Stochastic Stability of Delta Modulation

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Abstract
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.

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