Transient Envelope Analysis

The mathematical foundation for extracting the envelope without carrier-cycle distortion. The Hilbert transform creates a complex analytic signal by shifting the phase of the original real signal by 90 degrees.

• Instantaneous Magnitude: The absolute value of the analytic signal defines the precise envelope.
• Carrier Removal: This process separates the slow-varying amplitude contour from the fast-varying RF oscillations.
• Phase Trajectory: Simultaneously reveals the instantaneous phase, exposing phase discontinuities during the transient.

Borrowed from audio synthesis, the ADSR model segments the transient envelope into distinct, measurable phases.

• Attack: The duration and slope of the initial energy rise from the noise floor to peak power.
• Decay: The brief dip following the peak as the power amplifier bias settles.
• Sustain: The steady-state amplitude level, though often absent in very short bursts.
• Release: The ramp-down signature as energy storage elements discharge.

Quantifying the non-ideal amplitude excursions relative to the steady-state level reveals underdamped control loops.

• Overshoot: The percentage peak amplitude exceeding the nominal sustain level during the attack phase, caused by the power amplifier's control loop.
• Undershoot: The amplitude dip below the baseline immediately following the release phase, reflecting power supply reverse recovery.
• Damping Factor: The rate at which these oscillations decay is a direct hardware signature.

Parasitic inductance and capacitance in the transmitter's output matching network resonate upon switching, creating a damped sinusoidal artifact.

• Resonant Frequency: The specific frequency of the ringing is determined by the LC tank circuit formed by parasitic components.
• Decay Constant (Tau): The exponential decay envelope of the ringing is a unique identifier of the circuit's Q-factor.
• Feature Extraction: Wavelet transforms are often used to isolate this non-stationary, multi-scale feature from the main ramp.

The precise duration required for the transmitter to stabilize within a specified tolerance after activation.

• Amplitude Settling: The time until the envelope remains within ±1% of the final steady-state value.
• Frequency Settling: The trajectory of the instantaneous frequency as the Phase-Locked Loop (PLL) acquires lock.
• PLL Dynamics: The settling profile reveals the loop filter characteristics, including PLL overshoot and damping, which vary due to component tolerances.

The statistical distribution of the 10%-90% transition times across multiple bursts provides a stochastic hardware metric.

• Rise-Time Variance: Reflects the randomness in the power-up sequence, including clock jitter and bias network charging.
• Fall-Time Variance: Indicates the discharge path impedances and power supply holdup capacitance inconsistencies.
• Histogram Analysis: The shape of the variance distribution (Gaussian vs. skewed) is itself a discriminative feature for emitter identification.