SPECTRUM ANALYSIS OF WAVES 



365 



2/2 



fallen to 2EL X " I " being the average value of a half wave of unit ampli 



tude ). Similarly when the frequency is such that the length of the pulse 



is a full wavelength, the average will vanish, and when the pulse length is 

 one and a half times the wavelength, the average is negative, having two 

 negative and one positive half waves over the length of the pulse, and the 



2 

 relative magnitude is 2EL X ^. These products are shown graphically 



on Fig. 2. Since these amplitudes correspond to those given in Fig. 1, 

 for the spectrum components at/ = /o = 1/4Z, 2/o , and 3/o , it is apparent 

 that the spectrum could be determined in this way. 



WHERE f = 



WHERE f = Val 



<o 

 u 



TIME, t 



a 4 

 3 rr 



"^ 



_4 

 '3TT 



FREQUENCY, f, IN TERMS OF C (WHERE C= V^O 



Fig. 2 — Graphical derivation of spectrum of single pulse by averaging product of pulse 

 with sinusoidal waves of various frequencies. 



Basic Technique 



In the analysis presented here, the single pulse and its spectrum will be 

 used in such a way that the need for individual integral transforms for each 

 complex wave form under study is avoided. The theory is simple. 



A complex wave form may be approximated to any desired accuracy by a 

 series of pulses, varying with respect to time in length, in amplitude, and 

 in position. Now the spectra of these individual pulses are already known. 

 Therefore, to find the frequency spectrum of the complex wave in question, 

 it is necessary only to combine properly the spectra of the various pulses 

 representing the complex wave. 



Thus the process is theoretically complete. The procedure is first to 



