Transient Kurtosis

Transient kurtosis is the fourth standardized moment of the signal's amplitude probability density function (PDF) computed over the short-duration transient window. It measures the propensity of the signal to produce extreme amplitude excursions relative to a Gaussian distribution.

• Excess kurtosis = kurtosis − 3, where a Gaussian distribution has kurtosis = 3
• Leptokurtic (kurtosis > 3): Heavy-tailed, peaked distribution indicating impulsive artifacts
• Platykurtic (kurtosis < 3): Light-tailed, flat distribution indicating bounded or compressed amplitudes
• Mesokurtic (kurtosis = 3): Gaussian reference, rare in real transient hardware signatures

Transient kurtosis excels at detecting impulsive hardware artifacts that are invisible to second-order statistics like variance. Power amplifier turn-on spikes, DAC glitches, and synthesizer switching events produce heavy-tailed amplitude distributions.

• DAC glitch energy: Momentary code-dependent spikes at burst onset create extreme positive kurtosis values
• VCO ringing: Damped sinusoidal oscillations during PLL lock produce characteristic kurtosis signatures
• Ground bounce: Simultaneous switching noise injects impulsive transients onto the signal envelope
• Power supply inrush: The instantaneous current surge modulates the RF envelope, generating non-Gaussian tails

A critical advantage of transient kurtosis is its theoretical blindness to Gaussian noise. While additive white Gaussian noise (AWGN) corrupts variance and RMS measurements, it contributes a constant kurtosis value of 3, leaving the non-Gaussian hardware signature exposed.

• Cumulant equivalence: The 4th-order cumulant equals kurtosis minus 3×variance², zeroing out Gaussian components
• SNR robustness: Kurtosis-based features maintain discrimination at lower signal-to-noise ratios than envelope variance
• Channel resilience: Multipath fading, which often produces Rayleigh-distributed amplitudes, is separable from hardware-induced kurtosis deviations
• Interference suppression: Narrowband interferers with constant-envelope modulation exhibit low kurtosis, distinguishing them from transient artifacts

Rather than a single scalar value, transient kurtosis is computed over a sliding window to produce a time-varying profile that tracks the evolution of non-Gaussianity throughout the burst onset or offset.

• Attack-phase kurtosis: Peaks during the initial amplifier ramp-up, capturing slew-rate-induced nonlinearities
• Settling-phase kurtosis: Decays toward mesokurtic values as the PLL locks and the amplifier stabilizes
• Turn-off kurtosis spike: Often exhibits a sharp leptokurtic peak as the power supply collapses and the modulator loses bias
• Window size trade-off: Shorter windows provide better temporal resolution but higher estimator variance; typical windows range from 32 to 256 samples

Transient kurtosis can be computed across multiple frequency sub-bands using wavelet decomposition or filter banks, revealing hardware artifacts that are localized in both time and frequency.

• Wavelet kurtosis: Applying the discrete wavelet transform before kurtosis estimation isolates impulsive features at specific scales
• Sub-band kurtosis vector: A feature vector formed by concatenating kurtosis values from each decomposition level
• Scale-dependent signatures: DAC glitches dominate at fine scales (high frequencies), while power supply modulation appears at coarser scales
• Discriminant power: Multi-scale kurtosis features significantly outperform single-scale kurtosis for emitter classification tasks

Transient kurtosis is the time-domain counterpart to the bispectrum's frequency-domain analysis of non-Gaussianity. While the bispectrum detects quadratic phase coupling, kurtosis captures the aggregate amplitude distribution effects of all non-linear mechanisms.

• Complementary features: Kurtosis provides a compact scalar or vector feature; bispectrum provides a 2D frequency map
• Computational efficiency: Kurtosis estimation is O(N) versus O(N²) for bispectrum, making it suitable for real-time edge deployment
• Feature fusion: Combining kurtosis with bispectral features in a neural network classifier yields state-of-the-art emitter identification accuracy
• Pre-screening: Kurtosis can serve as a fast pre-screener to trigger more computationally intensive bispectral analysis only when significant non-Gaussianity is detected

A measure of the asymmetry of the transient signal's amplitude probability density function. While kurtosis measures tailedness, skewness reveals directional biases in the hardware's non-linear response.

• Positive skew: Indicates the tail on the right side is longer, often caused by overshoot artifacts.
• Negative skew: Indicates the tail on the left side is longer, typical of rapid power-supply sag.
• Hardware origin: Asymmetric slew rates in the power amplifier's N-type and P-type transistors.

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

• Second-order cumulant: Equivalent to variance (signal power).
• Third-order cumulant: Equivalent to skewness (asymmetry).
• Fourth-order cumulant: Related to kurtosis, but subtracts the Gaussian contribution, revealing the true non-Gaussian hardware artifact.
• Key advantage: Robustly separates the device's intrinsic non-linear fingerprint from additive white Gaussian channel noise.

A higher-order spectral analysis technique that reveals quadratic phase coupling within the transient signal. It is computed as the 2D Fourier transform of the third-order cumulant sequence.

• Phase coupling detection: Identifies harmonics that are phase-coherent, a hallmark of non-linear mixing in the power amplifier.
• Gaussian noise suppression: The bispectrum of Gaussian noise is theoretically zero, making it an excellent tool for extracting features in low signal-to-noise ratio conditions.
• Application: Used to detect subtle intermodulation products generated during the turn-on transient that are invisible to standard power spectral density analysis.

The collective set of statistical measures beyond second-order (variance), including skewness, kurtosis, and cumulants, used to characterize the non-Gaussian nature of transient hardware artifacts.

• Gaussian assumption: Linear circuits with ideal components produce Gaussian-distributed signals. Real transients violate this.
• Non-Gaussianity as a feature: The deviation from Gaussianity is the direct imprint of the transmitter's unique non-linear components.
• Feature vector: A concatenated vector of HOS features provides a robust, low-dimensional input for a device classifier.

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

• Stability: Lipschitz-continuous to small time-warping deformations, making it robust to jitter.
• Architecture: Alternates wavelet convolutions with a modulus operator to recover high-frequency information lost by pooling.
• Relationship to kurtosis: The scattering coefficients capture the multi-scale energy distribution that kurtosis summarizes in a single number, providing a much richer feature space for classification.

A specific type of transient distortion characterized by a very high kurtosis value, indicating a heavy-tailed amplitude distribution with sharp, sporadic spikes.

• Physical cause: Often generated by the rapid discharge of parasitic capacitances or the non-ideal switching of digital logic gates during the power-up sequence.
• DAC glitch impulse: A common source, where a momentary voltage spike occurs at the digital-to-analog converter output due to timing skews.
• Detection: Kurtosis is the primary statistical detector for these events, as they are too brief to significantly affect variance but dominate the fourth-order moment.