A Tighter Uncertainty Principle for Linear Canonical Transform in Terms of Phase Derivative

Abstract

This study devotes to uncertainty principles under the linear canonical transform (LCT) of a complex signal. A lowerbound for the uncertainty product of a signal in the two LCT domains is proposed that is sharper than those in the existing literature. We also deduce the conditions that give rise to the equal relation of the new uncertainty principle. The uncertainty principle for the fractional Fourier transform is a particular case of the general result for LCT. Examples, including simulations, are provided to show that the new uncertainty principle is truly sharper than the latest one in the literature, and illustrate when the new and old lower bounds are the same and when different.Index Terms-Complex signal, fractional Fourier transform, linear canonical transform (LCT), uncertainty principle.

Uncertainty principles associated with the offset linear canonical transform

“…In Gröchenig, 28 the classical Heisenberg uncertainty principle was introduced, which is based on the interpretation of the standard deviation △ f x as the size of the essential support of function f. Thus, if we use other different notions of the support, then we can obtain other different forms of the uncertainty principle. [29][30][31][32][33][34][35][36] For instance, Donoho and Stark 28,37 used the concept " -concentrated" to replace standard deviation, and derived the corresponding uncertainty principle. Therefore, in this paper, we first generalize the Donoho-Stark's uncertainty principle for the OLCT, in order to get a similar quantitative result for the essential support of function f and its OLCT.…”

A Simplified Proof of Uncertainty Principle for Quaternion Linear Canonical Transform

“…It can be regarded as a generalization of many mathematical transforms such as the Fourier transform, Laplace transform, the fractional Fourier transform, and the Fresnel transform. Many fundamental properties of this extended transform are already known, including shift, modulation, convolution, and correlation and uncertainty principle, for example, in [1][2][3][4][5][6].…”

New convolution structure for the linear canonical transform and its application in filter design

“…Particularly, the LCT is very useful and effective in non-stationary signal processing since it can be regarded as the decomposition of a signal based on a non-orthogonal basis [18][19][20][21]. Meanwhile, some fundamental theories and concepts associated with the LCT have been established, for example the convolution and product theorems [22][23][24][25][26][27][28], the uniform and nonuniform sampling theorems [29][30][31][32][33][34], and the uncertainty principles [35][36][37][38][39].…”