Inverse Modeling

Unlike the Indirect Learning Architecture (ILA) which swaps input and output, inverse modeling directly estimates the predistorter by minimizing the error between the desired linear output and the actual Power Amplifier (PA) output. This approach solves for the inverse function F⁻¹ such that the cascade of the predistorter and PA approximates an identity operation.

• Avoids the assumption that the post-inverse equals the pre-inverse
• Directly minimizes the Error Vector Magnitude (EVM)
• Requires a model of the PA's forward behavior or iterative optimization

Modern inverse modeling leverages deep neural networks as universal function approximators to capture the complex, non-linear inverse behavior of power amplifiers. Architectures like the Real-Valued Time-Delay Neural Network (RVTDNN) process I and Q components separately with tapped delay lines to model temporal dependencies.

• Feed-forward networks for static non-linearity
• Recurrent neural networks for memory effects
• Augmented with envelope-dependent terms for AM-AM and AM-PM distortion correction

The inverse model parameters are refined through iterative optimization algorithms that minimize a cost function, typically the mean squared error between the ideal linear output and the actual PA output. This process must converge rapidly to track time-varying PA behavior.

• Stochastic gradient descent for neural network training
• Least squares estimation for polynomial-based models
• Online training enables real-time adaptation to temperature drift and aging

A critical capability of inverse modeling is compensating for memory effects—the dependence of the PA's current output on past input values caused by thermal dynamics, biasing networks, and trapping effects. The inverse model incorporates tapped delay lines or recurrent structures to cancel these temporal distortions.

• Short-term memory from impedance matching networks
• Long-term memory from thermal time constants
• Volterra series and Generalized Memory Polynomial (GMP) provide theoretical foundations

Advanced inverse modeling frameworks simultaneously correct for multiple impairments beyond PA non-linearity, including I/Q imbalance, local oscillator leakage, and modulator non-linearity. This joint optimization eliminates the need for separate compensation stages.

• Single model corrects gain and phase mismatches
• Reduces overall system complexity
• Improves Adjacent Channel Leakage Ratio (ACLR) and EVM holistically

In Massive MIMO systems, each beam experiences a different composite amplifier distortion due to antenna mutual coupling and impedance variations. Inverse modeling for these arrays must generate beam-specific predistortion functions or a unified model conditioned on beamforming weights.

• Addresses spatial variation in non-linearity
• Enables over-the-air DPD with remote observation receivers
• Critical for maintaining spectral compliance across all beam directions

A contrasting approach to inverse modeling where the predistorter coefficients are estimated by training a post-distorter model and copying its parameters. The ILA assumes the inverse is commutable, which breaks down in the presence of significant memory effects. While computationally simpler, it often yields suboptimal linearization compared to direct inverse modeling.

Modern inverse modeling leverages deep neural networks to approximate the complex non-linear inverse function of a power amplifier. Architectures include:

• Real-Valued Time-Delay Neural Networks (RVTDNN) for modeling temporal dependencies
• Recurrent Neural Networks (RNNs) for long-term memory effects
• Convolutional Neural Networks (CNNs) for spectral pattern recognition These models outperform classical Volterra-based inverses for wideband signals.

The classical mathematical foundation for non-linear system inversion. A Volterra series models the PA as a sum of multi-dimensional convolution kernels, capturing both static non-linearity and memory effects. The inverse model is derived by solving for the kernels that cascade to a linear response. Pruned Volterra models, like the Generalized Memory Polynomial (GMP), are common in commercial DPD systems.

The real-time process of updating the inverse model's parameters to track time-varying PA behavior. Causes of drift include:

• Thermal dynamics altering transistor gain
• Aging effects in semiconductor materials
• Load impedance mismatch from antenna detuning Adaptive inverse modeling ensures consistent linearization without interrupting live transmission.

A precursor step to inverse modeling that creates an accurate behavioral model of the PA's non-linear transfer function. This forward model is used to:

• Generate training data for the inverse model
• Simulate system performance offline
• Validate the linearization efficacy Accurate forward modeling is critical; errors propagate directly into the inverse solution.