MA THEM A TIC A L ANAL YSIS OF RA NDOM NOISE 323 



This may be obtained immediately from Campbell's theorem by applying 

 Parseval's theorem. 



As an example of the use of these formulas we derive the power spectrum 

 of the voltage across a resistance R when a current consisting of a great num- 

 ber of very short pulses per second flows through R. Let F{i — tk) be the 

 voltage produced by the pulse occurring at time tk . Then 



where (p{t) is the current in the pulse. We confine our interest to relatively 

 low frequencies such that we may make the approximation 



s{f) = ! R^(i)e-'"^' dt 



J— 00 



ip{i) dt = Rq 





where q is the charge carried through the resistance by one pulse. From 

 (2.6-4) it follows that for these low frequencies the continuous portion of 

 the power spectrum for the voltage is constant and equal to 



w(/-) = 2vR\^ = 2lR^q (2.6-8) 



where I = vq is the average current flowing through R. This result is often 

 used in connection with the shot effect in diodes. 



In the study of the shot effect it was assumed that the probability of an 

 event (electron arriving at the anode) happening in dt was vdt where v is the 

 expected number of events per second. This probability is independent of 

 the time /. Sometimes we wish to introduce dependency on time. As an 

 example, consider a long interval extending from to T. Let the prob- 

 ability of an event happening in /, / + dt be Kp{t)dt where K is the average 

 number of events during T and pit) is a given function of / such that 



I" 



Jo 



p{t) dt = 1 



For the shot effect p(t) = 1/T. 



What is the probability that exactly A' events happen in T? As in the 

 case of the shot effect, section LI, we may divide (0, T) into N intervals 

 each of length At so that NAt = T. The probability of no event happening 

 in the first At is 



(f) 



1 - Kp y-J M 



1* A careful discussion of this subject is given by Hurwitz and Kac in "Statistical 

 Analysis of Certain Types of Random Functions." I understand that this paper wiU 

 soon appear in the Annals of Math. Statistics. 



