Memory Polynomial

The memory polynomial retains only diagonal terms from the full Volterra series, eliminating cross-terms between different delay taps. This reduces the number of coefficients from O(K^M) to O(K×M), where K is the nonlinearity order and M is the memory depth. The model output is expressed as a sum of polynomials applied to delayed input samples, making it inherently parallelizable for FPGA implementation.

The nonlinearity order K determines which odd-order intermodulation products the model can capture. For typical Class-AB power amplifiers:

• K=5 captures 3rd and 5th order intermodulation
• K=7 extends to 7th order products for strongly compressed amplifiers
• K=9 or higher addresses deep saturation regions

Higher orders increase spectral regrowth prediction accuracy but risk overfitting if the memory depth is insufficient.

The memory depth M defines how many past samples influence the current output. Selection depends on the amplifier's time constants:

• M=2-3: Sufficient for low-bandwidth signals with minimal thermal trapping
• M=4-6: Required for wideband LTE and 5G NR signals where electrical memory effects from bias networks become significant
• M>6: Addresses long-term thermal memory in GaN HEMT amplifiers

Excessive depth introduces coefficient noise amplification during least-squares extraction.

Memory polynomial coefficients are extracted by formulating a linear regression problem in the complex baseband domain. The data matrix is constructed from delayed and nonlinearly transformed input samples. Ordinary least squares (OLS) provides a closed-form solution, but for ill-conditioned matrices, ridge regression or LASSO regularization is applied to enforce coefficient sparsity and improve numerical stability. The condition number of the data matrix directly impacts extraction accuracy.

The generalized memory polynomial (GMP) extends the standard model by adding cross-terms between different delays and envelope-dependent terms. This captures more complex memory interactions, such as:

• Lagging envelope terms: Products of the current sample with delayed envelope powers
• Leading envelope terms: Products of delayed samples with the current envelope

GMP bridges the accuracy gap between the memory polynomial and the full Volterra series while maintaining significantly lower complexity.

The memory polynomial's data matrix can become ill-conditioned when:

• The input signal has insufficient peak-to-average power ratio (PAPR) to excite all nonlinear orders
• Memory depth is over-specified relative to the signal bandwidth
• Higher-order polynomial terms create near-linear dependencies between columns

Mitigation strategies include Tikhonov regularization, orthogonal polynomial basis functions (e.g., Chebyshev polynomials), and careful signal design for model extraction.

The comprehensive mathematical foundation from which the memory polynomial is derived. It models nonlinear dynamic systems using multi-dimensional convolution kernels, capturing all possible interactions between current and past inputs. The memory polynomial simplifies this by retaining only the diagonal terms, dramatically reducing computational complexity while preserving essential memory effects.

The dependence of a power amplifier's current output on past input values, not just the instantaneous input. Caused by:

• Thermal memory: Slow temperature changes in the transistor junction
• Electrical memory: Bias circuit impedance variations with frequency
• Trapping effects: Charge capture in semiconductor defects The memory polynomial explicitly models these time-dispersive behaviors through its delayed terms.

An extension of the standard memory polynomial that incorporates cross-terms between different time delays and nonlinear orders. While the standard model uses only diagonal terms (same delay for all nonlinear orders), the generalized variant adds off-diagonal terms like x(n-1)²x(n-2) to capture more complex memory interactions. This improves accuracy for amplifiers with strong memory effects at the cost of increased coefficient count.

Amplitude-to-Amplitude distortion is the nonlinear relationship between input signal amplitude and output signal amplitude. In a perfectly linear amplifier, this relationship is a straight line. Real amplifiers exhibit gain compression at high power levels and potentially gain expansion in certain regions. The memory polynomial's nonlinear terms (odd-order powers) are specifically designed to model and invert this amplitude-dependent gain variation.

Amplitude-to-Phase distortion converts input amplitude variations into output phase shifts. This is particularly damaging to spectrally efficient modulations like QAM and OFDM, where phase integrity is critical. The memory polynomial captures this through its complex-valued coefficients, where the phase of each coefficient represents the phase shift introduced at that nonlinear order and memory depth.

The primary coefficient extraction method for memory polynomial models. Given measured input-output data, least squares finds the coefficient vector that minimizes the sum of squared errors between the model prediction and actual output. The memory polynomial's linear-in-parameters structure makes this particularly efficient:

• Form the regressor matrix from delayed and exponentiated input samples
• Solve the normal equations or use QR decomposition
• Results in a globally optimal solution for the given data