We propose two methods for the estimation of sparse communication channels. In the first method, we consider the problem of channel estimation based on training symbols, and formulate it as an optimization problem. In this formulation, we combine the objective of fidelity to the received data with a non-quadratic constraint reflecting the prior information about the sparsity of the channel. This approach leads to accurate channel estimates with much shorter training sequences than conventional methods. The second method we propose is aimed at taking advantage of any available training-based data, as well as any "blind" data based on unknown, constant modulus symbols. We propose a semi-blind optimization framework making use of these two types of data, and enforcing the sparsity of the channel, as well as the constant modulus property of the symbols. This approach improves upon the channel estimates based only on training sequences, and also produces accurate estimates for the unknown symbols

Cetin, Mujdat

Sadler, Brian M.

Çetin, Müjdat

English

Sabanci University Research Database

SEMI-BLIND SPARSE CHANNEL ESTIMATION WITH CONSTANTMODULUS SYMBOLSMu¨jdat C¸etin† and Brian M. Sadler‡†Laboratory for Information and Decision SystemsMassachusetts Institute of Technology77 Massachusetts Ave., Cambridge, MA 02139, USA‡Army Research Laboratory, AMSRL-CI-CN, Adelphi, MD 20783, USAABSTRACTWe propose two methods for estimation of sparse commu-nication channels. In the first method, we consider theproblem of channel estimation based on training symbols,and formulate it as an optimization problem. In this for-mulation, we combine the objective of fidelity to the re-ceived data, with a non-quadratic constraint reflecting theprior information about the sparsity of the channel. Thisapproach leads to accurate channel estimates with muchshorter training sequences than conventional methods. Thesecond method we propose is aimed at taking advantage ofany available training-based data, as well as any “blind”data based on unknown, constant modulus symbols. Wepropose a semi-blind optimization framework making useof these two types of data, and enforcing the sparsity of thechannel as well as the constant modulus property of thesymbols. This approach improves upon the channel esti-mates based only on training sequences, and also producesaccurate estimates for the unknown symbols.1. INTRODUCTIONIn many wireless communication systems, the propagationchannels involved exhibit a large delay spread, but a sparseimpulse response consisting of a small number of dominantechoes. A primary example is terrestrial transmission ofhigh definition television (HDTV) signals [1,2]. In Fig. 1 weshow an example of such a sparse channel impulse response.Conventional least-squares channel estimation techniquesdo not exploit the sparse structure of such channels andrequire the transmission of many training symbols to gen-erate an accurate estimate. Recently, a matching pursuitalgorithm that exploits the sparse structure of such chan-nels has been proposed [3]. This approach has also been ex-tended to multiuser environments [4]. In the initial portionof our work presented in this paper, we propose a channelestimation technique that has a similar goal of exploitingsparsity. In contrast with the approach in [3], we formulatethe channel estimation problem as a non-quadratic opti-mization problem involving a data fidelity term, and an 1-norm-based, sparsity-enforcing regularization term. Bothmatching pursuit and 1-norm regularization (also calledThis work was supported by the Army Research Office underGrant DAAD19-00-1-0466.0 20 40 60 80 100 120−0.500.51TapAmplitudeFig. 1. A sample sparse channel impulse response (adaptedfrom [3]).basis pursuit) can be viewed as trying to solve a combina-torial sparse signal representation problem in a suboptimalfashion. Matching pursuit provides a greedy solution, while1-norm-based methods replace the original problem witha relaxed version for tractability. Recent work has illumi-nated interesting theoretical properties of both approaches.The 1-norm-based approach we propose for channel esti-mation has regularization built into it to provide robustnessagainst noise. Furthermore, the optimization-based natureof our framework makes it easy to extend these ideas to thesemi-blind case, which we discuss next.The discussion above was implicitly focused on channelestimation in the presence of training symbols. Most cur-rent wireless communication systems depend on the trans-mission of such known symbols for channel estimation andequalization. However, use of training symbols limits theeffective transmission bandwidth. Therefore, it is of inter-est to reduce the number of training symbols. On the otherhand, blind equalization techniques do not require train-ing. One of the most popular blind techniques is basedon the so-called constant modulus algorithms [5]. Whilesuch blind algorithms have good performance with longdata sequences, they may not achieve equalization in ashort burst.1 In order to alleviate this problem, a num-ber of researchers have recently proposed methods which1Although most constant modulus algorithms are applied tolong data sequences, there is also some recent work on finite-interval constant modulus algorithms [6].couple training-based and blind techniques, leading to so-called semi-blind methods [7,8]. These methods provide anattractive tradeoff between training-based and blind tech-niques. However we are not aware of any semi-blind tech-nique designed explicitly for, and exploit the characteristicsof, sparse communication channels. We propose a semi-blind sparse channel estimation technique for the case ofconstant modulus symbols. In particular, we formulate anoptimization problem which contains terms for fidelity tothe training-based as well as blind data, an 1-norm termfor enforcing channel sparsity, and a term enforcing the con-stant modulus property of the symbols. The solution ofthis optimization problem yields both an estimate for thechannel impulse response, and estimates for the transmit-ted unknown symbols. Our experiments on simulated datademonstrate both the improvements of our sparse chan-nel estimation framework over conventional techniques, andalso how our semi-blind approach improves over our channelestimation technique based only on training data.2. OBSERVATION MODELLet the training symbols sT (n), n = 0, ..., NT − 1 be trans-mitted through a channel with impulse response c(m), m =0, ..., Nc − 1. We can model the observed signal samplesyT (n) as:yT (n) =Nc−1∑m=0sT (n − m)c(m) + v(n), n = 0, ..., NT − 1 (1)where v(n) denotes the measurement noise. We can writethis equation in matrix form as follows:yT = AT c+ v (2)where AT is a Toeplitz matrix that depends on the trans-mitted symbols; and c, yT , and v are the channel impulseresponse, observed data, and measurement noise, respec-tively, column-stacked as vectors. The conventional chan-nel estimation method is based on the pseudo-inverse oper-ation: cˆLS = A†TyT , where “†” denotes the pseudo-inverse.We will refer to this method as least-squares, as is custom-ary, although we should note that when NT < Nc, this isactually a least-squares, min-norm solution.3. CHANNEL ESTIMATION BYSPARSITY-ENFORCING REGULARIZATIONWe propose estimating the channel by minimizing the fol-lowing cost function:ET (c) = ‖yT −AT c‖22 + λ‖c‖1 (3)where λ is a scalar parameter. The first term of ET (c)is a data fidelity term, and the second term is a sparsity-enforcing regularization term. It is well-known that mini-mizing the 1-norm leads to a preference for sparse struc-ture [9]. By incorporating the prior information that thechannel is sparse, we aim to achieve accurate channel es-timation with a much smaller number of training symbolsthan conventional methods. We solve the optimization prob-lem in Eqn. (3) by adapting and applying the numericalalgorithm of [10].4. SEMI-BLIND CHANNEL ESTIMATIONWITH CONSTANT MODULUS SYMBOLSThe technique we proposed in the previous section reliesentirely on known training symbols. In this section, wepropose an extension, where we take advantage of sometraining symbols as well as a block of transmitted unknownsymbols for channel estimation. The technique also pro-duces estimates of the unknown symbols. We focus on thecase where the transmitted symbols have constant modulusK, and build that information into our formulation as well.Let us assume that we have two data streams: yT refer-ring to received data associated with the training symbols,and yB referring to received “blind” data associated withthe unknown symbols s. Then we construct the following“semi-blind” cost function:ESB(s, c) = ‖yT −AT c‖22 + ‖yB −AB(s)c‖22+λ‖c‖1 + γNs∑i=1[|si|2 − K2]2(4)where γ is a scalar parameter, Ns denotes the number ofunknown symbols, and si denotes the i-th unknown sym-bol. The matrixAB(s) is constructed in a similar fashion toAT , however it depends on the unknown symbols s ratherthan the training symbols, and we make that dependenceexplicit in our notation. The first two terms of ESB(s, c) en-force fidelity to training-based and blind observations, thethird term is the channel sparsity constraint, and the lastterm enforces the constant modulus property of the sym-bols. The major novelty of this formulation as comparedto other semi-blind techniques is the channel sparsity con-straint. Although we weight the training-based and blinddata fidelity terms in ESB(s, c) equally, one could general-ize this cost function slightly to apply different weights, e.g.if the two data streams have different SNRs. We minimizeESB(s, c) by applying a block coordinate descent algorithmon s and c. In particular, we first find an initial channelestimate cˆ(0) based only on the training symbols, using thetechnique described in Section 3. Then we run the followingset of alternating minimizations at each iteration k, startingwith k = 0:sˆ(k+1) = argminsESB(s, cˆ(k))cˆ(k+1) = argmincESB(sˆ(k+1), c) (5)To solve the first minimization above, we use gradient de-scent; and to solve the second minimization, we use thesame numerical algorithm utilized in Section 3. We run thecoordinate descent iterations in (5) until‖cˆ(k+1) − cˆ(k)‖2/‖cˆ(k)‖2 < δ, where δ > 0 is a small con-stant.5. EXPERIMENTAL RESULTSWe present simulations where the channel to be estimatedand equalized is given in Fig. 1. This impulse response haslength Nc = 119, and 10 non-zero taps. For the trans-mitted symbols, we use BPSK sequences with equiprobablesymbols.5.1. Training-based Channel EstimationWe first present the results of our training-based sparsechannel estimation technique of Section 3. We start with aremark about the observation model. Note that in Sec-tion 2, we assumed we had the training symbols sT (n)for n = 0, ..., NT − 1. However, the observation model inEqn. (1) also depends on sT (n) for n < 0. As a result,the matrix AT in Eqn. (3) also depends on those symbols.These symbols may be obtained from previous decodingsin a data stream or can be assumed zero if this is the firstpacket received [3]. Which assumption is made can have animpact on the results, especially when short data sequencesare of interest. Therefore we present results based on bothassumptions.We compare our 1-norm-based approach with match-ing pursuit [3], as well as conventional least squares. We runthese techniques on 100 different realizations of the trainingsymbols and measurement noise. We repeat these experi-ments at a range of SNRs, where SNR is defined as the vari-ance ratio of the signal and noise components in Eqn. (1).Given an estimated channel impulse response produced byeach technique, we find the locations of the 10 largest mag-nitude taps for each technique. We then count how many ofthese locations match the actual 10 non-zero tap locationsof the channel in Fig. 1.In the first set of experiments, we assume that sT (n)for n < 0 are obtained from previous decodings. Fig. 2shows the number of matches averaged over 100 realiza-tions for each technique, as a function of SNR. Fig. 2(a),(b)and (c) correspond to different numbers of training sym-bols NT . We observe that both our 1-norm-based methodand matching pursuit provide much more accurate chan-nel estimates than least squares. The 1 method providesslightly better performance than matching pursuit, whichis more significant when the number of training symbolsis relatively small. We should note that we have not opti-mized the choice of the free parameter λ in Eqn. (3), and theperformance of the 1 technique might be improved furtherby better parameter choices. In Fig. 3, we present similarresults for the second set of experiments where sT (n) forn < 0 are assumed to be zero.5.2. Semi-blind Channel EstimationWe now present the results of the semi-blind algorithm pro-posed in Section 4. For the training portion of the experi-mental setup, we assume that sT (n) for n < 0 are availablefrom previous decodings. We consider the transmission oftwo BPSK sequences: a training sequence of NT symbolsand an unknown sequence s of Ns = 200 symbols. Based onthe training and the blind data, we minimize ESB(s, c) tofind estimates of both the channel and the unknown sym-bols. We consider a range of SNRs, as well as training se-quence lengths NT , and find the average number of matchesin the channel impulse response over 100 realizations, as inthe experiments of Section 5.1. We present these resultsin Fig. 4. The dashed curves correspond to the 1 resultspresented in Section 5.1, based only on training data. Thesolid curves are based on the semi-blind experiments. Weobserve that we are able to exploit the unknown symbolsin the semi-blind framework to improve upon the resultsof our training-based method. As a result, our semi-blind0 5 10 15 20 25 30246810SNRaveg. # matches LSMPL1(a)0 5 10 15 20 25 30246810SNRaveg. # matchesLSMPL1(b)0 5 10 15 20 25 30246810SNRaveg. # matchesLSMPL1(c)Fig. 2. Average number of matches between the locationsof the 10 largest magnitude taps estimated by the threemethods (1, matching pursuit (MP), least squares (LS))and the actual non-zero taps of the channel. It is assumedthat sT (n) for n < 0 are obtained from previous decodings.Each plot is based on 100 trials. (a) NT = 40. (b) NT = 80.(c) NT = 130.0 5 10 15 20 25 300510SNRaveg. # matchesLSMPL1(a)0 5 10 15 20 25 300510SNRaveg. # matches LSMPL1(b)0 5 10 15 20 25 300510SNRaveg. # matchesLSMPL1(c)Fig. 3. Average number of matches as in Fig. 2, but forthe case where sT (n) = 0 for n < 0. (a) NT = 120. (b)NT = 130. (c) NT = 140.framework can achieve the accuracy of our training-basedchannel estimates with a much smaller number of trainingsymbols, resulting in communication bandwidth savings.Finally, we demonstrate the performance of our semi-blind technique in estimating the unknown symbol sequencess of length Ns = 200. After quantizing the continuous-valued symbol estimates produced by the minimization of0 5 10 15 20 25 30246810SNRaveg. # matchesonly trainingsemi−blind(a)0 5 10 15 20 25 30246810SNRaveg. # matchesonly trainingsemi−blind(b)0 5 10 15 20 25 30246810SNRaveg. # matchesonly trainingsemi−blind(c)Fig. 4. Average number of matches obtained by minimizingthe semi-blind cost function ESB(s, c) compared with theresults of the training-based cost function ET (c). Each plotis based on 100 trials. (a) NT = 40. (b) NT = 80. (c)NT = 130.ESB(s, c), we compare them to the true symbols, and countthe number of bit errors. Dividing the number of bit errorsby Ns, and averaging over 100 realizations, we obtain anestimate of the bit error rate (BER). Fig. 5 shows plots ofBER for three different sizes of training sequences, and arange of SNRs. This result demonstrates that our semi-blind approach can provide reasonable estimates of the un-known symbols. We have not optimized the choice of thefree parameters λ and γ in Eqn. (4), and the performanceof our technique might be improved further by better pa-rameter choices.6. CONCLUSIONWe have developed new techniques for the estimation ofsparse communication channels for training-based and forsemi-blind scenarios. We have defined appropriate opti-mization problems taking advantage of both training-basedand blind data streams, and incorporating prior informa-tion we have about the structure of the propagation chan-nels and about the unknown symbols. This work providesa principled framework synthesizing recent ideas on semi-blind equalization with ideas on sparse channel estimation.Our preliminary experiments have demonstrated the promiseof this framework in generating accurate channel estimates,as well as symbol estimates with short data streams. How-ever, a much more detailed analysis of the proposed frame-work is needed, and is a subject of our current research.We are interested in comparing our approach to other semi-blind techniques, illuminating the role of sparsity-enforcingregularization. We are also interested in characterizing po-tential performance improvements provided by this frame-work over the fully-blind case. Finally, we are interestedin exploring further performance improvements of our ap-proach by utilizing effective automatic techniques for choos-ing the free parameters involved.0 2 4 6 8 10 12 14 16 18 2010−410−310−210−1100SNRBERNT = 40NT = 80NT =130Fig. 5. Performance of the semi-blind technique in esti-mating the unknown symbols, in terms of BER.7. REFERENCES[1] A. Rontogiannis and K. Berberidis, “Efficient deci-sion feedback equalization for sparse wireless chan-nels,” IEEE Trans. Wireless Comunications, vol. 2,no. 3, pp. 570–581, May 2003.[2] M. Ghosh, “Blind decision feedback equalization forterrestrial television receivers,” Proc. IEEE, vol. 86,pp. 2070–2081, Oct. 1998.[3] S. F. Cotter and B. D. Rao, “Sparse channel estimationvia matching pursuit with application to equalization,”IEEE Trans. Communications, vol. 50, no. 3, pp. 374–377, Mar. 2002.[4] S. Kim and R. A. 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Semi-blind sparse channel estimation with constant modulus symbols

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Semi-blind sparse channel estimation with constant modulus symbols

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