The autocorrelation function (K(s)) of a certain statistically stationary variable (y(t)) is given by (a) (K(s)=K(0) e^{-alpha s^{2}} cos left(2 pi f^{*} s ight)) or

The autocorrelation function \(K(s)\) of a certain statistically stationary variable \(y(t)\) is given by

(a) \(K(s)=K(0) e^{-\alpha s^{2}} \cos \left(2 \pi f^{*} s\right)\)

or by

(b) \(K(s)=K(0) e^{-\alpha|s|} \cos \left(2 \pi f^{*} s\right)\), where \(\alpha>0\). Determine, and discuss the nature of, the power spectrum \(w(f)\) in each of these cases and investigate its behavior in the limits (a) \(\alpha \rightarrow 0\), (b) \(f^{*} \rightarrow 0\), and (c) both \(\alpha\) and \(f^{*} \rightarrow 0\).