On the joint probability density function for the autocorrelation estimates in ultrasound color flow imaging

Autocorrelation estimates of the Doppler signal with zero and unity lag are commonly used for color coding algorithm in 2-D color flow imaging. Signal power, center frequency, and bandwidth are derived from these two autocorrelation estimates, and combined in a color coding scheme. In this paper computer simulations show that the joint probability density function (PDF) for the autocorrelation estimates has a form which differs significantly from a Gaussian distribution. However, the PDF for the normalized autocorrelation estimates is shown to have a form closer to a Gaussian distribution, which means that the first- and second-order moments gives an adequate statistical description of the set of estimators. Analytical approximations for the first- and second-order moments in the joint PDF for the normalized autocorrelation estimates of all lags are derived, and the expressions are verified by computer simulations. Using this results, an expression for the probability density function of the correlation angle estimate is derived, showing a close fit to simulation results. Numerical results show that the performance of the correlation angle estimate can be improved significantly by applying spatial averaging to the autocorrelation estimates.<>