Instantaneous Frequency Drift

The primary physical cause of instantaneous frequency drift is the rapid self-heating of the transistor junction during the high-current turn-on event. As the power amplifier draws inrush current, the localized temperature spike alters the semiconductor's electron mobility and parasitic capacitances. This thermal shock directly modulates the resonant frequency of the oscillator tank circuit, causing a characteristic, repeatable frequency trajectory. The drift profile is a direct map of the thermal impedance and thermal time constants of the die and its packaging.

Voltage-controlled oscillator (VCO) pulling is a load-induced frequency shift that occurs when the power amplifier's input impedance changes abruptly during turn-on. The sudden impedance mismatch reflects power back into the VCO, momentarily detuning its resonant circuit. This creates a distinct frequency pushing signature that is unique to the physical layout and impedance matching network of the transmitter. The effect is highly sensitive to the parasitic reactances of the PCB traces connecting the VCO to the amplifier.

The frequency settling profile is the complete time-domain trajectory of the carrier frequency as it converges from its initial unstable state to its steady-state nominal value. This profile is a direct revelation of the phase-locked loop (PLL) loop filter dynamics. Key features include:

• Settling Time: The duration to lock within a specified ppm error.
• Overshoot: The peak frequency excursion beyond the target.
• Damping Factor: The rate of oscillation decay, indicating component tolerances in the loop filter.

The transient inrush current during turn-on causes a momentary voltage sag on the regulated supply rail due to the equivalent series resistance (ESR) of the decoupling network. Since the VCO's frequency is a function of its tuning voltage, any ripple or sag on the power supply directly frequency-modulates the output carrier. This creates a drift component that is a direct signature of the power distribution network's impedance and the specific decoupling capacitor characteristics.

The instantaneous frequency drift is mathematically the first derivative of the transient phase trajectory. By plotting the signal's instantaneous phase in the complex IQ plane during the transient, the drift manifests as a non-linear, time-varying angular velocity. This trajectory reveals the underlying non-linear differential equations governing the oscillator's start-up dynamics, providing a high-dimensional feature space for deep learning-based fingerprinting models.

A precise time-domain technique for extracting instantaneous frequency drift involves zero-crossing analysis. By measuring the exact time intervals between consecutive positive-going zero-voltage crossings of the captured transient waveform, the instantaneous period and thus the instantaneous frequency can be calculated. This method provides a sample-by-sample frequency trajectory without the time-frequency resolution trade-offs inherent in Fourier-based methods, capturing the fine-grained dynamics of the drift.

The dynamic behavior of the voltage-controlled oscillator during start-up, encompassing both frequency pushing and pulling effects. As the power supply stabilizes and the PLL acquires lock, the VCO's instantaneous frequency traces a unique trajectory.

• Frequency pushing: Carrier shift due to supply voltage variation during inrush current
• Frequency pulling: Carrier shift due to changing load impedance as the power amplifier activates
• The combined response imprints a device-specific signature on the carrier's instantaneous frequency

The complete trajectory of the instantaneous carrier frequency as it converges to its steady-state value after activation. This profile reveals the loop filter characteristics of the phase-locked loop.

• Captures the entire dynamic response: initial error, overshoot, ringing, and final convergence
• The settling time to within a specified tolerance (e.g., ±1 kHz) is a key discriminative feature
• Heavily influenced by component tolerances in the loop filter: resistors, capacitors, and charge pump current

The peak frequency excursion beyond the target lock frequency during the phase-locked loop's acquisition process. This transient overshoot is a direct indicator of the loop's damping factor.

• Underdamped loops exhibit significant overshoot and ringing before settling
• The overshoot magnitude and the number of ringing cycles form a hardware-specific signature
• Caused by tolerance variations in loop filter components, particularly the damping resistor

The path traced by the instantaneous phase of the signal in the complex (I/Q) plane during the transient period. While frequency drift captures the rate of phase change, the phase trajectory visualizes the underlying oscillator dynamics.

• A spiral converging to a steady-state rotation represents a frequency settling event
• Phase discontinuities appear as abrupt angular jumps in the trajectory
• The trajectory shape reveals the order and type of the transmitter's synthesis chain

A temporary elevation in the phase noise spectrum of the local oscillator during the transient locking period. Before the loop stabilizes, the oscillator exhibits significantly higher short-term frequency fluctuations.

• The noise burst duration correlates with the PLL lock time
• The spectral shape of the burst differs from steady-state phase noise due to the loop's dynamic response
• Provides a rich, high-dimensional feature for distinguishing identical transmitter models

A time-domain technique for extracting instantaneous frequency information from a transient by measuring the precise intervals between consecutive zero-voltage crossing points of the waveform.

• The reciprocal of twice the zero-crossing interval gives the instantaneous frequency at that moment
• Highly sensitive to phase noise and jitter, making it effective for capturing subtle hardware variations
• Computationally efficient compared to full time-frequency transforms, suitable for real-time embedded implementation