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The continuous autocorrelation Rf(τ) is the continuous cross-correlation of f(t) with itself, at lag τ, and is defined as:
where f* represents the complex conjugate. For a real function, f* = f.
Formally, the discrete autocorrelation R at lag j for signal xn is
where m is the average value (expected value) of xn. Quite frequently, autocorrelations are calculated for zero-centered signals, that is, for signals with zero mean. The autocorrelation definition then becomes
which is the definition of autocovariance.
Multi-dimensional autocorrelation is defined similarly. For example, in three dimensions the autocorrelation would be defined as
In the following, we will describe properties of one-dimensional autocorrelations only, since most properties are easily transferred from the one-dimensional case to the multi-dimensional cases.
A fundamental property of the autocorrelation is symmetry, R(i) = R(-i), which is easy to prove from the definition.
Since autocorrelation is a specific type of cross-correlation, it maintains all the properties of cross-correlation, and has a few specific properties of its own:
The autocorrelation of a white noise signal will have a strong peak at τ=0 and will be close to 0 for all other τ. This shows that white noise has no periodicity.
The autocorrelation function is related to the Power Spectral Density via the Fourier transform:
One application of autocorrelation is the measurement of the duration of light pulses produced by lasers. Very short duration light pulses (with durations of less than approximately 100 femtoseconds) produced by modelocked lasers cannot easily be measured by optoelectronic methods, since the response time of photodiodes and oscilloscopes are at best of the order of 200 femtoseconds. Instead, an autocorrelation method is used. An optical autocorrelator is a device which splits a beam of light into two beams, and then (after passing one of the beams through an adjustable delay lineThe term delay line has multiple meanings: In telecommunication: A delay line is a transmission line, or equivalent device, used to delay a signal. A single- input- channel device, such as a single-input sequential logic element, in which the output chann) recombines the beams in a manner similar to a Michelson interferometerThe Michelson interferometer is the classic setup for optical interferometry and was invented by Albert Abraham Michelson and used for the famous Michelson-Morley experiment. See there for a detailed discussion of its principle. But Michelson had already. The recombined beams are mixed in a nonlinear opticalNonlinear optics is the branch of optics that describes the behaviour of light in nonlinear media that is, media in which the polarization P responds nonlinearly to the electric field E of the light. This nonlinearity is typically only observed at very hi device (usually a second harmonic generation crystalThis article is about the form of solid matter. For other uses of this word, see Crystal (disambiguation . Insulin Crystals A crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extend).
Plotting the resulting optical signal as a function of the delay between the two arms of the autocorrelator produces an autocorrelation signal of the original input. From this, the duration of the input pulses can be found (if a particular pulse shape is inferred). The technique allows the measurement of pulse durations down to approximately 3 femtoseconds, without needing high- bandwidthAnalog Bandwidth is the width, usually measured in hertz, of a frequency band f f. It can also be used to describe a signal, in which case the meaning is the width of the smallest frequency band within which the signal can fit. It is usually notated B, W, electronics, as long as the pulses are part of a repetitive pulse train.
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