Convolution and Multichannel Sampling for the Offset Linear Canonical Transform and Their Applications

The offset linear canonical transform (OLCT) plays an important role in many fields of optics and signal processing. In this paper, we address the problem of signal filtering and reconstruction in the OLCT domain based on new convolution theorems. Firstly, we propose new convolution and product theorems for the OLCT, which state that a modified ordinary convolution in the time domain is equivalent to simple multiplication operations for the OLCT and the Fourier transform (FT). Moreover, it is expressible by a one dimensional integral and is easy to implement in designing filters. The classical convolution theorem in the FT domain is shown to be a special case of our derived results. Then, a practical multichannel sampling expansion for band-limited signal with the OLCT is introduced. This sampling expansion constructed by the new convolution structure can reduce the effect of spectral leakage and is easy to implement. By designing OLCT filters, we can obtain derivative sampling and second-order derivative interpolation. Furthermore, potential applications of the multichannel sampling are discussed. Last, based on the new convolution structure, we investigate and discuss several applications, including swept-frequency filter analysis, image denosing and image encryption. Some illustrations and simulations are presented to verify the validity and effectiveness of the proposed method.