Numerical Stability

The algorithm must prevent the correlation matrix from becoming ill-conditioned. When basis functions are highly correlated—common in memory polynomial models—the condition number spikes, amplifying rounding errors during matrix inversion. Stable implementations use regularization parameters or orthogonal basis functions to keep the condition number low.

Solving least-squares problems via QR decomposition rather than direct matrix inversion provides superior numerical stability. By decomposing the data matrix into an orthogonal matrix Q and upper triangular matrix R, the algorithm avoids squaring the condition number—a critical advantage when operating on 16-bit or 18-bit fixed-point DSP slices.

A regularization parameter (often denoted λ) is added to the diagonal of the correlation matrix before inversion. This technique:

• Prevents singular matrices during low-signal periods
• Dampens coefficient oscillation in steady-state
• Acts as a soft constraint against overfitting to measurement noise Typical values range from 10⁻⁴ to 10⁻² of the matrix trace.

In Recursive Least Squares (RLS) implementations, the forgetting factor λ must stay within strict bounds—typically 0.95 ≤ λ ≤ 0.999. Values too close to 1.0 cause the inverse correlation matrix to grow unbounded (estimator windup), while values too low introduce excessive noise. Fixed-point implementations require careful scaling to prevent overflow in the gain vector computation.

Every arithmetic operation must be analyzed for quantization error propagation. Critical paths include:

• Accumulator width in multiply-accumulate (MAC) units for basis function evaluation
• Divider precision in normalized LMS step-size calculation
• Feedback receiver ADC bit-width determining the error signal resolution Typical stable implementations use 18×18 multipliers with 48-bit accumulators.

A numerically stable algorithm includes coefficient freeze logic that halts adaptation when:

• Input signal power drops below a programmable threshold
• The condition number exceeds a safety limit
• Feedback receiver reports saturation or fault Upon freeze, coefficients hold their last valid values, preventing divergence that could damage the PA or violate spectral emission masks.

A numerical state where the correlation matrix has a high condition number, making the coefficient estimation highly sensitive to small perturbations. In DPD, this leads to wildly fluctuating predistorter coefficients and spectral regrowth. It is often caused by highly correlated basis functions or insufficient signal excitation.

A matrix factorization technique that decomposes the correlation matrix into an orthogonal matrix Q and an upper triangular matrix R. Solving the least-squares problem using R instead of direct matrix inversion provides superior numerical stability, especially for ill-conditioned matrices common in memory polynomial DPD.

A scalar value added to the diagonal of the correlation matrix during estimation. This technique, known as Tikhonov regularization or ridge regression, improves numerical stability by preventing singular matrices and reducing the variance of coefficient estimates. It introduces a small bias to achieve a lower overall mean squared error.

A method of representing real numbers with a fixed number of digits after the radix point, standard on FPGAs and ASICs. Unlike floating-point, it requires careful word-length optimization to balance dynamic range against quantization error. Overflow and underflow must be managed to prevent catastrophic instability in the adaptation loop.

A weighting parameter (λ) in recursive algorithms like Recursive Least Squares (RLS) that exponentially discounts older data. While essential for tracking time-varying PA characteristics, a value too close to 1 can lead to covariance matrix wind-up and numerical instability, especially in fixed-point implementations.

A computationally efficient method for decomposing a symmetric, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. It is often used as a numerically stable alternative to direct matrix inversion in block-based coefficient estimation algorithms, requiring roughly half the operations of QR decomposition.