158 



BELL SYSTEM TECHNICAL JOURNAL 



Similarly for a linear rectifier: 



1 + 



2 2 



Au — aP — AiQ 

 ^01 = aQ — Aq\ 



Amn ^^ \ ) A 



mn J 



W + « > 1 



(.3.4) 



We have shown in Fig. 2 how an ideal limiting characteristic, which trans- 

 mits linearly between the upper and lower limits, may be synthesized from 

 two biased linear rectification characteristics. Equation (3.4) shows how 

 to calculate the corresponding modulation coefficients, when the coefficients 

 for bias of one sign are known. The limiter characteristic is equal to az— 

 h (2) - h (2), where 



/i (2) = oc 



z - bi, 



0, 



z > —bi 

 z < —bi 



z > bi 



1 /2 (2) = a I 

 0, z < bxj \z + 62 



The expression for/2 (zj may also be written: 



'z — ( — 62), 2 > —bi 



0, Z < -^2 



ji (z) = a (z + 62) — a 



) 



(3.5) 



(3.6) 



Hence the modulation coefficient A^n for the limiter may be expressed in 

 terms of y4„,„ (61) and A^n ( — 62) as follows: 



(61) + {-T^^'Amn (62), m ^ n 7^ \ (3.7) 



A -mn — A 1 



If the limiter is symmetrical {b\ = 62), the even -order products vanish and 

 the odd orders are doubled. The terms aP, aQ are to be added to the 

 dexter of (3.7) for .4 10, ^01 respectively. The odd Hnear-rectifier coefficients, 

 when multiplied by two, thus give the modulation products in the output 

 of a symmetrical limiter with maximum amplitude ^0, as may be seen by 

 substituting fti = 62 = —^0 in (3.7). For the fundamental components 

 aP and aQ respectively must be subtracted from twice the Aio and Aoi co- 

 efficients for ^n- 



Linear Rectifier Coefficients 



D.C. 



^00 

 2 



/aP = ko-\- \ f [Vl - (*o + ki cos 3-)^ 



(3.8) 



— (^0 + ki cos y) arc (cos ^0 + ^1 cos y)] dy 



