An analysis of sigma-delta modulators as continuous systems

Introduces a method of analysis for sigma-delta analogue to digital converters based on replacing the nonlinear quantiser with a continuous element. Conventionally, analysis of sigma-delta modulation has been in the discrete domain, often treating the quantiser as a simple additive noise source, which has led to limited success in understanding the processes involved. However, the only truly discrete element in the circuit is the quantiser so if this could be represented accurately enough as a continuous element then a new approach to the analysis of ΣΔ ADCs may be possible. A representation of the quantiser as a hyperbolic tangent function with a sufficiently steep gradient in the crossover region is proposed. A system of ordinary differential equations is the constructed to represent first, second and third order modulators and these equations are solved numerically, by the Runge Kutta method, for linear and nonlinear signal inputs. The results from this process are compared to those produced by a conventional simulation of sigma-delta modulation working on the same inputs. The results are presented in the form of time waveforms, showing the signal before quantisation, and the signal after decimation, for a range of input signals. It is shown that, for first and second order modulation at least, these results are compatible and that a basis for further analysis of sigma-delta modulation has been achieved