Direct Learning Architecture (DLA)

DLA forms a closed-loop adaptive system where the predistorter coefficients are updated by directly comparing the desired linear signal with the measured PA output.

• Minimizes the true linearization error rather than an intermediate modeling error
• The cost function is the Mean Squared Error (MSE) between the ideal output and the actual PA output
• Requires a feedback path to capture the PA output, introducing loop delay that must be compensated
• Unlike ILA, DLA does not assume the predistorter and postdistorter are interchangeable

Because the predistorter precedes the nonlinear PA in the signal chain, the error surface is nonlinear in the predistorter parameters.

• The PA nonlinearity makes the cost function non-convex, potentially containing local minima
• Gradient-based methods like LMS and NLMS are commonly used for iterative descent
• More advanced algorithms such as Recursive Least Squares (RLS) can accelerate convergence
• The nonlinearity necessitates careful step size selection to avoid divergence or slow convergence

DLA supports online training where coefficients are updated continuously as new signal samples arrive, enabling the system to track time-varying PA behavior.

• Adapts to thermal memory effects as the amplifier heats up during operation
• Compensates for aging and environmental drift in PA characteristics
• Requires forgetting factor or sliding window techniques to prioritize recent data
• Computational complexity must be managed for real-time FPGA or ASIC implementation

Direct learning implementations must address ill-conditioning that arises from the correlation structure of wideband communication signals.

• Regularization parameters are added to the diagonal of correlation matrices to prevent numerical instability
• QR-RLS algorithms using Givens rotations provide superior numerical robustness over standard RLS
• The condition number of the input correlation matrix directly impacts convergence behavior
• Singular Value Decomposition (SVD) can diagnose and mitigate ill-conditioned system identification problems

DLA and Indirect Learning Architecture (ILA) represent the two dominant adaptive DPD paradigms, each with distinct tradeoffs.

• ILA identifies a postdistorter by placing a copy after the PA, then copies coefficients to the predistorter — assumes commutativity
• DLA directly identifies the predistorter, avoiding the commutativity assumption but requiring nonlinear optimization
• ILA is simpler to implement but may be suboptimal when the PA exhibits strong nonlinearity
• DLA generally achieves superior linearization performance at the cost of increased algorithmic complexity
• The choice depends on PA characteristics, signal bandwidth, and available computational resources

A critical implementation challenge in DLA is the alignment of the reference and feedback signals in time, phase, and gain.

• Fractional delay filters correct for sub-sample timing misalignments in the feedback path
• Cross-correlation techniques estimate the integer and fractional delay between forward and feedback paths
• Phase rotation and gain normalization ensure the error signal accurately represents the distortion
• Uncompensated loop delay introduces bias in coefficient estimates and degrades linearization performance
• Modern implementations use automated calibration routines to maintain alignment during operation

The simplest stochastic gradient algorithm used within DLA for coefficient updates. LMS estimates the gradient of the mean squared error using only the instantaneous error and input vector.

• Update rule: w(n+1) = w(n) + μ · e*(n) · x(n)
• Complexity: O(N) per iteration, where N is the number of coefficients.
• Convergence: Slow for correlated input signals; rate depends on eigenvalue spread of the input autocorrelation matrix.
• Step size μ: Must satisfy 0 < μ < 2/λ_max for stability, where λ_max is the largest eigenvalue.
• Misadjustment: Proportional to μ; a fundamental tradeoff between convergence speed and steady-state error.

A deterministic recursive algorithm that minimizes the weighted sum of squared errors. RLS converges an order of magnitude faster than LMS at the cost of O(N²) complexity.

• Core mechanism: Recursively updates the inverse of the input autocorrelation matrix using the matrix inversion lemma.
• Forgetting factor λ: Typically 0.95–0.999; weights recent samples more heavily to track time-varying PA characteristics.
• Convergence: Independent of input signal statistics; achieves the Wiener solution in approximately 2N iterations.
• Numerical stability: Direct RLS can diverge due to covariance matrix ill-conditioning; QR-RLS variants use Givens rotations for robust implementation.

A practical enhancement of LMS that normalizes the step size by the instantaneous power of the input vector. This makes convergence behavior independent of signal level fluctuations.

• Update rule: w(n+1) = w(n) + (μ̃ / ||x(n)||² + δ) · e*(n) · x(n)
• Normalized step size μ̃: Typically 0 < μ̃ < 2 for stability; δ is a small regularization constant preventing division by zero.
• Advantage: Robust to crest factor variations in communication signals like OFDM.
• Comparison to LMS: Faster initial convergence when input power varies; same steady-state behavior.
• Implementation: Requires one additional multiply-accumulate for power computation per iteration.

The numerically superior implementation of RLS that operates directly on the square-root of the inverse correlation matrix. It avoids explicit matrix inversion and is immune to the instability of conventional RLS.

• Mechanism: Applies a sequence of Givens rotations to triangularize the augmented data matrix, propagating the Cholesky factor R^(1/2) directly.
• Systolic array mapping: Naturally maps to parallel hardware architectures for FPGA implementation.
• Dynamic range: Requires half the numerical precision of conventional RLS for the same accuracy.
• Application: Preferred for fixed-point DSP implementations where numerical stability is paramount.
• Cost: Higher constant factor than LMS but O(N²) complexity with guaranteed stability.