Spectrum of a Carrier Frequency Modulated by Poisson Pulse Train With Applications to Feedback Diversity

Modeling the occurrence times as a Poisson process, the spectrum of a carrier frequency modulated by a random pulse train is studied. In general, the corresponding phase modulation is nonstationary, but the modulated carrier is shown to be wide-sense stationary, so that its RF spectrum can be obtained as the Fourier transform of its correlation function. Closed-form asymptotic results show that RF spectrum is proportional to ω^-4if the frequency modulation is discontinuous, and as ω^-6if it is continuous, where ω is the radian frequency measured from the carrier frequency. Numerically evaluated spectra are obtained for the particular case of a stationary tone-burst modulation, a form used in a proposed mobile-radio feedback diversity system. These results show that even for modulation indices as small as 0.5, the spectrum of the tone bursts themselves is not a good approximation to the RF spectrum. Finally, the spectral broadening caused by adding tone-burst modulation to a carrier already modulated by a flat-band Gaussian process is evaluated numerically. These results show that the modulation index of the bursts should be kept to unity or less in order to restrict the spectral broadening to 10 percent or less if the rms phase deviation of the Gaussian modulation is less than three radians.