Hi all I am reading the Papoulis's book on Random Variable and getting confused with the following formula: X(t) is input of a linear system with impulse response h(t), Y(t) is its output. X(t) is a WSS Random Process ===> the Cross Correlation: Rxy(to)= Rxx(to) * h*(-to). where * is convolution, and h*(-to) is conjugate of h(-to). Now the density spectrum Sxy(omega)= Sxx(omega)H*(omega). I am confused, I think it should be Sxy(omega)= Sxx(omega)H*(-omega)., because the CTFT of x(-t) is X(-omega). AM I missing some point> Thanks
not a descrete case but a simple question about signal processing
Started by ●June 20, 2007
Reply by ●June 20, 20072007-06-20
On Jun 20, 10:59 am, VijaKhara <VijaKh...@gmail.com> wrote:> Hi all > > I am reading the Papoulis's book on Random Variable and getting > confused with the following formula: > > X(t) is input of a linear system with impulse response h(t), Y(t) is > its output. > X(t) is a WSS Random Process > ===> the Cross Correlation: Rxy(to)= Rxx(to) * h*(-to). where * is > convolution, and h*(-to) is conjugate of h(-to). > > Now the density spectrum Sxy(omega)= Sxx(omega)H*(omega). > > I am confused, I think it should be Sxy(omega)= Sxx(omega)H*(-omega)., > because the CTFT of x(-t) is X(-omega). > > AM I missing some point> > > ThanksIf your input signal is real-valued, then the Fourier transform is symmetric, and hence H(w) = H(-w). If your input signal is complex valued, then the transformation H'(w) = H(-w) is called spectral inversion => the real and imaginary parts of x(t) are interchanged in the time-domain. Regards, BERT.