26MISO2 - Conditioned Spectral Analysis Again for...

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Unformatted text preview: Conditioned Spectral Analysis Again, for simplicity, consider the two input/single output system, where the two inputs and are not necessarily uncorrelated. One of the inputs, say x , can be decomposed into the sum of two uncorrelated terms as ) ( 1 t x ) ( 2 t x ) ( 2 t ) ( ) ( ) ( 1 2 1 : 2 2 t x t x t x • + = where represents the part of the x linearly related to x and , which is called the conditioned (residual) record, represents the part of the , not due to , or equivalently with the linear effects of x removed from . The same relations apply when the linear effects of x is removed from x by interchanging for . Similarly we can define the conditioned output as ) ( 1 : 2 t x ) ( 2 t ) ( 1 t ) ( 1 t ) ( 2 t x ) ) ( 1 2 t x • ) ( 2 t x ) ( 1 t x ( 1 x ( 2 t x ) ( 2 t ) ( 1 t ) t ) ( ) ( ) ( ) ( ) ( 1 1 : 1 1 t y t y t y t v t y • • + = + = where the conditioned output y represents the part of the output y with the linear effects of x removed from , and v ) ( 1 t • ) ( t y ) ( t ) ( 1 t ) ( ) ( 1 : 1 t y t = is the part of the output due to the input . From the above relations, we can define, as before, the spectral quantities given by ) ( 1 t x 11 12 1 : 12 1 12 1 : 12 12 G L G G G G = = + = • . G 11 1 1 : 1 1 1 1 : 1 1 G L G G G y y y y y = = + = • from which we obtain 11 1 1 11 12 12 , G G L G G L y y = = ....
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26MISO2 - Conditioned Spectral Analysis Again for...

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