Misadjustment

Misadjustment M is formally defined as the normalized difference between the steady-state MSE J(∞) of the adaptive filter and the minimum MSE J_min achievable by the optimal Wiener filter:

M = [J(∞) − J_min] / J_min

• J(∞): The asymptotic mean squared error after convergence
• J_min: The irreducible error floor set by the Wiener solution
• M is typically expressed as a percentage or decimal fraction
• A misadjustment of 0.1 means the adaptive filter operates at 10% excess MSE above the theoretical minimum
• This metric isolates the cost of adaptation from the cost of the estimation problem itself

For the Least Mean Squares (LMS) algorithm, misadjustment is directly proportional to the step size μ and the input signal power:

M ≈ (μ/2) · tr[R]

where tr[R] is the trace of the input autocorrelation matrix.

• Smaller μ → lower misadjustment but slower convergence
• Larger μ → faster convergence but higher excess MSE
• This creates the fundamental convergence rate vs. steady-state accuracy tradeoff
• In practice, misadjustment values of 5-15% are common for LMS in DPD applications
• The linear relationship holds only for small μ; larger values introduce nonlinear effects

Recursive Least Squares (RLS) algorithms exhibit fundamentally different misadjustment behavior compared to gradient-based methods:

• RLS misadjustment is proportional to (1 − λ) · N, where λ is the forgetting factor and N is the filter order
• Unlike LMS, RLS misadjustment is independent of the input signal statistics
• For stationary environments (λ = 1), RLS theoretically achieves zero misadjustment
• In practice, finite-precision effects and regularization introduce small excess errors
• Typical RLS implementations achieve misadjustment values below 1%, making them attractive for high-precision DPD coefficient estimation

Misadjustment arises from multiple physical and algorithmic sources in adaptive DPD systems:

• Gradient noise: Stochastic approximation of the true gradient introduces variance in coefficient updates
• Lag error: When the algorithm cannot track time-varying amplifier characteristics quickly enough
• Finite-precision effects: Quantization errors in fixed-point implementations add numerical noise
• Model mismatch: When the predistorter model order is insufficient to capture the PA nonlinearity
• Measurement noise: Noisy feedback observations propagate into coefficient estimates
• Understanding these sources enables targeted mitigation strategies in system design

In digital predistortion systems, misadjustment directly impacts adjacent channel leakage ratio (ACLR) and error vector magnitude (EVM) performance:

• A misadjustment of 10% can degrade ACLR by 2-5 dB from the theoretical optimum
• Variable step-size algorithms dynamically reduce μ after initial convergence to minimize steady-state misadjustment
• Momentum-based variants of LMS can reduce misadjustment without sacrificing convergence speed
• For 5G wideband signals, misadjustment must typically be kept below 5% to meet spectral mask requirements
• Designers often specify a maximum allowable misadjustment as a key system requirement before selecting the adaptation algorithm

Misadjustment cannot be directly measured in a live system since J_min is unknown. Instead, engineers use proxy methods:

• Learning curve analysis: Plot MSE vs. iterations and extrapolate the steady-state asymptote
• Ensemble averaging: Run multiple independent trials and average the squared error trajectories
• Excess MSE estimation: Subtract the lowest observed MSE from the steady-state value
• Wiener solution benchmarking: Compute J_min offline using batch least squares on captured data, then compare against online algorithm performance
• In production DPD systems, ACLR and EVM measurements serve as practical proxies for misadjustment

The foundational stochastic gradient descent algorithm that directly influences misadjustment through its step-size parameter. LMS updates coefficients using the instantaneous gradient estimate:

• Step-size μ: Larger values accelerate convergence but increase steady-state misadjustment
• Trade-off: Misadjustment is approximately proportional to μ · tr[R], where R is the input correlation matrix
• Simplicity: Requires only O(N) operations per iteration, making it ideal for high-speed DPD applications

The misadjustment of LMS serves as the baseline against which more sophisticated algorithms are compared.

An adaptive variant that normalizes the step size by the input signal power to achieve more consistent convergence behavior across varying signal levels. Key properties:

• Power normalization: Step size becomes μ / (||x(n)||² + δ), where δ prevents division by zero
• Stability: Maintains bounded misadjustment even with non-stationary input power
• Convergence: Achieves faster initial convergence than standard LMS for colored inputs

NLMS is widely used in DPD systems where the peak-to-average power ratio of modern communication signals causes significant input power fluctuations.

A computationally intensive algorithm that achieves significantly lower misadjustment than gradient-based methods by recursively computing the exact inverse of the input correlation matrix:

• Misadjustment: Approximately N/(2λ) where λ is the forgetting factor, independent of input statistics
• Convergence: Typically converges in < 2N iterations, an order of magnitude faster than LMS
• Cost: O(N²) complexity limits real-time implementation in wideband DPD systems

The QR-RLS variant improves numerical stability using Givens rotations, making it suitable for fixed-point FPGA implementations where misadjustment must be minimized.

The theoretical minimum mean squared error (MMSE) benchmark against which misadjustment is measured. The Wiener-Hopf equation defines the optimal coefficient vector:

• Optimal weights: w_opt = R⁻¹p, where R is the autocorrelation matrix and p is the cross-correlation vector
• MMSE floor: Represents the irreducible error due to measurement noise and model mismatch
• Misadjustment definition: M = (J_ss - J_min) / J_min, where J_ss is steady-state MSE and J_min is the Wiener MMSE

In DPD applications, the Wiener solution provides the theoretical linearization limit that adaptive algorithms approach but cannot exceed.

A critical parameter in recursive estimation algorithms that directly controls the trade-off between tracking capability and steady-state misadjustment:

• Range: 0 ≪ λ ≤ 1, with values typically between 0.95 and 0.999 for DPD applications
• Memory: Effective window length is approximately 1/(1-λ) samples
• Misadjustment relationship: Smaller λ improves tracking but increases misadjustment proportionally to (1-λ)/2

In time-varying power amplifier scenarios, the forgetting factor must be tuned to balance adaptation to thermal drift against excess misadjustment from noise amplification.

The speed at which an adaptive algorithm approaches the Wiener optimum, which is inversely related to misadjustment through the fundamental bias-variance tradeoff:

• LMS: Convergence time constant τ ≈ 1/(2μλ_min), where λ_min is the smallest eigenvalue of R
• Eigenvalue spread: Large spreads in the input correlation matrix slow convergence dramatically
• Misadjustment-speed product: For LMS, M · τ_avg ≈ constant, meaning faster convergence necessarily increases misadjustment

This trade-off drives the selection of variable step-size algorithms that use large μ during initial acquisition and small μ during steady-state tracking.