Kernel Ridge Regression

Unlike neural networks that require iterative backpropagation, KRR computes predistorter coefficients analytically via matrix inversion. The solution is obtained by solving (K + λI)α = y, where K is the kernel matrix, λ is the ridge regularization parameter, and y represents the target predistorted signal. This eliminates convergence uncertainty and guarantees a globally optimal solution for the given regularization hyperparameter.

KRR implicitly maps input I/Q vectors into a high-dimensional feature space without explicitly computing the transformation. Common kernel functions include:

• Radial Basis Function (RBF): k(x, x') = exp(-γ||x - x'||²), capturing smooth nonlinearities
• Polynomial kernel: k(x, x') = (xᵀx' + c)ᵈ, modeling structured nonlinear interactions
• Rational quadratic kernel: Handles multi-scale PA memory effects The kernel trick enables modeling of severe AM/AM and AM/PM distortion without manual basis function engineering.

The ridge penalty term (λ) in KRR serves dual purposes in predistorter coefficient extraction:

• Overfitting prevention: Shrinks coefficient magnitudes to avoid fitting measurement noise rather than true PA behavior
• Numerical stability: Improves the condition number of the kernel matrix, ensuring reliable inversion even with correlated training samples
• Generalization control: Larger λ values produce smoother predistorter functions that generalize better across varying signal statistics and power levels Cross-validation on held-out PA measurement data determines the optimal λ.

KRR offers distinct trade-offs versus neural network-based predistorters:

• Advantages: No local minima, deterministic training, fewer hyperparameters to tune, strong theoretical guarantees via representer theorem
• Limitations: O(n³) training complexity scales poorly with large datasets; prediction requires storing support vectors, increasing inference memory
• Hybrid strategies: KRR often serves as an offline reference model for behavioral cloning, where a compact neural network is trained to imitate the KRR solution for real-time FPGA deployment For moderate dataset sizes, KRR frequently achieves superior ACLR improvement compared to equivalently sized neural networks.

KRR's effectiveness is grounded in the representer theorem, which proves that the optimal predistorter function in a reproducing kernel Hilbert space can be expressed as a finite linear combination of kernel functions evaluated at the training points: f(x) = Σᵢ αᵢ k(x, xᵢ). This guarantees that the infinite-dimensional optimization reduces to a finite-dimensional problem with a unique solution, providing the theoretical basis for KRR's reliability in PA linearization applications.

Deploying KRR for digital predistortion requires addressing several engineering factors:

• Kernel selection: RBF kernels with automated γ tuning via median heuristic typically perform best for wideband PA nonlinearities
• Memory incorporation: Input vectors are constructed using tapped delay lines of I/Q samples to capture PA memory effects before kernel computation
• Complex-valued extensions: Wirtinger calculus or separate real/imaginary processing handles complex baseband signals
• Computational mitigation: Nyström approximation or random Fourier features reduce O(n³) complexity for large training sets
• Hardware mapping: The resulting α coefficients can be implemented as look-up tables or polynomial evaluations on FPGA fabric