Memory Polynomial

The defining structural simplification of the Memory Polynomial is the elimination of all off-diagonal Volterra kernel terms. A full Volterra series includes cross-terms where, for example, the input at time n-1 multiplies with the input at time n-2. The Memory Polynomial sets all such cross-terms to zero.

• Mathematical Form: The output is a sum over nonlinear orders k and memory taps q, using only terms of the form x(n-q) * |x(n-q)|^(k-1).
• Complexity Reduction: This reduces the number of coefficients from exponential growth O(M^K) in a full Volterra model to a linear product O(K × M), where K is the nonlinear order and M is the memory depth.

The Memory Polynomial's structure is not just a mathematical convenience; it aligns with the causal physics of many power amplifier non-idealities. It inherently models the fact that a signal's envelope distorts itself over time.

• AM-AM and AM-PM: Each polynomial term x(n-q) * |x(n-q)|^(k-1) directly captures the amplitude-dependent gain and phase shift at a specific memory lag.
• Self-Distortion: It models how a signal's own past envelope power causes current distortion, which is the dominant mechanism for self-heating and bias network modulation effects.
• Limitation: It fails to capture cross-modulation between signals at different time lags, which can be significant in wideband Doherty amplifiers with complex combining networks.

A critical practical advantage is that the Memory Polynomial is linear with respect to its coefficients. This allows for robust, one-shot parameter extraction using standard linear algebra techniques.

• Least Squares (LS): The optimal coefficients can be found directly by solving the normal equations, c = (X^H X)^(-1) X^H y, where X is the regressor matrix of basis waveforms.
• No Local Minima: Unlike neural network-based predistorters, the LS estimation guarantees a global minimum for the given model structure, ensuring repeatable and deterministic training.
• Regularization: Techniques like Ridge regression or LASSO can be easily incorporated to handle ill-conditioned data and perform automatic coefficient pruning.

To address the Memory Polynomial's inability to model cross-terms, the Generalized Memory Polynomial adds lagging envelope terms. This captures the interaction between a signal and the delayed envelope of another signal.

• Envelope Cross-Terms: GMP introduces terms like x(n-q) * |x(n-q-r)|^(k-1), where r is a lag difference, modeling how a past signal's power affects the current signal's distortion.
• Complexity Trade-off: While more accurate for wideband signals, the GMP's coefficient count grows significantly, requiring careful pruning using algorithms like Orthogonal Matching Pursuit.
• Use Case: GMP is often preferred for Doherty PAs and envelope tracking systems where complex low-frequency memory effects dominate.

The Vandermonde-like structure of the Memory Polynomial's basis matrix can lead to severe ill-conditioning, especially with high nonlinear orders. This makes coefficient estimation sensitive to measurement noise.

• Ill-Conditioning: As the polynomial order K increases, the columns of the regressor matrix become highly correlated, causing the condition number to explode.
• Orthogonal Polynomials: A common mitigation is to replace the standard polynomial basis |x|^k with orthogonal polynomials (e.g., Chebyshev or Legendre polynomials) that are mutually uncorrelated for a given signal distribution.
• Impact: Using an orthogonal basis dramatically improves the convergence speed of adaptive algorithms and the numerical stability of the coefficient extraction.

The Memory Polynomial is the most common model paired with the Indirect Learning Architecture for DPD. Its linear-in-parameters nature makes it ideal for the post-distorter identification step.

• Post-Distorter Training: In ILA, a Memory Polynomial is first trained to invert the PA's response by swapping the input and output signals. The extracted coefficients are then copied directly to the pre-distorter.
• No Model Inversion: This elegantly avoids the difficult analytical inversion of a nonlinear model, as the training process directly identifies the inverse.
• Adaptation: The coefficient vector can be updated on a sample-by-sample basis using Recursive Least Squares (RLS) or block-by-block using Least Mean Squares (LMS) for tracking temperature-driven changes.