Random Process Review

Transcription

1 Rando Process Review Consider a rando process t, and take k saples. For siplicity, we will set k. However it should ean any nuber of saples. t () t x t, t, t We have a rando vector t, t, t. If we find the joint pdf of, f f ( x, x, x ) t t t then we know everything about the rando process t. Stationary rando processes o show t is stationary, we ust show the tie-shift invariance, that is, for any, ft, t, t ( x, x, x) ft, t, t ( x, x, x). ypically we will show that by expanding the joint pdf, ft, t, t ( x, x, x) ft ( x) ft, t t ( x, x x) f t ( x) f ( x x) f, ( x x, x) t t t t t and show all three ters are tie-shift invariant f t ( x ) does not depend on t. f ( x x) depends on t t t t f, ( x x, x) depends on tt and/or t t t t t Recall that the rando telegraph signal satisfies these conditions.

2 Wide-sense stationary rando processes t is wide-sense stationary if Mean: t ( ) for all t Auto-correlation: R ( t, t ) t t R ( ), t t for any t and t If t herefore and is wide-sense stationary, the auto-covarince is C ( t, t ) t t t t R ( ). C ( t, t ) C ( ) C ( t, t ) C (0) R (0) j j Mean Vector and Covariance Matrix he ean vector is = ( t ), ( t ), ( t ). and the covariance atrix of the rando vector is Λ,,,,,,,,, C t t C t t C t t C t t C t t C t t C t t C t t C t t If t and is wide-sense stationary, Λ =,, C t t C t t C t t C t t C t t C t t

3 Gaussian Rando Vector is a Gaussian rando vector if and only if its joint characteristic function is ω exp ωλω jω where is the ean and Λ is the covariance atrix. he pdf f f ( x) can be found by the inverse Fourier transfor: ( x) exp x k Λ x Λ,, are said to be jointly Gaussian rando variables if and only if,, us a Gaussian rando vector. Weighted Su of Gaussian Rando Variables Let be a Gaussian rando vector and define Y as a transforation of Y A b where di k, A is a kk atrix, and b is a k-diesional constant vector. hen Y is also a Gaussian rando vector, and Y Hoework and Y A b Λ AΛ A Suppose,, is a -di Gausssian rando vector. Show that, is a -di Gaussian rando vector. Proof Since,, is a -di Gaussian rando vector, its joint characteristic function is of the for where ω,,, ω exp ωλ ω jω is the -diensional ean vector and Λ is the covariance atrix. Let Λ = and.

4 4 On the other hand, the joint CF of the -di rando vector (, ) is,, j x j x e e f, x x dx dx j x j x, e e f,, x x x dx dx dx,, 0,,,, In eq., ωλ ω ωλ ω where ω (, ) and (, ) Likewise ω 0 0 ω We have shown exp ω ωλ ω jω and thus we have proved is a -di Gaussian rando vector.

5 5 Hoework Suppose,, is a -di Gausssian rando vector with covariance atrix and ean vector. Find the covariance atrix and the ean vector of the -di rando vector,. Solution., are jointly Gausssian with covariance atrix and ean vector Furtherore, is Gausssian with variance and ean..

6 6 Gaussian Rando Process ( t) is referred to as a Gaussian rando process if t, t, t is a Gaussian rando vector for any sapling instants t, t, t. Wide-sense Stationary Gaussian Rando Process t is a wide-sense stationary Gaussian rando process =,, if t, t, t is a Gaussian rando vector with and C t t C t t Λ C t t C t t C t t C t t for any sapling instants t, t, t. A wide-sense staionary Gaussian rando process is a staionary Gaussian rando process. Recall the MD centra liit theore. When the donor process is WSS, the su process is a stationary Gaussian rando process. he two have the sae covariance atrix. Gaussian Input/Output he output of an LI filter is a Gaussian rando process when the input is Gaussian. Wide-Sense Stationary Input/Output he output of a LI filter is a WSS rando process when the input is WSS.

7 7 Gaussian Rando Vector Wj N if are indenpendent 0,. k Ref. Gallager, Stochastic Processes. Definition. W W, W,, W is referred to as an noralized IID Gaussian k-rv Definition... each j rando variables,,, is a set of jointly Gaussian zero-ean rando variables if k can be expressed as a linear cobination of soe finte set of noralized IID Gaussian j W, W,, W a W, j,,, k. j he above definition can be written in a vector for below Definition. is a zero-ean Gaussian k-rv if for soe noralized IID Gaussian -rv W, can be expressed as A W where A aj, j k, is a given atrix of real nubers. More generally, Definition.. U, U,, U are jointly Gaussian or equivalently, U is a Gaussian k-rv if U μ where is a zero-ean Gaussian k-rv and μ is a real k vector. k h..5. Let be a zero-ean Gaussian n-rv with covariance atrix Λ. hen the joint characteristic function of is copletely deterined by Λ ω e ωλ ω h..5.ext Let U be a Gaussian n-rv with an arbitrary ean. Let U μ for soe zero-ean Gaussian n-rv. μ is the ean value vector. hen the joint characteristic function of U is copletely deterined by μ and Λ. U ωλω jωμ ωe U and have the sae covariance atrix Λ.