826 



BELL SYSTEM TECHNICAL JOURNAL 



constant pure resistance, R^, we easily find that it turns out to be 

 n ^ ^0 



1 + AiX" + AiX' + . . . + AnX"'' ' 



where n is the number of branches in the network. Odd powers of 

 X are missing, just as they were when the network was terminated in a 

 filter impedance. 



REACTANCE 



CORRECTING 



NETWORK 



IdaX 



I 



LdnX 



ia|X 



idax 



TERMINATING' 

 RESISTANCE 



Fig. 15 — Generalized schematic of second or "reverse" type of filter terminations. 



Our problem consists in matching this expression to the filter 



Upon assuming that i?o = ^o, for simplicity, 



. . ^„ which will 



impedance, ZoVl — .%•- 



we see that it reduces to the selection of values of ^i 



secure approximate satisfaction of the equation 



1 



1 +^i.v- + 



+ ^n.V-" 



Vi^ 



X- 



Two empirical ^ choices of these parameters have been made, one 



^ Our previous methods of approximation, in terms of Taylor's series and Legen- 

 drian harmonics, are of course available here also. In addition, if we rewrite the 

 expression as 



Vr^T^ I? (1 + ^i.r2 + . . . + Ar^x^-) = 1 

 -fvo 



the left hand side appears as a linear combination of the associated Legendrian 

 functions Pi'(x), Pz'ix), . . ., defined by the general formula 



Pn'{x) = Vl -X-^Pn{x), 



ax 



where P„(x) is the usual Legendrian function. The problem can therefore be con- 

 sidered as that of approximating unity by a series of the associated functions. These 

 methods of approach differ chiefly in the relative weights which they ascribe to various 

 portions of the frequency band. Judged by this criterion neither of the first two 

 methods is very satisfactory for practical applications. The Taylor's series expan- 

 sion, of course, is best in the neighborhood of x = 0. The "least squares" property 

 of the ordinary Legendrian functions, on the other hand, tends to produce rough 

 equality in the numerical values of the departures from the desired function in 

 various portions of the frequency band. From the engineering standpoint, however, 

 it is the percentage departure from the desired impedance, and not the numerical 

 departure, which is of interest. This type of approximation therefore lea ds to a 

 relative over-emphasis of the region near x = 1, where the desired function 1/V 1 — x- 

 is large. The approach by means of the associated functions, however, avoids this 

 objection, since the approximated function is in this case a constant, and leads to 

 characteristics substantially as good as those obtained by means of the empirically 

 determined parameters discussed in the text. 



