Polyspectra

Unlike the power spectrum, which discards all phase information, polyspectra retain both magnitude and phase of the Fourier transform. This is critical because hardware impairments manifest as consistent, non-random phase couplings. The bispectrum preserves quadratic phase coupling, while the trispectrum captures cubic interactions. This phase sensitivity enables discrimination between signals with identical power spectra but different non-linear distortion profiles.

A defining property of polyspectra is their theoretical insensitivity to Gaussian noise. All cumulants of order greater than two are identically zero for Gaussian processes. By computing the bispectrum or trispectrum, additive white Gaussian noise is suppressed, revealing the non-Gaussian signal components generated by transmitter hardware imperfections. This allows feature extraction at signal-to-noise ratios where conventional power spectrum analysis fails.

Polyspectra directly detect and characterize quadratic phase coupling (QPC) and higher-order non-linear interactions. When a power amplifier exhibits non-linear behavior, it generates harmonics and intermodulation products with specific phase relationships. The bicoherence—a normalized bispectrum—quantifies the proportion of energy at a bifrequency pair that is quadratically phase-coupled, providing a bounded metric between 0 and 1 for non-linearity assessment.

Full polyspectral representations are computationally intensive and high-dimensional. Several reduction strategies exist:

• Diagonal slice spectrum: A 1D projection along the bispectrum's diagonal axis
• Integrated polyspectrum: Radial or axial integration condensing higher-order information
• Cumulant tensor decomposition: Multi-linear factorization using HOSVD to extract compact feature vectors These techniques preserve discriminative non-Gaussian signatures while enabling real-time implementation on embedded platforms.

Higher-order cumulants and polyspectra form the mathematical foundation for Independent Component Analysis (ICA) and Joint Cumulant Diagonalization. When multiple emitters occupy the same frequency band, their statistical independence—measured through higher-order cross-cumulants—enables blind separation without prior knowledge of channel conditions. This property is essential for co-channel emitter identification in dense electromagnetic environments.

Polyspectra can be extended to incorporate cyclostationary analysis, producing cyclic cumulants and cyclic polyspectra. Communication signals exhibit periodicity in their statistics due to modulation, coding, and framing structures. By computing polyspectra as functions of both frequency and cycle frequency, the resulting representations capture both non-Gaussianity and temporal periodicity—doubly robust features for modulation recognition and device identification.

The third-order frequency-domain representation that detects quadratic phase coupling between signal components. Unlike the power spectrum, which discards phase information, the bispectrum reveals non-linear interactions where two frequencies combine to generate a third at their sum or difference. This property is critical for identifying transmitter hardware impairments that manifest as non-linear mixing products.

A normalized bispectrum that measures the proportion of signal energy at a bifrequency pair that is quadratically phase-coupled. Bicoherence provides a bounded metric between 0 and 1, enabling consistent comparison across signals of varying power levels. It is particularly useful for distinguishing genuine non-linear hardware interactions from spurious statistical correlations.

Statistical measures beyond second-order variance that quantify deviations from Gaussianity in signal distributions. Key cumulants include:

• Skewness (3rd order): asymmetry in amplitude distribution
• Kurtosis (4th order): tailedness of the distribution These form the mathematical foundation for polyspectral analysis and are theoretically insensitive to additive Gaussian noise.

A non-linear phenomenon where two frequency components interact to generate a third at their sum or difference frequency. This coupling arises from hardware non-linearities such as amplifier compression and mixer imperfections. The resulting phase-locked frequency triplets serve as distinctive, unclonable fingerprints that persist across varying modulation schemes.

A multi-dimensional array organizing higher-order cumulants that enables joint blind source separation and feature extraction through tensor decomposition techniques. By preserving the multi-linear structure of statistical dependencies, cumulant tensors allow algorithms like HOSVD to compress polyspectral information into compact, discriminative feature sets for emitter classification.

The exploitation of higher-order statistics' theoretical insensitivity to Gaussian processes to extract non-Gaussian signal features buried below the noise floor. Since the central limit theorem causes thermal noise to converge to a Gaussian distribution, polyspectral techniques naturally filter this noise while preserving the non-Gaussian signatures of hardware impairments, enabling operation at very low signal-to-noise ratios.