handout_week14_b

# handout_week14_b - Stochastic Signals and Systems Analysis...

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Stochastic Signals and Systems Analysis and Processing of Random Signals Virginia Tech Fall 2008 Linear Systems In this lecture we look at transformations of random processes. Linear. Let X 1 ( t ) and X 2 ( t ) be two arbitrary time functions and let a 1 and a 2 be two arbitrary constants. Let the linear system be described by the transformation Y ( t ) = T [ X ( t )] . Then the system is linear if T [ a 1 X 1 ( t ) + a 2 X 2 ( t )] = a 1 T [ X 1 ( t )] + a 2 T [ X 2 ( t )] for all admissible functions X 1 ( t ) and X 2 ( t ) and all constants a 1 and a 2 .

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Linear Systems Let Y ( t ) be the response to input X ( t ) , then the system is said to be time-invariant if the response to X ( t - τ ) is Y ( t - τ ) . The impulse response h ( t ) of a linear, time-invariant (LTI) system is deﬁned by h ( t ) = T [ δ ( t )] where δ ( t ) is a unit delta function input applied at t = 0. The response of the system to an arbitrary input
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handout_week14_b - Stochastic Signals and Systems Analysis...

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