Generalized Memory Polynomial

The defining feature of the GMP model is its inclusion of lagging and leading cross-terms. Unlike the standard memory polynomial which only uses diagonal terms, the GMP adds terms that multiply a delayed sample by the envelope of a differently delayed sample raised to a nonlinear order. This captures the complex interactions between short-term and long-term memory effects that arise from bias circuit dynamics, thermal trapping, and impedance mismatches.

GMP models explicitly use the complex baseband envelope to couple nonlinearity with memory. A typical cross-term takes the form:

• x(n - m) * |x(n - m - l)|^k where m is the memory tap, l is the lag offset, and k is the nonlinear order. This formulation allows the model to represent how the instantaneous gain compression at one time instant is influenced by the signal amplitude at a different, offset time instant.

The GMP can be understood as a pruned Volterra series that selectively retains only the most physically significant kernel slices. While a full Volterra series grows exponentially with memory depth and nonlinear order, the GMP imposes a structured sparsity by:

• Keeping all diagonal terms (standard memory polynomial)
• Adding a controlled set of off-diagonal terms with limited lag/lead offsets This dramatically reduces the coefficient count while preserving the ability to model strong nonlinear memory.

A typical GMP implementation decomposes the output into three parallel branches:

1. Aligned Envelope Branch: Terms where the complex input and envelope are at the same time delay — equivalent to the standard memory polynomial
2. Lagging Envelope Branch: Terms where the envelope is delayed relative to the complex input, capturing slow bias modulation effects
3. Leading Envelope Branch: Terms where the envelope leads the complex input, modeling precursor memory phenomena in charge trapping This decomposition maps directly to physical amplifier behavior.

Despite its nonlinear structure, the GMP model is linear in its coefficients. The output is expressed as a weighted sum of basis functions:

• y(n) = Σ c_ij * φ_ij(x(n)) where each φ_ij is a known nonlinear function of the input signal. This property enables closed-form least squares estimation of all coefficients simultaneously, avoiding the convergence issues and local minima associated with iterative nonlinear optimization. The coefficient vector can be extracted directly via pseudo-inverse computation.

The inclusion of cross-terms can lead to ill-conditioned data matrices during coefficient extraction, especially with highly correlated wideband signals. Practical GMP implementations employ:

• Tikhonov regularization to penalize large coefficient magnitudes
• Orthogonal basis function transformations to decorrelate the regressors
• Principal component analysis (PCA) to reduce the effective dimensionality These techniques ensure robust model extraction without overfitting to measurement noise.

The direct predecessor to the GMP. The Memory Polynomial includes only diagonal terms where the delay index and nonlinear order are aligned. While computationally efficient, it struggles to capture strong memory effects caused by complex impedance interactions or trapping. The GMP extends the MP by adding lagging and leading cross-terms to model these envelope-dependent memory dynamics.

The most general polynomial-based model for nonlinear systems with memory. It uses multi-dimensional convolution kernels to capture all possible interactions between delayed signal components. The GMP is a pruned Volterra series that retains only the most physically significant cross-terms. Full Volterra models are often impractical due to an exponential explosion of coefficients with increasing nonlinearity order and memory depth.

The standard batch algorithm for extracting GMP coefficients from measured input-output data. It solves for the coefficient vector by minimizing the sum of squared residuals between the model prediction and the observed amplifier output. For the GMP, the data matrix is constructed from the basis waveforms defined by the polynomial's cross-term structure. Numerical stability depends on the condition number of this matrix.

The primary fidelity metric used to validate GMP model accuracy. NMSE quantifies the average power of the modeling error normalized by the power of the reference signal, expressed in dB. A well-extracted GMP typically achieves NMSE values below -40 dB for a Class-AB amplifier. NMSE primarily reflects in-band modeling accuracy and should be complemented by ACEPR for out-of-band validation.

A critical pitfall in GMP extraction. Including too many cross-terms can cause the model to memorize measurement noise rather than learn the underlying amplifier physics. This produces excellent training NMSE but poor generalization to new signals. Ridge regression (L2 regularization) or LASSO (L1 regularization) adds a penalty on coefficient magnitude to enforce smoothness and promote sparsity in the GMP coefficient vector.

A model validation metric specifically designed to assess out-of-band prediction accuracy. While NMSE measures in-band error, ACEPR quantifies the error power in the adjacent channels where spectral regrowth occurs. A GMP model optimized solely for NMSE may exhibit poor ACEPR, indicating it fails to capture the distortion mechanisms responsible for spectral regrowth. Joint NMSE/ACEPR optimization is often required.