Basis Function Orthogonalization

The defining property of an orthogonalized set is that the inner product between any two distinct basis functions is zero. This eliminates the statistical interdependence that causes the normal equation matrix to become ill-conditioned. In practice, this means that the information contributed by each basis function is unique and non-redundant, allowing the estimator to assign coefficient values independently without interference from correlated regressors.

The condition number of the data matrix—the ratio of its largest to smallest singular value—is a direct measure of numerical stability. A raw polynomial basis can have condition numbers exceeding 10¹², making matrix inversion extremely sensitive to quantization noise. Orthogonalization reduces this to values near 10³–10⁴, ensuring that fixed-point arithmetic in FPGA implementations does not amplify rounding errors into coefficient divergence.

Because each orthogonal basis function contributes independently to the error reduction, adaptive algorithms like Recursive Least Squares (RLS) and Least Mean Squares (LMS) converge in fewer iterations. The eigenvalue spread of the autocorrelation matrix is minimized, eliminating the slow modes that dominate convergence time in correlated bases. This is critical for tracking thermal memory effects that drift on millisecond timescales.

The classic sequential procedure for constructing an orthogonal set from a polynomial basis:

• Start with the first basis function unchanged
• For each subsequent function, subtract its projection onto all previously orthogonalized functions
• Normalize each result to unit norm if orthonormality is required This method is deterministic and preserves the span of the original basis, but can accumulate numerical error in finite precision for high-order systems.

A numerically superior alternative to Gram-Schmidt that operates directly on the basis function matrix. By factoring the matrix into an orthogonal matrix Q and an upper triangular matrix R, the least-squares solution becomes a simple back-substitution. This method uses Householder reflections or Givens rotations, which are inherently stable and map efficiently to systolic array hardware architectures in FPGAs.

Orthogonalization is a critical preprocessing step for sparse model selection algorithms like Orthogonal Matching Pursuit (OMP). By decorrelating the candidate dictionary, each iteration's selection of the most significant basis function is unambiguous. Without orthogonalization, OMP may select redundant terms that explain the same distortion component, wasting computational resources on coefficients that provide no incremental linearization benefit.

In memory polynomial models, basis functions like x(n)|x(n)|^k and x(n-1)|x(n-1)|^k are highly correlated, especially for modern wideband signals with high sample rates. This correlation creates a near-singular autocorrelation matrix during least squares estimation. The condition number—the ratio of the largest to smallest eigenvalue—can exceed 10^12, making the solution hypersensitive to measurement noise and quantization errors. Small perturbations in the captured PA output data lead to wildly different coefficient vectors, preventing reliable linearization.

Classical Gram-Schmidt suffers from catastrophic cancellation in finite-precision arithmetic, where accumulated rounding errors destroy orthogonality. The modified Gram-Schmidt (MGS) algorithm restructures the computation to subtract projections one at a time:

• For each new basis vector, immediately remove its projection onto ψ₁, then onto ψ₂, and so on
• This reordering reduces error accumulation by a factor proportional to the condition number In FPGA implementations with 18-bit or 25-bit fixed-point multipliers, MGS is essential to maintain orthogonality to within -60 dB of the theoretical ideal, preventing the re-emergence of ill-conditioning during real-time adaptation.

Orthogonalization directly compresses the eigenvalue spread of the data covariance matrix. Before transformation, eigenvalues may span 10 orders of magnitude, with the smallest eigenvalues buried in the noise floor. After orthogonalization, the eigenvalue spread approaches unity (0 dB), meaning all basis functions contribute with equal weight to the solution. This has three critical effects:

• Convergence speed in LMS-type adaptive algorithms becomes uniform across all modes
• The misadjustment noise in the coefficient estimate is minimized
• Step-size bounds for gradient algorithms can be set aggressively without risking instability

In the indirect learning architecture, the postdistorter is trained on the PA output to estimate the inverse. Orthogonalization is applied to the regressor matrix formed from the PA output samples. The process:

• Step 1: Generate the full set of memory polynomial basis functions from the captured PA output
• Step 2: Apply MGS to create an orthogonal basis spanning the same subspace
• Step 3: Solve for coefficients in the orthogonal domain using simple back-substitution
• Step 4: Transform coefficients back to the original polynomial basis for predistorter implementation This avoids the need to compute a matrix inverse directly, reducing the O(N³) complexity to O(N²).

For online adaptation, orthogonalization must operate on a sliding window of recent samples to track changing PA characteristics due to temperature drift or supply voltage variation. Techniques include:

• Recursive QR decomposition: Updates the orthogonal factorization with each new sample using Givens rotations, avoiding full recomputation
• Exponential forgetting factors: Weight recent samples more heavily, with λ typically between 0.95 and 0.999
• Block-based re-orthogonalization: Periodically re-applies MGS to a fresh block of samples to purge accumulated numerical drift These methods maintain a condition number below 10³ even during rapid thermal transients in GaN Doherty amplifiers.