Recursive Least Squares (RLS)

The forgetting factor (λ, typically 0.95 < λ < 0.999) is a weighting parameter that exponentially discounts the influence of older data on the current coefficient estimate. This mechanism is essential for tracking non-stationary environments, such as a power amplifier whose characteristics drift due to thermal memory effects or changing signal statistics.

• A smaller λ provides faster tracking but increases steady-state misadjustment.
• A λ closer to 1 yields better steady-state performance but slower adaptation to change.
• The effective memory of the algorithm is approximately 1/(1-λ) samples.

RLS recursively updates the inverse of the correlation matrix using the Matrix Inversion Lemma (Woodbury matrix identity), avoiding the O(N³) computational cost of direct inversion at each iteration. This recursive formulation reduces the per-iteration complexity to O(N²), where N is the number of basis function coefficients.

• The gain vector is computed by applying the existing inverse matrix to the new data vector.
• This enables real-time coefficient updates in closed-loop DPD systems without block-based processing delays.

Unlike LMS, whose convergence is governed by the eigenvalue spread of the input correlation matrix, RLS converges at a rate independent of the input signal's statistical conditioning. This is because RLS effectively whitens the input data by inverting the correlation matrix.

• RLS typically converges in approximately 2N iterations, where N is the filter order.
• This is orders of magnitude faster than LMS for signals with high eigenvalue disparity.
• Critical for DPD applications where the predistorter must lock rapidly after a change in carrier frequency or average power.

A regularization parameter (δ) is added to the diagonal of the initial correlation matrix estimate to prevent ill-conditioning during the early stages of adaptation or during periods of low input excitation. This is mathematically equivalent to initializing the inverse correlation matrix as P(0) = δ⁻¹I.

• Prevents large coefficient transients at startup.
• Mitigates the effects of finite-precision arithmetic in fixed-point FPGA implementations.
• A typical value for δ is a small positive constant relative to the signal power.

RLS minimizes a weighted sum of squared errors over all past observations, rather than the instantaneous squared error used by LMS. The cost function is: J(n) = Σᵢ₌₁ⁿ λⁿ⁻ⁱ |e(i)|² where e(i) is the error signal at time i. This deterministic, batch-like objective gives RLS its superior steady-state performance and immunity to gradient noise.

• Produces the exact least-squares solution for stationary signals when λ=1.
• The recursive formulation computes this sum without storing the entire history.

The primary drawback of RLS is its O(N²) computational complexity per iteration, compared to O(N) for LMS. For a DPD system with hundreds of basis function coefficients, this can strain real-time processing budgets.

• Fast RLS variants (e.g., lattice-based, QR-decomposition RLS) reduce complexity to O(N) while preserving convergence properties.
• QR-RLS is particularly valued in DPD for its superior numerical stability in fixed-point hardware.
• The choice between RLS and LMS variants is a direct trade-off between convergence speed and implementation cost.