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# 24SIMO - MAE591 RANDOM DATA Single Input-Multiple Output...

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MAE591 RANDOM DATA Single Input-Multiple Output Relations 1 Consider the single input-multiple output system shown below. Here x is assumed to be the stationary input with zero mean which is uncorrelated with the uncorrelated extraneous output noises with zero mean, , i = 1,2, … r . Thus it holds ) ( t ) ( t n i i all for f G j i for f G i j i xn n n , 0 ) ( ; , 0 ) ( = = . ) ( τ r h # ) ( t x ) ( 1 τ h ) ( 2 τ h ) ( 1 t v ) ( 2 t v ) ( t v r ) ( 1 t n ) ( 2 t n ) ( t n r ) ( 1 t y ) ( 2 t y ) ( t y r From the relations , i = 1,2, … r, we obtain the auto and cross spectral densities as ) ( ) ( ) ( t n t v t y i i i + = i i i i i i i i n n xx i n n v v y y G G H G G G + = + = 2 xx k i xy k x y i v v y y G H H G H G H G G k i k i k i * * = = = = , k i xx i xv xy G H G G i i = = The optimal frequency response functions can be estimated from xx xy i G G H i = And the coherence functions become then i i i i i i i i y y v v y y xx xy xy G G G G G = = 2 2 γ )] ( 1 )][ ( 1 [ 1 } }{ { 2 2 2 2 2 2 2 f f G G H G G H G H H G G G k i n n xx k n n xx i xx k i y y y y y y y y k k i i k k i i k i k i β β γ + + = + + = = which can be reduced to 2 2 2 * 2 2 ) )( ( ) )( ( k i k k i i k k i i k k i i i k k k i i k i k i xy xy y y xx y y xx xy x y x y xy y y xx y y xx xx x y xy k i y y y y y y y y G G G G G G G G G G G G G G G H H G G G γ γ γ = = = = 1 J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures , 3rd ed., John Wiley & Sons, 2000, Section 6.2.

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MAE591 RANDOM DATA C. W. Lee where the power N/S ratios are given by xx i n n v v n n i G H G G G f i i i i i i 2 ) ( = = β . Single input-two output system: relative time delays 2 ) ( t x 1 1 τ α s e ) ( t x ) ( t v ) ( 1 t n ) ( 2 t n ) ( 1 t y ) ( 2 t y ) ( 2 1 f H y y Using the basic input-ouput spectral relations xx f j xv y y G e G G 1 2 1 2 τ π α = = , G 1 1 1 1 n n xx y y G G + = , G 2 2 2 2 2 2 2 n n xx n n vv y y G G G G + α = + = we obtain 1 2 2 1 2 1 τ π = = θ f G y y y y
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