Multidimensional sampling

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. f ^ s ( ξ )   = d e f ∑ y ∈ Γ f ^ ( ξ − y ) = ∑ x ∈ Λ | Λ | f ( x )   e − i 2 π ⟨ x , ξ ⟩ , {displaystyle {hat {f}}_{s}(xi ) {stackrel {mathrm {def} }{=}}sum _{yin Gamma }{hat {f}}left(xi -y ight)=sum _{xin Lambda }|Lambda |f(x) e^{-i2pi langle x,xi angle },}     (Eq.1) f ( x ) = ∑ y ∈ Λ | Λ | f ( y ) χ ˇ Ω ( y − x ) {displaystyle f(x)=sum _{yin Lambda }|Lambda |f(y){check {chi }}_{Omega }(y-x)} ,    (Eq.2) 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.