Bispectral Entropy

Bispectral entropy applies Shannon entropy to the normalized bispectrum magnitude distribution, treating the bifrequency plane as a probability density. A highly structured bispectrum with concentrated energy at specific bifrequency pairs yields low entropy, indicating deterministic non-linear interactions characteristic of a specific transmitter's analog impairments. Conversely, a flat, diffuse bispectrum produces high entropy, suggesting either Gaussian noise dominance or a lack of exploitable quadratic phase coupling. This single scalar value enables rapid comparison of signal complexity across emitters without requiring full bispectral template matching.

To isolate genuine phase coupling from broadband energy fluctuations, bispectral entropy is typically computed on the bicoherence rather than the raw bispectrum. Bicoherence normalizes each bifrequency magnitude by the product of the corresponding power spectrum values, yielding a bounded measure between 0 and 1 that reflects the proportion of quadratically phase-coupled energy. Computing entropy on this normalized representation ensures the metric responds to structural complexity rather than arbitrary amplitude scaling, making it robust to variations in received signal strength and enabling fair comparison across different emitters and channel conditions.

A defining property of bispectral entropy is its theoretical insensitivity to additive Gaussian noise. Because the bispectrum of a Gaussian process is identically zero across all bifrequency pairs, Gaussian noise contributes no structure to the bispectral distribution. The entropy computation therefore reflects only the non-Gaussian signal components arising from transmitter hardware non-linearities. This property makes bispectral entropy particularly valuable in low-SNR environments where traditional power-spectrum-based features are corrupted, allowing extraction of device-specific signatures buried below the noise floor.

Rather than computing a single global entropy, practitioners often partition the bifrequency plane into distinct regions of interest and compute localized entropy values. The bispectrum's symmetry properties define 12 sectors in the full plane, with the principal domain—the triangular region bounded by f1 ≥ 0, f2 ≥ 0, and f1 + f2 ≤ Nyquist—containing all non-redundant information. Computing entropy within specific sub-bands or along diagonal slices isolates non-linear interactions associated with particular hardware subsystems, such as power amplifier memory effects in low-frequency regions versus mixer intermodulation products at higher bifrequencies.

Bispectral entropy serves as a compact feature vector component for machine learning classifiers performing emitter identification. When combined with complementary higher-order statistics—such as kurtosis, skewness, and cumulant-based features—bispectral entropy provides a dimensionally efficient summary of non-linear signal structure. Typical feature engineering pipelines compute entropy across multiple bifrequency sub-regions and across successive signal segments, producing a time-series of complexity values that captures both static hardware signatures and dynamic non-linear behavior. This approach avoids the computational burden of full bispectral template matching while retaining strong discriminative power.

Bispectral entropy is uniquely sensitive to phase randomization—a deliberate countermeasure where a transmitter introduces random phase dithering to obscure its fingerprint. While such randomization preserves the power spectrum (second-order statistics), it disrupts the consistent quadratic phase coupling that produces structured bispectral peaks. The resulting bispectrum becomes more diffuse, causing a measurable increase in bispectral entropy. This property enables detection of active spoofing attempts and distinguishes between genuine hardware impairments and artificially injected randomness, making bispectral entropy a valuable metric within adversarial device spoofing detection frameworks.