A family of convolution-based generalized Stockwell transforms
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A family of convolution-based generalized Stockwell transforms H. M. Srivastava1,2,3 · Firdous A. Shah4
· Azhar Y. Tantary4
Received: 28 January 2020 / Revised: 25 July 2020 / Accepted: 4 August 2020 © Springer Nature Switzerland AG 2020
Abstract The main purpose of this paper is to introduce a family of convolution-based generalized Stockwell transforms in the context of time-fractional-frequency analysis. The spirit of this article is completely different from two existing studies (see D. P. Xu and K. Guo [Appl. Geophys. 9 (2012) 73–79] and S. K. Singh [J. PseudoDiffer. Oper. Appl. 4 (2013) 251–265]) in the sense that our approach completely relies on the convolution structure associated with the fractional Fourier transform. We first study all of the fundamental properties of the generalized Stockwell transform, including a relationship between the fractional Wigner distribution and the proposed transform. In the sequel, we introduce both the semi-discrete and discrete counterparts of the proposed transform. We culminate our investigation by establishing some Heisenberg-type inequalities for the generalized Stockwell transform in the fractional Fourier domain. Keywords Stockwell transform · Wavelet transform · Wigner distribution · Fractional Fourier transform · Time-fractional-frequency analysis · Uncertainty principle
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Firdous A. Shah [email protected] H. M. Srivastava [email protected] Azhar Y. Tantary [email protected]
1
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
2
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan 40402, Republic of China
3
Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan
4
Department of Mathematics, University of Kashmir, South Campus, Anantnag, Jammu and Kashmir 192101, India
H. M. Srivastava et al.
Mathematics Subject Classification Primary 65R10, 42A38, 42C40, 42C20, 47G10 · Secondary 44A15, 94A12
1 Introduction Undoubtedly, one of the most valuable and widely-used integral transforms is the Fourier transform (FT) which generally converts a signal from time versus amplitude to frequency versus amplitude. Hence, clearly, it can be viewed as the frequency representation of a signal. In spite of vast applicability, FT seems to be inadequate for studying non-stationary signals for at least two reasons: firstly, it does not give any information about the occurrence of the frequency component at a particular time; secondly, it enables us to investigate problems either in the time domain or the frequency domain, but not simultaneously in both domains. In order to circumvent these limitations, Gabor [1] introduced the short-time Fourier transform (STFT) by using a Gaussian distribution function as a window function with the aim of constructing efficient time-frequency localized expansions of finite energy signals. The STFT relies on the procedure of segmenting a signal by virtue of a rigid time-l
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