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The problem of stability of an adaptive differential pulse code modulation (ADPCM) system is considered from two viewpoints. First, it is demonstrated that the step-size adaptation algorithm of a certain class of ADPCM's can be characterized by the system's response to step: function inputs. The degree of "hunting" or "oscillation" provides an indication of the deterministic stability of the algorithm, and algorithms are presented that minimize these effects. Secondly, stationary random processes with rational spectral densities are used to study stochastic stability of ADPCM. It is shown that for certain ADPCM's the time-averaged mean-absolute quantizing noise is bounded, that the joint distribution of the input and decoded output processes possesses a stationary distribution, that this joint distribution converges to the stationary distribution from an arbitrary starting point, and that the mean-absolute quantizing noise is finite under the stationary distribution.