Multi-channel sampling of low-pass signals

A deterministic signal x(t) band limited to is passed through m linear time-invariant filters (channels) to obtain the m outputs . If the filters are independent in a sense to be defined, then It Is shown that the common input may be reconstructed from samples of the outputs , each output being sampled at samples per second or th the rate associated with the Input signal. A rigorous derivation of this result Is given which proceeds from a minimum error energy criterion and leads to a system of linear algebraic equations for the optimal reconstruction filters. The system of equations derived here, which differs from the system given recently by Papoulis [1], has the advantage of depending on only one parameter rather than on the two parameters and ; it also puts into evidence the fact that the spectra of the optimal reconstruction filters can be pieced together directly, without additional computation, from the elements of the system\´s inverse matrix. Lastly, the solutions of the system obtained in the Papoulis formulation are shown to be time-varying linear combinations of the simpler one-parameter solutions.