A Structured Fixed-Rate Vector Quantizer Derived from a Variable-Length Scalar Quantizer
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The well-known error propagation problem inherent in any variable-length coding operation limits the usefulness of variable-length encoded scalar quantizers for transmission over noisy channels. In the absence of channel noise however, these quantizers are known to perform better than error-minimizing fixed-rate Lloyd-Max quantizers for a wide class of memoryless sources. Motivated by this observation, in this paper we develop a low complexity fixed-rate structured vector quantizer in which the structure of the codebook is derived from a variable-length scalar quantizer. Consider an n-level variable- length (e.g., Huffman coded) scalar quantizer S quantizing samples from a memoryless stationary source. Out of all possible n**m-vectors (m consecutive samples) at the output of S, the 2 **(mr) vectors with the smallest total (encoded) length are chosen as the codebook of an m-dimensional rate r bits/sample structured vector quantizer derived from S. A procedure for the design of this quantizer along with fast algorithms for implementation are developed and bounds on the performance of the scheme are developed. The structured vector quantizer can be designed and implemented even for fine (high rate) quantization at relatively large block-lengths and can achieve a rate- distortion performance superior to that of implementable LBG vector quantizers. Numerical results demonstrating the efficacy of this scheme along with comparisons against Lloyd-Max quantizers and optimal entropy-constrained quantizers, both in the absence and presence of channel noise are rendered. These results show that performance close to that of optimal entropy- constrained scalar quantizers is possible with this fixed-rate quantizer. The structured vector quantizer is also robust against channel errors and outperforms both Lloyd-Max and entropy-constrained scalar quantizers for a wide range of channel error probabilities. Extension of this idea to other types of sources proves very useful and will be described in a sequel paper.