Window-Dependent Bases for Efficient Representations of the Stockwell Transform

TL;DR

This paper establishes the DOST basis as an orthonormal, time-frequency localized basis for L^2([0,1]) and introduces an efficient algorithm for computing Stockwell transform coefficients, enhancing its practical applicability.

Contribution

It proves the DOST basis is an orthonormal basis for L^2([0,1]) and develops a fast O(N log N) algorithm for Stockwell transform coefficient computation.

Findings

DOST basis is orthonormal and time-frequency localized.

Unified framework for Stockwell transform and orthogonal decomposition.

Fast algorithm extends previous methods with O(N log N) complexity.

Abstract

Since its appearing in 1996, the Stockwell transform (S-transform) has been applied to medical imaging, geophysics and signal processing in general. In this paper, we prove that the system of functions (so-called DOST basis) is indeed an orthonormal basis of L^2([0,1]), which is time-frequency localized, in the sense of Donoho-Stark Theorem (1989). Our approach provides a unified setting in which to study the Stockwell transform (associated to different admissible windows) and its orthogonal decomposition. Finally, we introduce a fast -- O(N log N) -- algorithm to compute the Stockwell coefficients for an admissible window. Our algorithm extends the one proposed by Y. Wang and J. Orchard (2009).