Transient Differential Constellation

The foundation of this technique is the calculation of ΔIQ, the complex difference between consecutive IQ samples. For a sampled complex signal s[n] = I[n] + jQ[n], the differential signal is d[n] = s[n] - s[n-1]. This operation acts as a high-pass filter, removing static DC offsets and emphasizing the rate of change of the signal state. The resulting vector magnitude |d[n]| represents the instantaneous speed of the state trajectory, while its angle ∠d[n] indicates the instantaneous direction of movement in the complex plane.

Plotting the imaginary part of d[n] against its real part creates the differential constellation diagram. Unlike a standard constellation, which shows static symbol targets, this plot reveals the continuous path the signal takes as it stabilizes. Key visual features include:

• Spiral convergence: A characteristic inward spiral as the oscillator locks and amplitude settles
• Radial spikes: Indicative of momentary phase discontinuities or DAC glitches
• Asymmetric lobes: Revealing I/Q imbalance during the transient period
• Origin crossings: Showing the trajectory passing through zero, highlighting DC offset dynamics

The differential constellation is a rich source of quantifiable, device-specific features. Common extracted metrics include:

• Centroid drift: The movement of the cluster center over time, reflecting transient DC offset settling
• Dispersion radius: The standard deviation of d[n] magnitude, quantifying the transient phase noise and amplitude jitter
• Angular velocity profile: The rate of phase change, directly mapping the instantaneous frequency drift of the VCO
• Kurtosis of the differential magnitude: A higher-order statistic that is highly sensitive to the impulsive ringing artifacts and non-Gaussian glitches unique to each transmitter's power supply and matching network

A critical characteristic of the differential operation is its amplification of high-frequency noise and non-ideal artifacts. While steady-state constellations average out random jitter, the differential view makes transient clock jitter and DAC glitch energy dominant features. This is a double-edged sword: it provides a magnified view of the very hardware impairments used for fingerprinting, but it also requires careful burst onset detection to avoid processing pure noise before the signal appears. Pre-filtering with a transient matched filter is often employed to maximize the signal-to-noise ratio of the differential trajectory.

The differential constellation offers inherent resilience to certain channel impairments. Since it operates on the difference between adjacent samples, slow-varying channel effects like frequency-selective fading are partially mitigated. The operation subtracts out the common phase rotation between closely spaced samples. However, it remains sensitive to co-channel interference and fast Doppler spread, which can introduce spurious vectors that distort the true hardware trajectory. This makes it a preferred feature space for line-of-sight or short-range device authentication scenarios.

It is vital to distinguish this from a standard steady-state constellation. A steady-state plot shows the final, settled symbol points after the transient has concluded. The transient differential constellation captures the journey to those points. For a QPSK signal, the steady-state shows four tight clusters. The differential view shows a continuous curve starting from the origin, spiraling and arcing as the PLL settling transient and power amplifier ramp signature converge toward those four target locations. This trajectory is a direct window into the control loop dynamics and is far more unique than the final static endpoint.