Parallel Hammerstein

The model consists of multiple independent Hammerstein branches operating in parallel on the same input signal. Each branch applies a different static nonlinearity (e.g., polynomial terms of varying order) followed by a linear time-invariant (LTI) filter with distinct memory characteristics. The outputs of all branches are summed to produce the final model output. This structure naturally separates the modeling of different nonlinear orders and their associated memory effects, providing greater flexibility than a single Hammerstein or Wiener model.

Each branch contains a memoryless nonlinear function that operates on the instantaneous input sample. Common choices include:

• Polynomial terms: odd-order powers (x, x³, x⁵) for bandpass nonlinearity
• Spline functions: piecewise polynomial segments for smooth nonlinear approximation
• Look-up tables: arbitrary nonlinear transfer curves By assigning different nonlinear orders to separate branches, the model avoids the numerical ill-conditioning that plagues single-branch high-order polynomial models.

Following the static nonlinearity, each branch includes a linear filter that captures the memory effects associated with that specific nonlinear order. Filter implementations include:

• Finite Impulse Response (FIR) filters: tapped delay lines with complex coefficients
• Infinite Impulse Response (IIR) filters: for efficient modeling of long time constants
• Orthonormal basis filters: Laguerre or Kautz functions for parsimonious representation This per-branch filtering allows different nonlinear orders to exhibit different memory depths and time constants, matching physical PA behavior.

The Parallel Hammerstein model is a structured subset of the Volterra series. A full Volterra series with diagonal kernel representation can be decomposed into an equivalent Parallel Hammerstein structure. This connection provides a theoretical foundation for the model's completeness while offering practical advantages:

• Fewer coefficients than an equivalent memory polynomial for the same accuracy
• Better numerical conditioning due to the separation of nonlinear orders
• Modular implementation enabling independent optimization of each branch This makes it particularly suitable for FPGA and ASIC implementations where parallel processing is natural.

The linear-in-parameters structure of the Parallel Hammerstein model enables direct least squares estimation. The output is a linear combination of filtered nonlinear basis functions, so standard techniques apply:

• Batch Least Squares (LS): for offline model extraction from captured data
• Recursive Least Squares (RLS): for adaptive online coefficient tracking
• QR Decomposition (QRD): for numerically stable solutions with ill-conditioned data
• Orthogonal Matching Pursuit (OMP): for sparse branch selection and complexity reduction The decoupled branch structure also supports per-branch regularization, allowing independent control of model complexity for each nonlinear order.

The Parallel Hammerstein architecture offers several practical benefits for digital predistortion:

• Modular hardware mapping: each branch maps naturally to parallel DSP slices in FPGAs
• Scalable complexity: branches can be added or removed to trade accuracy vs. resources
• Independent adaptation: branches can be updated at different rates based on time constants
• Robust numerical properties: separation of nonlinear orders prevents the ill-conditioning common in high-order single-branch polynomials
• Physical interpretability: each branch corresponds to a distinct nonlinear mechanism with its own memory profile