Regularization Parameter

The regularization parameter λ is added as a scalar value to the main diagonal of the correlation matrix (R) before inversion, transforming the solution from w = R⁻¹p to w = (R + λI)⁻¹p. This diagonal loading artificially increases all eigenvalues by λ, preventing near-zero eigenvalues from causing explosive coefficient magnitudes. The technique is mathematically equivalent to imposing an L2 norm penalty on the coefficient vector, constraining the solution energy and preventing overfitting to noise in the feedback observation path.

The condition number κ(R) measures the sensitivity of the matrix inverse to perturbations. An ill-conditioned correlation matrix with κ(R) >> 1 amplifies small measurement errors into large coefficient errors. Adding λ reduces the condition number to:

• κ(R + λI) = (σ_max² + λ) / (σ_min² + λ)
• As λ increases, the condition number approaches 1, dramatically improving numerical stability
• This is critical for fixed-point FPGA implementations where finite precision arithmetic exacerbates ill-conditioning effects

The regularization parameter directly controls the bias-variance trade-off in coefficient estimation:

• λ = 0: Unbiased solution with minimum mean squared error on training data, but high variance and sensitivity to noise
• λ → ∞: All coefficients shrink toward zero, producing a highly biased but low-variance solution
• Optimal λ: Balances the systematic error from regularization bias against the random error from noise amplification In online DPD training, this trade-off is dynamic—higher λ values are needed during startup when the correlation matrix is poorly conditioned due to limited data samples.

Fixed λ is suboptimal for tracking time-varying PA nonlinearity. Advanced strategies include:

• Sample-based scheduling: High λ during initial convergence, then decayed as the correlation matrix becomes well-conditioned
• SNR-dependent adjustment: Increasing λ when the feedback receiver reports low signal-to-noise ratio, preventing noise from corrupting coefficient updates
• Eigenvalue tracking: Monitoring the minimum eigenvalue of R and setting λ proportional to its magnitude to maintain a target condition number
• Per-band regularization: In multi-band DPD, applying different λ values to each band's sub-correlation matrix based on individual signal statistics

The regularization parameter in DPD coefficient estimation is directly analogous to the damping factor in the Levenberg-Marquardt algorithm for nonlinear least squares. Both methods interpolate between:

• Gauss-Newton (λ → 0): Fast quadratic convergence near the optimum, but unstable when far from the solution
• Gradient descent (λ → ∞): Robust but slow linear convergence In Direct Learning Architecture (DLA) implementations, the regularization parameter can be dynamically adjusted based on the error vector magnitude trend, increasing λ when the cost function reduction stalls due to an ill-conditioned Jacobian.

Implementing regularization on FPGA-based DPD requires careful numerical design:

• Cholesky decomposition of (R + λI) is preferred over direct matrix inversion for fixed-point arithmetic, as the added diagonal ensures the matrix remains positive definite
• λ should be chosen as a power-of-two to replace multiplication with bit-shifting, reducing DSP slice consumption
• The QR decomposition approach with Tikhonov regularization appends √λ·I as additional rows to the data matrix, integrating regularization directly into the orthogonalization process
• Coefficient freeze logic should monitor the effective condition number post-regularization to detect when λ is insufficient to stabilize the solution

The numerical state the regularization parameter is designed to mitigate. When the correlation matrix has a high condition number, small perturbations in the input signal or measurement noise cause massive swings in the estimated coefficients.

• Cause: Highly correlated basis functions or insufficient signal excitation.
• Effect: Coefficient divergence and spectral regrowth spikes.
• Solution: Adding a scalar to the diagonal (regularization) lowers the condition number.

An adaptive algorithm where the forgetting factor and regularization parameter jointly control stability. RLS recursively computes the inverse of the correlation matrix, making it particularly susceptible to numerical instability.

• Initialization typically uses a scaled identity matrix (regularization) to ensure the first inverse is non-singular.
• Balances the trade-off between rapid convergence rate and steady-state misadjustment.

A stochastic gradient algorithm that implicitly avoids the matrix inversion problem, making explicit regularization less critical than in RLS. However, the learning rate serves a functionally similar role.

• A small learning rate acts as implicit regularization by slowing updates and preventing overfitting to noise.
• Normalized LMS (NLMS) normalizes the step size by input power, providing inherent numerical scaling akin to diagonal loading.

A numerically stable alternative to direct matrix inversion for solving least-squares problems. QR decomposition operates directly on the data matrix, avoiding the squaring of condition numbers that occurs when forming the correlation matrix.

• Often preferred over explicit regularization for fixed-point FPGA-based DPD implementation.
• Techniques like Householder reflections or Givens rotations maintain numerical stability without requiring manual tuning of a regularization scalar.

The phenomenon the regularization parameter directly combats. A predistorter with too many degrees of freedom will fit the specific noise characteristics of the training capture rather than the true PA nonlinearity.

• Symptom: Excellent EVM on the training signal but poor ACLR on a different waveform.
• Mechanism: Regularization penalizes large coefficient magnitudes, enforcing a smoother predistorter transfer function that generalizes to unseen data.

The broader class of algorithms where the regularization parameter is applied. Whether using block-based least-squares or iterative stochastic gradient descent (SGD), the regularization term appears as a penalty on the L2 norm of the coefficient vector.

• In Direct Learning Architecture (DLA), regularization stabilizes the PA model extraction step.
• In Indirect Learning Architecture (ILA), it prevents the postdistorter from fitting feedback receiver noise.