Journal
IEEE TRANSACTIONS ON SIGNAL PROCESSING
Volume 53, Issue 4, Pages 1425-1438Publisher
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TSP.2005.843700
Keywords
continuous-time signal processing; convolution; differential operators; Green functions; interpolation; modulation; multiresolution approximation; splines
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Causal exponentials play a fundamental role in classical system theory. Starting from those elementary building blocks, we propose a complete and self-contained signal processing formulation of exponential splines defined on a uniform grid. We specify the corresponding B-spline basis functions and investigate their reproduction properties (Green function and exponential polynomials); we also characterize their stability (Riesz. bounds). We show that the exponential B-spline framework allows an exact implementation of continuous-time signal processing operators including convolution, differential operators, and modulation, by simple processing in the discrete B-spline domain. We derive efficient filtering algorithms for multiresolution signal extrapolation and approximation, extending earlier results for polynomial splines. Finally, we present a new asymptotic error formula that predicts the magnitude and the Nth-order decay of the L-2-approximation error as a function of the knot spacing T.
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