A Generalized Sampling Theorem for Frequency Localized Signals

• Published in: Sampling theory in signal and image processing Vol. 8; no. 2; pp. 127 - 146
• Main Author: Hammerich, Edwin
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.05.2009; Sampling Publishing
• Subjects: Abstract Harmonic Analysis; Chebyshev approximation; Estimation theory; Hilbert space; Image and Speech Processing; Interpolation; Machine Learning; Mathematics; Methods; Properties; Sampling (Acoustical engineering); Signal; Signal processing; error estimate; generalized Chebyshev inequality; interpolating function; critical sampling interval; Frequency localization; reproducing kernel Hilbert space; generalized sampling theorem; 41A15; 94A20
• ISSN: 1530-6429
• DOI: 10.1007/BF03549512

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.