Integrated Polyspectrum

Computes the integrated polyspectrum by summing bispectral or trispectral values along radial lines emanating from the origin in the bifrequency plane. This approach captures scale-invariant phase coupling information.

• Preserves non-Gaussian signatures while collapsing 2D data to 1D
• Each radial angle corresponds to a specific frequency ratio
• Particularly effective for detecting quadratic phase coupling at harmonically related frequencies
• Output is a function of radial angle, forming a compact feature vector

Projects polyspectral data onto principal axes by integrating along lines parallel to frequency axes. This technique isolates marginal non-Gaussian distributions that characterize specific transmitter impairments.

• Integrates along constant f1 or f2 lines in the bifrequency domain
• Reveals amplifier non-linearity patterns concentrated along axes
• Computationally efficient—requires only 1D integration operations
• Produces feature vectors directly compatible with standard classifiers

Extracts the diagonal slice of the bispectrum where f1 = f2, reducing the 2D bispectrum to a 1D function. This slice captures self-phase coupling—interactions where a frequency couples with itself.

• Most computationally efficient integration path
• Retains key non-Gaussian information for many emitter types
• Diagonal bispectrum is real-valued, simplifying feature storage
• Often used as a baseline before applying more complex integration schemes

Integrates bispectral values along concentric circles centered at the origin, producing a function of radial frequency. This representation is rotation-invariant in the bifrequency plane.

• Robust to certain classes of channel distortion
• Each circular integral captures phase coupling at a specific frequency magnitude
• Reduces sensitivity to frequency misalignment between training and test data
• Forms a compact 1D feature vector indexed by radius

Transforms the O(N²) bispectrum or O(N³) trispectrum into O(N) feature vectors through integration, enabling real-time implementation on resource-constrained platforms.

• Reduces bispectrum from N×N grid to N-point vector via diagonal slicing
• Radial integration produces M-point vectors where M << N²
• Enables deployment on FPGA and embedded SDR platforms
• Maintains classification accuracy while reducing feature dimensionality by orders of magnitude

The choice of integration path—radial, axial, diagonal, or circular—depends on the specific hardware impairment being targeted and the expected channel conditions.

• Diagonal slices: Best for self-phase coupling from power amplifier non-linearities
• Radial integration: Captures harmonic relationships from mixer imperfections
• Axial integration: Isolates I/Q imbalance signatures concentrated along frequency axes
• Circular integration: Preferred when frequency alignment cannot be guaranteed
• Path selection is often optimized via cross-validation on representative emitter datasets