Coefficient Sparsity

Full behavioral models like the Volterra series or Generalized Memory Polynomial (GMP) contain many redundant coefficients that contribute negligibly to modeling accuracy. This redundancy arises because:

• Higher-order nonlinear terms often have diminishing impact on output
• Cross-terms between widely separated time delays are typically weak
• Many basis functions are highly correlated in practice

Sparsity-aware modeling exploits this by identifying and retaining only the most significant coefficients, often achieving 70-90% reduction with minimal NMSE degradation.

Coefficient pruning applies a magnitude-based threshold to eliminate near-zero terms. Common strategies include:

• Hard thresholding: Remove all coefficients below a fixed absolute value
• Relative thresholding: Eliminate coefficients below a percentage of the maximum coefficient magnitude
• Statistical pruning: Retain only coefficients exceeding a confidence interval based on estimation variance

The optimal threshold balances model complexity against linearization performance, typically targeting less than 0.5 dB ACLR degradation after pruning.

Dedicated algorithms promote sparsity during coefficient extraction rather than as a post-processing step:

• LASSO (L1-regularized Least Squares): Adds an L1 penalty that drives small coefficients exactly to zero
• OMP (Orthogonal Matching Pursuit): Greedily selects the most correlated basis functions one at a time
• Elastic Net: Combines L1 and L2 penalties for both sparsity and stability

These methods produce inherently sparse models that require fewer multiply-accumulate operations in FPGA or ASIC implementations.

Sparse models directly translate to hardware efficiency gains:

• Reduced multiplier count: Each zero coefficient eliminates one complex multiplication per sample
• Lower memory bandwidth: Fewer coefficients to store and fetch from lookup tables or registers
• Simplified routing: Less interconnect complexity in FPGA fabric

For a 5G NR 100 MHz signal with a GMP model, sparsity can reduce real-time operations from thousands to hundreds per sample, enabling DPD on cost-constrained small cells.

Sparse models often exhibit superior generalization compared to dense counterparts:

• Reduced overfitting: Fewer free parameters mean less capacity to memorize training noise
• Improved numerical conditioning: Eliminating correlated basis functions lowers the condition number of the regression matrix
• Better extrapolation: Sparse models capture only the essential physics, avoiding spurious patterns

This is particularly valuable when the training signal statistics differ from operational waveforms, as occurs in multi-standard transmitters.

The distribution of significant coefficients reveals physical amplifier characteristics:

• Diagonal dominance: Strong memory polynomial terms indicate classic memory effects
• Clustered cross-terms: Localized clusters suggest specific trapping or thermal time constants
• Sparse higher-order terms: Indicates soft nonlinearity with limited high-order distortion generation

Analyzing sparsity patterns aids in model structure selection, guiding engineers toward the most appropriate reduced-complexity model architecture for a given PA technology.

The algorithmic process of identifying and removing near-zero coefficients from a behavioral model to reduce computational complexity. Common approaches include:

• Magnitude-based pruning: Eliminating coefficients below a threshold
• LASSO regularization: Driving coefficients to exactly zero during estimation
• Iterative pruning: Removing coefficients and retraining to recover fidelity Pruning a memory polynomial model can reduce the number of active terms by 60-80% with less than 0.5 dB degradation in NMSE.

Least Absolute Shrinkage and Selection Operator adds an L1-norm penalty to the least squares cost function during coefficient estimation. This forces many coefficients to exactly zero, producing a sparse solution directly. Key properties:

• Performs simultaneous coefficient estimation and variable selection
• The regularization parameter λ controls the sparsity level
• Particularly effective for generalized memory polynomial models with many cross-terms LASSO is preferred over ridge regression when model interpretability and hardware implementation efficiency are priorities.

The fundamental trade-off in behavioral modeling where increasing the number of coefficients improves accuracy but raises implementation cost. Sparse models exploit the fact that many Volterra or memory polynomial terms contribute negligibly to output prediction. Metrics for this trade-off include:

• Normalized Mean Square Error (NMSE) for in-band fidelity
• Adjacent Channel Error Power Ratio (ACEPR) for out-of-band accuracy
• Floating-point operations per sample for computational cost Optimal sparsity is found at the knee point where additional coefficients yield diminishing fidelity returns.

Beyond removing individual coefficients, structured sparsity enforces patterns that map efficiently to hardware. Examples include:

• Delay-line pruning: Removing entire tap delays that contribute minimally
• Order-based pruning: Eliminating all terms above a certain nonlinearity order
• Block sparsity: Grouping coefficients by physical significance and pruning entire blocks Structured sparsity is critical for FPGA and ASIC implementations where irregular memory access patterns from random sparsity negate the benefits of coefficient reduction.

The process of determining the optimal sparsity level by evaluating model performance on held-out validation data not used during coefficient estimation. This prevents overfitting where a sparse model memorizes training data noise rather than learning true amplifier dynamics. The standard approach:

• Partition measured PA data into training, validation, and test sets
• Train models at multiple sparsity levels
• Select the sparsity that minimizes validation error
• Report final performance on the untouched test set This ensures the pruned model generalizes to new signals and operating conditions.

Iterative algorithms that build sparse models by selecting the most correlated basis functions one at a time. Key methods include:

• Orthogonal Matching Pursuit (OMP): Selects the basis vector most correlated with the current residual, then orthogonalizes
• Compressive Sampling Matching Pursuit (CoSaMP): Selects multiple candidates per iteration with backtracking These are alternatives to LASSO when the number of candidate terms is extremely large, such as in full Volterra series models where exhaustive search is computationally prohibitive.