We want to estimate the vector H from the vector Z.We need to know the covariance matrix for Z and the covariance between H and Z.The covariance matrix for Z has the i-th row, j-th column being E[ZiZj].You should be able to complete this for each i and j.Similarly for the covariance between Hi and Zj. Use the first image to help do the problem.

A communication system transmits a sinusoidal signal at three different frequencies. The propagation of the signal causes fading. The receiver measures the signal strength at the three frequencies. The measurement results are 3'\" || H1 + N1: H2+Ns, 33 = H3 + N3, .5\" II where Hi, i = 1, 2, 3 represents the signal fading level at the three different frequencies and N}, i = 1, 2.3 represents the measurement noise. The fading levels at the three different frequencies are known to be zero mean Gaussian with covariance matrix lo ole: ol CH = where o: = 1/2. The noise N1, N2, N3 have mean zero and variance 1 and are indepen- dent of each other and also independent of the fading H = [H], H2, H3]'. (a) Find the best (linear or otherwise) estimate of H1, H2 and H3 from the observations Z 11 ZR: ZE- (h) Find the mean squared error of the estimates of each of the fading levels. Suppose that X and Y are jointly Gaussian vectors (column vectors). Suppose that X has covariance matrix Cy, Y has covariance matric Cy and (X, Y) has covariance Cx,v. Suppose that we observe the random vector X. Then the best estimate of Y is the conditional mean. EYX = x] = MY + CY.XC (X - HX) The covariance of the error is E[(Y - Y)((Y - Y)'] = CY - CY,xCX CX,Y The MSE for Y, is the i-th diagonal element of this matrix