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1 Sinewave with Random Phase and Amplitude (12 marks) Let $X(t)$ be a random process defined as $X(t) = A cos(omega_o t + heta)$ (1) The frequency $omega_o$ is constant, while the amplitude $A$ and $ heta$ are independent random variables. The amplitude distribution has a mean $mu_A$ and a variance $sigma_A^2$. The phase has a uniform probability density function (pdf) given by $f_ heta( heta) = frac{1}{2pi}$ for $ heta in [0, 2pi)$ (2) a) Find the autocorrelation function $R_{XX}(t_1, t_2)$ for $X(t)$. [7] b) A wide sense stationary (WSS) random process must satisfy two conditions. What are they? [2] c) Is $X(t)$ a WSS random process? [1] d) Using the autocorrelation, write an expression for the power of the random signal $X(t)$. [2]

          1 Sinewave with Random Phase and Amplitude (12 marks)
Let $X(t)$ be a random process defined as
$X(t) = A cos(omega_o t + 	heta)$ (1)
The frequency $omega_o$ is constant, while the amplitude $A$ and $	heta$ are independent random variables.
The amplitude distribution has a mean $mu_A$ and a variance $sigma_A^2$. The phase has a uniform
probability density function (pdf) given by
$f_	heta(	heta) = frac{1}{2pi}$ for $	heta in [0, 2pi)$ (2)
a) Find the autocorrelation function $R_{XX}(t_1, t_2)$ for $X(t)$.
[7]
b) A wide sense stationary (WSS) random process must satisfy two conditions. What are
they?
[2]
c) Is $X(t)$ a WSS random process?
[1]
d) Using the autocorrelation, write an expression for the power of the random signal $X(t)$. [2]

$1 Sinewave with Random Phase and Amplitude (12 marks) Let X(t) be a random process defined as X(t) = A cos(omegao t + heta) (1) The frequency omegao is constant, while the amplitude A and heta are independent random variables. The amplitude distribution has a mean muA and a variance sigmaA^2. The phase has a uniform probability density function (pdf) given by f heta( heta) = frac12pi for heta in [0, 2pi) (2) a) Find the autocorrelation function RXX(t1, t2) for X(t). [7] b) A wide sense stationary (WSS) random process must satisfy two conditions. What are they? [2] c) Is X(t) a WSS random process? [1] d) Using the autocorrelation, write an expression for the power of the random signal X(t). [2]$

Added by Bego-A M.

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