Inverse Modeling

The core objective is to synthesize a transfer function H⁻¹ such that the cascade of the predistorter and the power amplifier results in an ideal linear gain. This is fundamentally an ill-posed inverse problem because the PA's gain compression creates a non-injective mapping where multiple input amplitudes can map to the same output amplitude. Successful inversion requires regularization techniques like Tikhonov regularization to stabilize the solution and prevent the model from generating infinite gain to compensate for deep saturation.

Modern wideband signals require the inverse model to replicate not just static nonlinearity but also dynamic memory effects. These are caused by:

• Electrical memory: Bias circuit impedance variations and envelope frequency-dependent matching networks.
• Thermal memory: Die temperature fluctuations that change transistor transconductance on a microsecond scale. The inverse model must incorporate tapped-delay lines or recurrent structures to compensate for these time-dispersive phenomena, ensuring inter-symbol interference is cancelled.

Inverse modeling can be implemented through two distinct topological approaches:

• Direct Inverse: The model is trained directly to map PA output samples back to input samples. This is computationally simple but mathematically biased in the presence of noise.
• Postdistorter Training: Used in the Indirect Learning Architecture (ILA), a postdistorter is trained on the PA output, and its coefficients are copied to the predistorter. This leverages the commutativity assumption of linear operators, though it is strictly valid only for exact inverses.

The inverse model's parameters exhibit extreme sensitivity to numerical conditioning. The condition number of the regression matrix directly impacts stability; high condition numbers amplify estimation errors. To mitigate this:

• QR decomposition is preferred over direct matrix inversion for solving least squares.
• Regularization adds a penalty term to the cost function to suppress coefficient drift.
• Fixed-point quantization for FPGA implementation requires careful analysis of coefficient word length to prevent limit cycles.

The inverse model must generalize across signal statistics not seen during training. Overfitting occurs when the model memorizes the specific training waveform's peak-to-average ratio rather than learning the underlying semiconductor physics. This is detected when Normalized Mean Squared Error (NMSE) is excellent on training data but Adjacent Channel Power Ratio (ACPR) degrades with a different modulation scheme. Cross-validation using multiple signal types is essential to verify robust generalization.

In operational deployments, the inverse model must track time-varying nonlinearities caused by temperature drift, voltage sag, and device aging. This requires online coefficient estimation algorithms such as Recursive Least Squares (RLS) or Kalman filtering. Unlike offline extraction, online adaptation must balance convergence rate against misadjustment noise, often using a variable forgetting factor to lock coefficients during steady-state operation while remaining agile to environmental transients.

The dominant training topology for inverse modeling. Instead of directly training the predistorter, an identical postdistorter model is placed after the power amplifier and trained to replicate the input signal. The converged coefficients are then copied to the predistorter. This avoids the need for real-time closed-loop gradient computation through the PA, simplifying implementation but assuming commutability between the pre- and post-distorter.

A closed-loop topology that directly estimates the predistorter coefficients by minimizing the error between the desired ideal input and the actual PA output. This requires modeling or measuring the PA's response during training. DLA is more robust to PA nonlinearity order and measurement noise than ILA but demands more complex real-time optimization, often using iterative gradient-based methods.

A direct learning technique that mathematically inverts a pre-identified power amplifier behavioral model to derive the predistorter transfer function. If the forward PA model is expressed as a polynomial or neural network, the inverse is solved analytically or through iterative root-finding. This method decouples PA identification from predistorter training but is highly sensitive to modeling errors and ill-conditioning.

A foundational stochastic gradient descent algorithm for online inverse modeling. LMS updates predistorter coefficients sample-by-sample using the instantaneous gradient of the squared error. Its computational simplicity—requiring only O(N) operations per sample—makes it ideal for FPGA implementation, though convergence speed depends heavily on the eigenvalue spread of the input signal's autocorrelation matrix.

An adaptive filtering algorithm that recursively minimizes a weighted linear least squares cost function, offering significantly faster convergence than LMS at the cost of higher computational complexity—O(N²) per iteration. RLS is particularly effective for tracking time-varying PA nonlinearities in wideband applications where the input signal statistics change rapidly.

A ridge regression technique critical for stabilizing inverse modeling when the estimation problem is ill-conditioned. By adding a penalty term λ||w||² to the least squares cost function, Tikhonov regularization prevents coefficient drift and numerical instability caused by near-singular autocorrelation matrices. This is essential for wideband signals and high-order polynomial predistorters where the condition number can become extremely large.