Orthogonal Matching Pursuit

At each iteration, OMP selects the dictionary atom (Volterra basis function) with the highest absolute correlation to the current residual signal. This greedy approach avoids the combinatorial explosion of testing all possible subsets. The selected atom is the one that explains the most remaining variance in the power amplifier output, ensuring rapid convergence to a compact model.

After selecting a new atom, OMP recomputes all coefficients by projecting the original signal onto the subspace spanned by all selected atoms. This orthogonalization step ensures that the residual is always orthogonal to the chosen basis, preventing the algorithm from re-selecting previously chosen terms and guaranteeing optimal coefficient values for the current support set.

OMP terminates based on a predefined sparsity level or error threshold:

• Fixed sparsity K: Stops after selecting exactly K kernel terms
• Error tolerance: Halts when the residual norm falls below a noise floor estimate This explicit control over model size prevents overfitting and directly aligns with hardware implementation constraints where only a limited number of Volterra coefficients can be stored.

For a dictionary of N candidate Volterra terms and a target sparsity of K, OMP scales as O(K·N·M) where M is the number of samples. This is dramatically faster than exhaustive search methods like Basis Pursuit. The algorithm requires only matrix-vector multiplications and least-squares solves on small submatrices, making it suitable for offline model extraction from large measurement datasets.

While LASSO regression uses L1-norm regularization to simultaneously shrink and select coefficients, OMP builds the support set sequentially:

• OMP: Explicit control over the exact number of terms; no shrinkage bias
• LASSO: Continuous shrinkage can bias retained coefficients downward
• OMP: Faster for very sparse solutions (K << N)
• LASSO: Better when many small but non-zero coefficients exist OMP is preferred when a hard limit on kernel count is required for FPGA implementation.

After each orthogonal projection, OMP updates the residual by subtracting the contribution of all selected atoms. The algorithm then computes the correlation between this new residual and every remaining dictionary atom. Efficient implementations use recursive least-squares or QR decomposition updates to avoid recomputing the full pseudoinverse at each iteration, significantly accelerating the selection process for large Volterra dictionaries.

The mathematical framework that enables the reconstruction of a sparse signal from far fewer measurements than required by the Nyquist-Shannon sampling theorem. It relies on two principles: sparsity (the signal has few non-zero coefficients in some domain) and incoherence (the sensing basis is uncorrelated with the representation basis).

• Enables efficient Volterra kernel identification with minimal test data
• Solves underdetermined linear systems by exploiting prior knowledge of sparsity
• Foundational to modern medical imaging, radar, and channel estimation

A class of iterative algorithms that make the locally optimal choice at each step to find a global solution. In the context of sparse modeling, greedy algorithms like OMP select one dictionary atom at a time to build a representation.

• Computationally efficient compared to exhaustive search
• Does not guarantee a globally optimal sparse solution
• Includes variants like Matching Pursuit (MP) , Compressive Sampling Matching Pursuit (CoSaMP) , and Stagewise OMP (StOMP)

A Volterra model where the vast majority of coefficients are zero, retaining only the most significant kernel terms. This is critical because a full Volterra series scales exponentially with nonlinear order and memory depth, making it computationally intractable for real-time DPD.

• Reduces a candidate set of thousands of terms to a few dozen
• Maintains modeling accuracy while enabling FPGA implementation
• Achieved through L1-regularization (LASSO) or greedy selection (OMP)

Least Absolute Shrinkage and Selection Operator. A linear regression technique that adds an L1-norm penalty (sum of absolute coefficient values) to the ordinary least squares cost function. This penalty forces some coefficients to be exactly zero, performing automatic variable selection.

• min ||y - Xβ||²₂ + λ||β||₁
• The hyperparameter λ controls the sparsity level
• Produces a convex optimization problem, guaranteeing a unique global minimum

The systematic process of decreasing the number of parameters in a mathematical model while preserving its essential input-output behavior. For Volterra DPD, this means identifying and discarding redundant or negligible kernel terms.

• Truncation: Limiting nonlinear order and memory depth
• Pruning: Removing specific cross-terms via OMP or LASSO
• Tensor Decomposition: Factorizing the kernel tensor into low-rank components
• Balances the bias-variance tradeoff to prevent overfitting

A measure of the sensitivity of a linear system's solution to perturbations in the input data. It is defined as the ratio of the largest to smallest singular value of the measurement matrix.

• A high condition number indicates an ill-conditioned, numerically unstable problem
• Correlated Volterra regressors (columns of the dictionary) increase the condition number
• OMP's orthogonalization step helps mitigate this by projecting the residual onto a subspace orthogonal to previously selected atoms