What Is Higher order statistics?

Higher order statistics (HOS) is a branch of statistical signal processing that extends classical second-order analysis by incorporating moments and cumulants of order three and above to characterize non-Gaussian random processes. Standard power-spectral analysis depends entirely on second-order statistics, the autocorrelation function and its Fourier transform, which completely describe Gaussian signals but lose critical information when signals deviate from Gaussian distributions. HOS provides the mathematical tools needed to detect, identify, and exploit that additional information. The field draws from probability theory, Fourier analysis, and system identification, and has been actively developed since the 1980s through workshops and publications associated with the IEEE Signal Processing Society.

Moments, Cumulants, and Polyspectra

The nth-order moment of a stationary random process is the expected value of the nth power of the process, while cumulants are related quantities defined to suppress Gaussian components. A fundamental property of cumulants is that they are identically zero for any Gaussian process regardless of order, a fact that makes cumulant-based statistics blind to additive Gaussian noise. This property has substantial practical value: cumulant-domain processing suppresses sensor noise and interference that would otherwise contaminate estimates. The bispectrum (third-order spectrum) and trispectrum (fourth-order spectrum) are the Fourier transforms of the third and fourth-order cumulant sequences, respectively, and together form the body of polyspectral analysis. A foundational tutorial by Mendel, published in the Proceedings of the IEEE, documented the cumulant-polyspectra formulas and system-theoretic relationships that became standard references for the field. The full tutorial is publicly archived at SIPI, University of Southern California.

Non-Gaussian Signal Analysis and System Identification

Many signals of engineering importance are non-Gaussian: speech, biomedical waveforms, radar returns from distributed targets, and communications signals from non-linear sources all exhibit skewness or excess kurtosis that second-order methods cannot capture. HOS methods recover phase information lost in power spectral estimation, enabling reconstruction of non-minimum-phase systems that are invisible to conventional spectral approaches. In blind system identification, cumulants permit estimation of an unknown channel's impulse response from output observations alone, without requiring a known input signal. This capability is particularly valuable in underwater acoustics and seismic processing, where source characteristics are unavailable. The use of differential equations to model signal dynamics connects HOS to broader analysis frameworks, since cumulant evolution equations for linear systems driven by non-Gaussian inputs admit closed-form solutions analogous to those of the Wiener-Hopf equation for second-order problems.

Applications in Array Processing and Communications

HOS found early application in sensor array processing, where cumulants can separate signals from background noise in ways that second-order methods cannot achieve when noise itself is colored Gaussian. Research on cumulants applied to array processing for non-Gaussian noise suppression showed that spatial filtering based on fourth-order cumulants achieves source separation even when second-order methods fail due to noise coloring. In communications, HOS-based detectors characterize modulation type through cumulant features, supporting automatic modulation classification in spectrum monitoring and cognitive radio systems. Bispectral analysis has been used to identify harmonics and phase coupling in power system waveforms, motor vibration signals, and electroencephalogram recordings, each domain where quadratic nonlinearities leave a distinct signature in third-order statistics.

Applications

Higher order statistics has applications in a wide range of disciplines, including:

• Blind channel equalization and system identification in communications and underwater acoustics
• Radar and sonar signal processing for target detection in non-Gaussian clutter
• Biomedical signal analysis including EEG phase-coupling detection
• Power systems harmonic identification and power quality assessment
• Seismic deconvolution and geophysical exploration