Transient Scattering Transform

The scattering transform is designed to be Lipschitz-continuous to translations, meaning a small time-shift in the transient signal results in a proportionally small change in the feature vector. This is achieved through a cascade of wavelet convolutions followed by a modulus non-linearity and a low-pass averaging filter. For transient fingerprinting, this ensures that minor jitter in burst onset detection does not catastrophically alter the extracted identity signature. The representation naturally separates the invariant signal envelope from the high-frequency carrier details.

Unlike raw spectrograms, the scattering representation is stable to small diffeomorphisms—non-linear time-warping deformations. This is critical for analyzing turn-on transients where thermal effects or voltage sag can non-linearly stretch the amplitude ramp profile. The wavelet modulus operator demodulates the signal, pushing high-frequency information to coarser scales. This ensures that a slight change in the rise-time variance due to temperature does not map to a completely different feature vector, enabling robust device recognition under varying environmental conditions.

The scattering transform preserves signal energy while hierarchically separating it across paths. The zeroth-order coefficients capture the local translation-invariant average. The first-order coefficients, computed from wavelet modulus spectrograms, capture the amplitude of transients at specific scales and times. Critically, higher-order coefficients recover information lost by the modulus operator by re-applying wavelets. This captures transient bispectrum-like interactions and the characteristic ringing artifact frequencies without explicitly computing higher-order statistics, providing a complete, non-redundant signature of the damped oscillation profile.

The modulus non-linearity combined with the low-pass filtering provides inherent robustness to additive Gaussian noise, which is essential for capturing low-power transient spectral splatter. The transform contracts small-amplitude perturbations. For a signal contaminated by channel noise, the Euclidean distance between scattering coefficients is bounded by the signal-to-noise ratio. This property allows the transform to focus on the deterministic transient envelope structure—such as the overshoot characterization and settling time analysis—while suppressing the stochastic background noise floor that plagues raw time-domain correlation methods.

By using a filter bank of Morlet or Gammatone wavelets with constant-Q spacing, the scattering transform provides a logarithmic frequency decomposition that mirrors the physics of transient events. Fast components like transient DAC glitches and leading edge jitter are captured by high-frequency, short-duration wavelets. Slower phenomena like PLL settling transients and transient thermal signatures are isolated by low-frequency, long-duration wavelets. This multi-scale representation naturally separates the transient attack profile from the transient decay profile without requiring explicit segmentation.

The scattering transform can be viewed as a pre-defined, non-learned convolutional neural network. The wavelet filters are fixed, not trained, eliminating the need for large labeled datasets of transient fingerprints. The cascade of convolution, modulus, and pooling mirrors the architecture of deep networks, providing a feature space where linear classifiers can separate device identities. This makes it an ideal front-end for few-shot device enrollment scenarios, where only a handful of burst leading edge slope examples are available to train a classifier for a new emitter type.

A deep convolutional network with fixed, pre-defined filters derived from wavelet transforms. Unlike learned CNNs, scattering networks cascade wavelet modulus operators to extract stable, translation-invariant representations without training. The architecture consists of three stages:

• Linear wavelet convolution to detect localized oscillatory patterns
• Complex modulus non-linearity to demodulate and pool high-frequency information
• Averaging via low-pass scaling function to build translation invariance The resulting scattering coefficients form a Lipschitz-stable representation, meaning small signal deformations produce proportionally small changes in the feature vector—critical for robust transient fingerprinting.

The property ensuring that a time-shifted version of the same transient produces an identical or near-identical feature representation. Scattering transforms achieve this through cascaded averaging operations that progressively discard precise temporal localization in favor of stable signal descriptors. This is essential for transient analysis because:

• Burst onset detection algorithms may introduce small timing jitter in window placement
• The same device's turn-on transient may be captured at slightly different trigger points across acquisitions
• Without translation invariance, these micro-shifts would corrupt the fingerprint, causing false-negative authentication failures

A mathematical guarantee that the distance between scattering representations is bounded by a constant multiple of the original signal deformation magnitude. For transient fingerprinting, this means:

• Minor channel distortion (multipath, fading) does not catastrophically alter the feature vector
• Temperature-induced component drift produces proportional, not chaotic, changes in the representation
• The scattering transform acts as a non-expansive operator, preventing noise amplification This stability property distinguishes scattering transforms from raw spectrograms or learned features, which may exhibit brittle sensitivity to small input perturbations.

The most common mother wavelet used in transient scattering transforms, defined as a complex exponential modulated by a Gaussian envelope. Morlet wavelets provide optimal joint time-frequency resolution due to their adherence to the Heisenberg-Gabor uncertainty limit. Key properties for transient analysis:

• Sinusoidal oscillatory shape matches the damped ringing artifacts common in transmitter turn-on events
• Complex-valued output captures both amplitude and phase information simultaneously
• Adjustable center frequency and bandwidth via scale parameter, enabling multi-resolution decomposition The wavelet's center frequency is typically set to 0.8-1.2× the carrier frequency to maximize sensitivity to transient-induced modulation.

The critical demodulation operator applied after each wavelet convolution in the scattering cascade. By taking the complex magnitude of wavelet coefficients, the modulus:

• Removes the carrier phase while preserving the envelope structure of transient artifacts
• Contracts signal energy toward lower frequencies, enabling information propagation to deeper network layers
• Creates invariance to local phase shifts caused by oscillator jitter during the transient period Without the modulus, high-frequency wavelet coefficients would oscillate at twice the carrier rate, preventing effective averaging and destroying the translation-invariant representation. The modulus effectively rectifies the analytic signal, similar to envelope detection in classical RF circuits.

The depth of the scattering cascade, defined by the number of successive wavelet-modulus operations before final averaging. Each order captures different signal characteristics:

• Order 0 (S₀): Simple time-averaged signal, equivalent to the DC component—limited fingerprinting utility
• Order 1 (S₁): First-order scattering coefficients capture amplitude modulation spectra of the transient envelope
• Order 2 (S₂): Second-order coefficients encode interactions between frequency components, revealing non-linear mixing products from power amplifier memory effects Most transient fingerprinting applications use orders 1 and 2, as higher orders suffer from energy decay and diminishing discriminative returns. The total number of scattering paths scales as O(J^m) where J is the number of wavelet scales per octave and m is the order.