interference calculus a general framework for interference management and network utility optimization

There are indeed close connectionsbetween Yates' standard interference functions [1] and our framework of gen-eral interference functions, although they are dened by dierent axioms.Both

Foundations in Signal Processing, Communications and Networking

Series Editors: W Utschick, H Boche, R Mathar

Martin Schubert · Holger Boche

Authors:

Martin Schubert

Heinrich Hertz Institute for

Telecommunications HHI Einsteinufer 37

Germany E-mail: boche@tum.de

ISBN 978-3-642-24620-3 e-ISBN 978-3-642-24621-0

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In memory of my grandmother Maria Amende (1908-1996)

Holger Boche

In memory of my father Roland Schubert (19 -1997) and my

grandmother Soe Schubert (1911-2003)

Martin Schubert39

This book develops a mathematical framework for modeling and optimizinginterference-coupled multiuser systems At the core of this framework is theconcept of general interference functions, which provides a simple means ofcharacterizing interdependencies between users The entire analysis builds onthe two core axioms scale-invariance and monotonicity, which are introduced

in Section 1.2

Axiomatic approaches are frequently used in science An example is erative game-theory, where interactions between users (or agents, players) aremodeled on the basis of axioms (axiomatic bargaining) The proposed frame-work does indeed have a conceptual similarity with certain game-theoreticapproaches However, game theory originates from economics or social sci-ence, while the proposed interference calculus has its roots in power controltheory and wireless comunications It adds theoretical tools for analyzing thetypical behavior of interference-coupled networks In this way, it complementsexisting game-theoretic approaches (see e.g Chapter 4)

coop-The proposed framework should also be viewed in conjunction with mization theory There is a fruitful interplay between the theory of interferencefunctions and convex optimization theory By jointly exploiting the proper-ties of interference functions (notably monotonicity), it is possible to designalgorithms that outperform general-purpose techniques that only exploit con-vexity

opti-The title interference calculus refers to the fact that the theory of ference functions constitutes a generic theoretical framework for the analysis

inter-of interference coupled systems Certain operations within the framework are

closed, that is, combinations of interference functions are again, interferencefunctions Also, certain properties are preserved under such operations In-terference calculus provides a methodology for analyzing dierent multiuserperformance measures that can be expressed as interference functions or com-binations of interference functions

largely inuenced by problems from this area Among the most inuentialworks is Yates' seminal paper on power control [1], from which we haveadopted the term interference function There are indeed close connectionsbetween Yates' standard interference functions [1] and our framework of gen-eral interference functions, although they are dened by dierent axioms.Both frameworks are compared in Section 2.4.

Our rst results on general interference functions were published in themonograph [2] Additional properties were shown in a series of journal articles[311] These extensions provide a deeper and more complete understanding

of the subject An overview tutorial on interference functions and applicationswas given at ICASSP 2010 [12] Among the many comments we received, therewas the repeated requests for a comprehensive overview that summarizes theimportant facts and concepts of interference functions

The present book was written in response to these requests It provides

an overview on the recent advances [311] Particular emphasis is put onanalyzing elementary structure properties of interference functions Exploitingstructure is essential for the design of ecient optimization algorithms.Although the focus of this book is on wireless communication, the proposedaxiomatic framework is quite general Therefore, it is our hope that researchersfrom other disciplines will be encouraged to work in this area

The targeted audience includes graduate students of engineering and plied mathematics, as well as academic and industrial researchers in the eld

ap-of wireless communications, networking, control and game theory No ular background is needed for reading this book, except for some familiaritywith basic concepts from convex analysis and linear algebra A general will-ingness to carry out detailed mathematical analysis is, however, important.The proofs and detailed calculations should help the reader in penetratingthe subject Reading our previous book [2] is not a prerequisite, although itmight be helpful since it covers additional fundamental aspects of interferencefunctions

partic-Our scientic work was motivated and inuenced by many researchers.Among those who were most inuential, we would like to name Tansu Alpcan,Mats Bengtsson, Michael Joham, Josef Nossek, Björn Ottersten, ArogyaswamiPaulraj, Dirk Slock, Sªawomir Sta«czak, Sennur Ulukus, Wolfgang Utschick,and Roy Yates We thank them and their group members for their inspiringwork

We also thank the funding agencies that made the research possible.The work was funded by the Federal Ministry of Education and Research(Bundesministerium für Bildung und Forschung, BMBF) within the projectsEASY-C (01BU0631), TEROPP (01SF0708), Scalenet (01BU566), by theGerman Research Foundation (Deutsche Forschungsgemeinschaft, DFG)Over the last ten years, the authors have been involved in research onresource allocation for wireless comunication networks Hence, this book is

1 Introduction 1

1.1 Notation 3

1.2 Basic Axiomatic Framework of Interference Functions 4

1.3 Convexity, Concavity, and Logarithmic Convexity 5

1.4 Examples  Interference in Wireless Networks 7

2 Systems of Coupled Interference Functions 17

2.1 Combinations of Interference Functions 18

2.2 Interference Coupling 18

2.3 Strict Monotonicity and Strict Log-Convexity 21

2.4 Standard Interference Functions and Power Control 23

2.5 Continuity 26

2.6 QoS Regions, Feasibility, and Fixed Point Characterization 29

2.7 Power-Constrained QoS Regions 32

2.8 The QoS Balancing Problem 35

3 The Structure of Interference Functions and Comprehensive Sets 39

3.1 General Interference Functions 40

3.2 Synthesis of General Interference Functions 46

3.3 Concave Interference Functions 50

3.4 Convex Interference Functions 61

3.5 Expressing Utility Sets as Sub-/Superlevel Sets of Convex/Concave Interference Functions 69

3.6 Log-Convex Interference Functions 72

3.7 Application to Standard Interference Functions 82

3.8 Convex and Concave Approximations 89

4 Nash Bargaining and Proportional Fairness for Log-Convex Utility Sets 99

4.1 Nash Bargaining for Strictly Log-Convex Utility Sets 100

4.2 The SIR Region of Log-Convex Interference Functions 109

4.3 Proportional Fairness  Boundedness, Existence, and Strict Log-Convexity 122

4.4 SINR Region under a Total Power Constraint 138

4.5 Individual Power Constraints  Pareto Optimality and Strict Convexity 140

5 QoS-Constrained Power Minimization 155

5.1 Matrix-Based Iteration 157

5.2 Super-Linear Convergence 164

5.3 Convergence of the Fixed Point Iteration 171

5.4 Worst-Case Interference and Robust Designs 177

6 Weighted SIR Balancing 183

6.1 The Max-Min Optimum 184

6.2 Principal Eigenvector (PEV) Iteration 188

6.3 Fixed Point Iteration 192

6.4 Convergence Behavior of the PEV Iteration 195

A Appendix 197

A.1 Irreducibility 197

A.2 Equivalence of Min-Max and Max-Min Optimization 198

A.3 Log-Convex QoS Sets 199

A.4 Derivatives of Interference Functions 201

A.5 Non-Smooth Analysis 202

A.6 Ratio of Sequences 203

A.7 Optimizing a Ratio of Linear Functions 204

A.8 Continuations of Interference Functions 205

A.9 Proofs 208

References 227

Index 235

Introduction

A fundamental problem in the analysis and optimization of multi-user nication networks is that of modeling and optimizing performance tradeos.Tradeos occur when users share a limited resource or if they are coupled bymutual interference In both cases, the users cannot act independently If oneuser increases the performance by using more resources, then this generallycomes at the cost of reducing the available performance margin of other users

commu-A typical example is a wireless multi-user system, where the signal mitted by one user causes interference to other users Interference is not re-stricted to the physical layer of the communication system, it also aectsrouting, scheduling, resource allocation, admission control and other higher-layer functionalities In fact, interference is one of the main reasons why across-layer approach is often advocated for wireless systems [13]

trans-Interference may also be understood in a more general way, as the tition for resources in a coupled multi-user system Interference is not limited

compe-to wireless communication scenarios It is also observed in wireline networks.For example, interference occurs between twisted-pair copper wires used forDSL transmission There are many other forms of interference in dierentcontexts

The modeling and optimization of coupled multi-user systems can be cult Adaptive techniques for interference mitigation can cause the interference

di-to depend on the underlying resources in a complicated nonlinear fashion Inorder to keep the complexity of the resource allocation manageable, interfer-ence is often avoided by allocating resources to users in an orthogonal manner,and residual interference is treated as noise Then the system becomes a col-lection of quasi-independent communication links This practical approachgreatly simplies the analysis of multi-user systems However, assigning eachuser a separate resource is not always an ecient way of organizing the system

If the number of users is high, then each user only gets a small fraction of theoverall resource Shortages are likely to occur when users have high capacityrequirements This will become even more problematic for future wireless net-works, which are expected to provide high-rate services for densely populated

M Schubert, H Boche, Interference Calculus, Foundations in Signal Processing,

Communications and Networking 7,

© Springer-Verlag Berlin Heidelberg 201 2

user environments The system then might be better utilized by allowing users

to share resources

This development drives the demand for new design principles based onthe dynamic reuse of the system resources frequency, power, and space (i.e thedistribution and usage of transmitting and receiving antennas over the servicearea) Interference is no longer just an important issue, but rather emerges

as the key performance-limiting factor The classical design paradigm of pendent point-to-point communication links is gradually being replaced by anew network-centric point of view, where users avoid or mitigate interference

inde-in a exible way by dynamically adjustinde-ing the resources allocated to each user.Interference modeling is an important problem in this context, because thequality of any optimization strategy can only be as good as the underlyinginterference model

The development of sophisticated resource sharing strategies requires athorough understanding of interference-coupled systems It is important tohave theoretical tools which enable us to model and optimize the nonlineardependencies within the network The interdependencies caused by interfer-ence are not conned to the lower layers of the communication system Forexample, it was shown in [14] how the manipulability of certain resource al-location strategies depends on the interference coupling

This book proposes an abstract theory for the analysis and optimization

of interference-coupled multi-user systems At the core of this theory lies theconcept of an interference function, which is dened by a framework of axioms(positivity, scale-invariance, monotonicity), as introduced in Section 1.2 Thisaxiomatic approach has the advantage of being quite general It is applicable

to various kinds of interference-coupled systems

The proposed axiomatic framework was strongly inuenced and motivated

by power control theory It generalizes some known concepts and results.For example, linear interference functions are included as special cases

It will be shown later that certain key properties of a system with linear terference functions extend to logarithmically convex (log-convex) interferencefunctions In many respects, log-convex interference functions can be regarded

in-as a natural extension of linear interference functions This area of research isclosely linked with the Perron-Frobenius theory, which has numerous impor-tant applications [15]

The proposed framework is also useful for the analysis of SIR and SINRregions, with and without power constraints This includes the problem of

nding a suitable operating point within the region This typically involves acompromise between fairness and eciency among the users of the system.Often, these are conicting goals, and there is more than one denition offairness This will be studied from a game-theoretic perspective in Chapter 4.Finally, the results of this book also contribute to a deeper understanding

of standard interference functions [1] The framework of standard interferencefunctions is conceptually similar to the one used here Both follow an axiomatic

1.1 Notation 3approach The dierence will be discussed in Section 2.4 It will be shown thatstandard interference functions are included in the theory presented here.After introducing some notational conventions, we will introduce the ba-sic axiomatic framework in Section 1.2 These axioms are the basis for allfollowing derivations Additional properties, like convexity and logarithmicconvexity will be introduced in Section 1.3 In Section 1.4 we will discussexamples of interference functions.

1.1 Notation

We begin with some notational conventions

• The sets of non-negative reals and positive reals are denoted by R+ and

R++, respectively RKdenotes the Kg-dimensional Euclidean vector space

• Matrices and vectors are denoted by bold capital letters and bold lowercaseletters, respectively

• Let y be a vector, then yl = [y]l is the lth component Likewise, Amn =[A]mn is a component of the matrix A

• A vector inequality x > y means xk > yk, for all k The same holds forthe reverse directions

• y > 0 means component-wise greater zero

• y ≥ x means yl≥ xl for all components

• y x means y ≥ x and there is at least one component l such that

yl> xl

• y 6= x means that inequality holds for at least one component

• exp(y) and log(y) means component-wise exponential and logarithm, spectively

re-Some often used variables and quantities are as follows

r, p Throughout this book, r is a K-dimensional non-negative vector of

system resources, which are not further specied A special case is

r = p, where p is a Ku-dimensional vector containing the transmissionpowers of Ku users

I(r) General interference function (see Section 1.2)

J (p) Standard interference function (see Section 2.4)

p Extended power vector p = [p1, , pK u, σ2

n]T, where σ2

n is the noisepower Sometimes, we normalize σ2

n = 1

K, KuIndex sets of cardinality K and Ku, respectively

SIR Signal-to-interference ratio

SINR Signal-to-interference-plus-noise ratio

QoS Quality of Service, dened as a strictly monotone and continuous tion of the SIR or SINR

func-γ Vector of SIR or SINR targets

Γ Diagonal matrix Γ = diag{γ}

V Coupling matrix, V = [v1, , vK u]T, where vk contains the couplingcoecients of user k

W Coupling matrix for log-convex interference functions

1.2 Basic Axiomatic Framework of Interference

Functions

Axiomatic characterizations have a long-standing tradition in science known examples include the axiomatic bargaining theory introduced by Nash[16, 17] (see also [18, 19]) and the axiomatic characterization of the Shannonentropy by Khinchin [20] and Faddeev [21] (see also [22]) Analyzing the basicbuilding blocks of a theoretical model often provides valuable new insight intoits underlying structure

Well-In this book, interference is dened as a monotone scale-invariant geneous) function

(homo-Denition 1.1 Let I : RK

+ 7→ R+ We say that I is a general interferencefunction (or simply interference function) if the following axioms are fullled:A1 (positivity) There exists an r > 0 such that I(r) > 0A2 (scale invariance) I(αr) = αI(r) for all α ≥ 0

A3 (monotonicity) I(r) ≥ I(r0)if r ≥ r0

The framework A1, A2, A3 is related to the concept of standard ence functions introduced by Yates [1], where scalability was required instead

interfer-of scale invariance Scalability was motivated by a specic power control lem It will be shown in Section 2.4 that standard interference functions can

prob-be comprehended within the framework A1, A2, A3

Concrete examples of interference functions will be discussed in Section 1.4.Most of these examples focus on multi-user communication systems, where r

1.3 Convexity, Concavity, and Logarithmic Convexity 5

is a vector of transmission powers, and I(r) is the resulting interference atsome receiver For example, I(r) can measure the impact of some systemvariables that are collected in the vector r

If one component of r is increased then axiom A3 states that the resultinginterference increases or remains constant This property is closely related tothe game-theoretic concept of comprehensiveness, which will be discussed inChapter 3

Scale-invariance (A2) is best understood by studying the examples in tion 1.4

Sec-An immediate consequence of A2 and A3 is non-negativity, i.e., I(r) ≥ 0for all r ≥ 0 This follows from A2 and A3 by contradiction Suppose I(r) < 0.For 0 < λ ≤ 1 we have 0 > I(r) ≥ I(λr) = λI(r) Letting λ → 0 leads to acontradiction, thus proving non-negativity

Sometimes positivity is needed instead of non-negativity Axiom A1 itivity) states that there exists at least one r > 0 such that I(r) > 0 It wasshown in [2] that A1 is equivalent to the statement I(r) > 0 for all r > 0 Ifthis is not fullled, then we have the trivial interference function I(r) = 0 forall r > 0 Hence, the only purpose of A1 is to rule out this trivial case.The axiomatic framework A1, A2, A3 is analytically appealing Some ba-sic results were already shown in [2] But the case of real interest is whenthe framework is extended by additional properties It will be shown in thefollowing that under the assumption of certain monotonicity and convexityproperties, interference functions oer enough structure to enable ecient al-gorithmic solutions for dierent kinds of resource allocation problems

(pos-1.3 Convexity, Concavity, and Logarithmic Convexity

Convexity plays an important role in engineering, economics, and other entic disciplines [23] When investigating a problem, a common approach is

sci-to rst look whether it is convex or not Theoretical advances have given usnew tools that are successfully applied to the optimization of multi-user com-munication systems [24] Many examples can be found, for example, in thecontext of multi-user MIMO and robust optimization [2530]

1.3.1 Convex and Concave Interference Functions

Standard convex optimization strategies are applicable to any kind of convexproblem However, standard approaches typically ignore the particular ana-lytical structure of the problem at hand Thus, they are not necessarily a goodchoice when convergence speed and complexity matters

In this book we are interested in convex problems that arise from convexinterference functions, which are not just convex, but which also fulll thebasic axioms A1, A2, A3

Denition 1.2 A function I : RK

+ 7→ R+ is said to be a convex interferencefunction if A1, A2, A3 are fullled and in addition I is convex on RK

+ wise, a function I : RK

Like-+ 7→ R+ is said to be a concave interference function ifA1, A2, A3 are fullled and in addition I is concave on RK

+.Examples of nonlinear convex and concave interference functions will bediscussed in Section 1.4 Linear interference functions are both convex andconcave nonlinear concave interference functions typically occur when inter-ference is minimized This includes adaptive receive or transmit strategies,e.g beamforming [2, 26, 27, 3133], CDMA [34, 35], or base station assign-ment [36,37] Convex interference functions typically occur when interference

is maximized Such worst-case strategies are known from robust tion [29, 30]

optimiza-One of the important goals of this book is to show that convex interferencefunctions have a rich mathematical structure that can be exploited to yieldecient algorithmic solutions Examples are the SI(N)R-balancing algorithmsthat will be discussed in Chapters 5 and 6

1.3.2 Log-Convex Interference Functions

Sometimes, a problem is not convex but there exists an equivalent convexproblem formulation Then the original non-convex problem can be solved in-directly by solving the equivalent problem instead This is sometimes referred

in [3944]

Denition 1.4 A function I : RK

+ 7→ R+ is said to be a log-convex terference function if I(r) fullls A1, A2, A3 and in addition I(exp{s}) islog-convex on RK Log-concave interference functions are dened accordingly.Let f(s) := I(exp{s}) Then a necessary and sucient condition for log-convexity is [23]

in-1.4 Examples  Interference in Wireless Networks 7

f s(λ)

≤ f(ˆs)1−λf (ˇs)λ, ∀λ ∈ (0, 1); ˆs, ˇs∈ RK , (1.2)where

s(λ) = (1− λ)ˆs + λˇs , λ∈ (0, 1) (1.3)The corresponding vector r(λ) = exp s(λ) is

r(λ) = ˆr(1−λ)· ˇrλ (1.4)The change of variable r = exp{s} was already used by Sung [38] in thecontext of linear interference functions (see the following example), and later

in the sense of Denition 1.4 The converse is not true, however Therefore, theclass of log-convex interference functions is broader than the class of convexinterference functions Log-convex interference functions include convex inter-ference functions as special case Therefore, the requirement of log-convexity

is relatively weak

Log-convex interference functions oer interesting analytical possibilitiessimilar to the convex case, while being less restrictive In remainder of thisbook, we will discuss the properties of log-convex interference functions indetail It will turn out that log-convex interference functions preserve many

of the properties that are known for the linear case An example is the SIRregion studied in Chapter 4

For completeness, we also discuss the class of log-concave interference tions They were not studied in the literature so far This is because log-concave interference functions do not have the same advantageous properties

func-as the log-convex interference functions For example, it is not true that everyconcave interference function is a log-concave interference function A simpleexample is the linear interference function which is log-convex but not log-concave There are further dierences, e.g the sum of log-convex interferencefunctions is a log-convex interference function, however the same is not truefor log-concave interference functions

1.4 Examples  Interference in Wireless Networks

In this section we will discuss examples of interference functions satisfying theaxioms A1, A2, A3 These examples originate mainly from research in wirelesscommunication, especially power control theory However, the analysis of cou-pled multiuser systems is a broad and diverse eld (see e.g [15]), therefore,more application examples certainly exist

Consider a wireless communication system with Kuusers sharing the sameresource, thus mutual interference occurs The users' transmission powers arecollected in a vector

p = [p1, , pK u]T ∈ RKu

+ (1.6)The goal is to control the powers p in such a way that a good system per-formance is achieved The performance of user k is measured in terms of itssignal-to-interference ratio (SIR)

SIRk(p) = pk

Ik(p) , k∈ Ku (1.7)Here, Ik(p)is the interference (power cross-talk) observed at user k, for giventransmission powers p The functions I1, ,IK u determine how the users arecoupled by mutual interference (see Fig 1.1)

1.4.1 Linear Interference Function

The interference of user k is dened as

Ik(p) = pTvk, k∈ Ku, (1.8)where vk ∈ RKu

+ is a vector of coupling coecients By collecting all Ku pling vectors in a coupling matrix or link gain matrix

cou-V = [v1, , vK u]T , (1.9)

we can rewrite (1.8) as

Ik(p) = [V p]k, k∈ Ku (1.10)The popularity of the linear model is due to its simplicity, but also to itsclose connection with the rich mathematical theory of non-negative matrices(Perron-Frobenius theory) In the past, this has led to many theoretical results

1.4 Examples  Interference in Wireless Networks 9and power control algorithms, e.g [4852] The applicability of the Perron-Frobenius theory is not limited to power control There are many furtherexamples of systems characterized by a non-negative irreducible matrix For

an overview we refer to [15]

For practical applications, the signal-to-interference-plus-noise ratio (SINR)

is a typical performance measure The SINR is also dened as (1.7), where Ik

also depends on noise power σ2

n To this end, we introduce the extended powervector

p = [p1, , pK u, σn2]T ∈ RKu +1

+ (1.11)The resulting interference-plus-noise power is

by one dimension This notation will allow us later to investigate dierentproblems within a single unifying framework Some properties are shared byboth models, no matter whether there is noise or not For example, moststructure results from Chapter 3 readily extend to the case where there isadditional noise

However, noise clearly makes a dierence when investigating resource location algorithms in a power-constrained multi-user system Then it is im-portant to consider the special properties resulting from the assumption of aconstant noise component This will be studied in detail in Section 2.4.Linear interference functions are concave, convex, and also log-convex after

al-a chal-ange of val-arial-able (see Section 1.3.2) Hence, al-all results in this book holdfor linear interference functions

1.4.2 Beamforming

The linear model is well understood and there is a wealth of interesting sults and applications, not limited to communication scenarios (see e.g [15]).However, interference often depends on the transmission powers in a non-linear way, e.g., if adaptive receive and transmit strategies are employed toavoid or mitigate interference Using a linear model may oversimplify the realsituation Therefore, it is desirable to extend the linear model

re-An example is the following nonlinear interference function resulting frommulti-user beamforming This scenario was studied, e.g., in [2628, 3133].Consider an uplink system with Ku single-antenna transmitters and an

M-element antenna array at the receiver Independent signals S1, , SK u

with powers pk = E[|Sk|2] are transmitted over vector-valued channels

h1, , hK u ∈ CM, with spatial covariance matrices Rk = E[hkhHk] Thesuperimposed signals at the array output are received by a bank of linear

lters u1, , uK u (the `beamformers') The output of the kth beamformer is

yk= uHk X

l ∈K u

hlSl+ n

, (1.13)where n ∈ CM is an AWGN vector, with E[nnH] = σ2

nI The SINR of user kis

SINRk(p, uk) = E[|uH

k hkSk|2]E[|Pl ∈kuH

A special case occurs if the channels h1, , hK u are deterministic, then

Rl= hlhHl In this case, the interference resulting from optimum beamformerscan be written in closed form

lin-1.4.3 Receive Strategies

The next example shows that the interference functions (1.14) and (1.16)can be understood within a more general and abstract framework of adaptivereceive strategies

1.4 Examples  Interference in Wireless Networks 11For every user k, we dene an abstract receive strategy zk from a non-empty compact set Zk The receive strategy zk leads to coupling coecients

Ik(p) = min

z k ∈Z k

pTvk(zk) , ∀k ∈ Ku (1.17)Noise can be included by using the extended power vector (1.11) and theextended coupling vector, as in (1.12)

Ik(p) = min

z k ∈Z k

pTvk(zk) , ∀k ∈ Ku (1.18)

A special case is the previous example (1.14), where beamformers ukwere used

as receive strategies The beamformers were chosen from the unit sphere, i.e.,

kukk2= 1 Note, that the model (1.17) allows for arbitrary other constraints.For example, beamformers with shaping constraints were studied in [27] This

is included in the generic model (1.17), where we only require that the set

Zk is compact in order to ensure the existence of the minimum The set

Zk can also be discrete, for example when there is a choice between severalreceivers A special case is the problem of joint beamforming and base stationassignment [36, 37]

As in the previous example, the resulting interference function is concave.1.4.4 The Spectral Radius  Indicator of Feasibility

Consider again the example of linear interference functions from tion 1.4.1 The function Ik(p) = [V p]k is based on a non-negative and ir-reducible coupling matrix V Irreducibility means that each user depends onthe transmission power of any other user, either directly or indirectly (seeAppendix A.1 for a formal denition) The concept of irreducibility is fun-damental for the analysis of interference-coupled systems It will be used atseveral points throughout this book

Subsec-A fundamental question is, under what conditions can certain SIR valuesSIR1(p), ,SIRK u(p) be achieved jointly? This depends on how the usersare coupled by interference Let γk be the target SIR of user k The targets

of all Ku targets are collected in a vector

γ = [γ1, γ2, , γK u]T ∈ RKu

++ (1.19)

If all SIR targets γ can be achieved then we say that γ is feasible It wasalready observed in early work [48, 54] that the feasibility depends on thespectral radius

ρV(γ) = ρ(Γ V ) , where Γ = diag{γ} (1.20)

In the context of non-negative irreducible matrices, ρV is also referred to asthe Perron root

If ρV(γ) ≤ 1, then there exists a p > 0 such that SIRk(p) ≥ γk for all

k∈ Ku The feasible SIR region is dened as follows

S = {γ > 0 : ρV(γ)≤ 1} (1.21)The function ρV(γ) is an indicator for the feasibility of an SIR vector γ Itprovides a single measure for the system load caused by the Ku users It

is observed that ρV(γ) fullls the axioms A1, A2, A3 Thus, the SIR region(1.21) is a sublevel set of an interference function

The structure of the region S is directly connected with the properties of

ρV(γ) This was already exploited in [38], where it was shown that the SIRregion S is convex on a logarithmic scale Additional properties were shown

in [3941] The log-convexity of S can also be understood as a special case

of [5], where the SIR region was studied within the framework of log-convexinterference functions In fact, ρV(γ)is a log-convex interference function, inthe sense of Denition 1.4 That is, ρV(exp q) is log-convex on RK u, where

we use the change of variable γ = exp q The region S is a sub-level set ofthe log-convex (thus convex) indicator function ρV(exp q) Consequently, thelog-SIR region is convex

The spectral radius is an indicator function resulting from linear ence functions This is a special case of the min-max function C(γ) that will

interfer-be discussed in the next subsection

1.4.5 Min-Max Balancing and Feasible Sets

Consider arbitrary interference functions I1, ,IK u In contrast to the vious example we only require the basic axioms A1, A2, A3 We wish to knowwhether there exists a p > 0 such that SIRk(p) ≥ γk for all k ∈ Ku, orequivalently

C(γ) = inf

p>0

max

k ∈K u

γk· Ik(p)

pk

 (1.22)

The optimizer of this problem (if existent) maximizes the minimum SIR (seeAppendix A.2)

Some SIR vector γ > 0 is feasible if and only if C(γ) ≤ 1 If C(γ) = 1and the inmum (1.22) is not attained, then this means that γ is a boundarypoint that can only be achieved asymptotically Our denition of feasibility

1.4 Examples  Interference in Wireless Networks 13includes this asymptotic case, but for most practical scenarios, `inf' can bereplaced by `min', which means that γ is actually attained by some p > 0.The feasible SIR region is dened as

S = {γ > 0 : C(γ) ≤ 1} (1.23)

If the interference functions are linear, i.e., Ik(p) = [V p]k as in the previousexample (1.10), then C(γ) is simply the spectral radius (1.20) of the weightedcoupling matrix Γ V This can be seen from the Collatz-Wielandt type char-acterization [55] (see also [5658])

C(γ) = inf

p>0

max

In this example, interference functions occur on dierent levels The ical interference is modeled by I1, ,IK u On a higher level, the interferencefunction C(γ) provides a measure for the system load The properties of theresulting SIR region depends on the properties of C(γ), which depends on theproperties of I1, ,IK u These aspects will be studied in more detail in Sec-tion 2.6 In Section 2.7 we will discuss how to incorporate power constraints.1.4.6 Transmit Strategies and Duality

phys-Consider a system of Ku users with an irreducible coupling matrix G ∈

RK+u×Ku Assume that G depends on parameters z = (z1, , zK u)in a wise fashion That is, the kth column of G only depends on zk ∈ Zk As

column-a consequence, the interference [G(z)p]k of user k depends on all ters z = (z1, , zK u) This is typical for transmit strategies that optimizethe communication links at the transmitter side (e.g transmit beamforming).Thus, we refer to zk as a transmit strategy, in contrast to the receive strategydiscussed in the previous example

parame-However, the resulting interference values [G(z)p]k are dicult to handlesince each of them not only depends on p, but also on all transmit strategies

z1, , zK u The choice of any transmitter inuences the interference received

by all other users Thus, we cannot write the interference in terms of Ku

separate interference functions depending only on p, as in the previous amples When optimizing the system jointly with respect to p and z, then ajoint optimization approach is required An example is the problem of jointpower control and downlink beamforming, for which suboptimal heuristicswere proposed in early work [31, 59]

ex-Fortunately, there is a simple way of getting around the problem of coupledtransmit strategies We can exploit that the columns of G(z) are independentwith respect to z1, , zK u The key idea is to optimize the transpose system

V (z) = GT(z)instead of the original system G(z) Similar to (1.9) we dene

V (z) = [v1(z1), , vK u(zK u)]T ,The kth row of this virtual system V (z) only depends on the parameter zk.Hence, the resulting interference can be expressed in terms of the interferencefunctions (1.17) Introducing an auxiliary variable q ∈ RK u

Ku interference functions which can be optimized independently with respect

to the parameters zk The transmit strategy z becomes a virtual receivestrategy

It remains to show that the optimization of the virtual system V (z) leads

to the optimum of the original system G(z) Whether such a duality existsdepends on the optimization problem under consideration An example forwhich duality holds is the problem of SIR balancing, where the aim is tomaximize the worst SIR among all users This problem will be studied in detail

in Chapter 6 For the special case of transmit beamforming, the problem wasstudied in [8,59,60]

This duality between transpose systems was already observed in [61] in apower control context Duality was also observed in the context of the afore-mentioned downlink beamforming problem [62, 63] In this work, the matrix

V characterizes a downlink point-to-multipoint channel, whereas the pose VT has an interpretation as an uplink multipoint-to-point channel Thus,the term uplink-downlink duality was introduced to refer to this reciprocitybetween both channels

trans-Examples in the context of multi-antenna signal processing include [32,

33, 6466] A recent extension of this line of work is [67], where per-antennapower constraints were studied There is also an interesting relationship withthe MAC/BC duality observed in information theory [65, 66,68,69]

1.4.7 Robust Designs

Linear interference functions (1.8) can be generalized by introducing dependent coupling coecients vk(ck) Assume that the parameter ck standsfor some uncertainty chosen from a compact uncertainty region Ck A typicalsource of uncertainty are channel estimation errors or other system imperfec-tions Then, the worst-case interference is given by

parameter-Ik(p) = max

c ∈C pTvk(ck) , k∈ Ku (1.26)

1.4 Examples  Interference in Wireless Networks 15Performing power allocation with respect to the interference functions (1.26)guarantees a certain degree of robustness Robust power allocation was stud-ied, e.g., in [9,29, 30, 43].

As an example, consider again the downlink beamforming scenario cussed in the previous section In the presence of imperfect channel estimation,the spatial covariance matrices can be modeled as Rk = ˆRk+ ∆k, where ˆRk

dis-is the estimated covariance, and ∆k∈ Zkis the estimation error from a pact uncertainty region Zk In order to improve the robustness, the systemcan be optimized with respect to the worst case interference functions

Ik(p) = min

ku l k=1

max

The interference function (1.28) is neither convex nor concave in general, but

it also fullls the basic properties A1, A2, A3

1.4.8 Interference Functions in Other Contexts

The previous list of examples is by no means exhaustive It shows that terference is often nonlinear, and interference functions appear in dierentcontexts, not limited to power control

in-For example, a generic performance measure is as follows

Another example is the weighted sum utility

where w ∈ RK

+, with kwk1 = 1, are weighting factors and the utility vector

uis chosen from a compact set U ⊂ RK

++ For example, in a time-scheduledsystem uk could stand for a user rate and wk could be the queue backlog.The function Usum(w) is a convex interference function Consequently,

it is a log-convex interference functions after a change of variable (see tion 1.3.2)

Sec-More properties of interference functions will be studied in the remainder

of this book The analysis of interference functions is closely tied to the ysis of feasible sets (see e.g the example from Section 1.4.5) The properties

anal-of the feasible SIR sets are determined by the properties anal-of the underlying terference functions Thus, parts of the book are devoted to a detailed study

in-of the interdependencies between interference functions and feasible sets.Furthermore, the interference calculus is very closely connected with thetheory of monotone optimization (see e.g [72]), which is based on increasingpositively homogeneous functions This theory has been applied to the study

of models in Mathematical Economics [73] The dierences and similiaritiesbetween both theories have not yet been fully explored

Systems of Coupled Interference Functions

Consider a multi-user system characterized by K interference functions

I1(r),I2(r), ,IK(r),which all depend on the same resource vector r ∈ RK

+ We begin with themost general case where the interference functions are only characterized bythe axioms A1, A2, A3 (see p 4 in Section 1.2) The overall performance of thesystem is typically a function of all interference values, which depend on thesame underlying resource vector r Specic examples were already discussed

in Section 1.4

The analysis and optimization of such a system is complicated by thefact that the interference functions can be mutually coupled That is, theinterference value Ik(r)of some user k can depend on other users' resources

rl, l 6= k The users can also be coupled by sharing a common budget Thisleads to joint optimization problems that are often dicult to handle It istherefore important to have a thorough understanding of the properties ofinterference, and the structure of the optimization problems that result fromcombinations of interference functions

In this chapter we discuss some fundamental properties of coupled systems, and we show the connections with existing work in powercontrol theory In the context of power control, Yates [1] introduced the ax-iomatic framework of standard interference functions for modeling interfer-ence as a function of transmission powers The theory was further analyzedand extended in [11, 34, 74, 75] It will be shown in Section 2.4 that the ax-iomatic framework A1, A2, A3 with additional strict monotonicity provides

interference-an equivalent way of modeling stinterference-andard interference functions Hence, stinterference-an-dard interference functions can be regarded as a special case, and most resultsderived in this book immediately transfer to standard interference functions

stan-M Schubert, H Boche, Interference Calculus, Foundations in Signal Processing,

Communications and Networking 7,

© Springer-Verlag Berlin Heidelberg 201 2

2.1 Combinations of Interference Functions

Section 1.4.5 introduced the indicator function C(γ), which is an example

of an interference function being constructed as a combination of other terference functions Other possible combinations exist Consider interferencefunctions I1, ,IK, which fulll the axioms A1, A2, A3, then these propertiesare preserved by the following combinations

in-• The maximum of interference functions is again an interference function

I(r) = max

k∈KIk(r) (2.1)This remains valid when the maximum is replaced by the minimum

• Any linear combination of interference functions is an interference tion

For log-convex interference functions, the following properties hold:

• The sum of log-convex interference functions is a log-convex interferencefunction

• Let I(1) and I(2) be log-convex interference functions, then

I(r) = I(1)(r)
1 −α

· I(2)(r)
α

, 0≤ α ≤ 1 ,

is also a log-convex interference function

• Let I(n)(r)be a sequence of log-convex interference functions, which verges to a limit limn →∞I(n)(r) = ˆI(r) > 0 for all r > 0, then ˆI is also

con-a log-convex interference function

2.2 Interference Coupling

Interference coupling was well-dened for the specic examples of Section 1.4.For linear interference functions, the coupling between the users is charac-terized by a link gain matrix V ≥ 0, as dened by (1.9) This is a commonapproach in power control theory (see, e.g., [45] and references therein).However, the axiomatic framework A1, A2, A3 does not include the notion

of a coupling matrix It is a priori not clear whether the functions are coupled

2.2.1 Asymptotic Coupling Matrix

Independent of the choice of r, the interference coupling can be characterized

by an asymptotic approach To this end, we introduce el, which is the all-zerovector with the l-th component set to one

[el]n=

(

1 n = l

0 n6= l (2.4)

We have the following result

Lemma 2.1 Assume there exists a ˆr > 0 such that limδ→∞Ik(ˆr + δel) =+∞, then

lim

δ →∞Ik(r + δel) = +∞ for all r > 0 (2.5)Proof Let r > 0 be arbitrary There exists a λ > 0 such that λr ≥ ˆr Thus,A3 implies

lim

δ →∞Ik(λr + δel)≥ limδ

→∞Ik(ˆr + δel) = +∞ (2.6)With A2 we have Ik(λr + δel) = λIk(r +λδel) This implies limδ →∞Ik(r +

δ

λel) = +∞, from which (2.5) follows The interference function Ik is bounded and monotone increasing (axiom A3), thus the existence of the limits

un-is guaranteed uFor arbitrary interference functions satisfying A1-A3, condition (2.5) for-malizes the notion of user l causing interference to user k This enables us

to dene interference coupling by means of a matrix

Denition 2.2 The asymptotic coupling matrix is

on which Ik depends Notice that because of Lemma 2.1, the condition in(2.7) does not depend on the choice of r That is, AI provides a generalcharacterization of interference coupling for interference functions fullling

A1, A2, A3 The matrix AIcan be regarded as a generalization of the link gainmatrix (1.9) commonly used in power control theory In particular, [AI]kl =

1⇔ [V ]kl> 0and [AI]kl= 0⇔ [V ]kl= 0

With AI we dene the dependency set as follows

Denition 2.3 (dependency set) The dependency set L(k) is the indexset of transmitters on which user k depends, i.e.,

L(k) ={l ∈ K : [AI]kl = 1} (2.8)The set is always non-empty because we have ruled out the trivial caseI(r) = 0, ∀r, in our axiomatic interference model (see Section 1.2) AxiomA1 implies that each interference function depends on at least one transmit-ter, i.e the dependency set is non-empty and there is at least one non-zeroentry in each row of AI For some of the following results we need the addi-tional assumption that every column has at least one non-zero empty o themain diagonal This rather natural assumption means that every user causesinterference to at least one other user

2.2.2 The Dependency Matrix

The asymptotic coupling matrix AI is a general way of characterizing ference coupling It is applicable to arbitrary interference functions In thissection we will introduce another concept, namely the global dependency ma-trix DI It will turn out (Theorem 2.6) that DI= AI for the special case oflog-convex interference functions

inter-We begin with a local denition of dependency that depends on the choice

1 if there exists a r > 0 such that

Ik(r + δel)is not constant for somevalues δ > 0,

0 otherwise

(2.10)

2.3 Strict Monotonicity and Strict Log-Convexity 21Later, we will use DI in order to analyze how the interference couplingaects the structure of the boundary.

The following theorem connects AI and DI for the special case of convex interference functions Evidently, [AI]kl = 1 implies [DI]kl = 1, butthe converse is generally not true However, both characterizations are indeedequivalent if the underlying interference functions are log-convex

log-Theorem 2.6 Let I1, ,IK be log-convex interference functions, then bothcharacterizations are equivalent, i.e., AI= DI

Proof The proof is given in the Appendix A.9 u

2.3 Strict Monotonicity and Strict Log-Convexity

Consider an interference function Ik(r)with dependency set L(k) The tion depends on all rlwith l ∈ L(k) However, this does not necessarily meanthat Ik(r)is strictly monotone in these components Strict monotonicity onthe dependency set is a fundamental property, which is often needed to ensureunique solutions to certain optimization problems

func-Denition 2.7 (strict monotonicity) Ik(r)is said to be strictly monotone(on its dependency set L(k)) if for arbitrary r(1), r(2), the inequality r(1)

≥

r(2), with r(1)

l > rl(2) for some l ∈ L(k), implies Ik(r(1)) >Ik(r(2))

In other words, Ik(r)is strictly monotone increasing in at least one powercomponent Strict monotonicity plays a central role in this book, especiallyfor the result on power control

Whenever we address the problem of SINR optimiation in the presence

of power constraints, we can use an interference model I(p) that is based

on an extended power vector p ∈ RK u +1

+ An example was already given inSection 1.4.1 The component pKu+1 stands for the noise power, which isassumed to be equal for all users It will be shown in Section 2.4 that strictmonotonicity with respect to pKu+1 yields a framework which is equivalent toYates' framework of standard interference functions [1] This way, standardinterference functions can be comprehended within the framework A1, A2, A3.Next, we dene strictly log-convex interference functions

Denition 2.8 (strict log-convexity) A log-convex interference function

Ik is said to be strictly log-convex if for all ˆp, ˇp for which there is some

l∈ L(k) with ˆpl6= ˇpl, the following inequality holds

Ik p(λ)

< Ik(ˆ
1 −λ

· Ik(ˇ
λ

, λ∈ (0, 1) (2.11)where p(λ) = ˆp1−λ· ˇpλ

The following lemma shows that strict log-convexity implies strict tonicity

mono-Lemma 2.9 Every strictly log-convex interference function Ik is strictlymonotone on its dependency set (see Denition 2.7).

Proof Consider an arbitrary xed vector p ∈ RK

++, and an arbitrary l ∈ L(k)

We dene

p(l)(x) = p + xel, x > 0 (2.12)and

p(λ) = (p)1−λ· (p(l)(x))λ, λ∈ (0, 1) (2.13)Since l ∈ L(k), strict log-convexity implies

Ik(p(λ)) < (Ik(p))1−λ· (Ik(p(l)(x)))λ (2.14)

By denition (2.13) we have

pv(λ) = pv for all v 6= l (2.15)Also, x > 0 implies

pl(λ) = (pl)1−λ· (pl+ x)λ> pl (2.16)With A3 (monotonicity) we know that p ≤ p(λ) implies Ik(p) ≤ Ik(p(λ)).With (2.14) we have

Ik(p) < (Ik(p))1−λ· (Ik(p(l)(x)))λ,thus

(Ik(p))λ< (Ik(p(l)(x)))λ, (2.17)which shows strict monotonicity uNote that the converse of Lemma 2.9 is not true The following exam-ple shows a strictly monotone interference function which is not strictly log-convex That is, strict monotonicity is weaker than strict log-convexity.Example 2.10 Consider the interference function

2.4 Standard Interference Functions and Power Control 23

2.4 Standard Interference Functions and Power Control

A principal goal of power control is the selection of Ku transmit powers

p∈ RKu

++ to achieve a good system performance Optimization strategies aremostly based on the SIR or the SINR, depending on whether noise is part ofthe model or not Good overviews on classical results are given in [45, 46].Power control in the presence of noise and power constraints is an impor-tant special case of the axiomatic framework A1, A2, A3 The linear interfer-ence function (1.11) in Section 1.4.1 is an example that shows how noise can

be included by means of an extended power vector

p = [p1, , pK u, σ2

n]T

∈ RKu +1 + (2.19)While the impact of noise is easy to model in the case of linear interferencefunctions, it is less obvious for the axiomatic framework A1, A2, A3

In this section we discuss how noise can be included in the axiomaticframework This is closely connected with the concept of standard interferencefunctions The results apppeared in [11]

2.4.1 Standard Interference Functions

Yates [1] introduced an axiomatic framework of standard interference tions

ex-In [1] and related work, the following power control problem is addressed

If these targets are feasible, then the following xed point iteration convergesglobally to the unique optimizer of the power minimization problem (2.20)

p(n+1)k = γkJk(p(n)) , ∀k ∈ Ku, p(0)∈ RKu

+ (2.21)Properties of this iteration were investigated in [1,7,74,75] If a feasible solu-tion exists, then the axioms Y1Y3 ensure global convergence for any initial-ization p(0)

2.4.2 Comparison between Standard Interference Functions andGeneral Interference Functions

It was shown in [11] that standard interference functions can be understood

as a special case of the axiomatic framework A1, A2, A3 This framework isbased on the extended power vector (2.19), and the assumption that I(p) isstrictly monotone in the noise component pKu+1

Denition 2.12 (Strict monotonicity w.r.t noise) An interferencefunction I(p) is said to be strictly monotone with respect to pKu+1 > 0, iffor arbirary given vectors p and p0, with p ≥ p0, we have

pK

u +1> p0K

u +1 ⇒ I(p) > I(p0) (2.22)When comparing the axiomatic framework A1, A2, A3 (cf Section 1.2)with the framework Y1, Y2, Y3, it is observed that the only dierence is be-tween A2 (scale invariance) and Y2 (scalability) In order to establish a linkbetween both frameworks, we introduce the following denition

Denition 2.13 A function J : RK u

+ 7→ R++ is said to be a weakly standardinterference function if the following axiom Y2' is fullled together with Y1(positivity) and Y3 (monotonicity)

Y2' (weak scalability) αJ (p) ≥ J (αp) for all α ≥ 1

The following theorem shows how general interference functions I andstandard interference functions J are related To this end, we introduce thepower set

P =np = p

p

Ku+1

: p∈ RKu

+ , pK

u +1∈ R++

o (2.23)

In a power control context, p is a vector of transmission powers and pKu+1 isthe noise power For notational convenience, we dene I(p) = I(p, pKu+1).The following theorem [11] shows the connection between general and stan-dard interference functions

+ 7→ R+be a general interference function, then for any given

2.4 Standard Interference Functions and Power Control 253) Let IJ be dened as in (2.24) Then J is a standard interference function

if and only if IJ fullls A1, A2, A3, and for all p ∈ RK u

+ , the function

IJ(p, pK

u +1)is strictly monotone in the sense of Denition 2.12

Proof We begin by proving 1) Axiom A1 is fullled because IJ(p) > 0forall p ∈ P Axiom A2 (scale invariance) is fullled because for all λ > 0

It remains to show A3 (monotonicity) Consider two arbitrary vectors p(1), p(2)∈

P such that p(1)≥ p(2) With ˜λ = p(1)

K u +1/p(2)Ku+1 ≥ 1, we have

IJ(p(2)) = p(2)Ku+1· J p(2)1

p(2)Ku+1, , p

(2) Ku

p(2)Ku+1



= p(2)Ku+1· Jλ˜ p

(2) 1

p(1)Ku+1, , ˜λ p

(2) Ku

p(1)Ku+1



≤ p(2)K u +1· Jλ˜ p

(1) 1

p(1)Ku+1, , ˜λ p

(1) Ku

p(1)Ku+1



=IJ(p(1)) The rst inequality follows from Y 3 (monotonicity) and the second from Y 20

(weak scalability)

We now prove 2) Axiom Y3 follows directly from A3 Axiom Y1 holds on

RK++u because I(p) > 0 for all p > 0 This is a consequence of A1, as shown

in [2] Axiom Y2' follows from

J (αp) = I(αp, pKu+1)

≤ I(αp, αpKu+1) = αI(p, pKu+1) = αJ (p) Note that this inequality need not be strict because we did not made anyassumption on whether I depends on pKu+1 or not

We now prove 3) Let J be standard From 1) we know that IJ(p)fulllsA1, A2, A3 We now show strict monotonicity For arbitrary p(1), p(2)

∈ P,with p(1) = p(2) and ˜λ = p(1)

K u +1/p(2)Ku+1 > 1 the second inequality (2.27) isstrict This follows from Y2 (which holds for α > 1 because of continuity).Thus, IJ(p)is strictly monotone with respect to the component pK u +1 Con-versely, let IJ be strictly monotone and axioms A1, A2, A3 are assumed to befullled Then,

J (λp) = IJ(λp, 1) = λI(p,1) < λIJ(p, 1) = λJ (p) (2.28)

for all α > 0, thus Y2 holds Property Y3 follows directly from A3 Finally,

we show Y1 by contradiction Suppose that there exists a p ∈ P such that

J (p) = 0 Strict monotonicity of I implies

0 =J (p) = I(p) > I(αp) = αI(p), 0 < α < 1

Letting α → 0 we obtain a contradiction, thus proving Y1 uTheorem 2.14 shows that any standard interference J (p) can be expressedalternatively as I(p) with pKu+1 = 1 Both frameworks can be used inter-changeably Results that were derived for standard interference functions can

as well be derived by the axioms A1, A2, A3 plus the additional assumption

of strict monotonicity (2.22)

An example is the xed point iteration (2.21), which can be rewritten

in terms of I(p) An alternative convergence proof based on A1, A2, A3 wasgiven in [2]

Another example is the positivity of the functions J (p) It was observed

in [75] that Axiom Y1 (strict positivity) is actually redundant Strict positivity

J (p) > 0 already follows from Y2 and Y3 The same result can be shown forI(p) on the basis of axioms A2 and A3 It was shown in Section 1.2 thatA2 and A3 imply I(p) ≥ 0 for all p ≥ 0 Axiom A1 is only needed to ruleout the trivial case I(p) = 0 for all p With strict monotonicity we do notneed A1 anymore Then there always exists a p > 0 such that I(p) > 0,and hence I(p) > 0 for all p > 0 It is actually sucient that the strictlymonotone component is positive, i.e., pKu+1> 0 Assume an arbitrary p ≥ 0with pK

u +1> 0 The proof is by contradiction: Suppose that I(p) = 0, thenfor any α with 0 < α < 1,

0 =I(p) > I(αp) = αI(p) This would lead to the contradiction 0 = limα→0αI(p) < 0 Hence, strictmonotonicity (2.22) and positive noise pK

u +1 > 0 ensures that I(p) > 0 forarbitrary p ≥ 0

With Theorem 2.14 the result carries over to arbitrary standard ence functions Furthermore, the proof extends to arbitrary strictly monotoneinterference functions as introduced in Section 2.3 Strictly monotone inter-ference functions are positive whenever pk> 0, where pkis the component onwhich the function depends in a strictly monotone way

interfer-2.5 Continuity

Continuity is another fundamental property that will be needed throughoutthis book The following result was shown in [2]

2.5 Continuity 27Lemma 2.15 All interference functions I(r) satisfying A1, A2, A3 are con-tinuous on RK

++ That is, for an arbitrary p∗ ∈ RK

++, and an arbitrary quence p(n)∈ RK

se-++ such that limn →∞p(n)= p∗, the following holds

lim

n →∞I(p(n)) =I(p∗) (2.29)Lemma 2.15 shows continuity only on a restricted domain RK

++ instead of

RK+ That is, we exclude the zeros on the boundary of the set In many cases,this is sucient For example, when dealing with general signal-to-interferenceratios pk/Ik(p), we need to avoid possible singularities By restricting thedomain to RK

++, we ensure that Ik(p) > 0 This technical assumption is theprice we pay for generality of the interference model An example is (1.22),where the inmum is taken over all p > 0

However, interference functions are dened on RK

+ The case pk = 0can beinterpreted as user k being inactive The ability to model inactive users is animportant prerequisite for many resource allocation problems It is thereforedesirable to extend some of the results to RK

+ This motivates the continuationthat is introduced in the following subsection

2.5.1 Continuation on the Boundary

Certain key properties of interference functions are preserved on the boundary[11]

Assume that I(p) is dened on RK

++ Let p(n)

∈ RK ++ be an arbitrarysequence with limit limn →∞p(n)= p∈ RK

+ The interference function I has

a continuation Ic on the boundary, dened on RK

+

Ic(p) = lim

n →∞I(p(n)) (2.30)Certain properties of I are preserved when one or more coordinates pk tend

to zero This result is quite useful because it means that certain results shownfor RK

++ immediately extend to RK

+.The following theorem states that for any interference function, the prop-erties A1, A2, A3 are preserved on the boundary

Theorem 2.16 Let I be an arbitrary interference function dened on RK

++.Then, the continuation Ic(p)dened on RK

+ fullls the axioms A1, A2, A3.Proof We need the lemmas shown in Appendix A.8 Axiom A3 (monotonic-ity) follows from Lemma A.17 Axiom A2 (scale invariance) follows fromLemma A.14 Axiom A1 is also fullled since I(p) = Ic(p)for all p ∈ RK

++ ut

We can use this continuation to extend results that were previously shownfor RK

++to the non-negative domain RK

+ As an example, consider Lemma 2.15,which states continuity on RK

++ This is now extended to RK

+ by the followingtheorem More examples will follow