SHARING a common frequency band using multiple carriers

Transcription

1 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 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. INTRODUCTION SHARING 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 a 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 Manuscript received July 1, 2006; revised February 15, 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 ). This paper was presented in part at the IEEE International Conference on Communications, Istanbul, Turkey, June K. W. Shum and C. W. Sung are with the Dept. of Electronic Engineering, City University of Hong Kong ( kshum2007@gmail.com, albert.sung@cityu.edu.hk). K.-K. Leung is with Cluster Technology Limited, Units , Building 9, No. 5 Science Park West Ave, Hong Kong Science Park, Shatin, Hong Kong ( kinkwong@alumni.cuhk.net) Digital Object Identifier /JSAC /07/$25.00 c 2007 IEEE 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

2 1092 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 2007 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. SYSTEM MODEL 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. Theith transmitter-receiver pair divides the spectrum into L i orthogonal channels with the same bandwidth, say W i. We label these L i channels by L i := {1, 2,...,L i }. For i, j K, λ L i, µ L j,letg λµ 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 L i channels pertaining to user i are orthogonal, the link gain G λµ 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 n λ i in channel λ of receiver i. Weletp λ j 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 νi λ := G λµ ij pµ j + nλ i. (1) j i µ L j The signal to interference plus noise ratio (SINR) of receiver i in channel λ equals G λλ ii pλ i /νλ i. Without loss of generality, we normalize the link gains and noise powers so that G λλ ii =1 for all i Kand λ L i. The SINR can then be simplified to p λ i p λ i νi λ = j i µ L j G λµ ij pµ j +. nλ i 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 L 1 =... = L K, and the link gain G λµ ij is equal to zero whenever λ µ. TheSINRat channel λ of receiver i is then simplified to p λ i j i Gλλ ij pλ j +. nλ i We put the transmit powers of transmitter i together and form a power vector for user i, p i := [p 1 i,p2 i,...,pli i ] T, (2) which is a vector of dimension L i. Similarly, the interference vector ν i := [νi 1,ν2 i,...,νli i ] T (3) is the vector of interference (and noise) power. 1 The power of user i is subject to a total power constraint, λ L i p λ i p i, and individual power constraint p λ i m λ i, for all i, λ. We assume that p i λ L i m λ i,foralli, inorder to avoid trivial cases. The set of all feasible power vectors of transmitter i is denoted by P i, { P i := p i } [0, m λ i ]: p λ i p i. (4) λ L i λ L i We consider distributed power allocation problem where transmitter i allocates its power among the L i 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 L i ( ) C i = C i (p i, ν i ):=W i log 1+ pλ i νi λ. (5) λ=1 III. NASH EQUILIBRIUM A common equilibrium concept in distributed system is the Nash equilibrium, that will be defined as follows. Let P be the Cartesian product K P := P i = { } (p 1, p 2,...,p K ): p i P i. 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 C i (p i, ν i ) over all feasible power vector p i P i. The unique optimal power allocation is characterized in the following theorem. Theorem 1: A power vector ˆp maximizes C i (p i, ν i ) over all power vectors in P i if and only if there is a water-level ω 0 so that 0 if ω νi λ 0 ˆp λ i = m λ i if ω νi λ m λ i (6) ω νi λ otherwise, for all λ L i,and λ L i p λ i = p i. 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.

3 SHUM et al.: CONVERGENCE OF ITERATIVE WATERFILLING ALGORITHM FOR GAUSSIAN INTERFERENCE CHANNELS 1093 We define the waterfilling function as f(ν i ; p i, m i ):=[ˆp 1 i,...,ˆp Li i ] T, which is a function of interference vector ν i, total power constraint p i and individual power constraint m i := [ m λ i ]Li 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, C i (p i, ν i ) C i(p i, ν i ) for all p i P i where ν 1,...,ν K are the interference vectors corresponding to the power allocation p 1,...,p K, or equivalently, p i = f(ν i ; p i, m i ). 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)P i is compact and convex, (ii) C i : P 1 P K R is continuous in (p 1,...,p K ), and (iii) C i is concave in p i for any given p 1,...,p i 1, p i+1,...,p K, 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 E pi,ν i [C i (p i, ν i )] where the expectation is taken over p i and ν i.sincec i (p i, ν i ) is strictly concave in p i for any ν i, the expectation E νi [C i (p i, ν i )] taken over ν i only is strictly concave function in p i. No matter what the distribution of ν i is, there is a unique deterministic power allocation that maximize E pi,ν i [C i (p i, ν i )]. Therefore, there is no advantage in using mixed strategy. In the remaining of this paper, we will only consider deterministic strategy. IV. SYNCHRONOUS 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 to the waterfilling function. We let F : P P denote the function F (p 1,...,p K ):= ( f(ν i ; p i, m) i ) K i=1, where ν i is the interference vector computed as in (1) and (3). SIWA updates the power allocations (p (t) 1,...,p(t) K ) at time t by (p (t+1) 1,...,p (t+1) K )=F(p(t) 1,...,p(t) K ). (7) If SIWA converges to a point such that F (p 1,...,p K )=(p 1,...,p K ), 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 p 1,...,p K, 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)letφ(x, y) be a real-valued continuous function with domain X Y,whereX R m and Y R n 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 = C i (p i, 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,ahalf space is a subset {x V : a T x + b 0} for some vector a (a 0) and constant b. Apolyhedron 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 R 1,...,R S, such that for σ =1,...,S, the function g restricted to the region R σ is equal to an affine function, g(x) =A σ x + c σ, for all x R σ, for some suitable choice of matrix A σ and vector c σ. Let L denote K i=1 L i. We concatenate the power vectors to form a L-dimensional column vector, p := [p T 1...pT K ]T. The first L 1 components of p are the powers of user 1. The second block of L 2 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.

4 1094 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 2007 ω 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 T 1...ν T K] T. The system interference vector can be obtained by the system power vector by ν = Gp + n, (8) where n is the column vector n := [n 1 1,...,nL1 1,n1 2,...,nL2 2,...,n1 K,...,nLK K ]T, and G is an L L matrix whose entries are the link gains (and zeros). The matrix G can be interpreted as the system gain matrix. For example, when K =3and L =2, the interference vector [ ν 1 1 ν 2 1 ν 1 2 ν 2 2 ν 1 3 ν 2 3] T equals 0 0 G G G G G G G G G G G G G G G G G G G G G G G G p 1 1 p 2 1 p 1 2 p 2 2 p 1 3 p n 1 1 n 2 1 n 1 2 n 2 2 n 1 3 n 2 3. In general, the matrix G is a partitioned matrix with zero diagonal blocks. The (i, j)-block is an L i L j matrix whose (λ, µ)-entry is G λµ 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 λ i. 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 p i.furthermore, suppose that S i is the set of all saturated channels, and N i is the set of all active channels that are not saturated. It is straightforward to check that ω = ( p i + νi λ ) m λ i / N i, (9) λ N i λ S i 0 for λ L i \ (N i S i ) p λ i = m λ i for λ S i (10) ω νi λ for λ N i. satisfy the conditions in Theorem 1 and hence is the optimal solution. The solution can be expressed in matrix form as, p i = f(ν i ; p i, m i )=W (N i )ν i + b i (N i, S i ), (11) where b i (N i, S i ) is an L i -dimensional column vector, and W (N i ) is the L i L i matrix whose (k, l)-entry equals 0 if k N i or l N i, [W (N i )] kl := 1/ N i if i, j N i and i j, 1+1/ N i if i, j N i and i = j. The vector b i can be obtained from (9) and (10). It is noted that the submatrix of W (N i ) obtained by retaining rows and columns indexed by N i is equal to (1/ N i )J I, where J and I are the N i N i all-one matrix and identity matrix respectively. The next lemma gives a characterization of the optimal solution. Lemma 4: Let S i and N i be any two distinct subsets of L i. The followings are equivalent. 1) The interference powers satisfy νi λ + mλ i ω, for λ S i, νi λ <ω<νi λ + mλ i for λ N i, ω νi λ, for λ L i \ (S i N i ). where ω is computed as in (9). 2) The ω and p calculated by (9) and (10) is the optimal solution to max C i (p i, ν i ) subject to p i P i,sothat S i is the set of saturated channels and N i 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 S i are indeed saturated and channels in N i 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 (N 1 ), W (N 2 ),...,W (N K )), (12) is an L L block diagonal matrix, for some choice of subsets N i L i, i =1,...,K. Proof: If we pick two disjoint subsets N i and S i of L i,fori =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

5 SHUM et al.: CONVERGENCE OF ITERATIVE WATERFILLING ALGORITHM FOR GAUSSIAN INTERFERENCE CHANNELS 1095 Optimal power in channel ν Fig. 2. A two-channel waterfilling function. The maximum total power constraint is equal to 1, and there is no individual power constraint The x- and 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 N i and S i, i =1,...,K is finite. So the domain is partitioned into finitely many polyhedra. V. CRITERIA 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, andg 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 x n+1 = g(x n ) converges to x regardless of the choice of the initial point x 0, then we say that x is globally asymptotically stable. If there is a number α<1 such that 1 ν 1 d(g(x),g(x )) αd(x, x ) for all x, x X, theng 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 R n denote the n-dimensional real vector space, and a vector norm on R n. We have the following three examples 2 3 of vector norm: the l 1 norm, x 1 := n x i, i=1 ( n ) 1/2, the l 2 norm, x 2 := x 2 i the l norm, x := max x i. i=1,...,n It is clear that the distance function defined by d(x, y) := x y is indeed a metric. Given any vector norm on R n, the matrix norm defined on the set of all n n matrices by i=1 M := max{ Mx : x =1}, is called the matrix norm induced by the vector norm. 3 The matrix norm induced by the l 1 vector norm is the maximum column sum matrix norm, n M 1 := max m ij. j=1,...,n i=1 The matrix norm induced by the l vector norm is the maximum row sum matrix norm, n M := max m ij. 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 l 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 R 1,...,R S, so that in each region, say R σ,wehavef(p) =M σ p+b σ, for some matrix M σ and vector b σ.if M σ < 1 for all σ =1,...,S,thenF 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 α t p (0) p, where α is max σ { 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 R σ,then F (p ) F (p) =M σ (p p). Since Mx 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.

6 1096 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 2007 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 < β 1 < β 2 <... < β m < 1. For notational simplicity, we assume that the straight line starts from R 1, and goes through R 2, R 3,...,R m+1.alsoweletβ 0 = 0, β m+1 = 1,and = p p. So m F (p ( ) F (p) = F (p + βj+1 ) F (p + β j ) ). j=0 Taking the vector norm on both sides and using the triangular inequality, we have m F (p ) F (p) F (p + β j+1 ) F (p + β j ). j=0 Since F (p + β j+1 ) and F (p+β j ) are in the same region (on the boundary), by (13), F (p + β j+1 ) F (p + β j ) α(β j+1 β j ) for j =0,...,m. Whence m F (p ) F (p) α(β j+1 β j ) 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 L max be the maximum of L 1,...,L K.If the system link gain matrix satisfies any one of the following, L max Condition A 1 : G 1 < 2(L max 1), (14) L max Condition A : G < 2(L max 1), (15) Condition A 2 : G 2 < 1, (16) then the SIWA converges to the unique fixed point regardless of the initial condition, noise powers and maximum power constraints. Remarks: The intuition behind condition A 1 is that no transmitter generates excessive interference to others, while the intuition for condition A is that no receiver experiences excessive interference. Proof: We use the notation in the proof of Theorem 7. Condition A 1.Forσ = 1,...,S, the matrix M σ is the product A σ G where A σ is the matrix defined in (12). We will use the property of matrix norm that AB A B, and the following Claim: A σ 1 2(L max 1)/L max for all σ, (17) i.e., the maximum column sum of A σ is no greater than 2(L max 1)/L max. Assuming that the claim is true, we have A σ G 1 2(L max 1) G 1. L max If G 1 <L max /(2(L max 1)), then A σ G 1 < 1 for all i. By Theorem 7, we see that F is a contraction map if the condition in (i) is satisfied. We now prove the claim in (17). Since A σ is block diagonal, we only need to upper bound the column sums in each block of A σ. If we remove the zero columns and zero rows in A σ, we will obtain a square matrix W m := (1/m)J I, where J and I are the m m all-one matrix and the identity matrix respectively. It is straightforward to verify that the l 1 norm of this matrix is 2(m 1)/m. The largest possible value is thus 2(L max 1)/L max. So (17) holds and the claim is proved. Condition A. We only need to prove the claim in (17) with l 1 norm replaced by l norm. The proof is similar to condition A 1 and is omitted. Condition A 2. It is sufficient to show that each block of A σ has spectral norm equal to 1. To this end, we can verify that W T mw m = W 2 m = W m. From the above equality, we can conclude that the eigenvalues of W m are 0 and 1. We obtain another set of criteria when we apply Theorem 7 to some norms that are more complicated. To a vector w R n + with positive entries, we associate it with a weighted l norm, defined as x w := max x i /w i. (18) i=1,...,n The spectral radius of a matrix M is the maximal absolute value of the eigenvalues, and is denoted by ρ(m). The following lemma will be found useful. Lemma 9: If M is a nonnegative matrix, then the following are equivalent: (1) ρ(m) < 1, (2) there exists a vector w > 0 such that M w < 1. Proof: See [12] We consider the case where L 1 =... = L K = L. The system gain matrix G is partitioned into a K K block matrix, in which each block has dimension L L. Letthe(k, l)-th block be denoted by H kl.wedefineak K matrix H q whose (i, j)-entry is the matrix norm H kl q,where q may be the l 1, l 2 or l norm. It is noted that H q is a nonnegative matrix with zero diagonal. Theorem 10: Suppose that L 1 =... = L K = L. Ifany one of the following Condition B 1 : ρ( H 1 ) <L/(2(L 1)) (19) Condition B : ρ( H ) <L/(2(L 1)) (20) Condition B 2 : ρ( H 2 ) < 1 (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 B 2. The proofs of the other two conditions are similar. For p =(p 1,...,p K ),wedefine θ w (p) := ( p 1 2,..., p K 2 ) w. We can check that θ w is a vector norm. The outer norm is a weighted l norm for some w > 0, while the inner norm is the l 2 norm.

7 SHUM et al.: CONVERGENCE OF ITERATIVE WATERFILLING ALGORITHM FOR GAUSSIAN INTERFERENCE CHANNELS 1097 By Theorem 7, it suffices to show that the induced matrix norm of AG, wherea is defined as in (12), is strictly less than 1 for all possible choices of N i s. The vector AGp is partitioned into K blocks, each of length L; theith block is equal to K W (N j )H ij p j. j=1 Recall that the spectral radius of W (N j ) is 1 for any subset N j of L j.so, K 2 K W (N j )H ij p j H ij 2 p j 2. j=1 j=1 The right hand side is the ith component of H 2 p,wherep is the K-dimensional vector Therefore, p := ( p 1 2,..., p K 2 ). θ w (AGp) H 2 p w. If ρ( H 2 ) < 1, then by Lemma 9, we can choose w so that H 2 w < 1. As a result, we have θ w (AGp) H 2 w p w = H 2 w θ w (p), for all p. Hence the norm of AG induced by θ w is strictly less than 1. VI. ASYNCHRONOUS ITERATIVE WATERFILLING ALGORITHM 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. Let p (τ ) i 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 allocation of transmitter j available to transmitter i is p (τ i j (t)) j, where 0 τj i (t) t. The waterfilling function for transmitter i at time t is based on the system power vector p (τ i (t)) := [ p (τ j i (t)) ] K 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,...}.LetT i T be the set of time instants when transmitter i adjusts its power. Given the sets T 1,...,T K, the asynchronous iterative waterfilling algorithm (AIWA) is defined by { p (t+1) f(gp (τ i (t)) + n; p i, m i ), t T i, i = p (t) (22) i, otherwise. We assume lim t τj i (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 T i = {i, i+k, i+ 2K,...,} for all i, we have the sequential update. When T i = 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) =X 1 (n) X 2 (n) X K (n) for some subsets X i (n) X i. 2) (Inclusion Condition) X(0) X(1) X(2)... {q}, and all sequences {x k } such that x k 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, B 1, B 2 and B are sufficient conditions for global convergence of AIWA. Proof: We will prove that if condition B 2 holds, then AIWA will converge for any initial power allocation. The proofs of the other conditions are similar and omitted. Suppose that L 1 =... = L K = L and condition B 2 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 a weight vector w so that ρ( H 2 ) H 2 w =: α<1. Let the fixed point of F be p =[p T 1...p T K ]T.Define X(n) := { p P: θ w (p p ) α n θ w (p (0) p ) }. X(n) is equal to the cartesian product K i=1 X i(n), where X i (n) := { p i P i : p i p i 2 w i α n θ w (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 θ w. The synchronous convergence condition follows from the properties of the contraction map. VII. PARALLEL GAUSSIAN INTERFERENCE CHANNEL 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, diag(g 11 ij,g22 ij,...,gll ij ), for i j. For notational simplicity, we will write gij λ instead of gij λλ. The system can also be specified by the channel gain matrices G λ, λ =1,...,L,where { [G λ G λ ij for i j ] ij := 0 otherwise.

8 1098 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 2007 A. New Convergence Conditions Theorem 13: For parallel interference channels, if the channel gain matrices satisfy any one of the following, Condition C 1 : G λ L 1 < 2(L 1) λ, (23) Condition C : G λ L < 2(L 1) λ, (24) Condition C 2 : G λ 2 < 1 λ, (25) then SIWA globally converges to a unique fixed point. Proof: We can permutate the columns and rows of the system gain matrix G and obtain a block diagonal matrix diag(g 1, G 2,...,G L ) where each diagonal block is a channel gain matrix. The matrix norm of G does not change after such a permutation. Then G q =max λ Gλ q for q =1, 2, or. The theorem follows from Theorem 8. Theorem 14: Let Ḡ denote the entry-wise maximum of the channel gain matrices, If condition C or [Ḡ] ij := max λ=1,...,l [Gλ ] ij. Condition D 2 : ρ(ḡ) < 1 (26) is satisfied, then AIWA converges globally. In particular, the iterative waterfilling algorithm converges under both synchronous and sequential update if condition C or D 2 is true. Proof: For parallel Gaussian interference channel, condition C is equivalent to A, which is a sufficient convergence condition for AIWA by Theorem 12. Since H ij is diagonal, we have H 1 = H 2 = H = Ḡ. Hence, condition D 2 is equivalent to condition B 2. B. Comparison with Other Convergence Conditions Theorem 14 unifies and generalizes many convergence conditions available in the literature. Corollary 15 ([10]): If there exists a weight vector w such that one of the following conditions is satisfied, 1 (i) [max w j λ Gλ ij]w i < 1, for all j =1,...,K, (ii) i j 1 [max w i λ Gλ ij]w j < 1, for all i =1,...,K, j i then the iterative waterfilling algorithm converges globally under both synchronous and sequential update to a unique Nash equilibrium. Proof: The left hand side of the inequality in condition (i) can be rewritten as diag(w)ḡ diag(w) 1 1, which is the weighted l 1 matrix norm of Ḡ. Since the inequality Ḡ ρ(ḡ) holds for any matrix norm in general [21, Theorem 5.6.9], condition D 2 is satisfied and the iterative waterfilling algorithm converges under totally asynchronous model. In particular, it also converges synchronously and sequentially. The proof of (ii) is similar. Corollary 16 ([9]): If condition D 2 is satisfied, then the iterative waterfilling algorithm converges sequentially. Proof: By Theorem 14, it converges under totally asynchronous update, which contains sequential update as a special case. Corollary 17 ([11], [8]): If G λ ij < 1/(K 1) for all i, j and λ, then the iterative waterfilling algorithm converges both synchronously and sequentially. Proof: Let M be the K K matrix whose diagonal entries are zero and off-diagonal entries 1/(K 1). It can be verified that ρ(m) = 1.IfG λ ij < 1/(K 1) for all i, j and λ, then Ḡ < M, which implies that ρ(ḡ) < ρ(m) =1. Condition D 2 is satisfied and the results follow from Theorem 14. Theorem 2 in [10] is identical to condition C 2. In [9], it is shown that the iterative waterfilling algorithm converges sequentially if the channel gain matrices G λ are all symmetric. This result is not and cannot be implied by any conditions in this paper. If we assume that the gain matrices are symmetric, there may be more than one Nash equilibria. This case cannot be included in the framework of contraction mapping theorem. C. Numerical Examples Although the conditions A 1, A 2 and A are sufficient but not necessary, the constants on the right hand sides of condition A 1, A 2 or A cannot be replaced by larger constants. The first example illustrates this fact. Example 1: Suppose K =2and L =2. The link gain matrices are G 1 = [ ] 0 a, G 2 = b 0 [ ] 0 c, d 0 where a, b, c, d are strictly less than 1. Then it satisfies condition A 1, A 2 and A. Hence we have global asymptotic stability for any maximum power constraints and noise powers. If the link gain matrices are [ ] [ ] G ɛ =, G ɛ =, 1+ɛ 0 1 ɛ 0 where ɛ is an arbitrarily small positive number. This example fails all three conditions in Theorem 13. Suppose that the noise powers are constant for all channels, the maximum total power constraint is 10 for both transmitters and there is no individual power constraint. Suppose that both transmitters put all of the power in channel 2 initially. channel 1 channel 2 user user

9 SHUM et al.: CONVERGENCE OF ITERATIVE WATERFILLING ALGORITHM FOR GAUSSIAN INTERFERENCE CHANNELS 1099 In the next iteration, the power distribution under the SIWA is channel 1 channel 2 user 1 10 ɛ/2 ɛ/2 user 2 10 ɛ/2 ɛ/2 Then the system will go back to the initial condition and will oscillate between these two power allocations. We do not have convergence in this case. Since ɛ can be arbitrarily small, the three conditions in Theorem 8 are tight, meaning that the constant L max /(2(L max 1)) cannot be increased. A system that is stable under sequential updates may not be stable when the updates are performed in synchronous manner. The following is an example. Example 2: Suppose the setting is the same as in the last example, except that we adopt sequential power update instead. Suppose that transmitter 1 updates the power allocation while transmitter 2 remain unchanged. After the first iteration, we have channel 1 channel 2 user 1 10 ɛ/2 ɛ/2 user In the second iteration, it is transmitter 2 s turn to update the power distribution. However, transmitter 2 sees that the interference powers in channel 1 and 2 are (1 + ɛ)(10 ɛ/2) and (1 ɛ)ɛ/2 respectively. The difference (1 + ɛ)(10 ɛ/2) (1 ɛ)ɛ/2 =10+9ɛ is larger than 10. By the waterfilling process, transmitter 2 will continue putting all of its power in channel 2 and will not change the power allocation. The power allocation will remain unchanged from now on, and the system converges under sequential update. The next two examples illustrate that condition C and D 2 do not imply each other. Example 3: Suppose K = L =3, and the three channel gain matrices are 0 5/8 G 1 = 5/8 0 1/16, 1/4 1/ /16 5/8 G 2 = 1/4 0 1/4, 5/8 1/ /4 1/4 G 3 = 1/16 0 5/8. 1/16 5/8 0 The matrix Ḡ in Theorem 14 is 0 5/8 5/8 5/8 0 5/8. 5/8 5/8 0 Since ρ(ḡ) > 1, it fails condition D 2. However, the row sums of G 1, G 2 and G 3 are all less than 3/4. We can apply condition C in Theorem 14 and conclude that Power Fig Iteration User 1 User 2 User 3 Power trajectories in Example 3 with synchronous update. AIWA converges globally, and hence will also converge under sequential update. Figure 3 shows the convergence of SIWA in this example. The initial power allocation is uniform. The noise powers are arbitrarily chosen between 0 and 1. The maximal total power constraints for the three users are 2, 3, and 4. Example 4: Suppose that the three channel gain matrices are identical, G 1 = G 2 = G 3 = Ḡ = The spectral radius of Ḡ is less than 1. It satisfies condition D 2 but fails condition C. VIII. CONCLUSION When all users in a Gaussian interference channel use single-user detector, the system can be modeled as a noncooperative game, in which each user tries to maximize his total data rate. We consider the situation where users update their power allocations iteratively by waterfilling. Sufficient conditions for convergence to an equilibrium point is derived under totally asynchronous update, which includes synchronous and sequential update as special cases. Many existing convergence results are generalized. REFERENCES [1] T. S. Han and K. Kobayashi, A new achievable rate region for the interference channel, IEEE Trans. Inform. Theory, vol. 27, pp , Jan [2] S. T. Chung and J. M. Cioffi, The capacity region of frequency-selective Gaussian interference channels under strong interference, in Proc. IEEE ICC, 2003, pp [3] S. Vishwanath and S. A. Jafar, On the capacity of vector Gaussian interference channels, in Proc. IEEE ITW, Oct. 2004, pp [4] S. T. Chung, Transmission schemes for frequency selective Gaussian interference channel, Ph.D. dissertation, Stanford University, Nov [5] R. Cendrillon, M. Moonen, J. Verliden, T. Bostoen, and W. Yu, Optimal multiuser spectrum management for digital subscriber lines, in IEEE Int. Conf. on Comm., vol. 1, June 2004, pp [6] W. Yu, G. Ginis, and J. J. Cioffi, Distributed multiuser power control for digital subscriber lines, IEEE Journal on Selected Area in Comm., vol. 20, pp , 2002.

10 1100 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 2007 [7] W. Yu and J. M. Cioffi, Competitive equilibrium in the Gaussian interference channel, in Proc. IEEE Int. Symp. on Inform. Theory, Sorrento, Italy, June 2000, p [8] S. T. Chung, J. L. S. J. Kim, and J. M. Cioffi, A game-theoretic approach to power allocation in frequency-selective Gaussian interference channels, in Proc. IEEE Int. Symp. on Inform. Theory, Yokohama, Japan, 2003, p [9] Z.-Q. Luo and J.-S. Pang, Analysis of iterative waterfilling algorithm for multiuser power control in digital subscriber lines, EURASIP Journal on Applied Signal Processing, vol. 2006, 2006, article ID 24012, 10 pages, doi: /asp/2006/ [10] G. Scutari, D. P. Palomar, and S. Barbarossa, Simultaneous iterative water-filling for Gaussian frequency-selective interference channels, in Proc. IEEE Int. Symp. on Inform. Theory, Seattle, July 2006, pp [11] N. Yamashita and Z. Q. Luo, A nonlinear complementarity approach to multiuser power control for digital subscriber lines, Optimization Methods and Software, vol. 19, pp , [12] D. P. Bertsekas and J. N. Tsitsiklis, Parallel and distributed computation. Englewood Cliffs, New Jersey: Prentice Hall, [13] R. Etkin, A. Parekh, and D. Tse, Spectrum sharing for unlicensed bands, in First IEEE Int. Symp. on New Frontiers in Dynamic Spectrum Access Networks, DySPAN 2005, Baltimore, MD, Nov. 2005, pp [14] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, [15] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge: Cambridge university press, [16] G. Debreu, A social equilibrium existence theorem, in Proc. of the National Academy of Sciences, vol. 38, 1952, pp [17] K. Fan, Fixed points and minimax theorems in locally convex topological linear spaces, in Proc. of the National Academy of Sciences, vol. 38, 1952, pp [18] I. L. Glicksberg, A further generalization of the Kakutani fixed point theorem with application to Nash equilibrium points, in Proc. of the National Academy of Sciences, vol. 38, 1952, pp [19] C. Berge, Topological Spaces. New York: Dover, [20] W. Rudin, Principles of Mathematical Analysis, 3rd ed. Singapore: McGraw-Hill Inc., [21] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge: Cambridge University Press, [22] D. P. Bertsekas, Distributed asynchronous computation of fixed points, Mathematical Programming, vol. 27, pp , Kenneth W. Shum (M 00) received the B.Eng. degree in Information Engineering from the Chinese University of Hong Kong in 1993, and the M.S. and Ph.D. degree in Electrical Engineering from the University of Southern California in 1995 and 2000 respectively. He is now a post-doctoral fellow in the City University of Hong Kong. His research interests include information theory and resource allocation in wireless networks. Kin-Kwong Leung (M 04) received the B.Eng, M.Phil, and Ph.D. degrees in information engineering from the Chinese University of Hong Kong, in 1997, 2000 and 2003, respectively. He is currently a computational scientist in Cluster Technology Limited. His research interests include neural networks and transmitter adaptation in wireless networks. Chi Wan Sung (M 98) received the B.Eng, M.Phil, and Ph.D. degrees from the Chinese University of Hong Kong, in 1993, 1995, and 1998, respectively, all in information engineering. After obtaining his Ph.D., he became an Assistant Professor in the same university. In 2000, he joined the faculty at City University of Hong Kong, and is currently an Assistant Professor with the Department of Electronic Engineering. His research interests include multiuser information theory, and control and optimization of wireless networks.