Analysis of a delayed delta modulator

While delta modulation (DM) simply compares the current predictive estimate of the input with the current sample, delayed delta modulation (DDM) also compares with the upcoming sample so as to detect and anticipate slope overloading. Since this future sample must be available before the present output is determined and the estimate updated, delay is introduced at the encoding. The performance of DDM with perfect integration and step-function reconstruction is analyzed for each of three random input signals. In every case, the stochastic stability of the system is established. For a discrete time, independent and identically distributed input, the (limiting) joint distribution of input and output is derived, and the (asymptotic) mean-square sample point error mse(SP) is computed when the input is Gaussian. For a Wiener input, the joint distribution of the sample point and prediction errors is derived, and mse(SP) and the time-averaged mse (mse(TA)) are computed. For a stationary first-order Gauss-Markov input, the joint distribution of input and output is derived and mse(SP) and mse(TA) computed. Graphs of the mse\´s illustrate the improvement attainable by using DDM instead of DM. With optimal setting of parameters, mse(SP) (mse(TA)) is reduced about percent ( percent).