Bandwidth and Spectra of Phase-and-Frequency-Modulated Waves

The spectrum of a sinusoid, angle-modulated by a stationary Gaussian random process (GRP) of almost arbitrary spectrum, is investigated. Such a GRP provides a convenient model for voice signals, TV signals, and complex communication signals of other types. It is shown that the rms bandwidth of the modulated wave is the product of the rms bandwidth of the modulating wave and the rms angle modulation. In the frequency modulation case, this result says the rms bandwidth of the modulated wave is 1/2π times the rms frequency deviation. A general expansion for the shape of the spectrum of the modulated sinusoid is derived. In certain limiting situations, in addition to reducing to known results, this expansion provides a practical means of obtaining the spectrum in intermediate cases. Examples are presented, showing the spectrum of a sinusoid modulated by 1) a lowpass GRP with rectangular spectrum and 2) a bandpass GRP of Gaussian spectrum.