PROPERTIES OF SINE WA VE PLUS NOISE 151 



semi-log paper they tend to lie on a straight line whose slope suggests that 

 W{0) decreases as e^'' when p becomes large. 



The limiting form assumed by W(J) as p — > oo is given by equation (7.2). 

 When the normal law expression (8.1) assumed in this section for the power 

 spectrum of In is put in (7.2) we find that 



47rV 



^(/)-e7^({y^"'^^'^^^ (8.13) 



Fig. 9 shows that for p = 5 the limiting form (8.13) agrees quite well with 

 the exact form computed above. 



Both (7.2) and (8.13) show that, for small values of/, the power spectrum 

 of 6' varies as/^ when p > > 1. This is in accord with Crosby's* result 

 that the voltage spectrum of the random noise in the output of a frequency 

 modulation receiver is triangular when the carrier to noise ratio is large. 

 When this ratio becomes small he finds that the spectrum becomes rec- 

 tangular. Fig. 8 shows this effect in that the areas under the curves between 

 the ordinates at/ = and/ = Xc (where X is some number, generally less 

 than unity, depending on the ratio of the widths of the i.f. and audio bands) 

 become rectangles, approximately, as p decreases. 



APPENDIX I 

 The Integral le (k, x) 



The integraP^ 



Ie(k,x) = r e-''Io{ku)du, (AM) 



Jo 



where Io{ku) denotes the Bessel function of imaginary argument and order 

 zero, occurs in Sections 2 and 6. The following special cases are of interest. 



Ie{0, x) = 1 - e-'^ « 



Ie{l, x) = xe--[Io{x) + Ii{x)] (Al-2) 



The second of these relations is due to Bennett.^'^ 



* M. G. Crosby, " Frequency Modulation Noise Characteristics," Proc. I. R. E. Vol. 25 

 (1937), 472-514. See also J. R. Carson and T. C. Fry, "Variable Electric Circuit Theory 

 with Application to the Theory of Frequency Modulation," B. S.T.J. Vol. 16 (1937), 

 513-540. 



1® The notation was chosen to agree with that used by Bateman and Archibald (Guide 

 to Tables of Bessel Functions appearing in "Math. Tables and Aids to Comp.", Vol. 1 

 (1944) pp. 205-308) to discuss integrals used by Schwarz (page 248). 



^^ it is given in equation (62) of the reference cited in connection with our equation 

 (1.2) in Section ]. 



