A first-order autoregressive model is generated from the white noise series wt using the generating equations xt = xt1 + wt, where , for ||

A first-order autoregressive model is generated from the white noise series wt using the generating equations xt = φxt−1 + wt, where φ, for |φ| < 1, is a parameter and the wt are independent random variables with mean zero and variance σ2 w.

(a) Show that the power spectrum of xt is given by fx(ω) = σ2 w

1 + φ2 − 2φ cos(2πω)

.

(b) Verify the autocovariance function of this process is

γx(h) = σ2 w φ|h|

1 − φ2 , h = 0, ±1, ±2, . . ., by showing that the inverse transform of γx(h) is the spectrum derived in part (a).