Truncated Volterra Series

The canonical simplification of the truncated Volterra series. It retains only the diagonal kernels where all delayed samples are identical, reducing the number of coefficients from O(K^M) to O(K×M). This captures the dominant nonlinear memory effects while being computationally feasible for FPGA implementation. The model is expressed as a sum of polynomials applied to delayed versions of the input signal.

An extension of the memory polynomial that reintroduces select cross-terms between the signal and its lagging or leading envelope samples. This captures more complex memory effects than the MP while remaining far sparser than the full Volterra series. Key cross-term types include:

• Lagging envelope terms: |x(n-l)|·x(n-m)
• Leading envelope terms: |x(n+l)|·x(n-m) These terms model bias modulation and thermal trapping effects in GaN and GaAs PAs.

A systematic complexity reduction technique that starts with a full or near-full Volterra series and eliminates insignificant kernels based on a quantitative metric. Common pruning criteria include:

• Magnitude-based pruning: Remove kernels with coefficient magnitudes below a threshold
• Orthogonal matching pursuit (OMP): Greedy selection of the most correlated basis functions
• LASSO regularization: L1-norm penalty drives small coefficients to exactly zero The result is a sparse model that retains only the physically meaningful distortion mechanisms.

The two hyperparameters that define the truncation boundary of the Volterra series:

• Nonlinear Order (K): The highest polynomial power considered. Typically K=5, 7, or 9. Odd orders dominate because even-order distortion products fall out of band in narrowband systems. Higher K captures severe gain compression and high-order intermodulation products.

• Memory Depth (M): The number of past samples included. Typically M=2 to 5 for most PAs. Longer memory depths model thermal trapping and bias circuit dynamics in GaN devices.

A numerical conditioning process that transforms the correlated polynomial basis functions of the Volterra series into an orthogonal set. Raw polynomial terms like x(n), x(n)|x(n)|², and x(n)|x(n)|⁴ are highly correlated, leading to ill-conditioned data matrices and unstable coefficient estimation. Orthogonalization techniques include:

• Gram-Schmidt procedure applied to the basis function matrix
• Principal Component Analysis (PCA) to decorrelate and rank basis vectors
• Orthonormal basis functions like Laguerre or Kautz functions This dramatically improves least squares convergence and numerical stability.

The workhorse coefficient extraction algorithm for truncated Volterra series models. Given a matrix of basis function outputs Φ and a vector of measured PA output samples y, the LS solution is:

ĉ = (ΦᴴΦ)⁻¹Φᴴy

This minimizes the sum of squared errors between the model prediction and measured data. For the truncated Volterra series, the normal equations (ΦᴴΦ) can become ill-conditioned, requiring regularization via ridge regression or QR decomposition (QRD) for robust implementation in adaptive DPD systems.