Derivation of Extended Gaussian Functions Based on the Zak Transform

Abstract

Siohan and Roche introduced recently a new family of modulated filter banks, which was derived from a set of continuous-time orthogonal functions named extended Gaussian functions (EGFs). At first, these EGFs were obtained using the isotropic orthogonal transform algorithm (IOTA), i.e., a two-step orthogonalization procedure of the Gaussian function. As shown recently, using the Zak transform, the IOTA method yields the tight window function canonically associated with the Gaussian. In this letter, it is shown that the Zak transform can also be useful in recovering the series expansion of the EGFs. Practical guidelines are also provided in order to get accurate approximations of the orthogonal EGFs.