Multi-channel sampling of low-pass signals

Abstract

A deterministic signal x(t) band limited to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|\omega| &lt; \sigma</tex> is passed through m linear time-invariant filters (channels) to obtain the m outputs <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z_1(t),\cdots,Z_m(t)</tex> . If the filters are independent in a sense to be defined, then It Is shown that the common input <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> may be reconstructed from samples of the outputs <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(Z_k)</tex> , each output being sampled at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m \Pi</tex> samples per second or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(1/m)</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega</tex> rather than on the two parameters <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> ; 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.