QR-RLS

Instead of updating the inverse correlation matrix, QR-RLS maintains the Cholesky factor (upper triangular matrix R) of the data matrix. It applies Givens rotations to annihilate new data elements, directly solving the least squares problem without explicitly computing the inverse. This avoids squaring the condition number, a primary source of numerical instability in standard RLS.

QR-RLS uses Givens rotations—orthogonal plane rotations—to zero out elements in the incoming data row. Key properties:

• Orthogonality preservation: Rotations are numerically stable and preserve the norm of the matrix.
• Systolic array compatibility: The regular, localized data flow maps efficiently to parallel hardware architectures.
• Each rotation eliminates one element below the diagonal of R, maintaining the upper triangular structure.

The Inverse QR-RLS algorithm propagates the inverse Cholesky factor R⁻¹ directly, extracting the weight vector without back-substitution. This variant:

• Computes the Kalman gain vector directly from stored rotations.
• Eliminates the final triangular solve step, reducing latency.
• Is particularly suited for sample-by-sample update architectures where immediate coefficient availability is critical.

Standard RLS fails catastrophically when the input correlation matrix has a high condition number—common with narrowband or highly correlated training signals. QR-RLS operates on the square root of the data matrix, effectively halving the condition number sensitivity. This makes it the preferred choice for:

• Multi-carrier DPD with closely spaced tones.
• Wideband signals where sample correlation is high.
• Long training sequences where numerical drift accumulates.

QR-RLS incorporates an exponential forgetting factor λ (typically 0.97–0.999) to track time-varying PA characteristics. Implementation approach:

• Each existing row of R is multiplied by √λ before processing new data.
• This exponentially down-weights older observations, enabling thermal memory tracking.
• The square-root formulation ensures the forgetting operation remains numerically stable, unlike direct covariance matrix aging in standard RLS.

The QR-RLS algorithm maps naturally to systolic array architectures on FPGAs. The triangular array of processing elements:

• Boundary cells compute rotation parameters.
• Internal cells apply rotations to propagate updates.
• Data flows in a pipelined, rhythmic pattern with no global communication. This enables high-throughput, fixed-point implementations suitable for real-time closed-loop DPD in 5G massive MIMO systems.

The foundational adaptive filtering algorithm that QR-RLS numerically stabilizes. RLS recursively finds coefficients that minimize a weighted linear least squares cost function, offering an order of magnitude faster convergence than LMS algorithms.

• Utilizes the matrix inversion lemma to avoid explicit matrix inversion
• Susceptible to numerical instability and divergence due to covariance matrix ill-conditioning
• Forms the theoretical basis for the QR-decomposition-based implementation

A stochastic gradient descent algorithm that updates filter coefficients based on the instantaneous estimate of the mean squared error gradient. While computationally simpler than RLS, it suffers from slow convergence when the input signal is highly correlated.

• Computational complexity: O(N) per iteration vs. O(N²) for RLS
• Convergence rate depends on eigenvalue spread of the input autocorrelation matrix
• Often used as a baseline comparison for QR-RLS performance in DPD applications

A measure of a matrix's sensitivity to numerical errors, defined as the ratio of the largest to smallest singular value. A high condition number indicates ill-conditioning, which can cause standard RLS to diverge.

• QR-RLS directly addresses this by operating on the square root of the data matrix
• Doubling the condition number roughly halves the numerical precision
• Critical in DPD where input signals often exhibit high correlation and wide dynamic range

The gradual deviation of predistorter coefficients from their optimal values due to temperature changes, component aging, or numerical instability. QR-RLS's robust numerical properties make it inherently resistant to drift caused by finite-precision arithmetic.

• Long-term thermal memory effects can slowly shift the PA's optimal linearization point
• Accumulated round-off error in standard RLS can masquerade as physical drift
• QR decomposition maintains orthogonality in the data matrix, preventing error accumulation

A ridge regression technique that adds a penalty term to the least squares cost function to stabilize solutions for ill-posed inverse problems. Often combined with QR-RLS when the data matrix is rank-deficient.

• Adds a diagonal loading term λI to the normal equations
• Prevents overfitting to measurement noise in the observation path
• The regularization parameter λ trades bias for variance in the coefficient estimate

The standard metric for evaluating DPD linearization performance, representing the mean squared error between the ideal and linearized output normalized by the input signal power.

• Expressed in dB: NMSE = 10 log₁₀(E[|y_ideal - y_linearized|²] / E[|y_ideal|²])
• QR-RLS typically achieves NMSE values below -40 dB for well-conditioned signals
• Used alongside ACPR and EVM for complete linearization quality assessment