A New Algorithm for the Weighted Sum Rate Maximization in MIMO Interference Networks

Transcription

1 A New Agorithm for the Weighted Sum Rate Maximization in MIMO Interference Networks Xing Li, Seungi You 2, Lijun Chen, An Liu 3, Youjian Eugene Liu Abstract We propose a new agorithm to sove the non-convex weighted sum-rate maximization probem in genera MIMO interference networks. With the Gaussian input assumption, the previous state-of-the-art agorithms are the WMMSE agorithm and the poite water-fiing PWF agorithm. The WMMSE agorithm is provaby convergent, whie the PWF agorithm converges faster in most situations but sometimes osciates. Thus, it is highy desirabe to design an agorithm that takes advantage of the optima transmit signa structure to ensure fast convergence whie being provaby convergent. We present such an agorithm and prove its monotonic convergence. Moreover, our convergence proof uses very genera convex anaysis as we as a scaing invariance property of the weighted sum-rate maximization probem. We expect that the scaing invariance hods for and our proof technique appies to many non-convex probems in communication networks. Index Terms MIMO, Interference Network, Weighted Sumrate Maximization, Duaity, Scaing Invariance, Optimization I. INTRODUCTION One of the most effective approaches to accommodating the exposive growth in mobie data is to reduce the ce size and increase the base station/access point density. However, the path oss versus distance curve is fatter at shorter distance, as opposed to being steep at reativey ong distance. As a resut, the inter-ce interference becomes significant as the ce size/coverage area shrinks. Therefore, joint transmit signa design for interference networks is a key technoogy for the next generation wireess communication systems. In this paper, we consider the joint transmit signa design for a genera interference network caed MIMO B-MAC network [5]. The MIMO B-MAC network consists of mutipe mutuay interfering data inks between mutipe transmitters and mutipe receivers that are equipped with mutipe antennas MIMO. It incudes broadcast channe BC, mutiaccess channe MAC, interference channes, X networks, and many practica wireess networks as specia cases. Specificay, we study the probem of jointy optimizing the transmit signas of a transmitters in order to maximize the weighted sum-rate of the data inks for the MIMO B-MAC network, assuming Gaussian transmit signa and avaiabiity of a channe state information. It typifies a cass of probems that Coege of Engineering and Appied Science, University of Coorado at Bouder 2 Department of Contro and Dynamica Systems, Caifornia Institute of Technoogy 3 Department of Eectronic and Computer Engineering, Hong Kong University of Science and Technoogy are key to the next generation wireess communication networks where the interference is the imiting factor. This probem is non-convex, and various agorithms have been proposed for this probem or its specia cases over the years; see, e.g., [], [2], [3], [4], [5], [8], [9], [2], [3], [4], [5], [6]. In particuar, we have recenty proposed the poite waterfiing PWF agorithm for the MIMO B-MAC network [5]. It is based on the identification of a poite water-fiing structure that the optima transmit signa possesses. It is an iterative agorithm with the forward ink poite water-fiing foowed by the virtua reverse ink poite water-fiing. Because it takes advantage of the optima signa structure, the PWF agorithm has neary the owest compexity and the fastest convergence when it converges. For instance, it converges to the optima soution in haf iteration for parae channes. However, the convergence of the PWF agorithm is ony guaranteed for the specia case of interference tree networks [5] and it may osciate under certain strong interference conditions. Another state-of-the-art agorithm is the WMMSE agorithm in [9]. It is proposed for the MIMO interfering broadcast channes but coud be readiy appied to the more genera B-MAC networks. It transforms the weighted sum-rate maximization into an equivaent weighted sum-mse cost minimization probem, which has three sets of variabes and is convex when any two variabe sets are fixed. With the bock coordinate descent technique, the WMMSE agorithm is guaranteed to converge to a stationary point, though the convergence is observed in simuations to be sower than the PWF agorithm. Thus, it is highy desirabe to have an agorithm with the advantages of both PWF and WMMSE agorithms, i.e., fast convergence by taking advantage of the optima transmit signa structure and provabe convergence for the genera MIMO B- MAC network. One major contribution of this paper is to propose such an agorithm. A key finding in proving the poite water-fiing structure is to identify that a term invoving interference in the KKT condition of the weighted sum-rate maximization probem is equa to the reverse ink signa covariance of the dua network at a stationary point [5, Theorem 22]. Then, soving the KKT condition reveas that the optima transmit signa has the poite water-fiing structure, which reduces to the we known water-fiing structure for the MIMO parae channes. Instead of using this key finding to sove the KKT condition, our new agorithm uses it directy to aternativey update the forward ink and reverse ink signa covariance matrices. Numerica experiments demonstrate that the new agorithm coud be a few iterations or more than ten

2 2 T TX Fig.. T 2 TX T 3 TX R RX R 2 RX 2 R 3 RX 2 An exampe of B-MAC network Interfering ink Transmission ink iterations faster than the WMMSE agorithm, depending on the desired numerica accuracy. Note that being faster even by a few iterations wi be significant in distributed impementation where the overhead of each iteration costs significant signaing resources between the transmitters and the receivers. Indeed, the new agorithm is highy scaabe and suitabe for distributed impementation with distributed channe estimation because for each data ink, ony its own channe state and the aggregated interference pus noise covariance needs to be estimated no matter how many interferers are there. The distributed impementation of the new agorithm wi be deveoped in the future work. Besides proposing the new agorithm with fast convergence, another major contribution of this paper is to present an eegant proof of the monotonic convergence of the agorithm. The proof uses ony very genera convex anaysis, as we as a particuar scaing invariance property that we identify for the weighted sum-rate maximization probem. We expect that the scaing invariance hods for and our proof technique appies to many non-convex probems in communication networks that invove the rate or throughput maximization. The rest of this paper is organized as foows. Section II presents the system mode and formuates the probem. Section III briefy reviews the reated resuts on the poite water-fiing and presents the proposed new agorithm. Its monotonic convergence is estabished in Section IV. Section V presents numerica exampes and compexity anaysis. Section VI concudes. II. SYSTEM MODEL AND PROBLEM FORMULATION We consider a genera interference network named MIMO B-MAC network [5], [6]. A transmitter in the MIMO B-MAC network may send independent data to different receivers, ike in BC, and a receiver may receive independent data from different transmitters, ike in MAC. Assume there are totay L mutuay interfering data inks. The set of physica transmitter abes is T {TX, TX 2, TX 3,...} and the set of physica receiver abes is R {RX, RX 2, RX 3,...}. Define transmitter T of ink as a mapping from to ink s physica transmitter abe in T. Define receiver R as a mapping from to ink s physica receiver abe in R. The numbers of antennas at T and R are L T and L R respectivey. Fig. shows a B-MAC network with three data inks. When mutipe data inks have the same receiver or the same transmitter, techniques such as successive decoding and canceation or dirty paper coding can be appied [5]. The received signa at R is y H,k x k +w, k where x k C LT k is the transmit signa of ink k and is assumed to be circuary symmetric compex Gaussian; H,k C LR LT k is the channe matrix between T k and R ; and w C LR is a circuary symmetric compex Gaussian noise vector with identity covariance matrix. The circuary symmetric assumption of the transmit signa can be dropped easiy by appying the proposed agorithm to rea Gaussian signas with twice the dimension. Assuming the avaiabiity of channe state information at the transmitters, an achievabe rate of ink is I Σ :L ogi+h, Σ H, Ω 2 where Σ is the covariance matrix of x ; the interferences from other inks are treated as noise; and Ω is the interference-pusnoise covariance matrix of the th ink, which is Ω I+ H,k Σ k H,k. 3 k k If the interference from ink k to ink is competey canceed using successive decoding and canceation or dirty paper coding, we can simpy set H,k 0 in 3. It aows this paper to cover a wide range of communication techniques. The optimization probem that we want to sove is the weighted sum-rate maximization under tota power constraint: L WSRM_TP: max Σ:L w I Σ :L 4 s.t. Σ 0,, TrΣ P T where w > 0 is the weight for ink. The generaization to mutipe inear constraints can be done simiary to [7] and wi be considered in future works. III. THE OPTIMIZATION ALGORITHM In this section, we wi review briefy some reated resuts on the poite water-fiing and propose a new agorithm for the weighted sum-rate probem 4 that has fast, monotonic convergence. A. A Review on the Poite Water-fiing Structure and Agorithm Athough the probem 4 is non-convex and cannot be soved directy, the optima transmit signa has a poite water-fiing structure, based on which an efficient agorithm can be designed [5]. The resuts in [5] are briefy reviewed here. We first give the duaity resuts. Let

3 3 [H,k ], TrΣ P T denote a network with tota power constraint and channe matrices [H,k ] as in the probem 4. An achievabe rate region of 5 is defined as RP T Σ :L: L TrΣ P T r I Σ :L, L}. Its dua network or reverse inks is defined as [ ] H k,, Tr ˆΣ P T 5 { r R L + : 6 where the roes of a transmitters and receivers are reversed, and the channe matrices are repaced with their conjugate transpose. ˆ denote the corresponding terms in the reverse inks. Simiary, the interference-pus-noise covariance matrix of reverse ink is ˆΩ I+ k k 7 H k,ˆσ k H k, ; 8 the achievabe rate of reverse ink is Î ˆΣ:L ogi+h,ˆσ H,ˆΩ ; 9 the reverse ink achievabe rate region is defined as ˆRP T ˆΣ L :L: TrˆΣ P T ˆr Î ˆΣ:L, L {ˆr R L + : 0 }. The rate duaity states that the achievabe rate regions of the forward and reverse inks are the same [5, Theorem 9]. A covariance transformation [5, 8] cacuates the reverse ink input covariance matrices from the forward ones. The rate duaity is proved by showing that the reverse ink input covariance matrices cacuated from the covariance transformation achieves equa or higher rates than the forward ink rates under the same vaue of inear constraint P T [5, Lemma ]. The Lagrange function of probem 4 is where Θ :L and µ are Lagrange mutipiers. The KKT conditions are Σ L H, w H, Ω +H, Σ H, +Θ µi w k H k, Ω k Ω k +H k,k Σ k H k,k H k, k 0, µ P T TrΣ 0, trσ Θ 0, Σ, Θ 0, µ 0. The poite water-fiing structure is given as foows [5, Theorem 22]. A key finding eading to the poite water-fiing structure is that at a stationary point, the dua input covariance matrices ˆΣ :L cacuated from the covariance transformation satisfies ˆΣ w µ Ω Ω +H, Σ H,,,...,L. 2 We substitute it into to obtain the poite water-fiing structure, Q G D G, 3 D ν I 2 +, 4 whereq ˆΩ 2 2 Σ ˆΩ is the equivaent input covariance matrix of the ink ; G and are from SVD decomposition of the equivaent singe-user channe H F G ; H is given by H Ω 2 H 2,ˆΩ ; ˆΣ:L is obtained from Σ :L by the covariance transformation in [5, Definition 4] and is used to cacuate the corresponding ˆΩ :L ; ν w µ 0 is the waterfiing eve. That is the ink s equivaent input covariance matrix Q is a water-fiing over the equivaent channe H. In addition, at a stationary point, the ˆΣ :L obtained from the covariance transformation aso have the poite water-fiing structure [5, Theorem 2] and satisfy the foowing KKT conditions 6 of the foowing dua probem 5: Lµ,Θ :L,Σ :L w ogi+h, Σ H, Ω + +µ P T TrΣ, TrΣ Θ L WSRM_TP_D: max Σ:L w Î ˆΣ:L s.t. ˆΣ 0,, Tr ˆΣ P T, 5

4 4 whose KKT conditions are H w H, ˆΩ +H,ˆΣ H,, + ˆΘ ˆµI w k H k, ˆΩ k ˆΩk +H k,kˆσ k H k,k H k, k 0, 6 ˆµ P T TrˆΣ 0, tr ˆΣ ˆΘ 0, ˆΣ, ˆΘ 0, ˆµ 0. Simiary to 2, from the poite water-fiing structure on reverse inks, we have Σ w ˆΩ ˆΩ +H ˆµ,ˆΣ H,,,...,L. 7 It is we known that L TrΣ P T when Σ :L is a stationary point of probem 4, e.g., [6, Theorem 8 item 3]. This indicates that the fu power shoud aways be used. Since the covariance transformation preserves tota power, we P T [5, Lemma 8]. It can aso be proved that TrQ Tr ˆQ, where aso have L TrΣ L Tr ˆΣ ˆQ Ω 2 ˆΣ Ω 2 are the reverse ink equivaent covariance matrices [5, Lemma 20]. The poite water-fiing agorithm, Agorithm PP in [5], works as foows. After initiaizing the reverse ink interference pus noise covariance matrices ˆΩ :L, we perform a forward ink poite water-fiing using 3, 4 foowed by a reverse ink poite water-fiing, which is defined to be one iteration. The iterations stops when the change of the objective function is ess than a threshod or when a predetermined number of iterations is reached. Because the agorithm enforces the optima signa structure at each iteration, it converges very fast. In particuar, for parae channes, it reduces to the traditiona water-fiing and gives the optima soution in haf an iteration with initia vaues ˆΩ I,. Unfortunatey, this remarkabe agorithm sometimes does not converge and the objective function has some osciation, especiay in very strong interference cases. Agorithm Dua Poite Water-fiing Agorithm. Initiaize Σ s, s.t. L TrΣ P T 2. R L w I Σ :L 3. Repeat 4. R R 5. Ω I+ k H kσ k H k 6. ˆΣ PTw Ω Ω +H Σ H L w trω Ω +H Σ H 7. ˆΩ I+ k H kˆσ k H k 8. Σ PTw ˆΩ ˆΩ +H ˆΣ H L wtrˆω ˆΩ +H ˆΣ H 9. R L w I Σ :L 0. unti RR ǫ. This gives a new agorithm, Agorithm, that takes advantage of the structure of the weighted sum-rate maximization probem and, as confirmed by the anaytica anaysis and numerica experiments, has fast monotonic convergence. It converges to a stationary point of both probem 4 and probem 5 simutaneousy, and at the stationary point, both 2 and 7 achieve the same sum-rate. We wi anayze the convergence properties of Agorithm in the next section. IV. CONVERGENCE OF ALGORITHM In this section, we wi prove the monotonic convergence of Agorithm. As wi be seen ater, the proof uses ony very genera convex anaysis, as we as a particuar scaing invariance property that we identify for the weighted sum-rate maximization probem. We expect that the scaing invariance hods for and our proof technique appies to many non-convex probems in communication networks that invove the rate or throughput maximization. A. Preiminaries Lagrangian of the weighted sum-rate function: The weighted sum-rate maximization probem 4 is equivaent to the foowing probem: B. The New Agorithm Instead of using 2 and 7 to sove the KKT condition for the poite water-fiing structure, we can directy use these two equations to update ˆΣ :L and Σ :L. Note that, since the fu power is used, Lagrange mutipier µ can be chosen to satisfy the power constraint L TrΣ P T as µ L w tr ˆΩ ˆΩ +H P,ˆΣ H,. 8 T max Σ :L,Ω :L s.t. w ogω +H, Σ H,og Ω Σ 0,, TrΣ P T, Ω I+ k H,k Σ k H,k,,

5 5 which is sti non-convex. Consider the Lagrangian of the above probem FΣ,Ω,Λ,µ w ogω +H, Σ H,og Ω +µ{p T + TrΣ } Tr Λ Ω I H,k Σ k H,k, k Define ˆΣ µ Λ, F Σ 0, F Ω 0. ˆΩ I+ k nh k,ˆσ H k,, where the domain of F is {Σ,Ω,Λ,µ Σ H LR LR +,Ω H LR LR ++,Λ H LR LR,µ R + }. Here H n n, H+ n n, and H++ n n mean n by n Hermitian matrix, positive semidefinite matrix, and positive definite matrix respectivey. One can easiy verify that the function F is concave in Σ and convex in Ω. Furthermore, the partia derivatives are given by F H, w H, Ω +H, Σ H Σ, µi k H k, Λ H k,, then, F Ω w Ω +H,Σ H, Ω +Λ. Now suppose that we have the pair Σ,Ω such that TrΣ P T, Ω I+ k H,k Σ k H,k, FΣ,Ω,Λ,µ w ogω +H, Σ H,og Ω, which is the origina weighted sum-rate function. For notationa simpicity, denote the weighted sum-rate function by VΣ, i.e., VΣ w og I+ H,k Σ k H,k +H,Σ H, k og I+ H,k Σ k H,k. k 2 Soution of the first-order condition: Suppose that we want to sove the foowing system of equations in terms of Σ,Ω for given Λ,µ: the above system of equations becomes H, w H, Ω +H, Σ H, µˆω, w Ω Ω +H, Σ H, µˆσ. A soution to this system of equations is given by ˆΩ Σ Ω w µ w µ H, The proof can be found in []. B. Convergence Resuts ˆΩ +H,ˆΣ H, 9 H,ˆΣ H, + ˆΩ H,. 20 We are ready to present the foowing two main convergence resuts regarding Agorithm. Denote by Σ n the Σ vaue at the nth iteration of Agorithm. Theorem. The objective vaue, i.e., the weighted sum-rate, is monotonicay increasing in Agorithm, i.e., VΣ n VΣ n+. From the above theorem, the foowing concusion is immediate. Coroary 2. The sequence V n VΣ n converges to some imit point V. Proof: Since VΣ is a continuous function and its domain {Σ Σ 0,TrΣ P T } is a compact set, V n is bounded above. From Theorem, {V n } is a monotone increasing sequence, therefore there exists a imit point V such that im n V n V. If we define a stationary point Σ of Agorithm as, Σ n Σ impies Σ n+k Σ for a k 0,,, then we have the foowing resut. Theorem 3. Agorithm converges to a stationary point Σ. The proof of Theorems and 3 wi be presented ater in this section. Before that, we first estabish a few inequaities and identify a particuar scaing property of the Lagrangian F.

6 6 The first inequaity: Suppose that we have a feasibe point Σ n 0, and L Tr Σ n P T. In Agorithm, we generate Ω n such that Ω n I+ k H,k Σ n k H,k. Now we have a pair Σ n,ω n. Using this pair, we can compute Λ n,µ n such that Λ n w µ n P T Ω n Ω n Tr Λ n. Note that ˆΣ n in Agorithm is equa to ˆΣ n Λn µ. n +H, Σ n H,, From this, the gradient of F with respect to Ω at the point Σ n,ω n vanishes, i.e., FΣ n,ω,λ n,µ n Ω Ω n 0. Since F is convex in Ω, if we fix Σ Σ n, then Ω n is a goba minimizer of F. In other words, FΣ n,ω n,λ n,µ n FΣ n,ω,λ n,µ n 2 for a Ω 0. 2 Scaing invariance of F : We wi identify a remarkabe scaing invariance property of F, which pays a key roe in the convergence proof of Agorithm. For given Σ n,ω n,λ n,µ n, we have F α Σn, α Ωn,αΛ n,αµ n FΣ n,ω n,λ n,µ n for a α >0. To show this scaing invariance property, note that Ω n k H,k Σ n k H,k I, TrΣ n P T, P T µ n TrΛ n. Appying the above equaities and some mathematica manipuations, we have F α Σn, α Ωn,αΛ n,αµ n w og Ω n +H, Σ n +αµ n {P T α P T}+ w og Ω n H, +H, Σ n +αµ n P T +α w og Ω n +H, Σ n FΣ n,ω n,λ n,µ n, og Ω n Tr αλ n α II H, H, og TrΛ n og Ω n Ω n where the first equaity uses the fact that og Ω n +H, Σ n H α, og ogω n +H, Σ n ogω n. Furthermore, H, F α Σn,Ω,αΛ n,αµ n Ω α Ωn w α Ωn +H, α Σn H, +αλ n α FΣn,Ω,Λ n,µ n Ω 0. Ω n α Ωn α Ωn Therefore, α Ωn is a goba minimizer of F α Σn,Ω,αΛ n,αµ n, as F is convex in Ω. 3 The second and third inequaities: GivenαΛ n,αµ n, we generate Σ, Ω using equation 9 and 20. If we choose α so that L Σ P T, then Σ Σ n+ in Agorithm. Since Σ n+, Ω is chosen to make partia derivatives zero: FΣ, Ω,αΛ n,αµ n Σ Σ n+ 0, FΣ n+,ω,αλ n,αµ n Ω Ω 0, we concude that Σ n+ is a goba maximizer, i.e., FΣ, Ω,αΛ n+,αµ n+ FΣ n+, Ω,αΛ n,αµ n 22 for a Σ 0; and Ω is a goba minimizer, i.e., FΣ n+, Ω,αΛ n,αµ n FΣ n+,ω,αλ n,αµ n 23 for a Ω 0.

7 7 4 Proof of Theorem : With the three inequaities 2-23 obtained above, we are ready to prove Theorem. As in Agorithm Ω n+ I+ k H,kΣ n+ k H,k, we have VΣ n+ FΣ n+,ω n+,αλ n,αµ n FΣ n+, Ω,αΛ n,αµ n F α Σn, Ω,αΛ n,αµ n F α Σn, α Ωn,αΛ n,αµ n FΣ n,ω n,λ n,µ n VΣ n, where the first inequaity foows from 23, and the second and third inequaities foows from 22 and 2. 5 Proof of Theorem 3: We have shown in Coroary 2 that V n converges to a imit point under Agorithm. To show the convergence of the agorithm, it is enough to show that if VΣ n VΣ n+, then Σ n+ Σ n+k for a k,2,. Suppose VΣ n VΣ n+, then from the proof in the above, we have FΣ n+,ω n+,αλ n,αµ n FΣ n+, Ω,αΛ n,αµ n. Since Ω is a goba minimizer, the above equaity impies Ω n+ is a goba minimizer too. From the first order condition for optimaity, we have FΣ n+,ω,αλ n,αµ n+ Ω Ω n+ w Ω n+ +H, Σ n+ H n+, {Ω } +αλ n 0. On the other hand, we generate Λ n+ such that Λ n+ w αλ n. Ω n+ Ω n+ +H, Σ n+ H, This shows ˆΣ n+ ˆΣ n. However, since the trace of each matrix is same, we concude that ˆΣ n+ ˆΣ n. From this it is obvious that ˆΣn ˆΣ n+ and Σ n+ Σ n+2. V. NUMERICAL EXAMPLES AND COMPLEXITY ANALYSIS A. Numerica Exampes In this section, we provide numerica exampes to verify the anaysis in the previous sections and compare the proposed new Weighted SumRate / bits/s/hz forward ink reverse ink haf iterations Fig. 2. The monotonic convergence of the forward and reverse ink weighted sum-rates of the new agorithm with P T 00 and g,k 0dB,,k. agorithm against the PWF agorithm [5] and the WMMSE agorithm [9]. Consider a B-MAC network with L 0 inks among 0 transmitter-receiver pairs that fuy interfere with each other. Each ink has 3 transmit antennas and 4 receive antennas. For each simuation, the channe matrices are independenty generated by one reaization of H,k g,k H W,k, k,, where HW,k has zero-mean i.i.d. compex Gaussian entries with unit variance and g,k is the average channe gain. The weights w s are uniformy chosen from 0.5 to. The tota transmit power P T in the network is 00. Fig. 2 shows the convergence of the new agorithm for a network with g,k 0dB,,k. From the proof of Theorem, the weighted sum-rate of the forward inks and that of the reverse inks not ony increase monotonicay over iterations, but aso increase over each other over haf iterations. In Agorithm, the reverse ink transmit signa covariance matrices are updated in the first haf of each iteration ine 6, and the forward ink transmit signa covariance matrices are updated in the second haf ine 8. From Fig. 2, we ceary see that the weighted sum-rates of the forward inks and reverse inks increase in turns unti they converge to the same vaue, which aso confirms that probem 4 and its dua probem 5 reach their stationary points at the same time. We compare the performance of the new agorithm with the PWF and WMMSE agorithms under different channe conditions: weak, moderate, and strong interference. Fig. 3 shows a comparison under the weak interference condition. We see that the PWF agorithm converges sighty faster than the new agorithm, and it is cose to the stationary point in three iterations. The reason for this remarkabe convergence is that under the weak interference condition, the channes in the network are cose to parae channes and the PWF agorithm can converges to the optima soution in haf an iteration. Since the new agorithm is aso based on the poite water-fiing structure, it is not surprising that it has a fast convergence. In contrast, the WMMSE agorithm s convergence under such a

8 Weighted SumRate / bits/s/hz New Agorithm Poite WaterFiing Agorithm WMMSE Agorithm iterations Fig. 3. PWF agorithm vs. WMMSE agorithm vs. new agorithm under weak interference with P T 00, g, 0dB and g,k 0dB for k. Weighted SumRate / bits/s/hz New Agorithm Poite WaterFiing Agorithm WMMSE Agorithm iterations Fig. 6. An exampe that PWF, WMMSE, and new agorithms converge to different stationary points. P T 00, g, 0dB and g,k 0dB for k. Weighted SumRate / bits/s/hz New Agorithm Poite WaterFiing Agorithm WMMSE Agorithm iterations Fig. 4. PWF agorithm vs. WMMSE agorithm vs. new agorithm under moderate interference with P T 00 and g,k 0dB,,k. Weighted SumRate / bits/s/hz New Agorithm Poite WaterFiing Agorithm WMMSE Agorithm iterations Fig. 5. PWF agorithm vs. WMMSE agorithm vs. new agorithm under strong interference with P T 00, g, 0dB and g,k 0dB for k. channe condition is significanty sower; e.g., more than ten iterations sower to reach some high vaue of the weighted sum-rate i.e., if a high numerica accuracy is desired. When the gain of the interfering channes are comparabe to that of the desired channe, as shown in Fig. 4, the difference in the convergence speed between the PWF/new agorithm and the WMMSE agorithm is ess than that of the weak interference case. But around five iteration difference for some high vaue of the weighted sum-rate is sti significant. Under some strong interference conditions, as shown in Fig. 5, the PWF agorithm osciates around certain points and no onger converges, whie the other two agorithms sti have the guaranteed convergence. The new agorithm sti converges faster than the WMMSE agorithm and the difference is significant if high vaue of the weighted sum-rate is desired. Even in the case that one is satisfied with smaer weighted sum-rate, being faster by a few iterations wi be significant in distributed impementation where the overhead of each iteration costs significant signaing resources between the transmitters and the receivers. Combining the three cases together, we see that whie both agorithms are provaby convergent, the new agorithm outperforms the WMMSE agorithm in a situations, especiay in the weak interference case. Athough the PWF agorithm has better convergence over the new agorithm, it does not converge under certain strong interference channes. We can concude that the new agorithm preserves the fast convergence of the PWF agorithm and has the desired convergence property as the WMMSE agorithm as we. Note that given the same initia point, these three agorithms may converge to different stationary points, as shown in Fig. 6. Since the origina weighted sum-rate maximization probem is non-convex, a stationary point is not necessariy a goba maximum. In practica appications, we may run such agorithms mutipe times starting from different initia points and pick the stationary point that gives the argest weighted sum-rate. We may aso create a hybrid agorithm by combining the new agorithm and the PWF agorithm, by appying the PWF

9 9 agorithm unti the weighted sum-rate starts to drop, then switching to the new agorithm which wi converge to a stationary point. In this way, the hybrid agorithm wi retain the PWF agorithm s fast convergence and aso wi monotonicay converge under those strong interference channe conditions where the PWF agorithm may osciate. B. Compexity Anaysis We have evauated in the above convergence properties of the proposed new agorithm, the PWF agorithm and the WMMSE agorithm in terms of the number of iterations. We now anayze the compexity of each iteration for these agorithms. Reca that L is the number of users or inks, and for simpicity, assume that each user has N transmit and receive antennas, so the resuting Σ and ˆΣ is a N N matrix. Suppose that we use the straightforward matrix mutipication and inversion. Then the compexity of these operations are ON 3. For the new agorithm, at each iteration, Ω incurs a compexity of OLN 3 and Ω + H, Σ n+ H, incurs a compexity of OLN 3. To obtain ˆΣ, we have to invert a N N matrix, which incurs a compexity ofon 3. Therefore, the tota compexity for cacuating a ˆΣ is given by OLN 3, and the compexity of generating ˆΣ is given by OL 2 N 3. As cacuating Σ incurs the same compexity as cacuating ˆΣ, the compexity of the new agorithm isol 2 N 3 for each iteration. The PWF agorithm uses the same cacuation to generate Ω and incurs a compexity of OLN 3 for each Ω. Then, it uses the singuar vaue decomposition of Ω 2 H 2,ˆΩ, which incurs a compexity of ON 3. Since we need L of these operations, the tota compexity of the PWF agorithm is OL 2 N 3. For the WMMSE agorithm, it is shown in [9] that its compexity is OL 2 N 3. So, a three agorithms have the same computationa compexity per iteration if we use ON 3 matrix mutipication. Recenty, Wiiams [0] presents an ON matrix mutipication and inversion method. If we use this agorithm, then the new agorithm and the WMMSE agorithm have OL 2 N compexity since the N 3 factor comes from the matrix mutipication and inversion. However, in addition to L 2 number of matrix mutipications and inversions, the PWF agorithm has L number of N by N matrix singuar vaue decompositions. Therefore the compexity of the PWF agorithm is OL 2 N LN 3. VI. CONCLUSION We have proposed a new agorithm to sove the weighted sum-rate maximization probem in genera interference networks. Based on the poite water-fiing resuts and the rate duaity [5], the new agorithm updates the transmit signa covariance matrices in the forward and reverse inks in a symmetric manner and has fast and guaranteed convergence. We present an eegant convergence proof for this otherwise hard probem. Simuations demonstrate that the new agorithm has convergence speed cose to the state-of-the-art poite waterfiing agorithm, which is however not guaranteed to converge. 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