Convergence of Iterative Waterfilling Algorithm for Gaussian ... - CUHK

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 2007

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Convergence of Iterative Waterfilling Algorithm for Gaussian Interference Channels Kenneth W. Shum, Member IEEE, Kin-Kwong Leung, Member IEEE, and Chi Wan Sung, Member IEEE Abstract— Iterative waterfilling power allocation algorithm for Gaussian interference channels is investigated. The system is formulated as a non-cooperative game. Based on the measured interference powers, the users maximize their own throughput by iteratively adjusting their power allocations. The Nash equilibrium in this game is a fixed point of such iterative algorithm. Both synchronous and asynchronous power update are considered. Some sufficient conditions under which the algorithm converges to the Nash equilibrium are derived. Index Terms— Gaussian interference channel, iterative waterfilling, Nash equilibrium

I. I NTRODUCTION HARING a common frequency band using multiple carriers is a typical scenario in multi-user communication systems. For example, in digital subscriber line, several users are connected to a central office by copper wires. The crosstalk between the wires is known to be the dominant degradation factor. In wireless communication systems, the link quality is limited by multiple-access interference. Interference mitigation is important in both wireline and wireless communications. To this end, dynamic spectrum management plays a central role. The division of the spectrum can be done in many ways. We will mention two of them. In the first method, the spectrum is divided into many narrow frequency bands. This technique is called orthogonal frequency division multiplexing or discrete multitone, and is adopted in the asymmetric digital subscriber line standard and 802.11a wireless LAN standard. In the second method, direct-sequence code-division technique is employed, and each user is assigned a unique code sequence. The amount of inter-user interference is dictated by the cross-correlation between the code sequences. In multi-carrier system, this is extended so that each user is assigned a set of orthogonal sequences. There is no interference between any pair of sequences assigned to the same user, but any pair of sequences from two distinct users may interfere with each other. We model the above systems by Gaussian interference channel. For this channel, the optimal power and coding scheme

S

Manuscript received July 1, 2006; revised February 15, 2007. This work was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project no. CityU 119806). This paper was presented in part at the IEEE International Conference on Communications, Istanbul, Turkey, June 2006. K. W. Shum and C. W. Sung are with the Dept. of Electronic Engineering, City University of Hong Kong (email: [email protected], [email protected]). K.-K. Leung is with Cluster Technology Limited, Units 106-108, Building 9, No. 5 Science Park West Ave, Hong Kong Science Park, Shatin, Hong Kong (email: [email protected]) Digital Object Identifier 10.1109/JSAC.2007.070804.

is unknown. Indeed, the problem of finding the capacity region has been open for many years. The largest rate region currently known is achieved with superposition coding and interference cancelation [1]. For parallel interference channels, the capacity region was found only for some special cases [2], [3]. Performance of different power and coding schemes are compared in [4], [5]. We will follow a different approach in this paper. We minimize the complexity of the transceiver by treating interference as additive Gaussian noise. From a particular user’s viewpoint, the interference channel reduces to a parallel Gaussian channel. For fixed interference power, it is well known that the optimal power allocation is the so-called waterfilling solution. Since any power adjustment of a user will affect the interference towards the others, the users need to iteratively update their powers if there is no central coordination. This algorithm of iteratively adjusting the power allocation by waterfilling is called iterative waterfilling (IW), and is first suggested in [6]. As the users update their powers independently, it is important to ensure that the powers of all users will eventually converge. Sufficient conditions for convergence have been derived under different power update models. If the users take turn in a pre-specified order to update their powers, it is called sequential update. If the users change their powers at the same time, it is called synchronous update. Sufficient conditions for convergence under these two update models are discussed in [7], [8], [9], [10] and [10], [11] respectively. In this paper, we consider a more general model, called the totally asynchronous update model [12], which includes both sequential and synchronous updates as special cases. Under this model, the users may update at different rates, and the relative delay between updates may also vary. The convergence for synchronous update is an essential ingredient in proving convergence for totally asynchronous update. We will present some new sufficient conditions for synchronous update in this paper as well. Due to the lack of coordination, concepts from noncooperative game theory can be naturally applied to the system. The most important one is the Nash equilibrium, which can be interpreted as a limit point of the IW algorithm. Uniqueness of Nash equilibrium is discussed in [8], [9]. It turns out the uniqueness criteria in [8], [9] are also sufficient for convergence. Performance degradation due to the absence of central coordinator is discussed in [13]. Most of the existing works on the convergence of iterative waterfilling algorithm (e.g. [9], [10]) are based on the fact that waterfilling function can be viewed as a projection function, mapping vector of interference powers into the set of all

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feasible power allocations. In this paper, we analyze the waterfilling function in more details, and show that it is piece-wise affine, i.e., the domain can be partitioned into disjoint regions, so that in each region, we can represented the waterfilling function by an affine function. Since we know these affine functions explicitly for each region, more general sufficient convergence conditions can be derived. The paper is organized as follows. Section II describes the system model. In Section III, we show the existence of Nash equilibrium. The iterative waterfilling algorithm for power allocation is introduced in Section IV. Some sufficient conditions for convergence under synchronous and asynchronous update are discussed in V and VI respectively. We consider the special case of parallel interference channel in Section VII, and compare with other convergence conditions in the literature. Some numerical examples are presented in Subsection VII-C. The paper is concluded in Section VIII. II. S YSTEM M ODEL We consider K pairs of communicating terminals, and index both the transmitters and the receivers by K := {1, 2, . . . , K}. We will use the terms users and pairs of communicating terminals interchangeably. Transmitter i sends information only to receiver i. The ith transmitter-receiver pair divides the spectrum into Li orthogonal channels with the same bandwidth, say Wi . We label these Li channels by Li := {1, 2, . . . , Li }. For i, j ∈ K, λ ∈ Li , µ ∈ Lj , let Gλµ ij be the link gain from channel µ of transmitter j to channel λ of receiver i. We assume that the channel is time invariant and the link gains are all constant. Since the Li channels pertaining to user i are orthogonal, the link gain Gλµ ii is zero for λ = µ. In our system model, channels of different users are not assumed to be orthogonal. For example, two adjacent channels of user i may both overlap with a certain channel of user j due to misalignment in spectrum partitioning The noise is additive, white and Gaussian, with power nλi in channel λ of receiver i. We let pλj denote the transmit power of transmitter j in channel λ. The power of interference and noise seen by receiver i at channel λ is equal to

λµ µ Gij pj + nλi . (1) νiλ := j
=i µ∈Lj

The signal to interference plus noise ratio (SINR) of receiver λ λ i in channel λ equals Gλλ ii pi /νi . Without loss of generality, we normalize the link gains and noise powers so that Gλλ ii = 1 for all i ∈ K and λ ∈ Li . The SINR can then be simplified to pλi pλi =   . λ λµ µ λ νi j
=i µ∈L Gij pj + ni j

When all users divide the spectrum in the same fashion — the number of channels are identical, and overlapping channels share the same spectrum — we say that the channel is a parallel Gaussian interference channel. For parallel Gaussian interference channel, we have L1 = . . . = LK , and the link gain Gλµ ij is equal to zero whenever λ = µ. The SINR at channel λ of receiver i is then simplified to pλi λλ λ j
=i Gij pj



+ nλi

.

We put the transmit powers of transmitter i together and form a power vector for user i, i T pi := [p1i , p2i , . . . , pL i ] ,

(2)

which is a vector of dimension Li . Similarly, the interference vector (3) ν i := [νi1 , νi2 , . . . , νiLi ]T is the vector of interference (and noise) power.1 of user i is subject to a total power constraint, The power λ p ≤ p ¯i , and individual power constraint pλi ≤ m ¯ λi , λ∈Li i  λ ¯ i , for all i, in order for all i, λ. We assume that p¯i ≤ λ∈Li m to avoid trivial cases. The set of all feasible power vectors of transmitter i is denoted by Pi ,   
λ λ [0, m ¯i] : pi ≤ p¯i . (4) Pi := pi ∈ λ∈Li

λ∈Li

We consider distributed power allocation problem where transmitter i allocates its power among the Li channels, based on the feedback information from receiver i so as to maximize the total throughput. Each receiver considers the interference from other transmitters as additive white Gaussian noise. If the power allocation is given, the maximal data rate for user i is given by the Shannon formula Ci = Ci (pi , ν i ) := Wi

 pλ  log 1 + iλ . νi λ=1 Li


(5)

III. NASH E QUILIBRIUM A common equilibrium concept in distributed system is the Nash equilibrium, that will be defined as follows. Let P be the Cartesian product P :=

K 

 Pi = (p1 , p2 , . . . , pK ) : pi ∈ Pi .

i=1

In game-theoretic language, P is called the strategic space, and a point in P is called a strategy. The strategic space is the set of all possible configurations in the system. If the power of other users are fixed, then the interference vector ν i is also fixed. User i allocates powers in order to maximize his total throughput Ci (pi , ν i ) over all feasible power vector pi ∈ Pi . The unique optimal power allocation is characterized in the following theorem. ˆ maximizes Ci (pi , ν i ) over Theorem 1: A power vector p all power vectors in Pi if and only if there is a “water-level” ω ≥ 0 so that ⎧ ⎪ if ω − νiλ ≤ 0 ⎨0 λ pˆi = m (6) if ω − νiλ ≥ m ¯ λi ¯ λi ⎪ ⎩ λ ω − νi otherwise, for all λ ∈ Li , and




pλi = p¯i .

λ∈Li

In the case with no power constraint on each channel, Theorem 1 is proved in [14], [15]. We can prove Theorem 1 by adapting their proofs; details are omitted. 1 We

use the notation [· · · ] for row vector and [· · · ]T for column vector.

SHUM et al.: CONVERGENCE OF ITERATIVE WATERFILLING ALGORITHM FOR GAUSSIAN INTERFERENCE CHANNELS

We define the waterfilling function as i T ¯ i ) := [ˆ p1i , . . . , pˆL f (ν i ; p¯i , m i ] ,

which is a function of interference vector ν i , total power i ¯ i := [m constraint p¯i and individual power constraint m ¯ λi ]L λ=1 . It describes how to set the power allocation optimally, given that the powers of other terminals are fixed, and is thus also called the best-response function. Given the link gain matrices and noise powers, a power allocation p∗ = (p∗1 , . . . , p∗K ) ∈ P is called a Nash equilibrium or an equilibrium point if the following holds for all i = 1, . . . , K, Ci (p∗i , ν ∗i ) ≥ Ci (pi , ν ∗i )

for all pi ∈ Pi

where ν ∗1 , . . . , ν ∗K are the interference vectors corresponding to the power allocation p∗1 , . . . , p∗K , or equivalently, ¯ i ). p∗i = f (ν ∗i ; p¯i , m At a Nash equilibrium, given that the other terminals fix their power allocations, no terminal can further increase the data rate unilaterally, i.e., no terminal has incentive to change his power allocation at a Nash equilibrium. The absence of Nash equilibrium means that the distributed system is inherently unstable. The existence of Nash equilibrium in our system is a corollary to a fundamental theorem in game theory and mathematical economics, due to Debreu [16], Fan [17] and Glicksberg [18]. In our notation, it says, Theorem 2: If for each i = 1, . . . , K, (i) Pi is compact and convex, (ii) Ci : P1 × · · · × PK → R is continuous in (p1 , . . . , pK ), and (iii) Ci is concave in pi for any given p1 , . . . , pi−1 , pi+1 , . . . , pK , then Nash equilibrium exists. It is straightforward to show that all three conditions in the theorem are satisfied. Hence, we have at least one Nash equilibrium. The strategy in Theorem 2 is deterministic. In general, we can randomize the strategies, and model the powers as random variables. A randomized strategy is called a mixed strategy. The randomization induces a probability measure on the interference powers. Transmitter i maximizes the expected value Epi ,ν i [Ci (pi , ν i )] where the expectation is taken over pi and ν i . Since Ci (pi , ν i ) is strictly concave in pi for any ν i , the expectation Eν i [Ci (pi , ν i )] taken over ν i only is strictly concave function in pi . No matter what the distribution of ν i is, there is a unique deterministic power allocation that maximize Epi ,ν i [Ci (pi , ν i )]. Therefore, there is no advantage in using mixed strategy. In the remaining of this paper, we will only consider deterministic strategy. IV. S YNCHRONOUS ITERATIVE WATERFILLING ALGORITHM

In this section, we introduce the iterative waterfilling power allocation algorithm, where each terminal updates its power vector using the waterfilling function. In the synchronous IW algorithm (SIWA), all transmit terminals adjust their power allocations simultaneously, according

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to the waterfilling function. We let F : P → P denote the function  K ¯ i i=1 , F (p1 , . . . , pK ) := f (ν i ; p¯i , m) where ν i is the interference vector computed as in (1) and (3). (t) (t) SIWA updates the power allocations (p1 , . . . , pK ) at time t by (t+1) (t+1) (t) (t) , . . . , pK ) = F (p1 , . . . , pK ). (7) (p1 If SIWA converges to a point such that F (p1 , . . . , pK ) = (p1 , . . . , pK ), then it converges to a Nash equilibrium. In order to derive some convergence criteria for SIWA, we will investigate some basic properties of the function F in the remaining of this section. Theorem 3: The function F is a continuous function mapping from P to P. Proof: Since ν i is a continuous function of p1 , . . . , pK , it suffices to show that the waterfilling function f is continuous as a function of ν i . We will use the following “maximum theorem” [19, p.116]. (Maximum theorem) Let φ(x, y) be a real-valued continuous function with domain X × Y , where X ⊂ Rm and Y ⊂ Rn are closed and bounded sets. Suppose that φ(x, y) is strictly concave in x for each y. The functions M (y) = max{φ(x, y) : x ∈ X} and Φ(y) = arg max{φ(x, y) : x ∈ X} are welldefined for all y ∈ Y , and are both continuous. Apply the maximum theorem with φ = Ci (pi , ν i ). The waterfilling function is the function Φ in the theorem, and hence is a continuous function of ν i . The function f and F in SIWA are non-linear functions. We will show that the non-linearity is in fact piecewise linearity. This means that we can partition the domain into disjoint regions, and in each region the function behaves as an affine function. We first define piecewise affine function formally. In a finite-dimensional vector space V , a half space is a subset {x ∈ V : aT x + b ≥ 0} for some vector a (a = 0) and constant b. A polyhedron is the intersection of some finite collection of half spaces. A function g mapping from a domain in V to vector space W is called piecewise affine if the domain of g can be partitioned into finitely many polyhedra, say R1 , . . . , RS , such that for σ = 1, . . . , S, the function g restricted to the region Rσ is equal to an affine function, g(x) = Aσ x + cσ ,

for all x ∈ Rσ ,

for some suitablechoice of matrix Aσ and vector cσ . ˜ denote K Li . We concatenate the power vectors Let L i=1 ˜ to form a L-dimensional column vector, p := [pT1 . . . pTK ]T . The first L1 components of p are the powers of user 1. The second block of L2 components are the powers of user 2, etc. We call it the system power vector.2 The system interference 2 We will treat P as a subset of an L-dimensional ˜ vector space, and say that p ∈ P.

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straightforward to check that  

νiλ − m ¯ λi /|Ni |, ω = p¯i + λ∈Ni

ω

⎧ ⎪ ⎨0 pλi = m ¯ λi ⎪ ⎩ ω − νiλ

(9)

λ∈Si

for λ ∈ Li \ (Ni ∪ Si ) for λ ∈ Si for λ ∈ Ni .

(10)

satisfy the conditions in Theorem 1 and hence is the optimal solution. The solution can be expressed in matrix form as, pi = f (ν i ; p¯i , m ¯ i ) = W (Ni )ν i + bi (Ni , Si ), 1

2

3

4

5

6

7

8

Fig. 1. An example of waterfilling solution for 8 parallel channels. The shaded area is the allocated power. The empty area is the interference power. The water level is indicated by ω. Channel 4 is saturated, while channel 7 and 8 are inactive.

vector is the concatenation of the interference vectors in the same ordering as in the system power vector, ν := [ν T1 . . . ν TK ]T . The system interference vector can be obtained by the system power vector by ν = Gp + n,

(8)

where n is the column vector

˜ L ˜ matrix whose entries are the link gains (and and G is an L× zeros). The matrix G can be interpreted as the system gain matrix.For example, when K = 3 and T L = 2, the interference vector ν11 ν12 ν21 ν22 ν31 ν32 equals ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0 0 G11 21 G21 21 G11 31 G21 31

0 0 G12 21 G22 21 G12 31 G22 31

G11 12 G21 12 0 0 G11 32 G21 32

G12 12 G22 12 0 0 G12 32 G22 32

where bi (Ni , Si ) is an Li -dimensional column vector, and W (Ni ) is the Li × Li matrix whose (k, )-entry equals ⎧ ⎪ if k ∈ Ni or  ∈ Ni , ⎨0 [W (Ni )]k := 1/|Ni | if i, j ∈ Ni and i = j, ⎪ ⎩ −1 + 1/|Ni | if i, j ∈ Ni and i = j. The vector bi can be obtained from (9) and (10). It is noted that the submatrix of W (Ni ) obtained by retaining rows and columns indexed by Ni is equal to (1/|Ni |)J −I, where J and I are the |Ni | × |Ni | all-one matrix and identity matrix respectively. The next lemma gives a characterization of the optimal solution. Lemma 4: Let Si and Ni be any two distinct subsets of Li . The followings are equivalent. 1) The interference powers satisfy νiλ + m ¯ λi ≤ ω, for λ ∈ Si ,

L2 LK T 1 1 1 n := [n11 , . . . , nL 1 , n 2 , . . . , n2 , . . . , nK , . . . , nK ] ,

⎡

(11)

G11 13 G21 13 G11 23 G21 23 0 0

G12 13 G22 13 G12 23 G22 23 0 0

⎤ ⎡ 1⎤ ⎡ 1⎤ p1 n1 ⎥ ⎢p21 ⎥ ⎢n21 ⎥ ⎥ ⎢ 1⎥ ⎢ 1⎥ ⎥ ⎢p2 ⎥ ⎢n2 ⎥ ⎥ ⎢ 2⎥ + ⎢ 2⎥ . ⎥ ⎢p2 ⎥ ⎢n2 ⎥ ⎥ ⎢ 1⎥ ⎢ 1⎥ ⎦ ⎣p3 ⎦ ⎣n3 ⎦ p23 n23

In general, the matrix G is a partitioned matrix with zero diagonal blocks. The (i, j)-block is an Li × Lj matrix whose (λ, µ)-entry is Gλµ ij . We now look more closely at the waterfilling function f for one transmitter-receiver pair. In the remaining of this section, i is a fixed integer in K. We say that channel λ is active if the optimal power in channel λ is non-zero. A channel is saturated if the associated power equals the upper bound m ¯ λi . An example is illustrated in Fig. 1. Suppose that the interference vector ν i of transmitter i is given, and the resulting optimal power vector is pi . Furthermore, suppose that Si is the set of all saturated channels, and Ni is the set of all active channels that are not saturated. It is

νiλ < ω < νiλ + m ¯ λi for λ ∈ Ni ,

ω ≤ νiλ , for λ ∈ Li \ (Si ∪ Ni ).

where ω is computed as in (9). 2) The ω and p calculated by (9) and (10) is the optimal solution to max Ci (pi , ν i ) subject to pi ∈ Pi , so that Si is the set of saturated channels and Ni is the set of active but not saturated channel. Proof: We have already shown that the second statement implies the first one. For the converse, when the inequalities in the first statement are valid, then by Theorem 1, the channels in Si are indeed saturated and channels in Ni are positive but not saturated. The graph of waterfilling function f for two channels is plotted in Fig. 2. Theorem 5: The waterfilling function F in SIWA is piecewise affine. The domain of function F can be partitioned into finitely many polyhedral regions, and in each region, F (p) is equal to an affine function AGp + b, where A = diag(W (N1 ), W (N2 ), . . . , W (NK )),

(12)

˜ ×L ˜ block diagonal matrix, for some choice of subsets is an L Ni ⊆ Li , i = 1, . . . , K. Proof: If we pick two disjoint subsets Ni and Si of Li , for i = 1, . . . , K, the inequalities in Lemma 4 can be translated to inequalities with variables in p using (8), and define a (possibly empty) polyhedron in P. The restriction of

SHUM et al.: CONVERGENCE OF ITERATIVE WATERFILLING ALGORITHM FOR GAUSSIAN INTERFERENCE CHANNELS

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of vector norm: the 1 norm, x 1 :=

Optimal power in channel 1

1.5

the 2 norm, x 2 :=

1

n


|xi |,

i=1 n 


x2i

1/2

,

i=1

the ∞ norm, x ∞ := max |xi |. i=1,...,n

0.5

0 3 3

2

M := max{ M x : x = 1},

2

1 ν2

It is clear that the distance function defined by d(x, y) := x − y is indeed a metric. Given any vector norm · on Rn , the matrix norm defined on the set of all n × n matrices by

1 0

0

ν1

Fig. 2. A two-channel waterfilling function. The maximum total power constraint is equal to 1, and there is no individual power constraint The xand y-axis are the interference power in channel 1 and 2 respectively. The vertical axis is the optimal power allocated to channel 1. The optimal power for channel 2 can be obtained using the fact that sum of of powers equals 1.

every component of F on this polyhedron is an affine function by Lemma 4. So F is affine when restricted to this polyhedron. The number of choices of disjoint Ni and Si , i = 1, . . . , K is finite. So the domain is partitioned into finitely many polyhedra.

V. C RITERIA FOR GLOBAL CONVERGENCE OF SIWA The criteria that we will present are based on the Banach contraction theorem (see for example [20, p.220]). Let (X, d) be a metric space with metric d, and g be a function mapping X into itself. A point x ∈ X is called a fixed point of g if g(x) = x. If the sequence of points defined recursively by xn+1 = g(xn ) converges to x regardless of the choice of the initial point x0 , then we say that x is globally asymptotically stable. If there is a number α < 1 such that d(g(x), g(x )) ≤ αd(x, x ) for all x, x ∈ X, then g is called a contraction map or a contraction. Theorem 6 (Banach contraction theorem): If (X, d) is a complete metric space and g : X → X is a contraction, then g has a unique fixed point x∗ that is globally asymptotically stable. We will show in this section that, under some conditions, the function F in the synchronous IW algorithm is a contraction map. Hence, we have a unique Nash equilibrium and the algorithm converges for any initial condition. We first review some basic notions of vector norm and matrix norm [21, chapter 5]. Let Rn denote the n-dimensional real vector space, and · a vector norm on Rn . We have the following three examples

is called the matrix norm induced by the vector norm · .3 The matrix norm induced by the 1 vector norm is the maximum column sum matrix norm, n
M 1 := max |mij |. j=1,...,n

i=1

The matrix norm induced by the ∞ vector norm is the maximum row sum matrix norm, n
M ∞ := max |mij |. i=1,...,n

j=1

The spectral norm defined by √ M 2 := max λ : λ is an eigenvalue of M T M is the matrix norm induced by the 2 vector norm. The following theorem is the main theorem in this paper. A sufficient condition for global stability is proved for a general matrix norm, and we apply the general condition to some special matrix norms, which are computationally tractable. Theorem 7: Let · be the matrix norm induced by a vector norm · . Suppose that the domain of function F is partitioned into polyhedral regions R1 , . . . , RS , so that in each region, say Rσ , we have F (p) = M σ p + bσ , for some matrix M σ and vector bσ . If M σ < 1 for all σ = 1, . . . , S, then F is a contraction. In particular, we have (i) the synchronous IW algorithm converges for any initial power allocation, (ii) there is a unique fixed point p∗ , (iii) p(t) − p∗ ≤ αt p(0) − p∗ , where α is maxσ { M σ }. Proof: We will show that F is a contraction. Global stability will follow from the Banach contraction theorem. Suppose that p and p are in the same region, say Rσ , then F (p ) − F (p) = M σ (p − p). Since M x ≤ M · x , we obtain F (p ) − F (p) ≤ M σ · p − p ≤ α p − p .

(13)



Suppose that p and p belong to different regions. We connect p and p by a straight line, and by the convexity of the set of feasible power vectors, all points on this straight line are feasible power vectors. The straight line is parametrically described by p + β(p − p) for β ∈ [0, 1]. As we increase β from 0 to 1, we go across the boundary of the regions. 3 We

will use the same notation  ·  for both vector and matrix norm.

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The number of boundary crossings is finite since any straight line crosses a region at most two times and there are finitely many regions. Suppose that we cross the boundary at 0 < β1 < β2 < . . . < βm < 1. For notational simplicity, we assume that the straight line starts from R1 , and goes through R2 , R3 , . . . , Rm+1 . Also we let β0 = 0, βm+1 = 1, and ∆ = p − p. So m
  F (p + βj+1 ∆) − F (p + βj ∆) . F (p ) − F (p) = j=0

Taking the vector norm · on both sides and using the triangular inequality, we have F (p ) − F (p) ≤

m


F (p + βj+1 ∆) − F (p + βj ∆) .

j=0

Since F (p + βj+1 ∆) and F (p + βj ∆) are in the same region (on the boundary), by (13), F (p + βj+1 ∆) − F (p + βj ∆) ≤ α(βj+1 − βj ) ∆ for j = 0, . . . , m. Whence F (p ) − F (p) ≤

m


α(βj+1 − βj ) ∆ ≤ α p − p .

j=0

This proves that F is a contraction. In SIWA, the number of regions that partition the domain of F is very large, and it is impractical to check the conditions in Theorem 7 directly for all regions. We will derive some conditions which are easy to apply. Theorem 8: Let Lmax be the maximum of L1 , . . . , LK . If the system link gain matrix satisfies any one of the following, Lmax , 2(Lmax − 1) Lmax Condition A∞ : G ∞ < , 2(Lmax − 1) Condition A2 : G 2 < 1, Condition A1 : G 1
0 such that M w ∞ < 1. Proof: See [12] We consider the case where L1 = . . . = LK = L. The system gain matrix G is partitioned into a K ×K block matrix, in which each block has dimension L × L. Let the (k, )-th ¯q block be denoted by H k . We define a K × K matrix H whose (i, j)-entry is the matrix norm H k q , where · q ¯ q is a nonmay be the 1 , 2 or ∞ norm. It is noted that H negative matrix with zero diagonal. Theorem 10: Suppose that L1 = . . . = LK = L. If any one of the following ¯ 1 ) < L/(2(L − 1)) Condition B1 : ρ(H ¯ ∞ ) < L/(2(L − 1)) Condition B∞ : ρ(H ¯ 2) < 1 Condition B2 : ρ(H

(19) (20) (21)

holds, then F is a contraction under some suitably defined vector norm, and SIWA converges globally to a unique fixed point. Proof: We will only prove condition B2 . The proofs of the other two conditions are similar. For p = (p1 , . . . , pK ), we define θw (p) := ( p1 2 , . . . , pK 2 ) w ∞. We can check that θw is a vector norm. The “outer” norm is a weighted ∞ norm for some w > 0, while the “inner” norm is the 2 norm.

SHUM et al.: CONVERGENCE OF ITERATIVE WATERFILLING ALGORITHM FOR GAUSSIAN INTERFERENCE CHANNELS

By Theorem 7, it suffices to show that the induced matrix norm of AG, where A is defined as in (12), is strictly less than 1 for all possible choices of Ni ’s. The vector AGp is partitioned into K blocks, each of length L; the ith block is equal to K
W (Nj )H ij pj . j=1

Recall that the spectral radius of W (Nj ) is 1 for any subset Nj of Lj . So, K K  

  W (Nj )H ij pj  ≤ H ij 2 pj 2 .  j=1

2

j=1

¯ 2 p , where p The right hand side is the ith component of H is the K-dimensional vector p := ( p1 2 , . . . , pK 2 ). Therefore,

¯ 2 p w θw (AGp) ≤ H ∞.

¯ 2 ) < 1, then by Lemma 9, we can choose w so that If ρ(H ¯ H2 w ∞ < 1. As a result, we have
w ¯ 2 w ¯ w w θw (AGp) ≤ H ∞ p ∞ = H 2 ∞ θ (p),

for all p. Hence the norm of AG induced by θw is strictly less than 1. VI. A SYNCHRONOUS I TERATIVE WATERFILLING A LGORITHM The synchronization required in SIWA may not be available in practice. The users may update at different times, and may even update at different rates. In order to capture these ingredients, we introduce the totally asynchronous model [12] as follows. (τ ) Let pi be the power vector of user i at time τ . Suppose that when user i adjusts his power at time t, he only has delayed information about other users. At time t, the power (τ i (t)) allocation of transmitter j available to transmitter i is pj j , where 0 ≤ τji (t) ≤ t. The waterfilling function for transmitter i at time t is based on the system power vector  (τ i (t)) K i . p(τ (t)) := pj j j=1 Without loss of generality, we assume that the transmitters will update their power vectors only at the discrete time set T = {0, 1, 2, . . .}. Let Ti ⊆ T be the set of time instants when transmitter i adjusts its power. Given the sets T1 , . . . , TK , the asynchronous iterative waterfilling algorithm (AIWA) is defined by  i ¯ i ), t ∈ Ti , f (Gp(τ (t)) + n; p¯i , m (t+1) = pi (22) (t) pi , otherwise. We assume limt→∞ τji (t) = ∞ for all 1 ≤ i, j ≤ K, which guarantees that old information is eventually purged from the system. The totally asynchronous model includes the sequential and synchronous update as special cases. When Ti = {i, i+ K, i +

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2K, . . . , } for all i, we have the sequential update. When Ti = T for all i, we have the synchronous update. In order to derive sufficient conditions for the convergence of AIWA, we will apply the asynchronous convergence theorem (ACT) [12]. Theorem 11 (Asyn. convergence theorem): If there is a sequence of nonempty sets {X(n)} of P, n = 0, 1, 2, . . . , satisfying the following conditions: 1) (Box Condition) For every n, X(n) = X1 (n)×X2 (n)× · · · × XK (n) for some subsets Xi (n) ⊆ Xi . 2) (Inclusion Condition) X(0) ⊇ X(1) ⊇ X(2) ⊇ . . . ⊇ {q}, and all sequences {xk } such that xk ∈ X(k) for all k converge to q. 3) (Synchronous Convergence Condition) For all n and x ∈ X(n), F (x) ∈ X(n + 1). then the sequence obtained by the totally asynchronous update will converge to q provided that the initial condition is in X(0). A proof of this theorem can be found in [22]. Theorem 12: The condition A∞ , B1 , B2 and B∞ are sufficient conditions for global convergence of AIWA. Proof: We will prove that if condition B2 holds, then AIWA will converge for any initial power allocation. The proofs of the other conditions are similar and omitted. Suppose that L1 = . . . = LK = L and condition B2 is true, by Theorem 10, there exists a unique fixed point, and the SIWA converges. We want to define a sequence of set X(k) that satisfies the three conditions in ACT. We use the notation in the proof of Theorem 10. There is ¯ 2 w ¯ 2 ) ≤ H a weight vector w so that ρ(H ∞ =: α < 1. Let ∗ ∗T T ] . Define the fixed point of F be p = [p1 . . . p∗T K  X(n) := p ∈ P : θw (p − p∗ ) ≤ αn θw (p(0) − p∗ ) . K X(n) is equal to the cartesian product i=1 Xi (n), where  Xi (n) := pi ∈ Pi : pi − p∗i 2 ≤ wi αn θw (p(0) − p∗ ) . This implies the box condition. The inclusion condition holds because α < 1. By Theorem 10, F is a contraction under the metric induced by θw . The synchronous convergence condition follows from the properties of the contraction map.

VII. PARALLEL G AUSSIAN I NTERFERENCE C HANNEL We consider the case of parallel interference channel in this section. Each user partitions the spectrum into L channels in the same fashion. The (i, j)-th block in the system gain matrix is a diagonal matrix, 11 22 LL diag(gij , gij , . . . , gij ), λ instead for i = j. For notational simplicity, we will write gij λλ of gij . The system can also be specified by the channel gain matrices Gλ , λ = 1, . . . , L, where  Gλij for i = j λ [G ]ij := 0 otherwise.

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A. New Convergence Conditions Theorem 13: For parallel interference channels, if the channel gain matrices satisfy any one of the following, L ∀λ, 2(L − 1) L Condition C∞ : Gλ ∞ < ∀λ, 2(L − 1) Condition C2 : Gλ 2 < 1 ∀λ, Condition C1 : Gλ 1