Adaptation of memory depth in the gamma filter

Gamma filter is a special class of generalized feedforward filters where feedbacks are allowed only locally. We present the conditions for the selection of optimal parameters which are the weights and the memory depth of the filter. The conditions for these two set of parameters are decoupled from each other. This allows a matched filter implementation which gives an estimate of the memory depth. INTRODUCTION Gamma filter is a special case of a broader class of linear systems called the generalized feedforward filters [1 ]. Gamma filter is essentially an IIR filter where the recursion is kept local. It proves to be effective particularly in the identification of systems with long impulse responses. There is a close relationship between the gamma filter and the Laguerre filter such that the Laguerre filter is an orthogonalized form of the gamma filter [2]. Parameters of the gamma filter are the weights w’s and the memory depth λ. Their optimum values can be found by solving the extended Wiener-Hopf equations [1 ] either offline or adaptively. Due to the non-convex nature of the performance with respect to the memory depth, one faces the problem of local minima. In this paper we present a method to estimate the values of the local minima of the memory depth. This method decouples the solution of the memory depth λ from that of the weights. The method is based on constructing a matched filter for the optimal condition, and passing the desired signal through such a structure. The zero crossings of the output define the optimal values of the memory depth parameter. We present the theory that gave rise to the method, and present an example. GAMMA FILTER OPTIMALITY CONDITIONS Figure 1 shows the block diagram of the generalized feedforward filters. For the gamma filter G(s) is given by ( 1) which means that the system transfer function is