MATHEMATICAL ANALYSIS OF RANDOM NOISE 51 



For the first value of r, all we know is that h is distributed about zero with 

 II = i/'o . For the second value of r I2 is likely to be near — /i . This is 

 in line with the idea that the noise current through a narrow filter behaves 

 like a sine wave of frequency h(fb +/a) (and, incidentally, whose amplitude 

 fluctuates with an irregular frequency of the order of K/fc ~ fa))- The first 

 value of r corresponds to a quarter-period of such a wave and the second 

 value to a half-period. By drawing a sine wave and looking at points sepa- 

 rated by quarter and half periods, the reader will see how the ideas agree. 

 The characteristic function for the distribution of /i and /o is 



ave. e "■ - = exp — -- [u + v) — xp^uv (3.2-/) 



The three dimensional distribution in which 



h = m 



h = I(t -h ri) 



h = I(i + Ti + T2) 



where n and 72 are given and / is chosen at random is, as we might expect, 

 normal in three dimensions. The moments, from which the distribution 

 may be obtained by the method of Section 2.9, are 



Mil = i"22 = M33 = "Ao 



M23 = lAra 



M13 = ^(ri + T2) = hi+r2 



The characteristic function for /i , /o , Is is 



ave. e 



2-8) 



r lAo , 2 , 2 , 2, 1 (3.2 



= exp —— (Si -j- 2-2 + S3) — IJL12Z1Z2 — M23S2S3 — A(13 2l33 



3.3 Expected Number of Zeros per Second 



We shall use the following result. Let y be given by 



y = F(ai ,a2, • • • a^ ;x), (3.3-1) 



and let the c's be random variables. For a given set of o's, this equation 

 gives a curve of y versus x. Since the c's are random variables we shall call 

 this curve a random curve. Let us select a short interval .vi , xi + dx, 



