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A certain random signal has been sampled to form a sequence of random variables, X_n, n = 1,2,3... The samples have the property that the average power in the signal is E[X²_n] = 4 and the correlation between adjacent samples is E[X_nX_n+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= aX_n.

The goal is to choose the constant a so that the prediction error e_n+1= X_n+1- X'_n+1is minimized in the mean square sense.

(a) Find the value of a which makes E[e²_n+1] as small as possible.

(b) Find the resulting value of the mean square error, E[e²_n+1] How does this compare with the power in the original process E[X²_n].

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Engineering Mathematics, Engineering

Category:- Engineering Mathematics
Reference No.:- M91617263

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