  Sampling above the Nyquist rate is at the heart of sigma-delta modulation,
where the increase in sampling rate is translated to a reduction in the overall
(mean-squared-error) reconstruction distortion. This is attained by using a
feedback filter at the encoder, in conjunction with a low-pass filter at the
decoder. The goal of this work is to characterize the optimal trade-off between
the per-sample quantization rate and the resulting mean-squared-error
distortion, under various restrictions on the feedback filter. To this end, we
establish a duality relation between the performance of sigma-delta modulation,
and that of differential pulse-code modulation when applied to (discrete-time)
band-limited inputs. As the optimal trade-off for the latter scheme is fully
understood, the full characterization for sigma-delta modulation, as well as
the optimal feedback filters, immediately follow.
