Section 9.2

Linear prediction

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Problem 9.2.1

Let XX be a (weakly) stationary sequence with zero mean and autocovariance function c(m)c(m).

(i) Find the best linear predictor X^n+1\widehat{X}_{n+1} of Xn+1X_{n+1} given XnX_{n}.

(ii) Find the best linear predictor X~n+1\widetilde{X}_{n+1} of Xn+1X_{n+1} given XnX_{n} and Xn−1X_{n-1}.

(iii) Find an expression for D=E[(Xn+1−X^n+1)2]−E[(Xn+1−X~n+1)2]D = \mathbb {E}\left[\left(X_{n+1}-\widehat{X}_{n+1}\right)^{2}\right]-\mathbb {E}\left[\left(X_{n+1}-\widetilde{X}_{n+1}\right)^{2}\right], and evaluate this expression when:

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(a)

Xn=cos⁡(nU)X_{n} = \cos (n U) where UU is uniform on [−π,π][-\pi , \pi ],

(b)

XX is an autoregressive scheme with c(k)=α∣k∣c(k) = \alpha^{\left|k\right|} where ∣α∣<1\left|\alpha \right| < 1.

Problem 9.2.2

Does there exist a (weakly) stationary sequence {Xn:−∞<n<∞}\left\{ X_{n}:-\infty < n < \infty \right\} with zero means and autocovariance function

c(k)={1 if k=0a1+a2 if ∣k∣=10 if ∣k∣>1 c(k) = \begin{cases} 1 & \text{ if } k = 0 \\ \frac{a}{1+a^{2}} & \text{ if }\left|k\right| = 1 \\ 0 & \text{ if }\left|k\right| > 1\end{cases}

Assuming that such a sequence exists, find the best linear predictor X^n\widehat{X}_{n} of XnX_{n} given Xn−1X_{n-1}, Xn−2,…X_{n-2}, \ldots, and show that the mean squared error of prediction is (1+a2)−1\left(1+a^{2}\right)^{-1}. Verify that {X^n}\left\{ \hat{X}_{n}\right\} is (weakly) stationary.

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