Least Squares Estimation

Under the classical assumptions of linear regression, the Ordinary Least Squares (OLS) estimator is the Best Linear Unbiased Estimator (BLUE). This means that among all estimators that are both linear functions of the observed data and unbiased, OLS achieves the minimum possible variance. For PA modeling, this guarantees that coefficient estimates extracted via least squares are the most precise achievable using linear methods, assuming the noise on measurements is white and uncorrelated. The theorem does not require the errors to be normally distributed; it only requires zero mean, constant variance (homoscedasticity), and zero correlation.

The least squares solution is obtained by solving the normal equations: θ̂ = (XᵀX)⁻¹Xᵀy, where X is the regression matrix of basis functions evaluated at the input signal samples, and y is the vector of measured output samples. This closed-form expression is derived by setting the gradient of the sum of squared residuals to zero. The matrix XᵀX is the autocorrelation matrix of the basis functions, and its invertibility requires that the input signal is persistently exciting—containing sufficient spectral content to probe all model dynamics.

The cost function minimized is the Residual Sum of Squares (RSS): J(θ) = Σᵢ(yᵢ - ŷᵢ)² = ||y - Xθ||²₂. This L2-norm penalty heavily weights large errors, making the estimator sensitive to outliers in measurement data. In PA modeling, minimizing RSS directly minimizes the Normalized Mean Square Error (NMSE) between the model output and the measured amplifier output. The quadratic nature of the cost function guarantees a single global minimum, avoiding the local minima problems encountered in iterative optimization methods.

The condition number of the XᵀX matrix critically affects solution accuracy. When basis functions are highly correlated—common in polynomial models with high nonlinear orders—the matrix becomes ill-conditioned, amplifying measurement noise into large coefficient errors. Techniques to improve stability include:

• Tikhonov Regularization (Ridge Regression): Adds a penalty λ||θ||² to the cost function, improving conditioning at the cost of slight bias
• Orthogonal Basis Functions: Using orthogonal polynomials instead of standard monomials reduces correlation between columns of X
• Double-Precision Arithmetic: Essential for solving systems with condition numbers exceeding 10⁶

Least squares estimation can be implemented in two modes for PA modeling:

Batch Least Squares: Processes the entire captured data record at once using the normal equations. Provides the optimal estimate for stationary conditions but requires matrix inversion of size equal to the number of model coefficients.

Recursive Least Squares (RLS): Updates coefficient estimates sample-by-sample as new data arrives, enabling online adaptation to changing amplifier characteristics due to temperature drift or aging. RLS achieves convergence an order of magnitude faster than LMS algorithms by maintaining an estimate of the inverse autocorrelation matrix.

When measurement noise is not uniform across the data record, Weighted Least Squares (WLS) improves estimation accuracy by assigning different importance to each residual. The cost function becomes J(θ) = Σᵢ wᵢ(yᵢ - ŷᵢ)², where weights wᵢ are typically chosen inversely proportional to the noise variance at each sample. In PA modeling, WLS is particularly useful for:

• Emphasizing high-power regions where amplifier nonlinearity is most pronounced
• De-weighting low-SNR regions near the noise floor
• Compensating for non-uniform signal statistics in the training data

The primary cost function minimized by least squares estimation. NMSE quantifies the average power of the error signal (the difference between the model output and the measured data) normalized by the power of the reference signal.

• Expressed in decibels (dB), with lower values (e.g., -40 dB) indicating superior model fidelity.
• Directly drives the coefficient update mechanism in batch least squares extraction.
• A critical metric for validating the accuracy of a behavioral model against independent test data.

A mathematical technique added to the least squares cost function to prevent overfitting and improve numerical stability. It works by penalizing the magnitude of the model coefficients.

• Ridge Regression (L2): Adds a penalty proportional to the square of the coefficient magnitude, shrinking them toward zero without eliminating them.
• LASSO (L1): Adds a penalty proportional to the absolute value of coefficients, forcing some to exactly zero to create a sparse model.
• Essential when the data matrix is ill-conditioned, a common occurrence with highly correlated wideband signals.

An iterative, stochastic gradient descent algorithm that provides a low-complexity alternative to batch least squares for real-time adaptive tracking. Unlike the batch solution, LMS updates coefficients sample-by-sample.

• The update rule is: w(n+1) = w(n) + μ * e(n) * x(n), where μ is the step size.
• Trades off convergence speed against steady-state misadjustment via the step size parameter.
• Ideal for tracking slow changes in power amplifier behavior due to thermal memory effects or aging.

The reliability of a least squares solution is governed by the condition number of the data matrix. A high condition number indicates an ill-conditioned matrix, where small measurement errors are amplified into massive coefficient errors.

• The Moore-Penrose pseudoinverse is the standard solution, but its direct computation can be unstable.
• QR decomposition or Singular Value Decomposition (SVD) are preferred numerical methods for solving the normal equations robustly.
• Critical for FPGA implementations where finite-precision arithmetic can exacerbate instability.

A statistical method to ensure a model extracted via least squares generalizes to unseen data and does not merely memorize the training set. The measured dataset is partitioned into distinct subsets.

• Holdout Method: A simple split into a training set for coefficient extraction and a test set for NMSE validation.
• K-Fold Cross-Validation: The data is split into K folds; the model is trained on K-1 folds and validated on the remaining fold, rotating through all combinations.
• A rising NMSE on the validation set relative to the training set is a definitive indicator of overfitting.

A property of the extracted model where a significant number of coefficients are near-zero and contribute negligibly to the output. Least squares alone does not guarantee sparsity.

• LASSO regularization is explicitly used to enforce sparsity during estimation.
• A sparse model reduces the number of multiply-accumulate operations (MACs) required in a hardware predistorter, directly lowering power consumption and latency.
• Pruning near-zero coefficients after a standard least squares fit is a common heuristic for complexity reduction in Generalized Memory Polynomial models.