Chapter Summary

Chapter Summary

Key Points

• 1.

  Complex-valued networks preserve phase structure. Using complex weights halves parameter count compared to unconstrained real networks and provides phase equivariance, which is natural for wireless signals.

• 2.

  Real/imaginary stacking is the simplest approach. Concatenate real and imaginary parts as channels: (B, 2C, H, W). Compatible with all standard layers. Use complex-structured layers only when phase equivariance matters.

• 3.

  Wirtinger calculus enables complex optimisation. PyTorch computes the conjugate Wirtinger derivative automatically. The loss must be real-valued; use |z|^2 for MSE on complex outputs.

• 4.

  Differentiable forward models enable end-to-end learning. Implement the physics (FFT, channel, convolution) as PyTorch modules so gradients flow through the entire chain. This allows joint optimisation of transmitter and receiver.

• 5.

  Choose activations carefully. CReLU (split ReLU) is simple but not phase-aware. modReLU preserves phase and is better for phase-sensitive tasks. Always avoid applying standard ReLU directly to complex tensors.