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Cardinal Exponential Splines: Part I—Theory and Filtering Algorithms

M. Unser, T. Blu

IEEE Transactions on Signal Processing, vol. 53, no. 4, pp. 1425-1438, April 2005.



unser0503fig01.gif

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 L2-approximation error as a function of the knot spacing T.

Please consult also the companion paper by M. Unser, "Cardinal Exponential Splines: Part II—Think Analog, Act Digital," IEEE Transactions on Signal Processing, vol. 53, no. 4, pp. 1439-1449, April 2005.


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