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 as an integral of the power spectrum $G(\omega )$:

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

If we model the power spectrum as a single frequency $G(\omega )\equiv \delta (\omega -\omega _{0})$, 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 from the phase of the autocorrelation with lag one:

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

The variance of this frequency is calculated as follows:

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

External links

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