  This paper investigates the effect of sub-Nyquist sampling upon the capacity
of an analog channel. The channel is assumed to be a linear time-invariant
Gaussian channel, where perfect channel knowledge is available at both the
transmitter and the receiver. We consider a general class of right-invertible
time-preserving sampling methods which include irregular nonuniform sampling,
and characterize in closed form the channel capacity achievable by this class
of sampling methods, under a sampling rate and power constraint. Our results
indicate that the optimal sampling structures extract out the set of
frequencies that exhibits the highest signal-to-noise ratio among all spectral
sets of measure equal to the sampling rate. This can be attained through
filterbank sampling with uniform sampling at each branch with possibly
different rates, or through a single branch of modulation and filtering
followed by uniform sampling. These results reveal that for a large class of
channels, employing irregular nonuniform sampling sets, while typically
complicated to realize, does not provide capacity gain over uniform sampling
sets with appropriate preprocessing. Our findings demonstrate that aliasing or
scrambling of spectral components does not provide capacity gain, which is in
contrast to the benefits obtained from random mixing in spectrum-blind
compressive sampling schemes.
