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What is autocorrelation?

Module by: Brandon Hodgson. E-mail the author

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What is autocorrelation?

Correlation is a mathematical tool used frequently in signal processing for analysing functions or series of values, such as time domain signals (Wikipedia 2006). Correlation is the mutual relationship between two or more random variables (Ali). Autocorrelation is the correlation of a signal with itself (Parr 1999). This is unlike cross-correlation, which is the correlation of two different signals (Parr 1999).

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 (Wikipedia 2006).

Different definitions of autocorrelation are in use depending on the field of study which is being considered and not all of them are equivalent (Wikipedia 2006). In some fields, the term is used interchangeably with autocovariance(Wikipedia 2006).

In statistics, the autocorrelation of a discrete time series or a process Xt is simply the correlation of the process against a time-shifted version of itself (Wikipedia 2006). If Xt is second-order stationary with mean μ then this definition is

Figure 1
Figure 1 (b649019848cf9ea45681a1f74712992a)
(1)(Wikipedia 2006)

where E is the expected value and k is the time shift being considered (usually referred to as the lag) (Wikipedia 2006). 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 (Wikipedia 2006). It is common practice in many disciplines to drop the normalisation by σ2 and use the term autocorrelation interchangeably with autocovariance (Wikipedia 20006).

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

Figure 2
Figure 2 (f58e9aabd35490fe19ce6de9867d4e4c)
(2)(Wikipedia 2006)

where f* represents the complex conjugate and the circle represents convolution (Wikipedia 2006). For a real function, f* = f (Wikipedia 2006).

Formally, the discrete autocorrelation R at lag j for signal xn is

Figure 3
Figure 3 (5609ddb2f8b127e19aa3f396a3a12602)
(3)(Wikipedia 2006)

where m is the average value (expected value) of xn. Frequently, autocorrelations are calculated for zero-centered signals, that is, for signals with zero mean (Wikipedia 2006). The autocorrelation definition then becomes

Figure 4
Figure 4 (932daf9f6808bf90d219e97775e2a82c)
(4)(Wikipedia 2006)

Multi-dimensional autocorrelations are defined similarly.

Example: In three dimensions the autocorrelation would be defined as

Figure 5
Figure 5 (751bee19b4b334a0a5f88589a41c167b)
(5)(Wikipedia 2006)

The properties of autocorrelation are discussed in the following list. The properties are for one-dimensional autocorrelations only, but can easily be transferred to multi-dimensional cases (Wikipedia 2006). This list is extracted from (Wikipedia 2006).

  • 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

Figure 6
Figure 6 (abd8da04d1eaad633e29f41c9a74e655)
(6)(Wikipedia 2006)

when f is a real function, and an Hermitian function

Figure 7
Figure 7 (28b7dc68f25dac34520549659a1354d6)
(7)(Wikipedia 2006)

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,
    Figure 8
    Figure 8 (d99f67bdd96b68c50e9e65e622b82853)
    . 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:

Figure 9
Figure 9 (8cf3336cea1f2be4ae65647151af4cce)
(8)(Wikipedia 2006)

Figure 10
Figure 10 (e1104259b1b8ea4926308e3c01d26a40)
(9)(Wikipedia 2006)

Exercise: What does the following autocorrelation function depend on?

Figure 11
Figure 11 (clip_image002)
(10)(Ali 2000)

References:

Ali M. "Probabilistic Systems Analyses : Part II," University of the Witwatersrand, South Africa, 2000.

Parr J, Phillips C. "Signals, Systems and Transforms," Prentice Hall, New Jersey, second edition, 1999.

Wikipedia. "Autocorrelation", Wikimedia Foundation Inc,  http://en.wikipedia.org/wiki/Autocorrelation, Last accessed 14 February 2006.

Brandon Hodgson

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