Closed-form solution of granular quantization error for a class of sigma-delta modulators

This paper presents a closed-form solution for the granular quantization error (i.e. the quantization error under the no-overload condition) for a class of sigma-delta modulators. This class includes the conventional multi-loop sigma-delta modulators as a special case. The proposed closed-form solution is based on a nonlinear matrix difference equation relating the quantization error to its past values and the present and past values of the input signal via the arithmetic operation of the quantizer. The closed-form solution is established in terms of a theorem accompanied by a formal proof. It is then applied to the determination of the exact quantization error for a triple-loop sigma-delta modulator