System Identification

System identification for DPD focuses on building an inverse model of the power amplifier. Rather than modeling the PA's forward nonlinearity directly, the goal is to identify a predistorter function that, when cascaded with the PA, produces a linear overall response. This requires careful handling of causality constraints and stability considerations, as the inverse of a nonlinear system with memory may not always exist or be realizable. The identification process typically uses the indirect learning architecture (ILA) or direct learning architecture (DLA) to estimate the predistorter coefficients from measured input-output pairs.

System identification approaches divide into two categories:

• Parametric models: Represent the system with a finite number of parameters, such as memory polynomial coefficients or Volterra kernel weights. These are compact and interpretable but require selecting the correct model structure a priori.
• Nonparametric models: Make fewer structural assumptions, using techniques like neural networks or look-up tables to learn the mapping directly from data. These offer greater flexibility for capturing complex nonlinearities but may require more data and computational resources. The choice depends on the amplifier technology (e.g., Doherty, GaN), signal bandwidth, and available hardware resources.

The quality of an identified model depends critically on the persistence of excitation of the input signal. The training waveform must sufficiently excite all nonlinear modes and memory depths of the amplifier to produce a well-conditioned identification problem. Key considerations include:

• PAPR distribution: Must match the operational waveform to capture realistic nonlinear behavior.
• Bandwidth: Should span the full linearization bandwidth to characterize frequency-dependent memory effects.
• Statistical properties: Signals with Gaussian-like amplitude distributions provide better conditioning for correlation matrix inversion. Inadequate excitation leads to ill-conditioning and models that fail to generalize to real communication signals.

An identified model must be rigorously validated to ensure it generalizes beyond the training data. Standard practices include:

• Normalized Mean Square Error (NMSE): Quantifies the time-domain prediction accuracy between the model output and measured PA output.
• Adjacent Channel Leakage Ratio (ACLR): Verifies that the model accurately predicts spectral regrowth in adjacent channels.
• Error Vector Magnitude (EVM): Measures in-band distortion prediction fidelity.
• Cross-validation: Splitting measured data into training and test sets to detect overfitting, where the model memorizes noise rather than learning the true system dynamics. A model with excellent training fit but poor test performance indicates insufficient regularization or an overly complex model structure.

System identification algorithms operate in two fundamental modes:

• Batch identification: Processes an entire block of captured data at once, solving a least-squares problem using techniques like QR decomposition or singular value decomposition. This provides optimal estimates for stationary systems but cannot track time-varying behavior.
• Recursive identification: Updates model parameters sample-by-sample using algorithms like Recursive Least Squares (RLS) or Least Mean Squares (LMS). These methods incorporate a forgetting factor to track slowly varying PA characteristics due to temperature drift, aging, or changing operating conditions. Modern DPD systems often combine both: batch extraction for initial model acquisition and recursive updates for continuous background adaptation.

The identification process is only as accurate as the observed feedback signal. The feedback receiver introduces its own impairments that must be accounted for:

• IQ imbalance: Gain and phase mismatches in the feedback demodulator create image interference that corrupts the error signal.
• Feedback nonlinearity: The observation path itself may introduce compression or distortion, particularly if the coupled signal is strong.
• Loop delay: The propagation latency through the transmit chain and feedback path must be precisely estimated using cross-correlation techniques and compensated with fractional delay filters. Failure to calibrate these impairments leads to the predistorter learning to compensate for feedback artifacts rather than actual PA nonlinearity.

A DPD training architecture that estimates the predistorter coefficients by placing a copy of the predistorter in the feedback path. This approach avoids the explicit intermediate step of identifying a separate PA model.

• Mechanism: The post-distorter is trained to invert the PA's nonlinearity using the observed output.
• Advantage: Simpler implementation as it bypasses explicit PA model extraction.
• Limitation: Sensitive to measurement noise in the feedback path, which can bias the coefficient estimation.

A closed-loop architecture that iteratively minimizes the error between the desired linear output and the actual PA output. It requires an identified PA model to compute the error gradient for updating the predistorter.

• Process: Compares the ideal signal with the actual PA output to generate a precise error signal.
• Requirement: Depends on accurate System Identification of the PA's forward behavior.
• Benefit: Generally more robust to feedback noise than ILA, leading to superior linearization performance.

An adaptive filtering algorithm that recursively finds coefficients minimizing a weighted linear least squares cost function. It offers an order of magnitude faster convergence rate than LMS.

• Key Parameter: The forgetting factor (λ) exponentially weights past data, enabling tracking of non-stationary PA behavior.
• Trade-off: High computational complexity of O(N²) per iteration, making FPGA implementation challenging.
• Use Case: Ideal for rapidly changing signal conditions where fast re-convergence is critical.

A stochastic gradient descent algorithm that updates coefficients to minimize the instantaneous squared error. It is prized for its extreme simplicity and low computational overhead.

• Update Rule: Coefficients are adjusted by a step size proportional to the product of the error and the input vector.
• Complexity: O(N) per iteration, making it highly suitable for high-speed FPGA-Based DPD Implementation.
• Constraint: Convergence speed is highly dependent on the eigenvalue spread of the input correlation matrix.

A variant of LMS that normalizes the coefficient update step size by the power of the input signal vector. This significantly improves convergence stability for signals with fluctuating power levels, such as modern communication waveforms.

• Mechanism: Divides the standard LMS step size by the squared Euclidean norm of the input vector plus a small regularization parameter.
• Benefit: Prevents gradient noise amplification during high-power signal peaks.
• Application: The de facto standard for many real-time Closed-Loop DPD systems due to its balance of simplicity and robustness.

The mathematical core of the adaptation loop. The cost function (typically Mean Squared Error) quantifies the aggregate error between the desired linear output and the actual PA output.

• Error Signal: The instantaneous difference driving the coefficient update.
• Minimization: Algorithms like LMS and RLS iteratively adjust predistorter coefficients to find the global minimum of this cost surface.
• Metrics: Directly impacts EVM and ACLR performance; a lower cost function value correlates with better linearization.