Joint Coefficient Estimation

Unlike sequential methods that estimate per-band coefficients independently, joint estimation formulates a single least-squares problem encompassing all bands simultaneously. The composite error vector stacks residual errors from each band, and the optimizer minimizes the total squared error across the entire multi-band system. This guarantees a globally optimal solution for the given model structure rather than a locally optimized compromise.

Joint estimation inherently captures the interdependence between in-band and cross-band coefficients. When a 2D memory polynomial includes terms like |x₁(n-m)|²·x₂(n-l), the coefficient for this term affects distortion cancellation in both bands. Joint estimation resolves these coupled parameters simultaneously, preventing the parameter divergence that can occur when cross-terms are estimated independently and then combined.

The estimation problem is cast as Y = X·θ, where:

• Y is the stacked output vector from all bands
• X is the composite regression matrix containing basis functions for all bands and cross-terms
• θ is the vector of all unknown coefficients

The solution θ̂ = (XᴴX)⁻¹XᴴY is computed via QR decomposition or Cholesky factorization for numerical stability, especially when the regression matrix is ill-conditioned due to correlated basis functions.

Joint estimation of an N-band model with M coefficients per band and K cross-terms requires solving a system of size (N·M + K) × (N·M + K). For dual-band systems with memory depth 5 and nonlinearity order 7, this can exceed 500×500 matrices. The cubic complexity O(n³) of matrix inversion makes real-time adaptation challenging, driving research into reduced-complexity joint estimators using iterative methods like conjugate gradient or coordinate descent.

Joint estimation integrates naturally with the multi-band indirect learning architecture (MB-ILA). The post-distorter is trained jointly on the attenuated PA output, and the estimated coefficients are copied to the predistorter. This closed-loop structure ensures the joint solution adapts to time-varying PA characteristics including thermal drift and bias changes, maintaining ACLR suppression during continuous operation.

Joint estimation problems often exhibit multicollinearity between basis functions, particularly when cross-terms resemble scaled versions of in-band terms. Ridge regression (L2 regularization) adds a penalty term λ||θ||² to the cost function, stabilizing the matrix inversion. The regularization parameter λ is typically selected via cross-validation or set adaptively based on the condition number of XᴴX.

The signal processing procedure for estimating the parameters of a multi-band DPD model from observed input and output waveforms. Joint estimation is a specific strategy within this broader extraction process.

• Offline extraction: Uses captured time-domain waveforms in batch processing
• Online extraction: Adapts coefficients in real-time during transmission
• Joint extraction estimates all coefficients simultaneously, capturing cross-band interactions
• Sequential extraction estimates per-band coefficients first, then cross-band terms

A closed-loop DPD adaptation method where a post-distorter model is identified from the attenuated PA output and then copied to the predistorter in the forward path. Joint coefficient estimation is the mathematical core of the post-distorter identification stage.

• The post-distorter is trained to invert the PA's nonlinear response
• Joint estimation within MB-ILA captures cross-band memory effects
• Avoids the need for a direct PA model inversion
• Coefficients are periodically refreshed to track thermal and aging drift

A behavioral model that extends the memory polynomial to two dimensions by including cross-terms dependent on the envelope magnitudes of both concurrent bands. Joint coefficient estimation is essential for 2D-MMP because the cross-terms couple the bands mathematically.

• Basis functions include |x₁(n-m)|^k * |x₂(n-m)|^l terms
• Captures cross-band modulation and intermodulation effects
• Joint estimation solves a single least-squares problem for all coefficients
• Model complexity scales quadratically with nonlinearity order

An extension of the GMP model incorporating cross-band envelope and sample-crossing terms to capture complex nonlinear interactions. Joint coefficient estimation is mandatory for MB-GMP due to the dense coupling between all basis functions.

• Includes lagging and leading envelope terms across bands
• Models both short-term and long-term memory effects
• Coefficient vector estimated via regularized least squares (e.g., ridge regression)
• Pruning techniques reduce the basis set before joint estimation to manage complexity

A long-term memory effect in multi-band amplifiers where the nonlinear behavior in one frequency band is influenced by the past envelope history of a signal in a different band. Joint coefficient estimation explicitly models these cross-band memory paths.

• Caused by thermal coupling and bias circuit impedance at envelope frequencies
• Manifests as dynamic cross-modulation between bands
• Requires memory depth terms spanning both bands in the model structure
• Ignoring cross-band memory leads to incomplete linearization and residual IMD

The foundational numerical method underlying joint coefficient estimation. The multi-band DPD model is formulated as a linear-in-parameters regression problem, and the coefficient vector is solved by minimizing the sum of squared errors.

• Ordinary least squares (OLS): Direct solution via pseudo-inverse (XᴴX)⁻¹Xᴴy
• Regularized least squares: Adds λ||w||² penalty to prevent overfitting
• Recursive least squares (RLS): Updates coefficients sample-by-sample for online adaptation
• Matrix conditioning is critical; highly correlated basis functions degrade estimation accuracy