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. Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... This long range radar antenna, known as ALTAIR, is used to detect and track space objects in conjunction with ABM testing at the Ronald Reagan Test Site on the Kwajalein atoll[1]. Radar is a system that uses radio waves to detect, determine the distance of, and map, objects such...

The algorithm is both computationally faster and significantly more accurate compared to the Fourier transform, since the resolution is not limited by the number of samples used. In mathematics, the discrete Fourier transform (DFT), sometimes called the finite Fourier transform, is a Fourier transform widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal, solve partial differential equations, and to perform other operations such as convolutions. ...

Derivation

The autocorrelation of lag 1 can be expressed using the inverse Fourier transform of the power spectrum S(ω): Autocorrelation is a mathematical tool used frequently in signal processing for analysing functions or series of values, such as time domain signals. ...

If we model the power spectrum as a single frequency , this becomes:

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: Autocorrelation is a mathematical tool used frequently in signal processing for analysing functions or series of values, such as time domain signals. ...

The spectral variance is calculated as follows:

Applications

• Estimation of blood velocity and turbulence in color flow imaging used in medical ultrasonography.
• Estimation of target velocity in pulse-doppler radar

Medical ultrasonography (sonography) is an ultrasound-based diagnostic imaging technique used to visualize muscles and internal organs, their size, structure and any pathological lesions, making them useful for scanning the organs. ... Pulse-doppler is a radar system that functions by sending short pulses of radio energy and simultanously listens for the echo from objects using the same attenna. ...

External links

• A covariance approach to spectral moment estimation, Miller et al., IEEE Transactions on Information Theory.
• Doppler Radar Meteorological Observations Doppler Radar Theory. 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, May 1985