Generalized convolution concept extension to multidimensional signals

In this paper a concept generalizing the well known circular convolution to any linear invertible block-transformation is presented. The proposed approach is first summarized for chosen one-dimensional cases and then it is extended to multidimensional transformations. The definition of the generalized convolution and some its properties, illustrated with examples of selected transformations, are presented. The theorems regarding analysis with the use of random signals and stochastic processes, redefined for the generalized convolution are listed. Finally, the methodology of extending the technique so that it becomes suitable for multidimensional signals and separable transformations is shown. The presented concept, in which any number of dimensions may be considered, is supplemented with the example of the two-dimensional DCT-IU.