In digital signal processing, multidimensional sampling is the process of converting a function of a multidimensional variable into a discrete collection of values of the function measured on a discrete set of points. This article presents the basic result due to Petersen and Middleton on conditions for perfectly reconstructing a wavenumber-limited function from its measurements on a discrete lattice of points. This result, also known as the Petersen–Middleton theorem, is a generalization of the Nyquist–Shannon sampling theorem for sampling one-dimensional band-limited functions to higher-dimensional Euclidean spaces. In essence, the Petersen–Middleton theorem shows that a wavenumber-limited function can be perfectly reconstructed from its values on an infinite lattice of points, provided the lattice is fine enough. The theorem provides conditions on the lattice under which perfect reconstruction is possible. As with the Nyquist–Shannon sampling theorem, this theorem also assumes an idealization of any real-world situation, as it only applies to functions that are sampled over an infinitude of points. Perfect reconstruction is mathematically possible for the idealized model but only an approximation for real-world functions and sampling techniques, albeit in practice often a very good one.
Preliminaries
The concept of a bandlimited function in one dimension can be generalized to the notion of a wavenumber-limited function in higher dimensions. Recall that the Fourier transform of an integrable function f ( ⋅ ) {\displaystyle f(\cdot )} on n-dimensional Euclidean space is defined as:
f ^ ( ξ ) = F ( f ) ( ξ ) = ∫ ℜ n f ( x ) e − 2 π i ⟨ x , ξ ⟩ d x {\displaystyle {\hat {f}}(\xi )={\mathcal {F}}(f)(\xi )=\int _{\Re ^{n}}f(x)e^{-2\pi i\langle x,\xi \rangle }\,dx}
where x and ξ are n-dimensional vectors, and ⟨ x , ξ ⟩ {\displaystyle \langle x,\xi \rangle } is the inner product of the vectors. The function f ( ⋅ ) {\displaystyle f(\cdot )} is said to be wavenumber-limited to a set Ω {\displaystyle \Omega } if the Fourier transform satisfies f ^ ( ξ ) = 0 {\displaystyle {\hat {f}}(\xi )=0} for ξ ∉ Ω {\displaystyle \xi \notin \Omega } . Similarly, the configuration of uniformly spaced sampling points in one-dimension can be generalized to a lattice in higher dimensions. A lattice is a collection of points Λ ⊂ ℜ n {\displaystyle \Lambda \subset \Re ^{n}} of the form
Λ = { ∑ i = 1 n a i v i | a i ∈ Z } {\displaystyle \Lambda =\left\{\sum _{i=1}^{n}a_{i}v_{i}\;|\;a_{i}\in \mathbb {Z} \right\}}
where {v1, ..., vn} is a basis for ℜ n {\displaystyle \Re ^{n}} . The reciprocal lattice Γ {\displaystyle \Gamma } corresponding to Λ {\displaystyle \Lambda } is defined by
Γ = { ∑ i = 1 n a i u i | a i ∈ Z } {\displaystyle \Gamma =\left\{\sum _{i=1}^{n}a_{i}u_{i}\;|\;a_{i}\in \mathbb {Z} \right\}}
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