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Autocorrelation is the crosscorrelation of a signal with itself. Informally, it is the similarity between observations as a function of the time separation between them. It is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal which has been buried under 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.
Contents
Definitions
Different fields of study define autocorrelation differently, and not all of these definitions are equivalent. In some fields, the term is used interchangeably with autocovariance.
Statistics
In statistics, the autocorrelation of a random process describes the correlation between values of the process at different points in time, as a function of the two times or of the time difference. Let X be some repeatable process, and i be some point in time after the start of that process. (i may be an integer for a discretetime process or a real number for a continuoustime process.) Then X_{i} is the value (or realization) produced by a given run of the process at time i. Suppose that the process is further known to have defined values for mean μ_{i} and variance σ_{i}^{2} for all times i. Then the definition of the autocorrelation between any two time s and t is
where "E" is the expected value operator. Note that this expression is not welldefined for all time series or processes, because the variance may be zero (for a constant process) or infinite. If the function R is welldefined, its value must lie in the range [−1, 1], with 1 indicating perfect correlation and −1 indicating perfect anticorrelation.
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