[Random phases in a communication system] A particular communication system tran

Question

[Random phases in a communication system] A particular communication system transmits a perfectly sinusoidal carrier wave, X = sin(T), where T is measured in microseconds. A particular receiver samples the signal at a random time T, and measures just one output voltage, X = sin(T). T has the following pmf: pT(u) = {0.2 u {pi /2, 3pi /2, 5pi /2, 7pi /2, 9pi /2}, 0 otherwise Find E[T] and sigma 2 T Find E[X], E[X2], and sigma 2 X without finding the pmf of X. Suppose that a different receiver samples the signal at time S, where ps(v) = {1/N 0 v = n pi - pi /2,n {1, ..., N} 0 otherwise, where N is a constant positive integer. Let the measured signal sample be Y = sin(S). Find E[Y], E[Y2], and sigma 2 Y. Be sure to describe the outcome in the case when N is even, as well as the case when N is odd.

Explanation / Answer

mean= integral - to xp(x)dx ==> integral x* e^Xdx =





b) varience=E[x^2]-E[x]^2
E[x]= 0 to 1 xf(x)dx
E[x^2]=x^2*e^x

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