5. Advantages of Noise Shaping and Dither

We have shown in Chap. 3 that zeroth and first-order noise-shaping TDCs can be modelled by quantizers and first-order sigma-delta modulators, respectively. In this chapter, we consider the cases in which a dither signal \({\textit{dtr}}[n]\) is added to the input \(in[n]\) of a quantizer or a sigma-delta modulator followed by a moving average filter. In order to keep our analysis simple, we assume that \(\textit{dtr}[n]\) is uniformly distributed over the interval \([-\varDelta d/2, \varDelta d/2]\). Dither has the effect of making the quantization errors that are associated with the quantizer and the sigma-delta modulator more white and uniformly distributed over their intervals of definition. The moving average filters remove part of the noise associated with the dither and the quantization error. The removal of part of the power of the quantization error corresponds to an increase in the effective precision of a quantizer or a sigma-delta modulator when followed by a moving average filter. We determine analytically the precisions of these systems in terms of the maximum difference between the input and the output when this difference is bounded. In the cases where the maximum difference between the input and output is unbounded (because its distribution is Gaussian, for example), we assume that a measure of the precision of a quantizer or a sigma–delta modulator is the size of the interval \([-3\sigma , 3\sigma ]\) of the contributions of the quantization error and the dither to the output, where \(\sigma \) is the standard deviation.