Department of Electrical Engineering, University of Edinburgh

- Introduction
- What are HOS ?
- Why use HOS ?
- Frequency domain methods for detecting nonlinearities
- Conclusions
- Getting more information
- Some HOS jargon
- References

Speech or music signals are generated mechanically by systems with nonlinear dynamics. If the prediction and coding quality of such signals is to be improved then more of the information available in the signal must be used : the signal higher order statistics (HOS) must be exploited.

The aim of this article is to introduce HOS to a wide audience (assuming basic knowledge of signal processing and statistics), and to outline some of the reasons why they can be useful in practical applications.

In recent years the field of HOS has continued its expansion, and applications have been found in fields as diverse as economics, speech, seismic data processing, plasma physics and optics. Many signal processing conferences (ICASSP, EUSIPCO) now have sessions specifically for HOS, and an IEEE Signal Processing Workshop on HOS has been held every two years since 1989 (the most recent one took place in June 1995 in Spain).

`x(n)`

,
`R(m) = <x(n) x(n+m)> `

`<>`

The third-order
measure is called the **third-order moment**

`M(m1,m2) = <x(n) x(n+m1) x(n+m2)>`

Note that the third-order moment depends on **two** independent
lags `m1`

and `m2`

. Higher order moments can be
formed in a similar way by adding lag terms to the above equation.
The signal cumulants can be easily derived
from the moments.

`P(k)`

, and it can be calculated in two ways:
- Take a Discrete Fourier Transform (DFT) of the autocorrelation
function ;
.`P(k) = DFT[ R(m) ]`

*Or:*Multiply together the signal Fourier Transform`X(k)`

with its complex conjugate ;`P(k) = X(k) X*(k)`

`B(k,l)`

can be calculated in
a similar way:
- Take a
**Double**Discrete Fourier Transform (DDFT) of the third-order cumulant ;`B(k,l) = DDFT[ M(m1,m2) ]`

*Or:*Form a product of Fourier Transforms at**different**frequencies ;

`B(k,l) = X(k) X(l) X*(k+l)`

Now just as the second-order measures are related to the signal
**variance** so the third-order measures (third-order cumulant and
bispectrum) are related to the signal **skewness**, the
fourth-order measures (fourth-order cumulant and trispectrum) are
related to the signal **kurtosis**, and higher-order measures are
related to higher-order moments of the signal.

- Second-order measures (such as
the power spectrum and autocorrelation functions) contain no phase
information. As a consequence of this
- non-minimum phase signals cannot be correctly identified by 2nd-order techniques. This is important in the field of linear processes, which will not be discussed in this article.
- certain types of phase coupling (associated with nonlinearities) cannot be correctly identified using 2nd order techniques.

- Any Gaussian signal is completely characterised by its mean and
variance. Consequently the HOS of Gaussian signals are either zero (e.g. the
3rd order moment of a Gaussian signal is zero), or contain redundant
information. Many signals encountered in practice have non-zero HOS,
and many measurement noises are Gaussian, and so in principle the HOS
are less affected by Gaussian background noise than the 2nd order
measures. (e.g. the power spectrum of a deterministic signal
**plus**Gaussian noise is very different from the power spectrum of the signal alone. However the bispectrum of the signal + noise is, at least in principle, the same as that of the signal).

It is evident that the bicoherence has two independent frequency axes
*k* and *l*, and that the bicoherence magnitude emerges from
the frequency plane. We only consider the bicoherence in a small
region of the *k,l* plane because of symmetry relations. The
bicoherence has phase too, and under certain circumstances that needs
to be examined too.
The interpretation of the bicoherence varies depending on the type of
signal under study.

- For stochastic wide-band signals, the level of the bicoherence
gives a measure of the signal
**skewness**(a statistical test for skewness based on the bispectrum was devised by Hinich in 1982). - For deterministic signals, a peak in the bicoherence indicates
the presence of
**Quadratic Phase Coupling (QPC)**, which is a specific type of nonlinearity.

`http://www.ee.ed.ac.uk/~hos/`

. This page contains links to research groups around
the world with interests in HOS, as well as hos bibliographies and a
noticeboard of forthcoming HOS events.
In Edinburgh, our own special areas of interest include HOS in Speech Signals, HOS for seismic deconvolution, Chaos in Speech, Nonlinear signal prediction. For more information on these areas, or general questions concerning this article, please contact

- Steve McLaughlin (any area of nonlinear signal processing).
- Achilleas Stogioglou (seismic deconvolution using parametric HOS methods),
- Justin Fackrell (detecting nonlinearities in speech using HOS).

**moments**Moments are statistical measures which characterise signal properties. We are used to using the mean and the variance (the first and second moments, respectively) to characterise a signal's probability distribution, but unless the signal is Gaussian (Normal) then moments of higher orders are needed to fully describe the distribution. In practice in HOS we usually use the cumulants rather than the moments (see below).**cumulants**The*n*th order cumulant is a function of the moments of orders up to (and including)*n*. For reasons of mathematical convenience, HOS equations/discussions most often deal with a signal's*cumulants*rather than the signal's*moments*.**polyspectra**This term is used to describe the family of all frequency-domain spectra, including the 2nd order. Most HOS work on polyspectra focusses attention on the bispectrum (third-order polyspectrum) and the trispectrum (fourth-order polyspectrum).**bicoherence**This is used to denote a normalised version of the bispectrum. The bicoherence takes values bounded between 0 and 1, which make it a convenient measure for quantifying the extent of phase coupling in a signal. The normalisation arises because of variance problems of the bispectral estimators for which there is insufficient space to explain (see Hinich 1982 for more details).

**J M Mendel**
*Tutorial on Higher Order Statistics (Spectra) in signal processing and
system theory: theoretical results and some applications*
Proceedings of the IEEE, **79**(3),
pp 278-305, March, 1991.

**C L Nikias and A P Petropulu**
*Higher-Order Spectra analysis*
PTR Prentice Hall, New Jersey, 1993.

**M J Hinich**
*Testing for Gaussianity and linearity of a stationary time series*
Journal of Time Series Analysis,**3**(3),
pp 169-176, 1982.

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