Sampling and Reconstruction of Wave-Number-Limited Functions in N-Dimensional Euclidean Spaces

The well-known Whittaker-Kotel'nikov-Shannon sampling theorem for frequency-bandlimited functions of time is extended to functions of multidimensional arguments. It is shown that a function whose spectrum is restricted to a finite region of wave-number space may be reconstructed from its samples taken over a periodic lattice having suitably small repetition vectors. The most efficient lattice (i.e., requiring minimum sampling points per unit hypervolume) is not in general rectangular, nor is a unique reconstruction function associated with a given sampling lattice. The above results also apply to homogeneous wave-number-limited stochastic processes in the sense of a vanishing mean-square error. It is also found that, given a particular sampling lattice, the optimum (mean-square) presampling filter for nonwave-number-limited processes effects an ideal wave-number cutoff appropriate to the specified sampling lattice. Particular attention is paid to isotropic processes: minimum sampling lattices are specified up to eight-dimensional spaces, and a number of typical reconstruction functions are calculated.

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