Stochastic Stability of Delta Modulation

In spite of the great simplicity of delta modulation as an analog-todigital encoding technique, it has not yet succumbed to an adequate mathematical analysis. Although realistic inputs such as speech are extremely difficult to characterize, considerable insight could be obtained from a thorough analysis for the case of a stationary random input process with a prescribed spectral density. Yet no such results have been obtained because of the mathematical complexity of the nonlinear feedback loop. In fact, the presence of a feedback loop raises the possibility that instability in some sense could arise. The possibility that the decoded signal could "run away" or become unbounded, failing to track the original signal, has never been theoretically excluded. 821 822 T H E BELL SYSTEM TECHNICAL JOURNAL, APRIL 1972 Although experience with delta modulation shows that such an extreme form of instability never arises, it has never been shown analytically that the mean-square or mean-absolute quantizing noise has a finite upper bound. Another possibility which has never been theoretically excluded is erratic operation, where the statistical average of the quantizing noise magnitude continues to vary with time. In other words, although the input process is stationary, the decoded approximation process would be nonstationary with a time-varying probability distribution even after low-pass filtering. If this were the case, the decoded process would not be replicating the original process very effectively.