Novel Wigner distribution and ambiguity function associated with the linear canonical transform

Abstract

The Wigner distribution (WD) and ambiguity function (AF) associated with the linear canonical transform (LCT) attract much attention in the literature. For this, many generalized time-frequency distributions are currently derived, for example the WD and AF in the LCT domain (WDL, AFL), the unified WD and AF in the LCT domain (UWA, WAL), and the WD and AF based on the generalized convolution in the LCT domain (LWD, LAF). However, there are two issues associated with these generalized distributions, which the first one relates to the fact that the marginal properties of these distributions do not have the elegance and simplicity comparable to those of the WD and AF and the other issue relates to the fact that there has no affine transformation relationship between the LCT and WDL, AFL, LWD, and LAF, and the affine transformation relationships between the LCT and UWA as well as the LCT and WAL do not generalize very nicely and simply the classical results for the WD and AF. Focusing on the above issues, this paper deduces a kind of novel WD and AF associated with the LCT, which has the elegance and simplicity in marginal properties and affine relations associated with the LCT comparable to the WD and AF. Then some essential properties, and relations with other classical time-frequency representations of the newly defined WD and AF are investigated. Applications of the new WD and AF are also performed to show the advantage of the theory.