DISTORTION IN NOISE MODULATED FM SIGNAL 881 



with variable a. and /S//. The first transducer produces no distortion of 

 the signal, and if condition (1.5) is satisfied by the second, the con- 

 siderations of this paper will apply. 

 Equation (1.4) may be written as 



so that 



When we write 



e{t) =Imlog^. (1.6) 



Viit) 



Vo{t)/Vi(t) = 1 + [vo{t) - Vi(t)]/Vi(t), 



expand the logarithm in (1.6), and use (1.5), we obtain our approximate 

 expression for d{t) : 



e{t) = Im [voit) - Vi(t)]/Vi(t) = Im Vo(t)/vi{t) 



= Im [vm~' f_ ViiO g{t- t') dt' ^^ ^^ 



= Im / exp [ip{t' — t) -\- i(p(t') — i(p(t)]g(t — t') dt' . 



J — CO 



So far there is nothing essentially new in our work. 



II AUTOCORRELATION FUNCTION OF d{t) 



In Section I, (p'{t) could be any reasonable sort of signal. In the follow- 

 ing work we assume that it is a Gaussian noise whose power spectrum, 

 ^*''(/)) is given to us. The power spectrum of <p{t) is 



w,ij) = wAI)/i2wff, (2.1) 



and its autocorrelation function is 



^r = f W^if) cos 27r/T df. (2.2) 



We have written ^^ instead of i^(t) or R^ir) to simplify the appearance 

 of the formulas which occur in our work. 



Our problem is to find the power spectrum, iveif), of the distortion 

 d(t), given w^{f). The method of solution is much the same as that used 

 in Reference 4. We first find the autocorrelation function Rb{t) of d(t) 

 and then obtain we(f) by taking the Fourier cosine transform of Reir). 



