Complex-Valued Neural Network (CVNN)

CVNNs perform forward propagation using complex-valued weights and complex activation functions, such as the complex ReLU or modReLU. This preserves the algebraic structure of IQ samples, where a single complex neuron can represent both magnitude and phase. Unlike real-valued networks that treat I and Q as separate channels, CVNNs maintain the holomorphic relationship between components, enabling more compact representations of rotational and periodic phenomena common in RF signals.

Training requires Wirtinger calculus (CR-calculus) to compute gradients with respect to complex parameters. The backpropagation algorithm is extended using conjugate partial derivatives, ensuring that the phase information encoded in complex weights is updated correctly. This avoids the information loss that occurs when real-valued optimizers naively split complex numbers into real and imaginary parts. Key benefits:

• Preserves phase coherence across layers
• Enables stable gradient flow in deep architectures
• Supports standard optimizers like Adam with complex extensions

A complex-valued neuron with N complex weights has 2N real degrees of freedom but learns a richer representational geometry. Empirical studies in RF modulation classification show CVNNs achieve equivalent accuracy to real-valued networks with 50-75% fewer parameters. This parameter efficiency stems from the orthogonal decision boundaries formed by complex activation functions, which are better suited to separating signals in the complex plane than axis-aligned real-valued boundaries.

CVNNs inherently model rotational equivariance—a property critical for RF signals where phase rotation corresponds to time delay or Doppler shift. A complex weight multiplication applies both scaling and rotation, allowing the network to learn features invariant to absolute phase. This eliminates the need for explicit data augmentation with phase-shifted copies of training signals. Applications include:

• Doppler-robust radar target classification
• Carrier frequency offset tolerant modulation recognition
• Phase-agnostic RF fingerprinting

Standard batch normalization fails on complex data because it treats real and imaginary components independently, destroying their correlation. Complex batch normalization whitens the data using a 2×2 covariance matrix that captures the real-imaginary cross-correlation. This stabilizes training by ensuring that the circularly symmetric nature of complex distributions is preserved through normalization, leading to faster convergence and improved final accuracy in deep CVNN architectures.

In contested RF environments, CVNNs demonstrate inherent robustness to adversarial jamming and co-channel interference. The complex decision boundaries create non-linear phase-amplitude filters that can separate overlapping signals in the complex domain where real-valued networks see only a corrupted magnitude spectrum. This makes CVNNs particularly effective for:

• Classifying weak signals under strong co-channel interference
• Distinguishing spoofed vs. genuine transmitters
• Detecting low-probability-of-intercept waveforms

A deep learning architecture trained on raw IQ samples or spectrograms to categorize signals by modulation, protocol, or device identity. CVNNs represent a specialized subclass of these architectures, operating directly on complex-valued data to preserve phase relationships critical for distinguishing co-channel interference. Traditional real-valued networks require separate I and Q channels, doubling parameters and potentially discarding the structural information inherent in the complex plane.

A blind signal processing technique where a neural network identifies the modulation scheme of a received waveform without prior demodulation. CVNNs excel at AMC tasks because complex nonlinear activation functions like modReLU and complex batch normalization directly capture the rotational and scaling properties of modulated constellations. This contrasts with real-valued CNNs that must learn to reconstruct phase relationships from separate real and imaginary input channels.

A statistical signal processing method exploiting the periodic properties of modulated signals for robust classification in low signal-to-noise ratio (SNR) environments. CVNNs can implicitly learn cyclostationary signatures through complex convolutional layers that detect spectral correlation patterns. This provides resilience against noise and interference that traditional energy detectors miss, making CVNN-based classifiers effective below 0 dB SNR.

A deep learning technique identifying unique hardware-level imperfections in transmitter waveforms for device authentication. CVNNs enhance RFF by preserving the subtle phase noise and I/Q imbalance artifacts that constitute a device's fingerprint. Real-valued networks often discard these discriminative features during the I/Q-to-real conversion process, reducing authentication accuracy in dense emitter environments.

The hardening of RF machine learning models against evasion attacks where an intelligent jammer subtly manipulates its waveform to fool the classifier. CVNNs offer inherent robustness advantages because adversarial perturbations must simultaneously corrupt both magnitude and phase while maintaining physical realizability. Complex-valued adversarial training further constrains the perturbation space, yielding classifiers that resist over-the-air attacks more effectively than their real-valued counterparts.

A method converting raw time-domain signals into time-frequency images processed by Convolutional Neural Networks. While effective, this approach discards phase information during the magnitude-only spectrogram computation. CVNNs operating on raw IQ data avoid this information loss, capturing both transient phase shifts and frequency-domain signatures simultaneously. This dual-domain awareness is critical for classifying interference patterns that differ primarily in phase structure rather than spectral content.