Generalized Memory Polynomial (GMP)

The GMP introduces terms of the form x(n-m) · |x(n-m-l)|^k, where the signal sample and its envelope share the same time index n-m. This captures the interaction between a signal's instantaneous value and the memory of its own amplitude at different lags, modeling self-induced memory effects caused by impedance variations in the bias network.

• Mechanism: The lag l allows the envelope to influence the signal from a different temporal offset
• Benefit: Accurately models the dynamic AM-AM and AM-PM behavior that changes with signal bandwidth
• Typical range: l from 0 to 3 samples for most practical amplifiers

These terms use the structure x(n-m) · |x(n-m-l)|^k where l > 0, meaning the envelope value lags behind the signal sample. This models scenarios where the amplifier's thermal memory or charge trapping effects cause the current output to depend on past envelope values.

• Physical origin: Thermal time constants in the transistor junction and substrate
• Critical for: GaN HEMT amplifiers where trapping effects have millisecond-scale memory
• Implementation note: Lagging terms often require fewer polynomial orders k than aligned terms

The GMP uniquely includes terms x(n-m) · |x(n-m+l)|^k where l > 0, so the envelope leads the signal. While physically non-causal, these terms mathematically compensate for group delay variations and pre-ringing artifacts in the amplifier's frequency response.

• Purpose: Captures the inverse of lagging memory effects in the predistorter model
• Essential for: Wideband signals where the amplifier's phase response is non-linear
• Practical note: Leading terms are typically limited to small l values (1-2) to maintain model stability

The GMP is parameterized by three key dimensions: nonlinearity order K, memory depth M, and cross-term depth L. The total number of coefficients grows as O(M · K · L), requiring careful trade-offs between modeling accuracy and computational complexity.

• Typical values: K=5-7, M=3-5, L=2-4 for commercial Doherty PAs
• Coefficient count: A GMP with K=7, M=4, L=3 can have over 200 complex coefficients
• Optimization: Use LASSO regression or ridge regularization to prune redundant terms and prevent overfitting

GMP coefficients are typically extracted using least squares estimation in a matrix formulation. The basis functions are constructed into a regression matrix Φ, and the coefficients w are solved via w = (Φ^H Φ)^(-1) Φ^H y, where y is the observed PA output.

• QR decomposition or Cholesky factorization is used for numerical stability
• Indirect Learning Architecture (ILA) swaps input/output to avoid inverse modeling
• Computational cost: Matrix inversion scales as O(N³) where N is the number of coefficients
• Real-time adaptation: Recursive least squares (RLS) variants enable online coefficient updates

The standard memory polynomial uses only terms x(n-m) · |x(n-m)|^k, where the envelope and signal share the exact same lag. The GMP's cross-terms provide a superset of basis functions that capture inter-modulation between different time offsets.

• Standard MP: ~30-50 coefficients for typical configurations
• GMP: ~100-250 coefficients, offering 3-5 dB better ACLR suppression for wideband signals
• Trade-off: Higher computational cost vs. superior linearization for signals exceeding 100 MHz bandwidth
• When to use GMP: When the standard MP fails to meet ACLR targets for wideband or Doherty PAs

The defining innovation of the GMP model. These terms multiply the current complex baseband sample with the envelope magnitude of a delayed or advanced sample.

• Lagging Term: x(n) |x(n-q)|^{k-1} — captures the effect of past envelope values on the current output.
• Leading Term: x(n) |x(n+q)|^{k-1} — captures pre-distortion effects where the amplifier anticipates future signal peaks.
• Physical Origin: These terms model bias network dynamics and thermal memory that standard memory polynomials miss.

Complex interactions where the non-linear distortion at one instant depends on the signal envelope at a different instant. This is the core physical phenomenon that GMP addresses.

• Example: A high-power pulse at time (n-q) causes a thermal transient that shifts the amplifier's operating point, distorting the signal at time n.
• GMP Capture: The cross-terms x(n) |x(n-q)|^{k-1} directly model this coupling between time-separated samples.
• Impact: Neglecting these effects leads to incomplete linearization and residual spectral regrowth.

Even GMP can generate hundreds of coefficients. Pruning techniques identify and remove statistically insignificant terms to reduce computational load.

• Methods: Orthogonal matching pursuit, LASSO regression, or principal component analysis.
• Trade-off: Balancing normalized mean squared error (NMSE) against the number of active coefficients.
• Hardware Impact: A pruned GMP model with 20-30 coefficients can often match the performance of a full 100+ coefficient model, enabling real-time FPGA implementation.

Alternative block-structured behavioral models that also address memory effects, often compared against GMP in academic literature.

• Hammerstein: A static non-linearity followed by a linear filter.
• Wiener: A linear filter followed by a static non-linearity.
• Comparison: GMP generally outperforms these simpler cascaded structures for wideband signals where strong memory effects are present, as it captures non-linear memory interactions that separable block models cannot.