Adaptive Filtering

A stochastic gradient descent algorithm that updates filter coefficients using the instantaneous estimate of the mean squared error gradient.

• Update Rule: w(n+1) = w(n) + μ · e(n) · x(n)
• Complexity: O(N) per iteration—extremely low computational cost
• Convergence: Slow, highly dependent on eigenvalue spread of input correlation matrix
• Misadjustment: Inherent excess MSE due to gradient noise, proportional to step size μ

Best suited for resource-constrained FPGA implementations where simplicity outweighs convergence speed. The step-size parameter μ controls the trade-off between convergence rate and steady-state misadjustment.

An adaptive algorithm that recursively finds coefficients minimizing a weighted linear least squares cost function, offering an order of magnitude faster convergence than LMS.

• Core Mechanism: Updates the inverse autocorrelation matrix P(n) at each iteration
• Forgetting Factor λ: Controls memory length—smaller λ tracks non-stationarity faster
• Complexity: O(N²) per iteration due to matrix inversion lemma operations
• Convergence: Independent of input eigenvalue spread—dramatically faster than LMS

Ideal for tracking time-varying PA nonlinearities caused by thermal memory effects and supply voltage fluctuations. The forgetting factor λ (typically 0.95–0.999) determines the effective memory window.

A numerically stable implementation of RLS that operates on the square root of the inverse correlation matrix using QR decomposition, eliminating covariance matrix ill-conditioning.

• Method: Applies Givens rotations to maintain upper triangular Cholesky factor R(n)
• Stability: Superior to standard RLS in finite-precision arithmetic
• Complexity: O(N²) but with higher constant factor than conventional RLS
• Use Case: Critical for fixed-point FPGA/ASIC implementations where round-off errors can cause standard RLS divergence

Essential when the condition number of the input data matrix is high, which commonly occurs with wideband signals and closely spaced basis functions in memory polynomial models.

An optimal recursive estimator that tracks time-varying DPD coefficients by modeling parameter evolution as a dynamic system with process and measurement noise.

• State-Space Model: Treats DPD coefficients as state variables with random walk dynamics
• Process Noise Covariance Q: Controls tracking agility—higher Q enables faster adaptation to PA drift
• Measurement Noise Covariance R: Accounts for observation path SNR and ADC quantization
• Advantage: Provides minimum-variance unbiased estimates when noise statistics are known

Particularly effective for GaN Doherty PAs exhibiting significant thermal memory effects, where coefficient drift follows predictable thermal time constants. The Kalman gain optimally weights prediction versus measurement.

A stabilization technique that adds an L2 penalty term to the least squares cost function to solve ill-posed inverse problems in DPD coefficient estimation.

• Cost Function: J(w) = ||y - Xw||² + γ||w||²
• Regularization Parameter γ: Controls bias-variance trade-off
• Effect: Reduces condition number of (X^H X + γI), preventing numerical instability
• Application: Essential when basis functions are highly correlated, causing near-singular autocorrelation matrices

Prevents coefficient drift and wild parameter swings during burst training with limited samples. The Levenberg-Marquardt algorithm extends this concept for nonlinear least squares problems by adaptively adjusting γ.

Two fundamental update strategies governing when DPD coefficients are recalculated, representing a trade-off between latency, computational load, and tracking agility.

• Sample-by-Sample: Coefficients updated with each new sample—maximum tracking speed but highest computational throughput requirement
• Block Update: Coefficients recomputed after accumulating N samples—reduces update rate by factor N, enabling time-sharing of hardware resources
• Burst Training: Updates occur only during dedicated training intervals or specific transmission bursts
• Hybrid Approaches: Fast sample-by-sample LMS for fine tracking combined with periodic block RLS for coarse recalibration

Block size selection balances convergence rate against misadjustment—larger blocks provide better gradient estimates but slower response to PA drift.

A numerically stable implementation of the RLS algorithm using QR decomposition to solve the least squares problem directly without explicitly computing the inverse autocorrelation matrix. This approach is highly robust to ill-conditioning caused by high condition numbers.

• Method: Transforms the data matrix into an orthogonal matrix Q and an upper triangular matrix R
• Benefit: Prevents numerical instability and coefficient drift in fixed-point hardware
• Application: Essential for long-term, uninterrupted DPD operation

A ridge regression technique that adds a penalty term (L2 norm) to the least squares cost function to stabilize the solution of ill-posed inverse problems. This prevents overfitting and wild coefficient swings during direct learning architecture training.

• Cost Function: J(w) = ||y - Xw||² + λ||w||²
• Parameter: The regularization parameter λ controls the bias-variance trade-off
• Result: Reduces coefficient drift and improves spectral regrowth suppression

A fundamental trade-off in adaptive filtering. Convergence rate defines how quickly the algorithm reaches the optimal Wiener solution, while misadjustment is the excess mean squared error beyond the theoretical minimum caused by gradient noise.

• LMS: Larger step-size μ increases convergence but also increases misadjustment
• RLS: Achieves faster convergence with zero misadjustment in stationary environments
• Metric: Normalized Mean Squared Error (NMSE) quantifies the combined effect