compendium

A. Sandryhaila, J. Kovačević and M. Püschel, "Algebraic signal processing theory: 1-D nearest-neighbor models," IEEE Trans. Signal Proc., vol. 60, no. 5, May 2012, pp. 2247-2259.

[ pdf | @ IEEE Xplore | bibtex]

We present a signal processing framework for the analysis of discrete signals represented as linear combinations of orthogonal polynomials. We demonstrate that this representation implicitly changes the associated shift operation from the standard time shift to the nearest-neighbor shift introduced in this paper. Using the algebraic signal processing theory, we construct signal models based on this shift and derive their corresponding signal processing concepts, including the proper notions of signal and filter spaces, z-transform, convolution, spectrum, and Fourier transform. The presented results extend the algebraic signal processing theory and provide a general theoretical framework for signal analysis using orthogonal polynomials.

All necessary proofs are included in the paper.