Tikhonov Regularization

In DPD coefficient estimation, the autocorrelation matrix of the input signal often has a high condition number, making the least squares solution highly sensitive to noise and numerical errors. Tikhonov regularization adds a diagonal loading term λI to the matrix, effectively reducing the condition number and stabilizing the inversion. This prevents wild coefficient swings that would otherwise degrade linearization performance.

When training a digital predistorter on a limited set of captured waveforms, the model may fit noise and measurement artifacts rather than the true PA nonlinearity. This overfitting produces excellent training performance but poor generalization to new signals. Tikhonov regularization constrains the coefficient magnitudes, preventing the model from developing excessively complex transfer functions that memorize training data rather than learning the underlying distortion characteristic.

In QR-decomposition Recursive Least Squares implementations, Tikhonov regularization is applied by initializing the upper triangular matrix R with λI rather than zeros. This prevents the initial covariance matrix from being singular and ensures numerical stability during the early stages of adaptation. The technique is particularly valuable in fixed-point FPGA implementations where limited precision arithmetic can amplify ill-conditioning problems.

Tikhonov regularization has a Bayesian interpretation: it corresponds to placing a Gaussian prior with zero mean and variance proportional to 1/λ² on the DPD coefficients. The regularized solution is then the maximum a posteriori (MAP) estimate, combining prior knowledge that coefficients should be small with the observed data. This perspective guides principled selection of λ based on expected coefficient magnitudes.

Selecting the regularization parameter λ requires balancing two competing objectives:

• Too small λ: Insufficient regularization, solution remains ill-conditioned and noise-sensitive
• Too large λ: Excessive bias, the predistorter underfits and fails to cancel nonlinearity Common selection methods include L-curve analysis, generalized cross-validation (GCV), and the discrepancy principle, which chooses λ so the residual norm matches the expected noise level.

A measure of a matrix's sensitivity to numerical errors during inversion. In DPD coefficient estimation, a high condition number indicates an ill-posed problem where small measurement noise causes large coefficient fluctuations. Tikhonov regularization directly reduces the effective condition number of the autocorrelation matrix, trading a small bias for a dramatic reduction in variance. Monitoring this metric is essential for diagnosing numerical instability in real-time adaptive systems.

An adaptive filtering algorithm that recursively computes the least squares solution with exponential forgetting to track time-varying systems. RLS offers faster convergence than LMS but is susceptible to covariance matrix blow-up when excitation is insufficient. Incorporating Tikhonov regularization into RLS—often called regularized RLS—prevents numerical divergence by adding a diagonal loading term to the inverse correlation matrix at each iteration.

A numerically robust implementation of RLS that uses QR decomposition to solve the least squares problem without explicitly inverting the correlation matrix. QR-RLS operates directly on the data matrix using Givens rotations, providing superior numerical stability. When combined with Tikhonov regularization, the diagonal loading is incorporated into the upper triangular R matrix, ensuring the algorithm remains stable even with persistently low excitation signals.

A modeling failure where the predistorter memorizes noise and measurement artifacts rather than learning the true PA nonlinearity. Overfitting produces excellent training performance but poor generalization to new signals. Tikhonov regularization combats this through weight decay—penalizing large coefficient magnitudes forces the model to learn only the dominant, reproducible nonlinear patterns. The regularization parameter λ controls the bias-variance tradeoff.