A stationary process X (t) is applied to a linear time-invariant filter of impulse response b (t),

Question:

A stationary process X (t) is applied to a linear time-invariant filter of impulse response b (t), producing an output Y (t).

(a) Show that the cross-correlation function RYX (?) of the output Y (t) and the input X (t) is equal to the impulse response b (?) convolved with the autocorrelation function RX (?) of the input, as shown by, show that the second cross-correlation function RXY (?) equals

(b) Find the cross-spectral densities SYX (f) and SXY (f).

(c) Assuming that X (t) is a white noise process with zero mean and power spectral density N0/2, shows that. Comment on the practical significance of this result.