Orthogonal Matching Pursuit (OMP)

At each iteration, OMP selects the dictionary atom (basis function) that has the highest absolute correlation with the current residual error. This greedy approach ensures that the most impactful distortion-causing terms are identified first, building the model in order of significance. The residual is then updated by projecting the signal onto the orthogonal complement of the selected atoms, guaranteeing that no atom is selected twice.

OMP enforces sparsity by limiting the number of active coefficients to a predefined value K. This directly addresses the curse of dimensionality in Volterra and GMP models, where the full basis function set can contain hundreds of terms. By retaining only K critical terms, OMP produces a predistorter with dramatically reduced computational complexity, making it ideal for FPGA and ASIC implementations where multiply-accumulate operations are at a premium.

After selecting a new atom, OMP recomputes all coefficients by solving a least squares problem on the subspace spanned by the selected atoms. This orthogonal projection step ensures that the residual is always orthogonal to the chosen basis functions, leading to optimal coefficient values for the current support set. This avoids the suboptimal coefficient estimates that plague simpler matching pursuit variants.

OMP terminates based on one of three criteria:

• Sparsity target: A fixed number of coefficients K is reached
• Error threshold: The residual power falls below a specified noise floor
• Correlation threshold: No remaining atom exceeds a minimum correlation value

This flexibility allows engineers to balance linearization accuracy against computational budget for a given hardware platform.

Efficient OMP implementations use QR decomposition to maintain an orthonormal basis for the selected subspace. As each new atom is added, its component orthogonal to the existing basis is computed via Gram-Schmidt orthogonalization, and the QR factors are updated incrementally. This avoids solving ill-conditioned normal equations and provides superior numerical stability when the dictionary contains highly correlated basis functions.

The performance of OMP depends critically on the dictionary construction. For DPD applications, dictionaries typically include:

• Memory polynomial terms: x(n-m)|x(n-m)|^p
• GMP cross-terms: x(n-m)|x(n-m-l)|^p
• Envelope memory terms: x(n)|x(n-m)|^p

A well-designed dictionary must be overcomplete—containing more candidate functions than strictly necessary—to give OMP the freedom to select the most parsimonious representation of the PA's nonlinear dynamics.

A numerical conditioning process that transforms correlated polynomial basis functions into an orthogonal set. This is a critical preprocessing step for OMP because the algorithm's greedy selection relies on the magnitude of correlation between the residual and dictionary atoms. Without orthogonalization, highly correlated basis functions can confuse the selection step, leading to ill-conditioned intermediate matrices and suboptimal atom choices. Techniques like Gram-Schmidt or QR decomposition are applied to the candidate dictionary before OMP executes.

A complexity reduction technique that removes insignificant kernels from a full Volterra series model based on a significance metric. OMP serves as the primary algorithmic engine for this pruning process. Instead of testing all possible kernel combinations, OMP iteratively selects the most salient Volterra kernels from a massive candidate pool. This directly addresses the 'curse of dimensionality' inherent in Volterra models, yielding a sparse set of kernels that capture the dominant nonlinear memory effects without the prohibitive computational cost of the full series.

A batch coefficient extraction algorithm that minimizes the sum of squared errors between the power amplifier output and the model's prediction. OMP is intrinsically linked to LS estimation: after each greedy selection of a new basis function, OMP solves a least squares problem on the currently selected support set to update all coefficients. This orthogonal projection step ensures the residual is always orthogonal to the chosen atoms, preventing the re-selection of the same atom and guaranteeing convergence in at most M iterations for an M-sparse signal.

The systematic process of decreasing the number of coefficients in a behavioral model to minimize computational load while preserving linearization performance. OMP is a direct model order reduction algorithm because it explicitly trades off model fidelity against sparsity (the number of coefficients). The stopping criterion—often based on the residual error norm falling below a threshold or reaching a target sparsity level (K) —directly controls the final model order. This allows an FPGA designer to specify a hard upper bound on the number of multipliers required.

The systematic selection or pruning of cross-terms in a behavioral model to balance linearization accuracy against computational complexity. In a Generalized Memory Polynomial (GMP), the number of lagging and leading envelope cross-terms explodes combinatorially. OMP is ideally suited for cross-term management because it treats every candidate cross-term as an independent dictionary atom. The algorithm naturally selects only the cross-terms with the highest correlation to the residual distortion, automatically building a sparse GMP structure without manual engineering of which cross-terms to include.

A regularized least squares estimation technique that adds an L2 penalty on coefficient magnitude to the cost function. While standard OMP uses unregularized LS for the coefficient update step, a variant known as Regularized OMP incorporates ridge regression to prevent overfitting when the selected support set becomes large or the data matrix is ill-conditioned. The regularization parameter (λ) shrinks the coefficients toward zero, improving the robustness of the extracted sparse DPD model, especially when modeling noisy power amplifier measurements.