Higher-Order Statistics (HOS)

HOS measures are intrinsically zero for Gaussian processes. This property makes them powerful tools for signal detection and interference classification, as any non-zero HOS value immediately indicates a deviation from thermal noise. This is the foundational principle for using HOS in spectrum anomaly detection.

Unlike second-order statistics (autocorrelation, power spectrum), HOS preserves phase relationships within a signal. This allows for the identification of non-minimum phase systems and the detection of specific signal coupling mechanisms that are invisible to conventional spectral analysis.

A critical property for spectrum monitoring: HOS is theoretically immune to additive white Gaussian noise (AWGN). The cumulants of a Gaussian process are zero for orders greater than two. This enables the extraction of non-Gaussian signal features even at very low signal-to-noise ratios (SNR).

HOS is highly sensitive to non-linear transformations in the signal path. This allows for the detection of hardware impairments like amplifier compression, mixer intermodulation, or intentional jamming signals that introduce non-linear artifacts into the spectral environment.

The third-order statistic, skewness, measures the asymmetry of the probability distribution of the signal amplitude. A non-zero skewness indicates a departure from symmetric distributions, useful for identifying specific modulation types or asymmetric interference patterns.

The fourth-order statistic, kurtosis, quantifies the peakedness or tailedness of a distribution relative to a Gaussian. High kurtosis signals the presence of impulsive noise, bursty interference, or outlier events, making it a primary feature for anomaly scoring in real-time spectrum monitoring.

Leverages the third-order cumulant to quantify the asymmetry of a signal's amplitude probability distribution.

• A symmetric Gaussian process has a skewness of 0.
• Positive skewness indicates a tail extending toward higher amplitudes.
• Negative skewness indicates a tail toward lower amplitudes.
• Radar pulses and certain modulated interferers introduce measurable skewness into the noise floor.

The Fourier transform of the third-order cumulant, revealing quadratic phase coupling between frequency components.

• Suppresses Gaussian noise entirely (theoretical bispectrum of Gaussian noise is zero).
• Detects non-linear interactions generated by faulty power amplifiers or intentional jammers.
• Bicoherence normalizes the bispectrum to a [0,1] scale for robust thresholding.

Uses fourth-order cumulants to separate mixed signals without prior knowledge of the mixing matrix.

• Algorithms like JADE (Joint Approximate Diagonalization of Eigenmatrices) exploit non-Gaussianity.
• Isolates independent anomalous emitters overlapping in time and frequency.
• Critical for disaggregating co-channel interference in dense spectrum environments.

A statistical goodness-of-fit test that quantifies how much a sample's skewness and kurtosis deviate from a normal distribution.

• Computes a single scalar test statistic from the third and fourth moments.
• A high Jarque-Bera statistic strongly rejects the null hypothesis of Gaussianity.
• Used as a lightweight, real-time anomaly flag for streaming I/Q data.

An alternative representation of a probability distribution that encodes all statistical moments in a single function.

• The n-th derivative of the MGF evaluated at zero yields the n-th moment.
• Cumulant-generating functions (log of the MGF) directly produce cumulants.
• Provides a unified mathematical framework for deriving HOS-based detectors.