assignment5

# assignment5 - Assignment 5 1 Let Y = A cos(t c where A has...

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Assignment 5 1. Let Y = A cos( ωt ) + c where A has mean m and variance σ 2 and ω and c are constants. Find the mean and variance of Y . compare the results to those obtained in following example. Example: Expected Values of a Sinusoid with Random Phase Let Y = a cos( ωt + Θ) where a , ω , and t are constants, and Θ is a uniform random variable in the interval(0 , 2 π ). The random variable Y results from sampling the amplitude of a sinusoid with random phase Θ. Find the expected value of Y and expected value of the power of Y , Y 2 . E [ Y ] = E [ a cos( ωt + Θ)] = Z 2 π 0 cos( ωt + θ ) 2 π = - a sin( ωt + θ ) ± ± 2 π 0 = - a sin( ωt + 2 π ) + a sin( ωt ) = 0 The average power is E [ Y 2 ] = E [ a 2 cos 2 ( ωt + Θ)] = E ² a 2 2 + a 2 2 cos(2 ωt + 2Θ) ³ = a 2 2 + a 2 2 Z 2 π 0 cos(2 ωt + θ ) 2 π = a 2 2 Note that these answers are in agreement with the time averages of sinusoid: the time average (“dc” value) of the sinusoid is zero; the time-average power is a 2 / 2. 2. Find the mean and variance of the Gaussian random variable by applying the mo- ment theorem to the characteristic function:

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assignment5 - Assignment 5 1 Let Y = A cos(t c where A has...

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