Non-isotropic angular Stockwell transform and the associated uncertainty principles

ABSTRACT

For an efficient and directional representation of signals in higher dimensions, we propose the non-isotropic angular Stockwell transform in the context of time-frequency analysis. The proposed transform is aimed at rectifying the conventional Stockwell transform by employing an angular and scalable localized window which offers directional flexibility and thereby results in the multi-scale and directional analysis of signals in higher dimensions. The basic properties of the proposed transform such as orthogonality relation, reconstruction formula, derivation of the admissibility condition and characterization of the range are discussed using the machinery of operator theory and Fourier transforms. In addition, we introduce the discrete version of the non-isotropic angular Stockwell transform and establish a sufficient condition for the corresponding discrete family to be a frame in $L^2\left({\mathbb{R}}^2\right)$. Furthermore, some generalizations of the well-known Heisenberg-type inequalities are derived for the non-isotropic angular Stockwell transform in the Fourier domain.

KEYWORDS:

• Stockwell transform
• window function
• wavelet transform
• discrete Stockwell transform
• Fourier transform
• uncertainty principle

AMS SUBJECT CLASSIFICATIONS:

• 65R10
• 47G10
• 42C40
• 42C15
• 42A38
• 94A12

Acknowledgments

The authors are deeply indebted to the anonymous referees' for meticulously reading the manuscript, pointing out many inaccuracies and giving several valuable suggestions to improve the initial version of the manuscript to the present stage.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is supported by SERB (DST), Government of India, under Grant No. EMR/2016/007951.