888 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



Dividing by w^{f) = Po/(2Trff gives the ratio of the interference power 

 to the signal power 



{a^af /2)\2 - f/h) (5.6) 



where the relation Pq = (2x0-) //& has been used to ehminate Po • Here 

 (7 is the rms frequency deviation of the F]\I signal in cps. The expression 

 (5.6) agrees with results of some earlier work done at Bell Telephone 

 Laboratories. In that work the second order modulation products were 

 summed directly. 



It is interesting to apply the formulas given here to some of the 

 cases considered by Medhurst and Small. They have shown that when 

 (in our notation) a = —r cos 2TrjT and (3 = the power spectrum of the 

 distortion is 



W6-e{J) = sin' x/r[u;e-e(/)]echo , (5.7) 



and when a = and l3 = r sin 2TfT, 



we-e{f) = cos" 7r/T[w'9_e(/)]echo • (5.8) 



Here [it'fl_?(/)]echo is the power spectrum of the distortion due to a simple 

 echo of amplitude r and delay T (corresponding to a = —r cos 2irfT 

 and /3 = r sin 2'kJT). Expressions (5.7) and (5.8) may also be obtained 

 by setting the impulse response g{t) equal to 



m +^5(«- T) ±^5(f + T) 



in (3.2). 



The second order modulation approximation for the a = —r cos 

 2irjT, /3 = case may be obtained from (4.10) and turns out to be 



-co 



/ w^{u)w^{f — u)[2r sin 'pT sin irfT sin iruT sin 7r(/ — u)Tf du. (5.9) 



J— X 



It is seen that this contains the factor sin tt/T predicted by (5.7). When 

 (5.9) is applied to the FJVI case of (5.2) an integral something like (5.5) 

 (but more complicated) is obtained. The ratio of the second order 

 modulation interference power to the signal power is found to be 



2[r sin pT sin TrfT]\a/0lJK (5.10) 



where K is the quantity 



sin {y/2) sin {a — y) 2 

 y{a - ii) 



K = 2cr f 



dy (5.11) 



