The autocorrelation technique is a method for estimating the dominating frequency in a

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

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 Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, w ...

theory. The algorithm is both computationally faster and significantly more accurate compared to the

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

, since the resolution is not limited by the number of samples used.

Derivation

The

autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...

of lag 1 can be expressed using the inverse Fourier transform of the power spectrum

S(\omega)

: :

R(1) = \frac \int_^ S(\omega) e^ d\omega.

If we model the power spectrum as a single frequency

S(\omega) \ \stackrel\ \delta(\omega - \omega_0)

, this becomes: :

R(1) = \frac \int_^ \delta(\omega - \omega_0) e^ d\omega

R(1) = \frac e^

where it is apparent that the phase of

R(1)

equals the signal frequency.

Implementation

The mean frequency is calculated based on the

with lag one, evaluated over a signal consisting of N samples: :

\omega = \angle R_N(1) = \tan^\frac.

The spectral variance is calculated as follows: :

\text\ = \frac \left( 1 - \frac \right).

Applications

* Estimation of blood velocity and turbulence in ''color flow imaging'' used in

medical ultrasonography Medical ultrasound includes diagnostic techniques (mainly medical imaging, imaging techniques) using ultrasound, as well as therapeutic ultrasound, therapeutic applications of ultrasound. In diagnosis, it is used to create an image of internal ...

. * Estimation of target velocity in

pulse-doppler radar A pulse-Doppler radar is a radar system that determines the range to a target using pulse-timing techniques, and uses the Doppler effect of the returned signal to determine the target object's velocity. It combines the features of pulse radars an ...

External links

A covariance approach to spectral moment estimation
Miller et al., IEEE Transactions on Information Theory. * Doppler Radar Meteorological Observation
Doppler Radar Theory
{{full, date=November 2012 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. Radar theory Signal processing Autocorrelation