Least Mean Squares (LMS)

LMS operates as a stochastic approximation of the steepest descent method, replacing the true gradient of the mean squared error with an instantaneous estimate computed from a single sample. Unlike batch Least Squares (LS) which requires the full covariance matrix, LMS updates the weight vector w(n+1) = w(n) + μ e(n) x(n) at each time step, where μ is the step size, e(n) is the instantaneous error, and x(n) is the input regressor vector. This sample-by-sample recursion eliminates matrix inversion entirely, reducing computational complexity to O(N) per iteration where N is the number of filter taps.

The step size parameter μ (mu) governs the fundamental bias-variance tradeoff in LMS. A large μ accelerates convergence toward the Wiener solution but produces excessive misadjustment—the steady-state excess mean squared error above the theoretical minimum. A small μ yields precise coefficient estimates with low misadjustment but slows adaptation. The stability bound requires 0 < μ < 2/λ_max, where λ_max is the largest eigenvalue of the input autocorrelation matrix. In practice, μ is typically set to a fraction of 1/tr(R), where tr(R) is the total input power.

Unlike deterministic batch algorithms, LMS never converges to the exact Wiener solution. It exhibits perpetual coefficient jitter around the optimum due to gradient noise. The misadjustment M quantifies this penalty: M ≈ μ · tr(R) / 2. This reveals the direct proportionality between step size and steady-state error. For DPD applications, this means the linearization performance floor is set by the chosen μ. Techniques like step size annealing—gradually reducing μ during training—can achieve both fast initial convergence and low final misadjustment.

The Normalized LMS (NLMS) algorithm addresses LMS's sensitivity to input signal power fluctuations—a critical concern in DPD where signal crest factor varies. NLMS normalizes the step size by the instantaneous input power: μ(n) = μ̃ / (||x(n)||² + δ), where δ is a small regularization constant preventing division by zero. This power normalization ensures consistent convergence speed regardless of signal amplitude, making NLMS the preferred choice over standard LMS in predistortion applications where wideband signals exhibit high peak-to-average power ratios.

LMS requires only 2N+1 multiplications and 2N additions per iteration, where N is the number of predistorter coefficients. This minimal footprint enables direct hardware implementation on FPGAs and ASICs without complex matrix operations. Key operations:

• Filtering: y(n) = w^T(n) x(n) — N multiplications, N-1 additions
• Error computation: e(n) = d(n) - y(n) — 1 subtraction
• Weight update: w(n+1) = w(n) + μ e(n) x(n) — N+1 multiplications, N additions This deterministic, low-latency structure makes LMS the baseline algorithm against which more complex coefficient extraction methods are benchmarked.

LMS convergence speed is governed by the eigenvalue spread of the input autocorrelation matrix R—the ratio λ_max/λ_min. A large spread, common in DPD where memory polynomial basis functions are highly correlated, causes slow convergence along directions corresponding to small eigenvalues. This ill-conditioning motivates alternatives like Transform-Domain LMS or Affine Projection Algorithms for coefficient extraction. Pre-whitening the regressor data or applying Principal Component Analysis (PCA) can compress the eigenvalue spread and accelerate LMS convergence in practice.

An adaptive filtering algorithm that recursively updates the inverse of the input correlation matrix to achieve an order of magnitude faster convergence than LMS. Unlike LMS, which uses a stochastic instantaneous gradient, RLS minimizes a weighted least-squares cost function of all past errors.

• Convergence is typically an order of magnitude faster than LMS
• Computational complexity is O(N²) versus O(N) for LMS
• Insensitive to the eigenvalue spread of the input signal
• Ideal for rapidly time-varying PA characteristics where LMS tracking is too slow

A variant of LMS that normalizes the step size by the instantaneous power of the input vector. This prevents gradient noise amplification when input signal levels fluctuate, a common scenario in wideband communication signals with high peak-to-average power ratios.

• Step size μ is divided by ||x(n)||² + ε
• Maintains stability across varying signal envelopes
• Adds minimal computational overhead over standard LMS
• The regularization constant ε prevents division by zero during silent periods

A batch estimation algorithm that computes the optimal coefficient vector in a single step by solving the normal equations. Unlike LMS, which processes samples sequentially, batch LS uses the entire captured data record to minimize the sum of squared residuals.

• Solution: w = (XᴴX)⁻¹Xᴴy using the Moore-Penrose pseudoinverse
• Provides the global minimum for the given data record
• Used for offline model extraction during PA characterization
• Computationally intensive for large datasets due to matrix inversion

A scalar parameter in recursive estimation that exponentially weights recent data more heavily than past samples. When integrated with LMS-like updates, it enables the model to track slowly time-varying PA characteristics caused by thermal drift and aging.

• Typical range: 0.95 < λ < 0.999
• λ = 1.0 corresponds to infinite memory (stationary assumption)
• Lower λ values provide faster tracking but noisier estimates
• Essential for long-term operational stability in deployed DPD systems