Multi-Band Coefficient Extraction

Unlike single-band DPD, multi-band systems must solve for all coefficients simultaneously in a single optimization step. This joint estimation captures the coupling between bands.

• Single Least-Squares Problem: Formulates one large matrix equation incorporating all band signals and cross-terms
• Cross-Coupling: Correctly identifies that distortion in Band 1 depends on the instantaneous envelope of Band 2
• Computational Cost: Matrix dimensions scale quadratically with the number of bands and model memory depth
• Example: A 2D-MMP model with memory depth M=3 and nonlinearity order K=5 requires estimating hundreds of coefficients at once

The Multi-Band ILA is the dominant closed-loop extraction method. A post-distorter is identified from the attenuated PA output, then copied to the forward path.

• Post-Distorter Identification: The inverse model is trained on PA input/output data
• Copy to Predistorter: Assumes the post-inverse equals the pre-inverse (valid for mild nonlinearities)
• No Iterative Convergence: Avoids the convergence issues of direct learning in multi-dimensional spaces
• Limitation: Performance degrades if measurement noise is significant in the feedback path

The workhorse algorithm for coefficient extraction is the Least Squares (LS) estimator, which minimizes the squared error between the model output and the observed PA output.

• Batch Processing: Collects a block of I/Q samples and solves the normal equations
• Moore-Penrose Pseudoinverse: Used when the data matrix is ill-conditioned
• Regularization: Ridge regression (L2) or LASSO (L1) prevents overfitting to measurement noise
• QR Decomposition: Numerically stable implementation for solving the LS problem on FPGA/DSP hardware

Before extraction can occur, the regressor matrix must be constructed from the multi-band input signals. Each column represents a specific nonlinear basis function.

• 2D Envelope Indexing: Basis functions depend on |x₁(n)| and |x₂(n)| simultaneously
• Memory Cross-Terms: Includes delayed samples like x₁(n-m)·|x₂(n-m)|^k
• Sample-Crossing Terms: Captures interactions like x₁(n)·x₂(n-1)* for cross-band memory
• Pruning: Removes statistically insignificant basis functions to reduce the coefficient count

Accurate coefficient extraction requires sub-sample time alignment between the transmitted reference and the observed PA output. Misalignment destroys model fidelity.

• Cross-Correlation: Finds integer-sample delay between input and feedback paths
• Fractional Delay Interpolation: Farrow structure or Lagrange interpolation for sub-sample alignment
• Phase Coherence: Compensates for LO phase drift between capture events
• Impact: Even 0.1 sample misalignment can increase NMSE by 5-10 dB

For tracking time-varying PA behavior due to temperature and aging, recursive algorithms update coefficients sample-by-sample without full matrix inversion.

• Recursive Least Squares (RLS): Fast convergence but O(N²) complexity per iteration
• Least Mean Squares (LMS): Low complexity O(N) but slower convergence
• Forgetting Factor (λ): Controls how quickly old data is discounted (typically 0.95-0.999)
• Application: Essential for mobile handsets where PA characteristics drift with battery voltage and thermal state