The instantaneous frequency of a Gaussian signal: The one-dimensional density function

The density function of the instantaneous frequency of a Gaussian signal is derived without a narrow-band assumption. It is possible to express this density using two parameters of the Gaussian signal, viz., the mean frequency and the intensity of zero-crossings, or alternatively, the mean frequency and the bandwidth. It is shown that the probability of negative instantaneous frequencies is nonzero for all bandwidths except the limiting case of a purely sinusoidal wave.