MATHEMATICAL ANALYSIS OF RANDOM NOISE 327 



values of /. Since the values in these intervals are independent we have 



/(/;/(/ + r) = I[l) 1{1 + T) = 



for all values of / when t > //. 



To obtain the average when t < hwe consider / to Uc in the first interval 

 < / < //. Since all the intervals are the same frcm a statistical point 

 of view we lose no generality in doing this. If / + t < //, i.e., t < h — t, 

 both currents lie in the first interval and 



/(/)/(/ -}- r) = a' 



li t > h — T the current /(/ + t) corresponds to the second interval and 

 hence the average value is zero. 



We now return to (2.5-4). The integral there extends from to T. 

 When T > h, the integrand is zero and hence 



^(r) = 0, T> h (2.7-7) 



When T < h, our investigation of the interval < t < h enables us to write 

 down the portion of the integral extending from to h: 



\ I{i)I{t + T)dt = \ a'di + Odi 



Jo Jo Jh-T 



= a{h — t) 



Over the interval of integration (0, T) we have T/h such intervals each 

 contributing the same amount. Hence, from (2.5-4), 



n 7^ 



4^(t) = Limit ^-y- (// - t) 



= a^(l -j], < T < h 



The power spectrum of this type of telegraph wave is thus 



wif) = 4a / (l - j) cos 27r/r ^r 

 _ , fa sin TrfJ/V 



(2.7-8) 



(2.7-9) 



This is seen to have the same general behavior as wf/) for the first type 

 of telegraph signal given by (2.7-5), when we relate the average number, 

 n, of changes of sign per second to the interval length hhy nh = 1. 



