MATHEMATICAL ANALYSIS OF RANDOM NOISE 325 



/which make both it and s(f) large at the same time. On the other hand, 

 if both p{t) and /'"(/) fluctuate at about the same rate this term must be 

 considered. 



2.7 Second Example — Random Telegraph Signal^' 



Let /(/) be equal to either a or —a so that it is of the form of a fiat top 

 wave. Let the intervals between changes of sign, i.e. the lengths of the 

 tops and bottoms, be distributed exponentially. We are led to this dis- 

 tribution by assuming that, if on the average there are n changes of sign per 

 second, the probability of a change of sign in /, / + dt is fidl and is independ- 

 ent of what happens outside the interval /, / + dt. From the same sort of 

 reasoning as employed in section 1.1 for the shot effect we see that the 

 probabihty of obtaining exactly K changes of sign in the interval (0, T) is 



P{K) =i^%-- (2.7-1) 



We consider the average value of the product I{t)I{t + t). This product 

 is a if the two /'s are of the same sign and is — c if they are of opposite sign. 

 In the first case there are an even number, including zero, of changes of sign 

 in the interval (/, / + t), and in the second case there are an odd number of 

 changes of sign. Thus 



Average value of /(/)/(/ + r) (2.7-2) 



= a X probability of an even number of 

 changes of sign in /, / + t 



— a X probability of an odd number of 

 changes of sign in /, / + t 



The length of the interval under consideration is|/ + T — /| = \t\ seconds. 

 Since, by assumption, the probability of a change of sign in an elementary 

 interval of length A/ is independent of what happens outside that interval, 

 it follows that the same is true of any interval irrespective of when it starts. 

 Hence the probabilities in (2.7-2) are independent of / and may be obtained 

 from (2.7-1) by setting T = \t\ . Then (2.7-2) becomes, assuming t > 

 for the moment, 



/(/)/(/ + t) = a'[p{Q) + />(2) + p{A) + • • •] 



- a\p{\) +p{Z) +p{S) + ...] 



2 -M 



= a e 



[1—^4- ^^^ _ 1 

 1! "^ 2! "J 



= a'e-''' (2.7-3) 



" Kenrick, cited in Section 2.2, 



