  We explore two fundamental questions at the intersection of sampling theory
and information theory: how channel capacity is affected by sampling below the
channel's Nyquist rate, and what sub-Nyquist sampling strategy should be
employed to maximize capacity. In particular, we derive the capacity of sampled
analog channels for three prevalent sampling strategies: sampling with
filtering, sampling with filter banks, and sampling with modulation and filter
banks. These sampling mechanisms subsume most nonuniform sampling techniques
applied in practice. Our analyses illuminate interesting connections between
under-sampled channels and multiple-input multiple-output channels. The optimal
sampling structures are shown to extract out the frequencies with the highest
SNR from each aliased frequency set, while suppressing aliasing and out-of-band
noise. We also highlight connections between undersampled channel capacity and
minimum mean-squared error (MSE) estimation from sampled data. In particular,
we show that the filters maximizing capacity and the ones minimizing MSE are
equivalent under both filtering and filter-bank sampling strategies. These
results demonstrate the effect upon channel capacity of sub-Nyquist sampling
techniques, and characterize the tradeoff between information rate and sampling
rate.
