contradiction.

Using the 6M − 2N sets of beamformers obtained in this manner, we note that at each BS, we

have 18M − 6N dimensions occupied by signal, 12M − 4N dimensions occupied by interference with

12N − 30M unoccupied dimensions. We now pick 2N − 5M random beamformers for each user so as to

use all available dimensions at both the BSs. Since the second set of beamformers are chosen randomly,


they are linearly independent from the first set of 6M − 2N beamformers almost surely. We have thus

ensured each user achieves M DoF using a space extension factor of two.

Case iv: 25 ≤ γ ≤ 1

2 : In order to achieve N/5 DoF/user, we consider a space extension factor of

five and consider designing N transmit beamformers per user. Once again, 3 : 2 is the highest possible

packing ratio and there are 15M − 5N sets of three beamformers (one for each of three user in a cell)

having this packing ratio. If we are to use all such beamformers, we can at most cover 5(15M − 5N)

dimensions at each BS. Since 5(15M − 5N) ≥ 5N , we have sufficient number of such sets to use all

available dimensions at the two BSs. Choosing N such sets of beamformers achieves N DoF/user over

five space extensions.

Case v: 12 < γ < 5

9 : The goal here is to achieve 2M DoF/user using a space extension factor of five.

To keep the presentation simple, we assume M and N are divisible by five and achieve 2M/5 DoF/user.

Since 2M > N , many-to-one type of interference alignment becomes feasible and in fact, 2 : 1 is the

highest possible packing ratio. There are three ways to choose a pair of users from a cell, and for each

pair there exist 2M −N sets of beamformers having a packing ratio of 2 : 1. For users in cell 2, these

beamformers can be formed by solving equations of the form

[H(2i,1)H(2k,1)]

v2ij

v2kj

= 0, (B.7)

where i, k ∈ {1, 2, 3}, i 6= k. We thus have 2(2M −N) beamformers per user. Since we assume channels

to be generic and since 2(2M −N) < M , the set of 2(2M −N) beamformers are almost surely linearly

independent. When these 6(2M −N) beamformers are used by users in each cell, each BS has 4N − 6M

unused dimensions. We fill the unused dimensions using beamformers having the next best packing

ratio—3 : 2. In order to ensure the linear independence of this new set of beamformers from the set of

beamformers already designed, we multiply each channel matrixH(lm,n) with a matrixWlm on the right,

where Wlm is a M×(2N−3M) matrix whose columns are orthogonal to the 4M−2N beamformers that

have already been designed for user lm. Let the effective channel matrix H(lm,n)Wlm be denoted by

H(lm,n). Note that H(lm,n) is a N×2N−3M matrix and since 3(2N−3M) > N , there exist beamformers

having packing ratio 3 : 2. Similar to Case iv, we design 2N − 18M5 sets of such beamformers, ensuring

that all dimensions at the two BSs are used while achieving (2N− 18M5 )+2(2M−N) = 2M/5 DoF/user.

Case vi: 59 ≤ γ ≤ 2

3 : We need to achieve 2N DoF/user over 9 spatial extensions. To keep the

presentation simple, we simply assume that N is divisible by nine and present the arguments without

any spatial extensions. Since 2M > N , beamformers having packing ratios 2 : 1 exist. We have

3(2M − N) sets of such beamformers per cell, and using any N/3 (note that (N/3) < 3(2M − N)) of

them ensures that all dimensions at both the BSs are occupied by either interference or signal.

Case vii: 23 < γ < 3

4 : This case is discussed in detail in Section 2.4 and we only mention the exact

equations and transformations necessary to design the required beamformers. For users in cell 2, the

3M − 2N sets of beamformers having packing ratio 3 : 1 are designed by solving the system of equations

given by


H(21,1) H(22,1) 0

0 H(22,1) H(23,1)

v21j

v22j

v23j

= 0. (B.8)

We use an analogous set of equations for users in cell 1 and denote the set of beamformers designed

in this manner using the set {vikj}3M−2Nj=1 for all i ∈ {1, 2} and k ∈ {1, 2, 3}. We then multiply each

channel matrix Hik,l on the right by a matrix Wik, where Wik is a M × (2N − 2M) matrix whose

columns are orthogonal to the set {vikj}3M−2Nj=1 . Letting the effective channel matrix be denoted by

Hik,l, we see that we now have 2N − 2M effective antennas at each user and the best possible packing

ratio is 2 : 1. There exist 3(3N − 4M) pairs of beamformers having a packing ratio of 2 : 1, and solving

for any 3N − 4M pairs using equation (B.7) allows us to achieve the requisite number of DoF/user.

Case viii: 34 ≤ γ ≤ 1. Our goal is to achieve N/4 DoF/user. We assume N to be divisible by four

and present the arguments without any explicit reference to spatial extensions. Since 3M > N , packing

ratio of 3 : 1 is possible and there exists a total of 3M − 2N such sets of beamformers. Designing any

N/4 such sets through (B.8) gives us the requisite number of DoF/user.

Case ix: 1 < γ < 4/3 We need to design M/4 DoF/user, and we assume that M is a multiple of four.

Note that since M > N , the users can now zero-force the interfering BS. Each user can design M −N

transmit beamformers such that the interfering BS sees no interference. As before, we then multiply

the channel matrices Hik,l by a M × 2N −M matrix Wik that is orthogonal to the M − N transmit

beamformers obtained from zero-forcing. We now have 2N −M effective antennas at each user and it is

easy to see that there exist 4N − 3M sets of transmit beamformers having packing ratio of 3 : 1 for such

a system. Designing any N− 3M4 sets of such beamformers through (B.8) lets us achieve M/4 DoF/user.

Appendix C

Proof of Theorem 4.3.1*

C.1 Mathematical Background

The goal of this section is to present a concise introduction to the tools used in the proof of Theorem

4.3.1. Most of this material has been presented in various forms in earlier papers [21, 22, 78] and is

presented here for completeness and to bring more clarity to the concepts involved.

C.1.1 Transcendental Field Extensions

It the following, let F be a field and F [x1, . . . , xn] and F(x1, . . . , xn) denote the ring of polynomials and

rational functions over F respectively. Let K be a field containing F and denote the field extension by

K/F .

Definition An element α ∈ K is algebraic over F if there exists a nonzero f ∈ F [x] such that f(α) = 0.

If no such f exists, then α is transcendental over F . A set S = {α1, . . . , αn} ⊂ K is algebraically

dependent over F if there exists a nonzero f ∈ F [x1, . . . , xn] such that f(α1, . . . , αn) = 0. Otherwise S

is algebraically independent over F .

Clearly, algebraic independent elements over F are transcendental over F .

Example Consider Q ⊂ C. The element√2 ∈ C is algebraic over Q as it is a root of f(x) = x2 − 2.

The element π ∈ C is transcendental over Q. The elements π, π2 ∈ C (both transcendental) are not

algebraically independent over Q, as f(π, π2) = 0 for f(x1, x2) = x21 − x2.

Let S = {α1, . . . , αn} ⊂ K be an algebraically independent set over F . We can consider adjoining

the elements of S to F , denoted by F(S) = F(α1, . . . , αn). F(S) is defined to be the smallest field

extension of F containing all elements of S. The following lemma shows that the field F(S) has an easy

representation.

Lemma C.1.1. Let K/F be a field extension. If α1, . . . , αn ∈ K are algebraically independent over F ,

then F(α1, . . . , αn) and F(x1, . . . , xn) are isomorphic (as field extensions of F).

*This section is written in collaboration with Siyu Liu (graduate student, University of Toronto).

94

Appendix C. Proof of Theorem 4.3.1 95

Definition A subset S ⊂ K is a transcendence basis for K/F if S is algebraically independent over Fand K is algebraic over F(S).

Example Let K = F(x1, . . . , xn), then x1, . . . , xn is a transcendence basis for K/F .

We should expect any two bases to have the same size, and this is indeed the case. We shall define this

invariant.

Definition The transcendence degree trdeg(K/F) of a field extension K/F is the cardinality of any

transcendence basis of K/F .

The tools we developed so far gives us the following proposition, which we will use as a key step in the

necessary part of Theorem 4.3.1.

Proposition C.1.2. Let K = F(x1, . . . , xn). Any set S = {α1, . . . , αm} ⊂ K with m > n is algebraically

dependent over F .

Proof. Follows from the fact that trdeg(K/F) = n < m.

C.1.2 Zariski Topology and a Theorem of Chevalley

Let K be an algebraically closed field (e.g. C). Let S ⊂ K[x1, . . . , xn] be a set of polynomials. Define

the zero-locus Z(S) as:

Z(S) = {x ∈ Kn | f(x) = 0∀f ∈ S}.

A subset V of Kn is called an affine algebraic set if V = Z(S) for some S. The Zariski topology on Kn

is defined by specifying the closed sets to be the affine algebraic sets. Thus, open sets are of the form

Kn \Z(S) for some S ⊂ K[x1, . . . , xn]. Intuitively, open sets are “big” in Zariski topology. This is made

precise by the fact that open sets are dense (their closures are equal to Kn). Zariski open sets allow us

to define a property to be generic as follows.

Definition A property of Kn is said to be true generically if it is true over a non-empty Zariski open

set of Kn.

Closely related to open and closed sets is the concept of constructible sets.

Definition A set is locally closed if it is the intersection of an open set with a closed set. A finite union

of locally closed sets is called a constructible set.

Two important facts related to constructible sets that are used in the proof are as follows.

Proposition C.1.3. Every constructible set contains a dense open subset of its closure.

Theorem C.1.4 (Special case of Chevalley Theorem). Let f1, . . . , fn ∈ K[x1, . . . , xn], and define f =

(f1, . . . , fn) : Kn → Kn to be the corresponding polynomial map. Then the image of f (Im(f)) is a

constructible set.


A useful set of equivalent conditions that are satisfied by polynomial maps are presented in the

following proposition.

Definition A polynomial map f = (f1, . . . , fn) : Kn → Kn is dominant if Im(f) is dense in Kn.

Proposition C.1.5 ( [81], Prop. 5.2). For a polynomial map f = (f1, . . . , fn) : Kn → Kn, the following

conditions are equivalent.

1. f is a dominant map.

2. The function f1, . . . , fn are algebraically independent over K.

3. The Jacobian Jf = det

([∂fi∂xj

]

i,j

)

of f is not identically zero.

The above discussions give us the following proposition, which we will use as a key step in the sufficiency

part of Theorem 4.3.1.

Proposition C.1.6. Let f = (f1, . . . , fn) : Kn → Kn be a dominant polynomial map. Then Im(f)

contains a non-empty Zariski open set.

Proof. By Chevalley’s theorem, Im(f) is contructible. Since f is dominant, then the closure of Im(f) is

Kn. By Proposition C.1.3, Im(f) contains a dense open subset of Kn.

C.2 Proof of Theorem 4.3.1

This section proves a slightly more general form of Theorem 4.3.1 where the G-cell network is permitted

to have different number of users in each cell. Such networks are represented as (G, {Kg}, N × M)

networks. The new theorem statement follows.

Theorem C.2.1. Consider a (G, {Kg}, N×M) network where each user is served with one data stream.

Let ugk and vgk denote the transmit and receive beamformer corresponding to the (g, k)th user where the

set of beamformers {ug1,ug2, . . . ,ugKg} is linearly independent for every g. Further, let I ⊆ {(gk, i) :

g 6= i, 1 ≤ g, i ≤ G, 1 ≤ k ≤ Kg} be a set of user-BS pairs such that for each (gk, i) ∈ I the interference

caused by the ith BS at the (g, k)th user is completely nulled, i.e.,

vHgkH(i,gk)uij = 0, ∀j ∈ {1, 2, . . . ,Ki}. (C.1)

A set of transmit and receive beamformers {ugk} and {vgk} satisfying the polynomial system defined by

I exist if and only if

M ≥ 1 (C.2)

N ≥ Kg, ∀g. (C.3)

and

|Jusers|(M − 1) +∑

l∈JBS

(N −Kl)Kl ≥∑

(gk,l)∈JKl (C.4)

where J is any subset of I and Jusers and JBS are the set of user and BS indices that appear in J .


Proof. The proof closely follows the proof presented in [21] to establish a similar feasibility result.

Let the beamformers used by BS i be collectively represented as the matrix Ui, i.e., Ui =

[ui1,u12, . . . ,uiK ] . Let vgk and Ui be such that (gk, i) ∈ J . The interference alignment condition

implies that Ui must have rank Ki. Thus, we can apply invertible linear transformations to vgk and Ui

such that

vgk = Pvgk

1

vgk

Qvgk Ui = Vu

i

IKi×Ki

Ui

Qui .

We define H(gk,i) = Pvgk

−1Hgk,iQui−1, and partition it in the following way.

H(gk,i) =

H

(1)(gk,i) H

(2)(gk,i)

H(3)(gk,i) H

(4)(gk,i)

,

where H(1)(gk,i) has size 1×Ki. Note that H(gk,i) is still a generic matrix. With the above transformation,

we can rewrite the interference alignment condition as

[

1 vHgk

]

H

(1)(gk,i) H

(2)(gk,i)

H(3)(gk,i) H

(4)(gk,i)

I

Ui

= 0.

This can be expanded as the following equation.

H(1)(gk,i) + vH

gkH(3)(gk,i) + H

(2)(gk,i)Ui + vH

gkH(4)(gk,i)Ui = 0 ∀(gk, i) ∈ J (C.5)

To establish the necessity part of the theorem, first note that the total number of scalar equations

in C.5 is

∑

(gk,i)∈JKi,

and the total number of scalar variables (unknown entries in {vgk}’s and {Ui}’s) is

|Jusers|(M − 1) +∑

i∈JBS

(N −Ki)Ki.

Thus if

|Jusers|(M − 1) +∑

i∈JBS

(N −Ki)Ki <∑

(gk,i)∈JKi, (C.6)

then we would have more equations than unknowns in (C.5). We shall show that no solution (for {vgk}’sand {Ui}’s) can exist in this case.

Consider a transcendental field extension F of C with a transcendence basis given by entries of

{vgk, Ui}(gk,i)∈J . The transcendence degree of F is |Jusers|(M − 1) +∑

i∈JBS(N −Ki)Ki. Construct,

for each (gk, i) ∈ J ,

Fgk,i(vgk, Ui) = −(vHgkH

(3)(gk,i) + H

(2)(gk,i)Ui + vH

gkH(4)(gk,i)Ui). (C.7)


Note that Fgk,i is a 1×Ki vector with each entry in F . In particular, each entry of Fgk,i is a quadratic

polynomial of the entries in vgk and Ui. If (C.6) holds, then the total number of these entries (quadratic

polynomials) in {Fgk,i}(gk,i)∈J is strictly greater than the transcendence degree of F over C. Thus, by

Proposition C.1.2, these entries are algebraically dependent over C. In particular, there must exist a

nonzero polynomial p (in∑

(gk,i)∈J Ki variables with coefficients in C) such that

p({Fgk,i(vgk, Ui)}(gk,i)∈J ) = 0 ∀{vgk, Ui}(gk,i)∈J ,

where the notation p({Fgk,i(vgk, Ui)}(gk,i)∈J ) means that p takes on each entry of every Fgk,i as an input

in a specified order. Note that p is independent of {H(1)(gk,i)}(gk,i)∈J . Thus if we view p as a polynomial in

the variable X = ({H(1)(gk,i)}(gk,i)∈J ), then p can be expanded locally at X = ({Fgk,i(vgk, Ui)}(gk,i)∈J )

as

p({H(1)(gk,i)}(gk,i)∈J ) =p({Fgk,i(vgk, Ui)}(gk,i)∈J )

+∑

(gk,i)∈J(H

(1)(gk,i) − Fgk,i(vgk, Ui))Qgk,i({H(1)

(gk,i)}(gk,i)∈J ) ∀{vgk, Ui}(gk,i)∈J ,

where Qgk,i is some polynomial vector of size Ki × 1. Our assumption on p implies

p({H(1)(gk,i)}(gk,i)∈J ) =

∑

(gk,i)∈J(H

(1)(gk,i) − Fgk,i(vgk, Ui))Qgk,i({H(1)

(gk,i)}(gk,i)∈J ) ∀{vgk, Ui}(gk,i)∈J .

If equation (C.5) is satisfied, then there exists a choice of matrices {vgk, Ui}(gk,i)∈J such that

H(1)(gk,i) − Fgk,i(vgk, Ui) = 0 ∀(gk, i) ∈ J .

For this choice, we have

p({H(1)(gk,i)}(gk,i)∈J ) = 0. (C.8)

However, {H(1)(gk,i)}(gk,i)∈J is generic and independent of p. Thus (C.8) can only be satisfied if p is

identically the zero polynomial. This contradicts our assumption on p and proves the necessity part of

Theorem C.2.1.

For sufficiency, we focus on the case when the total number of variables equals the total number

of equations. All other cases follow easily. To establish the sufficiency part of the theorem, note that

it suffices to find a choice of {H(2)(gk,i), H

(3)(gk,i), H

(4)(gk,i)}(gk,i)∈I such that the Jacobian of the polynomial

map (C.7) (in variables {(vgk, Ui)}(gk,i)∈I) is nonzero. The condition that the Jacobian of a polynomial

map is zero is an algebraic condition on {H(2)(gk,i), H

(3)(gk,i), H

(4)(gk,i)}(gk,i)∈I . Thus, if there exist a choice

of {H(2)(gk,i), H

(3)(gk,i), H

(4)(gk,i)}(gk,i)∈I such that the Jacobian is nonzero, then the Jacobian is nonzero for

generic choices of {H(2)(gk,i), H

(3)(gk,i), H

(4)(gk,i)}(gk,i)∈I . After establishing the Jacobian of the polynomial

map is nonzero, Proposition C.1.5, tells us that the map (C.7) is in fact dominant. Then Proposition

C.1.6 tells us that the image of the map (C.7) contains a non-empty Zariski open set U of F . Thus,

equation (C.5) holds for all (H(1)(gk,i))(gk,i)∈I ∈ U and therefore holds generically.

We now establish a choice of {H(2)(gk,i), H

(3)(gk,i), H

(4)(gk,i)}(gk,i)∈I such that the Jacobian of the polynomial

map (C.7) is nonzero. The construction of the Jacobian mirrors the construction in [22] and only a brief

description is given below.


Figure C.1: Structure of the Jacobian matrix. Shaded regions depict non-zero partial derivatives.

Before constructing the Jacobian matrix, we create a single concatenated vector of variables by

ordering the variables {Ui} in a lexicographic manner followed by the variables {vgk} also listed in a

similar manner. A list of equations is created by first listing all equations (as seen in C.7) that involve

interference cancellation from the 1st BS, followed by the 2nd BS, and so on. Let this vectorized list

of variables and equations be denoted as λ and ψ respectively. Note that both vectors are of length∑G

i=1 ((N −Ki)Ki + (M − 1)Ki). The part of λ that corresponds to the {Ui} variables is denoted as

λu. Similarly define λv. The part of ψ that corresponds to equations involving Ui is denoted as ψUi.

Further, let the number of equations that involve the ith BS’s beamformers be given by ei.

The (i, j)th entry in the Jacobian matrix J is given by ∂ψi

∂λj. The notations

∂ψUi

∂Ui,

∂ψuij

∂uij,

∂ψuij

∂λv

all

refer to submatrices of J are straightforward to infer.

First, set {H(4)(gk,i)} to zero for all g, k, and i. We are left with choosing values for the {H(2)

(gk,i)} and

{H(3)(gk,i)} matrices. Setting {H(4)

(gk,i)} to zero results in a Jacobian matrix with a structure as illustrated

in Fig. C.1. The resulting Jacobian has a repeating structure on the left (corresponding to the∂ψ

uij

∂uij

that have the same partial derivatives for a fixed i as j is varied and a sparse structure on the right. It

can be shown that such a structure satisfies the conditions for Hall’s theorem and can hence be reduced

to a permutation matrix by setting certain entries to zero. Retain the permutation structure that results

from such a reduction only in the ∂ψ∂λv

submatrix of J (right side of Fig. C.1) by setting all non-zero

channel values to 1. With this submatrix fixed, it can now be shown that for any random choice of the

non-zero submatrices on the left, the resulting Jacobian is full-rank almost surely. To see this, first note

that the rank of the jacobian is now a sum of the ranks of the individual submatrices∂ψ

uij

∂λ . Note that

each such submatrix has non-zero entries in mutually exclusive columns. Now, each submatrix∂ψ

uij

∂λ has∑

l 6=iKl rows. Note that by construction, the right side of this submatrix (this is the∂ψ

uij

∂λv

submatrix)

has (∑

Kl −N) ones on distinct rows. Using a column transformation and eliminating non-zero entries

on (∑

Kl − N) rows on the left side (i.e., the∂ψ

uij

∂λu

submatrix), we are left with exactly (N − Ki)

non-zero rows in∂ψ

uij

∂λu

having non-zero entries in the same number of columns. It is easy to see that for

any random choice of the channel matrices {H(2)(gk,i)} and {H(3)

(gk,i)}, such a submatrix is full rank, thus


proving that each of the submatrices is full-rank and hence the Jacobian has a non-zero determinant.

For further details on such a construction, please refer [22]. Such a construction completes the proof of

sufficiency.

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