How to generate a simple dither

In dithered lattice quantization, an intentional noise - dither - is being added to the source before the quantization and subtracted after the quantization. If the dither is uniform over the basic Voronoi region of the lattice, then it is known that the resulting error is independent of the source. This independence property is attractive for reducing the perceptual effects of the quantization, as well as for analytical purposes. But in practice, it may be hard to generate a uniform distribution over the Voronoi region of a good multi-dimensional lattice. We show that simpler dithering schemes are sufficient to achieve the desired independence property. Specifically, the dither can be uniform over any basic cell of the lattice - not necessarily its Voronoi cell; e.g. a dither uniform over the parallelepiped cell (induced by the generator matrix) is simple to generate. For construction-A lattices, the dither can be i.i.d., where each sample is uniform over the construction-A period interval.