xi(t) X(t) h(t) | H(jw) Y(t) Ryy(t) Syy (jw) Figure 1 v2cos(21fot+0) The input signal X, (t) to the system shown in Figure 1 is a wide sense stationary zero mean Gaussian white noise process with power spectral density, Sx,x, (jw) = , (Watt/Hz) where N, is a constant. The signal is applied to a mixer followed by a low pass filter. The local oscillator produces a signal 2(cos(2nfot + 0)) where o is a continuous uniformly distributed (-11,1) random variable. The output of the mixer is X(t) = Xi(t) 2cos(21f.t + ). Let Y(t) be the output process. The filter has a transfer function given by Hom / 10002 (1000 + jw f. Find the cross spectral density function, Sxy(jw), and the cross-correlation function, Rxy(T) g. Find the output process power spectral density, Syy (jw). h. Determine the output signal autocorrelation function, Ryy(t). i. Is Y(t) wide sense stationary. j. What is the expected value of the total output signal power? Is there a DC component in the output process? k. Compare the expected value of total output signal power E[Y2(t)] to that of the input E[X(t)?]. Which one is higher? Does this result agree with the nature of the filter h(t) and the overall system? xi(t) X(t) h(t) | H(jw) Y(t) Ryy(t) Syy (jw) Figure 1 v2cos(21fot+0) The input signal X, (t) to the system shown in Figure 1 is a wide sense stationary zero mean Gaussian white noise process with power spectral density, Sx,x, (jw) = , (Watt/Hz) where N, is a constant. The signal is applied to a mixer followed by a low pass filter. The local oscillator produces a signal 2(cos(2nfot + 0)) where o is a continuous uniformly distributed (-11,1) random variable. The output of the mixer is X(t) = Xi(t) 2cos(21f.t + ). Let Y(t) be the output process. The filter has a transfer function given by Hom / 10002 (1000 + jw f. Find the cross spectral density function, Sxy(jw), and the cross-correlation function, Rxy(T) g. Find the output process power spectral density, Syy (jw). h. Determine the output signal autocorrelation function, Ryy(t). i. Is Y(t) wide sense stationary. j. What is the expected value of the total output signal power? Is there a DC component in the output process? k. Compare the expected value of total output signal power E[Y2(t)] to that of the input E[X(t)?]. Which one is higher? Does this result agree with the nature of the filter h(t) and the overall system