Generalized Memory Polynomial (GMP)

The GMP model augments the standard memory polynomial with two distinct cross-term types: lagging envelope terms (|x(n-m)|·x(n-k)) and leading envelope terms (|x(n-m)|·x(n+k)). These terms capture the interaction between the instantaneous signal sample and the envelope at different time offsets, modeling effects where the PA's nonlinear behavior depends on the signal's recent amplitude history. This structure effectively represents long-term thermal memory and bias circuit modulation that simpler models miss.

GMP provides a tunable balance between modeling fidelity and computational cost through independent control of three parameter sets:

• Nonlinear order (P): Controls polynomial degree for aligned terms
• Memory depth (M): Number of taps for standard memory terms
• Cross-term depths (Lb, Lc): Lagging and leading envelope memory spans This granularity allows engineers to allocate coefficients where they matter most, achieving comparable accuracy to a full Volterra series with significantly fewer parameters.

GMP excels at predicting asymmetric spectral regrowth patterns that arise from complex memory effects in GaN and GaAs power amplifiers. By including envelope-sample cross-products, the model captures the frequency-dependent AM/AM and AM/PM characteristics that cause unequal upper and lower sideband distortion. This makes GMP particularly effective for wideband signals (100+ MHz) in 5G NR applications where memory effects span many symbol periods.

Despite its enhanced modeling capability, GMP maintains a linear-in-parameters structure, meaning the output is a linear combination of basis functions weighted by complex coefficients. This property enables direct application of efficient estimation algorithms:

• Least Squares (LS) for batch coefficient extraction
• Recursive Least Squares (RLS) for adaptive tracking
• QR Decomposition (QRD-RLS) for numerically stable implementation The linear structure avoids the convergence challenges of nonlinear optimization methods.

The cross-term structure of GMP introduces correlation between basis functions, which can lead to ill-conditioned estimation matrices. Numerical stability is restored through orthogonalization techniques:

• Gram-Schmidt orthogonalization decorrelates basis functions sequentially
• Principal Component Analysis (PCA) identifies and retains dominant components
• Orthonormal basis transformations (Laguerre, Kautz) reparameterize the model Proper orthogonalization ensures robust coefficient extraction even with limited observation data.

GMP maps efficiently to FPGA and ASIC implementations through its structured computation pattern. The model can be decomposed into:

• Look-Up Tables (LUTs) indexed by signal magnitude for memoryless nonlinearity
• FIR filter structures for memory polynomial terms
• Parallel multiply-accumulate paths for cross-term evaluation This regularity enables high-throughput pipelined architectures operating at sample rates exceeding 491.52 MSPS for wideband DPD applications.

A variant that incorporates memory effects of the signal's envelope magnitude rather than just the complex baseband signal itself. This structure is particularly effective at capturing long-term thermal and bias-related memory.

• Models the slow-varying envelope separately from the fast-varying carrier
• Addresses self-heating effects in GaN and GaAs power amplifiers
• Often combined with GMP for comprehensive behavioral modeling

A complexity reduction technique that removes insignificant kernels from a full Volterra series model. The GMP can be viewed as a pruned Volterra series where only specific cross-term structures are retained.

• Uses significance metrics to rank kernel importance
• Retains only diagonal and near-diagonal kernel slices
• Reduces coefficient count from O(M³) to O(M²) compared to full Volterra

The systematic selection of which cross-terms to include in a GMP model. Poor cross-term selection leads to overfitting and numerical instability, while optimal selection maximizes linearization accuracy.

• Lagging cross-terms: Signal × delayed envelope magnitude
• Leading cross-terms: Signal × advanced envelope magnitude
• Selection criteria: Correlation analysis, mutual information, or sparse regression

A numerical conditioning process that transforms the correlated GMP basis functions into an orthogonal set. This dramatically improves the stability and convergence speed of coefficient estimation algorithms like Least Squares.

• Prevents ill-conditioned data matrices during extraction
• Enables fixed-point implementation on FPGA hardware
• Common methods: Gram-Schmidt, QR decomposition

Block-structured alternatives to GMP that separate nonlinearity and memory into distinct cascaded blocks. These models offer lower complexity but cannot capture the interleaved nonlinear-memory interactions that GMP handles.

• Hammerstein: Static nonlinearity → Linear filter
• Wiener: Linear filter → Static nonlinearity
• GMP advantage: Simultaneous nonlinearity and memory in cross-terms