Q. 4 A signal generator generates a random sinusoid, X cos(2t + ), whose amplitu

Question

Q. 4 A signal generator generates a random sinusoid, X cos(2t + ), whose amplitude is given by a random variable X, with an Gaussian density function X N(1,1), and whose phase is an independent random variable that equals one of {/2,+/2}
with equal probability. This signal’s amplitude is additively corrupted by independent noise, that is given by Y N(0,1). The output amplitude is

denoted by Z, where Z = (X +Y).
1. Compute Var(Z), E[XY], E[Z], and E[XZ].
2. In this part, determine an expression for a linear estimator of the form ˆ

X = cZ +d, which minimizes the mean-square error

between ˆ

X and X. Here, c,d are any real numbers that you can choose in order to minimize the mean-square error.

Explanation / Answer

Z=X+Y

Z~N(1,2)

Var(Z)=2 since X & Y independently distributed

E(XY)=E(X)*E(Y)

=0*1

=0

E(Z©)=E(Z)*E(©)

= 1*0=0 since indendency and E(©)=0

E(XZ)=E(X^2)+E(XY)

=1+0=1

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