On amplitude-weighted instantaneous frequencies

The instantaneous frequency of a signal may be uniquely defined from the analytical signal. However, for a Gaussian signal, this frequency has an infinite variance and is useless. Instead, amplitude-weighted instantaneous frequencies are often used. Here it is shown that if powers of the envelope A^k(t), k = 1, 2, 3,.. are used as weight factors, the first power minimizes the variance for a narrow-band signal.