Correlation Matrix

The DPD correlation matrix R is inherently Hermitian (equal to its own conjugate transpose) and positive definite for non-zero input signals. This guarantees that all eigenvalues are real and strictly positive, ensuring a unique global minimum exists for the least-squares cost function. This property is what makes Cholesky decomposition—a numerically efficient, triangular factorization—the preferred direct solver over Gaussian elimination in hardware implementations.

When the input signal is wide-sense stationary and the basis functions are memory polynomials, the correlation matrix exhibits a Toeplitz structure. Each descending diagonal is constant, drastically reducing the number of unique elements that must be computed and stored. This structure enables the use of Levinson-Durbin recursion, reducing the computational complexity of matrix inversion from O(N³) to O(N²), a critical optimization for real-time adaptation.

The condition number κ(R), defined as the ratio of the largest to smallest eigenvalue, quantifies the matrix's sensitivity to numerical errors. A high condition number (κ >> 1) indicates ill-conditioning, where small perturbations from fixed-point arithmetic or measurement noise are amplified, leading to wildly inaccurate coefficient estimates. This commonly arises from highly correlated basis functions or insufficient signal bandwidth to excite all nonlinear modes.

To combat ill-conditioning, Tikhonov regularization is applied by adding a small diagonal matrix to the correlation matrix: R_reg = R + γI. The regularization parameter γ biases the solution toward smaller coefficient magnitudes, trading a slight increase in bias for a dramatic reduction in variance. This is mathematically equivalent to incorporating a prior belief that the optimal coefficients are small, and is essential for stable fixed-point FPGA implementation.

In online training, the correlation matrix is not recomputed from scratch. Instead, it is updated recursively using a rank-1 outer product of the new basis function vector: R(k) = λR(k-1) + x(k)x(k)^H. The forgetting factor λ (0 < λ ≤ 1) exponentially discounts past data, allowing the matrix to track time-varying PA characteristics due to thermal drift or changing operating conditions.

Instead of explicitly inverting the correlation matrix, numerically stable implementations use QR decomposition to solve the normal equations. The basis function matrix X is factored into an orthogonal matrix Q and an upper triangular matrix R. The coefficients are then found by back-substitution: w = R⁻¹Q^H y. This avoids squaring the condition number, a pitfall of forming the normal equations directly via X^H X.