Orthonormal Basis Functions

The defining property where the inner product of any two distinct basis functions is zero. This eliminates cross-correlation between regressors in the data matrix, ensuring that each coefficient captures independent distortion information. For a set of functions {φ_k}, orthogonality means ⟨φ_i, φ_j⟩ = 0 for all i ≠ j. This property directly prevents the ill-conditioning that plagues standard polynomial models when memory depth increases.

Each basis function is scaled to have unit energy, meaning ⟨φ_k, φ_k⟩ = 1. This normalization ensures that all basis functions contribute with equal statistical weight during coefficient estimation, preventing functions with large dynamic range from dominating the least squares solution. Combined with orthogonality, this yields a perfectly conditioned identity autocorrelation matrix, enabling faster convergence in adaptive algorithms like Recursive Least Squares (RLS).

A classic orthonormal set constructed from polynomials weighted by an exponential decay factor. The Laguerre functions are defined as:

• φ_k(n) = L_k(n) · e^{-αn}
• where L_k(n) is the k-th Laguerre polynomial
• α controls the memory decay rate

These functions naturally capture exponentially fading memory effects typical of thermal and bias circuit dynamics in power amplifiers, making them more efficient than simple tapped delay lines for long memory spans.

A generalization of Laguerre functions that uses multiple distinct poles rather than a single repeated pole. Kautz functions are parameterized by a set of stable poles {ξ_i} inside the unit circle, allowing them to model resonant or oscillatory memory effects that simple exponential decays cannot capture. This makes them particularly effective for PAs with complex impedance interactions or Doherty amplifier architectures where memory exhibits non-monotonic behavior.

The condition number of the data matrix—the ratio of largest to smallest singular value—directly determines the sensitivity of coefficient estimates to measurement noise. Standard polynomial models often exhibit condition numbers exceeding 10^6, causing unstable solutions. Orthonormal basis functions reduce this to near unity (1.0), ensuring:

• Robust coefficient extraction via QR decomposition
• Insensitivity to finite-precision arithmetic
• Reliable operation in FPGA fixed-point implementations

A constructive algorithm that transforms any linearly independent set of functions into an orthonormal basis. Applied to DPD, it converts standard memory polynomial terms into an equivalent orthonormal set without changing the spanned subspace. The process:

• Start with the first basis function and normalize it
• For each subsequent function, subtract its projections onto all previously orthogonalized functions
• Normalize the result

This enables on-the-fly basis generation from conventional polynomial models.

The direct process that creates orthonormal basis functions. It transforms a set of correlated polynomial terms (e.g., x(n)|x(n)|^k) into a new set that is mutually uncorrelated. This is critical because standard polynomial basis functions in a memory polynomial become highly correlated for typical communication signals, leading to an ill-conditioned data matrix. Orthogonalization, often performed using Gram-Schmidt or QR decomposition, dramatically improves the numerical stability of coefficient extraction.

A measure of how sensitive a mathematical problem is to small errors in input data. In DPD, a model with poor conditioning causes coefficient estimation algorithms like Least Squares (LS) to amplify noise, resulting in wildly inaccurate or unstable predistorter weights. Orthonormal basis functions directly address this by ensuring the data matrix has a condition number close to unity, guaranteeing that the solution is robust and physically realizable in fixed-point FPGA hardware.

A specific class of orthonormal basis functions that are optimal for modeling systems with exponentially decaying memory. They are defined by a single Laguerre pole parameter that can be tuned to match the dominant time constant of the power amplifier's thermal or bias memory. Using Laguerre functions often requires far fewer coefficients than a standard memory polynomial to capture long memory effects, making them a highly efficient choice for wideband signal linearization.

A generalization of Laguerre functions that uses multiple, distinct poles to model more complex memory dynamics. While Laguerre functions capture a single dominant time constant, Kautz functions can represent systems with multiple resonances or disparate memory effects simultaneously. This makes them suitable for advanced Doherty amplifier optimization, where the main and peaking amplifiers may exhibit different thermal and trapping characteristics requiring a richer set of basis functions.

A matrix factorization method that decomposes the data matrix into an orthogonal matrix Q and an upper triangular matrix R. In the context of DPD, QRD is used to solve the Least Squares (LS) problem for coefficient extraction with superior numerical stability compared to directly inverting the normal equations. It is the standard method for implementing orthogonalization in adaptive systems, enabling robust online training algorithms that can run reliably on embedded processors.

A dimensionality reduction technique that identifies the most significant directions of variance in the basis function matrix. By projecting the data onto these principal components, PCA effectively creates a reduced set of orthonormal basis functions. This not only improves conditioning but also performs model order reduction by discarding components that contribute minimally to the output, resulting in a compact predistorter with fewer coefficients and lower computational complexity.