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 over-sampled 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 orderSigma-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.