Question

1. (20 Pts) Consider a pair of random processes $X(t)$ and $Y(t)$ that are related to a wide-sense stationary (WSS) process $W(t)$ as follow: $X(t) = W(t)\cos(2\pi f_c t + \Theta)$ $Y(t) = W(t)\sin(2\pi f_c t + \Theta)$ where $\Theta$ is uniformly distributed over $(0, 2\pi)$, and also independent of $W(t)$. a) (5 Pts) Find the cross-correlation $R_{XY}(\tau)$. b) (5 Pts) Find the autocorrelation of $X(t)$. c) (5 Pts) Find the power spectral density of $Y(t)$. d) (5 Pts) In what condition $X(t)$ and $Y(t)$ will be orthogonal?

          1. (20 Pts) Consider a pair of random processes $X(t)$ and $Y(t)$ that are related to a
wide-sense stationary (WSS) process $W(t)$ as follow:
$X(t) = W(t)\cos(2\pi f_c t + \Theta)$
$Y(t) = W(t)\sin(2\pi f_c t + \Theta)$
where $\Theta$ is uniformly distributed over $(0, 2\pi)$, and also independent of $W(t)$.
a) (5 Pts) Find the cross-correlation $R_{XY}(\tau)$.
b) (5 Pts) Find the autocorrelation of $X(t)$.
c) (5 Pts) Find the power spectral density of $Y(t)$.
d) (5 Pts) In what condition $X(t)$ and $Y(t)$ will be orthogonal?

1. (20 Pts) Consider a pair of random processes X(t) and Y(t) that are related to a
wide-sense stationary (WSS) process W(t) as follow:
X(t) = W(t)cos(2π fc t + Θ)
Y(t) = W(t)sin(2π fc t + Θ)
where Θ is uniformly distributed over (0, 2π), and also independent of W(t).
a) (5 Pts) Find the cross-correlation RXY(τ).
b) (5 Pts) Find the autocorrelation of X(t).
c) (5 Pts) Find the power spectral density of Y(t).
d) (5 Pts) In what condition X(t) and Y(t) will be orthogonal?

Added by Milagros Y.

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