A First-Order Markov Model for Understanding Delta Modulation Noise Spectra

A first-order Markov process is used to model the sequence of quantization noise samples in delta modulation. An autocorrelation parameter
C
in the Markov model controls the shape of the noise spectrum, and as
C
decreases from 1 to 0 and then to -1, the spectrum changes from a low-pass to a flat, and then to a high-pass characteristic. One can also use the Markov model to predict the so-called out-of-band noise rejection that is obtained when delta modulation is performed with an oversampled input, and the resulting quantization noise is lowpass filtered to the input band. The noise rejection
G
is a function of
C
as well as an oversampling factor
F
and an interesting asymptotic result is that
G=\frac{1-C}{1+C} \dot F
if
F \gg \frac{1+C}{1-C} \dot \frac{\pi}{2}
. Delta modulation literature has noted the importance of the special
G
values,
F
and
2F
. These correspond to autocorrelation values of 0 and -1/3.