© Copyright 2007 IEEEA potential usefulness of a priori time-of-arrival (TOA) information for asynchronous co-channel interference (CCI) cancellation is addressed. An interval-based maximum likelihood (ML) benchmark is developed and compared to the averaged ML benchmark and regularized second-order semi-blind (SB) solution that do not use the TOA information. It is found that in short burst scenarios the SB algorithm demonstrates performance that is close to the interval-based ML benchmark. Furthermore, it is shown that SB may outperform the ML benchmarks in the ML breakdown situation. For longer bursts, the known TOA information can significantly improve the performance.Kuzminskiy, A.M. and Abramovich, Y.I

Abramovich, Yuri

Kuzminskiy, Alexandr M.

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Interval-Based Maximum Likelihood Benchmark for Adaptive Second-Order Asynchronous CCI Cancellation

Adelaide Research &amp; Scholarship

   The following paper posted here is not the official IEEE published version. The final published version of this paper can be found in the Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (2007 : Honolulu, Hawaii):pp.II-865-II-868   Copyright © 2007 IEEE.   Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.  INTERVAL-BASED MAXIMUM LIKELIHOOD BENCHMARK FOR ADAPTIVESECOND-ORDER ASYNCHRONOUS CCI CANCELLATIONAlexandr M. Kuzminskiy*Bell LaboratoriesAlcatel-LucentThe Quadrant, Swindon SN5 7DJ, UKE-mail: ak9galcatel-lucent.comABSTRACTA potential usefulness of a priori time-of-arrival (TOA) informationfor asynchronous co-channel interference (CCI) cancellation is ad-dressed. An interval-based maximum likelihood (ML) benchmarkis developed and compared to the averaged ML benchmark and reg-ularized second-order semi-blind (SB) solution that do not use theTOA information. It is found that in short burst scenarios the SB al-gorithm demonstrates performance that is close to the interval-basedML benchmark. Furthermore, it is shown that SB may outperformthe ML benchmarks in the ML breakdown situation. For longerbursts, the known TOA information can significantly improve theperformance.Index Terms- Asynchronous co-channel interference, semi-blind second-order filtering, interference time-of-arrival, maximumlikelihood benchmark.1. INTRODUCTIONSpace-time (frequency) equalization and interference cancellationtechniques in wireless communications usually exploit known train-ing (pilot) symbols to estimate the weights of an antenna array. Theunderlying assumption for these techniques is that the training datais reliable since the co-channel interference (CCI) overlaps with thetraining symbols of the desired signal. Normally, this is the case forthe synchronous CCI, which has the same time-frequency structureas the desired user. Asynchronous cells, packed transmission, inter-ference avoidance schemes, and other techniques lead to more com-plicated asynchronous or intermittent CCI scenarios [1] - [3] and oth-ers, where the interference may partially overlap or not overlap withthe training data of the desired signal. For example, a cross-layeranalysis in [2] demonstrated that this kind of interference is espe-cially relevant for visiting (distant) users in an Open Access Network(OAN) built on wireless local area network (WLAN) with mediumaccess control (MAC) based on carrier sensing multiple access withcollision avoidance (CSMA/CA).A second-order statistic adaptive SB algorithm for asynchronousCCI cancellation is proposed and studied in [4], [5]. It is based onregularization of the conventional training-based least squares (LS)solution by means of the weighted covariance matrix estimated over*Part of this work has been done in the context of the IST FP6 MEM-BRANE project.Yuri I AbramovichIntelligence, Surveillance &Reconnaissance Division, DSTOPO Box 1500, Edinburgh SA 51 1 1, AustraliaEmail: Yuri.Abramovichgdsto.defence.gov.authe whole data interval. It is shown in [4], [5] that its performance intypical asynchronous CCI scenarios is close to the performance of anon-asymptotic ML benchmark jointly estimated over both the train-ing and working intervals. This benchmark is based on the stochasticML bounds developed in [6].Similarly to the SB algorithm, the benchmark in [4], [5] usesthe averaged estimates of the covariance matrices over the trainingand working intervals without recovering a temporal interferencestructure, i.e., TOA for different interference components. Poten-tially, TOA could be obtained by means of some detection of abruptchanges algorithms [7]. It is clear that this additional a priori infor-mation could be taken into account by interference cancellation al-gorithms. Obviously, in some scenarios the additional TOA informa-tion may be critically important. For example, ifnon-overlapping in-tervals affected by different interference components could be foundat the data interval, then these intervals could be processed using dif-ferent weight vectors instead of one set of weights trying to cancelall the interference components as in the average SB solution in [4],[5]. In other scenarios an advantage of such modified algorithm maybe less obvious.Taking into account that TOA estimation may be a difficult prob-lem especially for short intervals, it is important to develop a mod-ified interval-based non-asymptotic ML benchmark assuming theknown TOA information. This benchmark can be useful for as-sessment of the potential performance in particular scenarios. If theinterval-based ML benchmark demonstrates close to the averagedML benchmark or SB performance, then additional efforts for TOAobtaining may not be justified in such scenarios.In this paper we develop such an interval-based non-asymptoticbenchmark and apply it in the particular asynchronous CCI scenariotypical for visiting user collisions in an OAN WLAN system withCSMA/CA [2]. We show that for short data bursts the interval-basedand averaged benchmarks demonstrate close performance. Further-more, in the high signal-to-noise (SNR) case the SB algorithm mayoutperform the benchmarks because of the ML breakdown. On thecontrary, for relatively long bursts with the low probability to meetshort intervals with different interference scenarios, the additionalTOA information may significantly increase the potential perfor-mance.In Section 2 we describe the data model and formulate the prob-lem. In Section 3 an interval-based non-asymptotic ML benchmarkis developed. The simulation results are given in Section 4. Section5 concludes the paper.1-4244-0728-1/07/$20.00 ©2007 IEEE II- 865 ICASSP 2007Authorized licensed use limited to: Alcatel Lucent. Downloaded on November 10, 2008 at 04:43 from IEEE Xplore.  Restrictions apply.2. DATA MODEL AND PROBLEM FORMULATIONWe consider the following narrowband, e.g., for a separate OFDMsub-carrier, data model of the signal received by an antenna array ofK elements:Mx(n) = hs(n) + E gmum(n) + z(n), (1)m=lwhere n = 1 ... N is the time index; x(n) E CK 1 is the vectorof observed outputs of an antenna array; s(n) is the desired signal,E{lsl2} = p, E{s(q)s*(g)} = O, q g, E{ } denotes expectation;um(n), m = 1 ... M are the M < K -1 components of CCI:signal p, is unknown, the power of the training signal Pt is set to1. Note, that this modelling means that strictly speaking we haveto simulate different random-like training sequences to accurately fitinto the Gaussian assumption. In what follows for analytical deriva-tions we accept Gaussian random-like training signal model, whilefor actual simulations we model the desired signal and CCI as inde-pendent streams of random (+1 + jl)/V2 symbols. Another sim-plification is that we assume that all the intervals Tm ofTm symbols,m = O, ... , M with different interference scenarios are long enoughto form sufficcient statatistics for each interval, i.e., Tm > K. Thisassumption significantly simplifies Gaussian modelling and do notchange the overall results significantly.A signal estimate can be found as the output of a spatial filter:E{Lum(q)u4n(g)} {= vO,q = g E fmq = gDfmq gg(2)JVm is the appearance interval for the m-th interference compo-nent, z(n) E CK 1 is the vector of noise, E{z(n)z*(n)} PoIK,E{z(q)z*(g)} = 0, q :A g and hECKe 1 and gm E CK 1 are thevectors modelling linear propagation channels for the desired sig-nal and interference. All propagation channels are assumed to bestationary over the whole data slot and independent for different an-tenna elements and slots. The desired signal, noise, and all interfer-ence components are assumed to be independent circular Gaussianprocesses.The training interval of Nt > K + 1 samples: s(n), n=1, . . , Nt, is locaded in the preamble and known at the receiver. Theworking data interval n = Nt + 1, . . ., N of Nd N -Nt samplesis defined as the rest of the slot.The interference appearance intervals nfm......,N ofNm = N -m + 1 samples, where nnm+ > nm > Nt form = 1, ... , M -1, are assumed to be known at the receiver forthe interval-based ML benchmark and not known for the SB algo-rithm and averaged ML benchmark. All the propagation channels,interference and noise power are not known at the receiver.The introduced model is illustrated in Fig. 1. It correspondsto the situation where the CCI components in a "hidden terminal"environment appear with random delays because of the CSMA/CAprotocol as shown in [2]. The important feature of this model is thatsome of the intervals with different interference scenarios in Fig. 1may be quite short leading to estimation difficulties.Training interval Data interval'1 t |Desired signal,l Nt, 7n1 :nX2f ~~~~~~~~CCI1CCI2To T T2Fig. 1. Narrowband data model for two-component asyn-chronous CCISimilarly to [4], [5], for simplification of Gaussian modellingwe assume that the training samples are generated as i.i.d. com-plex Gaussian random values. While the actual power of the usefuls(n) = w* (n)x(n), n = Nt + 1, N. (3)In the case of averaged or "stationarized" processing a fixed weightvector w(n) = w is used for the whole data interval. Particularly,the SB solution [4], [5] can be expressed as follows:WSB = [(1 -)Rt +6Rdl rt, (4)where Rt= NtC 1Nt x(n)x* (n) and rt = NtC 1Nt s*(n)x*(n)are the covariance matrix and cross-correlation vector estimated overthe training interval, Rd= Nd 1NN t±1 x(n)x*(n) is the co-variance matrix estimated over the whole data interval and 0 < 6 <1 is the regularization coefficient.One can see that the SB estimator (4) contains the conventionalLS solution wLS Rt= 1rt as a particular case for 6 = 0.In the case of an interval-based processing, it is naturally to ap-ply different spatial filters at different intervals Tm ofTm symbols:w(n) =wm, n Tm,m=O,...,M. (5)The problem is to find a ML bound for such estimates and com-pare its efficiency with the average ML benchmark [4], [5] and theSB algorithm (4) that do not use the TOA information.3. INTERVAL-BASED NON-ASYMPTOTIC MLBENCHMARK3.1. Optimization criterionUnder the introduced data model 1, the covariance matrices at thetraining and data intervals are:Rt= VV* + POIK, (6)Rm = Rm-, + gmg: mT = 1, . .., M, (7)where Ro = Rt and v = vph.During the training interval we observe (K+ 1)-dimension inde-pendent Gaussian training vectors x(n) = [s(n), xT(n)I]T, n E Art.Taking into account that Nt > K + 1, the sufficient statistic at thetraining interval isNtRt = Nt-1 Ex(n)x*(n)n=lPt ritrt RtJ, (8)1 Some simplifications are adopted for the data model in Section 2 such asno CCI at the training interval, Tm > K and all the CCI components last tillthe end of the data burst. The proposed benchmark can be easily modifiedfor the general scenario, e.g., undersumpled case Tm < K can be addressedsimilarly to [8]II- 866Authorized licensed use limited to: Alcatel Lucent. Downloaded on November 10, 2008 at 04:43 from IEEE Xplore.  Restrictions apply.where Pt = NtC1Z nt1 s(n)*s(n), rt = NtC s(n)*x(n)Rt = Nt Z 1 x(n)x*(n) andE{Rt} [ v*Rt (9)During the working interval we observe K-dimension indepen-dent Gaussian vectors x(n), n C Tm and again because ofTm > K,the sufficient statistics at the working intervals are Rm=Tmn1Ir , x(n)x*(n), where E{RA} = R, form= 0,... ,M.Taking into account the structure of covariance matrices at dif-ferent intervals, the admissible set of the optimization parameterscan be introduced as follows:At [ b bb* + dIK 1>0 (10)Am =Am 1+cmcr >0, m =1,..., M, (11)Ao = bb* + dIK > 0, (12)where b and cm, m = 1, ... I M are a K x I vectors and d is apositive scalar.Since sufficient statistics Rt and Rm are non-degenerate ma-trices, the ML estimates could be obtained via maximization of amonotonic function of the product of the M + 2 likelihood ratios(LR):Find max -y(At, Am), (13)b,c, ,dM-y(At, Am) =yt(At) H 'ym(Am)T-'Nt, (14)m=O''t(At) =det(At Rt)exp(K + 1) (15)exp[tr(At 1Rt)1det(Am1Rtm)exp(K)tym(Am) exp[tr(A-1Rm)] 7 (16)3.2. InitializationFor Monte-Carlo simulations in Section 4 we adopt an approximaterecursive procedure that estimates vector cm assuming that matrixAm- 1 is known:K-1d[°] = (K -1)-'1 ci, b[°i=l[0]Co rt,cm AmxAm2 1Vmax, (18)Am + c[0]c[°]*m 1 Ml (19)where Amax and Vmax are the maximum eigenvalue and the cor-A_1 A A_1responding eigenvector of matrix Am2 iRmAm2 1 and oi are theeigenvalues of matrix Rt in ascending order.The ML estimate of a partially known covariance matrix used in(18), (19) is based on the algorithm developed in [9]. Other initial-ization procedures, e.g., similar to [4], [5], can be applied as well.3.3. Outliers selectionAs far as the optimization problem formulated in Section 3.1 is notconvex, any initialization and locally convergent optimization algo-rithm cannot guarantee finding the global solution. A technique thatallows us to decide if the local solution is far away from the globalmaximum, i.e. it can be classified as an outlier, is developed in [6].For benchmarking applications as in this paper, the outliers can bejust disregarded and performance can be estimated over remainingtrials [4], [5].The basic idea is that the global maximum of the LR function(13) always exceeds the LR function for the actual parameters, i.e.for the global solutions (Rt)ML and (Rm)ML we have'ymax -= Y [(Rt)ML, (Rm)ML] > -y(Rt, Rm)) = 2'o. (20)In this paper we apply the interval-based ML benchmark only tosimulated data. So, the selection procedure is as follows:- Find a local solution to the nonlinear optimization problem (13)using the presented initialization and calculate the LR value for thisI-solutionry = y(Rt, Rm).- If y > 'yo, accept this solution, otherwise classify it as anoutlier.4. SIMULATION RESULTSWe simulate a four-element antenna array and two-component inter-ference according to the asynchronous CCI scenario in Fig. 1. Thedesired signal and interference are generated as independent streamsofrandom symbols (+1 + 1) /XV. All propagation channels are sim-ulated as independent complex Gaussian vectors with unit varianceand zero mean. The training sequence of Nt = 8 symbols and vari-able total number of symbols N are considered. The interval-basedML benchmark presented in Section 3, the averaged ML benchmarkfrom [4], [5] and the SB algorithm (4) are compared by means ofthe mean square error (MSE) and bit error rate (BER) performanceestimated over 104 trials with independent channel and TOA real-izations.The MSE and BER results are presented in Fig. 2, 3 for variableSNR and different burst duration. The benchmark performance is es-timated over (70 - 95)% of trials depending on conditions using theoutlier selection procedure specified in Section 3.3. First of all, onecan see that for all considered scenarios the LS performance is verypoor compared to SB and both benchmarks as expected. Further-more, for shorter bursts the interval-based ML advantage over theaverage benchmark and SB is not significant: it is less than 2 dB forlow and medium SNR in Fig. 2. For longer bursts, the interval-basedML benchmark shows much better results, e.g., 4 dB and 6 dB im-provement at 0.lI% BER compared to the averaged ML benchmarkand SB as in Fig. 3.An important observation can be made for the high SNR in Fig.2, where the SB algorithm outperforms both the ML benchmarks.A possible reason is the ML breakdown for short overlapping inter-vals, where the ML solution becomes unreliable and leads to poorestimates in terms of MSE and BER. Obviously, short overlappingintervals can be met with higher probability in a short burst scenarioand their average impact is more significant for higher SNR. This iswhy the ML performance degradation in our simulations is visibleonly in Fig. 2 for SNR=20 dB.II- 867Authorized licensed use limited to: Alcatel Lucent. Downloaded on November 10, 2008 at 04:43 from IEEE Xplore.  Restrictions apply.This effect is further illustrated in Fig. 4 that presents the MSEand BER performance in the scenario shown in Fig. 1 with ni =n2 = Nt + 1, i.e., both interference components completely overlapwith the data interval. The fixed size of the training interval of Nt =8 symbols and variable duration of the data interval are assumedin Fig. 4. Clearly, both the ML benchmarks are equivalent in thiscase. Again, one can see that for very short data interval ofNd = 5symbols, SB outperforms the ML benchmark; for medium size datainterval of Nd = 10 symbols, their performance is almost the same;and for longer data interval ofNd = 42 symbols, the ML benchmarkclearly outperforms the SB solution. It is worth emphasizing thatin all these scenarios the SB solution is very poor in terms of MLestimation. For example, in the case of Nd = 5, VSB/ YO < 0.1 isobserved in more that 90% of trials for SNR=15 dB.5. CONCLUSIONAn asynchronous interference cancellation problem has been con-sidered. A potential usefulness of an a priori TOA information hasbeen studied. An interval-based ML benchmark has been developedand compared to the averaged ML benchmark and SB solution thatdo not use the TOA information. It has been found that in short burstscenarios the SB algorithm demonstrates performance that is closeto the interval-based ML benchmark. Furthermore, it may outper-form the ML benchmarks in the ML breakdown situation. For longerbursts, the known TOA information can significantly improve theperformance. Thus, new second-order semi-blind algorithms withTOA estimation may be required for such scenarios.Fig. 2. MSE and BER performance for N = 50K=4, M=2, Nt=8, N=10010lLUE/ lo6. REFERENCES[1] J. Karlsson, J. Heinegard, "Interference rejection combining forGSM," in Proc. ICUPC, pp.433-437, 1996.[2] A. M. Kuzminskiy, H. R. Karimi "Cross-layer design of up-link multiple-antenna interference cancellation for WLAN withCSMA/CA in open acess networks," to appear in ICC, 2007.[3] M. Nicoli, M. Sala, 0. Simeone, L. Sampietro, C. Santacesaria,"Adaptive array processing for time-varing interference mitiga-tion in IEEE 802.16 systems," in Proc. PIMRC, Helsinki, 2006.[4] A. M. Kuzminskiy, Y. I. Abramovich, "Adaptive second-orderasynchronous CCI cancellation: maximum likelihood bench-mark for regularized semi-blind technique," in Proc. ICASSP,vol. 4, pp. 453-456, Montreal, 2004.[5] A. M. Kuzminskiy, Y. I. Abramovich, "Second-order asyn-chronous interference cancellation: Regularized semi-blindtechnique and non-asymptotic maximum likelihood bench-mark," Signal Processing, vol. 86, no. 12, pp. 3849-3863, 2006.[6] Y. I. Abramovich, N. K. Spencer, A. Y. Gorokhov, "Bounds onmaximum likelihood ratio-part I: application to antenna arraydetection-estimation with perfect wavefront coherence," IEEETrans. Signal Processing, vol. 52, no. 6, pp. 1524 - 1536, 2004.[7] M. Basseville, I. V. Nikiforov, "Detection of abrupt changes.Theory and applications," Prentice Hall, 1993.[8] B. Johnson, Y. I. Abramovich, "GLRT-based detection-estimation of Gaussian signals in under-sampled training con-ditions," in Proc. EUSIPCO, Florence, 2006[9] A. Medles, "Coding and advanced signal processing for MIMOsystems," PhD Thesis, Telecom Paris (ENST) Institut Eurecom,2004.10K=4, M=2, Nt=8, N=1001010-lY,LUco1010SNR, dBFig. 3. MSE and BER performance for N = 100K=4, M=2, Nt=81010°LuCO10lo-'5IK=4, M=2, Nt=810010LU 1010I~~~~I105 20 -51 0 15 20 5SNR, dB10 15 20SNR, dBFig. 4. Illustration of the ML breakdownII- 868K=4, M=2, Nt=8, N=50 K=4, M=2, Nt=8, N=5010CO 1 01010010m10o-10-SNR, dB SNR, dBAuthorized licensed use limited to: Alcatel Lucent. Downloaded on November 10, 2008 at 04:43 from IEEE Xplore.  Restrictions apply.