Zero-Crossing Analysis

Zero-crossing analysis operates by detecting the exact temporal points where a waveform's amplitude transitions from negative to positive (or vice versa). The time interval between successive zero-crossings directly corresponds to the half-period of the signal, allowing for the calculation of instantaneous frequency as the reciprocal of twice this interval. This method is computationally efficient and provides a direct, sample-by-sample frequency trajectory without requiring a full Fourier transform.

An abrupt phase discontinuity during a turn-on or turn-off transient manifests as a sudden, non-uniform shift in the zero-crossing pattern. Instead of a smooth change in interval duration, a discontinuity creates an anomalously long or short half-period. By analyzing the instantaneous phase trajectory derived from cumulative crossing times, one can precisely quantify the magnitude and timing of these phase jumps, which are unique identifiers of the transmitter's switching hardware.

The primary vulnerability of zero-crossing analysis is its sensitivity to leading edge jitter and broadband noise. Spurious multiple crossings near the zero-axis, caused by noise, can create false triggers and corrupt the instantaneous frequency estimate. Mitigation strategies include:

• Hysteresis band: Implementing a Schmitt trigger logic to ignore small-amplitude noise fluctuations.
• Interpolation: Using linear or sinc interpolation between samples to refine the exact crossing instant to sub-sample precision.
• Pre-filtering: Applying a bandpass filter to isolate the carrier and suppress out-of-band noise.

The sequence of zero-crossing intervals forms a time series that can be mined for transient fingerprint features. Statistical moments calculated on this interval sequence are highly discriminative:

• Mean and variance of the instantaneous frequency during the transient.
• Transient kurtosis of the interval distribution, capturing the peakedness caused by ringing artifacts.
• Differential interval analysis, which examines the rate of change of the frequency trajectory to isolate the PLL's loop filter response.

While the Hilbert transform provides a continuous analytic signal for extracting instantaneous frequency, zero-crossing analysis offers a discrete, non-parametric alternative. Key trade-offs include:

• Computational load: Zero-crossing is far less computationally intensive, suitable for edge AI deployment.
• Bandwidth limitation: The Hilbert transform handles wideband signals more gracefully, whereas zero-crossing assumes a quasi-monochromatic signal.
• Transient attack profiles: Zero-crossing excels at capturing the rapid, non-linear frequency shifts during the initial VCO pulling phase of a burst onset.

The primary output of zero-crossing analysis, where the reciprocal of the interval between consecutive zero-crossings provides a sample of the signal's instantaneous frequency. This creates a time-varying frequency trajectory that reveals the dynamic behavior of a transmitter's oscillator during start-up. Unlike Fourier methods, this time-domain approach preserves the exact temporal evolution of frequency settling, making it ideal for capturing the unique phase-locked loop (PLL) lock-in profile of a specific device.

A complementary technique that computes the analytic signal to extract the instantaneous amplitude envelope without the distortion of carrier cycles. While zero-crossing analysis focuses on frequency, the Hilbert transform provides the precise amplitude ramp profile. Together, they form a complete picture of the transient's attack, decay, sustain, and release (ADSR) characteristics, capturing both the power amplifier's slew rate and the oscillator's settling behavior simultaneously.

The path traced by the instantaneous phase in the complex (I/Q) plane during the transient period. By integrating the instantaneous frequency derived from zero-crossing intervals, one can reconstruct the phase discontinuity and phase settling profile. This trajectory reveals the underlying dynamics of the transmitter's modulator and oscillator, exposing non-linear interactions that are invisible to amplitude-only analysis.

A critical pre-processing step that precisely locates the temporal boundary where a transmission transitions from the noise floor to an active state. Accurate onset detection is essential for zero-crossing analysis because the leading edge jitter and initial frequency error are among the most distinctive hardware-specific features. Algorithms typically use energy thresholding or matched filtering to isolate the exact start of the transient for subsequent zero-crossing measurement.

A feature extracted by decomposing the transient signal using a wavelet basis, providing joint time-frequency localization. While zero-crossing analysis offers high temporal precision, wavelet transforms capture the multi-scale nature of transient events, such as the simultaneous fast ringing artifact and slow thermal drift. This multi-resolution approach complements zero-crossing data by isolating the damped oscillation profile from the underlying frequency settling trajectory.

The complete time-domain response of a phase-locked loop as it acquires lock, including frequency overshoot and phase error convergence. Zero-crossing analysis is the primary tool for measuring this transient, as it directly quantifies the instantaneous frequency deviation from the target carrier. The resulting frequency settling profile is highly dependent on component tolerances in the loop filter, making it a robust, unclonable hardware signature for device fingerprinting.