The frequency spectrum of pulse width modulated signals

The determination of the frequency spectrum of a pulse width modulated (PWM) signal with general band-limited input x(t) has been an open problem for many years. We describe a new approach that gives exact analytical expressions for the spectra of uniform-sampling PWM signals and natural-sampling PWM signals with single-edge as well as with doubleedge modulation. For the special case of single tone modulating signals, our results reduce to those obtained previously using a double Fourier series method. We also show that if the maximum magnitude of the derivative of x(t) is smaller than twice the carrier frequency, then a PWM signal consists of a baseband signal y(t) together with y(t) phase-modulated onto each carrier harmonic, where, for uniform-sampling PWM, y(t) is a nonlinear function of the modulating signal x(t), while for natural-sampling PWM, y(t) is just x(t) itself, that is, there is no distortion in the baseband when natural sampling is used.