(2) The second input x^(t) is replaced by Xr,-,{t) obtained 

 by removing the linear effects of Xi(t) from Xp(t) by 

 optimum least-squares prediction techniques. 



(3) The third input x^(t) is replaced by x^, ^(t) obtained 

 by removing the linear effects of both x-|(t) and Xp(t) 

 from x^(t) by optimum least-squares prediction techniques. 



This procedure can be extended to any number of inputs. 



The systems in Figure 4 are denoted by {L.(f)} instead of by {H.(f)} 

 as in Figure 1 to distinguish these two distinct types of models. The 

 terms n(t) and y(t) are exactly the same in both models. Relationships 

 between these systems are derived in ll,2|. Note that the set of conditioned 

 inputs in Figure 4 will be mutually uncoherent. Optimum frequency response 

 functions for the three input case will now satisfy the equations listed 

 in Figure 5. The systems L-. and L^ are simpler to compute then H-, 

 and H„ in Figure 3, while L^ is the same as H^. 



4. Conditioned Spectral Density Functions 



Conditioned spectral density functions can be obtained by the iterative 



computational formulas shown in Figure 6. Observe that the formula for 



So ^ includes S^o -, and S , as special cases. This formula also 

 2y-l 22-1 yyl 



* 

 gives S^-..-, and S ^ -, = $2 _-,. Similarly the formula for S^ -| 2 '"^" 



eludes S03 T 2 ^^^ ^ V 1 2 ^^ special cases, and gives S ^.-i 2 ^ ^3y.l 2' 

 When there are only three inputs, as assumed here, the term S ., 23^ ^nn' 

 the spectral density function of the output noise n(t). 



5. Partial and [Multiple Coherence Functions 



Definitions for partial coherence functions are stated in Figure 7. 

 Specific formulas follow by substituting the particular conditioned spectral 



114 



