Indirect Learning Architecture (ILA)

ILA operates on the p-inverse estimation principle. Instead of directly identifying the predistorter, it places a copy of the predistorter model in the feedback path after the power amplifier (PA). The algorithm then minimizes the error between the output of this postdistorter and the input to the predistorter. If the PA is invertible, the postdistorter converges to the optimal predistorter. This approach transforms the nonlinear inverse problem into a standard system identification task with the PA output as the input and the predistorter input as the desired signal.

Unlike the Direct Learning Architecture (DLA), ILA performs coefficient estimation in an open-loop configuration. The training path is decoupled from the transmission path, meaning the PA output is captured and processed offline or in a parallel path without affecting the live signal. This decoupling eliminates the stability concerns associated with closed-loop adaptive systems. The extracted coefficients are then copied to the predistorter in the forward path. This architecture is particularly advantageous for batch estimation algorithms like Least Squares (LS) and QR Decomposition (QRD).

The theoretical validity of ILA rests on the assumption that the power amplifier is invertible. The method swaps the input and output roles during training, assuming the postdistorter will converge to the inverse of the PA. This holds for memoryless nonlinearities and weakly nonlinear systems with memory. However, for PAs with strong memory effects or non-invertible characteristics, the ILA estimate may be biased. The commutation error—the difference between the true predistorter and the ILA estimate—increases with PA nonlinearity order and memory depth.

In ILA, the noisy PA output serves as the input to the postdistorter model during training. This introduces noise coloring in the coefficient estimation process. Unlike DLA, where the clean reference signal drives the adaptation, ILA's estimate is influenced by measurement noise and ADC quantization errors in the feedback path. This can lead to biased coefficient estimates, especially at low signal-to-noise ratios. Regularization techniques, such as adding a regularization parameter to the diagonal of the correlation matrix, are often employed to mitigate noise amplification.

ILA supports both offline training and online training modes. In offline mode, a complete capture of PA input-output data is used with batch algorithms like LS, QRD, or Singular Value Decomposition (SVD) to extract coefficients. In online mode, adaptive algorithms such as Recursive Least Squares (RLS) or Least Mean Squares (LMS) update coefficients iteratively as new samples arrive. The open-loop nature makes ILA particularly well-suited for FPGA-based implementations where the training engine operates independently of the signal path, enabling non-real-time coefficient updates.

ILA estimation often involves solving linear systems with potentially ill-conditioned matrices, especially when using high-order polynomial models or wideband signals. The condition number of the data correlation matrix can become large, leading to numerical instability. Robust implementations employ QR decomposition with Givens rotations or Cholesky decomposition to maintain numerical precision. The forgetting factor in recursive implementations must be carefully chosen to balance tracking capability against noise enhancement and numerical stability.

A closed-loop estimation method that identifies predistorter parameters directly by minimizing the error between the desired linear output and the actual power amplifier output. Unlike ILA, DLA does not assume the PA inverse exists as a postdistorter.

• Key difference: DLA minimizes the linearized output error; ILA minimizes the postdistorter identification error
• Advantage: DLA is theoretically unbiased under noisy output measurements
• Challenge: Requires the PA model to be differentiable for gradient computation
• Use case: Preferred when the PA exhibits strong nonlinear dynamics that violate ILA's commutativity assumption

A batch estimation method that finds the optimal coefficient vector by minimizing the sum of squared errors between observed and desired signals. In ILA, LS is commonly used for offline training of the postdistorter copy.

• Mechanism: Solves the Normal Equation directly using matrix inversion or QR decomposition
• Advantage: Provides the optimal solution in a single computation for stationary systems
• Limitation: Computationally expensive for large datasets; does not track time-varying behavior
• ILA application: Used during initial calibration to extract predistorter coefficients from captured input-output data pairs

An adaptive algorithm that recursively updates the inverse of the input correlation matrix to achieve faster convergence than gradient-based methods. RLS is the online counterpart to LS batch estimation in ILA implementations.

• Convergence: Typically an order of magnitude faster than LMS-based algorithms
• Cost: O(N²) complexity per iteration due to matrix inversion lemma updates
• Forgetting Factor: A scalar parameter (typically 0.95–0.999) that exponentially weights recent data to track time-varying PA characteristics
• ILA integration: Enables real-time coefficient updates as the PA thermal state and signal statistics change during operation

A foundational stochastic gradient descent algorithm that updates filter coefficients iteratively based on the instantaneous estimate of the mean squared error gradient. LMS offers the lowest computational complexity for ILA coefficient adaptation.

• Update rule: w(n+1) = w(n) + μ·e(n)·x(n), where μ is the step size
• Complexity: O(N) per iteration — linear in the number of coefficients
• Tradeoff: Slow convergence for correlated input signals common in wideband DPD
• Variant: Normalized LMS (NLMS) normalizes the step size by input signal power to improve stability
• ILA use: Suitable for resource-constrained FPGA implementations where simplicity is prioritized over convergence speed

A numerically robust implementation of the Recursive Least Squares algorithm that uses Givens rotations to directly update the square-root of the inverse correlation matrix. This approach avoids the numerical instability of explicit matrix inversion in standard RLS.

• Mechanism: Maintains the Cholesky factor of the correlation matrix and updates it via orthogonal transformations
• Advantage: Excellent numerical conditioning even with ill-conditioned input signals
• Hardware mapping: Givens rotations map efficiently to systolic array architectures on FPGAs
• ILA relevance: Critical for wideband DPD where the input correlation matrix can become nearly singular due to high sample-rate signals with limited spectral diversity

A control methodology that improves the transient response of a repetitive system by updating the input signal based on the error trajectory from previous iterations. ILC shares structural similarities with ILA but operates in the time domain over repeated trials.

• Core idea: The input for trial k+1 is computed as u_{k+1} = u_k + L·e_k, where L is a learning filter
• Similarity to ILA: Both use an inverse model identified from error signals to generate a pre-distorted input
• Key difference: ILC is trial-based (repetitive operations); ILA is sample-based (continuous streaming)
• Cross-domain insight: ILC convergence theory informs ILA stability analysis for periodic signal types