Complex-Valued Neural Network (CVNN)

Unlike real-valued ReLU, CVNNs use complex activation functions that operate on both magnitude and phase independently. Complex ReLU, for instance, applies ReLU to the magnitude while preserving the phase angle. ModReLU introduces a learned bias radius, and zReLU passes values with phase angles in [0, π/2]. These functions prevent the loss of critical phase information that encodes modulation schemes and channel distortions.

Convolution in the complex domain uses complex-valued filters where each weight has a real and imaginary component. Proper initialization is critical—techniques like complex Xavier/Glorot initialization account for the variance of both components. The complex convolution operation performs four real-valued convolutions internally: (A + iB) * (C + iD) = (AC - BD) + i(BC + AD). This preserves the algebraic structure of the signal.

Standard backpropagation fails because complex loss functions are non-holomorphic. CVNNs use Wirtinger calculus, which computes gradients with respect to the complex variable and its conjugate independently. The chain rule splits into two partial derivatives: ∂L/∂z and ∂L/∂z̄. This enables gradient-based optimization while respecting the complex structure, allowing frameworks like PyTorch and TensorFlow to train CVNNs with complex autograd support.

Standard batch normalization whitens real and imaginary parts independently, destroying their correlation. Complex batch normalization whitens the combined 2D vector distribution by computing a 2x2 covariance matrix and applying its inverse square root. This centers the data to zero mean and unit circular variance while preserving the phase structure, leading to faster convergence and better generalization in deep CVNN architectures.

A single complex neuron with n inputs has 2n real-valued parameters (n complex weights), but its decision boundary operates in a 2n-dimensional space. This provides orthogonal decision surfaces that real-valued networks with equivalent parameter counts cannot replicate. Empirical studies show CVNNs achieve comparable accuracy to real-valued networks with roughly half the parameters for tasks like automatic modulation classification and channel estimation.

The complex domain naturally encodes rotational invariance—multiplying by e^(iθ) rotates a signal in phase without changing its information content. CVNNs learn representations that are inherently robust to phase rotations caused by carrier frequency offset and Doppler shift. This inductive bias leads to superior generalization on unseen channel conditions compared to real-valued networks that must learn rotation invariance from data.

CVNNs accept raw in-phase and quadrature (IQ) samples as complex numbers (z = I + jQ), eliminating the need to split or concatenate real and imaginary components. This preserves the holomorphic relationship between I and Q channels.

• Native handling of complex baseband representation
• Avoids information loss from magnitude/phase decomposition
• Reduces input dimensionality by 50% compared to real-valued equivalents
• Enables learning of complex-valued feature maps that capture rotational equivariance

Standard activation functions like ReLU cannot be directly applied to complex numbers without violating Cauchy-Riemann equations required for analyticity. CVNNs employ specialized activations:

• modReLU: Applies ReLU to magnitude while preserving phase
• zReLU: Passes through values in the first quadrant, zeros out others
• cReLU: Separate ReLU on real and imaginary parts (split approach)
• Complex Cardioid: Preserves phase continuity for angle-sensitive tasks

These functions maintain gradient flow while respecting the geometry of the complex plane.

Training CVNNs requires Wirtinger calculus, which treats the complex variable and its conjugate as independent quantities. The gradient is computed as:

• ∂L/∂z = ∂L/∂x + j·∂L/∂y (standard complex derivative)
• Conjugate gradient ∂L/∂z̄ enables optimization in complex space
• Enables full complex-valued chain rule for end-to-end training
• Compatible with modern optimizers (Adam, SGD) with complex parameter updates

This mathematical framework ensures proper gradient flow through every complex-valued layer.

CVNNs excel at identifying modulation schemes (QPSK, 16-QAM, 64-QAM) directly from IQ streams because phase rotations are intrinsic to modulation constellations.

• Achieves 3-5 dB SNR improvement over real-valued CNNs at low SNR
• Naturally invariant to carrier phase offset through complex weight rotation
• Learns constellation geometry in the complex plane without explicit feature engineering
• State-of-the-art accuracy on RadioML 2018.01A dataset at SNR < 0 dB

CVNNs model the wireless channel as a complex multiplicative distortion h·x + n, making them ideal for learning channel inverses.

• Direct estimation of complex channel state information (CSI)
• Complex-valued autoencoders learn optimal precoding matrices
• Outperforms MMSE equalizers in high-mobility scenarios
• Joint channel estimation and symbol detection in a single network
• Handles IQ imbalance and phase noise as learnable distortions

Hardware impairments manifest as subtle, unique complex-valued distortions in the transmitted waveform. CVNNs capture these complex-domain fingerprints with high fidelity.

• Detects I/Q origin offset, gain imbalance, and quadrature skew
• Learns device-specific non-linearities in the complex plane
• Robust to channel variation through complex convolutional filters
• Achieves >99% identification accuracy across 50+ devices in controlled settings
• Enables physical layer authentication without cryptographic overhead