210 BELL SYSTEM TECHNICAL JOURNAL 



Freedom from interchannel interference is obtained if the numerator 

 vanishes and the denominator does not vanish for all unequal integer values 

 of j and k less than .¥. We note that this can be accomplished by setting 



Nyp = (4w + Ah + N - 2)yp = lir, (32) 



in which case the numerator becomes 2 sin {j — ^)7r/2 cos (;' — ;fe)7r/2 =: 

 sin (j — k)T = 0. Solving (32) for h and \p, we find 



h = -{2n- l)/2l 



(33) 

 yP = 2t/N j 



The denominator thus becomes N sin (/ — k)ir/N, and since the largest 

 possible value of j — k'lsN— 1, the denominator cannot vanish for 7 9^ k. 

 The value of n remains arbitrary; hence we may start with any harmonic 

 we please. The value of a is also immaterial. The general form of switch- 

 ing function thus derived is, omitting the constant of proportionality: 



n+N/i-l 



FM = Z) cos [{v + mq)t - (j - I) (2m - 2w + l)7rAV + a], 



(34) 



n — 0, \, 2, ■ • • , N even 



Dependence of the phase angle upon the initial harmonic may be 

 avoided in certain special cases; for example, if we set n — rN/2, where r 

 may be zero or any positive integer, we obtain the result that a particular 

 switching function which satisfies the required conditions is: 



(r+l)\/2-l 



Fi(t) = E cos [(p + mq)t - (j - l)(2w + Dtt/X + a], 



m^rN/l. (35) 



r = 0, 1, 2, • . • , TV" even 



Specific methods of realizing satisfactory switching functions will now be 

 examined. 



Plus-and-Minus Switching with Commutator 



Instead of opening and closing the circuit between each individual 

 channel and the line, the commutator may be designed to reverse the 

 polarity of the connections on alternate contacts with a given channel. An 

 appropriate switching function for alternate polarity reversal is: 



