136 BELL SYSTEM TECHNICAL JOURNAL 



These lead to 

 z = (bo/hY^'B' = 31/2^7 (tt^) D/bl = hho/hl -1 = 9/5-1 = 4/5 



u=^h\zVD = Si 5 = 4(r^ (6.17) 



7 = 1 + 2^ 



and the coeflScient in (6.13) may be simphfied by means of 





From (6.14) we see that (6.18) is equal to Ne' when noise alone is present 

 (and is of constant strength in the band of width ^). The curves of Ne'/^ 

 versus z shown in Fig. 7 were obtained by setting (6.17) and (6.18) in (6.13). 

 Ne'/^ approaches ^"''/(z ■\/3) as 2 ^ oo . 



When the wandering point T of Fig. 4 passes close to the point 0, 

 changes rapidly by approximately tt and produces a pulse in d\ In dis- 

 cussions of frequency modulation 6' is sometimes regarded as a noise voltage 

 which is applied to a low pass filter. Although the closer T comes to O 

 the higher the pulse, the area under the pulse will be of the order of tt and 

 the response of the low pass filter may be calculated approximately. 



That the pulses in 0' arise in the manner assumed above may be checked 

 as follows. We choose a point relatively far out on the curve for p = 5 in 

 Fig. 7, say z = Vsoyifiw) = 1.6 or 6' = 2.9/3. The number of pulses per 

 second having peaks higher than 2.9/5 is roughly Ne' = .009)9, and half of 

 these have peaks greater than 0' = 3.8/8 which is obtained from Fig. 7 for 

 N9' = .0045/8. From Fig. 6 we see that 6' exceeds 2.9/8 about .0018 of the 

 time. Thus the average width at the height 2.9/3 of the class of pulses 

 whose peaks exceed this value is .00 18/ (.009/8) = .2//3 seconds. This figure 

 is to be checked by the width obtained from the assumption that the typical 

 pulse arises when T moves along a straight line with speed v and passes 

 within a distance b of 0. We take tan 6 = vt/b = at so that 

 0' = a/ (I + aH^). From this expression for 6' it follows that a pulse of 

 peak height 3.8/3 (the median height) has a width of .3//8 seconds at d' = 

 2.9/3. This agreement seems to be fairly good in view of the roughness of 

 our work. A siiTiilar comparison may be made for p = by using the 

 limiting forms of (5.15) and (6.18). Here it is possible to compute the 

 average width instead of estimating it from the median peak value. Exact 

 agreement is obtained, both methods leading to an average width of 7r/(4^') 

 seconds at height d'. 



