Gaussianity Test

Gaussianity tests operate on a formal null hypothesis (H₀): the data is drawn from a Gaussian distribution. The alternative hypothesis (H₁) asserts non-Gaussianity. A p-value is computed from the test statistic; if it falls below a predetermined significance level (α)—typically 0.05 or 0.01—H₀ is rejected. This rejection provides statistical evidence that exploitable non-Gaussian signal structure exists, validating further higher-order spectral analysis. The framework requires careful consideration of Type I errors (false rejection of Gaussianity) and Type II errors (failure to detect non-Gaussianity).

These tests evaluate deviations in the third standardized moment (skewness) and fourth standardized moment (kurtosis) from their Gaussian expectations of 0 and 3, respectively.

• Jarque-Bera Test: Combines sample skewness and excess kurtosis into a single chi-squared statistic with 2 degrees of freedom. Particularly powerful for symmetric non-Gaussian alternatives.
• D'Agostino-Pearson Test: An omnibus test that separately transforms skewness and kurtosis to approximate normality, then combines them. Robust for sample sizes as small as n ≥ 20.
• Sample Moments: Raw moment estimates are sensitive to outliers; robust alternatives use L-moments or trimmed estimators for heavy-tailed signal distributions.

EDF tests measure the distance between the empirical cumulative distribution function (ECDF) of the sample and the theoretical Gaussian CDF.

• Kolmogorov-Smirnov (KS) Test: Uses the supremum of the absolute difference between ECDF and CDF. Distribution-free under H₀ but less sensitive to tail deviations.
• Anderson-Darling Test: A weighted variant that emphasizes discrepancies in the tails of the distribution, making it more powerful for detecting non-Gaussianity caused by heavy-tailed hardware impairments.
• Cramér-von Mises Criterion: Integrates the squared difference across the entire distribution, providing balanced sensitivity across the support.

The Shapiro-Wilk test is widely regarded as the most powerful omnibus Gaussianity test for moderate sample sizes (n < 2000). It operates by regressing ordered sample values against the expected order statistics of a standard Gaussian distribution.

• W-statistic: The ratio of squared linear combination of order statistics to sample variance. Values near 1 indicate Gaussianity.
• Shapiro-Francia: A simplified approximation for larger samples using only the correlation between ordered data and Gaussian quantiles.
• Filliben's Test: Evaluates the correlation on a probability plot; particularly useful for visual diagnostics alongside formal hypothesis testing.

For RF fingerprinting applications, Gaussianity tests based on higher-order cumulants directly connect to the feature extraction pipeline. Under H₀, all cumulants of order k ≥ 3 are identically zero.

• Bispectral Flatness Test: Evaluates whether the sample bispectrum is consistent with zero across all bifrequency pairs, using a chi-squared statistic derived from bicoherence estimates.
• Hinich's Linearity Test: A frequency-domain test that checks for non-zero bispectral values, simultaneously testing Gaussianity and linearity of the generating process.
• Cumulant Variance Bounds: Asymptotic variance expressions for sample cumulants enable construction of confidence intervals; observed cumulants exceeding these bounds reject Gaussianity.

Real-world signal analysis introduces challenges that affect test selection and interpretation.

• Sample Size Sensitivity: KS and Shapiro-Wilk lose power with very small samples; moment-based tests require n > 50 for reliable kurtosis estimation.
• Pre-Whitening: Serial correlation in time-series data inflates Type I error rates. Apply AR-model pre-whitening before testing to ensure independent observations.
• Segment-Based Testing: For long signal recordings, apply Gaussianity tests to overlapping or contiguous segments to detect intermittent non-Gaussianity caused by transient impairments.
• Composite Hypotheses: Tests that assume known mean and variance (simple H₀) are more powerful than those estimating parameters; use Lilliefors-corrected critical values when parameters are estimated.

The fourth standardized moment measuring the tailedness of a signal's amplitude distribution. Excess kurtosis (deviation from the Gaussian value of 3) directly indicates the presence of rare, extreme amplitude events caused by hardware non-linearities.

• Leptokurtic distributions (kurtosis > 3) have heavy tails, often from amplifier saturation
• Platykurtic distributions (kurtosis < 3) have thin tails, characteristic of clipped or compressed signals
• Serves as a computationally efficient pre-screen before committing to full bispectral analysis

The third standardized moment that quantifies asymmetry in a signal's amplitude probability density function. Non-zero skewness reveals directional hardware biases, such as amplifier non-linearity that compresses one signal quadrant more than another.

• Positive skew: Distribution tail extends toward positive amplitudes
• Negative skew: Distribution tail extends toward negative amplitudes
• Highly sensitive to DC offset and I/Q imbalance in quadrature modulators

A non-linear phenomenon where two frequency components, f1 and f2, interact within a device's analog stages to generate energy at their sum (f1+f2) or difference (f1-f2) frequencies. This coupling is undetectable by power spectrum analysis but clearly visible in the bispectrum.

• Arises from second-order non-linearities in amplifiers and mixers
• The phase of the generated component is locked to the phases of the originating frequencies
• Forms the physical basis for bispectrum-based fingerprinting

A normalized bispectrum that provides a bounded metric (0 to 1) quantifying the proportion of signal energy at a bifrequency pair that is quadratically phase-coupled. Unlike the raw bispectrum, bicoherence is independent of signal amplitude.

• Value near 1.0: Strong, consistent phase coupling (likely hardware-induced)
• Value near 0.0: No phase coupling (likely stochastic noise)
• Enables threshold-based detection of non-linearities without amplitude calibration

A compact statistical fingerprint constructed from estimated higher-order cumulants (3rd, 4th, and sometimes 5th order) that serves as input to machine learning classifiers. These vectors are theoretically insensitive to Gaussian noise, making them robust in low-SNR environments.

• Typically includes cumulant combinations like C40/C21² for modulation-independent fingerprinting
• Dimensionality is far lower than raw bispectral data, enabling lightweight classifiers
• Forms the bridge between statistical signal processing and deep learning emitter identification

A dimensionality reduction technique that projects high-dimensional signal data onto directions that maximize non-Gaussianity, isolating the subspace containing hardware-specific fingerprint information. This separates the fingerprint from Gaussian thermal noise and Gaussian-like interference.

• Often implemented via Independent Component Analysis (ICA)
• The projected subspace dimensions correspond to individual non-Gaussian sources
• Enables blind separation of co-channel emitters without prior knowledge of their signatures