Generalized Feedforward Structures: A New Class of Adaptive Filters

In this paper we introduce a new class of filters, the generalized feedforward structures, that combine attractive properties of the moving average (MA) filters for adaptation (i.e. fast algorithms, trivial stability) with some of the power of autoregressive moving average (ARMA) filters (i.e. decoupling of the length of the impulse response with filter order). Preliminary results show that this class of filters is much more efficient than conventional MA filters (i.e. for a given minimum mean square error (MSE) the filter order is much smaller). We have extended the Wiener-Hopf solution for this class of filters and have developed some design tools. The generalized feedforward structures accept Widrow’s adaptive linear combiner as a special case. An identification example will be presented. 2 Generalized Feedforward Structures: A New Class of Adaptive Filters Introduction Computational Neuroengineering Laboratory Introduction Although the class of ARMA systems is much more effective for modeling, identification, and signal processing, the subclass of MA (moving average) filters is almost exclusively utilized in adaptive signal processing. The main reason is centered in the difficulty of ensuring stability during adaptation of ARMA systems. Moreover, the algorithms to search for the global optimum are computationally expensive and difficult to analyze. On the other hand MA filters have well understood adaptation frameworks (Wiener-Hopf optimal solution), there are fast algorithms to adapt the filter coefficients with a computational complexity of O(L) (L number of filter weights), and the MA filters are intrinsically stable. However, MA filters have the drawback that the length of the impulse response is coupled with the filter order. This means that in problems that require information from the distant past to adapt the filter weights, very large filter orders are necessary. Normally the number of degrees of freedom of the problem at hand would require much smaller filter orders, but the coupling between memory length and filter order imposes the large order. This means that the adaptation becomes too sensitive to noise, and that the adaptation time become too long, and computationally expensive. In this paper we present a new filter class that we call the generalized feedfoward structure that extends directly the results of Widrow’s LMS [1] to a special class of ARMA systems. The adaptation process can still be described by the Wiener-Hopf solution [2], the filters have trivial stability conditions, and can be adapted by a modified LMS algorithm with the same complexity as the LMS. Moreover, they decouple the length of the impulse response from the filter order. Therefore, the filter order can be set by the number of degrees of freedom of the problem at hand, irrespective of the length of the impulse response required to carry out the adaptation. We briefly describe the filter class, the adaptation process and present one example of identification. Generalized Feedforward Filters It is well known that the eigenmodes of the ARMA filter when implemented as a parallel structure are decoupled, i.e. one can write the output as where x(n) is the input, y(n) is the output, wk are the filter weights, xk are the state variables, and the {gk} are the K+1 eigenmodes of the linear system (Figure 1a). The eigenmodes {gk}, being linearly independent, can be considered as a set of basis functions. Therefore, (eq.1) is simply stating a projection operation on a particular set of basis functions (that is minimal when y(n) accepts a linear model). Since basis functions are not uniquely defined, we will be exploring here other decompositions of y(n), droping the condition of minimality (L>K) but imposing conditions for fast adaptation. y n ( ) wk gk n m − ( ) x m ( ) m 0 = n ∑ k 0 = K ∑ wkxk n ( ) k 0 = K ∑ = = (1)