Skewness

Skewness is the third standardized moment of a probability distribution, computed as the expected value of the cubed deviation from the mean divided by the cube of the standard deviation. For a signal sample x₁, x₂, ..., xₙ with mean μ and standard deviation σ, skewness γ₁ = E[(x - μ)³] / σ³.

• Zero skewness: Symmetric distribution (e.g., ideal Gaussian)
• Positive skewness: Right-tailed, mass concentrated on left
• Negative skewness: Left-tailed, mass concentrated on right

In RF fingerprinting, non-zero skewness directly indicates amplifier non-linearity favoring one quadrant of operation.

Skewness captures directional distortion in transmitter power amplifiers where compression or clipping occurs asymmetrically. Key hardware sources include:

• Class-AB amplifier biasing: Slight DC offset shifts the operating point, causing asymmetric waveform clipping in one polarity
• DAC integral non-linearity (INL): Code-dependent errors that skew the output distribution toward specific amplitude ranges
• I/Q modulator imbalance: Phase and gain mismatches between in-phase and quadrature paths produce distributional asymmetry

These manufacturing variances are unique per device and stable over time, making skewness a robust fingerprint feature.

Skewness belongs to the family of higher-order statistics that characterize non-Gaussian signal behavior:

• Second-order (variance): Measures spread, insensitive to asymmetry
• Third-order (skewness): Measures asymmetry, the simplest statistic capturing directional distortion
• Fourth-order (kurtosis): Measures tailedness, complementary to skewness

Skewness is the lowest-order moment that detects deviation from Gaussian symmetry. In cumulant terms, the third-order cumulant κ₃ equals the third central moment, making skewness the normalized cumulant: γ₁ = κ₃ / κ₂^(3/2).

Sample skewness estimation requires careful handling in RF applications:

• Adjusted Fisher-Pearson coefficient: g₁ = [√(n(n-1)) / (n-2)] × m₃ / m₂^(3/2), where mₖ are sample central moments
• Sensitivity to outliers: A single large transient can dominate the cubed deviation; robust alternatives include medcouple and quantile-based skewness
• Sample size requirements: Reliable skewness estimates typically need >100 samples due to the cubed term's variance
• Channel effects: Multipath fading can alter amplitude distributions; skewness features should be extracted after channel equalization or used with channel-robust normalization

Skewness serves as a discriminative feature in RF fingerprinting pipelines:

• Feature vector component: Combined with kurtosis, variance, and bispectral features to form a comprehensive statistical fingerprint
• Modulation-independent: Skewness captures hardware artifacts present regardless of the modulation scheme, enabling cross-protocol identification
• Quadrant-specific analysis: Computing skewness separately for positive and negative amplitude ranges isolates directional amplifier compression
• Real-time computation: O(n) complexity enables implementation on FPGA and SDR platforms for low-latency device authentication

While powerful, skewness has inherent limitations that necessitate complementary analysis:

• Symmetric non-linearities: Some impairments (e.g., symmetric clipping, crossover distortion) produce zero skewness despite significant non-linearity — requiring kurtosis or bispectral analysis for detection
• Gaussian noise masking: Additive white Gaussian noise dilutes skewness; higher-order cumulants (κ₃, κ₄) provide theoretical noise immunity
• Distributional ambiguity: Different underlying distributions can produce identical skewness values — bicoherence and quadratic phase coupling analysis resolve this ambiguity
• Temporal drift: Amplifier aging and temperature variations slowly alter skewness signatures, motivating drift compensation algorithms for long-term deployment

Skewness directly measures the asymmetry introduced by power amplifier compression in one quadrant of the I/Q plane. A positively skewed amplitude distribution often indicates that the amplifier saturates more aggressively on positive voltage swings than negative ones, creating a distinctive hardware fingerprint.

• Detects Class AB amplifier crossover distortion
• Quantifies AM/AM compression asymmetry
• Distinguishes between soft and hard clipping behaviors
• Provides a single scalar metric for amplifier health monitoring

Manufacturing variances in digital-to-analog converter (DAC) resistor ladders produce systematic code-dependent errors. The skewness of the reconstructed waveform's amplitude distribution captures the directional bias of these integral non-linearity (INL) errors, creating a unique, unclonable device signature.

• Correlates skewness with INL profile curvature
• Identifies most significant bit (MSB) segment mismatches
• Enables supply chain authentication of ADC/DAC chips
• Robust against additive white Gaussian noise due to higher-order moment properties

When in-phase and quadrature branches exhibit gain imbalance, the resulting constellation distortion produces an asymmetric amplitude distribution. Skewness computed separately on the I and Q rails quantifies the directional gain mismatch, revealing whether the I-branch amplifier is stronger than the Q-branch.

• Isolates gain imbalance from phase imbalance
• Tracks temperature-dependent analog front-end drift
• Provides a compact feature for few-shot device enrollment
• Complements kurtosis for complete non-Gaussian profiling

The brief power amplifier ramp-up period exhibits strong non-linear behavior as bias voltages stabilize. Skewness of the transient amplitude envelope captures the asymmetric charging characteristics of DC blocking capacitors and bias tee inductors, which vary subtly between devices due to component tolerances.

• Analyzes microsecond-duration turn-on transients
• Exploits capacitor dielectric absorption variations
• Provides a feature independent of steady-state modulation
• Enables identification even with identical chipset models

Carrier feedthrough caused by DC offsets in the quadrature modulator shifts the constellation origin, creating amplitude distribution asymmetry. The skewness of the envelope signal quantifies the magnitude and direction of this offset, which is unique to each mixer's semiconductor doping variations.

• Detects LO leakage without demodulation
• Maps skewness to mixer transistor mismatch
• Functions at very low SNR using cumulant-based estimation
• Enables physical layer authentication without cryptographic overhead

While multipath fading alters second-order statistics, skewness remains relatively stable under linear channel distortions. By computing skewness on the baseband complex envelope after coarse synchronization, the feature retains its discriminative power across diverse propagation environments.

• Invariant to linear time-invariant channel effects
• Pairs with cyclic cumulant analysis for joint robustness
• Reduces domain adaptation burden for deployed classifiers
• Validated in Rayleigh and Rician fading simulations