Autocorrelation technique

The autocorrelation technique is a method for estimating the dominating frequency in a complex signal, as well as its variance. It is significantly more accurate compared to the Fourier transform, since the resolution is not limited by the number of samples used.

Principle

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

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

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

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

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

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

Implementation

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

$\omega =phase\{R_{N}(1)\}=\tan ^{-1}{\frac {im\{R_{N}(1)\}}{re\{R_{N}(1)\}}}$

The variance of this frequency is calculated as follows:

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

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

• Radar reference, anyone ???
• Real-Time Two-Dimensional Blood Flow Imaging Using an Autocorrelation Technique, IEEE Transactions on sonics and ultrasonics, May 1985