Autocorrelation technique

The autocorrelation of lag 1 can be expressed using the inverse Fourier transform of the power spectrum ${\displaystyle S(\omega )}$:

${\displaystyle R(1)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }S(\omega )e^{i\,\omega }d\omega .}$

If we model the power spectrum as a single frequency ${\displaystyle S(\omega )\ {\stackrel {\mathrm {def} }{=}}\ \delta (\omega -\omega _{0})}$, this becomes:

${\displaystyle R(1)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\delta (\omega -\omega _{0})e^{i\,\omega }d\omega }$

${\displaystyle R(1)={\frac {1}{2\pi }}e^{i\,\omega _{0}}}$

where it is apparent that the phase of ${\displaystyle R(1)}$ equals the signal frequency.

The mean frequency is calculated based on the autocorrelation with lag one, evaluated over a signal consisting of N samples:

${\displaystyle \omega =\angle R_{N}(1)=\tan ^{-1}{\frac {{\text{im}}\{R_{N}(1)\}}{{\text{re}}\{R_{N}(1)\}}}.}$

The spectral variance is calculated as follows:

${\displaystyle {\text{var}}\{\omega \}={\frac {2}{N}}\left(1-{\frac {|R_{N}(1)|}{R_{N}(0)}}\right).}$

• Estimation of blood velocity and turbulence in color flow imaging used in medical ultrasonography.
• Estimation of target velocity in pulse-doppler radar

• A covariance approach to spectral moment estimation, Miller et al., IEEE Transactions on Information Theory.
• Doppler Radar Meteorological Observations Doppler Radar Theory. Autocorrelation technique described on p.2-11
• Real-Time Two-Dimensional Blood Flow Imaging Using an Autocorrelation Technique, by Chihiro Kasai, Koroku Namekawa, Akira Koyano, and Ryozo Omoto, IEEE Transactions on Sonics and Ultrasonics, Vol. SU-32, No.3, May 1985.