Basis Function

Each basis function applies a specific nonlinear operation to the input signal or its delayed versions. Common transformations include:

• Polynomial terms: (x(n) \cdot |x(n)|^k) for odd-order nonlinearities
• Memory terms: (x(n-m)) capturing delayed signal contributions
• Cross terms: (x(n) \cdot x(n-1)^2) modeling interactions between time-shifted samples

The complete predistorter output is a linear combination of these basis functions weighted by complex coefficients.

The most widely adopted basis set in DPD implementations, defined as:

(\phi_{k,m}(n) = x(n-m) \cdot |x(n-m)|^{k-1})

• k: nonlinearity order (odd values typically 1,3,5,7,9)
• m: memory depth (0 to M-1)
• Total basis functions = K × M, where K is number of odd orders

This structure captures both static nonlinearity and linear memory effects while remaining computationally tractable for FPGA implementation.

Standard polynomial bases suffer from ill-conditioning because higher-order terms are highly correlated. Orthogonal bases mitigate this:

• Orthogonal polynomials (Chebyshev, Legendre): Decorrelate nonlinear orders for improved numerical stability
• Gram-Schmidt orthogonalization: Adaptively constructs an orthonormal basis from the input signal statistics
• Principal Component Analysis (PCA): Reduces basis dimensionality by retaining only high-variance components

Orthogonal bases enable faster convergence in adaptive algorithms and reduce coefficient estimation variance.

Extends the standard memory polynomial by adding cross-memory terms that capture interactions between different delay taps:

(\phi_{k,m,l}(n) = x(n-m) \cdot |x(n-m-l)|^{k-1})

• Models nonlinear memory effects where the PA response depends on the envelope at multiple time instants
• Significantly increases basis count: complexity grows with (O(K \cdot M^2))
• Often pruned using LASSO regularization or greedy selection to retain only dominant terms
• Essential for wideband signals where memory effects span multiple symbol periods

Not all basis functions contribute equally to linearization performance. Pruning strategies reduce complexity:

• Magnitude-based pruning: Remove terms with coefficient magnitudes below a threshold
• Greedy forward selection: Iteratively add the basis function that most reduces residual error
• LASSO (L1 regularization): Drives unnecessary coefficients to exactly zero during estimation
• Cross-validation: Evaluates generalization performance to prevent overfitting

Pruned models can achieve 50-70% reduction in basis count with minimal ACLR degradation.

The condition number of the basis function correlation matrix directly impacts coefficient estimation accuracy:

• Ill-conditioned matrices (high condition number): Small measurement noise causes large coefficient errors
• Causes: Highly correlated polynomial terms, insufficient signal excitation, narrowband signals
• Mitigations:
  • Use orthogonal basis functions
  • Apply Tikhonov regularization (ridge regression)
  • Ensure persistently exciting training signals with sufficient PAPR
  • Implement QR decomposition with column pivoting for robust least-squares solutions