[IEEE Trans. on Information Theory, February 2006, pp. 640-661]
Quantizers with Uniform Decoders and Channel-Optimized Encoders
Benjamin Farber and Kenneth Zeger
Abstract
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
of the CNC can be strictly smaller than that of the NBC,
we also demonstrate that the NBC
is sub-optimal 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.