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| The '''autocorrelation technique''' is a method for estimating the dominating frequency in a [[Complex number|complex]] signal, as well as its variance. Specifically, it calculates the first two moments of the power spectrum, namely the mean and variance. It is also known as the '''pulse-pair algorithm''' in [[radar]] theory.
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| The algorithm is both computationally faster and significantly more accurate compared to the [[discrete Fourier transform|Fourier transform]], since the resolution is not limited by the number of samples used.
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| == Derivation ==
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| The [[autocorrelation]] of lag 1 can be expressed using the inverse Fourier transform of the power spectrum <math>S(\omega)</math>:
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| :<math> R(1) = \frac{1}{2\pi} \int_{-\pi}^{\pi} S(\omega) e^{i\,\omega\,1} d\omega. </math>
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| If we model the power spectrum as a single frequency <math>S(\omega) \ \stackrel{\mathrm{def}}{=}\ \delta(\omega - \omega_0)</math>, this becomes:
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| :<math> R(1) = \frac{1}{2\pi} \int_{-\pi}^{\pi} \delta(\omega - \omega_0) e^{i\,\omega} d\omega </math>
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| :<math> R(1) = \frac{1}{2\pi} e^{i\,\omega_0} </math>
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| where it is apparent that the phase of <math>R(1)</math> equals the signal frequency.
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| == Implementation ==
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| The mean frequency is calculated based on the [[autocorrelation]] with lag one, evaluated over a signal consisting of N samples:
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| :<math>\omega = \angle R_N(1) = \tan^{-1}\frac{im\{ R_N(1) \}}{re\{ R_N(1) \}}. </math>
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| The spectral variance is calculated as follows:
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| :<math>var\{ \omega \} = \frac{2}{N} \left( 1 - \frac{|R_N(1)|}{R_N(0)} \right). </math>
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| == Applications ==
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| * Estimation of blood velocity and turbulence in ''color flow imaging'' used in [[medical ultrasonography]].
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| * Estimation of target velocity in [[pulse-doppler radar]]
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| {{inline|date=February 2012}}
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| == External links ==
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| * [http://ieeexplore.ieee.org/xpl/abs_free.jsp?arNumber=1054886 A covariance approach to spectral moment estimation], Miller et al., IEEE Transactions on Information Theory. {{full|date=November 2012}}
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| * Doppler Radar Meteorological Observations [http://www.ofcm.gov/fmh11/fmh11partb/2005pdf/fmh-11B-2005.pdf Doppler Radar Theory].{{full|date=November 2012}} Autocorrelation technique described on p.2-11
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| * [http://server.oersted.dtu.dk/31655/documents/kasai_et_al_1985.pdf 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, May 1985 {{full|date=November 2012}}
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| [[Category:Radar theory]]
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| [[Category:Signal processing]]
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| [[Category:Time series analysis]]
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