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 ?_xixi[m] in terms of h[n] and ?_xrxr[m].

(b) Find an expression for the cross-correlation sequence ?_xrxi[m]. Show that in this case ?_xrxi[m] is an odd function of m.

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

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

Transcribed Image Text:

2 sin (an/2) n +0, n = 0. h[n] = 0.