Real-Valued Time-Delay Neural Network (RVTDNN)

Unlike complex-valued neural networks that process I and Q as a single complex entity, the RVTDNN decomposes the baseband signal into two independent real-valued streams. The I(t) and Q(t) components are fed into separate input nodes. This architectural choice simplifies gradient computation during backpropagation, as standard real-valued activation functions and optimizers can be used directly without modification. The network inherently learns the non-linear coupling between the amplitude and phase paths by combining these streams in subsequent hidden layers.

The defining feature of the RVTDNN is the tapped delay line (TDL) structure applied to both I and Q inputs. For a memory depth of M, the network receives not just the current sample x(n), but a vector of delayed samples: [x(n), x(n-1), ..., x(n-M)]. This transforms the static non-linear mapping into a dynamic model capable of capturing memory effects caused by:

• Thermal dynamics in the transistor substrate
• Bias network impedance variations at the modulation envelope rate
• Trapping effects in gallium nitride (GaN) devices Each delayed tap connects to all neurons in the first hidden layer, creating a fully connected temporal receptive field.

The RVTDNN employs a standard multi-layer perceptron (MLP) structure after the input TDL stage. Typical configurations use 1-2 hidden layers with hyperbolic tangent (tanh) or rectified linear unit (ReLU) activations. The feed-forward topology is sufficient because the tapped delay lines externalize the temporal processing, eliminating the need for recurrent connections. This results in:

• Stable training without vanishing or exploding gradients common in RNNs
• Parallelizable inference since no sequential state updates are required
• Deterministic latency ideal for real-time DPD in 5G NR physical layer pipelines

The output layer produces two real values: the predistorted I'(t) and Q'(t) components. These are recombined to form the complex baseband predistorted signal that drives the power amplifier. The training objective minimizes the mean squared error between the desired linear output and the actual PA output, effectively learning the inverse amplifier characteristic. Key design considerations include:

• Linear output activation to allow unbounded correction values
• Joint I/Q optimization where the loss function accounts for both amplitude and phase distortion simultaneously
• Coefficient extraction that maps directly to the modulator's digital-to-analog converter (DAC) inputs

The standard RVTDNN uses only the raw I and Q time-delayed samples as inputs. The Augmented RVTDNN (ARVTDNN) extends this by adding envelope-dependent terms: |x(n)|, |x(n)|², |x(n)|³, and their delayed versions. These additional inputs explicitly provide the instantaneous amplitude information, reducing the burden on the hidden layers to compute these non-linear transformations internally. The trade-off:

• Standard RVTDNN: Fewer input parameters, requires deeper/wider hidden layers to model envelope non-linearity
• ARVTDNN: Larger input dimensionality, but can achieve equivalent linearization with a smaller hidden layer structure
• Selection criteria: ARVTDNN is preferred for strongly non-linear Doherty PAs; standard RVTDNN suffices for mildly non-linear class-AB amplifiers

The RVTDNN is typically trained using the Indirect Learning Architecture (ILA). In this method, a post-distorter model is identified by swapping the PA input and output during training, then copied to the predistorter path. This avoids the need to compute a direct inverse of the PA model. The training process:

• Step 1: Capture PA input z(n) and normalized output y(n)/G (where G is linear gain)
• Step 2: Train the RVTDNN to map y(n)/G → z(n), learning the post-inverse
• Step 3: Deploy the trained weights as the predistorter: x(n) → z(n) This approach guarantees convergence to the optimal predistorter under the assumption that the PA is invertible and the model has sufficient capacity.

The defining structural element of an RVTDNN. A tapped delay line creates a finite memory window by feeding the network not just the current sample, but also P past samples of the I and Q components. This transforms a static non-linearity problem into a dynamic one the network can learn. The depth of the TDL (the number of taps) directly determines the maximum memory effect duration the model can capture, balancing model accuracy against computational complexity.

Unlike complex-valued neural networks that process in-phase (I) and quadrature (Q) data as a single complex entity, the RVTDNN treats them as two distinct real-valued input streams. Each stream has its own set of tapped delay lines and connections to the hidden layer. This separation allows the network to independently model AM-AM and AM-PM distortions without requiring complex backpropagation algorithms, simplifying implementation on standard deep learning frameworks.

The tapped delay line outputs feed into one or more fully connected hidden layers with non-linear activation functions. These layers learn the complex, high-dimensional mapping between the delayed I/Q inputs and the predistorted output. Common activation choices include hyperbolic tangent (tanh) for its smooth anti-symmetric properties, which suit the odd-symmetry of typical amplifier non-linearity. The depth and width of these layers define the model's capacity to fit severe non-linearities.

Training is performed using standard real-valued backpropagation with a mean squared error (MSE) cost function. The network is typically trained in an Indirect Learning Architecture (ILA), where the amplifier's input and output are swapped to directly identify the predistorter. The optimizer minimizes the difference between the predistorter output and the desired linear signal. This avoids the need for complex Wirtinger calculus required by complex-valued networks.

The RVTDNN is a direct neural extension of the Memory Polynomial (MP) model. While the MP model uses a fixed polynomial basis expansion of delayed terms, the RVTDNN replaces this rigid structure with a learned non-linear mapping. This gives the RVTDNN superior modeling capability for amplifiers with strong non-linear memory effects that cannot be easily captured by a truncated polynomial series, at the cost of increased computational load during inference.

To better capture complex envelope-dependent memory, the basic RVTDNN is often augmented with envelope-dependent terms. The instantaneous amplitude of the input signal is computed and fed as an additional input, sometimes with its own tapped delay line. This creates an Augmented RVTDNN (ARVTDNN) that explicitly provides the network with magnitude information, improving its ability to model envelope memory effects without requiring a deep network to first compute the magnitude internally.