The Convolution Theorem Involving Windowed Free Metaplectic Transform

The convolution product is widely used in many fields, such as signal processing, numerical analysis and so on; however, the convolution theorem in the domain of the windowed metaplectic transformation (WFMT) has not been studied. The primary goal of this paper is to give the convolution theorem of WFMT. Firstly, we review the definitions of the FMT and WFMT and give the inversion formula of the WFMT and the relationship between the FMT and WFMT. Then, according to the form of the classical convolution theorem and the convolution operator of the FMT, the convolution theorem in the domain of the WFMT is given. Finally, we prove the existence theorems of the proposed convolution theorem.