Augmented Hammerstein Model

The core architecture places a static nonlinearity block before a linear time-invariant (LTI) filter. This cascade reflects the physical reality of power amplifiers, where the nonlinear distortion generated by the transistor is subsequently shaped by the matching network's frequency-dependent memory. The static block typically models AM/AM and AM/PM conversion via a complex-valued polynomial or spline function, while the LTI filter captures electrical memory effects from bias circuits and impedance matching.

Unlike the standard Hammerstein model, the augmented version adds parallel nonlinear branches that operate on delayed and cross-multiplied signal terms. These branches capture nonlinear memory effects—interactions between the signal's envelope at different time instants—that a single cascade cannot represent. Common augmentations include:

• Lagging cross-terms: products of the current sample with delayed envelope values
• Leading cross-terms: products with future envelope values for improved phase alignment
• Higher-order Volterra cross-kernels restricted to dominant memory depths

The model operates directly on complex baseband I/Q signals, preserving both magnitude and phase information essential for predistortion. The static nonlinearity is typically expressed as a complex polynomial where odd-order terms dominate due to bandpass filtering. The LTI filter is implemented as a complex finite impulse response (FIR) structure, with coefficients that shape the frequency-dependent distortion. This complex-valued formulation avoids the information loss that occurs when separating and processing I and Q components independently.

A key mathematical property is the separability of static nonlinearity and linear memory, which simplifies parameter extraction. The static block can be identified first using single-tone or narrowband measurements, then the LTI filter and augmentation branches are fitted using wideband modulated signals. This two-stage identification reduces the dimensionality of the optimization problem compared to full Volterra series extraction, enabling faster convergence with fewer training samples.

When implemented as a neural network, the augmented Hammerstein structure maps to a custom feedforward topology. The static nonlinearity becomes a layer with polynomial activation functions, the LTI filter is realized as a tapped delay line with trainable weights, and the augmentation branches form parallel paths that merge at the output. This neural instantiation allows gradient-based training using backpropagation, enabling joint optimization of all parameters directly from measured PA input-output data.

The augmented structure generalizes well across varying signal bandwidths, peak-to-average power ratios (PAPR), and modulation formats because the static nonlinearity captures the device's intrinsic transfer characteristic while the dynamic blocks adapt to frequency-dependent effects. This contrasts with pure memory polynomial models that may overfit to specific training signal statistics. The model's physically motivated structure acts as an inductive bias, improving extrapolation to operating conditions not seen during training.