QR Decomposition (QRD)

QRD inherently orthogonalizes the columns of the basis function matrix A. The Q matrix forms an orthonormal basis for the column space of A, meaning Q^H Q = I. This transformation decouples the correlated polynomial and memory terms, preventing the amplification of rounding errors during coefficient solving. By projecting the observed PA output onto this orthonormal basis, the extraction process becomes immune to the numerical rank deficiency that plagues the normal equations approach.

The condition number for solving a least squares problem via QRD is proportional to the condition number of the original matrix A, whereas the normal equations method squares the condition number. For ill-conditioned data matrices common in wideband DPD with deep memory, this squaring effect can destroy all significant digits. QRD avoids explicitly forming the Gram matrix A^H A, preserving accuracy and ensuring robust coefficient extraction even when basis functions are highly correlated.

Once A = QR is computed, the optimal coefficient vector c is found via back-substitution:

• Compute the projection: y' = Q^H y
• Solve the triangular system: R c = y' Because R is upper triangular, solving for c is computationally cheap and numerically stable. This two-step process avoids matrix inversion entirely, making it ideal for fixed-point FPGA implementations where division and matrix inversion are resource-intensive.

The Modified Gram-Schmidt (MGS) algorithm computes the QR decomposition column-by-column. For each column of A, it subtracts its projection onto all previously computed q vectors, then normalizes the residual to produce the next orthonormal column of Q. MGS is preferred over Classical Gram-Schmidt in finite-precision arithmetic because it computes projections against the most recently updated vectors, significantly reducing the accumulation of orthogonalization errors.

Givens rotations selectively zero out individual elements below the diagonal of A using plane rotations. Each rotation is defined by a 2x2 unitary matrix that annihilates a single element. This method is highly amenable to systolic array hardware architectures because:

• It operates on pairs of rows independently
• It requires only multiply-accumulate and square root operations
• It can be deeply pipelined for high-throughput, real-time DPD coefficient updates

When the basis function matrix A is rank-deficient—common when model order is over-specified—standard QRD can be extended with column pivoting. Column pivoting reorders the columns of A by descending norm, ensuring the diagonal elements of R are non-increasing. This reveals the numerical rank and allows the algorithm to gracefully handle singular matrices by identifying which basis functions contribute negligible energy, enabling automatic model order reduction.

A preprocessing step that transforms correlated polynomial basis functions into an orthogonal set. QR decomposition is the computational engine that performs this orthogonalization.

• Gram-Schmidt: Classical method, but numerically unstable
• Householder Reflections: Preferred method within QR for superior stability
• Modified Gram-Schmidt: A more stable variant, mathematically equivalent to Householder QR
• Benefit: Orthogonal bases eliminate multicollinearity, accelerating coefficient convergence

A scalar metric quantifying how sensitive a matrix is to numerical errors during inversion. QR decomposition is preferred precisely because it mitigates the effects of high condition numbers.

• Definition: κ(A) = σ_max / σ_min, the ratio of largest to smallest singular value
• Ill-Conditioned Threshold: κ > 10^6 indicates severe numerical instability
• QR Mitigation: Operates directly on A rather than A^H A, effectively working with the square root of the condition number
• DPD Relevance: Memory polynomial basis matrices are notoriously ill-conditioned

A dimensionality reduction technique that identifies the most significant directions in the basis function space. QR decomposition can serve as a building block for PCA-based model pruning.

• Singular Value Decomposition (SVD): The full decomposition revealing principal components
• QR as Precursor: QR factorization is often the first step in iterative SVD algorithms
• Variance Thresholding: Retain only components explaining 99.9% of variance
• Outcome: A compact, well-conditioned model with minimal linearization performance loss