Blind Equalization Formulated as a Self-organized Learning Process

Transcription

1 Blind Equalization Formulated as a Self-organized Learning Process Simon Haykin Communications Research Laboratory McMaster University 1280 Main Street West Hamilton, Ontario, Canada L8S 4K1 Abstract In this paper we describe a procedure for building a blind equalizer, motivated by neural network theory. The procedure treats the blind equalization problem as a self-organized process. The network consists of an input layer, a single hidden layer, and a single output unit. The learning process proceeds in two stages. In Stage I the nonlinear transformation for the input layer to the hidden layer is computed in a self-organized manner, which is frozen once steady-state conditions are reached. Stage 11, building on Stage I, resembles a conventional Bussgang algorithm except for the fact that the output nonlinearity is adapted alongside the linear weights connected to the output unit. 1 Introduction The conventional form of adaptive equalization has two distinct characteristics: The equalizer is linear, and the equalization involves the use of a training sequence for the adaptation of the equalizer coefficients prior to data transmission. There are several practical situations where it is not feasible to use a training sequence. For example, in a mobile radio environment using digital communications, it is impractical to use a training sequence for two reasons: The system cost of repeatedly sending a known sequence to train the equalizer is too high. Severe fading, makes it difficult (if not impossible) to establish data transmission when outage occurs. To overcome these limitations, blind equalization (which operates without a training sequence) recommends itself for use in a mobile radio communication system. An overview of blind equalization, treated as a special case of blind deconvolution, is presented in Cl]. Two families of nonlinear adaptive filtering algorithms are reviewed in this reference: (1) the Bussgang family, and (2) the class of algorithms based on the use of higher-order statistics. In reference [ll it is pointed out (perhaps for the first time) that the basic structure of a Bussgang algorithm consists of a single neuron, and that the underlying blind equalization process is essentially a form of self-organized learning. In this paper we take a more sophisticated look at the blind equalization process, using ideas from neural networks. The paper is organized as follows. In Section 2, we describe the essence of blind equalization treated as a form of selforganized learning. In Section 3, we describe two possible methods for designing Stage I of the blind equalizer. In Section 4 we describe the procedure for designing Stage I1 of the equalizer. The paper concludes with final remarks in Section Blind Equalization as a Form of Self-organization In a study of perception using multilayer feedforward networks, Linsker [21 used a Hebbian learning rule to compute the synaptic weights of the network. Just as importantly, the computation in Linsker's model proceeds on a stage-by-stage basis. Basically, the Hebbian rule is used to learn the synaptic weights connecting the input layer to the first hidden layer of computation nodes; once steady-state conditions are reached, the synaptic weights are frozen. Next, the synaptic weights connecting the first hidden layer to the second $ IEEE 346

2 hidden layer are treated in a similar fashion, and so on. The important point to note is that in Linsker's model of perception, learning proceeds on a layer-by-layer basis. the 0 Inspired by Linsker's model, we may formulate first principle of blind equalization: Assuming the use of a multilayer feedfbrward network fir blind equalization, the adaptation of the qynaptic weights in the network should proceed in a self-organized fashion, and on a layer-by-layer basis. For blind equalization to be feasible, the input signal should have non-gaussian statistics. As such, some form of nonlinear transformation would have to be applied to the input space. Typically, the dimensionality ofthe transformed signal space is much higher than hat of the original input signal space. One way of applying this nonlainear transformation is to use a set of radial basis functions (e.g., Gaussian centres). Alternatively, a form competitive learning could be used to transform the input space into a feature map. In event, the for the use Of this nonlinear transformation is that linear filtering is more likely to succeed in the new transformed signal space than the original input signal space; the essence ofthis rationale is explained by Cover's theorem [31. We may thus formulate the second principle of blind equalization as follows: fairly open, the nonlinearity should be "hard", as would be the case in a decision-directed mode of operating a conventional adaptive equalizer. Accordingly, we formulate the third and tinal principle of blind equalization: The output layer consists of a single nonlinear unit that is fed by the outputs of the hidden layer in amordunce with the Bussgang algorithm with the sigmoidal nonlinearity of the unit in small steps (alongside the synaptic weights) between an "almost linear" condition and a "hard limiting" one. On the basis of these three principles, we may now formulate the structure of the blind equalizer as depicted in Fig Self-organization of the Hidden Layer For the self-organization of the hidden layer, we may use one of two suitable procedures. One procedure would be to fomulatt! the hidden layer as a self-organizing feature map (SOEM) and use ahonen's procedure based on competitive learning to compute the SOFM 151. Important advantages of this are the simplicity of &honenss sofm and the ~pological preserving of the map. A of this approach is that the SOFM algorithm deviates from an ideal density matching condition. A hidden layer of computation rwdes may be used to apply a nonlinear transformation to Alternatively, we may use the principle of input with the size Of the maximum information preservation by hidden layer being much larger than Linsker [e]. According to this principle, the hidden the input luyer. layer consists of a set of Gaussian centres. The advantages of this second approach are an ideal For the next Of a Bussgang density matching condition and the topologyme Of adaptation 141 is On the Outputs Of the hidden layer, but with an important modification' property. However, the operation of the Ensker algorithm is complicated than that of S~cifically, the nonlinearity Of the Kohonenps SoFM algorithm, and requires a output unit is made variable rather than fixed. in C13, it was shown that delicate treatment. such a zero-memory form of nonlinearity closely fits the conditional mean estimator that is an inherent part of the Bussgang 4. Self-organization of the algorithm. The rationale for adjusting the slope of output Unit the sigmoidal nonlinearity may be explained as follows: Initially, the nonlinearity should be "very soft" to the point where the unit may be viewed as essentially "linear". Ultimately, when the eye is For the self-organization of the output unit, two adjustments are applied to the synaptic weights and sigmoidal nonlinearity of the unit. The 347

3 adjustments of the synaptic weights are done in accordance with the least-mean-square (LMS) algorithm. For the desired response we simply use the output signal of the output unit, and for the actual response we use the corresponding input signal of the sigmoidal nonlinearity (i.e., the output of the linear combiner part of the output unit); see Fig. 2a for details. For the adjustment of the sigmoidal nonlinearity, we proceed as follows: Let dx) denote the input-output relation of this nonlinearity, where x is the input signal; dx) is varried in accordance with the formula (see Fig. 2b): where k is a variable parameter, defined by and n is the iteration number (n2l). 5. Concluding Remarks In this paper we have presented the basic principles underlying the operation of a blind equalizer, treated as a self-organized learning process. The structure of the equalizer may be viewed as an extension of a Bussgang equalizer in that it includes a hidden layer, the nodes of which are fully connected (at their inputs) to the source nodes of the input layer, and also fully connected to (at their outputs) to the single unit constituting the output layer of the network. Preliminary results confirm the experimental validity of this approach to the design of a blind equalizer. References S. Haykin, Adaptive Filter Theory, Second Edition, Chapter 20, Prentice-Hall, R. Linsker, From basic network principles to neural architecture" (Series), Proc. Nat. Acad. Sciences (USA), vol. 83, T.M. Cover, "Geometrical and statistical property of systems of linear inequalities with applications in pattern recomition", IEEE Trans. Electronic Computors, vol. EC14, pp , S. Bellini, "Bussgang techniques for blind equalization", Globecom, Houston, Texas, pp , T.Kohonen, Self-organization and Associative Memory, Third Edition, Springer-Verlag, R. Linsker, "Self-organization in a perceptual network", Computer, vol. 21, pp ,

4 Input layer Output laycr Hidden layer Fig. 1 Block diagram of blind equalizer 349

5 Adjustablc sigmoidal lincarity Actual response Dcsircd rcspon sc 1 Error signal output (desired response) Input (actual responsc) Fig- (b) Dctails of the output unit 350