Cardinal Exponential Splines: Part I—Theory and Filtering Algorithms

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.


Published in:
IEEE Transactions on Signal Processing, 53, 4, 1425–1438
Year:
2005
Publisher:
IEEE
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 Record created 2005-11-30, last modified 2018-10-07

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