SPECTRUM ANALYSIS OF WAVES 383 



quency spectrum is determined for AL = k sin vl. The spectrum thus 

 computed is shown in Fig. 12. L 



An example of this type of pulse modulator is given by a two position 

 switch or ideal limiter when the signal to be modulated is applied simul- 

 taneously to the limiter with an isosceles triangle wave as carrier. The 

 carrier should have a higher peak amplitude than the signal and a recurrence 

 rate based on the desired carrier frequency. Figure 12 is arranged to show 

 the output spectrum for such a limiter in terms of k, when k is the ratio 

 of the peak amplitudes of the sinusoidal signal and triangular carrier wave 

 inputs. 



A comparison of this spectrum with that on Fig. 11 shows that the 

 two spectra have almost the same form, c and v have the same amplitude 

 characteristics in each case. The c ± 2v and 2c ± v terms have differences 

 that are like those found before in comparing the pulse position modulated 

 wave on Fig. 8 and the phase modulated carrier on Fig. 9. As in that case, 

 when c becomes very much greater than v the differences vanish. 



Application of Pulse Width Modulator 



Practical interest in this case lies in the fact that the signal is present 

 in the output spectrum with a linear characteristic that makes such a 

 modulator a linear amplifier. The "on-off" or "hunting" servomechanism 

 is based on a modified form of such an amplifier in which the carrier is sup- 

 plied by the self oscillation of the system. The term modified form is used 

 because the self oscillations in general are more nearly sinusoidal than 

 triangular in form and so do not give a linear change in pulse length over 

 as wide a range of input amplitudes as does a triangular carrier. No 

 attempt will be made to analyze such a system here since it has been handled 

 elsewhere.^ However the above method is applicable to such problems 

 regardless of the shape of the carrier or the signal. 



Other Forms of Pulse Modulation 



Another form of pulse modulation of interest is that of pulse length modu- 

 lation in which either the start or the end of each pulse is fixed, so that the 

 centers of the pulses vary in position with the length. This is a combination 

 of both the pulse position and the pulse width modulations described above 

 and can be analyzed by a combination of the methods developed. 



These same methods are also applicable to the analysis of frequency and 

 phase modulated waves after they have been put through a limiter, as they 

 generally are before detection. 



9 See L. A. Macall, "The Fundamental Theory of Servomechanisms" D. Van Nostrand 

 Company, 1945. 



