The random process r[n] is a zero-mean, unit-variance, white process. The ran­dom process...

The random process r[n] is a zero-mean, unit-variance, white process. The ran­dom process y[n] is obtained by filtering r[n] through a filter with frequency response G(ejΩ),as depicted in Figure Assume all signals and system impulse responses are real-valued.

(a) What is the PSD of y[n], Syy(ejΩ),), expressed in terms of G(ejΩ),)?

The process x[n] is obtained from the multiplication of the process r[n] specified above and a process w[n] as shown in Figure The process w[·] is independent of r[n] and takes the value 1 with probability p, and 0 with probability (1 − p), independently for each n:

---

(b) Calculate the mean and autocovariance functions of x[n]. Is x[n] a white process?

---

(c) Design the LTI filter H1(ejΩ) in Figure  with the input x[n], so that the output process q[n] has the same PSD as y[n], your result from part (a).

---

(d) Design the LTI filter H2(ejΩ) in Figure  for which the input x[n]will produce an output  that at every instant is the LMMSE estimate of y[n].

---

(e) For your answer in part (d), calculate the resulting mean square error. What is the mean square error when p = 0 and when p = 1? Comment on these answers; do they seem reasonable to you?