Consider an LTI system whose impulse response is real and is given by h[n]. Suppose the...

Consider an LTI system whose impulse response is real and is given by h[n]. Suppose the responses of the system to the two inputs x[n] and v[n] are, respectively, y[n] and z[n], as shown in Figure P2.97.

The inputs x[n] and v[n] in the figure represent real zero-mean stationary random processes with autocorrelation functions Φ_xx[n] and Φ_vv[n], cross-correlation function Φ_xv[n], power spectra Φ_xx(e^jω) and Φ_vv(e^jω), and cross power spectrum Φ_xv(e^jω).

(a) Given the cross power spectrum of y[n] and z[n], where Φ_yz(e^jw) is defined by

with Φ_yz[n] = E{y[k]z[k − n]}.

(b) Is the cross power spectrum Φ_xv(e^jω) always nonnegative; i.e., is Φ_xv(e^jω) ≥ 0 for all ω? Justify your answer.