844 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1953 



lates pilot errors into shape errors and drives the appropriate regulating 

 networks to obtain the corrections. 

 Let the equaUzer shapes be given by functions of the form 



SM) = ^nF.(/), (1) 



where "n" is the subscript number identifying the particular equalizer. 



(The capital 'W is reserved for the total number of equalizers.) 



**^n(/)" is the equaUzer shape (on a "unit basis") as a function 



of the frequency "f\ 

 "kn' is the amount of shape introduced by adjustment, ^'kn' 



may be positive or negative. 

 "SnifY^ is the resultant shape put in the system by adjusting 

 Fn{f) by an amount fc„ . 

 The total shape introduced by all "iV" equalizers is obviously; 



StotM) = E &(/) = ZKFnifl (2) 



n=l n=l 



To obtain a match of Stotm to the given equalization error, iS^given , at 

 "M " frequencies from m = 1 to m = M, requires that; 



'S'total(/m) = A^givenC/m). (3) 



at each frequency from /i to fu - Or, in terms of equation (2) 



/Sgiven(/m) = Z KFnifm) (4) 



n=l 



again, at each frequency from /i to fm . 



All of the important conclusions regarding the action of an equaliza- 

 tion computer are implicit in the "Af " equations indicated by equation 

 (4). 



Consider a case where there are three shapes. Let the information as 

 to the difference between the system state and its desired state be deter- 

 mined by the deviation of three pilot levels which are observed to be 

 8i , 62 and 8z at the pilot frequencies /i , /2 and /s . The problem is to find 

 the values of fci , ^2 and ks that will give a match at these frequencies. 

 This means that the following equations must be satisfied: 



Sl(f^) + .S2(/i) + .S3(/,) = 5, (5) 



*Sl(/2) + *S2(/2) + S,(fo) = 62 (6) 



Slifz) + *S:2(/3) -f *S3(/3) = 8, (7) 



