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This paper discusses the use of symmetric convolution and the discrete sine and cosine transforms (DSTs and DCTs) 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 DTTs of those two sequences. There are 16 members in the family of DTTs. Each provides a representation for a corresponding distinct type of symmetric-periodic sequence. The author defines symmetric convolution, relates the DSTs and DCTs to symmetric-periodic sequences, and then use these principles to develop simple but powerful convolution-multiplication properties for the entire family of DSTs and DCTs. 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 DTTs. Conventional linear convolution is possible if one first zero-pad the input data. Symmetric convolution and its fast implementation using DTTs are now an alternative to circular convolution and the DFT