Forgetting Factor

The forgetting factor (λ) applies an exponential decay to historical data. Each past sample is weighted by λᵏ, where k is the sample age. A value of λ=0.99 means a sample 100 iterations ago contributes only 37% of its original weight. This creates a sliding window of relevance without the computational burden of storing a finite buffer, making it ideal for real-time embedded DSP systems.

The forgetting factor directly defines the asymptotic memory length of the algorithm. The effective window length N is approximated by N ≈ 1/(1-λ).

• λ=0.99 → N ≈ 100 samples
• λ=0.999 → N ≈ 1000 samples
• λ=0.9 → N ≈ 10 samples This relationship allows engineers to tune the algorithm's responsiveness to physical changes in the power amplifier's thermal state or aging characteristics.

Selecting λ embodies the classic bias-variance tradeoff in adaptive filtering:

• High λ (close to 1): Lower steady-state misadjustment and better noise rejection, but sluggish response to PA characteristic changes.
• Low λ (e.g., 0.95): Rapid tracking of dynamic nonlinearities, but increased parameter variance and sensitivity to measurement noise. The optimal value depends on the coherence time of the physical distortion process being linearized.

In Recursive Least Squares (RLS) for digital predistortion, λ is embedded directly in the inverse correlation matrix update: P(k) = λ⁻¹[P(k-1) - K(k)uᵀ(k)P(k-1)]. This prevents P(k) from collapsing to zero as data accumulates, ensuring the algorithm maintains persistent excitation and can continuously adapt predistorter coefficients. Without λ, RLS would eventually become insensitive to new measurements.

Advanced DPD implementations use time-varying λ(k) to optimize performance:

• Burst detection: Temporarily lower λ when a sudden change in signal statistics or PA operating point is detected.
• Covariance-trace monitoring: Automatically adjust λ based on the trace of the inverse correlation matrix to maintain numerical stability.
• Error-driven adaptation: Reduce λ when the linearization error exceeds a threshold, enabling rapid re-convergence after abrupt changes.

In fixed-point FPGA implementations, λ must be carefully quantized. Values too close to 1.0 can cause accumulator overflow in the recursive updates. Common practice uses λ in the range [0.95, 0.999] with Q15 or Q31 fixed-point representation. QR-RLS implementations using Givens rotations offer superior numerical stability by directly operating on the square-root of the correlation matrix, mitigating round-off error amplification.

The canonical algorithm where the forgetting factor is most prominently applied. RLS recursively updates the inverse of the input correlation matrix to achieve an exact solution at each iteration. The forgetting factor (λ) exponentially weights past data, enabling the algorithm to track time-varying systems by discarding obsolete information. Without it, RLS would suffer from estimator wind-up in non-stationary environments.

A foundational stochastic gradient algorithm that serves as a low-complexity alternative to RLS. Unlike RLS, LMS does not use an explicit forgetting factor; its tracking ability is governed solely by the step size (μ). A larger step size provides faster tracking but higher misadjustment, creating a direct trade-off analogous to tuning λ in RLS for non-stationary environments.

An optimal recursive state estimator that generalizes RLS. When the state transition matrix is set to the identity divided by the square root of the forgetting factor, the Kalman filter reduces to a forgetting factor RLS algorithm. This connection reveals that λ corresponds to an assumption about process noise covariance, providing a stochastic state-space interpretation of exponential weighting.

A scalar added to the diagonal of the correlation matrix to prevent ill-conditioning. While the forgetting factor controls temporal weighting, the regularization parameter controls initial bias. In the initialization of RLS, the inverse correlation matrix is set to δ⁻¹I, where δ is a small positive constant. This ensures the matrix is invertible before sufficient data has been collected.

The fundamental trade-off in adaptive filtering. A forgetting factor close to 1.0 provides low misadjustment (steady-state error) but slow convergence when the system changes. A smaller λ provides rapid tracking of non-stationary dynamics but increases noise sensitivity. This mirrors the bias-variance tradeoff in machine learning and is the central design tension in coefficient estimation.