  A Delta-Sigma modulator that is often utilized to convert analog signals into
digital signals can be modeled as a static uniform quantizer with an error
feedback filter. In this paper, we present a rate-distortion analysis of
quantizers with error feedback including the Delta-Sigma modulators, assuming
that the error owing to overloading in the static quantizer is negligible. We
demonstrate that the amplitude response of the optimal error feedback filter
that minimizes the mean squared quantization error can be parameterized by one
parameter. This parameterization enables us to determine the optimal error
feedback filter numerically. The relationship between the number of bits used
for the quantization and the achievable mean squared error can be obtained
using the optimal error feedback filter. This clarifies the rate-distortion
property of quantizers with error feedback. Then, ideal optimal error feedback
filters are approximated by practical filters using the Yule-Walker method and
the linear matrix inequality-based method. Numerical examples are provided for
demonstrating our analysis and synthesis.
