Autocorrelation technique

Last updated May 31, 2019

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

A complex number is a number that can be expressed in the form $a + bi$ , where $a$ and $b$ are real numbers, and $i$ is a solution of the equation $x 2 = -1$ . Because no real number satisfies this equation, $i$ is called an imaginary number. For the complex number $a + bi$ , $a$ is called the real part, and $b$ is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.

Radar is a detection system that uses radio waves to determine the range, angle, or velocity of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain.A radar system consists of a transmitter producing electromagnetic waves in the radio or microwaves domain, a transmitting antenna, a receiving antenna and a receiver and processor to determine properties of the object(s). Radio waves from the transmitter reflect off the object and return to the receiver, giving information about the object's location and speed.

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous, and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle.

Derivation

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

Autocorrelation, also known as serial correlation, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals.

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 $S(\omega )\ {\stackrel {\mathrm {def} }{=}}\ \delta (\omega -\omega _{0})$ , 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 mean frequency is calculated based on the autocorrelation with lag one, evaluated over a signal consisting of N samples:

\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:

{\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, Miller et al., IEEE Transactions on Information Theory. ^{[ full citation needed ]}
Doppler Radar Meteorological Observations Doppler Radar Theory.^{[ full citation needed ]} 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.

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