STATISTICS OF INSTANTANEOUS FREQUENCY AND AMPLITUDE AS RELATED TO DOPPLER SPECTRUM

The distribution of the instantaneous frequency and power of the signal associated with a given Doppler spectrum is investigated for two cases: (case A) where the power at any frequency of the Doppler spectrum is exponentially distributed and (case B) where the power at each frequency is determinate. It is found that in case A the joint distribution of the instantaneous power and frequency depends only on (1) the total power, (2) the mean, and (3) the variance of the Doppler power spectrum. This result holds to a first approximation for case B also. The distributions of (1) instantaneous power, (2) instantaneous frequency, (3) average power as a function of instantaneous frequency, (4) average frequency as a function of instantaneous power, (5) variance of frequency as a function of instantaneous power, and (6) variance of power as a function of instantaneous frequency are also obtained. In particular, it is shown that the mean of the instantaneous frequency is equal to the mean power‐weighted frequency of the Doppler spectrum in case A. In case B, this result is only approximately valid. Copyright 1969 by the American Geophysical Union.