Abstract:

The classical sampling theorem permits reconstruction of a bandlimited function from its values on a set of equidistant points on the real line R. This theory has been extended in many directions. In one of these extensions, one considers sampling sets, which are unions of cosets of one subgroup, hence periodic sampling. We consider the case where the sampling set is a union of cosets of possibly different subgroups, hence noperiodic sampling. In our theory, a function f can be reconstructed from its samples provided the sampling set and the support of the Fourier transform of f satisfy certain compatibility conditions. New results show that these conditions can be relaxed. An explicit reconstruction formula is given for sampling sets which are unions of two shifted lattices. While explicit formulas for unions of more than two lattices are possible, it is more convenient to use a recursive algorithm. The analysis is presented in the general framework of locally compact abelian groups, but several specific examples are given on the real line. Examples in finite cyclic groups Z_N and Z_N X Z_N are presented and implemented in MATLAB.