Gaussian Noise Suppression

All higher-order cumulants of order k > 2 are identically zero for any Gaussian random process. This is the foundational property exploited in HOS-based noise suppression. When a non-Gaussian signal of interest is corrupted by additive Gaussian noise, the cumulant of the mixture equals the cumulant of the signal alone. The noise component simply vanishes in the higher-order domain, providing a theoretical noise-rejection mechanism that the power spectrum (second-order) cannot achieve.

HOS processing enables feature extraction below the noise floor where conventional second-order methods fail. In practical RF fingerprinting scenarios, a signal buried in Gaussian noise with a negative SNR (e.g., -5 dB) can still yield a usable bispectrum or trispectrum. This is because the HOS domain effectively filters the noise energy, reconstructing a representation where the signal's non-Gaussian structure dominates. The recovery is not amplification but selective statistical filtering.

Unlike the power spectrum, which discards all phase information, the bispectrum and trispectrum preserve Fourier phase relationships. This is critical for RF fingerprinting because many hardware impairments manifest as non-linear phase couplings between frequency components. Gaussian noise suppression in the HOS domain retains these phase signatures, allowing classifiers to exploit both magnitude and phase features that uniquely identify a transmitter's analog chain.

Higher-order whitening is a pre-processing technique that decorrelates data beyond second-order statistics. By applying a linear transformation derived from the fourth-order cumulant tensor, the signal is rendered statistically independent up to fourth order. This simultaneously suppresses Gaussian noise and prepares the signal for subsequent Independent Component Analysis (ICA) or Joint Cumulant Diagonalization, enabling blind separation of co-channel emitters even in low-SNR environments.

Before applying HOS-based noise suppression, a Gaussianity test (e.g., D'Agostino's K-squared test or the Shapiro-Wilk test on estimated cumulants) validates that exploitable non-Gaussian structure exists. If a signal segment fails to reject the Gaussian null hypothesis, its higher-order features are statistically unreliable. This gate ensures that downstream fingerprinting models only process segments where kurtosis and skewness confirm the presence of hardware-induced non-Gaussian signatures.

The bispectrum and trispectrum of a signal-plus-Gaussian-noise mixture are mathematically equivalent to the polyspectra of the signal alone. This additive invariance means that HOS-based feature vectors extracted from a device's emissions remain stable regardless of the instantaneous noise power, as long as the noise remains Gaussian. This property is essential for channel-robust feature learning, where varying thermal noise conditions would otherwise corrupt second-order fingerprints.

Standard power spectrum analysis fails when signals drop below the noise floor. Higher-Order Spectral Analysis (HOSA) exploits the fact that Gaussian noise has zero theoretical bispectrum and trispectrum. By computing the bispectrum or trispectrum of a received signal, non-Gaussian signal components—such as transmitter-induced harmonics and phase couplings—are recovered intact while additive white Gaussian noise is mathematically suppressed. This enables device identification at signal-to-noise ratios (SNRs) where conventional energy detection is impossible.

Higher-order cumulants (third-order and fourth-order) are theoretically zero for Gaussian processes. This property is exploited to construct cumulant-based feature vectors that are inherently noise-robust. Key techniques include:

• Kurtosis estimation: Measures tailedness of amplitude distribution; non-zero excess kurtosis indicates non-Gaussian hardware signatures
• Skewness analysis: Detects directional amplifier non-linearity asymmetries
• Cyclic cumulant computation: Combines cyclostationary periodicity with non-Gaussianity for doubly-robust features These features remain stable even when the signal is deeply embedded in thermal noise.

When multiple transmitters occupy the same frequency band, their signals mix with Gaussian noise. Independent Component Analysis (ICA) and Joint Cumulant Diagonalization leverage higher-order statistics to separate these mixtures without prior knowledge of the source signals. The process:

• Estimates the fourth-order cumulant tensor of the mixture
• Diagonalizes cumulant matrices to maximize statistical independence
• Recovers individual emitter signatures from the noise floor This enables attribution of specific transmissions to specific devices even in dense spectral environments.

Raw signal data is high-dimensional and dominated by Gaussian noise components. Non-Gaussian subspace projection identifies and isolates the low-dimensional manifold containing hardware-specific fingerprint information. The technique:

• Projects data onto directions maximizing non-Gaussianity
• Discards Gaussian-dominated dimensions that contain only noise
• Retains the subspace where quadratic phase coupling and amplifier non-linearities reside This serves as a pre-processing step before classification, dramatically improving signal-to-noise ratio in the feature space.

Bicoherence normalizes the bispectrum to produce a bounded metric (0 to 1) that quantifies the proportion of quadratically phase-coupled energy at each bifrequency pair. Unlike raw bispectral magnitude, bicoherence is independent of signal amplitude and provides a consistent detection statistic across varying noise conditions. Applications include:

• Detecting weak non-linearities from power amplifier compression
• Identifying intermodulation products generated by mixer imperfections
• Serving as a Gaussianity test to validate the presence of exploitable hardware fingerprints before classification

Cumulant tensors organize higher-order statistics into multi-dimensional arrays that capture joint relationships across time, frequency, and statistical order. Higher-Order Singular Value Decomposition (HOSVD) factorizes these tensors to separate signal and noise subspaces. The decomposition:

• Compresses the cumulant tensor into a core tensor and orthogonal factor matrices
• Identifies components dominated by Gaussian noise for removal
• Preserves multi-linear interactions characteristic of specific transmitter impairments This provides a principled framework for joint blind source separation and feature extraction in low-SNR environments.

Statistical measures beyond second-order variance that quantify deviations from Gaussianity. Third-order cumulant (skewness) reveals directional asymmetry in amplifier distortion, while fourth-order cumulant (kurtosis) captures the 'tailedness' of a signal's amplitude distribution. These form the mathematical foundation for robust RF fingerprint extraction, as they are theoretically zero for Gaussian noise, enabling feature recovery below the noise floor.

A third-order frequency-domain representation that detects quadratic phase coupling between signal components. Unlike the power spectrum, the bispectrum preserves phase information and suppresses Gaussian noise. Key derived metrics include:

• Bicoherence: A normalized measure of phase coupling consistency
• Diagonal Slice Spectrum: A 1D projection reducing computational complexity
• Bispectral Entropy: Quantifies the complexity of non-linear interactions

A non-linear phenomenon where two frequency components (f1, f2) interact within a transmitter's analog chain to generate a third component at their sum or difference frequency (f1±f2). This coupling is a distinctive hardware-induced fingerprint caused by amplifier non-linearity and mixer imperfections. It is invisible to standard power spectrum analysis but clearly detectable via bispectral processing.

Statistical hypothesis tests that determine whether a signal's amplitude distribution deviates from Gaussian. Common methods include:

• Jarque-Bera test: Based on sample skewness and kurtosis
• Kolmogorov-Smirnov test: Measures distance from Gaussian CDF
• D'Agostino's K-squared test: Combines skewness and kurtosis measures Validating non-Gaussianity confirms the presence of exploitable hardware fingerprints before committing to higher-order processing pipelines.

A compact statistical fingerprint constructed from estimated higher-order cumulants that serves as input to machine learning classifiers. Typical construction involves:

• Estimating cumulants up to 4th order from signal segments
• Assembling into a fixed-length vector
• Applying Higher-Order Whitening for normalization
• Feeding into classifiers like SVMs or neural networks These vectors are inherently robust to Gaussian noise due to the theoretical insensitivity of cumulants.

The unsupervised recovery of individual emitter signals from mixtures using statistical independence criteria. Independent Component Analysis (ICA) exploits the fact that signals from different transmitters are statistically independent and non-Gaussian. Joint Cumulant Diagonalization simultaneously diagonalizes multiple cumulant matrices to achieve separation without gradient-based optimization, enabling isolation of co-channel emitters in dense spectral environments.