{"title": "A Hybrid Linear/Nonlinear Approach to Channel Equalization Problems", "book": "Advances in Neural Information Processing Systems", "page_first": 674, "page_last": 681, "abstract": "", "full_text": "A Hybrid Linear/Nonlinear Approach to Channel \n\nEqualization Problems \n\nWei-Tsih Lee \n\nJohn Pearson \n\nDavid Sarnoff Research Center \n\nCN5300 \n\nPrinceton, NJ 08543 \n\nAbstract \n\nChannel equalization problem is an important problem in high-speed \ncommunications. The sequences of symbols transmitted are distorted by \nneighboring symbols. Traditionally, the channel equalization problem is \nconsidered as a channel-inversion operation. One problem of this \napproach is that there is no direct correspondence between error proba(cid:173)\nbility and residual error produced by the channel inversion operation. In \nthis paper, the optimal equalizer design is formulated as a classification \nproblem. The optimal classifier can be constructed by Bayes decision \nrule. In general it is nonlinear. An efficient hybrid linear/nonlinear \nequalizer approach has been proposed to train the equalizer. The error \nprobability of new linear/nonlinear equalizer has been shown to be bet(cid:173)\nter than a linear equalizer in an experimental channel. \n\n1 INTRODUCTION \nIn a typical communication system, a sequence of symbols {ld are transmitted though a \nlinear time-dispersive channel h(t). Let x(t) be the received signal, it can be written as \n\nx (t) = L/jh (t-n1) + W (I) \n\nj \n\n(1) \nwhere h(t) denotes the elementary pulse waveform, and wet) represents the random noise \nwith iid Gaussian distribution. In a Quadrature Amplitude Modulation (QAM), symbols \n(Id are represented by complex numbers. During the transmission, interferences from \nneighboring symbols may distort the received signals. It is called Intersymbol Interference \n(lSI). It mainly because following reasons: nonideal channel which introduces phase or \namplitude distortions, phase jitter, and impulse noise. Thus, equalization techniques are \nused to reduce the lSI. \n\n674 \n\n\fA Hybrid Linear/Nonlinear Approach to Channel Equalization Problems \n\n675 \n\n2 ADAPTIVE LINEAR/RADIAL BASIS FUNCTION APPROACH \nTO EQUALIZER DESIGN \nTraditionally, the channel equalization problem is considered as a channel-inversion oper(cid:173)\nation. The idea is that an equalizer is constructed as to undo the interference from neigh(cid:173)\nboring symbols as they passing through a linear dispersive channel. It can be used to \nexplain different equalizer structures (Zero-forcing, Least mean square, and decision feed(cid:173)\nback) and their performance [Proakis, 1989]. One problem of this approach is that in gen(cid:173)\neral there is no direct correspondence between error probability and residual error \nproduced by the channel inversion operation. In [Gibson, etal, 1991], authors proposed a \nclassification viewpoint for the equalizer design. They suggested that the optimal equal(cid:173)\nizer should be a classifier whose decision boundary is constructed according to Bayes \ndecision rule. Compared with the channel inversion approach, the outputs of receiver are \nused as features for a classifier. The decision is made solely based on the classifier output, \nhence, on feature distribution. As it is well-known in [Fukunaga, 1978], the optimal deci(cid:173)\nsion boundaries can rapidly be computed if the features are Gaussian distributed. How(cid:173)\never, there is no idea about the structure of the optimal equalizer (classifier)for time(cid:173)\ndispersive channel outputs. In next section, we prove that for a linear channel, the optimal \nequalizer is nonlinear. \n\n2.1 THE OPTIMAL EQUALIZER OF A LINEAR TIME-DISPERSIVE CHANNEL \nLet us first consider a two-value equalization problem. Symbols with two possible values \n( -1, I) are transmitted. Let the channel be represented in a discrete form as a FIR of (hi), \ni=O,N-l. The output Xi can be written as \n\nN-l \nX\u00b7 = ~ I .\nL \n, \nj=o \n\n. h .+w . \n, \n\n'-I I \n\n(2) \n\nThe optimal equalizer design is equivalent to the following Bayes decision problem. \nGiven (xi}, decide Ii by \n\n1 \n/. = { \n, \n\n-1 \n\nif P(lj=-II X j,xj+l'\u00b7\u00b7\u00b7\u00b7\u00b7,xi+N-l) >P(lj= lI X j,xj+l'\u00b7\u00b7\u00b7\u00b7\u00b7,xj+N-l) \n\n(3) \nwhere P (lj = 11 xi,xi + 1 , \u2022\u2022.\u2022. ,xi+N-l) is the posterior probability of the transmitted symbol \nIi being 1 given channel output (xd. \nBy Bayes theorem, expression(3) can be expanded to the following form: \n\ni+N-l \n\nIT \n\nj = j \n\nL \n\nkl' k;z \u2022.... ,kj _ N + I e {I. -1 } \n\nP(xjl Ii = 1,\u00b7\u00b7\u00b7Jj _ N+ 1 = ki - N+ 1) P(lj = 1, ... J j - N+ 1 = kj _ N+ 1) \n\n(4) \n\n(5) \nSince conditional probability P (xii I j = 1, ... J j -N + 1 = kj _ N + 1) in (5) is a Gaussian distri(cid:173)\nbution, the numerator in (5) is a mixture of Gaussian distribution. Plugging (5) into (3), \nBayes decision rule determines the optimal decision boundary as the solution of eqUality. \nSince denominator is the same on both sides, it can be ignored. Rearranging the equation, \n\n\f676 \n\nLee and Pearson \n\nit can be written as summation of exponential functions. The solution of this equation is \nnonlinear function of {xil. In general, no analytical form can be found. However, it can be \nsolved by numerical methods. Thus, the optimal decision boundary can be determined. \nThe result can be extended to multi-class problems. \nBased on the result established above, we provide a theoretical justification of a nonlinear \nequalizer approach to linear time-dispersive channel. The theoretical comparison of per(cid:173)\nformances of linear and optimal equalizers can be found in [Gibson, et.al, 1991]. They \nconcluded that performance of linear equalizers can not be improved by increasing tap \nlength. This also suggests that a nonlinear equalizer approach is necessary. Another reason \nfor nonlinear equalization approach is due to channels with spectrum hulls [Proakis, \n1989]. In this case, the linear equalizer can not achieve the desired performance due to \n\"noise enhancement\". \n\n2.2 NONLINEAR EQUALIZER DESIGN PROBLEM \nThere are several approaches to nonlinear equalizer design. To reduce the Least Mean \nSquare (LMS) error, Voterra-series approach uses high-order product terms of input as \nnew features. The tree-structured linear equalizer method [Gelfand, et.al., 1991] partitions \nthe feature-space, and makes a piecewise linear approximation to the optimal nonlinear \nequalizer. As reported in [Gelfand, et.al., 1991], the tree-structured linear equalizer \napproach provides reasonable fast convergence and lower error probability as compared \nwith linear and Voterra series approaches. The problem of this approach is that a lot of \ntraining samples are needed to achieve good performance. A neural network approach, \nMultiLayer Perceptron(MLP) [Gibson, et.al, 1991], trains 3 or 4 layers interconnected \nPerceptrons to form the nonlinear decision boundary. It is observed in [Gibson, et.al, \n1991] that the performance of a MLP equalizer is close to optimal Bayes classifier. How(cid:173)\never, the training time is long and a fine-turing procedure is used. A nonlinear equalizer \napproach using radial basis functions is also reponed in [Chen, et.al., 1991]. \nTo put equalizers into use, the long training time is unpractical, and a fine-adjusting proce(cid:173)\ndure is not allowed. Hence, it is desired to have an efficient, automatic procedure for non(cid:173)\nlinear equalize design. To achieve this goal, we propose a hybrid linear and radial basis \nfunctions approach for automatic nonlinear equalizer design. \nAlthough the optimal equalizer should be nonlinear, all these nonlinear design methods \nrequire long training time or large amount of training samples. Linear equalizers are not \noptimal, but with following advantages: easy training, fast convergence. It is also reported \nthat the linear equalizer is relatively robust [Fukunaga, 1978]. Hence, it is desirable to \ncombine the advantages of both linear and nonlinear equalizers. However, the hybrid \nstructure should provide desired properties: fast convergence, automatic training proce(cid:173)\ndure, and low error rate. To satisfy these constraints, we propose a feature-space partition(cid:173)\ning approach to hybrid equalizer design. \n\n2.3 FEATURE\u00b7SPACE PARTITIONING APPROACH TO HYBRID EQUALIZER \nDESIGN \nTo design a hybrid linear/nonlinear equalizer, we adopt the feature-space partitioning con(cid:173)\ncept. The idea is similar to the one developed in [Gelfand, et.al, 1991]. Here, we consider \na partitioning method based on geometrical reasoning for eqUalization problems. The idea \nis based on the fact that linear equalizers can recover distorted signals, except the cases \nwhen strong noise push samples into boundaries where two classes overlaid with each \n\n\fA Hybrid Linear/Nonlinear Approach to Channel Equalization Problems \n\n677 \n\nother. We consider the \"confused\" samples as these samples near decision boundaries. The \nseparation of \"confused\" samples can be accomplished based on the output values of lin(cid:173)\near equalizers. If the distance between output value and the closest point in signal constel(cid:173)\nlation [Proakis, 1989] is greater than a threshold, then we consider current sample is \n\"confused\". This means that the sample is the one close to decision boundary. To achieve \nan accurate classification, we classify it by a nonlinear equalizer, which is constructed for \nseparating the samples near Bayes decision boundary. \nThe hybrid structure consisted of a linear equalizer, followed by a radial basis function \n(RBS) network, as shown in Fig. 1. A RBS network (Fig.2) is a two-layered network with \nradial_basis_function nodes in first layer, and a weighted linear combination of outputs of \nthese nodes. \nEach feature vector consisted of a collection of consecutive data from the channel. It is \nassumed that these data are properly time and carrier synchronized [Proakis, 1989]. For \nthe QAM, a complex-valued linear equalizer is adopted. The distance between output \nvalue of linear equalizer and the closest point is then computed and compared with thresh(cid:173)\nold as described before. The \"confused\" samples are classified by a nonlinear RBS equal(cid:173)\nizer. The output of a RBS network can be written as weighted summation of outputs of \nnodes as follows: \n\n(6) \n\nwhere f(x) is the output of network. The oUfut value of each node is computed according \nto the bell-shaped function centered at ci' (J is the width of a node. Wi is the weight asso(cid:173)\nciated with ith node. In our experiments, the width of all nodes are fixed. The first N train(cid:173)\ning samples are assigned to the centers of N -nodes network. The weights are adjusted \naccording to stochastic gradient decent rule: \n\n(7) \n\nwhere 11 is the learning rate. d k is the desired output of network \nTo train a hybrid LE/RBS equalizer, a collection of training samples is used to adjust the \nparameters of linear equalizer. The training samples for RBS network are collected \naccording to the distance rule described above. They are used to adjust weights of RBS \nnetwork only. \nThe classification of a unknown sample follows a similar rule. The output value of the lin(cid:173)\near equalizer is computed. If the distance between output value and the closest point in \nsignal constellation is smaller than the threshold, then the closest point is considered as the \nrecovered signal. If not, the output of the RBS network is used to classify the sample. The \nclosest point in signal constellation from output of RBS network is then used for sample \nclass. Note, however, that there is a different interpretation for the output of linear and \nRBS equalizer. The function of linear equalizers can be considered as an approximation of \nchannel inversion. Hence, it is similar to a deconvolution computation [Proakis, 1989]. \nHowever, for a RBS network, the output is the summation of weighted local Gaussian \nfunctions. For closely located points, the network is asked to give same output by then \ntraining procedure. Thus, it is more a classification approach than a deconvolution \nmethod. \nThe approach provides a design method for hybrid LE/RBS equalizer. The linear equaliz-\n\n\f678 \n\nLee and Pearson \n\n{Xn } \n\nline:lI' Equalizer \n\nyes \n\nRadial_basis _ function \nnetwork \n\nFig. 1 System Diagram of Hybrid Linear/Nonlinear Equalizer \n\nI(x) \n\nFig. 2 A RadiaCBasis_Function Network \n\n\fA Hybrid Linear/Nonlinear Approach to Channel Equalization Problems \n\n679 \n\ners perfonn the channel inversion or partitioning of the feature space, depending on the \noutput value. More complicated tree-structured equalizer [Gelfand, et.al., 1991] can be \nadopted for this proposes. The nonlinear RBS networks are used for classifying \"con(cid:173)\nfused\" samples. They can be replaced by MLPs. Hence, the approach provides a general \nmethod for designing hybrid structure eqalizers. However, the trade-off between complex(cid:173)\nity and efficiency of these combinations has to to be considered. For example, a multilayer \ntree-structured equalizer can divide the space into smaller regions for finer classification. \nHowever, the small amount of training samples in practice can be a problem for this \nmethod. A MLP network can be used for nonlinear classifier. Nevertheless, convergence \ntime will be a major concern. \n\n3 EXPERIMENT \nWe have applied our hybrid design method to a 4-QAM system. The channel is modeled \nby \n\nX j _ 1 = 0.406/j +0.814/j _ 1 +0.407/j _ 2 +wj \n\n(8) \n\nwhere Ij = {- 1 - j,- 1 + j,l- j,1 + j} . \nA 7-tapped complex linear equalizer is used for classifying the input. Threshold for non(cid:173)\nlinear equalizer is 0.1. We use 4,000training samples and 5,000 testing samples. A 400 \nnodes RBS network is used for nonlinear equalizer. The first 400 \"confused\" training sam(cid:173)\nples are used for the center of network. The network is trained according to (6). Learning \ncoefficient T) is chosen to be 0.01. The width of a RBS node is 1.0. \nFig. 3 shows the symbol error probability vs. SNR. The error probability is evaluated over \n5,000 testing samples. The hybrid LE/RBS network produces nearly 10% reduction of \nerror rate compared with linear equalizer. This shows that a hybrid linear/RBS network \nequalizer can reduce the error rate by classifying \"confused\" samples near decision \nboundaries. No comparison with Bayes classifier has been made. In our experiments, it is \nobserved that the error rate can be reduced further by increasing the number of RBS \nnodes. This seems imply that a large-size RBS network will in general produce better clas(cid:173)\nsification result. However, since there is always a limitation of the computation resources: \ncomputation time and memory storage, the perfonnance of the hybrid linear/RBS network \nis limited, especially in high signal constellation case discussed below. \nEqualization in high signal constellation, 16 and 64-QAM, have been tried. The result \nshows no significant improvement This can be explained by the increasing of complexity. \nRecall that the RBS network is to separate the samples near the boundary. To deal with the \nincreasing of number of classes due to high signal constellation, the number of nodes of \nnetwork must increase proportionally. Since the increasing rate is exponential in terms of \nnumber of classes, it implies a straight-forward implementation of RBS network method \ncan not be used for high signal constellation. In [Chen, et.al., 1991], authors suggest a \ndynamical RBS network with adjustable center location and width. The algorithm runs in \nbatch mode. It is reported that the size of network can be reduced dramatically by the \ndynamical RBS network method. However, for equalizer application, on-line version of \nthe algorithm is needed. \n\n\f680 \n\nLee and Pearson \n\n\u00b7 \u00b7 \u00b7 : \n_CUlt \u00b7 1 .. \n\n:< \n\nom \n\nSNR \n\nFig. 3 Error Probability of Hybrid Linear/Radial_Basis_Function Network Equalizer for a \n\nLinear Channel xi - 1 = 0.406Ii + 0.814/i _ 1 + 0.407li _ 2 +wi with 4-QAM. \n\n4 CONCLUSION AND DISCUSSION FOR HYBRID LE/RBS \nEQUALIZER DESIGN \nBy combining feature-space partitioning and nonlinear equalizers, we have developed a \nhybrid linear/nonlinear equalization approach. The major contribution of this research is \nto provide a theoretical justification of nonlinear equalization approach for linear time-dis(cid:173)\npersive channels. A feature-space partitioning method by linear equalizer is proposed. \nRBS networks for nonlinear equalizers are integrated into the design to separate the sam(cid:173)\nples near decision boundary. The experiments for 4-QAM equalization have demonstrated \nthe feasibility of the approach. For high signal constellation modulation, a dynamical RBS \nnetwork method [Chen, etal., 1991] has been suggested to overcome the problem of \nincreasing complexity. \nThe hybrid Linear/nonlinear equalization approach combines the strength of linear and \nnonlinear equalizers. It offers a framework to integrate the deconvolution and classifica(cid:173)\ntion methods. The approach can be generalized to include complicated partitioning \n\n\fA Hybrid Linear/Nonlinear Approach to Channel Equalization Problems \n\n681 \n\nscheme and other nonlinear networks, such as :MLP, as well. \nMore researches need to be conducted to make this approach practical for general use. The \nrelationship between the performance of hybrid equalizer and taps length of linear equal(cid:173)\nizers, the width and the number ofRBS nodes need to be investigated. The on-line version \nof dynamical RBS network [Chen, et.al., 1991] need to be developed. \n\nReference: \nProakis, J. G., Digital Communications, McGrwa-Hill company, New York, 1989. \nGibson, GJ., Siu, S., Cowan, C.F.N., \"The Application of Nonlinear Structures to Recon(cid:173)\nstruction of Binary Signals,\" IEEE. Trans. on Signal Processing, vol. 39, No.8, Aug.,pp. \n1877-1884, 1991. \nFukunaga, K., Introduction to Statistical Pattern Recognition, Academic Press, New York, \n1978. \nGelfand, S.B., Ravishankar, C.S., and Delp, EJ., \"Tree-structured Piecewise Linear Adap(cid:173)\ntive Equalization,\" ICC91, 001383, 1386. \nChen, S., Gilbson, GJ., Cowan, C.F.N., and Grant, P.M., \"Reconstruction of binary sig(cid:173)\nnals using an adaptive radial-bas is-function equalizer,\" Signal Processing, 22, pp. 77-93, \n1991. \nChen, S., Cowan, C.F.N., and Grant, P.M., \"Orthogonal Least Squares Learning Algorithm \nfor Radial Basis Function Networks,\" IEEE. Trans. on Neural Networks, vol. 2, no., 2, \nMarch, pp.302-309, 1991. \n\n\f", "award": [], "sourceid": 673, "authors": [{"given_name": "Wei-Tsih", "family_name": "Lee", "institution": null}, {"given_name": "John", "family_name": "Pearson", "institution": null}]}