Recursive Least Squares (RLS)

The forgetting factor (λ, typically 0.95 < λ < 1) assigns exponentially decreasing weights to older data, enabling the algorithm to track time-varying systems. A smaller λ provides faster tracking of non-stationary PA characteristics due to thermal drift, while a λ closer to 1 yields better steady-state accuracy in stationary conditions. This mechanism directly addresses coefficient drift in long-running DPD systems.

RLS avoids the O(N³) computational cost of direct matrix inversion by applying the Woodbury matrix identity to recursively update the inverse of the input autocorrelation matrix. This reduces per-iteration complexity to O(N²), where N is the number of DPD coefficients. The recursion propagates both the gain vector and the inverse correlation matrix, maintaining numerical precision in sample-by-sample update modes.

RLS converges in approximately 2N iterations, where N is the filter order, compared to LMS which requires 10N–30N iterations. This is because RLS is a second-order method that whitens the input signal using the inverse correlation matrix, decorrelating the tap inputs. For wideband DPD with 50+ memory polynomial coefficients, this translates to convergence within microseconds rather than milliseconds.

Standard RLS suffers from covariance matrix ill-conditioning when the condition number grows large, leading to divergence. QR-RLS mitigates this by operating directly on the square-root of the covariance matrix using Givens rotations or Householder reflections. This doubles the dynamic range and guarantees positive-definiteness, making it the preferred variant for fixed-point FPGA implementations in closed-loop DPD systems.

The O(N²) complexity of RLS is significantly higher than the O(N) complexity of Least Mean Squares (LMS). For a predistorter with 100 coefficients, RLS requires approximately 10,000 multiply-accumulate operations per sample versus 100 for LMS. This computational burden often restricts RLS to burst training modes or offline coefficient estimation, rather than continuous sample-by-sample adaptation in resource-constrained FPGAs.

When the input signal has low persistent excitation—common during silent intervals or narrowband transmissions—the autocorrelation matrix becomes singular. Tikhonov regularization adds a diagonal loading term (δI) to the initial covariance matrix, stabilizing the inversion. This prevents the gain vector from exploding and ensures robust coefficient updates during signal dropouts in open-loop DPD architectures.

A stochastic gradient descent algorithm that updates filter coefficients based on the instantaneous estimate of the mean squared error gradient. Unlike RLS, LMS has a significantly lower computational complexity of O(N) per iteration.

• Convergence: Slower than RLS, especially for highly correlated input signals
• Sensitivity: Convergence rate is highly dependent on the eigenvalue spread of the input autocorrelation matrix
• Misadjustment: Introduces excess error due to gradient noise from instantaneous estimates
• Use Case: Preferred in resource-constrained embedded systems where simplicity outweighs convergence speed

A numerically stable implementation of the RLS algorithm that uses QR decomposition via Givens rotations to solve the least squares problem directly on the data matrix. This avoids explicit computation of the inverse autocorrelation matrix.

• Stability: Eliminates the positive-definiteness loss that plagues standard RLS in finite-precision arithmetic
• Implementation: Well-suited for systolic array hardware architectures
• Complexity: Higher than standard RLS at O(N²) but essential for ill-conditioned problems
• Application: Critical for wideband DPD where the input correlation matrix may become nearly singular

An optimal recursive estimator for the state of a dynamic system from noisy measurements. In DPD contexts, the Kalman filter provides a Bayesian framework for tracking time-varying PA nonlinearities.

• Relationship to RLS: RLS is a special case of the Kalman filter with a specific state-space formulation
• Process Noise: Explicitly models coefficient drift due to thermal effects and aging
• Advantage: Provides both coefficient estimates and their uncertainty covariance
• Trade-off: Requires knowledge of the process noise covariance matrix, which is often tuned heuristically

A measure of the sensitivity of a matrix to numerical errors, defined as the ratio of the largest to smallest singular values. A high condition number indicates ill-conditioning that can destabilize RLS coefficient estimation.

• Impact on RLS: The inverse autocorrelation matrix becomes highly sensitive to rounding errors when the condition number is large
• Causes: Highly correlated input signals, narrowband training data, or insufficient excitation
• Mitigation: Tikhonov regularization adds a diagonal loading term to improve conditioning
• Monitoring: Real-time condition number tracking is essential for adaptive DPD robustness

The gradual deviation of predistorter coefficients from their optimal values over time due to environmental changes or numerical instability in the adaptation algorithm.

• Thermal Drift: PA characteristics shift as the transistor junction temperature changes during operation
• Aging Effects: Semiconductor degradation over months and years alters the nonlinear transfer function
• RLS Vulnerability: The forgetting factor λ creates a finite memory window; if excitation is insufficient, coefficients can drift toward zero
• Countermeasure: Periodic retraining or leaky LMS modifications that penalize coefficient magnitude

A ridge regression technique that adds a penalty term proportional to the squared L2-norm of the coefficient vector to the least squares cost function. This stabilizes RLS solutions for ill-posed inverse problems.

• Mechanism: Adds a diagonal matrix δI to the autocorrelation matrix before inversion
• Effect: Biases the solution toward smaller coefficient magnitudes, trading some accuracy for stability
• Selection: The regularization parameter δ is chosen via cross-validation or the L-curve method
• DPD Application: Essential when the PA model order is high relative to the available training samples