Supervisor: David H. Staelin.

The problem of scalar quantization of certain memoryless sources with entropy coding is considered. The work is divided into two parts.

In the first part, the problem of scalar quantization is treated, under the assumption that ideal entropy coding can be performed on the output of the quantizer. Special attention is given to Gaussian, Laplacian, and gamma memoryless sources, as these are often used to model various signals that arise in speech- and image-coding systems. A procedure for finding locally optimal quantizers is provided, and results are provided for $\nu$-power distortion measures, where $\nu = 1,2,...,6.$ Evidence from a large-scale random search is provided to support the hypothesis that the locally-optimal quantizers are, in most of the cases considered, globally optimal. Based on the same evidence, it is hypothesized that in some cases globally optimal quantizers are asymmetric, despite the symmetry of the source distributions and lack of alphabet constraint. Uniform quantization, which is well-known to be optimum in the limit of vanishingly small distortion, is shown to be nearly optimal for all three sources, over a range of entropies and values of $\nu$. However, it is also shown that locally-optimal quantizers are occasionally substantially nonuniform. It is further demonstrated that the higher the order of the distortion measure, the better entropy-constrained quantization appears, relative to alphabet-constrained quantization. The first part of this thesis also discredits the widely held view that uniform quantizers are exactly optimal with respect to mean-square error in the case of Laplacian sources, and conjectures about how this mistaken idea might have arisen.

The second part of the thesis justifies the assumption of the availability of optimal entropy coding by describing a suitable, near-optimal code. In particular, an efficient arithmetic code well-suited to the quantization problem is devised. It is shown that the performance of this code is very nearly optimal for the sources of interest, over a useful range of entropies. Although the code is based in principle on previously reported work, many details not provided elsewhere are worked out and presented, and the derivation of the code presented is original. Also original is an optimal procedure for rounding-off source letter probabilities to admissible, finite-precision values.