358 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



The gain and phase for this infinite array of zeros are obtained from the 

 function 



= kp'" + log (1 + e-^'"""") + const. 



Thus the correct function kp^''^ is obtained modified by a term of the order 

 e~'^^^ representing ''granularity error". 



The solution as it stands is impractical for three reasons: 

 (i) an infinite number of singularities are used, 

 (ii) the singularities are all zeros so that one cannot satisfy the physical 



realizability requirements, 

 (iii) the granularity error becomes appreciable at low frequencies. 



Objections (ii) and (iii) may be avoided by choosing two numbers ki , 

 ^2 such that k = k2 — ki and making lumped charge approximations for 

 kip^''^ and kip^'"^ separately. That is, we put zeros at — (/^ — J) VV^2 and 

 poles at — {n — ^Yir/ki. By choosing ki and ^2 large enough we obtain a 

 very fine-grained approximation to the ideal (continuous) charge distribu- 

 tion and can make the frequency at which granularity effects become bother- 

 some as low as desired. Moreover since poles as well as zeros are used, we are 

 now in a better position to satisfy the physical realizability requirements. 

 When designing the network in this way it is convenient to make ki/ki a 

 rational number with numerator and denominator as small as possible. If 

 the numerator and denominator are q^ and qi then every zero pn , for which 

 2n — 1 is a multiple of ^2 , is cancelled by a pole which falls at the same 

 place. 



The most obvious way to remedy defect (i) is to use just the first .V zeros 

 and the first N poles, picking N large enough so that the infinite set of zeros 

 and poles which are being ignored produce only a negligible effect in the 

 frequency band of interest | co | < 1. To get an idea of how large N must 

 be, we evaluate the integral 



fip) = I 



k log (1 + p/r) 

 2. Vr '^'■' 



which represents the gain and phase contributed by all the charge from 

 p = —R to p = — 00 in the continuous distribution. The substitution 

 r = x^ transforms the integral into an easily handled form and we find 



/(/>) = - * [VR log (1 + 1) - 2V^ tan-' V/TAr], 

 so that/(/>) is about k p/ir\/ R when | R/p \ is large. 



