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# Tutorial5_solution - 1 Tutorial 5 Random Signal Analysis...

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Principles of Communications 1 Tutorial 5 Random Signal Analysis

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Principles of Communications 2 Problem 1 A random process is defined by where Θ is a random variable uniformly distributed on (0, 2 π ). Determine the mean and the autocorrelation function of X ( t ). X ( t ) = A cos(2 π f 0 t + Θ ) R X ( t 1 , t 2 ) μ X ( t )
Principles of Communications We note that at time t is a function of the random variable Θ . Therefore, we have By definition, the mean of X ( t ) is given by 3 Solution μ X ( t ) We know from probability theory that a function g ( X ) of a random variable X is itself a random variable. The expected value of g ( X ) is μ X ( t ) = E [ X ( t )] = E [ A cos(2 π f 0 t + Θ )] E [ g ( X )] = g ( x ) f X ( x ) dx −∞ E [ A cos(2 π f 0 t + Θ )] = A cos(2 π f 0 t + θ ) f Θ ( θ ) d θ 0 2 π A cos(2 π f 0 t + Θ )

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Principles of Communications 4 Solution Given that μ X ( t ) = A cos(2 π f 0 t + θ ) 1 2 π d θ 0 2 π = 0 We observe that in this case is independent of t. f Θ ( θ ) = 1 2 π 0 < θ < 2 π 0 otherwise we obtain μ X ( t )
Principles of Communications 5 Solution R X ( t 1 , t 2 ) = E [ X ( t 1 ) X ( t 2 )]

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