A.C. den Brinker

Optimality conditions for truncated Kautz series

Kautz functions constitute a complete orthonormal basis for square-summable functions both on a continuous as well as a discrete semi-infinite axis. A special case of the Kautz functions are the well-known Laguerre functions. The Kautz functions can be used as series expansions for causal impulse responses. Convergence of such series depends on the parameters in the Kautz functions. The conditions for the optimal parameters in a truncated Kautz series are derived.

Adaptive modified Laguerre filters

Abstract When designing an adaptive filter, decisions have to be made about the signal description, choice of the filter, its parameterization and the optimization criterion. These choices are interdependent in order to attain a computationally simple adaptive mechanism. We investigate appropriate signal descriptions and filters starting from an exponentially weighted squared-error criterion. Simple variations of (generalized) Laguerre filters are used. The weights in these filters can be tracked by an RLS-like algorithm. Convergence properties are discussed.

Calculation of the local cross-correlation function on the basis of the Laguerre transform

A simple and computationally efficient mechanism for calculating a running or local cross-correlation function of two time-domain signals is presented. In order to obtain a running cross-correlation function, the signals must be windowed. It is argued that an appropriate window for a local cross correlation is an exponential function. To obtain a computationally efficient mechanism, the windowed functions are decomposed in a series of orthogonal functions. The set or orthogonal functions is matched to the chosen window and is a Laguerre-Fourier series. The cross correlation of the windowed functions is equal to a weighted summation of cross-correlated pattern functions. The weights are determined by cross correlating the Laguerre coefficients. >

Laguerre filters with adaptive pole optimization

An algorithm for an adaptive Laguerre filter is described. The Laguerre weights are updated using standard RLS. As a new feature the Laguerre pole is adapted at each sampling instant for more flexibility. The additional complexity that is required to calculate the gradient w.r.t. the pole is small. A sign algorithm is used in order to have complete control of the step size with which the Laguerre pole is varied. The step size in the sign algorithm is pole dependent and chosen such that the cumulative transient signal caused by the persistent tuning of the Laguerre pole remains small enough. Also, the step size is chosen such that the Laguerre filter always remains stable. A computer experiment shows that the adaptive filter performs well.

Optimal parametrization of truncated generalized Laguerre series

In this paper we address the problem of approximating functions on a semi-infinite interval by truncated series of orthonormal generalized Laguerre functions. The generalized Laguerre functions contain two parameters, namely a scale factor and an order of generalization. The rate of convergence of a generalized Laguerre series depends on the choice of these parameters. Results concerning the determination of the two parameters are presented.

An iterative solution for the optimal poles in a Kautz series

Kautz series allow orthogonal series expansion of finite-energy signals defined on a semi-infinite axis. The Kautz series consists of orthogonalized exponential functions or sequences. This series has as free parameters an ordered set of poles, each pole associated with an exponential function or sequence. For reasons of approximation and compact representation (coding), an appropriate set of ordered poles is therefore convenient. An iterative procedure to establish the optimal parameters according to an enforced convergence criterion is introduced.

Optimal parameters in modulated Laguerre series expansions

The harmonically modulated Laguerre functions constitute an orthonormal basis in the Hilbert space of square-integrable functions on R/sup +/. This basis comprises three free parameters, namely a modulation, a scale and an order factor. In practice, we are interested in series expansions that are as compact as possible. The free parameters can be used as the means to obtain a compact series expansion for a given function. As the compactness criterion the first-order moment of the energy distribution in the transform domain is chosen. In that case, the optimum compactness parameters can be given in a simple analytic form depending on signal measurements only.

Adaptive Orthonormal Filters

Abstract Several sets of orthonormal functions on a semi-infinite discrete interval are reviewed. These sets of functions provide series expansions to arbitrary square-summable causal impulse responses. Truncated sets can be incorporated in adaptive filters, and the discussed cases can all be seen as HR-filter extensions to the well-known tapped-delay-line. The sets depend on one or more parameters. If sufficient a priori knowledge is available on the impulse response of the ‘unknown’ system, these parameters can be estimated beforehand. The different series expansions and their free parameters provide the possibility to make choices for the basis functions, and thus to keep the number of relevant expansion terms low. The latter has the practical implication that, although using the discussed sets in adaptive filters instead of the tapped-delay-line increases the complexity of the partial filtering operations itself, a far greater reduction of complexity in the control loop is potentially achievable.

Adaptive line enhancement using a second-order IIR filter

A second-order IIR filter is considered as the basic component of an adaptive line enhancer (ALE). As a new feature, the bandwidth of the proposed ALE is adapted simultaneously with the center frequency. This leads to the possibility of combining the convergence speed and accuracy. The adaptation of the filter poles is controlled by a sign algorithm. The step sizes are chosen such that transients caused by the retuning of the filter are ensured to remain much smaller in amplitude than the response of the filter to the input signal. When the input signal consists of a sinusoid corrupted by wideband noise, an accurate frequency parameter estimate can be obtained with the algorithm given in the paper.

Construction of local bases

A local signal analysis represents the properties of a signal on selected bounded domains. In the one-dimensional case, the local signal analysis can be based on orthogonal projections which are related to the subdivision into intervals. The multidimensional situation is more difficult to analyze. Some ideas of how to transfer the one-dimensional case to the multidimensional one are considered, replacing intervals by polytopes. As an example, the hexagonal subdivision of the two-dimensional space is treated.