Numerical Stability

The algorithm must monitor the condition number of the regression matrix, which quantifies sensitivity to input perturbations. A high condition number—common with strongly correlated wideband signals—indicates that small measurement noise can cause large coefficient swings. Stable implementations compute this metric and trigger countermeasures when thresholds are exceeded.

Adding a penalty term proportional to the squared L2-norm of coefficients constrains the solution space and dramatically improves stability. The regularization parameter λ controls the bias-variance tradeoff:

• Small λ: Near-least-squares solution, higher variance
• Large λ: Smoother coefficients, lower variance, slight bias increase
• Optimal λ: Typically selected via cross-validation or the L-curve method

For online adaptation, Recursive Least Squares (RLS) updates coefficients incrementally without full matrix inversion. A forgetting factor λ (typically 0.95–0.999) exponentially weights recent samples more heavily, allowing the algorithm to track time-varying PA behavior while maintaining numerical conditioning through controlled memory depth.

Instead of solving the normal equations directly—which squares the condition number—stable implementations use QR decomposition to solve the least-squares problem. Adding column pivoting identifies and ranks linearly independent basis functions, allowing the algorithm to discard near-dependent terms that contribute to ill-conditioning without meaningful modeling improvement.

While fixed-point FPGA implementations are the deployment target, coefficient extraction should use IEEE 754 double-precision (64-bit) floating point during training. The extended mantissa (53 bits) provides sufficient dynamic range to accumulate inner products without catastrophic cancellation, especially for high-order polynomial terms where single-precision rounding errors can destabilize convergence.

Truncated SVD decomposes the regression matrix and discards singular values below a threshold relative to the largest singular value. This directly addresses rank deficiency caused by oversampled or band-limited signals. Typical thresholds range from 10⁻⁴ to 10⁻⁶ of the maximum singular value, eliminating noise-dominated dimensions while preserving signal-bearing components.

A statistical method that adds a penalty equivalent to the square of the magnitude of coefficients to the loss function. In DPD, this shrinks model parameters toward zero, directly combating ill-conditioned matrices by adding a constant to the diagonal of the normal equations. This prevents wild coefficient swings due to noise amplification, ensuring stable indirect learning architecture (ILA) convergence.

A measure of how sensitive a function's output is to errors in its input. For a matrix, it is the ratio of the largest to smallest singular value. A high condition number indicates an ill-conditioned problem where small measurement noise in the feedback path leads to large errors in extracted DPD coefficients, causing the linearization to fail or become unstable.

A matrix factorization technique that decomposes the observation matrix into its fundamental components. For numerical stability, SVD allows the direct identification and truncation of small, noise-dominated singular values. This pseudo-inversion technique provides a robust solution for least-squares coefficient extraction even when the normal equations are nearly singular.

An orthogonalization method that factors the observation matrix into an orthogonal matrix (Q) and an upper triangular matrix (R). Solving for DPD coefficients using QR decomposition avoids explicitly computing the squared matrix, which squares the condition number. This provides superior numerical stability over direct normal equation solving, especially for wideband signal linearization with high sample rates.

An iterative optimization technique that adaptively interpolates between Gauss-Newton and gradient descent. It is highly effective for nonlinear DPD models like Generalized Memory Polynomials (GMP). The algorithm dynamically adjusts a damping factor to handle ill-conditioned Jacobian matrices, ensuring robust convergence even when initial coefficient estimates are poor.

A numerical representation system crucial for FPGA-based DPD implementation. Unlike floating-point, fixed-point has limited dynamic range, making it highly susceptible to overflow and quantization errors. Maintaining numerical stability requires careful word-length analysis and scaling to prevent the accumulation of rounding errors in recursive online training algorithms.