Let the observed series xt be composed of a periodic signal and noise so it can be written as xt=ß1cos(2p?kt)+ß2sin(2p?kt)+wt where wt is a white noise process with variance sw. The frequency ?k is...

Let the observed series xt be composed of a periodic signal and noise so it can be written as xt=ß1cos(2p?kt)+ß2sin(2p?kt)+wt

where wt is a white noise process with variance sw. The frequency ?k is assumed to be known and of the form k/n in this problem. Suppose we consider estimating ß1,ß2 and sw2 by least squares, or equivalently, by maximum likelihood if the wt are assumed to be Gaussian.

(a) Prove, for a fixed?k, the minimum squared error is attained by where the cosine and sine transforms (4.31) and (4.32) appear on the right-hand

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