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

The convolution and product theorem for the Fourier transform (FT) plays an important role in signal processing theory and application. The linear canonical transform (LCT), which is a generalization of the FT and the fractional Fourier transform (FRFT), has found many applications in optics and non-stationary signal processing. Recently, some scholars have formulated a series of convolution and product theorems for the LCT, however, both of them do not maintain the convolution theorem for the FT. The purpose of this paper is to present a new convolution structure for the LCT having the elegance and simplicity in both time and LCT domains comparable to that of the FT. We also show that with the new convolution theorem it is convenient to implement in the designing of multiplicative filters through both the new convolution in the time domain and the product in the LCT domain.