Bispectrum

The bispectrum of a Gaussian process is theoretically zero. This property allows the bispectrum to completely suppress additive Gaussian noise, revealing non-Gaussian signal components that would otherwise be buried below the noise floor. For RF fingerprinting, this means hardware-induced non-linearities remain detectable even at very low Signal-to-Noise Ratios (SNR).

The bispectrum detects quadratic phase coupling (QPC), a phenomenon where two frequency components, f1 and f2, interact non-linearly to generate a third at their sum or difference frequency with a consistent phase relationship. This is a direct signature of non-linear analog components like power amplifiers. Key indicators:

• A peak at bifrequency (f1, f2) indicates energy at f1+f2 is phase-coupled.
• The bicoherence normalizes this to a 0-1 bounded metric.
• QPC patterns serve as unique, unclonable device identifiers.

The bispectrum exhibits 12 symmetry regions in the bifrequency plane, meaning the full bispectrum is completely determined by its values in a triangular principal domain. These symmetries arise from the permutation of frequency arguments and the conjugate symmetry of the Fourier transform. Exploiting these symmetries:

• Reduces computation by approximately 92%.
• Ensures consistent feature extraction regardless of frequency ordering.
• The non-redundant region is the triangle defined by 0 ≤ f2 ≤ f1 ≤ f1+f2 ≤ fs/2.

Unlike the power spectrum, which discards all phase information by squaring the Fourier magnitude, the bispectrum retains the Fourier phase of the signal. This is critical for RF fingerprinting because many hardware impairments manifest as phase distortions rather than amplitude distortions. The bispectrum's complex-valued output captures:

• Phase relationships between harmonically related components.
• Non-minimum phase system characteristics.
• Asymmetric distortion profiles caused by amplifier non-linearity in specific quadrants.

The full bispectrum is a 2D function, which can be computationally expensive. The diagonal slice, B(f, f), is a 1D projection that retains significant non-Gaussian signature information while dramatically reducing complexity. Other reduction techniques include:

• Radially integrated bispectrum (RIB): Integrates along radial lines in the bifrequency plane.
• Axially integrated bispectrum (AIB): Integrates along axes parallel to f1 or f2.
• Circularly integrated bispectrum (CIB): Integrates along concentric circles. These compressed representations form compact cumulant-based feature vectors for machine learning classifiers.

Bispectral entropy quantifies the irregularity and complexity of the bispectrum distribution. It is derived by treating the normalized bispectral magnitude as a 2D probability density function and computing its Shannon entropy. For emitter identification:

• High entropy: Indicates a complex, distributed non-linearity pattern typical of multi-stage amplifiers.
• Low entropy: Indicates concentrated non-linearities, such as a single dominant mixing product.
• This metric provides a single scalar feature that discriminates between device classes with different hardware architectures.

A normalized bispectrum that provides a bounded, frequency-by-frequency measure of quadratic phase coupling strength. Values range from 0 to 1, where a bicoherence near 1 indicates a high proportion of signal energy at a bifrequency pair is consistently phase-coupled. This normalization makes it an excellent detection statistic for identifying hardware-induced non-linearities independent of signal amplitude.

The statistical building blocks of polyspectral analysis. Key cumulants for RF fingerprinting include:

• Third-order cumulant (skewness): Captures asymmetric amplitude distributions caused by amplifier non-linearity.
• Fourth-order cumulant (excess kurtosis): Measures the 'peakedness' of a distribution relative to Gaussian, sensitive to mixer intermodulation products.
• Cumulants of order >2 are theoretically zero for Gaussian processes, making them natural Gaussian noise rejectors.

A one-dimensional projection of the bispectrum along its diagonal axis where f1 = f2. This slice, B(f, f), captures self-phase coupling and significantly reduces computational complexity from O(N²) to O(N). While it discards off-diagonal coupling information, it retains key non-Gaussian signatures and is often used as a computationally efficient feature vector for real-time emitter classification on edge hardware.

A fundamental advantage of higher-order statistics: all polyspectra of order >2 are theoretically zero for Gaussian processes. Thermal noise in receivers is typically Gaussian, while transmitter impairments produce non-Gaussian signal components. By operating in the bispectral or trispectral domain, analysts can extract hardware-specific signatures buried well below the noise floor, dramatically improving identification range and robustness compared to second-order power spectrum analysis.

A compact statistical fingerprint constructed from estimated higher-order cumulants that serves as input to machine learning classifiers. A typical vector might include:

• 1D diagonal bispectral slices for computational efficiency
• Integrated polyspectral values along radial paths
• Bispectral entropy as a complexity measure These features are concatenated and fed to classifiers such as SVMs or deep neural networks for emitter identification.