728 BELL SYSTEM TECHNICAL JOURNAL 



is used, Jn{x) being the Bessel coefficient of the first kind, of the nth order and 

 with the argument x. For brevity, J nix) will be written Jn . Also, the 

 first term of (28) is to be omitted, because, as was shown in the preceding 

 example, it represents frequency sum terms which are filtered out by the 

 diode output circuit. Then we have 



~ a{t) + ih{t) = hke-^"'''-''' f e''"-^''' E jy""'' dr. (42) 



•'—00 n=—oo 



When the integration is carried out, the result is 



ao J inqt 00 j inqt 



a{t) + ibit) = hk Z ^ i-^ --^ = ^^ Z -^rr4-r--- (43) 



n=-oo iCOo + T + ^^9 n=-oo « + tA + tfiq 



To obtain the magnitude of a + ib, which is the envelope required, we multi- 

 ply the above Fourier series for a -\- ib hy that for a — ib, obtaining a double 

 summation, which can be written as follows: 



c\i) = a\t) + b\i) = Co + E (<;„«•"" + e„e-'"") 



n=l 



00 



= flo + 2 2^ a^ cos nqt 



n=l 



where an is the real part of the complex coefficient Cn and ao = co . 

 The coefficients are given by 



h^ Jl, T T 



(44) 



z 



4C2 m=-oo (a — tA — imq)[a + ^A + i{m + n)q] 



fi V^ JmJn 



(45) 



Cn — . ^« ■/ ■■ 



4C2w=-oo (a + iA + imq)[a — tA — i(^ + w)^]' 

 Obtaining this result involves use of the relation 



J^n{x) = i-T Jn{x). (46) 



From (45) 



^ J^ y^ JraJm^nW + (A + ^g) (A + Wg + ^g)] /.ys 



" 4C2 m=-^ [a^ + (A + wg)2][a2 + (a + ;«g + nqy\ ^ 



Finally, we have to obtain, as before, the difference between the en- 

 velopes of the voltages across the two impedances of the balanced frequency 

 detector; that is, we have to determine 



V,{t) = cM - ci{t) (48) 



where Ci(l) is given by (44) as it stands and 62(1) is obtained from the same 

 expression merely by reversing the sign of A. The complete solution for 



