Transient Wavelet Coefficient

Wavelet analysis decomposes a transient signal into approximation and detail coefficients at multiple scales. This reveals hardware artifacts that exist at different time durations simultaneously:

• Low scales (high frequencies): Capture fast ringing artifacts and phase discontinuities
• High scales (low frequencies): Characterize the amplitude ramp profile and settling behavior
• Each scale isolates a specific physical mechanism in the transmitter chain

Unlike the Fourier transform, which loses all temporal information, wavelet coefficients preserve when a spectral event occurs. This is critical for transient analysis because:

• Phase-locked loop settling occurs in a specific temporal window after burst onset
• Power amplifier overshoot is localized to the first few microseconds of the ramp
• The coefficient magnitude at a given scale and translation directly maps to a specific hardware impairment at a specific moment

The choice of mother wavelet directly impacts the discriminability of extracted features. Key considerations include:

• Daubechies wavelets: Excellent for capturing the abrupt discontinuities at burst onset and offset
• Symlets: Provide near-symmetric shape matching for ringing artifacts with minimal phase distortion
• Morlet wavelets: Offer optimal joint time-frequency resolution for analyzing damped oscillation profiles
• Matching the wavelet shape to the expected transient morphology maximizes coefficient sparsity

The distribution of wavelet energy across scales forms a compact device fingerprint. This is computed as the squared magnitude of detail coefficients at each decomposition level:

• Energy concentration at fine scales indicates a transmitter with fast switching transients and high-frequency ringing
• Energy spread across coarse scales reveals slow power supply modulation and thermal settling effects
• The normalized energy vector is translation-invariant, making it robust to burst onset detection jitter

Wavelet coefficient thresholding separates the deterministic hardware fingerprint from stochastic channel noise. The process involves:

• Hard thresholding: Zeroing coefficients below a noise floor estimate, preserving transient edges
• Soft thresholding: Shrinking coefficients toward zero for smoother reconstruction
• Level-dependent thresholds: Applying different cutoff values per scale, as noise energy is typically concentrated at fine scales
• This preprocessing step ensures that subsequent classification models train on hardware-intrinsic features rather than environmental artifacts

Transient wavelet coefficients exhibit stability to small time shifts, a critical property for real-world capture scenarios where burst onset detection may have microsecond-level jitter:

• Scattering transforms extend this by cascading wavelet decompositions with modulus operations, achieving full translation invariance
• Coefficient decimation at coarser scales naturally provides shift-tolerance
• This stability ensures that the same device produces consistent feature vectors across multiple captures, even with imperfect triggering