Forward Modeling

Forward modeling is a system identification approach that treats the power amplifier as a black-box dynamic system. The objective is to find a mathematical function f(·) such that ŷ(n) = f(x(n)) closely approximates the measured output y(n). This contrasts with inverse modeling, where input and output roles are swapped to directly estimate the predistorter. Forward models are essential for offline simulation, allowing engineers to evaluate linearization strategies without continuous hardware access. The model captures both static nonlinearity (AM-AM, AM-PM conversion) and dynamic memory effects caused by trapping, thermal phenomena, and bias network impedance.

The extraction process is formulated as a linear regression problem when using polynomial-based models. The output is expressed as a weighted sum of basis functions:

• y = X·θ, where X is the regression matrix of basis functions evaluated on the input signal
• θ is the vector of unknown model coefficients
• The system is typically overdetermined, with far more measurement samples than parameters

This structure enables efficient solution via least squares (LS) estimation. The Moore-Penrose pseudoinverse provides the optimal coefficient vector in a single batch computation. For real-time adaptation, recursive least squares (RLS) updates coefficients iteratively as new samples arrive.

The fidelity of an extracted forward model depends critically on the training waveform used to excite the amplifier. Effective stimuli must:

• Exercise the full dynamic range of the PA, including compression and saturation regions
• Possess a peak-to-average power ratio (PAPR) representative of the target modulation scheme
• Provide sufficient spectral richness to identify frequency-dependent memory effects
• Maintain persistent excitation to ensure the regression matrix is well-conditioned

Common choices include OFDM signals, noise-like waveforms, and multi-tone stimuli. Insufficient excitation leads to models that fail to generalize to operational signals.

Choosing the appropriate model structure balances accuracy against computational complexity. Key considerations include:

• Nonlinearity order: Higher orders capture severe compression but increase parameter count
• Memory depth: Longer memory taps model low-frequency thermal and trapping effects
• Basis function pruning: Removing correlated terms reduces ill-conditioning without sacrificing fidelity

Tools like the Akaike Information Criterion (AIC) and cross-validation guide model order estimation. Overly complex models risk overfitting to measurement noise, while overly simple models fail to capture essential nonlinear dynamics.

Forward model extraction frequently encounters ill-conditioned regression matrices due to high correlation among polynomial basis functions. This manifests as:

• Extreme sensitivity of coefficients to minor measurement noise
• Unstable solutions where small data changes produce wildly different parameter estimates
• Elevated condition numbers indicating near-singular covariance matrices

Mitigation strategies include ridge regression (L2 regularization), principal component analysis (PCA) for basis orthogonalization, and QR decomposition for numerically stable pseudoinverse computation. Proper time alignment between reference and captured signals to sub-sample accuracy is a critical pre-processing step.

A forward model's utility is measured by its ability to generalize beyond the training data. Validation employs:

• Normalized mean squared error (NMSE) between modeled and measured outputs
• Adjacent channel power ratio (ACPR) prediction accuracy for spectral regrowth
• Cross-validation using held-out signal segments not seen during training
• Time-domain waveform comparison to verify envelope fidelity

A well-extracted forward model serves as a digital twin of the physical amplifier, enabling rapid prototyping of predistortion algorithms, efficiency optimization studies, and system-level simulations without repeated hardware measurements.