Orthonormal Bases for Adaptive filtering

In the field of adaptive filtering the most commonly applied filter structure is the transversal filter, also referred to as the tapped-delay line (TDL). The TDL is composed of a cascade of unit delay elements that are tapped, weighted and then summed. Thus, the output of a TDL is formed by a linear combination of its input signal at various delays. The weights in this linear combination are called the tap weights. The number of delay elements, or equivalently the number of tap weights, determines the duration of the impulse response of the TDL. For this reason, one often speaks of a finite impulse response (FIR) filter. In a general adaptive filtering scheme the adaptive filter aims to minimize a certain measure of error between its output and a desired signal. Usually, a quadratic cost criterion is taken: the so-called mean-squared error. There exists an extensive literature on the subject of adaptive optimization of the tap weights. A problematic aspect of the TDL is that its associated basis functions, delayed versions of a unit pulse function, are extremely localized in time. As a consequence, a large number of delay elements or tap weights is required to mimic the dynamic behavior of systems that exhibit a large memory. For adaptive filtering a large number of weights may be undesirable for the following reasons. First, a lot of costly memory is needed to store the previous input samples and the values of the tap weights. Second, a large number of weights implies a large computational burden to perform the convolution with the input signal. Third, once the adaptive filter has reached its optimum, each adaptive weight fluctuates around its steady state value in this way causing a certain noise contribution at the output of the adaptive filter. The cumulative effect of the noise contributions of a large number of weights is a limiting factor in terms of the performance of the adaptive filter. In this thesis alternative adaptive filter structures are studied. We specifically investigate linear regression models of which the output is a weighted summation of filtered versions of the input signal, were the tap filters are Infinite Impulse Response (IIR) filters. Due to the similarities between the TDL and the linear regression model, we may use adaptation algorithms such as the Least-Mean-Square (LMS) or the Recursive Least-Squares (RLS) algorithms that were initially developed for the TDL, to optimize the weights of the linear regression model. When the tap filters are chosen appropriately for a specific application, one can suffice with less adaptive weights compared to the TDL. We investigate candidate sets of filters for linear regression models. We find that for the purpose of adaptive filtering specifically those sets of filters are interesting, of which the the impulse responses form (complete) systems of orthonormal functions. We consider (generalized) Laguerre, Kautz, Jacobi, Legendre, and Meixner-like filters and treat their relevant properties. These filters contain parameters (for example poles) that need to be chosen beforehand and therefore we consider some methods to find good values for them. As a by-product of the treated methods we propose a way to find good values for the free parameters in truncated expansions of finite support signals with orthonormal Hermite functions. The behavior of the LMS algorithm for the optimization of the weights in linear regression models is studied, where we consider the rate of convergence and the misadjustment. Further, we propose an adaptive optimization algorithm for the free parameter (a multiple pole) of a Laguerre filter. Finally, the adaptive optimization of a complex pole pair is considered for a specific second-order IIR adaptive filter, namely an adaptive line enhancer.