density functions computed from Figure 6. Note that partial coherence 

 functions are really ordinary coherence functions of conditioned variables 

 and hence are bounded between zero and unity. 



A general definition for the multiple coherence function is given 

 at the top of Figure 8 that applies both to arbitrary inputs as in Figure 

 1 and to ordered conditioned inputs as in Figure 4. Formulas for cases of 

 one, two or three inputs are listed in Figure 8 which reveal new relation- 

 ships between multiple and partial coherence functions. 



6. Decomposition of Three/Output Model 



The preceding analysis yields the frequency domain decomposition of 

 the conditioned three input/output model as shown in Figure 9 where the 

 {L.} are the optimum frequency response functions of Figure 5. It follows 

 that the total output spectral density function S is decomposed here 

 into four distinct terms where: 



Y-| S = spectral output at y(t) due to linear effects 



of x-|(t) as Xi(t) passes through L-, 



spectral output at y(t) due to linear i 



of x^.-ilt) as Xp.-ilt) passes through L^; 



2 

 Y2 .-1 S .-, = spectral output at y(t) due to linear effects 



2 

 Yo T o S 1 o = spectral output at y(t) due to linear effects 

 '3yl,2 yy 1,2 ^ ^ -^ ' ' 



of x^.-| ^(t) as x^ 1 ^{t) passes through L^; 



S T „ T = S = spectral output at y(t) due to unknown 

 yy -1 ,2,3 nn ^ ^ -^ 



independent terms n(t). 

 This procedure can be extended to any number of inputs. 



115 



