Least Mean Squares (LMS)

A direct extension of LMS that normalizes the step size by the input signal power to eliminate gradient noise amplification. When the input signal exhibits significant power fluctuations—common in modern communication waveforms with high Peak-to-Average Power Ratio (PAPR)—standard LMS becomes unstable. NLMS divides the step size μ by the squared Euclidean norm of the input vector plus a small regularization constant, ensuring consistent convergence behavior regardless of signal amplitude. The update equation is: w(n+1) = w(n) + [μ / (||x(n)||² + δ)] · e(n) · x(n). This normalization makes NLMS the practical default choice for DPD applications where signal statistics vary over time.

An adaptive algorithm that minimizes a weighted linear least squares cost function recursively, offering an order-of-magnitude faster convergence than LMS at the cost of O(N²) computational complexity. RLS maintains and updates the inverse of the input correlation matrix using the matrix inversion lemma, eliminating the need for explicit matrix inversion at each step. A forgetting factor λ (typically 0.95–0.999) exponentially weights past data, enabling the algorithm to track time-varying PA characteristics. In DPD applications, RLS is preferred when rapid adaptation to changing amplifier behavior is critical, such as during temperature transients or channel reconfiguration events.

The foundational optimization method from which LMS is derived. SGD updates model parameters using the gradient of a loss function computed on a single sample or small minibatch rather than the full dataset. Key properties relevant to DPD:

• Unbiased gradient estimates: The instantaneous gradient is a noisy but unbiased estimator of the true gradient
• Learning rate annealing: Gradually reducing the step size over time improves convergence to the optimal solution
• Momentum variants: Adding a velocity term (e.g., Adam, RMSprop) accelerates convergence in ravines and plateaus LMS is essentially SGD applied to a mean squared error cost function with an identity preconditioner.

A self-adjusting digital filter that automatically modifies its transfer function according to an optimization algorithm driven by an error signal. In the DPD context, the adaptive filter implements the predistorter, with coefficients updated to invert the PA's nonlinear characteristics. Core components:

• Filter structure: FIR, IIR, or nonlinear architectures (Volterra, memory polynomial)
• Adaptation algorithm: LMS, NLMS, RLS, or their variants
• Error computation: Difference between desired linear output and observed PA output The adaptive filter operates in a closed loop, continuously monitoring the feedback receiver and adjusting coefficients to maintain linearity as amplifier conditions change.

A fundamental trade-off in adaptive filtering that governs DPD performance. Convergence rate measures how quickly the algorithm reaches steady-state—critical for tracking rapid PA changes due to thermal effects or supply voltage variations. Misadjustment is the excess mean squared error above the theoretical Wiener optimum, caused by gradient noise from the stochastic updates.

• Large step size μ: Fast convergence, high misadjustment, potential instability
• Small step size μ: Slow convergence, low misadjustment, better steady-state accuracy The learning rate parameter directly controls this trade-off. Advanced techniques like variable step-size LMS dynamically adjust μ based on error signal statistics to balance both objectives.

The mathematical function that quantifies the aggregate error between desired and actual system output, which the adaptation algorithm minimizes. In standard LMS, the cost function is the instantaneous squared error: J(n) = |e(n)|² = |d(n) − w^H(n)x(n)|². For DPD applications, cost functions may be extended to include:

• Weighted error norms: Emphasizing in-band EVM vs. out-of-band ACLR
• Regularization terms: Adding ||w||² to prevent coefficient drift during low-signal periods
• Multi-objective formulations: Simultaneously optimizing multiple metrics like efficiency and linearity The choice of cost function directly shapes the predistorter's behavior and must align with system-level performance requirements.