Recursive Least Squares (RLS)

RLS incorporates a forgetting factor (λ, typically 0.95 < λ < 1) that applies exponential decay to historical data. This mechanism ensures the algorithm prioritizes recent observations over stale ones, enabling it to track time-varying amplifier behavior caused by thermal drift, bias shifts, and aging. A smaller λ provides faster tracking but increases steady-state misadjustment, creating a fundamental trade-off between tracking agility and estimation variance.

Unlike LMS which uses a scalar step size, RLS recursively computes the inverse of the input autocorrelation matrix (P(n)) using the Woodbury matrix identity. This second-order approach whitens the input signal in the parameter space, ensuring uniform convergence rates across all eigenmodes. The result is convergence within 10-20 samples for stationary signals, compared to hundreds or thousands for LMS, making RLS ideal for rapid DPD coefficient extraction during training bursts.

The primary drawback of RLS is its quadratic computational complexity relative to the number of filter taps (N). Each iteration requires:

• Gain vector computation: O(N²) operations
• Inverse matrix update: O(N²) via the Riccati equation
• Coefficient update: O(N) operations For a 50-tap memory polynomial DPD model, this translates to approximately 2,500 multiply-accumulate operations per sample, making real-time FPGA implementation challenging without algorithmic optimizations or resource sharing.

RLS requires careful initialization of the inverse correlation matrix P(0). The standard approach sets:

• P(0) = δ⁻¹I, where δ is a small positive constant (typically 0.01 to 0.001)
• w(0) = 0 for the coefficient vector This initialization acts as implicit regularization, preventing numerical instability during the first few iterations when the autocorrelation estimate is rank-deficient. For ill-conditioned DPD basis functions, explicit L2 regularization can be incorporated by adding a diagonal loading term to the correlation matrix.

RLS exhibits a deterministic convergence trajectory that is independent of the input signal's eigenvalue spread—a critical advantage over LMS when modeling power amplifiers with correlated nonlinear basis functions. The steady-state misadjustment (excess MSE) is approximately:

• M = (1-λ)/(1+λ) × N This reveals that misadjustment grows linearly with the number of taps (N) and inversely with λ. For a 30-coefficient GMP model with λ=0.99, the theoretical misadjustment is approximately 0.15%, demonstrating RLS's superior steady-state accuracy compared to stochastic gradient methods.

Standard RLS can suffer from covariance matrix blow-up due to the accumulation of round-off errors, causing P(n) to lose positive definiteness. Mitigation strategies include:

• Square-root RLS: Propagates the Cholesky factor of P(n) to guarantee numerical stability
• QR-decomposition RLS: Uses Givens rotations to maintain orthogonality
• Periodic reinitialization: Resets P(n) when condition number exceeds a threshold These variants are essential for fixed-point FPGA implementations where limited precision arithmetic exacerbates numerical issues.