Complex-Valued Neural Network (CVNN)

CVNNs process I/Q signals as single complex entities z = I + jQ rather than separate real channels. Each neuron performs complex multiplication (W·z) and applies a complex activation function. This preserves the amplitude-phase relationship inherent in modulated signals, eliminating the need for the network to learn the Hilbert transform relationship between I and Q components. Real-valued networks processing concatenated I/Q vectors can introduce cross-term distortion that CVNNs inherently avoid.

Training uses Wirtinger calculus to compute gradients with respect to complex parameters. The loss function J(W) is non-holomorphic, so gradients are computed using conjugate partial derivatives:

• ∂J/∂W = (∂J/∂W_re) + j(∂J/∂W_im)
• The update rule becomes: W_new = W_old - η(∂J/∂W*)

This ensures proper gradient descent on the complex plane, avoiding the phase ambiguity that occurs when real-valued optimizers are applied to complex parameters independently.

Standard real activations (ReLU, tanh) cannot be directly applied to complex values without destroying phase information. CVNNs use specialized functions:

• modReLU: Applies ReLU to magnitude while preserving phase: f(z) = ReLU(|z| + b)·e^(j∠z)
• Complex tanh: tanh(z_re) + j·tanh(z_im), bounded and fully complex-differentiable
• Cardioid function: f(z) = 0.5(1 + cos(∠z))·z, introducing phase-dependent nonlinearity

These preserve the complex differentiability required for Wirtinger backpropagation.

Power amplifier memory effects manifest as phase distortion across time delays. CVNNs with complex-valued tapped delay lines capture this directly:

• Complex input vector: [z(t), z(t-1), ..., z(t-M)] where each element retains its phase
• Complex weights multiply delayed samples, preserving relative phase relationships between time steps
• This enables the network to model AM/PM memory effects with fewer parameters than equivalent real-valued architectures

Real-valued TDNNs require separate delay lines for I and Q, doubling parameters without capturing cross-phase dynamics.

CVNNs demonstrate superior convergence for PA linearization tasks due to reduced parameter space:

• A complex weight w = w_re + jw_im represents two degrees of freedom in one parameter
• Equivalent real-valued networks require 4x the connections to model the same I/Q interactions
• CVNNs achieve comparable ACLR improvement with 40-60% fewer trainable parameters

This compactness reduces overfitting risk on limited PA measurement datasets and accelerates gradient descent convergence by operating in the natural signal domain.

The CVNN produces a complex predistorted signal z_pd(t) that directly feeds the transmitter's I/Q modulator without post-processing. This end-to-end complex pipeline:

• Eliminates I/Q reconstruction errors from separate real-valued network outputs
• Maintains sample-level phase coherence critical for wideband signals
• Integrates seamlessly with Direct Learning Architectures (DLA) where the error signal e(t) = z_desired(t) - z_pa(t) is naturally complex

The output can be directly applied to the Memory Polynomial predistorter structure or used as the complete predistortion function.

A preprocessing strategy that bridges real-valued networks and complex signals. The complex I/Q sample is decomposed into alternative representations before being fed to a real-valued neural network:

• Cartesian: Separate I and Q branches
• Polar: Magnitude and phase components This technique allows standard real-valued optimizers to operate on phase-sensitive data, but the decomposition itself is a non-learned, fixed transform that may not be optimal for the PA's specific nonlinearity.

The learning algorithm that makes CVNNs trainable. It extends standard backpropagation to the complex domain by computing gradients with respect to complex weights using Wirtinger calculus. The key insight is treating a complex variable and its complex conjugate as independent entities, allowing the calculation of conjugate partial derivatives that define the steepest descent direction on the complex loss surface. This enables direct optimization of phase-sensitive cost functions.

A family of activation functions that apply real-valued nonlinearities separately to the real and imaginary parts of a complex pre-activation:

• Split-ReLU: ReLU(ℜ(z)) + i·ReLU(ℑ(z))
• Split-tanh: tanh(ℜ(z)) + i·tanh(ℑ(z)) These functions are bounded but not analytic, meaning they violate the Cauchy-Riemann equations. However, they are differentiable in the real sense and work well in practice for PA linearization where fully complex analytic activations are too restrictive.

The mathematical framework that enables gradient-based optimization of non-holomorphic complex functions. It defines two partial derivatives:

• R-derivative: Derivative with respect to z treating z̄ as constant
• Conjugate R-derivative: Derivative with respect to z̄ treating z as constant The conjugate gradient defines the direction of steepest descent for real-valued loss functions of complex variables, forming the theoretical backbone of complex backpropagation used to train CVNN predistorters.