SYMMETRIC CONVOLUTION AND THE DISCRETE SINE AND COSINE TRANSFORMS

This paper discusses the use of symmetric convolution and the discrete sine and cosine transforms (DST's & DCT's) for general digital signal processing. The operation of symmetric convolution is a formalized approach to convolving symmetrically extended sequences. The result is the same as that obtained by taking an inverse discrete trigonometric transform (DTT) of the product of the forward DTT's of those two sequences. There are 16 members in the family of DTT's. Each provides a representation for a corresponding distinct type of symmetric-periodic sequence. In this paper, we define symmetric convolution, relate the DST's and DCT's to symmetric-periodic sequences, and then use these principles to develop simple but powerful convolution-multiplication properties for the entire family of DST's and DCT's. Symmetric convolution can be used for discrete linear filtering when the filter is symmetric or antisymmetric. The filtering will be efficient because fast algorithms exist for all versions of the DTT's. Conventional linear convolution is possible if we first zero-pad the input data. Symmetric convolution and its fast implementation using DTT's are now an alternative to circular convolution and the DFT.