334 



BELL SYSTEM TECHNICAL JOURNAL 



2t 

 n = -— m 







m = 1, 2, 3, 4, 



(3) 



Thus for a rectangular pulse of 5° (tt/S 6 radians) pulse width the 72nd, 

 144th, 216th, etc. harmonics vanish, and harmonics in the vicinity of these 

 missing harmonics have lower amplitudes as can be seen from the curves 

 of Fig. 2. 



As the pulse width of a rectangular wave increases, the number of har- 

 monics which vanish increases. For a pulse width of 90° every fourth 

 harmonic is missing. For a pulse width of 180°, the familiar square wave, 



0.05 

 0.04 



I 0.01 



IS 



UJ 0.008 



X 



UJ 0.006 

 f] 0.005 

 Q. 0.004 



16 20 24 



HARMONIC NUMBER 



(n) 



Fig. 4 — Comparison of the harmonic content of waves consisting of rectangular and o 



trapezoidal pulses 



every even harmonic vanishes and the wave contains only odd harmonics. 

 As the pulse width is increased beyond 180° the number of harmonics in- 

 creases and it can be shown that a wave having a pulse width greater than 

 180° will have the same harmonic content^ as a wave of pulse width (360° — 

 b). Thus for a large harmonic content it is desirable to have a wave having 

 either extremely narrow pulses or pulses lasting nearly 360°. 



True rectangular pulses are never obtained in practice. One common 

 type of distortion in such pulses when obtained by the "limiter" action of a 

 vacuum tube consists in the pulses having sloping rather than vertical sides. 

 The sloping sides arise from the fact that the pulses are essentially sine waves 



* This statement is correct for absolute magnitude of the harmonics only. 

 the harmonics in the two waves will be 180° out of phase. 



Certain of 



