Nonlinear function generation using oversampled Sigma-Delta-modulators

In this paper we present a technique for nonlinear function generation which uses a novel sparse look-up table approach. This novel approach can be derived from the simple look-up table concept and utilizes the interpolating properties of oversampled Sigma-Delta-modulators. We show that the approximation process with oversampled Sigma-Delta-modulators can be generally understood as a weighted sum of piecewise-continuous polynomial B-splines. The weights are digitally stored in a sparse look-up table memory which can be used for the nonlinear function mapping. The shape of the polynomial B-splines depends on the order of the loop-filter which is used in the Sigma-Delta-modulator topology. Detailed investigations of approximation behaviour show that a spectrum of linear, square, and cubic piecewise-continuous interpolation can be achieved if we use up to 3rd order Sigma-Delta-modulators. With the increased order of modulator topology the approximation quality is also improved under the assumption that we use the same weight set for function mapping. The fact that polynomial B-splines delivers only nonzero values within a limited range of input arguments makes this approximation method much less sensitive to local errors than classical approximation methods like Lagrange or Newton interpolation.