Methods for Model Complexity Reduction for the Nonlinear Calibration of Amplifiers Using Volterra Kernels

Abstract

Volterra models allow modeling nonlinear dynamical systems, even though they require the estimation of a large number of parameters and have, consequently, potentially large computational costs. The pruning of Volterra models is thus of fundamental importance to reduce the computational costs of nonlinear calibration, and improve stability and speed, while preserving accuracy. Several techniques (LASSO, DOMP and OBS) and their variants (WLASSO and OBD) are compared in this paper for the experimental calibration of an IF amplifier. The results show that Volterra models can be simplified, yielding models that are 4–5 times sparser, with a limited impact on accuracy. About 6 dB of improved Error Vector Magnitude (EVM) is obtained, improving the dynamic range of the amplifiers. The Symbol Error Rate (SER) is greatly reduced by calibration at a large input power, and pruning reduces the model complexity without hindering SER. Hence, pruning allows improving the dynamic range of the amplifier, with almost an order of magnitude reduction in model complexity. We propose the OBS technique, used in the neural network field, in conjunction with the better known DOMP technique, to prune the model with the best accuracy. The simulations show, in fact, that the OBS and DOMP techniques outperform the others, and OBD, LASSO and WLASSO are, in turn, less efficient. A methodology for pruning in the complex domain is described, based on the Frisch–Waugh–Lovell (FWL) theorem, to separate the linear and nonlinear sections of the model. This is essential because linear models are used for equalization and cannot be pruned to preserve model generality vis-a-vis channel variations, whereas nonlinear models must be pruned as much as possible to minimize the computational overhead. This methodology can be extended to models other than the Volterra one, as the only conditions we impose on the nonlinear model are that it is feedforward and linear in the parameters.

High-Accuracy Low-Cost Generalized Complex Pruned Volterra Models for Nonlinear Calibration

“…On the other hand, the Optimal Brain Surgeon (OBS) algorithm [18], [21], [22] starts from the full model and removes one variable at the time, selecting the one that influences the residual error the less. The impact of each variable on the error depends on the curvature of the error curve and the value of the parameter, and variables which are either small or have small curvature are removed first.…”

“…The two techniques operate in opposite directions: the OMP from the simplest to the most complex model, and the OBS from the complete to the simplest model. The best technique is the one that selects the minimum-error pruned model for a given complexity: in general, none of the two techniques outperforms the other for all desired pruning levels, and combining the two allows finding a better approximation of the complexity-accuracy trade-off [18].…”