A Study of Subtractive Digital Dither in Single-Stage and Multi-Stage Quantizers

The effect of subtractive digital dither on single-stage and multi-stage quantizers is examined. In the single stage case, it is proved that the conditional density of the quantization error becomes asymptotically independent of the input and uniformly distributed as the number of digital dither levels increases. For a fixed number of levels, the conditional expectation of a class of piecewise constant functions is shown to be independent of the input. A Fourier series representation of quantization error densities is used to construct a linear periodically time varying model of multistage quantization and dithering systems. For a homogeneous multi-stage quantization system, it is shown that when the least common multiple of the numbers of digital dither levels at successive stages is large, the statistical dependence of the quantization noise on the input disappears almost entirely. Computer simulations confirm this analysis.