Technical Summary: Sparse Solution Trade-offs in GMP DPD via Least Squares Thresholding

Problem Statement

Satellite communication systems face a critical challenge in balancing high data throughput with strict Size, Weight, and Power (SWaP) constraints inherent to space-borne hardware. A primary bottleneck is the Power Amplifier (PA), which must operate near saturation to maximize power efficiency. However, this operation induces significant nonlinear distortions, specifically AM/AM (amplitude-to-amplitude) and AM/PM (amplitude-to-phase) nonlinearities. These effects cause spectral regrowth into adjacent frequency bands, violating regulatory spectral masks (e.g., FCC, ITU) and degrading constellation diversity at the receiver.

To mitigate these issues, Digital Pre-Distortion (DPD) is employed, typically modeling the PA using Generalized Memory Polynomials (GMP), a specialized subset of the Volterra series. While GMP effectively models memory effects and cross-terms, utilizing the full set of GMP kernels is computationally prohibitive for SWaP-constrained environments. The selection of an optimal subset of kernels is an NP-Hard problem. Existing dynamic search methods, such as Orthogonal Matching Pursuit (OMP) and Matching Pursuit (MP), address this but incur high computational overhead. Furthermore, the paper notes that MP relies on an orthogonality assumption that breaks down when applied to the highly correlated feature matrices characteristic of GMP models.

Methodology

The authors propose a static search approach called Least Squares Thresholding (LST) to identify a sparse subset of GMP kernels. The methodology operates within an Iterative Learning Architecture (ILA), where the inverse of the PA is estimated and applied to the pre-distortion block.

1. Model Formulation: The PA output is modeled as a linear combination of GMP basis functions. The goal is to solve for the coefficient vector $x$ in the equation $Ax = b$, where $A$ is the GMP feature matrix and $b$ is the target signal.
2. Sparse Optimization: Instead of solving for all coefficients, the objective is to minimize the $L_0$ norm (number of non-zero elements) of the solution vector $\hat{x}$ subject to the constraint $A\hat{x} = b$.
3. Algorithm Comparison:
   • OMP (Orthogonal Matching Pursuit): A greedy algorithm that iteratively selects the column most correlated with the residual and re-solves the least squares problem for the entire selected set at every step. It is computationally expensive due to repeated matrix inversions.
   • MP (Matching Pursuit): Simplifies OMP by updating coefficients via scalar projection, assuming the selected kernels are orthogonal to the previously selected subspace. The paper argues this assumption is invalid for GMP due to high inter-kernel correlation.
   • LST (Least Squares Thresholding): A static approach that computes initial correlations between all columns and the target signal once. It sorts these correlations in descending order and greedily adds kernels to the support set based on this fixed order, solving a least squares problem only for the current set. This avoids the repeated re-correlation and matrix operations of OMP.

Key Contributions

• Systematic Comparison: This work presents the first systematic comparison of a static search approach (LST) against dynamic greedy algorithms (OMP and MP) specifically for sparse GMP-DPD under SWaP constraints.
• Theoretical Insight: The paper provides a theoretical demonstration of why MP's underlying orthogonality assumption fails in the context of highly correlated GMP feature matrices, leading to suboptimal performance.
• Complexity Reduction: The authors demonstrate that LST achieves a significant reduction in computational complexity compared to OMP while maintaining near-identical linearization performance.
• Novel Metric Application: The study utilizes Normalized Mutual Information (NMI) as a fidelity metric alongside standard RF metrics (ACPR, EVM, NMSE) to capture statistical dependence and nonlinear residual distortions that linear correlation might miss.

Results

Experiments were conducted using a GaN-based power amplifier (Cree CGH40006-TB) with a 200 MHz bandwidth 64-QAM signal. The GMP model utilized 322 total kernels ($h=11, l=6, j=4$).

• Performance: At approximately 20–25 non-zero kernels, both LST and OMP achieved performance metrics (NMSE, ACPR, EVM, NMI) nearly identical to the full ridge regression solution.
  • EVM: LST achieved an EVM of -27.48 dB, significantly outperforming the 3GPP 64-QAM limit of -21.94 dB.
  • NMI: LST marginally outperformed Ridge Regression (0.7365 vs. 0.7335), suggesting implicit regularization.
  • MP Performance: MP showed notably poorer performance across all metrics (e.g., EVM of -24.27 dB), confirming the negative impact of the invalid orthogonality assumption.
• Complexity: LST demonstrated a 2.77× reduction in computational operations compared to OMP.
  • LST Operations: $\approx 2.67 \times 10^{11}$
  • OMP Operations: $\approx 7.39 \times 10^{11}$
  • MP Operations: $\approx 4.72 \times 10^{11}$

Significance and Claims

The paper claims that LST offers a practical solution for implementing sparse GMP-DPD in space-borne hardware where real-time processing and low power consumption are critical. By replacing dynamic, iterative search methods with a static thresholding approach, the authors achieve a drastic reduction in FPGA computational load without sacrificing linearization quality.

The significance of this work lies in its validation that a static, pre-computed ordering of kernels is sufficient for GMP-DPD, challenging the necessity of more complex dynamic algorithms like OMP in this specific domain. The authors conclude that future work will focus on integrating this sparse kernel selection into an Iterative Learning Control (ILC) loop and validating the approach on actual FPGA hardware.