QR-RLS

QR-RLS operates on the Cholesky factor (square-root) of the inverse correlation matrix rather than the full matrix itself. This fundamental design choice doubles the effective numerical precision and guarantees that the correlation matrix remains positive-definite throughout adaptation.

• Propagates the square-root factor S(n) where P(n) = S(n)S^H(n)
• Eliminates matrix inversion operations that amplify round-off errors
• Maintains stability even with ill-conditioned input signals common in wideband predistortion
• Condition number of S(n) is the square root of P(n), reducing sensitivity to perturbations

The core computational kernel of QR-RLS uses Givens rotations to annihilate individual elements of the augmented data matrix. Each rotation is an orthogonal transformation that selectively zeros out one element while preserving the norm of the affected rows.

• A Givens rotation matrix G(i,j,θ) applies a plane rotation between rows i and j
• Parameters c = cos(θ) and s = sin(θ) are computed to zero a specific element
• Orthogonality ensures G^T G = I, preserving the least-squares solution exactly
• Enables systolic array implementations where processing elements operate in parallel

QR-RLS maps naturally onto systolic array architectures, where a grid of processing elements performs Givens rotations in a pipelined, rhythmic fashion. Each cell communicates only with its immediate neighbors, eliminating global routing bottlenecks.

• Boundary cells compute rotation parameters (c, s) from diagonal elements
• Internal cells apply rotations to off-diagonal elements using received parameters
• Achieves O(n²) throughput with O(n) processing elements for n coefficients
• Ideal for FPGA and ASIC implementation in real-time DPD systems operating at hundreds of MHz

After the QR decomposition updates the upper triangular matrix R(n), the optimal coefficient vector is extracted through back-substitution rather than matrix inversion. This exploits the triangular structure for efficient computation.

• Solves R(n)w(n) = p(n) where R is upper triangular
• Starts from the last coefficient and works backward: w_k = (p_k - Σ R_{k,j}w_j) / R_{k,k}
• Requires only O(n²/2) operations versus O(n³) for full matrix inversion
• Numerically stable because division is by diagonal elements of well-conditioned R

QR-RLS incorporates an exponential forgetting factor λ (typically 0.95–0.999) to track time-varying power amplifier characteristics. The forgetting factor is applied directly in the square-root domain by scaling the existing triangular factor before incorporating new data.

• Applies √λ scaling to the existing R matrix before each update
• Smaller λ provides faster tracking of thermal memory effects and bias point drift
• Larger λ improves steady-state accuracy for static or slowly varying channels
• Enables tracking of Doherty amplifier load modulation during envelope variation

QR-RLS provides mathematically identical results to conventional RLS in infinite precision, but dramatically outperforms it in fixed-point embedded implementations where word lengths are limited to 16–32 bits.

• Conventional RLS: O(n²) complexity but prone to covariance matrix blow-up
• QR-RLS: O(n²) complexity with 2× effective precision due to square-root domain
• Conventional RLS may diverge after 10³–10⁴ iterations in 16-bit arithmetic
• QR-RLS maintains convergence indefinitely, critical for continuous online training in deployed DPD systems

The parent algorithm of QR-RLS that recursively updates the inverse correlation matrix to achieve an order of magnitude faster convergence than gradient-based methods. RLS minimizes a weighted least squares cost function using a forgetting factor λ to track time-varying systems. The standard RLS update involves the matrix inversion lemma (Woodbury identity) to avoid explicit matrix inversion, but suffers from numerical instability in finite-precision arithmetic due to loss of symmetry and positive-definiteness in the inverse correlation matrix.

A matrix factorization that decomposes the data matrix A into an orthogonal matrix Q and an upper triangular matrix R (A = QR). The orthogonality of Q preserves the Euclidean norm, making QRD the gold standard for solving ill-conditioned least squares problems. In QR-RLS, the decomposition is maintained and updated directly, avoiding the squaring operation that doubles the condition number in conventional RLS. This provides twice the numerical precision in fixed-point implementations.

A numerically stable orthogonal transformation that selectively zeroes out individual elements of a matrix through a plane rotation. Each rotation is defined by two parameters (cos θ, sin θ) computed to annihilate a specific element below the diagonal. In QR-RLS, Givens rotations are the core update mechanism applied to the square-root of the inverse correlation matrix when new data arrives. Unlike Householder reflections, Givens rotations allow selective element elimination without recomputing the entire decomposition, making them ideal for recursive updates.

The simplest adaptive filtering algorithm that updates coefficients using the instantaneous gradient estimate of the mean squared error. LMS requires only O(N) operations per iteration but converges significantly slower than RLS, especially for correlated input signals common in wideband predistortion. The step size μ controls the convergence-misadjustment tradeoff. LMS serves as the baseline against which QR-RLS convergence speed is measured, often requiring 10–100× more iterations to reach equivalent steady-state error.

A scalar parameter (0 < λ ≤ 1) that applies exponential weighting to past data in the recursive cost function. Data k samples old is weighted by λᵏ, creating an effective memory of approximately 1/(1-λ) samples. In DPD applications, λ balances tracking capability (lower values for rapidly changing amplifier characteristics due to thermal effects) against estimation accuracy (higher values for stationary conditions). QR-RLS implements the forgetting factor through exponential weighting of the Cholesky factor before applying Givens rotations.

The ratio of the largest to smallest singular value (σₘₐₓ/σₘᵢₙ) of the input correlation matrix, quantifying sensitivity to numerical errors. High condition numbers indicate ill-conditioning where small input perturbations cause large solution variations. Standard RLS squares the condition number by working with the inverse correlation matrix directly. QR-RLS operates on the square-root factor, halving the effective condition number and enabling stable operation on fixed-point hardware with limited precision, critical for FPGA-based DPD implementations.