Indirect Learning Architecture (ILA)

The core mechanism of ILA is the identification of a post-distorter in the feedback path, not the predistorter directly. A copy of the predistorter is placed after the PA. The adaptation algorithm then solves for coefficients that make this post-distorter's output equal to the original predistorter's input. By the p-inverse assumption, if the post-distorter linearizes the PA, the identical predistorter placed before the PA will also linearize it. This clever architectural trick avoids the need for a separate PA model extraction step, simplifying the training process significantly.

Unlike Direct Learning Architecture (DLA), ILA performs coefficient estimation in an open-loop fashion. The adaptation algorithm does not require an error signal derived from comparing the PA output to a desired linear reference in real-time. Instead, it solves a system identification problem: finding the post-distorter coefficients that map the observed PA output back to the predistorter input. This makes ILA inherently more stable during training, as there is no closed feedback loop that can oscillate or diverge. The trade-off is that ILA is sensitive to measurement noise in the feedback path, which directly corrupts the coefficient estimate.

ILA coefficient estimation is typically formulated as a least squares (LS) problem. Given a block of N samples of the predistorter input u(n) and the normalized PA output y(n)/G (where G is the linear gain), the algorithm constructs a data matrix from the basis function outputs of the post-distorter. The optimal coefficients are found by solving the normal equations or using QR decomposition for numerical stability. This block-based approach provides a one-shot estimate, making it suitable for initial calibration. For online tracking, recursive formulations like Recursive Least Squares (RLS) are employed to update coefficients iteratively as new samples arrive.

A critical limitation of ILA is its sensitivity to measurement noise and feedback path impairments. In the least squares formulation, the regressor matrix is constructed from the noisy PA output observation. This violates the assumption that the independent variables are noise-free, leading to biased coefficient estimates—a phenomenon known as errors-in-variables. In contrast, DLA uses the clean reference signal to construct its regressor matrix. For ILA, this bias manifests as residual nonlinearity in the linearized output, particularly at high signal-to-noise ratios where the noise floor becomes the dominant error source.

ILA relies on the p-inverse assumption: that the post-distorter identified in the feedback path is identical to the required predistorter. This holds exactly only if the PA is a one-to-one mapping with a unique inverse. For PAs with strong memory effects or hysteresis, the inverse may not be unique or commutative. In practice, the assumption is valid for most memory polynomial models, but can break down for architectures like Doherty PAs with complex nonlinear memory. When the assumption fails, iterating the ILA process (Iterative Learning Control) can progressively refine the predistorter coefficients.

ILA is computationally attractive because it avoids the PA model extraction step required by DLA. In DLA, the error gradient requires backpropagation through an identified PA model, adding latency and complexity. ILA solves a single system identification problem directly. For a memory polynomial with K coefficients, the computational complexity is dominated by the matrix inversion or QR decomposition, which scales as O(K³) for block-based LS or O(K²) per iteration for RLS. This makes ILA well-suited for FPGA implementation using systolic arrays for matrix operations.

A foundational stochastic gradient descent algorithm frequently paired with ILA for coefficient estimation. LMS updates predistorter coefficients to minimize the instantaneous squared error between the post-distorter output and the predistorter input. Key characteristics:

• O(N) complexity per iteration — ideal for real-time FPGA implementation
• Simple structure with no matrix inversion required
• Convergence rate depends on the eigenvalue spread of the input correlation matrix
• Susceptible to slow convergence with ill-conditioned signals

An adaptive algorithm offering faster convergence than LMS at the cost of O(N²) complexity. RLS recursively minimizes a weighted linear least squares cost function using a forgetting factor to discount older data. In ILA contexts, RLS excels at tracking time-varying PA nonlinearities caused by thermal drift. The trade-off: superior steady-state performance versus significantly higher computational and memory requirements that challenge tight FPGA resource budgets.

The core mechanism of ILA. A copy of the predistorter is placed in the feedback path as a post-distorter, trained to invert the observed PA output back to the predistorter input. The key insight: if the post-distorter can undo the PA's nonlinearity, its coefficients are directly transferable to the forward predistorter. This elegantly avoids the need for explicit PA model extraction, simplifying the adaptation loop at the cost of assuming the PA is invertible.

A critical prerequisite for ILA coefficient estimation. The transmitted reference signal and the observed feedback signal must be precisely synchronized in the digital domain. Loop delay — the total propagation latency through the TX chain, PA, coupler, and feedback receiver — must be estimated and compensated. Fractional delay filters achieve sub-sample alignment, as even nanosecond misalignment introduces phase errors that corrupt the error signal and degrade linearization performance.

A safeguard mechanism that halts the adaptation loop to lock predistorter coefficients during adverse conditions. Triggers include:

• Signal absence: no input to drive meaningful adaptation
• Feedback path failure: unreliable observation data
• High PAPR events: transient conditions that could cause coefficient divergence Without freeze logic, ILA can drift into unstable regions, producing spectral regrowth worse than the uncorrected PA. Essential for production-grade deployments.