Stationary phase approximations of FM spectra

An approximation of the Fourier transform of an FM signal with modulation index
m
is calculated by a stationary phase method. When
m \rightarrow \infty
the error of amplitude and phase converges to zero,
O(m^{-1/2})
. If the spectrum obtained by the approximation is smoothed over a frequency interval equal to several bandwidths of the modulation, then the well-known adiabatic approximation of the FM spectrum is obtained. If the signal has hidden periodicities, the adiabatic approximation destroys the spectral fine structure, but the spectral lines can be calculated from the approximation formula when used without smoothing.