Multidimensional sampling: Difference between revisions

Revision as of 21:26, 25 April 2012

Revision as of 08:40, 26 April 2012

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==Preliminaries==

Analogous to theThe concept of a [[Bandlimiting|bandlimited]] function in one dimension, one can definebe generalized to the notion of a wavenumber-limited function in higher dimensions as a generalization of the concept of a [[Bandlimiting|bandlimited]] function in one dimension. Recall that the [[Fourier transform]] of an integrable function ''ƒ(.)'' on ''n''-dimensional space is defined as:

:<math>\hat{f}(\xi) = \mathcal{F}(f)(\xi) = \int_{\Re^n} f(x) e^{-2\pi i \langle x,\xi \rangle} \, dx</math>

where ''x'' and ''ξ'' are ''n''-dimensional [[vector (mathematics)|vectors]], and <math>\langle x,\xi \rangle</math> is the [[inner product]] of the vectors. The function ''ƒ(.)'' is said to be wavenumber-limited to a set <math>\Omega</math> if the Fourier transform satisfies <math>\hat{f}(\xi) = 0</math> for <math>\xi \notin \Omega</math>.