System Identification

System identification inverts the traditional modeling paradigm. Instead of deriving equations from first principles (physics), it infers the mathematical relationship directly from measured stimulus-response pairs. The process involves:

• Selecting a model structure (e.g., memory polynomial, neural network)
• Applying an excitation signal (input data) that is persistently exciting
• Using an estimation algorithm to minimize the error between the model's predicted output and the actual measured system output This is critical for power amplifiers, where complex electro-thermal dynamics make pure physics-based modeling intractable.

Every system identification problem is defined by three fundamental components:

• Data (Z^N): A finite set of N input-output samples collected from the system.
• Model Set (M):* A parameterized family of candidate models (e.g., all 5th-order memory polynomials).
• Criterion of Fit: A scalar function, typically Mean Squared Error (MSE), that quantifies the prediction error. The identification task is to find the member of the model set that minimizes the criterion given the data. The Bias-Variance Tradeoff governs the choice of model set complexity.

System identification techniques bifurcate into two distinct approaches:

• Parametric Methods: Assume a finite-dimensional parameter vector (e.g., filter coefficients). Algorithms like Least Squares (LS) and Recursive Least Squares (RLS) estimate these specific parameters. This is the dominant approach for DPD.
• Non-Parametric Methods: Estimate the system's impulse or frequency response directly without a fixed parameterization (e.g., correlation analysis, spectral analysis). These are often used for initial system analysis before committing to a parametric model structure. Parametric methods are preferred for real-time DPD due to their compact representation.

A model is useless if it only works on the data it was trained on. Rigorous model validation is non-negotiable:

• Cross-validation: Hold back a portion of the measured data (test set) not used during training.
• Residual Analysis: The prediction error (residuals) should be white noise, uncorrelated with the input. Any remaining correlation indicates unmodeled dynamics.
• Generalization: Test the model with a completely different excitation signal (e.g., a different modulation scheme) to ensure it captures the system's invariant physics, not just the training signal's statistics.

To uniquely identify all parameters of a model, the input signal must be persistently exciting of sufficient order. This means the input must contain enough frequency components to excite all the dynamic modes of the system. A pure sine wave cannot identify a nonlinear system with memory. In DPD, this necessitates using wideband, noise-like communication signals (e.g., OFDM) or specifically designed multi-tone signals during the model extraction phase to ensure the resulting model is valid across the entire operational bandwidth.

System identification exists on a spectrum between white-box (pure physics) and black-box (pure data) models. Gray-box modeling integrates partial physical knowledge into the model structure. For a power amplifier, this might mean:

• Constraining a neural network to enforce a known AM-AM/AM-PM characteristic shape.
• Using a Volterra series structure, which is physically motivated by the Taylor expansion of a nonlinear system with memory. This hybrid approach improves generalization and reduces the amount of training data required compared to a purely black-box model.