Real-Valued Time-Delay Neural Network (RVTDNN)

Unlike Complex-Valued Neural Networks (CVNNs), the RVTDNN operates on real-valued scalar inputs. The complex baseband signal is split into its In-phase (I) and Quadrature (Q) components before entering the network. This allows the use of standard real-valued backpropagation and activation functions (like tanh or ReLU), simplifying implementation on standard digital hardware and FPGA fabric without requiring complex arithmetic units.

To model the memory effects of a power amplifier (thermal trapping, bias circuit dynamics), the RVTDNN employs Tapped Delay Lines (TDLs) at the input. The current sample and its P past samples are concatenated to form the input vector. This transforms the static nonlinear mapping into a dynamic one, allowing the network to learn how past signal envelopes influence current distortion. The memory depth P is a critical hyperparameter.

The core of the RVTDNN is a standard Multi-Layer Perceptron (MLP). It consists of an input layer (sized by memory depth), one or more hidden layers, and a linear output layer. The hidden layers provide the nonlinear mapping capability required to approximate the inverse of the PA's AM/AM and AM/PM distortion curves. The universal approximation theorem guarantees that a sufficiently wide network can model the continuous nonlinearity.

Standard TDL inputs only capture linear memory. To model nonlinear memory effects, the input vector is often augmented with envelope-dependent cross-terms. For example, terms like |x(n)| * x(n-m) or |x(n-m)|^2 * x(n) are concatenated to the input. This manually injects domain knowledge into the feature space, reducing the learning burden on the hidden layers and improving extrapolation for higher-order nonlinearities.

The RVTDNN is classically trained using the Indirect Learning Architecture (ILA). A postdistorter network is trained to map the PA output back to the PA input. Once converged, this postdistorter is copied directly to the predistorter path. This avoids the need for real-time closed-loop adaptation during training and assumes the commutability of the nonlinear blocks, which holds well for mild-to-moderate nonlinearities.

Training the RVTDNN involves minimizing a cost function (typically Mean Squared Error between the desired and actual output) using gradient descent. The real-valued nature of the network allows the use of standard backpropagation algorithms (e.g., Levenberg-Marquardt or Adam optimizer) without the need for Wirtinger calculus required by CVNNs. This simplifies the training software stack significantly.

A preprocessing step that splits the complex baseband I/Q signal into its real and imaginary components before feeding them into the RVTDNN. This allows a standard real-valued network to process complex communication signals.

• I/Q Separation: The in-phase (I) and quadrature (Q) branches are treated as independent real-valued input streams.
• Envelope-Dependent Terms: The instantaneous amplitude |x(n)| is often computed and fed as an additional input to model AM/AM and AM/PM distortion.
• Phase Preservation: By processing I and Q jointly, the network implicitly learns the phase relationships critical for distortion cancellation.

The defining structural element of an RVTDNN. A finite number of past input samples are stored and presented to the network simultaneously to model the power amplifier's memory effects.

• Memory Depth (M): The number of delayed taps determines the temporal span the model can capture.
• Short-Term Effects: Captures electrical memory caused by bias circuit impedance and matching networks.
• Tap Spacing: Typically uniform at the sampling period, but non-uniform spacing can be used to capture long-term thermal memory efficiently.

The standard supervised learning algorithm used to optimize RVTDNN weights by minimizing the error between the desired predistorted signal and the network's output.

• Loss Function: Typically Mean Squared Error (MSE) between the target and predicted I/Q samples.
• Levenberg-Marquardt: Often preferred over standard SGD for small-to-medium RVTDNNs due to faster convergence on nonlinear least-squares problems.
• Gradient Calculation: Error gradients flow backward through the tapped delay lines, updating weights to minimize both instantaneous and memory-dependent distortion.

A critical trade-off in RVTDNN design. The network must learn the true underlying PA nonlinearity, not memorize the specific training signal's characteristics.

• Overfitting Symptoms: Excellent performance on the training signal but degraded linearization on new modulation schemes or power levels.
• Regularization: Techniques like weight decay and early stopping prevent the network from fitting measurement noise.
• Validation Strategy: Training should use a different signal type (e.g., LTE) than validation (e.g., WCDMA) to ensure true generalization.

The process of converting the trained 32-bit floating-point RVTDNN weights to fixed-point integers for efficient deployment on FPGA or ASIC hardware.

• Precision Reduction: Weights and activations are mapped to 8-bit or 16-bit integers to reduce logic utilization and increase throughput.
• Quantization-Aware Training: Simulates quantization effects during training to minimize performance loss after conversion.
• Throughput Gain: A quantized RVTDNN can process samples at multi-GSPS rates required for wideband 5G signals.