The discrete-time model of delta modulation is considered for a stationary random input process with a rational spectral density, and an auto-covariance that goes to zero as the lag approaches infinity. For leaky integration, the joint distribution of input and decoded approximation processes is shown to approach a unique stationary distribution from any initial condition. Under the stationary distribution, the decoded process may take on all values in a bounded interval that is independent of the input process. For the often-studied ideal integration model of delta modulation, it is shown that the successive distributions at even parity time instants converge to a limiting stationary distribution, while at odd parity time instants the distributions converge to a different limiting distribution. Under these limiting distributions, the decoded process is assigned a positive probability for each level of a (discrete) lattice of amplitudes. The mean-absolute approximation error and mean-absolute amplitude of the decoded process are shown to be finite under the limiting distributions. For both ideal and leaky integration cases, an explicit upper bound on mean-absolute approximation error is given, which is independent of the spectral density of the input process.
[1]
J. Doob.
Asymptotic properties of Markoff transition prababilities
,
1948
.
[2]
S. Foguel.
Existence of invariant measures for Markov processes
,
1962
.
[3]
Terrence L. Fine,et al.
The response of a particular nonlinear system with feedback to each of two random processes
,
1968,
IEEE Trans. Inf. Theory.
[4]
David J. Goodman,et al.
Delta modulation granular quantizing noise
,
1969
.
[5]
H. R. Schindler.
Delta modulation
,
1970,
IEEE Spectrum.
[6]
Emmanuel N. Protonotarios,et al.
Application of the Fokker-Planck-Kolmogorov equation to the analysis of differential pulse code modulation systems
,
1970
.