Power Spectrum Optimization for Interference Mitigation via Iterative Function Evaluation

Power Spectrum Optimization for Interference Mitigation via Iterative Function Evauation Hayssam Dahrouj, Wei Yu, Taiwen Tang, and Steve Beaudin Eectrica and Computer Engineering Dept., University of Toronto, Canada BinQ Networks Inc., Kanata, Ontario, Canada Emais: {hayssam,weiyu}@comm.utoronto.ca, eric.tang@binqnetworks.com, steve.beaudin2@yahoo.com Abstract This paper proposes practica methods for and examines the benefit of dynamic power spectrum optimization for interference mitigation in wireess networks. The paper envisions a distributed antenna system, depoyed as a means to increase the network capacity for areas with high data traffic demand. The network comprises severa access nodes AN), each serving a fixed set of remote radio units caed remote terminas RT). The RTs beonging to each AN are separated from each other using orthogona frequency division mutipe access OFDMA) over a fixed bandwidth, where ony one RT is active at each frequency tone. The system performance is thus imited by internode interference soey, and no intranode interference. This paper proposes methods for power spectrum optimization based on the idea of iterative function evauation. The proposed methods provide a significant improvement of the overa network throughput, as compared to conventiona wireess networks with fixed transmit power spectrum. The proposed methods are computationay feasibe and fast in convergence. They can be impemented in a distributed fashion across a access nodes, with reasonabe amount of internode information exchange. Some of the proposed methods can be further impemented asynchronousy at each AN, which makes them amenabe to practica utiization. I. INTRODUCTION Interference is a major botteneck in wireess network design. Deveoping and optimizing advanced, yet practica, interference mitigation techniques is particuary important nowadays, due to the rapid pace of growth of wireess networks with enormous data usage, and the scarcity of the avaiabe radio resources, e.g. bandwidth and transmit power. Dynamic power spectrum optimization is an important cass of interference mitigation methods that seek to increase the network capacity and reiabiity via power contro. The present paper aims to deveop nove, feasibe, practica methods for power spectrum optimization. Dynamic power spectrum optimization is especiay appicabe to distributed antenna system DAS) where the basestation transmit capabiity is enhanced by adding mutipe remote radio units. The setup under discussion assumes a ceuar network comprising severa remote terminas RT), each covering a reativey sma area as a means to increase the network capacity for areas with dense data traffic. The RTs are then connected to access nodes ANs) via wireess inks which are meant to repace the expensive optica fiber inks. The ANs are responsibe for the transmission strategies and radio resource management for the different RTs. From a design perspective, the interest of this paper is to mitigate the internode interference, thereby maximizing the aggregate data capacity of the RTs, via practica power spectrum optimization methods. The main chaenge in power contro remains the probem of finding computationay efficient methods to aocate the power of the different transmitters across the different frequency tones. The power spectrum optimization probem is especiay we studied in the iterature for digita subscribers ines DSL) [1], [2], [3], [4], [5]. For wireess networks, power contro can be performed using the concept of interference price [6], [7], [8], [9]) which quantifies the effect of interference between the mutipe transmitter-receiver pairs in the wireess medium. The power contro methods can aso be incorporated with scheduing [1], [11], [12]. This goa of this paper is to study a cass of iterative function evauation based methods for power management. These methods have an advantage of ow computationa compexity, whie showing a significant gain compared to the conventiona maximum power transmission poicy. Further, these methods end themseves to distributed impementation with reasonabe amount of inter-access-node information exchange. Some of the methods can aso be impemented asynchronousy, which makes them amenabe to practica utiization. The proposed methods make use of channe measurements done on a per-tone basis for every AN-RT pair. The measurements are subsequenty provided to either a centra server for further centraized processing, or to each of the severa access nodes for distributed processing. These measurements are of particuar interest in fixed depoyment scenarios, where the channes are sow varying. Further, the measurement can be done periodicay, thus aowing the adaptation of radio resource aocations with the dynamicay changing environment. In this paper, the measurements are utiized for internode interference mitigation via joint dynamic power spectrum adaptation and scheduing. II. SYSTEM MODEL AND PROBLEM FORMULATION A. System Mode Consider a wireess mutice network with L ANs and K RTs per AN, with singe antennas at both the ANs and RTs. The RTs beonging to each AN are separated from each other using orthogona frequency division mutipe access OFDMA) of N subcarriers over a fixed bandwidth, where ony one RT is active at each frequency tone. This paper

6 4 AN RT done assuming a fixed schedue. The focus of this paper is the power aocation step, which is essentiay a weighted rate-sum maximization probem on a per-tone basis: 2 y m) max w k r n k s.t. 3) 2 4 where r n k = og ) hn k 2 Γσ 2 + j Pn j hn jk 2 ) 4) Fig. 1. 6 6 4 2 2 4 6 x m) A distributed antenna system with seven 7 ANs and 4 RTs per AN. aims to use power management methods to aeviate internode interference. In particuar, the th AN may aocate its power at each tone n {1,,N}, depending on the scheduing assignment of RTs and the channe gains between the AN-RT pairs. Let k = f,n) and k = fj,n) be the schedued RTs of the th AN and the jth RT respectivey, both at the nth tone. The received signa at the kth RT is a summation of the intended signa and the internode interference: y n = h n kx n + j h n jkx n j +z n 1) where x n is a compex scaar denoting the information signa for the kth RT, h n jk C is the channe from the jth AN to the kth RT, and z n is the additive white Gaussian noise with variance σ 2 /2 on each of its rea and imaginary components. Fig. 1 iustrates the system mode for seven ANs and four RTs per AN. B. Probem Formuation For competeness, the overa network optimization probem can be stated as a maximization of the og utiity function: max og Rk ),k s.t. R k = ogsinr n k) {n k=f,n)},n SINR n = hn k 2 Γσ 2 + j Pn j hn jk 2 ) where R k is the ong term average rate of the kth RT of the th AN, is the maximum power constraint imposed on each AN at each tone, and where the maximization is over the scheduing assignment k = f, n), and the power spectra density eves. The Γ is the signa-to-noise ratio SNR) gap. This paper adopts an iterative scheduing and power contro poicy simiar to that in [1], [11], [12], in which the scheduing is done assuming fixed power, and power optimization is 2) is the instantaneous rate of the schedued kth RT for the th AN at the nth tone, the weights comes from the scheduing objective e.g. w k = R 1 k for proportiona fairness), and where the maximization is over the set of powers. The rest of the paper examines practica numerica methods to sove 3) and quantifies their performance. III. POWER SPECTRUM OPTIMIZATION The weighted sum-rate maximization probem 3) is a nonconvex optimization probem, whose goba optima soution is not easy to find. Like many previous approaches, this paper aso aims at oca optima soutions, but with a focus on reduced computationa compexity. In particuar, the current paper considers a cass of strategies based on an iterative function evauation approach. The nove methods, described beow, are simpe to impement, fast in convergence, and simiar in performance to traditiona fu-bown Newton s method. Some of the proposed methods can be impemented in a distributed fashion, and asynchronousy at each AN, with no need for step size choices. A. Iterative Function Evauation Methods IFEM) 1) Fu-IFEM: We begin by taking the gradient of the objective function R of the probem 3) with respect to : R rk n = w k P n + rjk n w jk j i = w k SINR n ) P n SINR n h n jk w 2 jk i j Pn i hn ijk 2 SINR n 5) j j where SINR n j is defined as: SINR n Pj n j = hn jjk 2 Γσ 2 + i j Pn i hn ijk 2 ) A oca optima soution must be such that the above gradient is zero. The main idea of the iterative function evauation method IFEM) is to rewrite the above as: SINR w n k SINR = n j w h n jk jk 2 i j Pn i hn ijk 2 SINR n j 6) 7)

It is now possibe to compute a the terms on the right-handside of the equation using the current power aocation, and update the new power aocation as above. This step is a simpe function evauation, which can be done iterativey, hence this method is caed Fu-IFEM. More formay, the power eve of every AN at every tone,, is updated from step t to t+1 according to the foowing: t+1) = w k SINR n t) SINR n t) j w jk h n jk 2 σ 2 + i j Pn i t) hn ijk 2 SINR n j t) SINR n j t) where the maximum power constraint is aso taken into account. 2) θ-ifem: The above fu-ifem agorithm requires finding the individua signa-to-interference-and-noise ratios SINRs) at every iteration. To further simpify the computationa compexity, we propose the foowing heuristic method that repaces the per-iteration SINR s with the vaues of SINR s cacuated under the initia maximum power transmission strategy. Athough this method does not guarantee oca optimaity, it has the advantage that its convergence is easy to prove. Further, it provides significant gain as compared to the maximum power transmission strategy, as the simuation resuts show. This method, caed θ-ifem, finds the power eve according to the foowing update equation: t+1) = where w k j w h n jk jk 2 θ n j = σ 2 + i j Pn i t) hn ijk 2 θ n j SINR n j SINR n j SINR n SINR n 8) 9) 1) is a fixed factor cacuated from the maximum power transmission strategy. 3) IFEM: To further simpify the power update equations, one can set θj n to 1. This is in fact a high-sinr approximation of the probem. The resuting update equation becomes: t+1) = w k j w jk h n jk 2 σ 2 + i j Pn i t) hn ijk 2 11) For physica patforms that ony permit each AN to aocate one vaue for the power across a tones, the power can be found by taking the average of the power vaues of IFEM. The resuting method is caed AP IFEM. B. Convergence Anaysis The convergence of fu-ifem is difficut to estabish in fu generaity. But the foowing convergence resut is avaiabe for both θ-ifem and IFEM under both the synchronous and asynchronous modes. Proposition 1. Starting from any initia ), both θ-ifem and IFEM agorithms converge to a unique fixed point. Furthermore, the convergence is sti guaranteed under a totay asynchronous mode. Proof: The proof is based on coroary 1 in [13], as both update equations of θ-ifem and IFEM, written as t + 1) = [ g Ψ n t) )], satisfy the foowing standard function properties: 1) If,n, then g Ψ n) >. 2) If P n,n, then g Ψ n) g Ψ n). 3) For ρ > 1, we have ρg Ψ n) > gρψ n ),n. where the variabes, = 1,,L, are stacked into one vector Ψ n. The convergence to the unique fixed point and the asynchronous convergence foow as a consequence. C. Connection with SCALE [3]) In [3], a power contro agorithm named SCALE is proposed. The agorithm is motivated by geometric programming. SCALE is a two-stage agorithm, and in the notation of this paper can be thought of as having a θ-ifem-ike agorithm in the inner oop, and a θj n update in the outer oop. The SCALE agorithm definesαj n = SINRn j SINR, so thatθ n j n = αn j /αn. It runs j iterative function evauation with fixed αj n in the inner oop, then update αj n based on the resuting SINRs in the outer oop. The fu-ifem proposed in this paper is essentiay a simpification of SCALE. Instead of the two-stage process in which θj n is updated in an outer oop, fu-ifem impicity updates the power vector and θj n at the same time. In addition, as shown in the simuation resuts, the use of a singe fixed θj n derived from the maximum transmit power eve may aready be near optimum eading to θ-ifem), thus θj n may not need to be updated at a. Further, at high SINR, can be set to 1, eading to IFEM. θ n j D. Comparison with Newton s Method As a baseine comparison, we aso describe the foowing Newton s method NM) update equation as in [12]: t) = t+1) = [ t)+µ t)] Smax, 12) where µ is the step size, and w k t) w k t))2 1 SINR n ) 1 1 SINR n where τj n is the interference price defined as h n jk 2 j τ n j τj n = w jk i j Pn i t) hn ijk 2 SINR n j ) 2 13) 14) where k = fj,n). The main disadvantage of the Newton s method is that the choice of step size cannot be easiy done in a distributed fashion, and certainy not asynchronousy.

Ceuar Layout Hexagona Number of ANs 7 Frequency Reuse 1 Number of RTs per AN 4 AN-to-AN Distance d 1 AN-to-RT Distance d 2 Dupex TDD Channe Bandwidth 1 MHz AN Max Tx Power per Subcarrier -32.7 dbw SINR Gap 12 db Tota Noise Power Per Subcarrier -158.61 dbw Distance-dependent Path Loss 128.37.6og 1 d) Samping Frequency 11.2 MHz FFT Size 124 TABLE I SYSTEM MODEL PARAMETERS To simpify the computations required at each iteration of Newton s method, we aso propose a high-sinr Newton s method HSNM) where the update equation 13) is approximated as t) = w k t) j w jk IV. SIMULATIONS h n jk 2 σ 2 + i j Pn i t) hn ijk 2 w k t))2 15) This section evauates the benefit of the proposed power spectrum optimization methods on a wireess network comprising seven ANs, and 4 RTs per AN, over 1MHz bandwidth. The transmission of each AN to its own RTs interferes with the other ANs transmissions. RTs beonging to one AN are separated from each other using OFDMA with 124 subcarriers, where ony one RT is active at each frequency tone. The parameters used in simuation are as outined in Tabe I. The AN-to-AN distance is set to d 1 ; the AN-to-RT distance is set to d 2. Both d 1 and d 2 vary so as to study the performance of the proposed methods for various topoogies. For iustration purposes, the weighting factors w k in probem 3) are set to 1,,k), which aows a sum-rate comparison. Fig. 2 shows the sum-rate performance over a ANs for a network with AN-to-AN distance d 1 =.5km and ANto-RT distance d 2 =.15km over different reaizations of the channe. Fig. 2 shows that there is a sma performance oss due to the high SINR approximation i.e. IFEM and HSNM have a ower performance as compared to fu-ifem). Nevertheess, IFEM and HSNM outperform the maximum power method significanty. We aso observe that AP IFEM, which aocates one power vaue for each AN across a the tones, is aways superior to the maximum power method. Tabes II, III, and IV iustrate the performance of IFEM for different network topoogies. Tabe II considers the effects of ce sizes, and shows that the benefit of power optimization is more pronounced in a sma-ce setting, where the interference eve is higher. Tabes III and IV examine the effect of RT Tota Sum Rate Across A Hubs in bps/hz 7 68 66 64 62 6 58 56 54 52 IFEM, HSNM Fu IFEM AP IFEM Max Power 5 1 2 3 4 5 6 7 8 9 1 Number of Reaizations Fig. 2. Sum-rate in bps/hz over 7 ANs, 4 RTs per AN. AN-to-AN distance is.5km. AN-to-RT distance is 15m. Sum Rate bps/hz) Sma-ce d 1 =.5km) Large-ce d 1 = 1km) IFEM 6.68 91.33 HSNM 6.68 91.33 Fu-IFEM 62.61 91.58 Max Power 53.1 86.22 Fu-IFEM Gain 18.1% 6.2% TABLE II 7 ANS, 4 RTS PER AN. d 1 IS THE AN-TO-AN DISTANCE. AN-TO-RT DISTANCE d 2 IS 15M. ocations within each ce. It is shown that the benefit of power optimization is noticeaby higher for ce-edge users, where the interference is arger. It can be observed from Tabes II, III, and IV that IFEM and HSNM aways have the same performance, as both empoy a high-sinr appoximation. However, at the ce-edge of sma ces, where the SINR eve is not sufficienty arge to justify the high-sinr approximation, IFEM and HSNM become inferior to fu-ifem. This is, however, not the case for ceedge users of arger ces, i.e. d 1 = 1km, shown in Tabe IV, where SINR vaues are arger, and where IFEM, HSNM and fu-ifem again have simiar performance. In a cases, IFEMs aways remain superior to the maximum power method. Figs. 3 and 4 compare the convergence performance of IFEM agorithms with the Newton s method. For fair comparison, the Newton s method is potted here with a constant step size of 1. As seen in Fig. 3, because of the constant step size, the Newton s method has a poor performance initiay, and IFEM converges faster overa. Note that the convergence speed comparison depends on the SINR. Fig. 3 corresponds to a high SINR situation, where IFEM outperforms the Newton s method. Fig. 4 shows an opposite situation, at a reativey ow SINR, where the convergence of the Newton s method is faster than the fu-ifem. Note that at high SINR, the achievabe sum-rate performances of IFEM, θ-ifem, fu-

Sum Rate bps/hz) Ce-edge d 2 = 333m) Ce-center d 2 = 125m) IFEM 34.84 78.39 HSNM 34.84 78.39 Fu-IFEM 41.11 78.77 Max Power 3.54 71.91 Fu-IFEM Gain 34.6% 9.5% TABLE III 7 ANS, 4 RTS PER AN. AN-TO-AN DISTANCE IS.5KM. Sum Rate bps/hz) Ce-edge d 2 = 667m) Ce-center d 2 = 25m) IFEM 44.86 83.55 HSNM 44.86 83.55 Fu-IFEM 46.86 84.24 Max Power 41.49 8.18 Fu-IFEM Gain 12.9% 5.1% TABLE IV 7 ANS, 4 RTS PER AN. AN-TO-AN DISTANCE IS d 1 = 1KM. IFEM and Newton s method are simiar, whie at ow SINR, fu-ifem and Newton s method outperform θ-ifem, which in term outperforms IFEM. V. CONCLUSION Given the scarcity of the avaiabe radio resources, the performance of future wireess networks is expected to widey depend on the feasibiity of the dynamic power spectrum optimization methods. This paper presents nove and practica methods to manage interference in wireess systems. The proposed methods represent efficient ways of updating the power spectra density eves for a transmitters, based on the frequency domain channe measurements. The proposed methods, fu-ifem, θ-ifem and IFEM, are simpe methods, with ow computationa compexity and fast convergence. Their performance is simiar to the fu-bown Newton s method, but without the need for step size choices. They can aso be impemented in a distributed fashion, and asynchronousy at each transmitter, and are therefore exceent fits for practica appicabiity. REFERENCES [1] R. Cendrion, W. Yu, M. Moonen, J. Verinden, and T. Bostoen, Optima mutiuser spectrum baancing for digita subscriber ines, IEEE Trans. Commun., vo. 54, no. 5, pp. 922 933, May 26. [2] W. Yu and R. Lui, Dua methods for nonconvex spectrum optimization of muticarrier systems, IEEE Trans. Commun., vo. 54, no. 6, pp. 131 1322, Jun. 26. [3] J. Papandriopouos and J. S. Evans, Low-compexity distributed agorithms for spectrum baancing in muti-user DSL networks, in Proc. IEEE Inter. Conf. Commun. ICC), Istambu, Turkey, Jun. 26. [4] W. Yu, Mutiuser water-fiing in the presence of crosstak, in Inform. Theory and App. Workshop, San Diego, U.S.A., 27. [5] P. Tsiafakis, M. Dieh, and M. Moonen, Distributed spectrum management agorithms for muti-user DSL networks, IEEE Trans. Signa Process., vo. 56, no. 1, pp. 4825 4843, Oct. 28. [6] J. Huang, R. A. Berry, and M. L. Honig, Distributed interference compensation for wireess networks, IEEE J. Seect. Areas Commun, vo. 24, no. 5, May 26. [7] J. Yuan and W. Yu, Distributed cross-ayer optimization of wireess sensor networks: A game theoretic approach, in Proc. IEEE Goba Teecommun. Conf. Gobecom), San Francisco, U.S.A., 26. Tota Sum Rate across a Hubs in bps/hz 92 91 9 89 88 87 Newton Method 86 IFEM θ IFEM Fu IFEM 85 5 1 15 2 25 3 Iterations Fig. 3. Sum-rate in bps/hz versus the number of iterations, over 7 ANs, 4 RTs per AN. AN-to-AN distance is 1km. AN-to-RT distance is 15m. It shows the convergence of the different methods at high SINR. Tota Sum Rate across a Hubs in bps/hz 42 4 38 36 34 Newton Method 32 Fu IFEM θ IFEM IFEM 3 1 2 3 4 5 6 Iterations Fig. 4. Sum-rate in bps/hz versus the number of iterations, over 7 ANs, 4 RTs per AN. AN-to-AN distance is.5km. AN-to-RT distance is 333m. It shows the convergence of the different methods at ow SINR. [8] C. Shi, R. A. Berry, and M. L. Honig, Distributed interference pricing for OFDM wireess networks with non-separabe utiities, in Proc. Conf. on Inform. Sciences and Systems CISS), Princeton, NJ, Mar. 28, pp. 755 76. [9] F. Wang, M. Krunz, and S. Cui, Price-based spectrum management in cognitive radio networks, IEEE J. Se. Top. Signa Processing, vo. 1, no. 2, pp. 74 87, Feb. 28. [1] L. Venturino, N. Prasad, and X. Wang, Coordinated scheduing and power aocation in downink mutice OFDMA networks, IEEE Trans. Veh. Techno., vo. 58, no. 6, pp. 2835 2848, Ju. 29. [11] A. L. Stoyar and H. Viswanathan, Sef-organizing dynamic fractiona frequency reuse for best-effort traffic through distributed inter-ce coordination, in INFOCOM, Apr. 29. [12] W. Yu, T. Kwon, and C. Shin, Joint scheduing and dynamic power spectrum optimization for wireess mutice networks, in Conference on Information Science and Systems CISS), Princeton, NJ, Mar. 21. [13] R. Yates, A framework for upink power contro in ceuar radio systems, IEEE J. Se. Areas Commun., vo. 13, no. 7, pp. 1341 1347, Sep. 1995.