Quantizers with uniform decoders and channel-optimized encoders

Scalar quantizers with uniform decoders and channel-optimized encoders are studied for a uniform source on [0,1] and binary symmetric channels. Two families of affine index assignments are considered: the complemented natural code (CNC), introduced here, and the natural binary code (NBC). It is shown that the NBC never induces empty cells in the quantizer encoder, whereas the CNC can. Nevertheless, we show that the asymptotic distributions of quantizer encoder cells for the NBC and the CNC are equal and are uniform over a proper subset of the source's support region. Empty cells act as a form of implicit channel coding. An effective channel code rate associated with a quantizer designed for a noisy channel is defined and computed for the codes studied. By explicitly showing that the mean-squared error (MSE) of the CNC can be strictly smaller than that of the NBC, we also demonstrate that the NBC is suboptimal for a large range of transmission rates and bit error probabilities. This contrasts with the known optimality of the NBC when either both the encoder and decoder are not channel optimized, or when only the decoder is channel optimized.