470 BELL SYSTEM TECHNICAL JOURNAL 



Eq. (I — 7) applied to 0(/) gives the result: 



<f>\nto) = At). (1—9) 



By combining (I — 8) and (I — 9), we obtain 



AnQ = <f>V). (I-IO) 



APPENDIX II 



Fundamental Theorem on Aperture Effect in Sampling 



If we sample the wave Q cos gt at a rate/s , and multiply each sample by 

 a short rectangular pulse of unit height and duration r centered at the 

 sampling instants, we obtain by reference to Eq. (1-4) replacing lirfs by 



F{t) = Q cos qt S{t, t) = ^Q cos qt 



-\- QZl [co3 {mcos + q)^ + cos (mcos — q)t]. 



m=l miT ■ 



The fact that pulse modulation is similar to the more familiar carrier 

 modulation processes is brought out by this equation; the sampling frequency 

 is in fact the carrier. The writer has found that the method of calculation 

 he published in 1933/^ in which the signal and carrier frequencies are taken 

 as independent variables, is ideally suited for calculations of pulse-modulated 

 spectra. Artificial and cumbersome devices such as assuming the signal 

 and sampling frequencies to be harmonics of a common frequency are thereby 

 avoided. 

 A unit impulse 5(/) has zero duration and unit area; hence we may write: 



b{t) = Lim "^^ . (II— 2) 



T-»o r 



A train of samples in which each sample is multiplied by a unit impulse 

 may therefore be written as 



5Z Q cos qth{t — tn) = Lim ^ cos ql 



+ (?^ ^ [cos (mcos + ql) + cos {mcos — q)i\ . 



m=l WTTT J 



Suppose we apply the train of waves (II-3) to a linear electrical network 

 which delivers the response g{t) when the input is a unit impulse 5(/). The 

 steady state admittance of the network is given by. 



YoM = f giOe-'-'' dt (II— 4) 



J— 00 



(II-3) 



