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Derivation

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

R ( 1 ) = 1 2 π ∫ − π π S ( ω ) e i ω d ω . {\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 S ( ω )   = d e f   δ ( ω − ω 0 ) {\displaystyle S(\omega )\ {\stackrel {\mathrm {def} }{=}}\ \delta (\omega -\omega _{0})} , this becomes:

R ( 1 ) = 1 2 π ∫ − π π δ ( ω − ω 0 ) e i ω d ω {\displaystyle R(1)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\delta (\omega -\omega _{0})e^{i\,\omega }d\omega } R ( 1 ) = 1 2 π e i ω 0 {\displaystyle R(1)={\frac {1}{2\pi }}e^{i\,\omega _{0}}}

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

Implementation

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

ω = ∠ R N ( 1 ) = tan − 1 ⁡ im { R N ( 1 ) } re { R N ( 1 ) } . {\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:

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

Applications

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

External links

• A covariance approach to spectral moment estimation[dead link], 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.