New convolution and product theorem for the linear canonical transform and its applications

The convolution theorem for the linear canonical transform (LCT) is of importance in signal processing theory and application. Recently, some attempts at extending the convolution theorem in the Fourier transform (FT) domain to the LCT domain have derived many important results, which are very useful and effective in filter design and signal reconstruction, but none of them generalize very nicely and simply the classical result for the FT. In this paper, we formulate a new kind of convolution structure for the LCT, which has the elegance and simplicity in both time and LCT domains comparable to that of the FT and preserves the commutative and associative properties. Then with the new convolution theorem, it is easy to implement in the designing of multiplicative filters through both the new convolution in the time domain and the product in the LCT domain, and it is convenient to deduce the Shannon-type reconstruction formula for bandlimited signals in the LCT domain. Theoretical analyses and numerical simulations are also presented to show the correctness and effectiveness of the proposed techniques.