Please solve step by step

4.11 Let the observed series * be composed of a periodic signal and noise so it can be written as X, = 81 cos(2xwx!) + 82 sin(2mux!) + Wr where w, is a white noise process with variance ow. The frequency wx is assumed to be known and of the form & in this problem. Suppose we consider estimating B1, By and ow by least squares, or equivalently, by maximum likelihood if the w, are assumed to be Gaussian. (a) Prove, for a fixed wx, the minimum squared error is attained by = 2n-1/2 |de(x) where the cosine and sine transforms (4.31) and (4.32) appear on the right-hand side. (b) Prove that the error sum of squares can be written as SSE = > x, - 21(WE) so that the value of ox that minimizes squared error is the same as the value that maximizes the periodogram /,(wx) estimator (4.28). (c) Under the Gaussian assumption and fixed up , show that the F-test of no regression leads to an F-statistic that is a monotone function of 1(wx )