Transient Higher-Order Statistics

A statistical measure of the asymmetry of the transient signal's amplitude probability density function. It reveals directional biases in the hardware's non-linear response.

• Positive skewness: Indicates the amplitude distribution has a longer tail on the positive side, often caused by overshoot in the power amplifier ramp.
• Negative skewness: Indicates a longer tail on the negative side, potentially revealing asymmetrical slew rates in the modulator.
• Zero skewness: Suggests a symmetric distribution, typical of ideal Gaussian noise, but rare in real hardware transients.

Skewness is a third-order statistic, making it inherently sensitive to the non-linearities that distinguish one transmitter from another.

A higher-order statistical measure quantifying the peakedness and tailedness of the transient signal's amplitude distribution relative to a Gaussian distribution.

• High kurtosis (leptokurtic): Indicates a sharp central peak and heavy tails, characteristic of impulsive, non-Gaussian artifacts like DAC glitches or ringing.
• Low kurtosis (platykurtic): Suggests a flatter, more uniform distribution, potentially from saturation effects in the amplifier.
• Excess kurtosis: The standard metric, where a Gaussian distribution has an excess kurtosis of 0. Transient hardware signatures typically exhibit positive excess kurtosis.

Kurtosis is a fourth-order statistic, effectively capturing the impulsive energy of transient events.

The specific use of cumulants to isolate the deterministic non-linear signatures of transmitter hardware during the transient. Cumulants are higher-order statistics that are theoretically blind to Gaussian noise.

• Second-order cumulant: Equivalent to variance, measuring signal power.
• Third-order cumulant: Related to skewness, revealing asymmetric non-linearities.
• Fourth-order cumulant: Related to kurtosis minus 3, isolating non-Gaussian impulsive components while suppressing Gaussian background noise.

This property makes cumulant analysis exceptionally powerful for extracting a clean transient fingerprint from noisy real-world captures, as thermal noise and many channel effects are Gaussian.

A higher-order spectral analysis technique that reveals quadratic phase coupling within the transient signal. It is computed as the two-dimensional Fourier transform of the third-order cumulant sequence.

• Effectively suppresses Gaussian noise, as the bispectrum of a Gaussian process is identically zero.
• Detects non-linear interactions where two frequency components interact to generate a third, a hallmark of power amplifier non-linearity and memory effects.
• Serves as a rich, two-dimensional feature map for device identification, capturing the non-linear signal structure that is invisible to standard power spectral density analysis.

The bispectrum provides a robust, noise-immune representation of the transient's non-Gaussian hardware signature.

A feature vector derived from a cascade of wavelet transforms and modulus non-linearities. It provides a translation-invariant and stable representation of the transient signal's structure.

• Built by iteratively applying wavelet decompositions, taking the modulus, and averaging, which preserves high-frequency information lost in simple averaging.
• Invariant to small time-shifts and deformations, making it robust to leading edge jitter and minor alignment errors in burst onset detection.
• Captures the multi-scale, non-linear structure of the transient in a mathematically rigorous framework, bridging the gap between handcrafted features and learned deep learning representations.

It is particularly effective for characterizing the complex, hierarchical nature of transient artifacts like ringing and damped oscillations.