Autocorrelation and spectral density in MATLAB

Question

This question is twofold.

We have an LTI system that is a first degree Butterworth LP filter with the power TF

where fu = 110Hz and f1 = 90Hz

The input X(t) has the autocorrelation: R_X(\tau) = 5e^{-600|\tau|}

1) How can I calculate the power spectral density of the output in MATLAB? FFT? How do I represent the autocorrelation as a vector?

2) How can I simulate the system and plot the output in MATLAB?

Answer 1

To calculate the Autocorrelation of the output:

R_y = xcorr(y); % with Y(f) = X(f).H(f) or y(t) = x(t)*h(t)

To calculate the P.S.D of the output:

PSD_y = fft (R_y);

To reprensent them graphically, you can use plot

if you do not have the x(t) or X(f) expressions, you could use this relation:

PSD_y (f) = PSD_x (f) . |H(f)|^2  % with PSD_x (f) = fft(R_x);