Problem 5. Estimation. Suppose we are trying to estimate the power. Po, of a sinusoid in AWGN. The received signal is of the form r(t) = s(t) +n(t) where, s(t)=2P cos(2 fet + 0) and n(t) is a Gaussian process such that for any value of t, n(t) is a mean 0 finite variance random variable. At the demodulator r(t) is mixed with local signals si(t) and 82(f) separately where si(t) = 2 cos(2 fet) and s2(t) = 2sin(2 f.). The results of this mixing are then each averaged over a time period T, where T spans an integer number of periods of the sinusoid, i.e., the mixed results are integrated from 0 to T and then scaled by 1/T. This yields two random variables: X1 and X2. Note that E[X] =P cos(0), E[X2] = Po sin(0). =V Let us say the variance of X, is 02/2 and the variance of X2 is 02/2 so the total variance is . Note that each part of this problem is an analytic exercise only (no sim- ulation). a. Let Y = X + X2. Show b. Show E[Y] Po+o, Var[Y] = 2Poo + 01. P (E[Y]) Var[Y]. c. In an actual implementation we can empirically estimate the mean and variance of Y respectively as Y= N k=1 N 1 N-1 Yk, s= (1 - 1) k=1 where, Yk, k = 1, 2,.... N are independent realizations of Y. We now let Show d. We next let P = (Y) - s. E[P] P. P = (Y) - CNS. Find the constant cN (that depends on N) so that E[P] = p. e. The result of part (d) yields an unbiased estimator of P. We can take the square root now to estimate Po. Will this estimator of Po also be unbiased? You do not have to do any calculations in this part but you may just say in words why you think this estimator for Po is biased or unbiased.