In this contribution we present the solution of the utility greedy discrete bit loading for interference limited multicell OFDM networks. Setting the utility as the sum of consumed power proportions, the algorithm follows greedy way to achieve the maximum throughput of the system. Simulation has shown that the proposed algorithm has better performance and lower complexity than the traditional optimal algorithm. The discussion of the results is provided

Li, Yonghua

Lv, Tiejun

Su, Xinghui

Yang, Shaoshi

Yu, Xuefen

English

Southampton (e-Prints Soton)

Utility Greedy Discrete Bit Loading for interferencelimited Multi-Cell OFDM systemXinghui Su, Tiejun Lv, Member, IEEE, Xuefen Yu, Shaoshi Yang, Yonghua LiKey Lab of Universal Wireless Communications, Ministry of Education,Beijing University of Posts and Telecommunications, Beijing, China 100876Abstract—In this contribution we present the solution of theutility greedy discrete bit loading for interference limited multi-cell OFDM networks. Setting the utility as the sum of consumedpower proportions, the algorithm follows greedy way to achievethe maximum throughput of the system. Simulation has shownthat the proposed algorithm has better performance and lowercomplexity than the traditional optimal algorithm. The discussionof the results is provided .I. INTRODUCTIONCooperative resource allocation is an effective method toavoid the inter-cell interference in multi-cell OFDMA sys-tems. Resource allocation for OFDMA networks has threemajor tasks: subchannel assignment, rate allocation, and powercontrol. Since the inter-cell interferences are affected by thesubchannel assignment, the corresponding rate adaptation andpower control, any change of the resource allocation in aspecific cell will affect the other nearby cells. Thus, thecooperative resource allocation is a very complex process inthe multi-cell OFDMA systems. This paper only considersthe situation when the subchannel allocations have been done.More specifically, this paper considers the cooperative resourceallocation in the multi-cell OFDM systems.The relevant researches could be found in the fields ofDSL systems, cognitive radio networks [1], [2], and [3]. Theoptimal spectrum balancing (OSB) algorithm has been shownin [4]. The OSB algorithm converts the maximum problemto the weighted problem and employs the dual decompositionto solve it. The iterative spectrum balancing algorithm whichfollows the idea of OSB is proposed and obtains much smallercomplexity [5], [6]. However, the weighted optimization needsto predetermine each user’s weight and their target rate, whichmakes the optimized result can’t be obtained directly. Thus,the weighted optimization algorithms has to check all thepossible weight combination in order to get the maximum,which make the algorithm become too complex. Further more,there are some internal disadvantages for OSB that couldbring performance degradation. These disadvantages will bediscussed in the follows.This research has been supported by Program for New Century ExcellentTalents in University (NCET) under Grant No. NCET-06-0091, NationalNatural Science Foundation of China (NSFC) under Grant No.60972075,Specialized Research Fund for the Doctoral Program of Higher Education(SRFDP) grant No. 20070013018 and the Fundamental Research Funds forthe Central Universities.Xinghui Su, Tiejun Lv (xhui.su@gmail.com, lvtiejun@tsinghua.org.cn)Our work invokes the greedy method and provides the utilitygreedy discrete bit loading for the interference limited multi-cell OFDM systems. The simulation results prove that theoptimal discrete bit loading algorithm performs better thanOSB. Further more, the utility greedy discrete bit loading hasmuch smaller complexity than OSB.II. SYSTEM MODELWe consider an uplink multiuser OFDMA Multi-Homecellsystem. Suppose that there are N (N ≥ 2) pairs of transmitterand receiver, each of them occupies the same span of tonesand is equipped with one transmit and one receive antenna.We assume that one receiver gets desired information fromone predetermined transmitter. Then the received signal-to-interference-plus-noise ratio (SINR) for the ith receiver at tonek isSINR[k]i =p[k]i gi,iN∑j=1,j =ip[k]j gi,j + σ, (1)where p[k]i is the transmitted power spectrum density (PSD)at tone k by the ith transmitter, gi,j is the channel gain fromthe jth to the ith receiver and σ is the additive, white andGaussian noise (AWGN) PSD.The goal of this paper is to maximize system’s throughputby allocating power for each transmitter at each tonemax⇀PN∑i=1K∑k=1r[k]is.t.K∑k=1p[k]i ≤ Pi,max,∀i,(2)where Pi,max is the total power constraint for the ith trans-mitter and r[k]i is the achievable bit loading of the ith receiveron tone [7]. r[k]i is obtained byr[k]iΔ= log2(1 +1ΓSINR[k]i), (3)where Γ denotes the SNR-gap to capacity, which is a functionof the desired BER, coding gain and noise margin. Formulation(3) could be rewritten asp[k]i − Γ(2r[k]i − 1) N∑j=1,j =ip[k]jgi,jgi,i= Γ(2r[k]i − 1) σgi,i(4)978-1-4244-3709-2/10/$25.00 ©2010 IEEEAnd since the modem supports only integer bitloading, thesearch space could be reduced according to exact bitloadings.Thereby these can be written in matrix form as[I−ΨG]P = ΨΩ (5)where PN×1 =[p[k]1 · · · p[k]N]T,ΩN×1 =[σ · · · σ ]T , ΨN×N =diag⎧⎨⎩Γ(2r[k]1 −1)g1,1, . . . ,Γ(2r[k]N −1)gN,N⎫⎬⎭ and GN×N =⎡⎢⎢⎢⎣0 g1,2 · · · g1,Ng2,1 0 · · · g2,N............gN,1 gN,2 . . . 0⎤⎥⎥⎥⎦.Then this cooperative resource allocation problem on tone kis feasible for all simultaneous transmitting-receiving pairs aslong as the matrix (I−Ψ ·G) is non-singular, i.e. invertible.Then the power allocated for rk isP = [I−ΨG]−1 ΨΩ. (6)This simple power allocation scheme, however, suffers fromlarge complexity O((bmax + 1)NK), where bmax is the max-imal bitloading that one modem could support. Consideringand K is large in OFDMA system, this problem becomesintractable. The OSB algorithm below reduces this complexityto some extent. And Utility Greedy Discrete Bit Loadingalgorithm we proposed further edges that complexity.III. UTILITY GREEDY DISCRETE BIT LOADINGThis work provides the utility greedy discrete solutionswhen such constraints are taken into account. For eachtone k (1 ≤ k ≤ K), we assume that the sum of bit loadingR[k] could be transmitted using power function ek (RK)Δ=[e[k]1 , . . . , e[k]N]. Actually, there are numerous rk which couldsatisfy R[k] =N∑i=1r[k]i while under the power constraint.Define the function γ asγ(R[k])={rk|N∑i=1r[k]i = R[k], r[k]i < rmax, k = 1 . . .K}(7)where rmax is the maximum bit loading constrained by thesystem . If rk is determined and available (I−ΛkG) isinvertible), the power function ek (RK) could be calculatedasek(R[k]) = pk = (I−ΛkG)−1 Λkσ. (8)Introduced the incremental power functionΔek(R[k]) Δ= [Δe[k]1 , . . . ,Δe[k]N ]={ek(R[k])− ek (R[k] − 1) , if R[k] ≥ 1ek(R[k]), if 0 ≤ R[k] ≤ 1.(9)Suppose the remaining power to be allocated for each pairis ΔP1, . . .ΔPN when the sum of bit loading on tone k isWall10mAHome Cell 1BHome Cell 2C DFig. 1. Homecell simulation environmentR[k]− 1. Then when the sum of bit loading increases to R[k],the utility function is set asu(R[k])=n∑i=1Δe[k]iΔPi(10)The proposed optimal discrete bit loading algorithm is asfollows:Initialization:Initialize rkΔ= [0, . . . , 0]T , F Δ= [0, . . . , 0]T for k = 1, ...,K.{F}k = 1 is the flag which indicates the feasibility ofadding more bits to the tone k under the power constraint,whereas {F}k = 0 means that the tone k is saturated. LetpkΔ=[p[k]1 , . . . , p[k]N]. The total bit loading of the system isinitialized as R =K∑k=1R[k] = 0.Bit loading Iteration:Step 1. Check the possibility to add one more bit to the totalbit loading of the system.1). Check that if it is possible to add one more the bit loadingto each tone. For the tone k,a) Generate the available rk set when add one more bit asγ(R[k]+1). For each rk ∈ γ(R[k] + 1), calculate the ek (RK)using (12). If the ek (Rk + 1) satisfies the power constraint,which could be denoted asK∑q=1,q =k(p[q]i + {ek (Rk + 1)}i)≤Pi,max, i = 1...N , set {F}k = 1.b) Search for r∗k = argr∗k∈γ(R[k]+1)min{u(R[k] + 1)}andrecord the relevant ek (Rk + 1) and utility u(R[k] + 1).2). If F equals to [0, . . . , 0]T , stop the iteration. On thecontrary, go to Step 2.Step 2. Updating the bit loading.1). Collect all the subchannels that could be added to one morebit as ν = {k| {F}k = 1, k ∈ {1, ...,K}} .2). Search for k∗ = argk∈νmin{u(R[k] + 1)}.3). Let rk∗ = r∗k∗ and pk∗ = ek∗ (Rk∗ + 1). Thus, Rk∗ =Rk∗ + 1 and R = R + 1, which means the total bit loadingincreases by one bit.4) Re-initialize F Δ= [0, . . . , 0]T for k = 1, ...,K.5). Go to Step1.Note that the complexity of the utility greedy discrete bitloading algorithm (UG) is VUG = O(K2bmax+1∑b=1bN).IV. SIMULATION AND DISCUSSIONThe utility greedy discrete bit loading algorithm is nu-merical simulated using the channel model of homecells.TABLE IHOMECELL SIMULATION PARAMETERSParameters ValueCell radius 10 mAntenna gain 0 dBiUE antenna gain 0 dBiDistance dependent pathloss 39 + 20log(r)+10n dBTx power [-10, 20] dBmBandwith (W) 100 MHzSubcarrier bandwidth 15 KHzNoise power on each subcarrier -174 dBm/HzNoise Figure 9 dBReferred to TR25.814 and R1-084026 of 3GPP, the simulationparameters are shown in TABLE I.The two homecells model is invoked for the simulations ofthis section. Assume that it contains only one user in eachhomecell. A wall exists between the two homecells which is avery common scene in the office building. Then the simulationenvironment is shown as Figure 1.In order to evaluate the performance of utility greedydiscrete bit loading algorithm, the comparison with the OSBwould be given. Because of the limit of conditions and hugecomplexity of OSB, this paper only compares the performanceof both algorithms in three typical pairs of locations, e.g. whenthe users of Homecell 1 and Homecell 2 locate respectivelyin A and B, A and D, C and D. When the users of the twohome cells locate in A and B, the interference for both twousers is very small. When the users of the two home cellslocate in A and D, the interference for user in home cell 1 issmall while the interference for user in home cell 2 is large.When the users of the two home cells locate in C and D, theinterference for both users are large. The simulations in thesethree pairs of locations could fully reveal the comparison theutility greedy discrete bit loading algorithm (UG Algorithm)and optimal spectrum balancing algorithm (OSB Algorithm).Suppose that the two homecells utilize the same TX powers.The total throughput of the system with different TX poweris depicted in Figure 2, Figure 3 and Figure 4.It’s obvious that the utility discrete bit loading algorithmalways behaves better performance than the optimal spectrumbalancing algorithm. The reason will be discussed in the nextsection. Considering the complexity of UG, it is easily knownthat the UG discrete bit loading algorithm has much smallercomplexity than the optimal spectrum balancing algorithm.V. RESULTS’ DISCUSSIONThe simulation results would be discussed in this section.It is not difficult to discover that the curve of the OSBalgorithm maintains on a value for some TX powers in Figure2and Figure3. Actually, this phenomenon is caused by somedisadvantage of the OSB algorithm. In order to make it clear,we first introduce the Lemma1.Lemma1: In order to maximize the throughput of the sys-tem, the two pairs i, j should share the tone k (k = 1, . . . ,K)-10 -5 0 5 10 15 2090010001100120013001400150016001700Throughput with powerThroughput /MbpsTx Power /dBmOSB AlgorithmUG Alogrithm?Fig. 2. Users of Home Cell 1 and Home Cell 2 loacte in A and B respectively.-10 -5 0 5 10 15 2070080090010001100120013001400150016001700Throughput with powerThroughput /MbpsTX Power /dBmOSB AlgorithmUG Algorithm?Fig. 3. Users of Home Cell 1 and Home Cell 2 loacte in A and D respectively.-10 -5 0 5 10 15 2060080010001200140016001800Throughput with powerThroughput /MbpsTX Power /dBmOSB AlgorithmUG Algorithm?Fig. 4. Users of Home Cell 1 and Home Cell 2 loacte in C and D respectively.simultaneously when satisfyp[k]i gi,ip[k]j gi,j≥ p[k]j gi,jZ[k]i, i, j = 1, . . . , N (11)where Z [k]i = σ + I[k]i , I[k]i is the interference for pair i ontone k.Otherwise, the pairs i, j should not transmit on the tone ksimultaneously whenp[k]i gi,ip[k]j gi,j<p[k]j gi,jZ[k]i, i, j = 1, . . . , N. (12)The proof of Lemma1 could be found in Appendix.With Lemma1, the cooperative resource allocation problemcould be divided into three states: spectrum sharing state,spectrum partitioned state and spectrum mixed state. Thedefinitions of each state are describe as follows.Spectrum sharing state, which is defined as all the pairsshare all the tones simultaneously.Spectrum partitioned state, which is defined as all thetones are only occupied by one pair at a time.Spectrum mixed state, which is defined as that sometones are utilized by only one pair while others are sharedby different pairs.The OSB algorithm drops its optimum when the dualproblem is decomposed into a set of sub-problems. After thedecomposition, the cooperative optimization on different tonescouldn’t be done which cause the performance degradation.Particularly, when the channel gain is considered in this paper,a pair has the same channel station on all the tones. Thenthe OSB algorithm is only effective on the spectrum sharingstate and spectrum partitioned state. When the TX power is inthe field of the spectrum mixed state, the OSB algorithm onlystays at a value which is the maximum in the spectrum sharingstate. Thus the curve of the OSB algorithm always maintainson a value for some TX powers in Figure2 and Figure3.Proof:In the spectrum partitioned state, the tone k is only occupiedby a pair of transmitter and receiver. Assume the pair i occupythe tone when the bit loading is Rk, thenRk = r[k]i = log2(1 +1Γp[k]i gi,jσ). (13)The power need to allocated by pair i isp[k]i =(2Rk − 1)Γσgi,j. (14)Suppose the R[k] +1 is achievable on tone k. When 2Rk 1, the utility could be obtained asu(Rk+1) =⎧⎨⎩p[k]i(Rk)ΔPiif r[k]i = Rk + 1p[k]j(Rk+1)ΔPj− p[k]i(Rk)ΔPiif r[k]j = Rk + 1, j = i.(15)Compare the different result of (20),p[k]j (Rk + 1)ΔPj− p[k]i (Rk)ΔPi− p[k]i (Rk)ΔPi≈ 2Δr[k]i − 2Δr[k]j2Δr[k]j+Δr[k]i−1, (16)where Δr[k]j is the bit loading could be added on tone kpotentially. It is obvious that the minimum of u (Rk + 1)will choose the pair who could obtain the most bit loading.According to [8], [9], it is optimum when choose the mini-mum u (Rk + 1). Thus, the utility greedy discrete bit loadingalgorithm is the optimal algorithm in the spectrum partitionedstate.VI. CONCLUSIONIn this paper, we propose the utility greedy discrete bit load-ing algorithm for the interference limited multi-cell OFDMsystems. And three kinds of spectrum states are proposed. TheUG algorithm is also proved to be the optimal algorithm inthe spectrum partitioned state. In our next research, we go onsearching for the optimal algorithm for the interference limitedmulti-cell OFDM systems as well as the multi-cell OFDMAsystems.VII. APPENDIXProof of Lemma1:Suppose two pairs i, j could transmit their data on tone kand l. Assume that these two tones has the same channel state,then Z [k]i = Z[L]i , Z[k]j = Z[L]j . When they both transmit theirdata simultaneously on the two tones, the PSD of each pairon each tone is p[k]i and p[k]j respectively. The total bit loadingisRk+l = 2 log2(1 + 1Γp[k]ig[k]i,jp[k]ig[k]i,j+Z[k]i)+ 2 log2(1 + 1Γp[k]jg[k]j,jp[k]ig[k]j,i+Z[k]j) . (17)Otherwise, when the each pair transmits it data onits unique tone, the total bit loading become R′k+l =log2(1 + 1Γ2p[k]igi,iZ[k]i)+ log2(1 + 1Γ2p[k]jgj,jZ[k]j).In order to compare these two data rate, let Rk+l minusR′k+l, we haveRk+l −R′k+lSINRΓ,pgZ⇒log2(p[k]igi,i/(p[k]igi,j)p[k]igi,j/Z[k]i)+ log2(p[k]jgj,j/(p[k]igj,i)p[k]igj,i/Z[k]j).(18)As seen from the above formulation, the two pairs shouldtransmit data on tone k simultaneously in order to achieve themaximum bit loading when p[k]igi,ip[k]jgi,j≥ p[k]jgi,jZ[k]iand p[k]jgj,jp[k]igj,i≥p[k]igj,iZ[k]j. Otherwise, when p[k]igi,ip[k]jgi,j<p[k]jgi,jZ[k]iand p[k]jgj,jp[k]igj,i<p[k]igj,iZ[k]j, the two pairs should be frequency orthogonal toachieve the maximum data rate.REFERENCES[1] L. Cao, L. Yang, and H. Zheng, “The impact of frequencyagility ondynamic spectrum sharing,” in Proc. of IEEE DySPAN, 2010.[2] C. Raphael, J. Huang, C. Mung, and M. Marc, “Autonomous SpectrumBalancing for Digital Subscriber Lines,” IEEE Transactions on SignalProcessing, vol. 55, no. 8, pp. 4241–4257, 2007.[3] J. Lee, R. Sonalkar, and J. Cioffi, “Multi-user discrete bit-loading forDMT-based DSL systems,” GLOBECOM-NEW YORK-, vol. 2, pp. 1259–1263, 2002.[4] R. Cendrillon, W. Yu, M. Moonen, J. Verlinden, and T. Bostoen, “Optimalmulti-user spectrum management for digital subscriber lines,” IEEETransactions on Communications, vol. 54, no. 5, pp. 922–933, 2006.[5] R. Cendrillon and M. Moonen, “Iterative spectrum balancing for digitalsubscriber lines,” in 2005 IEEE International Conference on Communi-cations, 2005. ICC 2005, vol. 3, 2005.[6] J. Papandriopoulos and J. Evans, “Low-complexity distributed algorithmsfor spectrum balancing in multi-user DSL networks,” in IEEE Interna-tional Conference on Communications, 2006. ICC’06, vol. 7, 2006.[7] P. N. X. W. Venturino, L., “An Improved Iterative Water-Filling Algorithmfor Multi-Cell Interference Mitigation in Downlink OFDMA Networks,”in Asilomar Conference on Signals, Systems and Computers, 2007.[8] A. Fasano, G. Di Blasio, E. Baccarelli, and M. Biagi, “Optimal discrete bitloading for DMT based constrained multicarrier systems,” in 2002 IEEEInternational Symposium on Information Theory, 2002. Proceedings,2002.[9] D. Welsh, “Matroid theory,” New York, 1976.

An Improved Iterative Water-Filling Algorithm for Multi-Cell Interference Mitigation

Autonomous Spectrum Balancing for Digital Subscriber Lines,”

Iterative spectrum balancing for digital subscriber lines,”

Low-complexity distributed algorithms for spectrum balancing in multi-user DSL networks,”

Matroid theory,”

Multi-user discrete bit-loading for DMT-based

Optimal discrete bit loading for DMT based constrained multicarrier systems,”

Optimal multi-user spectrum management for digital subscriber lines,”

The impact of frequencyagility on dynamic spectrum sharing,” in

Utility greedy discrete bit loading for interference limited multi-cell OFDM system

Utility greedy discrete bit loading for interference limited multi-cell OFDM system

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