Convolution theorems for the free metaplectic transformation and its application

The free metaplectic transformation (FMT) is widely used in several fields, including filter design, pattern recognition, image processing, and optics. This study investigated two new convolution theorems in the FMT domain to obtain a more concise and intuitive convolution form. First, on the basis of expressing the generalized translation, we derived the convolution theorem (of the first kind) in the FMT domain, which has the elegance and simplicity of the classical Fourier transform (FT) results. Second, we provide the convolution theorem (of the second kind) in the FMT domain to further study the diversity of the convolution theory. The provided solution can be represented using a simple integral that is easy to implement in multiplying filter designs. Finally, based on this simple form of the convolution theorems, we designed a multiplicative filter in the FMT domain, demonstrating the feasibility and efficiency of the filters designed in this study through simulations.