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handout_week4_b - Stochastic Signals and Systems Random...

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Stochastic Signals and Systems Random Variables Virginia Tech Fall 2008 Characteristic Function The characteristic function of the rv X is, except for the sign of j , the Fourier transform of its probability density function, Φ X ( ω ) = E [ e j ω X ] = Z -∞ f X ( x ) e j ω x dx . From the Fourier transform inversion formula, the probability density function of X is given by f X ( x ) = 1 2 π Z -∞ Φ X ( ω ) e - j ω x d ω Note: As we shall see later, the characteristic function of the sum of a number of independent random variables is simply related to the characteristic functions of each.
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Characteristic Function If X is a discrete random variable, the characteristic function is given by Φ X ( ω ) = X k p X ( x k ) e j ω x k (Note that the characteristic function in this case is a periodic function of ω with period 2 π .) The pdf can be obtained from the characteristic function by using p X ( k ) = 1 2 π Z 2 π 0 Φ X ( ω ) e - j ω k d ω k = 0 , ± 1 , ± 2 , . . .
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