Behavioral Modeling

Behavioral models operate purely on observed input-output data, ignoring internal device physics such as electron transport, thermal dynamics, and trapping effects. This abstraction enables rapid model development using only measured waveform datasets. The approach treats the amplifier as a non-linear dynamic system characterized solely by its terminal behavior, making it agnostic to the underlying technology—whether LDMOS, GaN, or GaAs.

Unlike static non-linearity models, behavioral models explicitly capture memory effects—the dependence of current output on past input values. These arise from:

• Thermal dynamics: Die temperature changes with signal envelope
• Bias network impedance: Frequency-dependent biasing modulation
• Trapping effects: Charge capture/release in semiconductor defects Memory depth is typically modeled using tapped delay lines or recurrent structures spanning several symbol periods.

Behavioral models represent the non-linear system as a weighted sum of basis functions applied to the input signal. Common expansions include:

• Memory polynomials: Simple delayed envelope terms
• Volterra kernels: Multi-dimensional convolution for full non-linear dynamics
• Generalized Memory Polynomials (GMP): Cross-terms between signal and lagging/leading envelope values
• Neural network activations: Learned non-linear transformations replacing fixed polynomial bases The choice of basis determines the model's expressiveness vs. complexity trade-off.

Model parameters are estimated by solving an optimization problem that minimizes the error between the model's predicted output and measured amplifier output. Key approaches:

• Least Squares (LS): Direct matrix inversion for linear-in-parameters models
• Recursive Least Squares (RLS): Online adaptation with forgetting factors
• Stochastic Gradient Descent: Iterative optimization for neural network models
• Regularization techniques: Ridge regression or LASSO to prevent overfitting The identification signal must be persistently exciting across the amplifier's operating range.

Behavioral model accuracy is quantified using standardized RF metrics:

• Normalized Mean Squared Error (NMSE): Time-domain waveform fidelity, typically below -35 dB for acceptable models
• Adjacent Channel Error Power Ratio (ACEPR): Frequency-domain spectral regrowth prediction accuracy
• Error Vector Magnitude (EVM): Constellation distortion after applying the model Cross-validation on unseen test signals is essential to verify generalization beyond the training dataset.

A fundamental challenge in behavioral modeling is balancing model fidelity against generalization. Over-parameterized models may memorize training data noise rather than learning the true system dynamics. Mitigation strategies include:

• Cross-validation on held-out signal segments
• Regularization to penalize excessive coefficient magnitudes
• Model order reduction via pruning or principal component analysis
• Diverse training signals covering amplitude, bandwidth, and PAPR variations The goal is a parsimonious model that accurately predicts behavior for any valid input signal, not just the training set.

The theoretical bedrock of non-linear dynamic system modeling. Represents a system's output as a sum of multi-dimensional convolution integrals, capturing both static non-linearity and memory effects.

• Kernels: Each order (1st, 3rd, 5th) has an associated kernel describing its contribution
• Truncation: Practical models limit the series to odd-order terms and finite memory depth
• Complexity: Full Volterra models suffer from a curse of dimensionality, making them impractical for real-time DPD without pruning

A workhorse behavioral model that extends the standard memory polynomial by introducing cross-terms between the signal and its lagging or leading envelope values.

• Captures complex memory effects missed by simpler models
• Lagging cross-terms: Model thermal trapping effects
• Leading cross-terms: Model bias circuit dynamics
• Balances modeling accuracy with coefficient count, making it suitable for hardware implementation

A coefficient identification strategy that avoids the need to compute a direct inverse of the power amplifier model.

• Mechanism: Swaps the input and output of the PA model to estimate the post-inverse
• Assumption: Relies on the commutability of the pre-inverse and post-inverse, which holds for static non-linearities but can degrade with strong memory effects
• Advantage: Computationally simpler than direct methods and widely used in commercial DPD systems

An iterative optimization approach that directly minimizes the error between the desired linear output and the actual PA output.

• Gradient-based: Updates predistorter coefficients using the chain rule through the PA model
• No commutability assumption: Handles strong memory effects more robustly than ILA
• Computational cost: Requires a forward PA model and iterative solver, increasing complexity
• Often implemented with real-time coefficient adaptation loops

Replaces polynomial-based models with deep neural networks to capture complex non-linear and memory behaviors that exceed the representational capacity of Volterra-derived structures.

• RVTDNN: Real-Valued Time-Delay Neural Network processes I and Q components with tapped delay lines
• Architectures: Feed-forward, recurrent (LSTM/GRU), and convolutional networks are all explored
• Advantage: Learns arbitrary non-linear functions without manual basis function selection
• Challenge: Higher computational complexity requires careful model compression for real-time deployment

The aggregate quality metric that quantifies the deviation of transmitted constellation points from their ideal positions, capturing the combined impact of all impairments.

• Definition: The root-mean-square of the error vector normalized to the ideal symbol magnitude
• Constellation collapse: High EVM indicates symbols are smeared, increasing bit error rate
• DPD target: Effective linearization directly reduces EVM by suppressing AM-AM and AM-PM distortion
• Typical 5G requirements demand EVM below 3.5% for 256-QAM