240 BELL SYSTEM TECHNICAL JOURNAL 



yljo and Aqi we must add cP/2 and cQ/2 respectively. The value of 

 Amn is zero for m -\- n even and greater than two; the other even order 

 products are listed in Table I. Another form of the result for odd order 

 products is 



^..=i(-)'^r^-<q^^.x (33) 



TT Jo A'' 



or 



m-\-n-\-l K Jr 1 



■Amn — \ 



lirY^n + l)r 



2 



^ (m -\- n — 2 n — m — 2 , . , „\ , , , . 



X F [—^ . 2 ' ^^ + 1: ^ )• (^'^) 



A three term recurrence formula for odd order products is: 



. _ _ 2l{m—l)k-j-n—\2^m--i, „-i + (m->rn — 6)kAm-2. n-2 ..rx 

 ^'"^ ~ {m + n-\-2)k ' ^ ^ 



When P = Q, and in + n is odd, 



^ _ 64 (-) ^+'P' . . 



Other Applications and Results 

 The solution for any full wave rectifier can be obtained from the 

 solution for the corresponding half wave rectifier. Thus we may easily 

 show that the output of a full wave linear rectifier contains neither of 

 the fundamentals and that the amplitudes of all other modulation 

 products are twice as large as the corresponding amplitudes in the 

 output of a half wave linear rectifier. It is also evident that by super- 

 posing the solutions for the linear and square law rectifiers we can 

 obtain the solution for a quadratic law rectifier having an output equal 

 to aie(t) + a2[e(/)]- when e{t) is positive and no output when e{t) is 

 negative. Biased rectifiers, peak choppers, and saturating devices 

 can be solved by the same methods used above, the solution of course 

 becoming more complicated for the more complicated kinds of char- 

 acteristics. Nor is the method restricted to " cut off " type modula- 

 tion. Curvature type modulators can be treated in the same way and 

 in many cases solution by the above method is simpler than by the 

 usual power series expansion. The method also appears to have 

 promise in the solution of magnetic modulation problems, where the 

 effect of hysteresis must be considered. 



