Least Squares Estimation

Least squares provides a deterministic analytical solution rather than requiring iterative gradient descent. The normal equation XᵀXβ = Xᵀy directly computes the optimal coefficient vector β that minimizes the squared error. This is critical for burst training in DPD where coefficients must be extracted rapidly during dedicated calibration intervals. The solution is globally optimal for linear-in-parameters models like memory polynomials and generalized memory polynomials.

The cost function J(β) = ||y - Xβ||² penalizes larger errors quadratically, making the estimator highly sensitive to outliers but optimal for Gaussian noise distributions. In DPD applications, this directly minimizes the Normalized Mean Squared Error (NMSE) between the ideal predistorted signal and the PA output. The quadratic nature ensures a smooth, differentiable error surface with a single global minimum when the regressor matrix X is full rank.

Least squares achieves the Best Linear Unbiased Estimator (BLUE) status when key assumptions hold: zero-mean errors, homoscedasticity (constant error variance), and uncorrelated residuals. In PA linearization, violations occur when measurement noise is colored or when the observation path introduces correlated distortions. Violations motivate alternatives like weighted least squares or generalized least squares for improved coefficient accuracy.

When the regressor matrix X has a high condition number, the normal equation solution becomes numerically unstable. This occurs in DPD when basis functions are highly correlated or when the observation bandwidth is insufficient. Tikhonov regularization (ridge regression) adds a penalty term λ||β||² to stabilize the inversion, trading a small bias for significant variance reduction. The Levenberg-Marquardt algorithm extends this for nonlinear least squares problems.

The Recursive Least Squares (RLS) algorithm reformulates the batch solution for sample-by-sample updates without full matrix inversion. At each time step k, the gain vector K(k) = P(k-1)x(k) / (λ + xᵀ(k)P(k-1)x(k)) updates coefficients incrementally. The forgetting factor λ (typically 0.95-0.999) controls the memory horizon, enabling tracking of thermal memory effects and coefficient drift in real-time closed-loop DPD systems.

QR-RLS decomposes the regressor matrix X into an orthogonal matrix Q and upper triangular matrix R using Givens rotations or Householder reflections. This avoids explicitly computing the ill-conditioned XᵀX product, dramatically improving numerical robustness. In FPGA-based DPD implementations, systolic array architectures efficiently pipeline QR decomposition for high-throughput coefficient updates, essential for wideband signal linearization in 5G systems.

An adaptive filtering algorithm that recursively finds the coefficients minimizing a weighted linear least squares cost function. Unlike block-based least squares, RLS updates the coefficient vector with each new sample, offering faster convergence than gradient-based methods at the cost of higher computational complexity.

• Converges in approximately 2N iterations where N is the number of coefficients
• Computational complexity of O(N²) per iteration
• Particularly effective for tracking time-varying PA nonlinearities
• Forms the basis for many closed-loop DPD training architectures

A stochastic gradient descent algorithm that updates filter coefficients based on the instantaneous estimate of the mean squared error gradient. LMS trades convergence speed for computational simplicity, making it attractive for resource-constrained FPGA implementations.

• Complexity of only O(N) per iteration
• Requires careful step-size tuning to balance convergence and misadjustment
• Susceptible to slow convergence when the input signal has a high eigenvalue spread
• Often used in sample-by-sample update DPD architectures

A measure of the sensitivity of a matrix to numerical errors during inversion. In least squares estimation, the condition number of the autocorrelation matrix determines how small perturbations in the data amplify into large errors in the coefficient solution.

• A condition number near 1 indicates a well-conditioned, stable problem
• Values above 10⁴ signal potential numerical instability
• High condition numbers arise from correlated input signals or insufficient excitation
• Mitigated through Tikhonov regularization or QR decomposition methods

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 the solution of ill-posed inverse problems common in DPD coefficient extraction.

• Adds a regularization parameter λ that controls the bias-variance tradeoff
• Effectively reduces the effective condition number of the system
• Prevents coefficient drift and overfitting to measurement noise
• Equivalent to assuming a Gaussian prior on the coefficients in a Bayesian framework

The primary metric for evaluating least squares estimation quality in DPD applications. NMSE represents the mean squared error between the ideal linear output and the actual linearized output, normalized by the input signal power to provide a scale-invariant measure.

• Expressed in decibels (dB) for intuitive interpretation
• Values below -40 dB typically indicate excellent linearization
• Directly reflects the minimization objective of the least squares cost function
• Used alongside ACPR and EVM for comprehensive DPD validation

A numerically stable implementation of the Recursive Least Squares algorithm that uses QR decomposition to solve the least squares problem without explicitly computing the matrix inverse. This approach dramatically improves robustness to ill-conditioning.

• Operates directly on the data matrix using orthogonal transformations
• Avoids the numerical issues of the standard RLS Riccati equation
• Can be implemented efficiently using Givens rotations on systolic arrays
• Preferred for fixed-point FPGA implementations where numerical precision is limited