Question

where \(\theta\) is a uniformly distributed random phase and \(\Omega > a\). (a) Define a WSS RP. (b) Define a strict-sense stationary RP. (c) Define a realizable linear system. (d) Is the following an autocorrelation function? \(\psi(\tau) = \begin{cases} 1 - |\tau|, & |\tau| < 1, \\ 0 & \text{otherwise.} \end{cases}\) Explain. 4.10 Assume \(X(t)\) is a stationary RP with autocorrelation function \(\psi_x(\tau) = \begin{cases} 1 - |\tau|, & -1 \le \tau \le 1, \\ 0 & \text{otherwise.} \end{cases}\) 162 Find the spectral density \(\Psi_y(\omega)\) for RANDOM PROCESSES \(y(t) = x(t)\cos(\omega_0 t + \lambda)\) when \(\omega_0\) is a constant and \(\lambda\) is an RV uniformly distributed on the interval \([0, 2\pi]\). 4.11 An RP \(X(t)\) is defined by \(x(t) = \cos(t + \theta),\) where \(\theta\) is an RV uniformly distributed on the interval \([0, 2\pi]\). Calculate the autocorrelation function \(\psi_{xx}(\tau, s)\) for \(y(t) = \int_0^t x(u) du.\) 4.12 Let \(\psi_1\) and \(\psi_2\) be two arbitrary continuous, absolutely integrable autocorrela- tion functions. Are the following necessarily autocorrelation functions? Briefly explain your answer. (a) \(\psi_1 \cdot \psi_2\). (b) \(\psi_1 + \psi_2\). (c) \(\psi_1 - \psi_2\). (d) \(\psi_1 * \psi_2\) (the convolution of \(\psi_1\) with \(\psi_2\)).

          where \(\theta\) is a uniformly distributed random phase and \(\Omega > a\).
(a) Define a WSS RP.
(b) Define a strict-sense stationary RP.
(c) Define a realizable linear system.
(d) Is the following an autocorrelation function?
\(\psi(\tau) = \begin{cases} 1 - |\tau|, & |\tau| < 1, \\ 0 & \text{otherwise.} \end{cases}\)
Explain.
4.10 Assume \(X(t)\) is a stationary RP with autocorrelation function
\(\psi_x(\tau) = \begin{cases} 1 - |\tau|, & -1 \le \tau \le 1, \\ 0 & \text{otherwise.} \end{cases}\)
162
Find the spectral density \(\Psi_y(\omega)\) for
RANDOM PROCESSES
\(y(t) = x(t)\cos(\omega_0 t + \lambda)\)
when \(\omega_0\) is a constant and \(\lambda\) is an RV uniformly distributed on the interval
\([0, 2\pi]\).
4.11 An RP \(X(t)\) is defined by
\(x(t) = \cos(t + \theta),\)
where \(\theta\) is an RV uniformly distributed on the interval \([0, 2\pi]\). Calculate the
autocorrelation function \(\psi_{xx}(\tau, s)\) for
\(y(t) = \int_0^t x(u) du.\)
4.12 Let \(\psi_1\) and \(\psi_2\) be two arbitrary continuous, absolutely integrable autocorrela-
tion functions. Are the following necessarily autocorrelation functions?
Briefly explain your answer.
(a) \(\psi_1 \cdot \psi_2\).
(b) \(\psi_1 + \psi_2\).
(c) \(\psi_1 - \psi_2\).
(d) \(\psi_1 * \psi_2\) (the convolution of \(\psi_1\) with \(\psi_2\)).

where θ is a uniformly distributed random phase and Ω > a.
(a) Define a WSS RP.
(b) Define a strict-sense stationary RP.
(c) Define a realizable linear system.
(d) Is the following an autocorrelation function?
ψ(τ) = 1 - |τ|, |τ| < 1,
0 otherwise.
Explain.
4.10 Assume X(t) is a stationary RP with autocorrelation function
(τ) = 1 - |τ|, -1 ≤τ≤ 1,
0 otherwise.
162
Find the spectral density (ω) for
RANDOM PROCESSES
y(t) = x(t)cos(ω0 t + λ)
when ω0 is a constant and λ is an RV uniformly distributed on the interval
[0, 2π].
4.11 An RP X(t) is defined by
x(t) = cos(t + θ),
where θ is an RV uniformly distributed on the interval [0, 2π]. Calculate the
autocorrelation function ψxx(τ, s) for
y(t) = ∫0^t x(u) du.
4.12 Let ψ1 and ψ2 be two arbitrary continuous, absolutely integrable autocorrela-
tion functions. Are the following necessarily autocorrelation functions?
Briefly explain your answer.
(a) ψ1 ·ψ2.
(b) ψ1 + ψ2.
(c) ψ1 - ψ2.
(d) ψ1 * ψ2 (the convolution of ψ1 with ψ2).

Added by Leslie C.

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