25MISO1 - Multiple Input Single Output Relationships...

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Multiple Input / Single Output Relationships Consider the multiple input-multiple output system shown below. Here are assumed to be the stationary inputs with zero mean which are uncorrelated with the uncorrelated extraneous output noise with zero mean, . Thus it holds q i t x i , , 2 , 1 ), ( " = ) ( t n q i f G n x i , , 2 , 1 , 0 ) ( " = = . ) ( τ q h # ) ( 2 t x ) ( 1 τ h ) ( 2 τ h ) ( 1 t v ) ( 2 t v ) ( t v q ) ( 1 t x ) ( t n ) ( t x q ) ( t y From the relation , we obtain the auto and cross-spectral densities as ) ( ) ( ) ( 1 t n t v t y q i i + = = nn q i q j x x j i nn q i q j v v yy G G H H G G G j i j i + = + = ∑∑ ∑∑ = = = = 1 1 * 1 1 = = = = = q i x x i q i v x y x q j G H G G i j i j j 1 1 , , 2 , 1 , " Similarly to the previous single input-single output relationship, we can obtain the optimal frequency response functions by minimizing the output noise power spectral density, which can be expressed as ∑∑ ∑∑ = = = = = = = = + = + = q i q j x x j i q i y x i q i yx i yy q i q j v v q i y v q i yv yy nn j i i i j i i i G H H G H G H G G G G G G 1 1 * 1 * 1 1 1 1 1 Then, from 0 * = k nn H G , we derive the relation = = = q j kj j ky q k G H G 1 , , 2 , 1 , " or, in matrix form, = qy y y qq q q q q q G G G G G G G G G G G G H H H # " # # # " " # 2 1 1 2 1 2 22 21 1 12 11 2 1
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MAE 591 RANDOM DATA Note that, for the optimal frequency response function, the output noise will automatically be
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