On the Delta Modulation of a First-Order Gauss-Markov Signal

Consider the delta-modulation (DM) of a first-order Gauss-Markov signalX. Let the adjacent-sample correlation inXbec, and let the (first-order) DM predictor coefficient bea. We express the quantizer input Qrin the formS_{r} - aE_{r - 1} + (c - a)X_{r - 1}], where Sris an "innovations" term,aE_{r - 1}denotes the effect of quantizationerror(E)feedback and(c - a)X_{r - 1}reflects the effect of using ana neq c. For the important case ofc rightarrow 1(which models over-sampled DM inputs), we propose the simplifying assumption 7] of uncorrelatedXandE; with this assumption, our formalization of quantizer input leads very simply to interesting results in linear (LDM) and adaptive delta modulation (ADM). The LDM results are generalizations of known expressions for optimum values ofa, and the step-size Δ, and the value of signal-to-noise ratio SNR. For ADM, we derive optimum multiplier values for step-size adaptations with a one-bit memory, using the case ofc sim a = 1for simplicity. Our results depend on modeling instantaneous step-size adaptation as a mechanism for tracking the expected magnitude ofQ; existing literature has formalized such adaptation models only for the case of multi-bit quantizers.