Optimal Sampling of a Bandlimited Noisy Multiple Output Channel

Abstract

This paper deals with an extension of Papoulis' generalized sampling expansion (GSE) of bandlimited signals to a case where a white noise, filtered to the signal's band, is added before sampling. We look for the best sampling scheme that maximizes the capacity or minimizes the mean-squared error (MSE) of the sampled channel between the input signal and the M sampled outputs signals, e.g. channels. Due to the noise it is beneficial to sample at a total rate higher than Nyquist, yet there is no gain in sampling each signal above Nyquist; thus the total rate considered, normalized by Nyquist rate, is between 1 to M. To avoid preference to any channel, the paper assumes that the channels are composed of all-pass linear time-invariant (LTI) systems. For the case where the total normalized rate is between M-1 and M, we show that the best scheme samples M-1 outputs at Nyquist rate and the last output at the remaining rate. Surprisingly, equal sampling is suboptimal in general, but when there is an integer relation between M and the total normalized rate, equal sampling achieves the optimal performance. We also make a conjecture on other rates that the best scheme either samples some channels at Nyquist and one another channel at the remaining rate, or sample the channels equally. This conjecture is supported by simulation for M ≤ 5. Finally, we discuss the relation between maximizing the capacity and minimizing the MSE.