Parameter Extraction

Parameter extraction operates on a complete captured dataset rather than sample-by-sample. The entire input-output record is processed simultaneously to solve for the optimal coefficient vector. This contrasts with online training, which updates coefficients incrementally. Batch processing enables the use of numerically stable matrix factorization methods like QR decomposition and Singular Value Decomposition (SVD) that require access to the full data matrix. The trade-off is that batch extraction cannot track time-varying amplifier behavior without periodic re-extraction cycles.

The dominant mathematical framework for parameter extraction is the least squares (LS) criterion. The objective is to find the coefficient vector θ̂ that minimizes the sum of squared errors between the measured amplifier output and the model prediction:

• Cost Function: J(θ) = ||y - Xθ||²
• Normal Equation Solution: θ̂ = (XᴴX)⁻¹Xᴴy
• X is the regression matrix constructed from input signal samples
• y is the vector of measured output samples

This closed-form solution provides the minimum variance unbiased estimator under the assumption of white Gaussian measurement noise.

The regression matrix X in power amplifier modeling is often ill-conditioned due to high correlation between basis functions, especially for memory polynomial models with closely spaced taps. The condition number κ(X) quantifies this sensitivity:

• High condition number → small measurement noise causes large coefficient errors
• Tikhonov Regularization adds a penalty term λ||θ||² to the cost function
• The regularized solution becomes: θ̂ = (XᴴX + λI)⁻¹Xᴴy
• The regularization parameter λ trades bias for variance reduction

Without regularization, extracted parameters may exhibit large magnitudes with alternating signs that cancel mathematically but amplify noise.

Parameter extraction is inseparable from model structure selection—the choice of which basis functions to include. Key considerations:

• Memory depth: Number of past samples to include, determined by the amplifier's memory time constant
• Nonlinearity order: Highest polynomial degree, limited by the amplifier's saturation characteristic
• Cross-term inclusion: Whether to include products of delayed samples (full Volterra) or restrict to aligned terms (memory polynomial)
• Sparsity pursuit: Using LASSO or OMP algorithms to select only significant terms

Overly complex models lead to overfitting, where extracted parameters fit noise rather than the underlying amplifier dynamics.

Direct solution of the normal equations via matrix inversion is numerically unstable for ill-conditioned systems. Robust extraction employs orthogonal decomposition techniques:

• QR Decomposition: Factor X = QR, then solve Rθ = Qᴴy via back-substitution. Avoids forming XᴴX entirely.
• SVD-Based Pseudoinverse: Decompose X = UΣVᴴ, then θ̂ = VΣ⁺Uᴴy. Allows truncation of small singular values.
• Cholesky Decomposition: For symmetric positive-definite XᴴX, factor as LLᴴ for efficient solution.

These methods maintain numerical accuracy even when the condition number exceeds 10¹⁰, common in wideband DPD applications.

Extracted parameters must be validated on data not used during extraction to assess generalization. Standard practices include:

• Holdout validation: Reserve a portion of captured data for testing
• Normalized Mean Squared Error (NMSE): Primary metric, typically expressed in dB
• Adjacent Channel Power Ratio (ACPR): Verify spectral regrowth suppression on validation signals
• Error Vector Magnitude (EVM): Assess in-band distortion on modulated test signals

A model achieving NMSE < -40 dB on training data but only -30 dB on validation data indicates overfitting and requires regularization or model complexity reduction.

The foundational batch estimation method that computes the optimal coefficient vector by minimizing the sum of squared errors between the observed PA output and the desired linear response. The solution is obtained via the Normal Equation, which requires inverting the input autocorrelation matrix. LS provides the minimum variance unbiased estimator under white Gaussian noise assumptions but can become ill-conditioned with highly correlated input signals. Used extensively in offline training for initial DPD coefficient extraction.

An adaptive algorithm that recursively updates the inverse of the input correlation matrix using the matrix inversion lemma, achieving an order of magnitude faster convergence than gradient-based methods. RLS minimizes a weighted least squares cost function where a forgetting factor exponentially discounts past data, enabling tracking of time-varying PA characteristics. The trade-off is O(n²) computational complexity per iteration, making it suitable for applications where convergence speed is critical.

A stochastic gradient descent algorithm that updates coefficients using the instantaneous estimate of the mean squared error gradient. LMS requires only O(n) operations per iteration, making it the most computationally efficient adaptive method. However, its convergence rate depends heavily on the eigenvalue spread of the input correlation matrix. The step size parameter controls the trade-off between convergence speed and steady-state misadjustment. Widely used in resource-constrained FPGA implementations.

A numerically robust implementation of RLS that avoids explicit matrix inversion by maintaining the square-root of the inverse correlation matrix. Uses Givens rotations to perform orthogonal triangularization, providing superior numerical stability in fixed-point hardware. QR-RLS is the preferred algorithm when implementing adaptive DPD on FPGAs with limited precision arithmetic, as it resists round-off error accumulation that can cause standard RLS to diverge.

A variant of LMS that normalizes the step size by the instantaneous power of the input signal vector. This normalization makes convergence behavior independent of input signal scaling, solving the gradient noise amplification problem of standard LMS. The update includes a small regularization parameter to prevent division by zero during silent intervals. NLMS is the default choice for online training in direct learning architectures where robustness to varying signal levels is essential.

The theoretical foundation underlying all linear estimation methods. Defines the optimal weight vector as w_opt = R⁻¹p, where R is the input autocorrelation matrix and p is the cross-correlation vector between the input and desired signal. Solving this equation yields the Wiener filter, which achieves the minimum mean squared error. In practice, R and p are unknown and must be estimated from data, leading to the LS, RLS, and LMS algorithmic families.