Phase Trajectory

The phase trajectory between two consecutive symbols is not instantaneous. Each transmitter exhibits a unique path governed by its pulse-shaping filter and power amplifier memory effect. Key characteristics include:

• Overshoot and ringing: Damped oscillations after a phase change reveal filter Q-factor and component tolerances
• Slew rate asymmetry: The rate of phase change often differs between clockwise and counter-clockwise transitions due to I/Q imbalance
• Settling time: The duration required for the phase to stabilize within a defined error band of the target constellation point

These microsecond-scale dynamics are highly repeatable and form a robust, unclonable fingerprint.

Power amplifiers introduce AM/PM conversion, where the output phase shift varies as a function of the instantaneous signal envelope. This creates a characteristic distortion pattern in the phase trajectory:

• Compression near constellation boundaries: As the signal envelope peaks during symbol transitions, the amplifier operates in its non-linear region, causing a phase lag
• Memory effect hysteresis: Due to thermal and electrical time constants, the phase trajectory for a given transition differs depending on the prior symbol sequence
• Asymmetric trajectories: The phase path from Symbol A to Symbol B is not simply the reverse of the path from B to A

This non-reciprocal behavior provides a rich source of distinguishing features.

During rapid phase changes, the phase-locked loop (PLL) in the local oscillator experiences transient perturbations. These manifest as:

• Micro-fluctuations in the instantaneous frequency during the first few microseconds of a new symbol
• Damped oscillatory convergence toward the target phase, with a frequency and decay rate unique to the PLL loop filter components
• Phase noise pedestal: A temporary broadening of the phase noise profile immediately following a transition

These PLL dynamics are determined by passive component values that vary slightly between nominally identical devices due to manufacturing tolerances.

Rather than analyzing the absolute phase path, differential phase trajectory examines the derivative of the phase over time. This approach:

• Amplifies subtle discontinuities that are invisible in the raw phase plot
• Normalizes carrier frequency offset, removing a confounding variable that varies with temperature
• Highlights inflection points where the trajectory curvature changes sign, creating a sparse set of highly discriminative landmarks

The differential representation is particularly effective when combined with dynamic time warping (DTW) for robust similarity measurement between trajectories of varying durations.

Modern fingerprinting systems treat the phase trajectory as a 2D image—a phase portrait—and apply convolutional neural networks for classification:

• Recurrence plots encode the trajectory into a texture image where pixel intensity represents the distance between phase states at different times
• Gramian angular fields transform the 1D phase sequence into a 2D matrix preserving temporal correlations
• Transfer learning with pre-trained vision models (e.g., ResNet) fine-tuned on phase portraits achieves state-of-the-art accuracy with limited training samples

This image-based approach automatically learns hierarchical features, from low-level trajectory curvature to high-level transition motifs.

Multipath propagation distorts the received phase trajectory, posing a significant challenge for fingerprinting. Mitigation strategies include:

• Channel equalization applied before trajectory extraction to remove linear distortion
• Domain-adversarial neural networks that learn features invariant to channel impulse response variations
• Higher-order statistical features derived from the trajectory's shape statistics (e.g., skewness of curvature distribution) that remain stable across moderate multipath
• Cyclostationary alignment: Synchronizing trajectory extraction to the symbol rate's cyclic frequency to suppress random channel effects

Effective channel-robust feature engineering is the critical differentiator between laboratory demonstrations and field-deployable systems.