Damped Oscillation Profile

The amplitude of the ringing artifact diminishes according to a precise exponential function, characterized by the time constant (τ). This constant is defined by the relationship τ = 2L/R, where L is the parasitic inductance and R is the total loop resistance in the resonant circuit. The exact decay rate reveals the quality factor (Q) of the unintended tank circuit, with higher Q values indicating a longer, more sustained oscillation that is highly specific to the physical geometry and component tolerances of the transmitter's output stage.

The oscillation frequency is determined by the parasitic inductance (L) and capacitance (C) in the circuit, following the formula f = 1/(2π√LC). This frequency typically falls in the range of tens to hundreds of megahertz, independent of the carrier frequency. Key contributors include:

• Bond wire inductance in the power amplifier package
• Junction capacitance of the output transistor
• PCB trace parasitics in the impedance matching network This frequency serves as a highly discriminative feature for emitter identification.

The ringing artifact is a classic second-order underdamped response to a step excitation (the turn-on or turn-off event). The damping ratio (ζ) is less than 1, causing the signal to overshoot and oscillate before settling. The mathematical model is:

• Peak overshoot is inversely proportional to ζ
• Settling time is proportional to 1/(ζωₙ), where ωₙ is the natural frequency
• Number of observable cycles before the signal falls below the noise floor is a direct function of ζ These parameters are uniquely shaped by the transmitter's power supply decoupling network.

The precise shape of the damped oscillation is a direct manifestation of manufacturing variances in passive components. Even within a 5% tolerance range, the specific combination of actual inductance, capacitance, and resistance values creates a unique signature. This includes:

• Capacitor dielectric absorption affecting the decay profile
• Inductor core material hysteresis introducing non-linearities
• Resistor temperature coefficients causing minor thermal modulation during the ring These microscopic differences are impossible to clone, making the profile a physically unclonable function (PUF).

Because the ringing artifact is a non-stationary event, its analysis requires joint time-frequency techniques. A wavelet transform or short-time Fourier transform (STFT) reveals how the resonant frequency's amplitude decays over time. Key observations include:

• The spectral centroid shifts slightly during the decay due to non-linear damping
• The instantaneous frequency may chirp if the parasitic capacitance is voltage-dependent
• The transient spectral splatter caused by the ringing can interfere with adjacent channels, providing an additional identifying characteristic for spectrum monitoring systems.

The damped oscillation profile is not entirely static; it exhibits a memory effect dependent on the excitation source. The amplitude and initial phase of the ringing are modulated by:

• The slew rate of the turn-on/turn-off edge—a faster edge injects more broadband energy, exciting higher-order modes
• The preceding steady-state power level, which determines the initial energy stored in the parasitic reactances
• Thermal state of the power amplifier, as junction temperature shifts the semiconductor capacitances This history-dependent behavior adds a layer of complexity to the fingerprint, requiring robust feature extraction algorithms.

A damped sinusoidal oscillation superimposed on the transient envelope. It is typically caused by parasitic inductance and capacitance resonating in the transmitter's output matching network or power supply decoupling path. The specific resonant frequency and decay rate of this artifact serve as a distinct hardware signature, directly reflecting the physical geometry and component values of the circuit.

The quantification of the transient amplitude excursion beyond the steady-state level during the ramp-up phase. This is caused by an underdamped response in the power amplifier control loop or bias network. The percentage of overshoot and the subsequent ringing frequency are directly tied to the damping factor (ζ) of the circuit, providing a unique identifier for the transmitter's feedback loop components.

A higher-order spectral analysis technique that reveals quadratic phase coupling within the transient signal. Unlike the power spectrum, the bispectrum effectively suppresses Gaussian noise and highlights non-linear hardware interactions. It is particularly effective for analyzing damped oscillations because it can detect the harmonic coupling generated by the non-linear mixing of the resonant frequency with the carrier.

A feature extracted by decomposing the transient signal using a wavelet basis. This provides joint time-frequency localization, capturing the multi-scale nature of transient events. Wavelet analysis is ideal for damped oscillation profiles because it can precisely localize the time-varying spectral content of the ringing, distinguishing the short-duration, high-frequency burst from the slower, steady-state signal components.

The dependence of the current transient shape on the previous operating state of the transmitter. This is caused by thermal trapping and charge storage in semiconductor materials. A damped oscillation profile can exhibit memory effects where the resonant frequency or decay constant shifts slightly based on the duty cycle of previous bursts, creating a history-dependent signature that adds another layer of uniqueness.