Cyclic Feature Vector

The core of a cyclic feature vector lies in its selection of cycle frequencies (α) . Different modulation schemes exhibit unique periodicities in their statistics.

• BPSK signals show cyclostationarity at α = 2f_c ± k/T_s (twice the carrier frequency plus multiples of the symbol rate).
• QPSK/OQPSK lack second-order features but reveal strong patterns at fourth-order cyclic cumulant frequencies.
• OFDM signals produce distinct features at the cyclic prefix rate, enabling blind identification.

A well-constructed vector samples the Spectral Correlation Function (SCF) or Cyclic Cumulant magnitudes precisely at these discriminating α values, forming a fingerprint unique to each modulation type.

A primary advantage of cyclic feature vectors is their inherent immunity to stationary Gaussian noise and interference.

• Stationary noise has no cyclic features (α = 0 only). By selecting features at non-zero cycle frequencies, the vector automatically rejects background noise.
• This property makes cyclic feature vectors far more robust than simple spectral or constellation-based features in low-SNR environments.
• Higher-order cyclic cumulants extend this robustness to non-Gaussian, symmetrically distributed interference.

The result is a feature set that captures only the structured, modulated signal energy, dramatically improving classifier performance in contested spectrum.

The raw Cyclic Domain Profile is a dense 2D function S(α, f) that is computationally prohibitive to use directly as a classifier input. The cyclic feature vector solves this through strategic reduction:

• Alpha Profile Extraction: Collapsing the SCF along the spectral frequency (f) axis at specific α values produces a 1D vector of correlation magnitudes.
• Peak-Picking: Selecting only the magnitudes at known, modulation-specific cycle frequencies reduces dimensionality from thousands of points to a handful of scalars.
• Statistical Summarization: Features like the Degree of Cyclostationarity or integrated power within a cycle frequency band provide single-value summaries.

This transforms an intractable pattern recognition problem into a low-dimensional, real-time classification task.

Advanced cyclic feature vectors fuse information from multiple statistical orders to resolve ambiguities between modulation schemes.

• Second-order features (from the SCF) efficiently separate BPSK, MSK, and OQPSK but fail to distinguish QPSK from 16-QAM.
• Fourth-order features (from the Cyclic Bispectrum or Cyclic Cumulants) differentiate square QAM constellations by capturing the distribution of signal energy in higher-order moment space.
• Sixth-order features can separate 64-QAM from 256-QAM.

A fused vector concatenating normalized features from orders 2, 4, and 6 provides a hierarchically discriminative signature that a single-order analysis cannot achieve.

Generating cyclic feature vectors in real-time requires efficient spectral correlation estimators. The FAM (FFT Accumulation Method) and SSCA (Strip Spectral Correlation Analyzer) algorithms make this feasible.

• FAM Algorithm: Uses a channelizer and short-time FFTs to compute the SCF with complexity O(N log N), trading spectral resolution for speed.
• SSCA Algorithm: Offers an alternative complexity-resolution trade-off, often preferred when fine frequency resolution is required.
• Both algorithms output the Cyclic Domain Profile from which the feature vector is extracted via peak detection at pre-computed cycle frequencies.

These algorithms enable streaming extraction of cyclic feature vectors on FPGAs and embedded processors for tactical SIGINT applications.

Cyclic feature vectors can be engineered for invariance to nuisance parameters like carrier frequency offset (CFO) and phase rotation, eliminating the need for perfect synchronization before classification.

• The Spectral Coherence Function normalizes the SCF by the signal power, producing a magnitude bounded between 0 and 1 that is insensitive to amplitude scaling.
• Cyclic Cumulant magnitudes are inherently phase-invariant, as the cumulant operation cancels phase rotations.
• CFO shifts the cycle frequencies by a known offset. Feature vectors can be extracted from a bank of cycle frequency bins around the nominal α values to maintain detection despite offset.

This invariance dramatically simplifies the preprocessing pipeline and improves classifier generalization across hardware platforms.