Overdetermined System

An overdetermined system takes the form Ax = b where A is an M × N matrix with M > N. The system has more equations than unknowns, making an exact solution generally impossible. Instead, the goal is to find the coefficient vector x̂ that minimizes the residual error ||Ax - b||². In DPD extraction, A is the basis function matrix constructed from input samples, b is the vector of measured amplifier output samples, and x contains the model coefficients.

The optimal coefficient vector is computed using the Moore-Penrose pseudoinverse: x̂ = (AᵀA)⁻¹Aᵀb. This yields the unique minimizer of the squared error norm. Key properties:

• The matrix AᵀA must be invertible (full column rank)
• The pseudoinverse generalizes the matrix inverse to non-square matrices
• The solution is the maximum likelihood estimator when measurement noise is Gaussian
• Computational complexity is O(N²M) for the normal equations approach

Overdetermined DPD systems are susceptible to ill-conditioning when basis functions are highly correlated. The condition number κ(AᵀA) = λₘₐₓ/λₘᵢₙ quantifies this sensitivity. A high condition number (>10⁶) causes:

• Coefficient estimates to become unstable and sensitive to measurement noise
• Small perturbations in captured data to produce wildly different model parameters
• Numerical precision loss during matrix inversion
• Mitigation requires regularization or basis function orthogonalization

To stabilize ill-conditioned overdetermined systems, regularization adds a penalty term to the cost function:

• Ridge Regression (L2): Minimizes ||Ax - b||² + λ||x||², shrinking coefficient magnitudes
• LASSO (L1): Promotes sparsity by driving unnecessary coefficients to exactly zero
• Elastic Net: Combines L1 and L2 penalties for balanced regularization
• The regularization parameter λ controls the bias-variance tradeoff
• Cross-validation selects optimal λ to prevent underfitting or overfitting

A typical 5G NR DPD extraction with a memory polynomial model (nonlinear order K=7, memory depth M=4) produces:

• N = K × M = 28 unknown coefficients
• M = 100,000+ captured I/Q sample pairs
• The basis function matrix A is 100,000 × 28 — highly overdetermined
• Solving via QR decomposition or SVD provides numerical stability
• The residual NMSE typically converges below -40 dB for a well-conditioned system

Overdetermined formulation is the standard approach in PA behavioral modeling because:

• Capturing more data than parameters averages out measurement noise
• Provides statistical confidence in extracted coefficients
• Enables model validation on held-out data not used in fitting
• Supports information criteria (AIC, BIC) for model order selection
• Contrasts with underdetermined systems where M < N, requiring compressed sensing or sparse recovery techniques