Algebraic Signal Processing Theory: 1-D Nearest Neighbor Models

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.