Postdistorter

The postdistorter's primary role is to learn the post-inverse of the power amplifier. By placing the nonlinear model after the PA and training it to reproduce the original input signal, the postdistorter effectively identifies the mathematical inverse of the amplifier's distortion characteristic. This extracted inverse model is then copied directly to the predistorter placed before the PA.

• Trains on the PA output signal to reconstruct the original input
• The converged coefficients represent the inverse nonlinearity
• Enables offline training without interrupting transmission

In the Indirect Learning Architecture (ILA), the postdistorter is positioned in a parallel training branch after the power amplifier. This placement is critical because it allows the postdistorter to observe the actual distorted output of the PA and learn the correction mapping without affecting the forward transmission path.

• Located in the observation/training path, not the transmission path
• Receives attenuated and downconverted PA output samples
• Operates independently of the main signal chain during training

Once the postdistorter converges to an optimal solution, its coefficients are directly copied to the predistorter. This copy-exact approach assumes that the post-inverse identified after the PA is mathematically equivalent to the pre-inverse needed before the PA. This assumption holds for systems where the PA characteristics are time-invariant over the training interval.

• No mathematical inversion of the model is required
• Coefficients transfer is a simple memory copy operation
• Assumes commutativity of the nonlinear system (valid for memoryless or weakly nonlinear PAs)

The postdistorter requires a reference signal for supervised learning. The ideal input to the PA serves as the target, while the attenuated PA output serves as the input to the postdistorter. The error between the postdistorter output and the original input drives coefficient adaptation.

• Reference: Original baseband input signal (pre-PA)
• Input: Feedback signal from the transmit observation receiver (TOR)
• Requires precise time alignment between reference and feedback paths
• Sensitive to IQ imbalance and feedback path nonlinearities

The postdistorter can implement the same nonlinear model structures used for predistorters, including memory polynomial, generalized memory polynomial (GMP), or Volterra series models. The choice of model structure determines the types of nonlinearity and memory effects that can be captured.

• Memory polynomial: Captures diagonal memory effects
• GMP: Adds cross-term memory for improved accuracy
• Neural network models: For highly complex nonlinearities
• Model complexity must balance linearization performance against coefficient extraction time

The postdistorter approach has inherent limitations. The commutativity assumption may not hold for strongly nonlinear PAs with significant memory effects. Additionally, feedback path impairments such as noise, nonlinearity, and IQ imbalance in the observation receiver directly corrupt the postdistorter training and degrade the final predistorter performance.

• Feedback path noise introduces coefficient estimation bias
• PA output SNR limits achievable linearization
• Non-commutative behavior in deep saturation causes residual distortion
• Requires high-quality observation receiver with better linearity than the PA under test

The architectural framework that defines the postdistorter's role. In ILA, a postdistorter is placed after the power amplifier (PA) and trained to replicate the PA's inverse transfer function. Once training converges, the learned coefficients are copied directly to the predistorter placed before the PA. This avoids the need for a real-time PA model but assumes the PA characteristic is invertible and time-invariant during the training block.

The primary alternative to ILA. DLA directly estimates the predistorter coefficients by minimizing the error between the desired input signal and the actual PA output, without an intermediate postdistorter. This closed-loop approach requires a real-time PA behavioral model to compute error gradients. DLA is more robust to measurement noise but computationally more intensive than the postdistorter-based ILA method.

A foundational stochastic gradient descent algorithm frequently used for adaptive postdistorter coefficient estimation. LMS updates coefficients sample-by-sample based on the instantaneous gradient of the squared error:

• Complexity: Very low, O(N) per iteration
• Convergence: Slow, highly dependent on step size selection
• Use Case: Suitable for slowly varying PA characteristics where computational resources are constrained

An adaptive filtering algorithm offering significantly faster convergence than LMS at the cost of higher computational complexity (O(N²)). RLS recursively minimizes a weighted linear least squares cost function, making it ideal for postdistorter training in rapidly changing signal conditions. Key variants include QR-RLS, which uses QR decomposition for improved numerical stability when the input correlation matrix is ill-conditioned.

The mathematical process of solving for the postdistorter's parameters. When the PA's inverse modeling problem is ill-conditioned—common with wideband signals—regularization techniques are essential:

• Tikhonov Regularization: Adds an L2 penalty to stabilize matrix inversion
• Levenberg-Marquardt: Interpolates between Gauss-Newton and gradient descent for robust nonlinear least squares
• Condition Number: Monitored to detect numerical instability before coefficients diverge

After postdistorter training, these metrics quantify linearization effectiveness before coefficients are deployed to the predistorter:

• NMSE (Normalized Mean Squared Error): Measures residual distortion power relative to input signal power
• EVM (Error Vector Magnitude): Quantifies in-band signal quality degradation
• ACPR (Adjacent Channel Power Ratio): Validates spectral regrowth suppression in adjacent frequency channels