QR Decomposition

The matrix Q is orthogonal, meaning its columns are orthonormal vectors. This property ensures that Q^TQ = I, preserving the Euclidean norm of vectors during transformation.

• Key benefit: Multiplying by Q^T does not amplify noise or rounding errors
• Condition number: Orthogonal matrices have a perfect condition number of 1
• Stability: Avoids the numerical instability inherent in directly inverting the normal equations

The matrix R is upper triangular, with all entries below the main diagonal equal to zero. This structure enables efficient solution of linear systems through back substitution.

• Back substitution: Solve Rx = b by computing the last variable first, then substituting upward
• Computational cost: O(n²) for back substitution versus O(n³) for general matrix inversion
• Hardware efficiency: Triangular structure maps well to systolic array implementations on FPGAs

QR decomposition solves the least-squares problem min ||Ax - b||² by transforming it into min ||Rx - Q^Tb||², avoiding the explicit formation of the normal equations A^TA.

• Normal equations: Forming A^TA squares the condition number, amplifying ill-conditioning
• QR approach: Works directly on A, preserving numerical accuracy
• DPD relevance: Critical when the correlation matrix of basis functions becomes ill-conditioned due to highly correlated polynomial terms

The condition number of R equals the condition number of the original matrix A, unlike the normal equations approach where the condition number is squared. This makes QR decomposition the preferred method for ill-conditioned problems.

• Condition number κ(A): Measures sensitivity of the solution to small perturbations in the data
• QR preserves κ: κ(R) = κ(A), maintaining the problem's intrinsic difficulty
• Practical impact: Enables stable coefficient estimation even when basis functions are nearly linearly dependent

Multiple algorithms exist for computing the QR decomposition, each with distinct numerical properties and computational requirements.

• Classical Gram-Schmidt: Intuitive but numerically unstable; orthogonality degrades with ill-conditioned matrices
• Modified Gram-Schmidt: Reorders operations to improve stability, widely used in adaptive filtering
• Householder reflections: Most numerically stable method, using orthogonal reflections to zero out sub-diagonal elements
• Givens rotations: Efficient for sparse matrices or when only selected elements need zeroing, ideal for streaming implementations

For online DPD training, the QR decomposition can be updated recursively as new data arrives, avoiding full recomputation. This enables real-time coefficient adaptation.

• QR-RLS algorithm: Maintains the Cholesky factor R of the inverse correlation matrix, updating it with each new sample
• Sliding window: Old data can be removed using hyperbolic rotations, implementing a finite memory
• Forgetting factor integration: Exponential weighting is applied directly to R through scaling operations
• FPGA implementation: Systolic array architectures exploit the triangular structure for pipelined execution

A numerical state where the correlation matrix has a high condition number, making the solution hypersensitive to small perturbations in the input data. In DPD coefficient estimation, ill-conditioning arises when basis functions are highly correlated, causing standard normal equations to amplify noise and produce wildly inaccurate predistorter coefficients. QR decomposition directly addresses this by avoiding the explicit formation of the normal equations, operating on the original data matrix instead.

The robustness of an algorithm to rounding errors and finite-precision effects. QR decomposition using Householder reflections or Givens rotations offers superior numerical stability compared to direct matrix inversion or Cholesky decomposition. This is critical for FPGA-based DPD implementations where fixed-point arithmetic with limited word lengths can cause unstable algorithms to diverge. QR-based solvers maintain accuracy even with ill-conditioned matrices.

An adaptive filtering algorithm that recursively minimizes a weighted linear least squares cost function. The core computational step in RLS involves updating the inverse correlation matrix. Implementing RLS via QR decomposition—known as QR-RLS—replaces the explicit matrix inversion with a numerically stable orthogonal triangularization process, enabling robust real-time coefficient tracking for closed-loop DPD systems.

A matrix formed by the autocorrelation of basis function outputs in the DPD model. When this matrix is inverted during block-based coefficient estimation, any ill-conditioning directly degrades the solution. QR decomposition factorizes the original data matrix A into Q and R, allowing the least-squares solution to be computed via back-substitution on the triangular system Rx = Qᵀb, bypassing the numerically fragile step of forming and inverting AᵀA.

A weighting parameter λ (0 < λ ≤ 1) that exponentially discounts older data in recursive algorithms. In QR-RLS, the forgetting factor is incorporated by applying exponential weighting to the triangular factor R during the update step. This enables the DPD system to track time-varying PA characteristics caused by thermal memory effects, bias shifts, and aging, ensuring the predistorter adapts to non-stationary operating conditions.

A scalar δ added to the diagonal of the correlation matrix to improve numerical conditioning and prevent overfitting. In QR-based estimation, regularization is applied by augmenting the data matrix with a scaled identity matrix before decomposition. This technique is essential during initial DPD training when the correlation matrix may be rank-deficient due to insufficient signal excitation, ensuring a stable initial coefficient vector.