On Lattice Quantization Noise

W e present several results regarding the propert ies of a random vector, uniformly distributed over a lattice cell. This random vector is the quantization noise of a lattice quantizer at high resolution, or the noise of a dithered lattice quantizer at all distortion levels. W e find that for the optimal lattice quantizers this noise is wide-sense-stat ionary and white. Any desirable noise spectra may be realized by an appropriate linear transformation (“shaping”) of a lattice quantizer. As the dimension increases, the normalized second.moment of the optimal lattice quantizer goes to 1/2xe, and consequent ly the quantization noise approaches a white Gaussian process in the divergence sense. In entropycoded dithered quantization, which can be modeled accurately as passing the source through an additive noise channel, this limit behavior implies that for large lattice dimension both the error and the bit rate approach the error and the information rate of an Additive White Gaussian Noise (AWGN) channel.