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Lecture16

# Lecture16 - Stationarity of Normal Processes If X(t is a...

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•1 Stationarity of Normal Processes •I f X ( t ) is a Gaussian process, then by definition are jointly Gaussian random variables • Their joint characteristic function is 11 2 2 ( ), , ( ) nn XX tXX t t == = " 1, () (,) / 2 12 (, ,, ) XX kk i ki k kl k X jt C t t n e μω ωω φωω ω = ∑∑ = " Gaussian Processes Properties f X ( t ) is wide-sense stationary, then we have • If we now shift the time indices to get • Their joint characteristic is as above for all n and c establishing strict stationarity from WSS for Normal processes 1 2 1 1 XX n k i k X jC t t n e μ = −− = " 112 2 , , , X Xt c X Xt c X ′′ = += + = + " Systems with Stochastic Inputs A deterministic system transforms each input waveform into an output waveform through time (t) operation Thus a set of realizations at the input corresponding to a process X ( t ) generates a new set of realizations at the output associated with a new process Y ( t ). (, ) i X t ξ [ (, )] ii Yt TXt = )} , ( { t Y Our goal is to study the output process statistics in terms of the input process statistics and the system function. ] [ T ⎯ → ) ( t X ⎯→ ) ( t Y t t ) , ( i t X ) , ( i t Y Stochastic Input/Output

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•2 Deterministic Systems Systems with Memory Time-Invariant systems Linear systems Linear-Time Invariant (LTI) systems Memoryless Systems )] ( [ ) ( t X g t Y = )] ( [ ) ( t X L t Y = Time-varying systems . ) ( ) ( ) ( ) ( ) ( + + = = τ d t X h d X t h t Y () ht Xt LTI system Memoryless Systems:The output Y ( t ) in this case depends on the present value of the input X ( t ). i.e., )} ( { ) ( t X g t Y = Memoryless system Memoryless system Memoryless system Strict-sense stationary input Wide-sense stationary input X ( t ) stationary Gaussian with ) ( XX R Strict-sense stationary output. Need not be stationary in any sense. Y ( t ) stationary,but not Gaussian with ). ( ) ( η XX XY R R = Theorem: If X ( t ) is a zero mean stationary Gaussian process, and Y ( t ) = g [ X ( t )], where represents a nonlinear memoryless device then Proof: where are jointly Gaussian random variables, and hence ) ( g )}. ( { ), ( ) ( X g E R R XX XY = = 2 1 2 1 2 1 ) , ( ) ( )}] ( { ) ( [ )} ( ) ( { ) ( 2 1 dx dx x x f x g x t X g t X E t Y t X E R X X XY ∫∫ = = = ) ( ), ( 2 1 = = t X X t X X * 1 12 /2 1 2 * * 1 2| | (0) ( ) ( 0 ) (, ) ( , ) , ( , ) { } XX XX XX XX XX xA x TT A RR fx x e X xx x A EXX L L π = == ⎛⎞ = ⎜⎟ ⎝⎠ Δ where L is an upper triangular factor matrix with positive diagonal entries. i.e., • Consider the transformation so that Z 1 , Z 2 are hence zero mean independent Gaussian random variables. Also and hence . 0
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Lecture16 - Stationarity of Normal Processes If X(t is a...

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