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An optimum adaptive delta modulator-demodulator configuration is derived. This device utilizes two past samples to obtain a step size which minimizes the mean square error for a Markov Gaussian source. The optimum system is compared using computer simulations with the linear delta modulator and an enhanced Abate delta modulator. In addition the performance is compared to the rate distortion bound for a Markov source. It is shown that the optimum delta modulator is neither quantization nor slope-overload limited. The highly nonlinear equations obtained for the optimum transmitter and receiver are approximated by piecewise-linear equations in order to obtain system equations which can be transformed into hardware. The derivation of the experimental system is presented. The experimental "optimum" system, an enhanced version of the Abate delta modulator and a linear delta modulator were tested and compared using sinusoidal, square-wave, and pseudorandom binary sequence inputs. The results show that the output signal-to-noise (SNR) ratio is approximately independent of the input signal power and is subject only to the limitations of the hardware employed. In addition, voice was recorded using these systems. The demodulated voice indicates negligible degradation is caused by the optimum system and by the enhanced Abate system while the linear delta modulator suffers significant degradation at a sampling frequency of 56 k/s. The systems were also tested at 19.2 k/s. At this bit rate, speech recognition, using the experimental "optimum" system, remained completely intelligible.