56 BELL SYSTEM TECHNICAL JOURNAL 



where the unit impulse function 5 is used to represent the very narrow band. 

 The power spectrum of the narrow band is approximately the same as that 

 of the wave Ay/ 2 cos 27r/2/. 



The integrals occurring in our formula are 



I w(J)df= I wi{f)df+A' 

 Jo Jo 



= W + A"" 



I w[f)fdf= I fw,U)df+AY2 

 Jo Jo 



= U + A'fl 

 We suppose that A and /^ are such that 



W» A^ 



U « AYi . 

 Then our formula (3.3-11) gives us the expected number of zeros 



2 4^2 



PP^l/2 



We may give a qualitative explanation of this formula if we regard our 

 noise current as composed of a small component 



h = 2^'"^ A cos lirfit 



due to the narrow band superposed on a large, slowly varying component 

 due to the lower band. Since the r.m.s. value of the second component is 

 W we may assign it a representative frequency /i and write it approxi- 

 mately as 



h = (2W)^'' cos lirfit 



The zeros of the noise current are clustered around the zeros of the second 

 wave. Near such a zero 



/i = ±(2WO'''27r/iA/ 



where A^ is the distance from the zero. The oscillations of I\ produce zeros 

 when I /i I is less than the amplitude of h or when 



A > W"^27rfi I A^ I 



and the interval over which zeros are produced is given by 



2A/ = il!I^ 

 ir/i 



