Kautz Functions Research Articles

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.

Optimality conditions for a specific class of truncated Kautz series

Kautz functions constitute a basis in L/sup 2/ and can be used for filter synthesis starting from a prescribed transient response. In a practical situation, only truncated series are used and thus the speed of convergence of such series is of importance. This convergence speed is a function of the parameters. The Kautz functions that are considered can be generated by a line of identical Nth-order allpass sections. The optimality conditions for the parameters of this specific class of truncated Kautz series are established solely on the basis of the orthonormality and the polynomial character of the basis functions.

A generalized orthonormal basis for linear dynamical systems

In many areas of signal, system, and control theory, orthogonal functions play an important role in issues of analysis and design. In this paper, it is shown that there exist orthogonal functions that, in a natural way, are generated by stable linear dynamical systems and that compose an orthonormal basis for the signal space l/sub 2sup n/. To this end, use is made of balanced realizations of inner transfer functions. The orthogonal functions can be considered as generalizations of, for example, the pulse functions, Laguerre functions, and Kautz functions, and give rise to an alternative series expansion of rational transfer functions. It is shown how we can exploit these generalized basis functions to increase the speed of convergence in a series expansion, i.e., to obtain a good approximation by retaining only a finite number of expansion coefficients. Consequences for identification of expansion coefficients are analyzed, and a bound is formulated on the error that is made when approximating a system by a finite number of expansion coefficients.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Laguerre and Kautz Models

The decomposition of signals and systems into appropriate basis components is a central problem in many fields of engineering and science. It would be an overwhelming task to give a general survey on modeling using general orthonormal basis functions. Since the impulse response of a stable finite dimensional linear system can be represented by a sum of exponentials (times polynomials), it seems reasonable to use basis functions of the same type. This leads to what we here will refer to as Kautz functions, which are the Laplace transforms of a class of orthogonalized exponentials. The well-known Laguerre functions are a special case of Kautz functions. In particular, we will discuss the following topics: Optimal basis functions given certain a priori information about the system; Properties and construction of orthonormal basis functions for approximation of dynamical systems; Use of orthonormal basis functions in system identification.