2.12. Given the autocorrelation function for the random phase sinusoid in the previous Problem 2.11 compute the 2 x 2 autocorrelation matrix.

v2.13. The autocorrelation sequence of a zero mean white noise process is rv (k) =σ v2δ (k) and the power spectrum is ( ) v2 j

Pv e σ θ = , where 2

σ v is the variance of the process. For the random phase sinusoid the autocorrelation sequence is, rx (m) = A cos(m ) 1 2

2

ω 0 and the power spectrum is,

[ ( ) ( )]

2 1

( ) 0 0 0 0 P e jθ = π A2 u ω −ω + u ω +ω

x where:

u0 (ω −ω 0 ) represents an impulse at frequency ω 0 .

What is the power spectrum of the first-order autoregressive process that has an autocorrelation sequence of, m rx (m) =α

where:

|α| < 1 2.14. Let x[n] be a random process that is generated by filtering white noise w[n] with a first-order LSI filter having a system transfer function of, 1 0.25 1 1

( ) − −

=

z H z If the variance of the white noise is 2w

σ = 1 what is the power spectrum of x[n], Px (z) ? Find the autocorrelation of x[n] from Px (z) .