  Representing a continuous-time signal by a set of samples is a classical
problem in signal processing. We study this problem under the additional
constraint that the samples are quantized or compressed in a lossy manner under
a limited bitrate budget. To this end, we consider a combined sampling and
source coding problem in which an analog stationary Gaussian signal is
reconstructed from its encoded samples. These samples are obtained by a set of
bounded linear functionals of the continuous-time path, with a limitation on
the average number of samples obtained per unit time available in this setting.
We provide a full characterization of the minimal distortion in terms of the
sampling frequency, the bitrate, and the signal's spectrum. Assuming that the
signal's energy is not uniformly distributed over its spectral support, we show
that for each compression bitrate there exists a critical sampling frequency
smaller than the Nyquist rate, such that the distortion in signal
reconstruction when sampling at this frequency is minimal. Our results can be
seen as an extension of the classical sampling theorem for bandlimited random
processes in the sense that it describes the minimal amount of excess
distortion in the reconstruction due to lossy compression of the samples, and
provides the minimal sampling frequency required in order to achieve this
distortion. Finally, we compare the fundamental limits in the combined source
coding and sampling problem to the performance of pulse code modulation (PCM),
where each sample is quantized by a scalar quantizer using a fixed number of
bits.
