Generalized Memory Polynomial

The GMP introduces lagging envelope cross-terms of the form x(n-m) · |x(n-m-l)|^k. Unlike the standard memory polynomial, which only uses aligned samples, these terms capture the interaction between a signal sample at one delay and the envelope at a different delay. This explicitly models long-term memory effects caused by bias network impedance and thermal dynamics that simpler diagonal-only models cannot represent.

A GMP model is defined by three sets of coefficients:

• Aligned terms: Standard memory polynomial basis (memory depth M, nonlinear order K)
• Lagging cross-terms: Signal sample at delay m multiplied by envelope at delay m+l
• Leading cross-terms: Signal sample at delay m multiplied by envelope at delay m-l

The total coefficient count scales as O(M · K · L) where L is the cross-term depth. While more complex than a memory polynomial, it remains far more compact than a full Volterra series.

For wideband signals (e.g., 100 MHz 5G NR carriers), the GMP significantly outperforms the standard memory polynomial. The cross-terms capture the frequency-dependent behavior of the PA's bias network, which causes asymmetric intermodulation distortion sidebands. Measured ACLR improvements of 2-4 dB over memory polynomial models are typical when linearizing GaN Doherty PAs with instantaneous bandwidths exceeding 200 MHz.

GMP coefficients are estimated using least squares (LS) estimation on the linear-in-parameters model structure. The regressor matrix is constructed by applying the cross-term basis functions to the input data. Due to the expanded basis set, ill-conditioning is a concern:

• Moore-Penrose pseudoinverse or QR decomposition is used for numerical stability
• LASSO regularization can prune insignificant cross-terms, yielding a sparse GMP
• Indirect learning architecture (ILA) is commonly used for DPD coefficient extraction

GMP-based DPD is well-suited for FPGA and ASIC implementation due to its feedforward structure:

• Look-up tables (LUTs) can replace polynomial evaluations for |x|^k terms
• Cross-terms are computed by multiplying delayed signal samples with LUT outputs
• Typical implementations use 3-5 memory taps and 3-5 cross-term lags
• Resource utilization scales linearly with the number of retained coefficients after pruning
• Latency is deterministic and bounded by the maximum memory depth

The GMP occupies a sweet spot in the complexity-performance tradeoff:

• Full Volterra: O(M^K) coefficients — intractable for K > 3
• Memory Polynomial: O(M · K) coefficients — misses cross-memory effects
• GMP: O(M · K · L) coefficients — captures dominant cross-terms

For a typical 5G PA with M=5, K=7, L=3, a full Volterra would require thousands of coefficients, while a GMP needs only ~100-200, yet achieves comparable linearization performance.