Fast reconstruction methods for bandlimited functions from periodic nonuniform sampling

Thomas Strohmer and Jared Tanner

A well-known generalization of Shannon's sampling theorem states that a bandlimited function can be reconstructed from its periodic nonuniformly spaced samples if the effective sampling rate is at least the Nyquist rate. Analogous to Shannon's sampling theorem this generalization requires that an infinite number of samples is available, which however is never the case in practice. Most existing reconstruction methods for periodic nonuniform sampling yield very low-order (often not even first-order) accuracy when only a finite number of samples is given. In this paper we propose a fast, numerically robust, root-exponential accurate reconstruction method. The efficiency and accuracy of the algorithm is obtained by fully exploiting the sampling structure and utilizing localized Fourier analysis. We discuss applications in analog-to-digital conversion where nonuniform periodic sampling arises in various situations. Finally we demonstrate the performance of our algorithm by numerical examples.

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