Coefficient Estimation

The central objective of coefficient estimation is to minimize a cost function that quantifies the error between the desired linear output and the actual power amplifier output. Common formulations include:

• Mean Squared Error (MSE): Minimizes the average squared difference between ideal and observed signals
• Least Squares (LS): Finds coefficients that minimize the sum of squared residuals in a batch
• Weighted Least Squares: Applies frequency-dependent weighting to prioritize out-of-band distortion reduction

The choice of cost function directly impacts Adjacent Channel Power Ratio (ACPR) and Error Vector Magnitude (EVM) performance.

Coefficient estimation strategies fall into two fundamental categories based on update timing:

Batch Estimation

• Accumulates a block of samples before computing coefficient updates
• Uses techniques like Least Squares or QR-RLS for stable, accurate solutions
• Suitable for offline calibration and burst training modes

Adaptive Estimation

• Updates coefficients incrementally with each sample or small batch
• Algorithms include LMS, RLS, and Stochastic Gradient Descent (SGD)
• Essential for tracking time-varying PA behavior due to temperature drift and aging

The convergence rate determines how quickly the estimator reaches optimal coefficients, while misadjustment represents the excess error beyond the theoretical minimum. Key trade-offs include:

• LMS: Simple implementation with slow convergence but low computational complexity
• RLS: Fast convergence at the cost of higher computational load and potential instability
• Kalman Filtering: Optimal tracking of time-varying coefficients with known state transition models

Condition number of the data covariance matrix critically affects stability. Ill-conditioned matrices require Tikhonov regularization or Levenberg-Marquardt optimization to prevent numerical instability.

Coefficient drift occurs when predistorter parameters gradually deviate from optimal values due to:

• Thermal memory effects: Temperature-induced changes in PA transistor characteristics
• Aging effects: Long-term degradation of semiconductor materials
• Numerical precision: Accumulation of quantization errors in fixed-point implementations

Mitigation strategies include sample-by-sample updates with leakage factors, periodic block updates triggered by performance monitoring, and closed-loop DPD architectures that continuously compare output feedback against the reference signal.

Overfitting in coefficient estimation occurs when the predistorter model captures noise and measurement artifacts rather than the true PA nonlinearity. Consequences include:

• Degraded generalization to unseen signal conditions
• Amplification of out-of-band noise rather than cancellation
• Poor performance under varying signal statistics

Prevention techniques include cross-validation with held-out data, regularization methods like ridge regression, and constraining model complexity through basis function selection in Volterra or memory polynomial models. The Normalized Mean Squared Error (NMSE) metric on validation data provides a reliable overfitting indicator.

Practical coefficient estimation must balance algorithmic sophistication against hardware limitations:

• Latency budget: Coefficient updates must complete within the coherence time of PA behavior changes
• Computational complexity: LMS requires O(N) operations per update while RLS demands O(N²)
• Numerical precision: Fixed-point FPGA implementations require careful scaling to avoid instability
• Memory bandwidth: Block update methods need buffer storage proportional to block length

QR-RLS offers a numerically stable alternative to standard RLS using Givens rotations, enabling systolic array implementations suitable for high-throughput FPGA-based DPD systems.

DLA is a closed-loop architecture that directly estimates the predistorter coefficients by minimizing the error between the desired ideal input and the actual PA output. This requires a model inversion step to backpropagate the error through the PA model.

• Key Benefit: Theoretically optimal solution, robust to measurement noise.
• Challenge: Requires an accurate PA behavioral model and a stable inversion algorithm.

RLS is an adaptive filtering algorithm that recursively finds the coefficients minimizing a weighted linear least squares cost function. It offers an order of magnitude faster convergence rate than LMS, making it suitable for tracking rapid changes in PA behavior.

• Trade-off: Higher computational complexity (O(N²)) compared to LMS.
• Enhancement: QR-RLS decomposition is often used to improve numerical stability and prevent coefficient drift.

LMS is a stochastic gradient descent algorithm that updates filter coefficients based on the instantaneous estimate of the mean squared error gradient. It is a foundational algorithm for adaptive filtering in DPD systems due to its simplicity and low computational cost.

• Key Parameter: The step size controls the trade-off between convergence rate and steady-state misadjustment.
• Use Case: Ideal for resource-constrained FPGA implementations requiring sample-by-sample updates.

Also known as ridge regression, Tikhonov regularization adds a penalty term to the least squares cost function to stabilize the solution of ill-posed inverse problems. In DPD, it prevents overfitting and numerical instability when the data matrix has a high condition number.

• Mechanism: Adds a small diagonal loading factor (λI) to the autocorrelation matrix.
• Result: Improves the robustness of least squares estimation and prevents wild coefficient swings.

NMSE is the primary metric for evaluating DPD linearization performance. It represents the mean squared error between the ideal linear output and the actual linearized output, normalized by the input signal power.

• Formula: NMSE = 10*log₁₀( E[|y_ideal - y_measured|²] / E[|y_ideal|²] )
• Target: Values below -40 dB typically indicate excellent linearization. It is closely related to EVM, which measures in-band distortion quality.