Communication in Gaussian noise. In a communication system, a transmitter wants to send some signal to a receiver over a noisy medium. Suppose the signal ? is a Gaussian distributed random variable ? ? N (0, ??2 ). The noise in the medium is modeled as a Gaussian distributed randomvariableW?N(0,?W2 )independentofthesignal.ThereceiverobservesX=2?+W.

(a) Find the estimator of ? given X that minimizes the MSE E[(?? ? ?)2]. (b) Find the MSE for the estimator you found in part (a).

(c) Find the linear LMS estimator of ? given X. (d) Find the MSE for the estimator you found in part (c).

(e) Find the LMS estimator of ?2 given X. (f) Find the linear LMS estimator of ?2 given X.

Communication in Gaussian noise. In a communication system, a transmitter wants to send some signal to a receiver over a noisy medium. Suppose the signal 9 is a Gaussian distributed random variable 9 ~ N(0, 0%). The noise in the medium is modeled as a Gaussian distributed random variable W N N(0, 0%,) independent of the signal. The receiver observes X = 28+ W. (a) Find the estimator of 9 given X that minimizes the MSE E[( (9)2]. (b) Find the MSE for the estimator you found in part (a). (c) Find the linear LMS estimator of 8 given X. (01) (e) Find the LMS estimator of 62 given X. (f) Find the linear LMS estimator of 82 given X. Find the MSE for the estimator you found in part (c)