Spectra of Digital FM

Formulas are derived for the spectral density function of an ensemble of continuous-phase, constant-envelope FM waves. The modulation signals are random time series of the form $sum_{n}a_{n}g(t-nT),$ where g(t) is an arbitrary pulse of finite duration rT, r ≧ 1. The an's are independent random variables possessing identical but otherwise arbitrary probability distribution. The derived results are general and are presented in terms of averages of elementary functions. When the an's are discrete random variables, both continuous and discrete spectra are treated, and conditions in terms of the modulation parameters are given under which discrete spectral lines are present. Several of our specialized formulas are applicable in the study of multilevel FM data transmission systems as well as in pulse frequency modulation.