An ideal Hilbert transformer with impulse response has input xr[n] and output x...

An ideal Hilbert transformer with impulse response

has input x_r[n] and output x_i [n] = x_r [n] ∗ h[n], where x_r [n] is a discrete-time random signal.

(a) Find an expression for the autocorrelation sequence Φ_xixi [m] in terms of h[n] and Φ_{xr xr} [m].

(b) Find an expression for the cross-correlation sequence Φ_{xr xi} [m]. Show that in this case, Φ_{xr xi} [m] is an odd function of m.

(c) Find an expression for the autocorrelation function of the complex analytic signal x[n] = x_r [n] + jx_i [n].

(d) Determine the power spectrum P_xx(ω) for the complex signal in part (c).