autocorrelation

Autocorrelation is a mathematical tool used frequently in signal processing for analysing functions or series of values, such as time domain signals. It is the cross-correlation of a signal with itself. Autocorrelation is useful for finding repeating patterns in a signal, such as determining the presence of a periodic signal which has been buried under noise, or identifying the fundamental frequency of a signal which doesn't actually contain that frequency component, but implies it with many harmonic frequencies.

Definitions

Different definitions of autocorrelation are in use depending on the field of study which is being considered and not all of them are equivalent. In some fields, the term is used interchangeably with autocovariance.

Statistics

In the field of statistics, given a discrete time series or a process X_t then the autocorrelation is simply the correlation of the process against a time shifted version of itself. If X_t is second order stationary with mean μ then this definition is

}{\sigma^2}

where E is the expected value and k is the time shift being considered(usually referred to as the lag). This function has the attractive property of being in the range -1,1 with 1 indicating perfect correlation (the signals exactly overlap when time shifted by k) and -1 indicating perfect anti-correlation. It is common practice in many disciplines to drop the normalisation by σ² and use the term autocorrelation interchangeably with autocovariance.

Signal processing

In signal processing, given a signal f(t), the continuous autocorrelation R_f(τ) is the continuous cross-correlation of f(t) with itself, at lag τ, and is defined as:

R_f(\tau) = f^*(-\tau) \circ f(\tau) = \int_{-\infty}^{\infty} f(t+\tau)f^*(t)\, dt = \int_{-\infty}^{\infty} f(t)f^*(t-\tau)\, dt

where f^* represents the complex conjugate and the circle represents convolution. For a real function, f^* = f. Formally, the discrete autocorrelation R at lag j for signal x_n is

R(j) = \sum_n (x_n-m)(x_{n-j}-m )

where m is the average value (expected value) of x_n. Quite frequently, autocorrelations are calculated for zero-centered signals, that is, for signals with zero mean. The autocorrelation definition then becomes

R(j) = \sum_n x_n x_{n-j}.

Multi-dimensional autocorrelation is defined similarly. For example, in three dimensions the autocorrelation would be defined as

R(j,k,\ell) = \sum_{n,q,r} (x_{n,q,r}-m)(x_{n-j,q-k,r-\ell}-m).

Properties

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. In the continuous case, the autocorrelation is an even function ( $R_f(-\tau) = R_f(\tau)$ ) when f is a real function, and an Hermitian function ( $R_f(-\tau) = R_f^*(\tau)$ ) when f is a complex function.

The continuous autocorrelation function reaches its peak at the origin, where it takes a real value, i.e. for any delay τ, $|R_f(\tau)| \leq R_f(0)$ . This is a consequence of the Cauchy-Schwarz inequality. The same result holds in the discrete case.

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 Wiener-Khinchin theorem relates the autocorrelation function to the power spectral density via the Fourier transform:

R(\tau) = \int_{-\infty}^\infty S(f) e^{j 2 \pi f \tau} \, df

S(f) = \int_{-\infty}^\infty R(\tau) e^{- j 2 \pi f \tau} \, d\tau.

Applications

One application of autocorrelation is the measurement of optical spectra and the measurement of very-short-duration light pulses produced by lasers, both using optical autocorrelators.

In signal processing, autocorrelation can give information about repeating events like musical beats or pulsar frequencies, even though it cannot tell the position in time of the beat.

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