326 BELL SYSTEM TECHNICAL JOURNAL 



From (2.5-5), this gives the correlation function for /(/) 



^(r) = aV"'^' (2.7-4) 



The corresponding power spectrum is, from (2.5-1), 



w(/) = 4a^ [ e"^"' cos 27r/r dr 



2a IX 



(2.7-5) 



T^y^ + M^ 



Correlation functions and power spectra of this type occur quite fre- 

 quently. In particular, they are of use in the study of turbulence in hydro- 

 dynamics. We may also obtain them from our shot effect expressions if we 

 disregard the d.c. component. All we have to do is to assume that the 

 effect F{t) of an electron arriving at the anode at time / = is zero for 

 t < 0, and that F(t) decays exponentially with time after jumping to its 

 maximum value at / = 0. This may be verified by substituting the value 



F(l) = 2a a/^ e""", / > (2.7-6) 



for F{t) in the expressions (2.6-2) and (2.6-4) (after using 2.6-5) for the 

 correlation function and energy spectrum of the shot effect. 



The power spectrum of the current flowing through an inductance and a 

 resistance in series in response to a very wide band thermal noise voltage is 

 also of the form (2.7-5). 



Incidentally, this gives us an example of two quite different /(/)'s, one a 

 flat top wave and the other a shot effect current, which have the same corre- 

 lation functions and power spectra, aside from the d.c. component. 



There is another type of random telegraph signal which is interesting to 

 analyze. The time scale is divided into intervals of equal length h. In an 

 interval selected at random the value of /(/) is independent of the values in 

 the other intervals, and is equally likely to be ^-c or —a. We could con- 

 struct such a wave by flipping a penny. If heads turned up we would set 

 /(/) = a in < / < A. If tails were obtained we would set /(/) = — a in 

 this interval. Flipping again would give either -fa or —a for the second 

 interval h < t < 2h, and so on. This gives us one wave. A great many 

 waves may be constructed in this way and we denote averages over these 

 waves, with / held constant, by bars. 



We ask for the average value of /(/)/(/ + t), assuming t > 0. First 

 we note that \i t > h the currents correspond to different intervals for all 



