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Consider the abstraction of a point—to—point Wireless communication system. The trans—
mitted signal X is an N (0, 0:) random variable. The channel output is H X , and the receiver
input is: Y = HX + V, Where V is an N (0, 03,) random variable that is independent of X
and represents thermal noise in the amplifier circuitry. Suppose the channel is subject to in—
termittent failure, e.g., due to fading. For simplicity, suppose H is a random variable taking the values 0 and 1 With probabilities (1 — p) and 19, respectively. Assume H is independent
of both X and V. 0 Find the linear least squares estimate of X based on the observation Y: E‘[X|Y], the
variance of the estimate, and the mean—square estimation error. . Show that E[X|Y = y] = 23:0P(H = i|Y = y)E[X|Y = y, H = i]

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