Behavioral Model

A behavioral model is derived exclusively from observed data rather than from first-principles physics or equivalent circuit schematics. It treats the power amplifier as a 'black box,' learning the nonlinear transfer function directly from measured complex baseband waveforms. This approach bypasses the need for proprietary semiconductor doping profiles or intricate electromagnetic field solvers, enabling rapid model extraction from standard vector network analyzer measurements.

Unlike static nonlinearities, behavioral models capture dynamic memory effects where the current output depends on past inputs. This is critical for modern GaN and GaAs Doherty amplifiers where thermal trapping and bias network reactance cause long-term dependencies. Models incorporate memory through:

• Tapped delay lines in feedforward structures
• Recurrent connections in neural network architectures
• Volterra kernel diagonals in polynomial frameworks Without memory modeling, wideband signals exhibit significant prediction error in the adjacent channels.

Behavioral models operate in the complex baseband equivalent domain, representing signals as in-phase (I) and quadrature (Q) components. This eliminates the high-frequency carrier from computations, dramatically reducing the required sampling rate to just the modulation bandwidth. The model maps complex input envelopes to complex output envelopes, inherently capturing both AM-AM distortion (gain compression/expansion) and AM-PM distortion (phase shift vs. amplitude) simultaneously.

A fundamental tension exists between model fidelity and robustness. A model with excessive degrees of freedom may achieve near-zero Normalized Mean Square Error (NMSE) on training data but fail catastrophically on unseen modulation schemes. Key mitigation strategies include:

• Cross-validation with held-out signal datasets
• Regularization (L1/L2 penalty) during coefficient extraction
• Coefficient sparsity analysis to prune redundant terms A well-regularized model maintains consistent Adjacent Channel Error Power Ratio (ACEPR) across diverse test signals.

The choice of model architecture balances accuracy, computational complexity, and numerical stability. Common structures form a complexity hierarchy:

• Memory Polynomial: Diagonal Volterra terms only; efficient for mild memory
• Generalized Memory Polynomial: Adds cross-terms for strong memory effects
• Volterra Series: Full nonlinear dynamic description but suffers from coefficient explosion
• Neural Networks: Universal approximators requiring careful architecture design to avoid overfitting
• LSTM Networks: Specialized for long-range temporal dependencies in thermal memory

Model extraction follows a rigorous workflow:

1. Stimulus design: Apply wideband signals (e.g., OFDM) that exercise the amplifier's nonlinear range
2. Data acquisition: Capture synchronized input-output complex envelope pairs
3. Parameter estimation: Solve for coefficients using Least Squares (LS) or iterative Least Mean Squares (LMS)
4. Validation: Evaluate NMSE, ACEPR, and Error Vector Magnitude (EVM) on independent test signals Numerical stability, quantified by the condition number of the regression matrix, dictates whether direct inversion or iterative solvers are required.

Quantitative measures to assess behavioral model fidelity. Key metrics include:

• Normalized Mean Square Error (NMSE): Measures average prediction error power relative to reference. Values below -35 dB indicate excellent model fit
• Adjacent Channel Error Power Ratio (ACEPR): Specifically evaluates prediction accuracy in adjacent channels, critical for DPD applications
• Error Vector Magnitude (EVM): In-band signal quality metric measuring constellation distortion

Cross-validation is essential: partition data into training and test sets to ensure the model generalizes to unseen signals rather than memorizing noise.