A Generalized Sampling Theorem for Frequency Localized Signals - Hammerich, Edwin (Sampling theory in signal and image processing, 2009)
A generalized sampling theorem for frequency localized signals is presented. The generalization in the proposed model of sampling is two-fold: (1) It applies to various prefilters effecting a “soft” bandlimitation, (2) an approximate reconstruction from sample values rather than a perfect one is obtained (though the former might be “practically perfect” in many cases). For an arbitrary finite-energy signal the frequency localization is performed by a prefilter realizing a crosscorrelation with a function of prescribed properties. The range of the prefilter, the so-called localization space, is described in some detail. Regular sampling is applied and a reconstruction formula is given. For the reconstruction error a general error estimate is derived and connections between a critical sampling interval and notions of “soft bandwidth” for the prefilter are indicated. Examples based on the sinc-function, Gaussian functions, and B-splines are discussed.