Multidimensional sampling: Difference between revisions

Revision as of 15:19, 10 May 2012

Revision as of 02:40, 23 May 2012

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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>.

Similarly, the configuration of uniformly spaced sampling points in one-dimension can be generalized to a [[Lattice (group)|lattice]] in higher dimensions. A lattice is a collection of points <math>\Lambda \subset \Re^n</math> of the form

<math>

\Lambda = \left\{ \sum_{i=1}^n a_i v_i \; | \; a_i \in\Bbb{Z} \right\}