DISTORTION IN NOISE MODULATED FM SIGNAL 883 



It may be verified that the expression (2.7) for Reir) is an even func- 

 tion of T, as it should be. Expression (2.7) is the autocorrelation func- 

 tion we set out to find. 



The distortion d(t) has an average value, 6, whose square is i?9(oc). 

 Since (p(t) is a noise function, its autocorrelation function \I/t goes to 

 zero as r approaches qo. Hence, Re('^) is given by the expression ob- 

 tained from (2.7) by setting 7?^ = 0. The autocorrelation function of 



e{t) - d is 



R,-ei.T) = Reir) - Re{^) 



= i Re [" dx f dy ^(a:)e-'^"-'^°+^^+^^ (2.8) 



■g{y)W''{e''' - 1) - e-'^'ie-''^ - 1)]. 



III POWER SPECTRUM OF THE DISTORTION 



Since d{t) has an average value which is generally not zero, its power 

 spectrum, we{f), has a spike of infinite height at/ = corresponding to 

 the power in the dc component d. When this spike is subtracted from 

 weif) the remainder is the power spectrum of d{t) — 6 given by 



we-eif) = 4 f Re-eir) COS 2wfT dr. (3.1) 



Jo 



When we use (2.8) and note that Re-eir) is an even function of t, we 

 obtain 



We-eif) = r dx r dyg{x)giy)e-''^^''^^'^ f [cos (p.r - pv) 



J-oa J~oo J-x, (32) 



•ie"" - 1) - COS ipx + py)ie~'''' - 1)] cos 27r/r dr. 



Reasoning similar to that given in Reference 4 shows that the inter- 

 channel interference spectrum, wdf), (i.e., M'c(/)A/is the average amount 

 of distortion power received in an idle channel of width Af centered on 

 the frequency /, all other channels being busy) may be obtained from 

 (3.2) by replacing (e^''" - 1) by (e^"" T R, - 1). 



The power spectrum of 9it) —6 may be regarded as made up of 

 modulation products of all orders. It turns out that the contribution of 

 n order products is given by the integral of the i?„" terms obtained 

 from the power series expansions of exp [dzRv]- 



IV FIRST AND SECOND ORDER MODULATION TERMS 



Here we shall study the first and second order modulation terms. 

 These arise from the first and second powers of R^^ in the expansion of 



