Maximum likelihood and maximum a posteriori estimation of Gaussian spectra. Application to attenuation measurements and color Doppler velocimetry

We address the problem of tracking slowly varying mean frequency and spectral width from a series of data vectors y
_p
(p=1, 2, ... P). When the y
_p
are small-sized, usual methods (periodogram, correlation lags...) suffer from strong variability. Moreover, the discrete nature of observed signals results in ambiguity in spectral domain. First in a Gaussian setting both for the shape of the spectra and for the statistics of observed signals, a maximum likelihood (ML) technique is provided for spectral moments estimation. Second, we propose a maximum a posteriori (MAP) method in order to account also for the slowly varying character of the spectral moments. Noticeable improvement are obtained with respect to usual methods: strong variance reduction in the field of ultrasound attenuation measurement (AM), and correct estimation beyond the Shannon limit in the field of color flow mapping (CFM)