228 BELL SYSTEM TECHNICAL JOURNAL 



return. This results in a frequency difference between the direct and 

 reflected signals which is equal to the product of the time delay IHjC 

 and the rate of change of frequency, and is given by the equation, 



Fd = 4Ai^ FmH/C cycles per second. 



The difference is plotted again at the bottom of the diagram and 

 appears as a series of trapezoids of height Fd. The time delay, 

 IH/C, has been greatly exaggerated in comparison with l/Fm, the 

 time interval corresponding to one cycle of frequency modulation, 

 in order to make the difference, Fd, large enough to show on the dia- 

 gram. Fd is actually only a few cycles in hundreds of millions. It 

 will be noted that Fd drops momentarily to zero twice for each com- 

 plete sawtooth variation of the transmitter frequency. This is due 

 to the necessity of varying the transmitter frequency first up and then 

 down, instead of forever in one direction. Hence the theory must be 

 considered from the standpoint that one altitude measurement is 

 made for each upward and another for each downward sweep, AF, of 

 transmitter frequency so that a total of 2Fm measurements are made 

 per second. The number of cycles of frequency Fd, occurring during 

 one frequency sweep, is 



Fs = FdX^= 2AFHIC, 



since TTv;- is the time of one sweep, AF. F^ is directly proportional to 



both the height and to the amount of transmitter frequency change, 

 AF. 



The fact that 2Fm separate measurements are made per second is 

 important only when considering small altitudes. The height which 

 gives a value of unity for F^ corresponding to a frequency meter signal 

 of IFm cycles per second is the minimum height which can be indicated 

 since lower altitudes give the same reading. Lower altitudes cause 

 only a fraction of a cycle of frequency, Fd, to be generated per sweep, 

 but since this fraction is repeated 2Fm times per second, it constitutes 

 a signal of the same frequency 2Fm and is so counted by the frequency 

 meter. In order to make this minimum altitude small, it is necessary 

 that AF be large, since they are inversely proportional to each other. 

 A frequency sweep of approximately 25 megacycles is required to 

 provide measurements down to the present minimum of about twenty 

 feet. If a high antenna efficiency is to be obtained over a band 25 

 megacycles wide, it is necessary that the percentage variation from the 

 average frequency during the modulation cycle be small. This 



