Andrzej Kubaszek
Rzeszów University of Technology,
Rzeszów, Poland

Discrete representations of generalized time domain function in symbolic-numerical analysis

This presentation is available at   https://pei.prz.edu.pl/~kubaszek/smacd06/

  1. INTRODUCTION
  2. GENERALIZED TIME DOMAIN FUNCTION SAMPLING
  3. INTERACTIVE GRAPHS
  4. CONCLUSIONS

INTRODUCTION

Computer simulation sometimes leads to the results, which can be supposed as the numerical instability. It is shown, that some of that results should be accepted – they are generalized time domain functions in Mikusinski operational calculus, and after some operations they can be useful solutions.

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GENERALIZED TIME DOMAIN FUNCTION SAMPLING

The functions of continuous time defined for t ≥ 0, can be expressed in Mikusinski's operational calculus in form of functions of Heaviside's operator p, where X(p) denote Laplace transform formulas: { x (t) } = X (p) p {1}

Replacing continuous time function {1} by sequence of samples (~0.5~, 1~1~1~...~), and replacing Heaviside operator utilizing algorithm of numerical integration we obtain expression for approximate sequence of samples of { x(t) } function.
[ 

Laplace - Z - PN

See the numerical example ...

This way we get discrete equivalence of abstract functions, like derivative of non-smooth function or function containing negative delay.

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CONCLUSIONS

  1. Repetitive symbolic-numerical time domain analysis which take into account generalized functions illustrates abstract functions introduced 50 years ago by Mikusiński, like derivative of non-smooth function or function containing negative delay operator [>>]
  2. Although series of samples of these functions pretend numerical instability case, although sampling theorem seems to be broken, discrete equivalences of such functions act the same way as in continuous time domain.
  3. Repetitive analysis and interactive graphs demonstrate unusual properties of discrete approximations of these functions.
  4. Knowledge regarding special properties of sequences of samples of such functions can be utilized to test or to improve DSP algorithms.

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Discrete representations of generalized time domain function in symbolic-numerical analysis.
Andrzej Kubaszek, Rzeszów University of Technology, Rzeszów, Poland.
https://pei.prz.edu.pl/~kubaszek/smacd06/