A certain random signal has been sampled to form a sequence of random variables, Xn, n = 1,2,3... The samples have the property that the average power in the signal is E[X2n] = 4 and the correlation between adjacent samples is E[XnXn+1]=2. We wish to digitize this signal using DPCM and hence we need to design a linear predictor. To keep it simple we will design a simple one-tap linear predictor which forms an estimate of the next sample by forming a constant times the previous sample X'n+1= aXn.
The goal is to choose the constant a so that the prediction error en+1= Xn+1- X'n+1is minimized in the mean square sense.
(a) Find the value of a which makes E[e2n+1] as small as possible.
(b) Find the resulting value of the mean square error, E[e2n+1] How does this compare with the power in the original process E[X2n].