Generalised Fractional Indexes Approximation with Application to Discrete-Time Generalised Weyl Symbol Computation

P Orlowski

doi:10.5755/j01.eee.123.7.2367

Abstract

Unique description of discrete-time, linear time-invariant systems on infinite time horizon requires only definition of finite number of coefficients, usually relatively small. In contradistinction the description of discrete-time linear time-varying systems requires in general definition of an infinite number of coefficients. Nevertheless neither analyzing nor processing data with infinite dimensional size is impossible. The main aim of the paper is to develop new generalised fractional indexes – computational method which allows to determine generalised Weyl symbol for arbitrary real a not only for a=±0.5 (integer indexes) and a=0. Parameter a allow to shape the set of parameterised impulse responses. The selection of the parameter a in the generalised Weyl symbol enable selection of the best accuracy region for the time-frequency transform. Numerical examples illustrates how the approximation of the system response with generalised fractional indexes increase accuracy for the computation of the discrete-time, time-frequency transformation calculated on finite time horizon. Ill. 4, bibl. 18 (in English; abstracts in English and Lithuanian).DOI: http://dx.doi.org/10.5755/j01.eee.123.7.2367

Highlights

Linear discrete-time system can be described by finite set of coefficients of difference equations or state space model
In contradistinction the description of discrete-time linear time-varying systems requires in general definition of an infinite number of coefficients
Linear timevarying systems can be classified with respect to the simplifying assumption

Summary

Introduction

Linear discrete-time system can be described by finite set of coefficients of difference equations or state space model. The set define dynamics of the linear time-invariant system for all times k (infinite time horizon). In contradistinction the description of discrete-time linear time-varying systems requires in general definition of an infinite number of coefficients. On the class of the system, but especially for time-varying systems in the general form analysis can be realized only on finite time horizon. It mean that accessible system data is limited by two constraints for indexes kmin and kmax that define range for variable k k k:k Z\ k kmin k kmax. Discrete-time formula of the Generalised Weyl Symbol can be written using digital set of parameterised impulse responses (5)

Generalised fractional indexes approximation

Conclusion