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 statistics, the autocorrelation function (ACF) of a discrete time series or a process X_t describes the correlation between the process at different points in time. If X_t has mean μ and variance σ² then the definition of ACF is

}{\sigma^2}\, ,

where E is the expected value. Note that this is not well-defined for all time-series or processes since the variance may be zero (for a constant process) or infinity. If the function is well defined then this definition has the attractive property of being in the range ^* with 1 indicating perfect correlation and −1 indicating perfect anti-correlation.

If X_t is second-order stationary then the ACF depends only on the difference between t and s and can be expressed as a function of a single variable. This gives the more familiar form,

}{\sigma^2}\, ,

where k is lag (|t - s|). It is common practice in many disciplines to drop the normalisation by σ² and use the term autocorrelation interchangeably with autocovariance. For a sample time series of length n, X₁, X₂ ... X_n with known mean and variance then an estimate may be obtained from

\hat{R}(k)=\frac{1}{(n-k) \sigma^2} \sum_{t=1}^{n-k} (f(t)-\mu)(f(t+k)-\mu)

for

k \in \mathbb{N}

If the true mean or variance for the process are not known then μ and σ²may be replaced by the standard formulae for sample mean and sample variance although this leads to a biased estimator Spectral analysis and time series, M.B. Priestley (London, New York : Academic Press, 1982).

Signal processing

In signal processing, the above definition is often used without the normalisation, that is, without subtracting the mean and dividing by the variance.

Given a signal f(t), the continuous autocorrelation R_ff(τ) is most often defined as the continuous cross-correlation integral of f(t) with itself, at lag τ.

R_{ff}(\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.

The discrete autocorrelation R at lag j for a discrete signal x_n is

R_{xx}(j) = \sum_n (x_n)(x^*_{n-j} )\, .

The above definitions work for signals that are square integrable, or square summable, that is, of finite energy. Signals that "last forever" are treated instead as random processes, in which case different definitions are needed, based on expected values. For wide-sense-stationary random processes, the autocorrelations are defined as

R_{ff}(\tau) = E\left

R_{xx}(j) = E\left x^*_{n-j}\right

For processes that are not stationary, these will also be functions of t, or n.

Alternatively, signals that last forever can be treated by a short-time autocorrelation function analysis, using finite time integrals. (See short-time Fourier transform for a related process.)

Multi-dimensional autocorrelation is defined similarly. For example, in three dimensions the autocorrelation of a square-summable discrete signal would be

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

When mean values are subtracted from signals before computing an autocorrelation function, the resulting function is usually called an auto-covariance function.

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 the autocorrelation is a 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.

The autocorrelation of a periodic function is, itself, periodic with the very same period.

The autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all τ) is the sum of the autocorrelations of each function separately.

Since autocorrelation is a specific type of cross-correlation, it maintains all the properties of cross-correlation.

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 a sampled instance of a white noise signal is not statistically correlated to a sample instance of the same white noise signal at another time.

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.