384 BELL SYSTEM TECIIMCAL JOURNAL 



APPENDIX A 



Fourier Transforms For Single Pulse 



The amplitude g{f) of the component of frequency/ in the spectrum of the 

 Complex Magnitude-time function e{t) is given by the d-c component of the 

 Moduhition products of c{t) and cos IttJI, found by averaging the product 

 over the period of the comi)lex wave. 



Thus, for non-periodic waves, where the period is from — x to + x , the 

 ampHtude of the spectrum at / is 



g(f) ^ f e(l) cos 2x/7 dt. (1) 



For the single pulse, where e{l) = £ for — L < / < L and e{l) = for all 

 other values of /, equation (1) reduces to 



gif) ~ f E cos lirft dt. (2) 



Integrating, 



g(/) ^ :—. sin lirfi 

 IttJ 



g{f)^. -.sin Itt/L. (3) 



Equation (3) is the expression for g(f) plotted on Fig. 1. 



Similarly, in the case of the single pulse, each increment in frequency df 

 contributes a factor proportional to g{f) cos 27r// df to the composition of 

 e{t), so that 



e(l) = f g(f) cos 27r// df. (4) 



Substituting in (4) the expression for g{f) given by equation (3), this becomes 



/A ^. -E /""sin 27r/Z, ^ ,^ ,. ,_, 



e(/) ^ - / -^^ cos 27r// df. (5) 



7r J-oo / 



APPENDIX B 



Frequency Spectrum Or Phase Modulated Wave 



The Pliase Modulated Wave in this case is given by 



cos ((■/ — k sin vl) = cos {ct) cos (k sin vt) -f sin (ct) sin (k sin vt) 

 Now cos (ct) cos (k sin ct) = Jo (k) cos {ct) 



+ Jo (k) cos (c - 2v) t 



