Coefficient Vector

Each element in a coefficient vector is a complex number, representing both a magnitude scaling and a phase rotation applied to a specific basis function. This dual control is essential because power amplifier distortion affects both the AM-AM (amplitude) and AM-PM (phase) characteristics of the signal. A purely real-valued vector cannot correct for phase distortion, making complex arithmetic a fundamental requirement for any DPD processor.

The length of the coefficient vector directly corresponds to the number of basis functions in the behavioral model. For a Memory Polynomial with nonlinear order K and memory depth M, the vector length is K × M. A Generalized Memory Polynomial (GMP) with cross-terms expands this significantly. This dimensionality drives the computational cost of the Least Squares (LS) estimation, which involves a matrix inversion of complexity O(N³), where N is the vector length.

Polynomial basis functions are highly correlated, leading to an ill-conditioned data matrix. This causes coefficient vectors estimated via standard Least Squares to be sensitive to noise and slow to converge in adaptive loops. Techniques like Basis Function Orthogonalization or Ridge Regression transform the problem to produce a more robust coefficient vector with smaller weight magnitudes, preventing overfitting and improving the stability of the predistorter.

The coefficient vector is not static; it must adapt to changing conditions such as temperature drift, channel frequency changes, and power supply variations. The update rate is a critical design parameter:

• Slow Tracking: Batch updates every hundreds of milliseconds for thermal effects.
• Fast Tracking: Recursive Least Squares (RLS) updates on a sample-by-sample basis for rapid envelope changes. The vector's temporal adaptation defines the DPD system's ability to maintain linearity in dynamic environments.

Not all coefficients contribute equally to linearization. Many have magnitudes near zero and can be pruned to reduce computational load. Techniques like Orthogonal Matching Pursuit (OMP) or Principal Component Analysis (PCA) identify the most significant coefficients, creating a sparse coefficient vector. This is critical for FPGA implementation, where a smaller vector directly translates to fewer multipliers and lower power consumption.

The foundational batch algorithm for computing the coefficient vector. It solves the overdetermined system of linear equations by minimizing the sum of squared errors between the observed power amplifier output and the model's prediction.

• Produces the optimal coefficient vector in a single computation for a given data block.
• The solution is given by the normal equation: w = (XᴴX)⁻¹Xᴴy, where X is the basis function matrix and y is the target vector.
• Highly sensitive to the numerical conditioning of the XᴴX matrix.

An adaptive filtering algorithm that updates the coefficient vector sample-by-sample, making it ideal for tracking time-varying power amplifier behavior due to thermal drift or bias changes.

• Minimizes a weighted linear least squares cost function, giving more importance to recent data.
• Offers significantly faster convergence than gradient-based methods like LMS.
• The computational complexity is O(N²), where N is the number of coefficients, driven by the recursive update of the inverse autocorrelation matrix.

A numerical preconditioning process that transforms the raw polynomial basis functions into an orthogonal set before coefficient extraction. This directly improves the stability of the coefficient vector.

• Reduces the condition number of the data matrix, preventing numerical instability in the LS solution.
• Common methods include Gram-Schmidt and QR decomposition.
• An orthogonal basis ensures that each coefficient can be estimated independently, accelerating convergence in adaptive systems.

A regularized form of least squares that adds an L2 penalty on the magnitude of the coefficient vector to the cost function. This prevents overfitting and improves the robustness of the extracted model.

• The cost function becomes: ||y - Xw||² + λ||w||², where λ is the regularization parameter.
• Shrinks the coefficients toward zero, which is particularly effective when the basis functions are highly correlated.
• The regularization parameter (λ) controls the bias-variance trade-off; a higher λ increases bias but reduces variance in the coefficient estimates.

A matrix factorization method used to solve the least squares problem for the coefficient vector with superior numerical stability compared to the direct normal equation approach.

• Decomposes the basis function matrix X into an orthogonal matrix Q and an upper triangular matrix R (X = QR).
• The coefficient vector is then solved via back-substitution: Rw = Qᴴy.
• Avoids the squaring of the condition number inherent in forming XᴴX, making it the preferred method for ill-conditioned polynomial models.

A greedy sparse approximation algorithm that builds a compact coefficient vector by iteratively selecting the most significant basis functions from a large dictionary.

• Starts with an empty model and adds the basis function most correlated with the current residual at each step.
• Produces a sparse coefficient vector where most entries are zero, drastically reducing the number of active terms.
• The iteration stops when a target sparsity level or error threshold is reached, balancing model complexity against linearization accuracy.