Forgetting Factor

The forgetting factor λ (lambda) applies an exponential decay to past data, where each older sample is weighted by λ^k (k = sample age). This creates a sliding window of effective memory without storing historical data. A value of λ = 0.99 means a sample 100 iterations ago contributes only 37% of its original weight, enabling the algorithm to gracefully forget stale information while maintaining numerical stability.

The forgetting factor directly controls the bias-variance tradeoff in adaptive systems:

• λ → 1.0: Near-infinite memory, low steady-state misadjustment, but slow tracking of parameter changes
• λ < 0.99: Fast adaptation to time-varying amplifier characteristics, but increased noise sensitivity
• λ < 0.95: Risk of estimator instability and coefficient divergence

For power amplifier linearization, λ is typically tuned to track thermal memory effects occurring over millisecond timescales while rejecting measurement noise.

In Recursive Least Squares (RLS) algorithms, the forgetting factor is embedded in the inverse correlation matrix update:

P(k) = (1/λ) × [P(k-1) − K(k) × u^T(k) × P(k-1)]

Where P(k) is the inverse autocorrelation matrix and K(k) is the Kalman gain vector. The division by λ effectively inflates the covariance, preventing P(k) from collapsing to zero and maintaining the algorithm's ability to adapt. Without this mechanism, RLS would become frozen after processing sufficient data.

Fixed forgetting factors cannot simultaneously achieve fast transient response and low steady-state error. Variable forgetting factor (VFF) schemes dynamically adjust λ based on:

• Prediction error magnitude: Increase forgetting when error spikes indicate system change
• Gradient-based adaptation: Adjust λ to minimize mean-square error
• Change detection: Reset λ temporarily when abrupt parameter shifts are detected

In Doherty amplifier DPD, VFF schemes excel because the amplifier's nonlinear characteristic shifts dramatically during load modulation transitions, requiring rapid re-estimation followed by stable operation.

As λ decreases below 0.99, the condition number of the correlation matrix can deteriorate, leading to:

• Covariance blow-up: P(k) grows exponentially during low-excitation periods
• Coefficient bursting: Parameter estimates oscillate wildly when input signal power drops

Mitigation techniques include:

• Regularized RLS: Adding a diagonal loading term to stabilize P(k)
• Covariance reset: Periodically reinitializing P(k) to prevent divergence
• Leakage factor: Combining forgetting with coefficient leakage toward zero

These safeguards are essential for FPGA-based DPD implementations where fixed-point arithmetic amplifies numerical issues.

The forgetting factor RLS is mathematically equivalent to a Kalman filter with a specific state-space model where:

• Process noise covariance is scaled by (1/λ − 1)
• The forgetting mechanism models parameter drift as a random walk

This connection provides a Bayesian interpretation: λ encodes the engineer's prior belief about how quickly amplifier characteristics change. In massive MIMO DPD, where each antenna element experiences different thermal dynamics, this probabilistic framework enables systematic tuning of forgetting factors across the array.

The primary adaptive filtering algorithm that utilizes the forgetting factor (λ). Unlike LMS, RLS minimizes a weighted least-squares cost function where past errors are exponentially de-weighted by λ. This provides significantly faster convergence for non-stationary signals at the cost of higher computational complexity (O(N²)).

• Directly implements exponential forgetting
• Optimal for tracking thermal memory drift
• Requires regularization to prevent covariance matrix blow-up

The effective memory horizon created by the forgetting factor. The time constant τ = 1/(1-λ) defines how many samples contribute 63% of the total weight. For a sample period Ts, the effective memory length is approximately Ts/(1-λ) seconds.

• λ=0.99: ~100 sample memory
• λ=0.999: ~1000 sample memory
• Balances tracking agility vs. estimate variance

An adaptive DPD architecture that iteratively updates predistorter coefficients by minimizing the error between the desired ideal signal and the actual PA output. DLA often employs RLS with a forgetting factor to track PA characteristic changes during transmission without interrupting the signal.

• Closed-loop coefficient update
• Forgetting factor enables continuous online adaptation
• Contrasts with Indirect Learning Architecture's post-distorter copy method

A protective mechanism used alongside the forgetting factor in RLS. When the input signal has low excitation (low power or narrow bandwidth), the covariance matrix P(n) can grow exponentially due to continuous multiplication by 1/λ. A conditional reset or regularization prevents numerical instability.

• Prevents estimator wind-up during silence
• Often triggered by input power threshold
• Maintains numerical conditioning for stable DPD

A simpler alternative to RLS that does not use a forgetting factor in its classical form. LMS employs a fixed step size μ rather than an exponential window, resulting in slower convergence for correlated inputs. However, normalized LMS (NLMS) can approximate forgetting behavior by adapting the step size to input power.

• O(N) complexity vs. RLS O(N²)
• No matrix inversion required
• Suitable for resource-constrained FPGA implementations

The physical phenomenon that necessitates a forgetting factor in DPD. GaN and GaAs power amplifiers exhibit slow-changing electrical memory due to self-heating and thermal time constants (milliseconds to seconds). The forgetting factor must be tuned so the estimator's memory horizon matches these thermal dynamics.

• Long-term memory: substrate temperature changes
• Short-term memory: junction-level transients
• λ selection directly impacts ACLR stability over temperature