Recursive Least Squares (RLS)

The forgetting factor (λ) is the critical tuning parameter that weights recent data more heavily than past observations. A value close to 1 (e.g., 0.999) provides a long memory suitable for quasi-static channels, while a smaller value (e.g., 0.95) enables rapid tracking of fast-changing power amplifier thermal dynamics. The effective memory window is approximately 1/(1-λ) samples, allowing engineers to explicitly trade steady-state misadjustment against tracking agility.

RLS avoids explicit matrix inversion at each step by applying the Woodbury matrix identity to recursively update the inverse of the input autocorrelation matrix. This reduces the computational complexity from O(N³) for direct inversion to O(N²) per iteration, where N is the number of predistorter coefficients. For a memory polynomial with 50 basis functions, this transforms an impractical real-time operation into a feasible one for FPGA-based DPD implementations.

Unlike Least Mean Squares (LMS) algorithms that converge stochastically, RLS minimizes the exact cumulative squared error up to the current time instant. This deterministic cost function ensures:

• Convergence in approximately 2N iterations, independent of the input signal's eigenvalue spread
• Insensitivity to the conditioning of the basis function matrix
• Superior performance when linearizing wideband signals where the input correlation matrix is ill-conditioned

The inverse correlation matrix is typically initialized as P(0) = δ⁻¹I, where δ is a small positive constant. This diagonal loading acts as implicit regularization, preventing numerical instability during the first few iterations when the data matrix is rank-deficient. For online DPD training, this initialization ensures the algorithm remains stable even before sufficient signal samples have been collected to fully excite all Volterra kernel dimensions.

While RLS converges an order of magnitude faster than LMS, it incurs significantly higher computational cost:

• LMS: O(N) multiplications per iteration
• RLS: O(N²) multiplications per iteration For a Generalized Memory Polynomial (GMP) with 200 coefficients, RLS requires approximately 40,000 operations per sample. This often necessitates QR-decomposition (QRD-RLS) systolic array implementations in hardware to parallelize the computation and maintain throughput for 5G NR wideband signals.

In the Indirect Learning Architecture (ILA) for DPD, RLS is employed to identify the postdistorter coefficients by minimizing the error between the attenuated PA output and the predistorter input. The algorithm's fast convergence is particularly advantageous during burst-mode transmission or when the PA characteristics shift rapidly due to thermal memory effects. Once converged, the postdistorter coefficients are copied directly to the predistorter block.

A numerical conditioning process that transforms correlated polynomial basis functions into an orthogonal set before coefficient estimation. This dramatically improves the convergence speed and stability of adaptive algorithms. Benefits for RLS:

• Reduces eigenvalue spread of the input correlation matrix
• Accelerates initial convergence from hundreds to tens of samples
• Prevents numerical instability caused by correlated regressors
• Enables the use of simpler, lower-precision arithmetic in hardware implementations

A greedy sparse approximation algorithm that iteratively selects the most correlated basis function from a large dictionary to build a compact predistorter model. While RLS estimates all coefficients simultaneously, OMP takes a fundamentally different approach. Key distinctions:

• Builds the model one term at a time based on correlation
• Produces sparse models with far fewer coefficients than full RLS solutions
• Reduces computational complexity for the running predistorter
• Often used offline to determine which terms to include before online RLS adaptation

A one-dimensional array containing the complex-valued weights for each basis function in the predistorter model. This vector fully defines the linearization transfer characteristic and is the direct output of the RLS algorithm. Implementation considerations:

• Stored as I/Q pairs in hardware memory
• Updated sample-by-sample in online RLS implementations
• Length directly determines predistorter computational complexity
• Must be protected from corruption during transmission to prevent spectral regrowth spikes