DISTORTION IN NOISE MODULATED FM SIGNAL 885 



When ^0 — ^x is so small that exp ( — i^o + 1^2) may be replaced by- 

 unity, as it is in some important practical cases, approximations may 

 be obtained for (4.5) and (4.7). The integral in x may be expressed as 

 the sum of integrals of the type 



/ N —ipx—2iriax j r —a—tS-t 



g{x)e "^ dx = [e ^If^a+f,, 



-00 



= Ga-{- iB, , (4.8) 



/" . _ . ' 



g{x)e'^'^ ^'^"^ dx = G-a — iB^a . 

 -00 



The values of the integrals follow from (1.2) and the Fourier integral 

 theorem. G and B are, respectively, even and odd functions of frequency, 

 and Ga , Ba are their values at the frequency f — fp -\- a where fp = p/2ir 

 is the carrier frequency: 



G at frequency fp-\-a = Ga, 



B at frequency fp-{-a = Ba. 

 In this way we get the approximation 



^~'wM) KGf - 2(?o + G^ff + {B, - B_,f\ (4.9) 



for the first order modulation term, and 



1 r" 



— - / duw^(u)w^{f - u)[{Gu - G-u + (j/-u - G-f+u - Gf 



218J-00 (4.10) 



+ G-ff + {Bu + B_„ + Bj.u + 5_/+„ - Bj - B^f - 2BoY] 



for the second order modulation term. 



Expression (4.10) is an approximation to the second order modulation 

 term (4.7). When most of the interchannel interference is due to second 

 order modulation products, (4.10) is also an approximation to wdf), the 

 interchannel interference spectrum. The following remarks may be of 

 some help in deciding whether (4.10) may be used. 



1. For the case of phase modulation and a "flat" signal band, the 

 first of equations (5.3) shows that \po and \//t may be made as small as 

 we please b,y choosing the signal power (as measured by Po) small 

 enough. Since /?,. is proportional to Po , Po may be chosen small enough 

 to make P^. and higher order terms negligible in the expansion of the 

 integrand of (3.2) (unless there is some sort of symmetry which causes 

 the second order terms to vanish). In this case the interference is mostly 

 second order modulation and (4.7) is a good approximation to wdf)- 

 Fvu'thermore, as Po approaches zero, exp ( — i^n + v^x) approaches unity 



