A white Gaussian noise process X(t) with power spectral density (PSD) N0/2=0.1 is input to an LTI filter with a transfer function H(f) given by
H(f) = 2, |f| <= W and 0, otherwise
The output is denoted Y(t).
(1) Find the autocorrelation function RX(t) of X(t).
(2) Find E[Y(t)].
(3) Find the PSD of Y(t).
(4) Determine a W such that E[Y2(t)] = 10.
(5) Find the first-order PDF of Y(t), i.e., fY(t)(y), when E[Y2(t)] = 10.
(6) Find the probability P(Y(t)>3), when E[Y2(t)] = 10.
Sx(f) = integral(-infinity to +infinity) [Rx(t)*exp(-i.2.pi.f.t)*dt]
=>Rx(t) = (1/(2.pi))*integral(-inf to +inf)[Sx(f)*exp(i.2.pi.f.t)*df]
Sx(f) = 0.1
=>Rx(t) = (0.1/(2.pi))*integral(-inf to +inf)[exp(i.2.pi.f.t)*df]
=> Rx(t) =(0.1/(2.pi))*2*integral(0 to +inf)[cos(2.pi.f.t)*df]
=> Rx(t) =(0.1/(2.pi))*2*pi*delta(t) = 0.1*delta(t)
where, delta(t) is the sampling function. --Answer
Y(f) = ...
Solution is 260 words, equations and explanations.