Cumulant-Based Streaming Classification

The core mechanism enabling streaming operation. Instead of recomputing statistics over an entire block, the classifier updates running estimates of moments and cumulants with each new IQ sample using exponential forgetting or sliding window recursions.

• Exponential weighting: Applies a decay factor λ (e.g., 0.99) to prioritize recent samples
• Constant memory: O(k²) storage for k-th order statistics, independent of stream length
• Update rule: μₙₑᵧ = (1-λ)·xₙₑᵧ + λ·μₒₗₔ

Maintains scale-invariant features by continuously normalizing higher-order cumulants with recursively estimated signal power. This ensures the classifier remains robust to time-varying gain control and fading amplitude.

• Running variance tracker: Updates σ² recursively alongside cumulants
• Division on read: Normalized cumulant = Ĉ₄₀ / (σ²)² computed only when classification is triggered
• Avoids batch dependency: No need to buffer samples for variance computation

Streaming classifiers incorporate change detection mechanisms that trigger re-evaluation only when the cumulant feature vector deviates significantly from the current hypothesis, conserving computational resources.

• Cumulative sum (CUSUM) detectors: Monitor cumulant trajectories for abrupt modulation changes
• Sequential probability ratio test (SPRT): Accumulates evidence before switching classification labels
• Hysteresis thresholds: Prevents rapid toggling between modulation hypotheses at decision boundaries

Advanced streaming classifiers dynamically adjust the exponential forgetting factor λ based on detected channel conditions or environment dynamics, balancing responsiveness against estimator variance.

• Fast attack, slow decay: Low λ when a modulation change is suspected; high λ during steady-state tracking
• Channel-aware scheduling: Increases forgetting during deep fades to discard unreliable samples
• Stability guarantees: Bounded λ prevents estimator collapse in low-SNR conditions

Monitors the statistical distribution of cumulant features over time to detect concept drift—when the signal environment changes fundamentally—triggering model adaptation or alerting operators.

• Distribution distance metrics: Kullback-Leibler divergence between recent and historical cumulant distributions
• Windowed hypothesis testing: Compares cumulant statistics across sliding time windows
• Application: Detects emergence of new modulation types or jamming waveforms in electronic warfare scenarios

Hardware realization of recursive cumulant estimators on FPGAs achieves microsecond-latency classification by pipelining multiply-accumulate operations for each incoming IQ sample without CPU intervention.

• Parallel moment engines: Dedicated hardware blocks compute 2nd, 3rd, and 4th-order moments simultaneously
• Fixed-point optimization: Custom bit-width allocation preserves numerical precision while minimizing DSP slice usage
• Throughput: Processes sample rates exceeding 1 GSPS on modern RFSoC platforms

The core mathematical engine that updates cumulant estimates incrementally with each new IQ sample, avoiding batch reprocessing. Online algorithms maintain running sums of moments (up to 8th order) and combine them to compute C40 and C42 on the fly.

• Uses exponential forgetting factors to track non-stationary signals
• Employs Welford-style numerically stable recurrence relations
• Enables constant-memory, O(1) per-sample computational complexity

A continuously updated structured input formed by concatenating recursively estimated cumulants and their ratios. The vector typically includes normalized C40, C42, C63, and cumulant ratios like |C40|/|C42|.

• Maintains a sliding window of recent feature values for temporal context
• Feeds directly into lightweight online classifiers (e.g., SGD-based SVMs)
• Provides inherent scale and phase invariance through normalization

A decision framework that updates the likelihood of each candidate modulation as new cumulant estimates arrive, rather than waiting for a full block. Wald's Sequential Probability Ratio Test (SPRT) is adapted to compare observed cumulants against theoretical values.

• Makes decisions with minimum average sample size
• Controls both Type I and Type II error probabilities explicitly
• Naturally handles early termination when confidence thresholds are met

A monitoring layer that tracks the statistical distribution of streaming cumulant features to detect concept drift in the signal environment. When the cumulant trajectory deviates from an expected profile, it signals potential jamming, spoofing, or a change in the transmitter's modulation scheme.

• Uses CUSUM or Page-Hinkley change-point detection on cumulant streams
• Triggers model adaptation or alerts without interrupting classification
• Distinguishes transient anomalies from genuine regime shifts

A hardware realization of the streaming cumulant classifier on Field-Programmable Gate Arrays for true real-time operation at the network edge. The pipeline parallelizes moment accumulation and cumulant combination using fixed-point arithmetic.

• Achieves sub-microsecond latency per sample update
• Implements deep pipelining for high-throughput IQ streams (100+ MSPS)
• Integrates directly with ADC front-ends for instantaneous classification

A decision tree architecture that applies specific cumulant thresholds at each node as estimates refine over time. The root node separates PSK from QAM using kurtosis, while subsequent nodes discriminate specific orders (e.g., QPSK vs. 8PSK) using higher-order cumulants.

• Reduces multi-class problem to a sequence of binary decisions
• Each node can declare confidence independently as samples accumulate
• Naturally supports open set rejection at leaf nodes for unknown modulations