Q3) When did you start your simulations: Consider the simulation of a first order auto- regressive, or AR(1), process. A wide-sense stationary real white noise process x[n] with autocorrelation Rxx[m] = 58[m] is passed through the AR(1) filter with a = 0.5, such that the output is y[n] = x[n] + 0.5y[n 1]. 1 x[n] H(2) y[n] 1 -0.5z-1 3.1) (3.1.a) Assume the above filter operates at all times. Then y[n] is jointly wide-sense stationary. Find the cross-correlation between x[n] and y[n], Rxy[m] = E{x[n ]y[n2]}, where m = n2 - 12. (Hint: There are many ways to solve this problem. You may use the frequency domain or the z domain approach, finding Sxy(w) or Sxy(2) first, and then taking the inverse Fourier or z transform. You may express y[n] as a convolution of x[n] with h[n] (impulse response), multiply the equation by x[nl], and take the expectation.] for m > 0 Rxy[m] = form so (4 marks) (3.1.b) Find the autocorrelation of the AR(1) process, Ryy[m]. Ryy[m] = (2 marks) H(2) n=0 1 1 -0.5z-1 y[n] Q3) When did you start your simulations: Consider the simulation of a first order auto- regressive, or AR(1), process. A wide-sense stationary real white noise process x[n] with autocorrelation Rxx[m] = 58[m] is passed through the AR(1) filter with a = 0.5, such that the output is y[n] = x[n] + 0.5y[n 1]. 1 x[n] H(2) y[n] 1 -0.5z-1 3.1) (3.1.a) Assume the above filter operates at all times. Then y[n] is jointly wide-sense stationary. Find the cross-correlation between x[n] and y[n], Rxy[m] = E{x[n ]y[n2]}, where m = n2 - 12. (Hint: There are many ways to solve this problem. You may use the frequency domain or the z domain approach, finding Sxy(w) or Sxy(2) first, and then taking the inverse Fourier or z transform. You may express y[n] as a convolution of x[n] with h[n] (impulse response), multiply the equation by x[nl], and take the expectation.] for m > 0 Rxy[m] = form so (4 marks) (3.1.b) Find the autocorrelation of the AR(1) process, Ryy[m]. Ryy[m] = (2 marks) H(2) n=0 1 1 -0.5z-1 y[n]