SPECTRUM ANALYSIS OF WAVES 361 



another practical application, of which the self oscillating or hunting servo- 

 mechanism is an example. 



The quantitative analysis of such systems depends on the ratio of the 

 pulse repetition rate to the signal frequency. When this ratio is low, the 

 solution can be obtained by a method shown here for resolving the modulated 

 waves into selected groups of effectively unmodulated components. This 

 technique is powerful since it can be done by graphical means whenever the 

 complexity of either the system or the signal warrants it. When the ratio of 

 pulse rate to signal frequency becomes high enough, such methods are no 

 longer practical. However, under these conditions other methods become 

 available, especially in cases like those mentioned above where the spectrum 

 of the modulation approaches one of the more familiar forms. An important 

 example of this occurs in the case of the pulse position modulator where, as 

 the spectrum approaches that of phase modulated waves, the solution can 

 often be found by the conventional Bessel's function technique used in 

 analyzing phase and frequency modulators. 



The method proposed here for obtaining the spectrum analysis of pulse 

 modulated waves is based on the use of the magnitude-time characteristic 

 of the single pulse and its frequency spectrum as a pair of interchangeable 

 building blocks, so that the analysis will develop this relationship. Before 

 doing this the elementary theory of spectrum analysis will be reviewed 



Review or the Elementary Theory of Spectrum Analysis 



A complex wave may be represented in two ways. One way is by its 

 magnitude at each instant of time. The other way is by its frequency 

 spectrum, that is, by the various sinusoidal components that go to make up 

 the wave. The two representations are interchangeable. 



The transformation from a given frequency spectrum to the corresponding 

 magnitude vs. time function is straight-forward, for it is apparent that the 

 various components in the frequency spectrum must add up to the desired 

 magnitude-time function. The necessary additions may be difficult to 

 make in some cases but they are not hard to understand. 



The reverse process of finding the frequency spectrum when the magni- 

 tude-time characteristic is given is more involved, though using Fourier anal- 

 ysis, the problem can generally be formulated readily enough. Furthermore 

 the mathematical procedures involved can be interpreted physically in 

 broad terms by modulation theory. However, these procedures become 

 more difficult to perform, and the physical relationships more obscure, as the 

 wave form under analysis becomes more complex. This is particularly 

 true when general or informative solutions rather than specific answers are 

 required. Pulse modulated waves are sufficiently new and complex to give 

 such difficulties. 



