A contribution to advanced theory of discrete signals and systems

The paper presents basic concepts of the discrete system theory from the viewpoint of Banach algebras and shows that some Banach algebras of sequences are not only suitable mathematical framework for the multidimensional discrete system theory but also open ways to new models of discrete systems and lead to hitherto unknown methods of homomorphic signal processing and deconvolution. Three Banach algebras of multidimensional sequences are investigated: the algebra of absolutely summable sequences, the algebra of periodic sequences, and the algebra of sequences with finite support. A general formula for the Gelfand transform is shown to be a generalization of the traditional Fourier transforms. It becomes a theoretical basis for investigation of deconvolution in the three algebras. Two basic operations of homomorphic signal processing, cepstrum and inverse cepstrum are shown to be equivalent to the logarithmic and exponential functions; their computation is facilitated by a close relationship among logarithmic functions (and also exponential functions) in the three Banach algebras.