The Power Spectrum of a Carrier Frequency Modulated by Gaussian Noise

General expressions for the power spectra of carriers frequency and phase modulated by Gaussian noise are given as functions of the shapes of the power spectra of the modulating voltages. Specific asymptotic (closed-form) expressions are obtained for the special case when the modulating voltage has a rectangular low-pass power spectrum. This particular case corresponds to the use of maximally flat and similar video amplifier tuning schemes. The half-power bandwidths of the power spectra of the modulated carriers are obtained from these asymptotic expressions and are plotted as functions of the root-mean-square frequency and phase deviations. The delta function at the carrier frequency (representing power remaining in the carrier) is evaluated for phase and frequency modulation where, in the latter case, the delta function exists only if the power spectrum of the modulating voltage does not extend to zero frequency.