Multidimensional sampling: Difference between revisions

Revision as of 10:54, 10 May 2012

Revision as of 11:02, 10 May 2012

Line 37:

The theorem gives conditions on sampling lattices for perfect reconstruction of the sampled. If the lattices are not fine enough to satisfy the Petersen-Middleton condition, then the field cannot be reconstructed exactly from the samples in general. In this case we say that the samples may be [[Aliasing|aliased]].

The generalization of the [[Poisson summation formula]] to higher dimensions shows<ref name="stewei71">E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces", Princeton University Press, Princeton, 1971.</ref> can be used to show that the samples, <math>\{f(x): x \in \Lambda\} </math>, of the function ''f(.)'' on the lattice <math>\Lambda</math> are sufficient to create a [[periodic summation]] of the function <math>\hat f(.)</math>. The result is:

{{NumBlk|:|<math>\hat f_s(\xi)\ \stackrel{\mathrm{def}}{=} \sum_{y \in \Gamma} \hat f\left(\xi - y\right) = \sum_{x \in \Lambda} |\Lambda|f(x) \ e^{-i 2\pi \langle x, \xi \rangle},</math>|{{EquationRef|Eq.1}}}}