Question

Consider a wide-sense stationary process zlnl] with power spectrum Pole'") = 1 13 12 cos w 10+6 cosw 1. Find the autocorrelation sequence ra[k] of zIn). 2. Find a stable and causal whitening filter H(z) that produces unit variance white noise when the input is rn]. Determine the impulse response hin] of this whitening filter

Show work for every step please

Answer 1

In a wide -Sense stationary process x (h) with power spectrum (1) P_x (e^jw) = 13 - 12 Cos w/10 + 6 Cos w the Autocorrelation and power spectral ferity form the Fourier transform pair P_x (e^jw) = 13 - 12 Cos w/10 + 6 Cos w = 13 - 12 (e^jw + e^-je/2/10 + 6 (e^jw + e^-jw/2) P_x (e^jw) = 13 - 6e^jw - 6e^-jw/10 + 3e^jw + e^-jw the relationship between 3 - transform and furrier transform is z = re^jw put r = 1 implies z = e^jw r_x (K) P_x (z) = 13 - 6z -6z^-1/10 + 3z + 3z^-1 = (3 - 2z) (3 - 2z^-1)/(3 + z) (3 - z^-1) r_x (K) = P_x (z) = G (z) G (z^-1) G (z) = 3 - 2z/3 + z H (z) = 1/G (z) = 3 + z/3 - 2z \r\n H (z) = z (1 + 3/z)/2z (- 1 + 3/(+2z) = 1/2 (1 + 3z^- 1)