Non-Gaussian Subspace Projection

The core principle replaces variance maximization (PCA) with non-Gaussianity maximization. The algorithm searches for projection vectors w such that the projected data wᵀx exhibits maximum statistical independence from a Gaussian distribution.

• Negentropy: Uses differential entropy to measure distance from Gaussianity, as Gaussian variables have maximum entropy for a given variance.
• Kurtosis-based contrast: Maximizes the absolute value of the fourth-order cumulant to find directions with the heaviest tails.
• Fixed-point iteration: FastICA and similar algorithms converge on independent components by iteratively updating projection vectors.

Non-Gaussian subspace projection is mathematically equivalent to ICA when the mixing matrix is square and the sources are independent. The projection finds the linear transformation that yields maximally independent non-Gaussian components.

• Blind source separation: Recovers individual emitter signals from co-channel mixtures without prior knowledge of the mixing process.
• Cumulant tensor diagonalization: Joint diagonalization of fourth-order cumulant matrices identifies the non-Gaussian subspace.
• Whitening pre-processing: Data is decorrelated and normalized to unit variance before projection, simplifying the search to an orthogonal rotation.

A critical advantage is the theoretical insensitivity to additive Gaussian noise. Higher-order cumulants of Gaussian processes are identically zero, so the projection automatically filters thermal noise and interference.

• Third-order and fourth-order cumulants: Gaussian distributions have zero skewness and zero excess kurtosis, making these statistics blind to Gaussian noise.
• Subspace partitioning: The signal space is decomposed into a Gaussian subspace (noise) and a non-Gaussian subspace (fingerprint information).
• Below-noise-floor extraction: Hardware impairments buried beneath the noise floor in power spectrum analysis become detectable in the non-Gaussian projection.

The projection serves as a dimensionality reduction engine that compresses raw IQ samples or spectral data into a compact feature vector retaining only the non-Gaussian fingerprint information.

• Cumulant-based feature vectors: The projected components' higher-order statistics form a discriminative signature for each emitter.
• Reduced classifier complexity: Lower-dimensional inputs enable simpler, faster neural network or SVM classifiers.
• Diagonal slice extraction: One-dimensional projections of polyspectral representations along specific axes further reduce dimensionality while preserving key non-linear coupling signatures.

When combined with channel-robust training, non-Gaussian subspace projection can isolate hardware-specific non-linearities that are invariant to linear channel distortions.

• Non-linear transfer function isolation: Amplifier compression and mixer non-linearities create higher-order statistics that survive multipath fading.
• Contrastive learning integration: Projections from the same device under different channel conditions are pulled together in the learned subspace.
• Domain adversarial training: A gradient reversal layer ensures the projection discards channel-specific variations while preserving emitter-specific non-Gaussian structure.

The FastICA algorithm is the standard implementation, using a fixed-point iteration scheme that converges quadratically to the non-Gaussian projection directions.

• Symmetric orthogonalization: All independent components are estimated simultaneously, preventing accumulation of estimation errors.
• Deflationary approach: Components are extracted one at a time, with each subsequent search constrained to be orthogonal to previously found directions.
• Non-linearity selection: The contrast function g(u) (e.g., tanh, cube, or exponential) is chosen based on the expected distribution of the source signals.

A pre-processing transformation that decorrelates and normalizes data beyond second-order statistics. While standard whitening removes mean and variance correlations, higher-order whitening additionally orthogonalizes the cumulant tensor, forcing cross-cumulants to zero. This prepares signals for cumulant-based feature extraction by ensuring that non-Gaussian structures are not masked by lower-order dependencies.

• Extends PCA whitening to third and fourth orders
• Essential pre-processing for robust cumulant estimation
• Reduces sensitivity to Gaussian noise contamination

A multi-dimensional array organizing higher-order cumulants that captures the complete non-Gaussian statistical structure of a signal. For a zero-mean random vector, the fourth-order cumulant tensor is a four-dimensional array where each element represents the joint cumulant of four signal components. Tensor decomposition techniques factor this structure to extract the most discriminative fingerprint features.

• Fourth-order tensor has O(n⁴) elements for n-dimensional data
• Symmetries reduce the effective degrees of freedom
• Serves as input to HOSVD and other multi-linear factorization methods

An information-theoretic measure of irregularity in the bispectrum distribution that quantifies the complexity of non-linear signal interactions. High bispectral entropy indicates distributed, complex phase coupling patterns characteristic of specific transmitter impairments, while low entropy suggests concentrated, deterministic non-linearities. This scalar metric provides a compact feature for device discrimination.

• Derived from the normalized bicoherence distribution
• Invariant to signal amplitude scaling
• Captures the richness of quadratic phase coupling in a single value

A multi-linear generalization of SVD that decomposes cumulant tensors into a core tensor and orthogonal factor matrices. HOSVD compresses the high-dimensional statistical information into a lower-dimensional subspace while preserving the multi-way interactions critical for fingerprinting. The resulting factor matrices span the non-Gaussian subspace where hardware-specific signatures reside.

• Extends matrix SVD to N-way arrays
• Produces orthogonal bases for each tensor mode
• Enables efficient statistical feature compression without information loss