Wavelet Scattering Network

WSNs achieve translation invariance through a final global averaging operation, ensuring that the representation of a signal remains stable even if the signal is shifted in time. This is critical for RF fingerprinting, where the precise start time of a captured burst is often arbitrary. Unlike adaptive max-pooling in CNNs, this invariance is a deterministic, mathematically guaranteed property derived from the low-pass filtering of the scattering domain.

The representation is Lipschitz continuous with respect to small diffeomorphisms, meaning minor time-warping deformations caused by channel Doppler effects or local oscillator drift produce linearly bounded changes in the feature space. This property, proven mathematically, ensures that the subtle, device-specific hardware impairment signatures are not corrupted by small, non-linear variations in the signal's temporal structure, providing robustness that learned CNN filters must approximate from data.

The scattering transform is a contactive operator that conserves signal energy while discarding non-informative phase information at each layer. The network's output coefficients decompose the signal's energy across paths of different scales and orders, providing a complete, interpretable representation. This ensures that the discriminating power of a power amplifier's non-linear memory effects is captured and distributed across multiple scattering paths without being lost.

Unlike deep CNNs, the filters in a WSN are fixed wavelet filters (typically Morlet or Gammatone wavelets), not learned from data. This provides a powerful, generic signal representation that does not require backpropagation or large labeled datasets for training the front-end. The resulting scattering coefficients can be fed into a simple, shallow classifier like an SVM, making the system highly data-efficient and resistant to overfitting on small Specific Emitter Identification (SEI) datasets.

The cascaded architecture computes a deep convolutional network by iteratively applying wavelet transforms and modulus non-linearities. Each layer recovers high-frequency information lost by the previous layer's averaging, creating a hierarchical representation:

• Layer 1: Captures amplitude modulation and carrier information.
• Layer 2: Captures interactions between frequency components, revealing non-linear intermodulation products.
• Higher Layers: Recover progressively finer, higher-order signal structures, analogous to the higher-order statistics used in Volterra series models.

The first-order scattering coefficients are computed by averaging the signal's wavelet modulus, which effectively acts as a robust, non-linear smoothing operation. This makes the low-order coefficients highly resilient to additive white Gaussian noise (AWGN). The signal's energy is concentrated in stable scattering paths, while noise energy is dispersed, allowing the extraction of a clean Radio Frequency DNA signature even at low signal-to-noise ratios (SNRs) without explicit denoising pre-processing.