Recursive Least Squares (RLS)

RLS employs a forgetting factor (λ, typically 0.95 < λ < 1) that applies exponential weighting to historical data. This mechanism ensures the algorithm can track time-varying systems by progressively discarding stale observations.

• Recent samples dominate the cost function
• Older data decays geometrically in influence
• Enables tracking of thermal memory effects in power amplifiers
• A λ of 1.0 disables forgetting for stationary systems

The core computational engine of RLS is the recursive update of the inverse autocorrelation matrix (P). Using the matrix inversion lemma (Woodbury identity), RLS avoids explicit matrix inversion at each step.

• Updates P(n) from P(n-1) in O(N²) operations
• Maintains full statistical information of the input signal
• Directly computes the Kalman gain vector
• Susceptible to numerical instability without proper regularization

The Kalman gain vector (k) determines how prediction errors are distributed across filter coefficients. It represents the optimal weighting of new information based on the current uncertainty in the parameter estimates.

• Computed as k(n) = P(n-1)u(n) / (λ + uᵀ(n)P(n-1)u(n))
• Large gain → high uncertainty → aggressive updates
• Small gain → high confidence → conservative updates
• Directly analogous to the Kalman filter gain

RLS converges in approximately 2N iterations where N is the filter order, compared to 10N–20N iterations for LMS-based algorithms. This quadratic convergence property makes RLS ideal for applications requiring rapid adaptation.

• Independent of input signal eigenvalue spread
• No step-size tuning required
• Achieves the theoretical Wiener solution asymptotically
• Particularly effective for highly correlated input signals

Standard RLS requires O(N²) operations per iteration due to the matrix update, compared to O(N) for LMS. This quadratic complexity limits its use in resource-constrained embedded systems.

• Standard RLS: ~4N² + 4N multiplications per sample
• Fast RLS variants reduce to O(N) using lattice structures
• QR-RLS offers numerical stability at similar complexity
• FPGA implementations often use systolic array architectures

The recursive update of the inverse correlation matrix can suffer from loss of symmetry and loss of positive definiteness due to finite-precision arithmetic. This can cause catastrophic divergence in fixed-point implementations.

• Regularization parameter (δ) stabilizes initial conditions
• QR-RLS uses Givens rotations for guaranteed stability
• Square-root RLS propagates matrix square-roots
• Double-precision floating-point recommended for standard RLS

A foundational stochastic gradient descent algorithm that updates filter coefficients iteratively based on the instantaneous estimate of the mean squared error gradient.

• Computational Cost: O(N) per iteration, significantly lower than RLS
• Convergence: Slow, highly dependent on eigenvalue spread of input correlation matrix
• Trade-off: Sacrifices convergence speed for extreme computational simplicity
• Use Case: Ideal for resource-constrained embedded systems where RLS is too expensive

A numerically robust implementation of the Recursive Least Squares algorithm that uses Givens rotations to directly update the square-root of the inverse correlation matrix.

• Stability: Avoids the numerical instability of standard RLS by propagating matrix square-roots
• Mechanism: Operates on the Cholesky factor rather than the full correlation matrix
• Precision: Maintains positive-definiteness even with finite-precision arithmetic
• Application: Preferred for fixed-point FPGA implementations of DPD coefficient estimation

An optimal recursive state estimator for linear dynamic systems, interpretable as a generalization of RLS for time-varying parameter estimation.

• State-Space Model: Explicitly models parameter evolution as a dynamic process
• Process Noise: The covariance matrix Q captures expected parameter drift over time
• Relationship: RLS is a special case of the Kalman filter with identity state transition and zero process noise
• Advantage: Superior tracking of time-varying PA characteristics due to thermal memory effects

A scalar parameter λ (0 < λ ≤ 1) that exponentially weights recent data more heavily than past data, enabling RLS to track time-varying systems.

• Effective Memory: The asymptotic sample window is approximately 1/(1-λ)
• Typical Range: 0.95 to 0.999 for DPD applications
• Trade-off: Smaller λ provides faster tracking but increases steady-state misadjustment
• Critical Role: Determines the algorithm's ability to follow PA characteristic changes due to temperature, aging, and bias drift

The ratio of the largest to smallest singular value of the input correlation matrix, quantifying the sensitivity of the RLS solution to numerical perturbations.

• Ill-Conditioning: High condition numbers amplify round-off errors in matrix inversion
• Causes: Highly correlated input signals or insufficient spectral excitation
• Mitigation: Regularization via diagonal loading adds a small constant δ to the correlation matrix diagonal
• Impact: Directly affects the numerical stability of standard RLS implementations in fixed-point hardware

The fundamental linear equation Rw = p that defines the optimal weight vector for a Wiener filter, where R is the autocorrelation matrix and p is the cross-correlation vector.

• Theoretical Foundation: RLS recursively solves this equation without explicit matrix inversion
• Optimal Solution: Represents the theoretical minimum mean squared error achievable
• Steady-State: RLS converges to the Wiener solution as the number of samples approaches infinity
• Benchmark: The Wiener solution serves as the reference for evaluating RLS convergence and misadjustment performance