Definition
In terms of statistics, autocorrelation is defined as the pearson correlation between the two random processes.
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signal processing
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The average value of the same time function is a function of the two values of transient T and T + A as a function of the delay time T, which is a metric between the signal and the delay signal. When the delay time is zero, it becomes a mean square value of the signal, and it is the largest.
In short, the autocorrelation function is an expression signal and its multipath signal. A signal is similar to the similar degree of similarity after the delay of other conditions such as reflection, refraction, and the like.
Properties
The following uses one-dimensional autocorrelation function as an example to indicate its nature, and multi-dimensional conditions can be easily promoted from one-dimensional situation.
Symmetry: From the definition obvious, R (I) = R (-i) can be seen. Continuous autocorrelation function is an even function.
When f is a real function,:
When F is a reply function, the autocorrelation function is Ehe Rice function, satisfaction:
in which the asterisk represents the conjugate.
The peak value of the continuous indexed function is acquired in the origin, that is, for any delay τ, all have | R_F (\ TAU) | \ LEQ R_F (0). This conclusion can be directly available in Kexi-Schwatz. There is also this conclusion of discrete autocorrelation functions. The autocorrelation function of the
periodic function is a function having the same cycle as the original function.
Two mutually-independent functions (ie, the sum of all τ, two functions of the two functions is 0) is equal to the sum of the respective autocorrelation functions.
Due to the self-correlation function is a special mutual correlation function, it has the latter all properties.
The autocorrelation function of the white noise signal is a delta function, and all points except τ = 0 are 0.
Wiener-khinchin theorem indicates that the autocorrelation function and power spectral density function is a pair of Fourier transform pairs:
real value, the symmetrical autocorrelation function has a fairly called transformation function, so the reputation in Vina-Chensin theorem at this time. The index item can be written as the following cosine form:
< P> The autocorrelation function of white noise is δ function:
The autocorrelation function of the color noise with Romorez power spectrum is:
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Application
signal processing, self-correlation can be provided Information about duplicate events, such as music festival (for example, determining rhythm) or pulse star frequency (although it can't tell us the position of the beat). In addition, it can also be used to estimate the pitch of the tone.