Forgetting Factor

The forgetting factor, typically denoted as λ (lambda), applies an exponential decay to historical data in the cost function. Each past error sample is weighted by λ^k, where k is the sample age. This creates a sliding window with an effective memory length of approximately 1/(1-λ) samples. For example, a λ of 0.99 gives an effective memory of ~100 samples, while λ of 0.999 extends this to ~1000 samples. The exponential profile ensures smooth, gradual de-emphasis rather than abrupt truncation.

Selecting the forgetting factor involves a fundamental engineering trade-off:

• Smaller λ (e.g., 0.95): Faster tracking of PA characteristic changes due to temperature drift or aging, but higher steady-state misadjustment and noise sensitivity.
• Larger λ (e.g., 0.999): Lower steady-state error and better noise averaging, but sluggish response to genuine system changes. This directly parallels the learning rate trade-off in gradient-based methods, making λ a critical hyperparameter for closed-loop DPD stability.

Advanced DPD implementations employ adaptive forgetting factors that dynamically adjust λ based on system conditions:

• Burst detection: Temporarily reduce λ when a sudden PA characteristic change is detected, then restore it for steady-state operation.
• Signal power gating: Increase λ (or freeze adaptation entirely) during low-power or no-transmission periods to prevent noise-driven coefficient drift.
• Gradient-based adjustment: Vary λ proportionally to the magnitude of the error signal, accelerating adaptation when distortion is high. These strategies prevent the wind-up and blow-up phenomena common in fixed-λ implementations.

When λ is very close to 1 (e.g., 0.9999) and the input signal has low excitation, the correlation matrix can become ill-conditioned. The repeated multiplication by λ⁻¹ in the RLS recursion amplifies rounding errors, potentially causing the inverse matrix P(k) to lose positive-definiteness. Mitigation strategies include:

• Adding a regularization parameter δ to the initial P(0) matrix.
• Using square-root RLS formulations based on QR decomposition for superior numerical robustness.
• Implementing periodic re-initialization of the covariance matrix in fixed-point FPGA implementations.

The forgetting factor works in concert with coefficient freeze mechanisms in production DPD systems. When the feedback receiver detects low signal-to-noise ratio or the PA is operating in deep back-off, the adaptation loop is frozen to prevent λ from needlessly discounting valid historical data. Upon resumption of normal operation, the forgetting factor smoothly re-engages, preventing transient spikes in distortion. This gated adaptation is essential for maintaining ACLR compliance during discontinuous transmission scenarios like TDD in 5G NR.

The canonical algorithm where the forgetting factor is most prominently applied. RLS recursively finds coefficients that minimize a weighted linear least squares cost function. The forgetting factor, typically denoted as λ (lambda), exponentially weights past errors, giving more importance to recent samples. This allows RLS to achieve an order of magnitude faster convergence than LMS-based algorithms in non-stationary environments, at the cost of O(N²) computational complexity.

A stochastic gradient descent algorithm that serves as the low-complexity alternative to RLS. Unlike RLS, standard LMS does not use an explicit forgetting factor; its inherent tracking ability is governed by a fixed step-size parameter (μ). However, variants like Leaky LMS introduce a leakage factor that is conceptually analogous to a forgetting factor, preventing coefficient drift in poorly conditioned scenarios. LMS is prized for its O(N) complexity and straightforward hardware implementation.

A closed-loop DPD training architecture where the forgetting factor plays a vital role in the system identification step. DLA requires an identified PA behavioral model to compute the error gradient for predistorter adaptation. When this model is estimated online using RLS, the forgetting factor enables it to track thermal memory effects and PA aging. A poorly chosen λ can cause the internal PA model to diverge from the physical device, leading to catastrophic DPD failure.

An alternative DPD architecture that estimates predistorter coefficients by placing a copy of the predistorter in the feedback path. In ILA, the forgetting factor is applied directly to the postdistorter coefficient estimation. Because ILA assumes the PA is invertible and minimizes the error between the predistorter input and postdistorter output, the forgetting factor's role in tracking time-varying PA characteristics is identical to its role in RLS-based system identification.

The forgetting factor directly influences the condition number of the correlation matrix in RLS. A smaller λ (e.g., 0.95) provides faster tracking but reduces the effective data window, potentially leading to an ill-conditioned matrix when the input signal lacks persistent excitation. This is countered by adding a regularization parameter (δ) to the diagonal of the correlation matrix during initialization. The interplay between λ and δ is critical for numerical stability on fixed-point FPGA implementations.

The forgetting factor embodies a fundamental trade-off in adaptive systems. A value close to 1 (e.g., 0.999) averages over a long data history, yielding low steady-state misadjustment but sluggish response to PA behavioral changes. A smaller value (e.g., 0.98) provides rapid convergence to track envelope-dependent nonlinearity shifts but amplifies gradient noise. This trade-off is formally analogous to the learning rate selection problem in stochastic gradient descent.