Bicoherence

Unlike the raw bispectrum, bicoherence is normalized to a [0,1] range, where 1.0 indicates perfect quadratic phase coupling and 0.0 indicates complete absence. This normalization divides the bispectrum by the power spectrum product at the constituent frequencies, removing amplitude dependence and isolating phase relationships. The bounded scale enables threshold-based detection of non-linearities without amplitude bias, making it directly comparable across different signals and devices.

Bicoherence specifically identifies quadratic phase coupling—the phenomenon where two frequency components f1 and f2 interact non-linearly to generate a third at f1+f2 with a consistent phase relationship. This coupling is a hallmark of non-linear system behavior and manifests in transmitters through:

• Power amplifier saturation effects
• Mixer intermodulation products
• Oscillator harmonic generation Coupled components exhibit high bicoherence, while spontaneously generated frequencies remain near zero.

Bicoherence theoretically vanishes for Gaussian processes, providing inherent noise immunity. Since Gaussian noise has zero bispectrum, the normalization drives bicoherence toward zero in noise-dominated regions. This property enables detection of non-Gaussian signal features buried below the noise floor. In RF fingerprinting, this means hardware-induced non-linearities remain detectable even at low signal-to-noise ratios where conventional power spectrum analysis fails.

Bicoherence maps coupling across the bifrequency plane (f1, f2), revealing which frequency pairs participate in non-linear interactions. Key regions include:

• Diagonal (f1=f2): Harmonic generation coupling
• Off-diagonal: Intermodulation product coupling
• Sum-frequency axis: Direct quadratic interaction evidence This spatial resolution allows identification of specific non-linear mechanisms within transmitter hardware, creating distinctive two-dimensional signatures for emitter classification.

Bicoherence estimation requires segment averaging to achieve statistical consistency. The variance scales inversely with the number of averaged segments, creating a fundamental trade-off:

• More segments: Lower variance, higher reliability
• Fewer segments: Higher frequency resolution, more noise Typical implementations use Welch's overlapped segment averaging with 50-75% overlap. For transient signal analysis, the limited observation window constrains averaging, requiring careful bias-variance optimization.

A critical validation technique applies phase randomization to distinguish true coupling from statistical artifacts. By destroying phase relationships while preserving the power spectrum, surrogate data sets establish a null hypothesis of no coupling. Bicoherence values exceeding the 95th percentile of surrogate distributions indicate statistically significant quadratic phase coupling. This non-parametric test prevents false positives in automated emitter identification pipelines.

Bicoherence directly quantifies the quadratic phase coupling generated by power amplifier non-linearities. When a transmitter's amplifier operates near saturation, it produces harmonic and intermodulation products with consistent phase relationships. The bicoherence magnitude at specific bifrequency pairs serves as a hardware-specific signature because manufacturing variances in transistor characteristics create unique, repeatable coupling patterns. Unlike the raw bispectrum, bicoherence normalizes by signal power, making it robust to varying transmission amplitudes and enabling cross-session device matching.

A critical advantage of bicoherence in RF analysis is its theoretical insensitivity to additive Gaussian noise. While second-order statistics like power spectral density are corrupted by Gaussian interference, bicoherence converges to zero for Gaussian processes. This property allows analysts to detect weak non-linear signatures buried below the noise floor. In practical emitter identification, bicoherence extracts hardware-induced phase coupling even at low signal-to-noise ratios where conventional fingerprinting methods fail, making it invaluable for passive signals intelligence collection.

Bicoherence-based features exhibit modulation transparency, meaning the quadratic coupling patterns remain consistent regardless of the information-bearing modulation scheme. This occurs because phase coupling originates in the analog transmitter chain—amplifiers, mixers, and oscillators—rather than the digital baseband modulation. Analysts can therefore identify emitters even when they switch between QPSK, 16-QAM, or OFDM waveforms. This property is essential for tracking agile radios that employ multiple modulation formats across frequency-hopping patterns.

The bicoherence spectrum creates a two-dimensional bifrequency map where each coordinate pair (f1, f2) represents a potential quadratic coupling interaction. Peaks in this map indicate frequencies where phase coupling is statistically significant. Analysts construct bicoherence feature vectors by extracting peak locations, magnitudes, and contour statistics from these maps. These compact representations serve as input to machine learning classifiers, enabling automated emitter sorting with reduced dimensionality compared to raw bispectral data while preserving discriminative non-linear information.

Bicoherence excels at analyzing transient signal periods—the brief turn-on and turn-off intervals of burst transmissions. During these transitions, amplifier bias circuits and oscillator stabilization loops exhibit pronounced non-linear behavior before reaching steady-state. The bicoherence computed over short-duration transients reveals unique coupling patterns specific to each device's power supply and thermal characteristics. These transient bicoherence signatures are particularly difficult to spoof because they depend on analog component physics rather than configurable digital parameters.

Bicoherence values require statistical validation to distinguish genuine phase coupling from estimation variance. Analysts apply threshold tests based on the number of averaged segments: for N segments, the 95% confidence threshold for zero bicoherence approximates √(6/N). Values exceeding this threshold indicate statistically significant quadratic coupling. This rigorous statistical framework prevents false emitter associations caused by random phase alignments, ensuring that bicoherence-based fingerprinting maintains high specificity in dense electromagnetic environments with multiple co-channel emitters.