Implementations of Shannon's sampling theorem, a time-frequency approach

Thomas Strohmer and Jared Tanner

Shannon's sampling theorem quantifies the Fourier domain periodization introduced by the equidistant sampling of a bandlimited signal when the sampling rate is at least as fast as the Nyquist rate dictated by the signal's bandwidth. If sampled faster than the Nyquist rate, i.e., oversampling, a reconstruction composed of highly localized atoms is possible, allowing for practical applications where only a truncated set of samples is available. More specifically, it is known that root-exponential accuracy can be achieved by constructing atoms whose Fourier transform (filter) is infinitely differentiable and compactly supported in the appropriate bandwidth. Unfortunately, there is no known compactly supported infinitely smooth filter whose corresponding atom has a known explicit representation. By considering filters with Gevrey regularity, we obtain root-exponential localization for the atom, and an efficient truncated Gabor approximation of the filter and atom. Furthermore, we present an alternative error decomposition that allows for the complete rigorous analysis of the error in truncating the signal, and of the error introduced in approximating the filter and atom. By scaling the approximation order appropriately, the root-exponential convergence rate is not adversely effected by the filter's approximation.

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