Poisson Summation Formulae Associated with the Special Affine Fourier Transform and Offset Hilbert Transform

Abstract

This paper investigates the generalized pattern of Poisson summation formulae from the special affine Fourier transform (SAFT) and offset Hilbert transform (OHT) points of view. Several novel summation formulae are derived accordingly. Firstly, the relationship between SAFT (or OHT) and Fourier transform (FT) is obtained. Then, the generalized Poisson sum formulae are obtained based on above relationships. The novel results can be regarded as the generalizations of the classical results in several transform domains such as FT, fractional Fourier transform, and the linear canonical transform.

Uncertainty Principles for Quaternion Windowed Offset Linear Canonical Transform of Two Dimensional Signals

“…The OLCT is a six-parameter class of linear integral transform which encompasses a number of well known unitary transforms including the classical Fourier transform, fractional Fourier transform, Fresnel transform, Laplace transform, Gauss-Weierstrass transform, Bargmann transform and the linear canonical transform [3,4,5]. Due to the extra degrees of freedom, OLCT has attained a respectable status within a short span and is being broadly employed across several disciplines of science and engineering including signal and image processing, optical and radar systems, electrical and communication systems, pattern recognition, sampling theory, shift-invariant theory and quantum mechanics [6,7,8,9,10]. Recently, in [11] we introduce a hybrid integral transform namely, windowed special affine Fourier transform which is capable of providing a joint time and frequency localization of non-stationary signals with more degrees of freedom.…”

Special Affine Wavelet Transforms and the Corresponding Poisson Summation Formula

“…We also note that the phasespace transform (1.1) is lossless if and only if | det M | = 1, that is; AD − BC = 1 and for this reason, SAFT is also known as the inhomogeneous canonical transform [2]. Due to the extra degrees of freedom, SAFT has attained a respectable status within a short span and is being broadly employed across several disciplines of science and engineering including signal and image processing, optical and radar systems, electrical and communication systems, pattern recognition, sampling theory, shift-invariant theory and quantum mechanics [10,11,12,13,14,15,16]. Despite the versatile applicability, the SAFT surfers from an undeniable drawback due to its global kernel involved in (1.1) and thereby is incompetent in situations demanding a joint analysis of time and spectral characteristics of a signal.…”