response functions {H.(f)}, i = 1, 2, ..., q. The output record y(t) 

 is assumed to be the sum of the individual outputs due to passage of the 

 individual inputs {x. (t)}, plus an unknown independent noise record n(t) 

 which accounts for all unknown nonlinear operations as well as extraneous 

 noise effects. 



Note in Figure 1 that q! different configurations are possible 

 depending upon which record is chosen as Xi(t), which is then selected 

 as x^(t), and so on. The analysis to follow is based on choosing a par- 

 ticular ordering of the inputs and sticking with this order. Similar results 

 apply to any other desired ordering. Special attention will be given in this 

 paper to the case of three inputs since the formulas can be listed for this 

 case without difficulty and it is representative of the general case. 



Figure 2 gives the result for the total output spectral density function 



S = S (f) in terms of other quantities for the three input case, where 

 yy yy^ ^ ^ , 



the dependence upon frequency f has been omitted in all these terms for 

 simplicity in notation. This will also be done in later equations of this 

 paper. Note that S can be either a power or an energy spectral density 

 function depending upon whether the data is either stationary random data 

 or transient data. The output noise spectral density function S repre- 

 sents the difference between S and results predicted from x, , x^ and 



yy r 2 



Xo by passage through any linear systems, H-, , H„ and H^. Because of 

 the cross-terms between inputs, it is not clear here how much of the output 

 is due to any particular input. 



Optimum linear systems are defined by least-squares prediction techniques 

 as those systems which produce minimum mean square system error. This will 

 occur if S is minimized as a function of H. for all i = 1, 2, ...» q, 

 leading, in general, to a set of complicated equations with many interacting 



112 



