LUT Coefficient Extraction

Coefficient extraction fundamentally relies on post-distorter identification, where the input and output of the power amplifier (PA) are mathematically swapped. The algorithm solves for the predistorter function that, when cascaded with the PA, produces a linear overall response. This is achieved by treating the PA output (attenuated) as the model input and the original clean signal as the desired output, effectively learning the inverse nonlinear characteristic of the amplifier.

The workhorse batch extraction method that minimizes the sum of squared errors between the desired linear output and the actual PA output. For a memory polynomial model, the LS solution is computed as:

• w = (X^H X)^(-1) X^H y, where X is the regressor matrix of basis functions
• Provides optimal coefficients in a single computation for stationary conditions
• Computationally intensive for large matrices but yields the minimum variance unbiased estimate under Gaussian noise assumptions

An online extraction algorithm that updates LUT coefficients sample-by-sample without requiring matrix inversion. RLS maintains an inverse correlation matrix and applies a forgetting factor (λ) to track time-varying PA nonlinearity:

• Convergence is typically achieved within 50-200 samples
• Computational complexity of O(N²) per iteration, where N is the number of coefficients
• Superior tracking capability compared to LMS for rapidly changing signal statistics

The dominant closed-loop extraction topology where the predistorter is trained in a post-distorter configuration. The PA output is fed through an identical copy of the predistorter model, and the error between this post-distorted signal and the original predistorter output drives coefficient updates. Key advantage: avoids the need to compute the PA inverse directly, instead iteratively converging to the optimal predistortion function through successive approximation.

An alternative extraction approach that explicitly models the PA forward characteristic first, then mathematically inverts this model to obtain predistorter coefficients. DLA requires:

• Accurate PA behavioral modeling as a prerequisite step
• Numerical inversion of the nonlinear model, often using iterative root-finding
• Advantageous when the PA characteristic changes slowly relative to signal dynamics, as the forward model can be updated independently of the inverse computation

Coefficient extraction quality depends critically on the regressor matrix conditioning. Basis functions are constructed from the input signal using:

• Memory polynomial terms: x(n-m)|x(n-m)|^k for memory depth m and nonlinearity order k
• Cross terms: x(n-m)|x(n-l)|^k capturing interactions between delayed samples
• Proper normalization and orthogonalization (e.g., Gram-Schmidt) prevents numerical instability during matrix inversion, especially for wideband signals with high peak-to-average ratios

The Least Mean Squares algorithm is the foundational iterative method for extracting LUT coefficients. It minimizes the mean squared error between the desired linear output and the actual PA output by recursively adjusting each table entry. The update equation is: LUT_i(n+1) = LUT_i(n) + μ * e(n) * conj(x(n)), where μ is the step size controlling convergence speed versus steady-state jitter. Its computational simplicity makes it ideal for real-time hardware implementation.

A closed-loop coefficient extraction method where the postdistorter is trained on the PA output signal. The key insight: if a postdistorter can linearize the PA in reverse, its coefficients can be copied directly to the predistorter. This avoids the need to compute the PA's exact inverse model analytically. The architecture uses two identical LUTs—one in the forward path for predistortion and one in the feedback path being trained.

This architecture directly estimates the predistorter coefficients by computing the error between the original input signal and the attenuated PA output. It requires an explicit PA model to backpropagate the error through the nonlinearity. While more computationally intensive than indirect methods, it offers superior performance when the PA characteristics are well-characterized and can adapt more robustly to measurement noise in the feedback path.

A mathematical technique for estimating predistortion values between discrete table entries to reduce quantization error. Common methods include:

• Linear interpolation: Simple weighted average between two adjacent entries
• Polynomial interpolation: Higher-order curve fitting for smoother transitions
• Cubic spline: Minimizes spectral regrowth from coefficient discontinuities Proper interpolation allows smaller LUT sizes while maintaining ACLR performance, reducing memory footprint and power consumption in FPGA implementations.

The offline or online process of populating and iteratively refining LUT entries using measured PA input-output data. Offline training uses captured waveforms in a lab environment to compute optimal coefficients before deployment. Online training continuously adapts during live operation using a feedback receiver. Key considerations include:

• Training signal statistics: Must match operational waveforms
• Convergence criteria: Error vector magnitude thresholds
• Coefficient smoothing: Prevents adaptation noise from causing spectral regrowth

A predistortion table architecture that stores a single complex-valued coefficient per entry to simultaneously correct both AM-AM (amplitude) and AM-PM (phase) distortion. Each coefficient is of the form G = |G| * e^(jφ), where the magnitude compensates for gain compression and the phase cancels the PA's input-power-dependent phase shift. This unified approach eliminates the need for separate amplitude and phase correction tables, reducing memory requirements by half.