Gaussianity Test

The test exploits the theoretical property that Gaussian processes have zero fourth-order cumulants (C40 = C42 = 0). For a received signal block, the sample estimate of |C40| or |C42| is compared against a statistically derived threshold. If the normalized cumulant magnitude exceeds the threshold, the signal is classified as non-Gaussian (e.g., QAM, PSK, ASK), triggering further subclassification. Values near zero indicate either noise or OFDM-like signals with high subcarrier counts, whose central limit theorem convergence produces near-Gaussian statistics.

A critical application is separating OFDM signals from pure Gaussian noise. While both exhibit near-zero kurtosis, OFDM waveforms with a finite number of subcarriers retain detectable non-Gaussianity in their sixth-order and eighth-order cumulants. The test can be extended to higher cumulant orders to exploit this residual structure:

• Fourth-order test: Both OFDM and noise appear Gaussian
• Sixth-order test: OFDM exhibits non-zero C60/C63 values, enabling discrimination
• Eighth-order test: Provides additional separation margin at high SNR

The reliability of the Gaussianity test is governed by the variance of the sample cumulant estimator, which depends on both the number of observed samples (N) and the signal-to-noise ratio. Below the Cumulant SNR Wall, the estimator variance exceeds the bias, making the test statistically unreliable regardless of observation length:

• High SNR, large N: Tight confidence intervals, reliable decisions
• Low SNR, small N: Estimator variance dominates, increasing false classifications
• SNR Wall: The theoretical boundary where cumulant-based tests fundamentally fail

Practical implementations require adaptive thresholding based on estimated SNR and sample count.

To ensure the test is independent of received signal power, cumulants are normalized by the signal variance raised to the appropriate power. The normalized fourth-order cumulant is computed as:

C40_norm = C40 / σ⁴

This normalization provides:

• Amplitude invariance: Test result unchanged by gain variations
• Scale-independent thresholds: Same decision boundary works across power levels
• Robustness to AGC: Automatic gain control fluctuations do not affect classification

The normalized test statistic enables direct comparison against theoretical values for ideal constellations without requiring prior power estimation.

The Gaussianity test serves as the root node in hierarchical cumulant-based classifiers. The decision flow proceeds as:

1. Gaussianity Test: Is the signal non-Gaussian?
   • Yes: Proceed to linear modulation branch (PSK/QAM/ASK)
   • No: Signal is either noise or OFDM
2. OFDM Detection: Apply higher-order cumulant tests to separate OFDM from noise
3. Modulation Subclassification: Use C42 and other cumulant ratios to identify specific modulation orders

This hierarchical approach reduces computational complexity by pruning the candidate set at each node, avoiding exhaustive multi-class comparison.

The Gaussianity test is formalized as a binary hypothesis test:

• H₀ (Null Hypothesis): The signal samples follow a Gaussian distribution
• H₁ (Alternative Hypothesis): The signal samples are non-Gaussian

The test statistic T = |Ĉ40| is compared to a threshold γ derived from the Neyman-Pearson criterion or constant false alarm rate (CFAR) approach. The threshold is set based on:

• Desired probability of false alarm (classifying noise as a signal)
• Asymptotic distribution of the sample cumulant under H₀
• Estimated noise power and sample size

This statistical rigor provides quantifiable confidence levels for each classification decision.

The standardized fourth central moment of a distribution, measuring the tailedness of a signal's amplitude. Used as a direct non-Gaussianity measure:

• Kurtosis = 3: Gaussian distribution (e.g., noise, OFDM)
• Kurtosis < 3: Sub-Gaussian (e.g., PSK modulations)
• Kurtosis > 3: Super-Gaussian (e.g., sparse signals)

Serves as the simplest single-statistic Gaussianity test, though less robust than multi-cumulant approaches for distinguishing specific linear modulation types.

The standardized third central moment measuring the asymmetry of a signal's probability distribution. Critical for detecting non-symmetric constellations:

• Zero skewness: Symmetric modulations (QPSK, 16QAM)
• Non-zero skewness: Asymmetric modulations (8PAM, 4+12-APSK)
• Also detects IQ imbalance impairments in the receiver chain

Often paired with kurtosis in a joint Gaussianity test to capture both asymmetry and tailedness deviations from the normal distribution.

A specific higher-order statistic measuring the normalized fourth-order moment minus the squared second-order moment. Provides robust Gaussianity deviation features:

• C42 for single-channel signals: C42 = E[|x|⁴] - |E[x²]|² - 2E²[|x|²]
• C40 for complex signals: captures fourth-order phase relationships
• Zero for Gaussian processes, non-zero for linear modulations

Forms the mathematical foundation of the Gaussianity test, enabling hierarchical separation of QAM, PSK, and ASK families from noise-like signals.

A likelihood-ratio or goodness-of-fit test that uses theoretical cumulant values for candidate distributions to formally accept or reject the Gaussian hypothesis:

• Null hypothesis H₀: Signal distribution is Gaussian
• Test statistic: Mahalanobis distance between sample and theoretical cumulants
• Threshold: Derived from chi-squared distribution under H₀

Provides a statistically rigorous alternative to simple threshold-based Gaussianity tests, with controllable false-alarm and miss probabilities for mission-critical classification.

The theoretical signal-to-noise ratio threshold below which the variance of a sample cumulant estimator exceeds its mean, rendering Gaussianity testing fundamentally unreliable:

• Below the wall: Cumulant estimates dominated by estimation noise
• Above the wall: Reliable Gaussian vs. non-Gaussian discrimination
• Depends on: Sample size, cumulant order, and true distribution

Critical for determining operational limits of cumulant-based Gaussianity tests in low-SNR environments like deep-space communications or passive SIGINT.

A classification framework that uses the compactness of known cumulant feature clusters to reject unknown modulation types:

• Gaussian cluster: Centered at origin in cumulant space (noise, OFDM)
• Known modulation clusters: Compact regions for PSK, QAM families
• Rejection rule: Samples falling outside all known clusters are flagged as unknown/novel

Extends the Gaussianity test beyond binary Gaussian/non-Gaussian decisions to multi-class open-set recognition, essential for spectrum monitoring in dynamic electromagnetic environments.