System Identification

System identification inverts the traditional first-principles modeling approach. Instead of deriving equations from physics, it infers the mathematical structure and parameters directly from measurements.

• Requires carefully designed training waveforms that excite all relevant dynamics
• Captures both linear and nonlinear behaviors without explicit physical knowledge
• The model is a functional approximation, not a physical replica
• Example: A memory polynomial model extracted from a 5G NR waveform captures PA nonlinearity without knowing the transistor physics

At its core, system identification solves an optimization problem: find the model parameters that minimize the discrepancy between predicted and measured outputs.

• Cost function: Typically mean squared error between model output and measured data
• Overdetermined systems: More measurement equations than unknown parameters require least-squares solutions
• Parameter estimation algorithms: LS, RLS, LMS, and NLMS each trade convergence speed against computational complexity
• The Moore-Penrose pseudoinverse provides the least-squares solution when direct matrix inversion is impossible

Choosing the right model structure balances bias against variance. An overly simple model underfits; an overly complex model memorizes noise.

• Model order estimation determines the optimal nonlinearity order and memory depth
• Akaike Information Criterion (AIC) penalizes parameter count to prevent overfitting
• Basis function selection prunes redundant terms to reduce complexity
• Cross-validation partitions data into training and validation sets to test generalization
• The bias-variance tradeoff governs all model selection decisions

The stability of parameter estimates depends critically on the condition number of the regression matrix. Ill-conditioned problems produce unreliable coefficients.

• Ill-conditioning occurs when basis functions are highly correlated, making the correlation matrix nearly singular
• Regularization techniques like ridge regression add an L2 penalty to stabilize solutions
• Principal Component Analysis (PCA) transforms correlated basis functions into uncorrelated components
• The covariance matrix reveals pairwise correlations that cause numerical instability
• A high condition number amplifies measurement noise into large coefficient errors

System identification operates in two fundamental modes depending on whether the system characteristics are time-invariant or slowly changing.

• Batch estimation (e.g., Least Squares) processes an entire captured data record at once, suitable for offline model extraction
• Recursive estimation (e.g., RLS, LMS) updates coefficients sample-by-sample as new data arrives
• A forgetting factor in recursive algorithms exponentially weights recent data to track time-varying behavior
• Direct learning architectures iteratively update predistorter coefficients in closed-loop operation
• Indirect learning architectures train a post-distorter model and copy it to the predistorter

Accurate system identification demands rigorous signal pre-processing before any coefficient extraction can begin.

• Time alignment synchronizes reference and captured waveforms to sub-sample accuracy
• Loop delay estimation measures propagation delay through the transmit and observation feedback paths
• Misalignment by even a fraction of a sample destroys model fidelity
• The training waveform must have specific statistical properties (e.g., sufficient peak-to-average ratio) to excite the full nonlinear dynamic range
• Post-distortion error quantifies residual nonlinearity after applying the identified predistorter