658 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



Table II — Z-Plane Natural Modes for Linear Phase 

 n DwcZf 



1 -2 



2 -3 ± z Vs" 



3 -4.64438 

 -3.67782 ± i 3.50876 



4 -5.79242 ± i 1.73446 

 -4.20758 ± i 5.31484 



5 -7.29348 

 -6.70392 ± i 3.48532 

 -4.64934 zh i 7.14204 



6 -8.49668 ± i 1.73510 

 -7.47142 ± i 5.25256 

 -5.03190 ± i 8.98532 



The error in the first mismatched Tchebycheff coefficient is a rough 

 measure of accuracy. It may be shown to be 



(-)"(i)co.)=^ "+^ 

 4"[l-3-5 ••• (2n - l)f(2n + 1) 



Cin+l — Cin+l — ,„r^ r. i- TTTT ^^^,/r.,_ 1 TV (113) 



This measure of accuracy is plotted in Fig. 11, for various numbers of 

 natural modes n. A sample detailed curve of |8 — ^ is shown in Fig. 12, 

 with dotted Unes corresponding to the estimated error (113). 



If delay D is such that the error is reasonable, all the zeros may be 

 natural modes. If these are combined with a Hke number of infinite 

 loss points, such that Za = —z^, an all-pass network will be obtained, 

 instead of one which approximates D without regard to gain. The all 

 pass network \\dll produce twice the delay, and twice the nonlinearity 

 of phase. In other words, for an all pass network, both coordinates in 

 Fig. 11 must be doubled. 



27. SIMPLIFICATION OF SINGULARITY ARRAYS 



In complex communication systems, a single equalizer may be re- 

 quired to correct for a number of effects. In a coaxial cable system, for 

 instance, a single network in the standard repeater may be required to 

 compensate for the following: Cable attenuation, characteristics of input 

 and output networks, effects of interstages (significant because the feed- 

 back is limited), and distortion due to variable controls at mean settings. 

 Tchebycheff polynomial methods may not be efficient when applied 



