Sampling representations and the optimum reconstruction of signals

This paper treats the problem of the representation and reconstruction of signals from several sets of sampled values using a multiple-channel sampling and reconstruction scheme. Realizable and unrealizable solutions are presented for the optimum linear postfiltering and prefiltering operations. It is shown that the number of "independent" sets of samples which are necessary for the exact reconstruction of a signal is equal to the maximum number of overlappings of its sampled spectrum. This enables many different sampling expansions to be derived. The simultaneous optimization of the unrealizable linear prefilter and postfilter combination is carried out for the case where two sets of sampled values are taken. It is shown that with the optimum combination of filters, the angular frequency range of the input signal is limited by the prefilters to a total width of
4\pi \rho
, which is a natural extension of the single-channel result.