input terms. However, for mutually uncoherent inputs, these equations 

 simplify greatly since each optimum linear system can be determined from its 

 own particular input independently of the other inputs. 



For the three input case of arbitrary inputs, the three optimum fre- 

 quency response functions will satisfy the equations listed in Figure 3. 

 The terms shown in the numerators and denominators are conditioned (residual) 

 spectral density functions found by computational algorithms developed in 



[-4 



In Figure 3, the particular conditioned quantities in H^ are defined 

 as follows with similar definitions for H-, and H^. 



Soo.-i 2 ^ spectral (power or energy) density function of x^(t) 



when the linear effects of both Xi(t) and Xp(t) are 

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

 techniques. 



So 1 p = cross-spectral density function between x^(t) and y(t) 

 when the linear effects of both Xi(t) and Xp(t) are 

 removed from either xJt) or y(t), or from both, by 

 optimum least-squares prediction techniques. 



3. Multiple Input/Output Models for Conditioned Inputs 



Figure 4 shows a multiple input/output model for conditioned inputs 

 Xi(t), Xp -|(t) and x^-, 2^^^' ^""^ ^° °'^' which are obtained from the 

 original inputs Xi(t), X2(t) and x^(t), and so on, shown in Figure 1. 

 These conditioned inputs are defined in the following ordered way: 



(1) The first input Xi(t) is left alone. 



113 



