Circuits, Systems, and Signal Processing

, Volume 38, Issue 11, pp 5212–5235

# Convolution Theorem with Its Derivatives and Multiresolution Analysis for Fractional S-Transform

Article

## Abstract

Fractional S-transform (FrST) is a time–frequency representation of signals with frequency-dependent resolution. FrST is also an advantageous technique for non-stationary signal processing applications. Till now, only linearity, scaling, time reversal, time marginal condition, and inverse FrST properties are documented. In this paper, some remaining properties of FrST are proposed to establish it as a complete transform technique. The proposed properties are convolution theorem, correlation theorem, and Parseval’s theorem. To expand the applicability of FrST as a mathematical transform tool, the multiresolution analysis concept is also documented. The multiresolution analysis has shown significant performance to develop the orthogonal kernel for FrST. Finally, the applications of proposed convolution theorem are demonstrated on multiplicative filtering for electrocardiogram signal and linear frequency-modulated signal under AWGN channel.

## Keywords

Fractional S-transform Convolution theorem Correlation theorem Parseval’s theorem Multiresolution analysis Multiplicative filtering

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