The Algebraic Structure in Signal Processing: Time and Space

The assumptions underlying linear signal processing (SP) produce more structure than vector spaces. We capture this structure by describing the space of filters as an algebra and the space of signals as the associated module. We formulate an algebraic approach to SP that is axiomatically based on the concept of a signal model. Signal models for time are visualized as directed graphs. We construct corresponding models for undirected graphs, which we hence call space models, and show that, in particular, the 16 DCTs and DSTs are Fourier transforms for these finite space models. Finally, we discuss the extension of our theory to separable and nonseparable 2-DSP