880 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



tortion. Unfortunately, the integrals are difficult to evaluate. However, 

 it is possible to obtain approximations for the second and third order 

 modulation terms. These are given in Section IV. Some miscellaneous 

 comments are made in Section V. 



I APPROXIMATE EXPRESSION FOR THE DISTORTION 6(t) 



Let the FM signal be <p'{i) = dcp/dt (for phase modulation the signal 

 would be (p(t)). Then the FM wave is the real part of 



v^{t) = e'>'+^>''^ (1.1) 



where p = 2x/p is the carrier frequency. Let this wave pass through a 

 transducer having attenuation a and phase shift /3, where a and ^ are 

 even and odd functions, respectively, of the frequency /. When a unit 

 impulse of voltage 8{t) is applied to the transducer input, the output is 



g(t) = r e-«-'-^+2.i/t ^^^_ ^2) 



J — 00 



For physical systems, g{t) is zero for negative /. 



When Vi{t) is applied to the transducer input, the output is 



vo(t) = f vAt')g{t - t') dt'. (1.3) 



When V(i{t) is applied to an FM receiver, the detector output consists of 

 the original signal <f'{t) plus the distortion d'{t) introduced by the trans- 

 ducer. Comparison with (1.1) shows that d{i) may be obtained by 

 solving 



when p, <p{t), vo{t) are assumed to be known, and V(t), 6{t) unknown. 

 When V{t) is taken to be positive, (1.4) determines d{t) except for an 

 additive term of 2Tn where n is an integer. 



We now assume that the transducer acts like a good transmission 

 medium in that the output differs but little from the input. More pre- 

 cisely, we assume 



U'o(0 - v^Q) I «i- (1-5) 



Since | Vi{t) | = 1, it follows that | i'o(0 1^1- Transducers having ap- 

 preciable attenuation and delay may be regarded as two transducers in 

 tandem, one with constant (independent of /) values of a and j8// which 

 are roughly equal to those of the original transducer, and the second 



