Generalized convolution concept based on DCT

A generalized approach to the so-called product filtering of digital signals valid for a wide class of linear invertible transformations is presented in this paper. Product type of digital filtering consists in multiplication of the transformed signal with some selectivity function in the transform domain and is in this paper interpreted as a generalized convolution process in the primary domain. Our considerations are based on the observation that the block-wise product filtering of digital signals can be performed by means of multiplication of a block of samples of the transformed signal with some function in a domain of any invertible transformation just in the same way as it is usually done in the frequency domain after the Fourier transformation. The only (sufficient) condition for a suitable forward transformation is the existence of the inverse transformation. The presented idea of the generalized product filtering and the generalized convolution has been confronted with a family of the DCT transformations and the Karhunen-Loeve transformation. For the DCT -III the convolution formula has been derived.