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BELL SYSTEM TECHNICAL JOURNAL 



llie transmission over each non-linear section, where the transmission is 

 dependent only on the amplitude, and the frequency spectrum used over 

 each linear section, where the transmission is dependent only on the fre- 

 quency. This a technique can be used on most pulse modulating systems 

 because such non-linear elements as the modulators and limiters generally 

 encountered are substantially independent of frequency. 



Frequency Spectrum of the Single Pulse 



The single pulse is a non-periodic function of time and so has a continuous 

 frequency spectrum. In this case the Fourier transforms are simple. They 

 are derived in Appendix A. Figure 1 gives a graphical representation of 

 the magnitude-time function and the frequency spectrum of the pulse. 

 The expressions are general and hold for pulses of any length or amplitude. 



It is instructive to note that the frequency spectrum in this case can be 



MAGNITUDE-TIME 

 FUNCTION, e (t) 





 TIME, 



FREQUENCY SPECTRUM, g (f) 





-6C -4C -2C 2C 4C 6C 



FREQUENCY,!, IN TERMS OF C (WHERE C = VaO 



Fig. 1 — Magnitude time and frequency spectrum representations of a single pulse. 



determined by using the graphical technique mentioned previously. For 

 example, consider the product of the magnitude-time function of the single 

 pulse with a sinusoidal wave of given frequency and unit amplitude, so 

 arranged in phase that its peak coincides with the center of the pulse. 

 Theoretically the average of this product taken over the infinite period will 

 give the relative magnitude of the component in the frequency spectrum 

 of the pulse having the same frequency as the sinusoidal wave. In this 

 case however, the average need only be taken over the length of the pulse, 

 since the product vanishes everywhere else. Thus at very low frequencies, 

 where the period of the sinusoidal wave is very much greater than the length 

 of the pulse, the average is proportional to 2EL where E is the amplitude 

 and 2L the length of the pulse. Then as the frequency increases, the average 

 of the product, and hence the relative amplitude of the component in the 

 spectrum, will first decrease. For the particular frequency such that the 

 length of the pulse is one half the period, the relative ami)litude will have 



