732 BELL SYSTEM TECHNICAL JOURNAL 



reciprocated, approximate the transfer characteristic to the specified degree 

 of precision. 



The choice of these approximating functions begins by finding a poly- 

 nomial 



fix"-) = ^0 + Axx"^ + Aox' + • • • + Anx"" (14) 



which approximates e^"" g-^'^/^^^ ^-q ^\^q required degree of precision through- 

 out the useful band and has an out-band variation subject to the initial 

 requirements that /(.v-) be positive and that the slope of fix^) not vary 

 rapidly in the immediate out-band region (approxknately 1 < a: < 1.5). 

 For values of x greater than about 1.5, the Tchebycheff polynomial is the 

 determining function, and variations in /(.r-) are no longer of importance. 

 A precise statement of these conditions and the exact frequency range in 

 which they are valid depend on the degree of equalization and the desired 

 resistance efficiency in a particular design. However, a more critical ex- 

 amination of Figs. 16 and 17 indicates that the generalized conditions stated 

 above are a reasonable guide in the choice of f{x~) for most applications. 

 The main criteria for judging the acceptability of a particular out-band 

 variation which accompanies the choice of in-band variation of /(.v-) to 

 produce optimum precision are physical realizability and the attainment of a 

 desired resistance efficiency. Considering first the condition for physical 



realizability, , ^. 2 t/V ^ > for < a; < ^, and referring to Fig. 16, 



J\X ) ~r C V n\^) 



a negative value of f{x~) in the immediate out-band region might be of 

 sufficient magnitude to cancel the positive effect of e-F„(.v) and, hence, 

 produce a negative value oif{x^) + €-Vn{x). However, at higher frequencies, 

 the squared Tchebycheff polynomial takes on very large positive values. 

 Thus, negative values and variations in f{x^) are effectively reduced in the 

 magnitude of their effect on 



K 



Znijx) 



/(/) + e Vi(x) 



in direct relation to the increase in the magnitude of €-F„(.t). 



In order that an accurate prediction of the resistance efficiency may be 

 made, it is necessary that the slope oif(x-) + eWl{x) increase in a uniform 

 manner in the immediate out-band region. Since variations in the slope of 

 f{x^) have their largest effect ui the region just outside the useful band, it is, 

 of course, best to prevent rapid variations in this region. 



The remaining condition on the form oifix"^) is that Ao should be adjusted 

 so that Ao < e"". By providing the transfer specification with a less steep 

 slope requirement at low frequencies it is possible to obtain over the valuable 



