On completeness and multi-channel sampling

Abstract

A bandlimited input x is applied as a common input to m linear time-invariant filters (channels) which are ‘independent’ in a certain sense. We investigate, from a unified point of view using the concept of completeness, the problem of uniquely reconstructing the input x for all time values from samples of the m outputs, each output being sampled at the uniform rate of σ/ gqm samples/second, where σ is the positive frequency bandwidth of the input signal. For the lowpass input case and independent channels, perfect reconstruction is always possible; similarly for the case of a bandpass input and an even number of channels, recovery of the input can also be accomplished at a rate of σ/ mπ samples/second. However, for an odd number m of channels and a bandpass input, it is shown that the rate σ/ mπ samples/second at the outputs no longer suffices to determine the input uniquely unless 2 ω 0 m/ σ is an odd integer, a constraint on the relation between the center frequency of the band, ω 0, and the bandwidth σ. To obtain a sampling theorem when the number of channels is odd and in the absence of such a positioning constraint, a higher sampling rate per channel must be employed. In those cases which permit a unique determination of the input from samples of the output channels, it is shown that a linear scheme involving m linear time-invariant post-filters can be used to effect the input reconstruction.