What is autocorrelation?
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 seriesor 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 |
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 |
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 |
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 |
Multi-dimensional autocorrelations are
defined similarly.
Example: In three dimensions the
autocorrelation would be defined as
Figure 5 |
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 |
when f is a real function, and an Hermitian
function
Figure 7 |
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,
. This is a consequence of the Cauchy-Schwarz inequality. The same result holds in the discrete case.Figure 8
- 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 10 |
Exercise: What does the following
autocorrelation function depend on?
(10)(Ali 2000)
Answer.
Figure 11 |
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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