Cyclic Feature Vector

The raw Spectral Correlation Function (SCF) is a dense 2D matrix. The cyclic feature vector compresses this into a manageable 1D vector by sampling the Cyclic Domain Profile (CDP) or spectral coherence magnitude only at specific, informative cyclic frequencies. This reduces computational load for downstream classifiers while preserving the unique cyclostationary signature of the emitter.

The vector's non-zero entries correspond directly to the theoretical cyclic frequencies of the signal's modulation format:

• BPSK: Strong features at twice the carrier offset plus/minus the symbol rate.
• QPSK: Features emerge at four times the carrier offset after a fourth-order nonlinearity.
• OFDM: Distinct peaks at the symbol rate and subcarrier spacing due to the cyclic prefix.

Stationary Gaussian noise has no cyclostationary signature. By constructing the feature vector from spectral coherence (a normalized measure) or cyclic cumulants, the representation becomes inherently robust to additive white Gaussian noise. The signal's periodic statistical structure stands out clearly in the cyclic domain, enabling reliable identification even at low signal-to-noise ratios.

Beyond modulation-induced cyclostationarity, the vector captures subtle periodicities caused by transmitter hardware imperfections:

• I/Q imbalance creates conjugate correlation at specific cyclic frequencies.
• Power amplifier non-linearity generates higher-order cyclic features.
• Clock jitter modulates the symbol rate cyclic frequency. These hardware-specific signatures form the basis for RF fingerprinting and physical layer authentication.

The cyclic feature vector serves as a fixed-length, real-valued input vector ideal for standard classifiers:

• Support Vector Machines (SVMs) for closed-set emitter identification.
• Deep neural networks for learning hierarchical representations from raw cyclic features.
• Open set recognition algorithms that use the vector's statistical distance metrics to reject unknown emitters. The structured nature of the vector ensures consistent feature alignment across training and inference.

While multipath fading distorts the raw SCF magnitude, the spectral coherence function normalizes out flat-fading channel effects. Additionally, the relative ratios between cyclic feature magnitudes at different cyclic frequencies remain stable across varying channel conditions. Domain adversarial training can further enforce channel-invariant representations, ensuring the cyclic feature vector remains a robust identifier regardless of the propagation environment.

The foundational two-dimensional transform for cyclostationary analysis. The SCF measures the spectral correlation density between frequency-shifted versions of a signal, revealing hidden periodicities. A cyclic feature vector is typically a sampled or projected representation of the SCF magnitude at key cyclic frequencies (alpha). It provides a complete map of a signal's second-order periodicity, distinguishing between stationary noise and modulated signals.

A compact, one-dimensional feature vector formed by projecting the Spectral Correlation Function (SCF) magnitude along the cyclic frequency (alpha) axis. The CDP collapses spectral information to focus purely on the strength of periodicity at each cyclic frequency, making it an ideal input for machine learning classifiers. It directly represents the cyclostationary signature used to distinguish between modulation schemes like BPSK, QPSK, and OFDM.

The time-domain counterpart to the SCF. The CAF computes the correlation of a signal with a frequency-shifted version of itself at a specific cyclic frequency (alpha). A non-zero CAF value confirms the presence of cyclostationarity. Feature vectors can be constructed by sampling the CAF magnitude across a range of lag and alpha values, capturing the temporal structure of the signal's periodic statistics.

The FFT Accumulation Method (FAM) is a computationally efficient channelized algorithm for estimating the SCF. It works by decimating the input signal into narrowband frequency bins using an FFT, then correlating the complex envelopes of these bins. The FAM is the standard practical implementation for generating the high-resolution SCF data from which a cyclic feature vector is extracted in real-time systems.

A higher-order statistical function that extracts the purely non-Gaussian periodic components of a signal. Unlike the SCF, cyclic cumulants are theoretically immune to additive Gaussian noise. A feature vector built from cyclic cumulant values at specific orders and cyclic frequencies provides a highly robust identifier for modulation classification, especially in low-SNR environments where second-order features may be obscured.

A normalized version of the SCF that quantifies the degree of spectral correlation on a scale from 0 to 1. By removing the signal's power spectrum bias, spectral coherence provides a scale-invariant feature for a cyclic feature vector. This normalization is critical for ensuring that a machine learning model focuses on the structural cyclostationary signature rather than the received signal strength, improving generalization across varying channel conditions.