An ideal Hilbert transformer with impulse response
has input xr[n] and output xi [n] = xr [n] ∗ h[n], where xr [n] is a discrete-time random signal.
(a) Find an expression for the autocorrelation sequence Φxixi [m] in terms of h[n] and Φxr xr [m].
(b) Find an expression for the cross-correlation sequence Φxr xi [m]. Show that in this case, Φxr xi [m] is an odd function of m.
(c) Find an expression for the autocorrelation function of the complex analytic signal x[n] = xr [n] + jxi [n].
(d) Determine the power spectrum Pxx(ω) for the complex signal in part (c).
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