Indirect Learning Architecture

The core mechanism of ILA is training a post-distorter block placed after the power amplifier (PA). The PA output signal, scaled by the desired linear gain, serves as the input to this post-distorter. The model coefficients are estimated by minimizing the error between the post-distorter's output and the original, un-distorted input signal. This effectively solves a system identification problem where the post-distorter learns to invert the PA's nonlinear characteristic. Once trained, the identical coefficient set is copied directly to the predistorter block placed before the PA, assuming the cascade of the predistorter and PA will yield a linear output.

A direct inverse model requires training a model to map the PA's output back to its input, which is a non-causal problem for systems with memory effects. ILA elegantly sidesteps this by training a forward model of the inverse. The post-distorter is trained as a forward behavioral model where the input is the attenuated PA output and the target is the original input signal. This transforms the problem into a standard system identification task solvable with linear regression techniques like Least Squares (LS) or adaptive algorithms like Recursive Least Squares (RLS). The architecture inherently assumes the PA's nonlinearity is invertible and that the post-distorter and predistorter operate under equivalent conditions.

ILA relies on the theoretical concept of the pth-order inverse, which states that if a post-inverse is placed after a nonlinear system, the same model can be placed before the system to achieve linearization. This holds perfectly for static nonlinearities but is an approximation for systems with memory. The accuracy of the coefficient copy depends on the assumption that the PA's characteristics are time-invariant during the training interval and that the signal statistics remain consistent. Any mismatch between the training environment and the operational environment introduces residual distortion, which is why ILA is often paired with periodic coefficient updates in adaptive implementations.

A critical vulnerability of ILA is its sensitivity to the observation receiver used to capture the PA output. The feedback path introduces its own non-idealities:

• IQ imbalance in the downconverter creates mirror-frequency interference that corrupts the training signal.
• DC offset introduces a bias in the captured waveform.
• Feedback path nonlinearity adds distortion that the post-distorter will attempt to compensate for, leading to incorrect predistorter coefficients.
• Loop delay must be precisely estimated and compensated; even sub-sample misalignment degrades model accuracy. These impairments necessitate rigorous calibration of the observation path before ILA training can yield optimal linearization performance.

In operational deployments, PA characteristics drift due to thermal memory effects, aging, and changing bias conditions. An adaptive ILA continuously updates predistorter coefficients by running the post-distorter training loop in the background. Algorithms like Normalized Least Mean Squares (NLMS) or Recursive Least Squares (RLS) with a forgetting factor enable sample-by-sample or block-by-block coefficient updates. The forgetting factor exponentially weights recent data, allowing the model to track slow variations while maintaining stability. This closed-loop adaptation is essential for maintaining ACLR compliance in base stations operating across wide temperature ranges and traffic loads.

ILA contrasts with the Direct Learning Architecture (DLA), which directly optimizes predistorter coefficients by minimizing the error between the desired ideal signal and the actual PA output. Key differences:

• ILA: Trains a post-distorter and copies coefficients. Computationally simpler per iteration but assumes commutativity of the nonlinear cascade.
• DLA: Requires a PA model to backpropagate the error through the nonlinearity. More computationally intensive but can achieve superior performance by directly targeting the linearization objective. ILA is often preferred for initial coefficient extraction and rapid prototyping, while DLA is favored for high-performance adaptive systems where the commutativity assumption limits achievable linearization depth.

A system identification approach that constructs a mathematical replica of the power amplifier by fitting a model to map input signals to measured output signals. This is a prerequisite step for many DPD architectures. Key characteristics:

• Models the PA's forward behavior, not its inverse
• Used to generate training data for offline DPD design
• Often built with Memory Polynomial or Volterra Series structures
• Evaluated using Normalized Mean Square Error (NMSE) metrics

A direct predistorter extraction technique that swaps input and output data during model training to estimate the inverse nonlinear characteristic of the power amplifier. The PA's output becomes the model input, and the PA's input becomes the desired output. This is the foundational principle behind ILA's post-distorter training step. The method assumes the Pth-order inverse theorem holds, which may not be strictly valid for all amplifier classes.

The residual nonlinear distortion measured after applying a predistorter, calculated as the difference between the ideal linear output and the actual amplifier output. In ILA, this metric is used to:

• Validate the copying accuracy from post-distorter to predistorter
• Monitor performance degradation over time
• Trigger coefficient re-extraction when exceeding thresholds
• Quantify Adjacent Channel Leakage Ratio (ACLR) improvement

A batch estimation algorithm that finds model coefficients by minimizing the sum of squared errors between the model's prediction and the measured output in a single computation. In ILA, LS is commonly used for the post-distorter training phase because:

• It provides a closed-form solution via the Moore-Penrose pseudoinverse
• It is computationally efficient for offline training
• It yields the minimum variance unbiased estimator under Gaussian noise assumptions
• It requires the full training dataset to be available at once

An adaptive filtering algorithm that updates model coefficients iteratively as new data arrives, offering faster convergence than LMS at the cost of higher computational complexity. RLS is often paired with ILA for online tracking of time-varying PA characteristics. Key properties:

• Incorporates a forgetting factor to weight recent data
• Converges in approximately 2N iterations (N = number of coefficients)
• Computational complexity of O(N²) per iteration
• Enables continuous adaptation without full retraining