MATHEMATICAL ANALYSIS OF RANDOM NOISE 57 



The number of zeros is this multiphed by 2/2 . Since there are 2/i such 

 intervals per second the number of zeros per second is 



TT 



This differs from the result given by our formula by a factor of 2/7r. 

 This discrepancy is due to our representing the two bands by the sine waves 

 h and I2. 



From this example we obtain the picture that when the integral for \J/o 

 converges corresponding to .1 — ^ 0, while at the same time the integral for 

 ^0 diverges, corresponding to /o ^ =c in such a way that Afo — > =c ^ the 

 noise current behaves something like a continuous function which has no 

 derivative. It seems that for physical systems the integrals will always 

 cGn\-erge since parasitic effects will have the effect of making w(f) tend to 

 zero rapidly enough. The frequency which represents the region where 

 this occurs is of the order of the frequency of the microscopic wiggles. 



So far we have been considering the formulas of this section in the most 

 favorable light possible. There are experiments which indicate the possi- 

 bihty of the formulas breaking down in some cases. Prof. Uhlenbeck has 

 pointed out that if a very broad band fluctuation current be forced to flow 

 through a circuit consisting of a condenser, C, in parallel with a series com- 

 bination of inductance, L, and resistance, R, equation (3.3-11) says that the 

 expected number of zeros per second of the current, 7, flowing through R 



(and L) is independent of R. It is simply -(LC)~^^. The differential 



TT 



equation for / is the same as that which governs the Brownian motion of a 

 mirror suspended in a gas^°, the gas pressure playing the role of R. Curves 

 are available for this motion and it is seen that their character depends 

 greatly upon the pressure^\ Unfortunately, it is difficult to tell from the 

 curves whether the expected number of zeros is independent of the pressure. 

 The differences between the curves for various pressures indicates that there 

 may be some dependence*. 



3.4 The Distribution of Zeros 

 The problem of determining the distribution function for the distance 

 between two successive zeros seems to be quite difficult and apparently 



^^ For example, by putting the circuit in series with a diode. 



^^ This problem in Brownian motion is discussed by G. E. Uhlenbeck and S. Goudsmit, 

 Phys., Rev., 34 (1929), 145-151. 



31 E. Kappler, Annalen d. Phys., 11 (1931) 233-256. 



* Since this was written M. Kac and H. Hurwitz have studied the problem of the ex- 

 pected number of zeros using quite a different method of approach which employs the 

 "shot-effect" representation (Sec. 3.11). Their results confirm the correctness of (3.3-11) 

 when the integrals converge. When the integrals diverge the average number of elec- 

 trons, per sec. producing the shot effect must be considered. 



