Design of Asymmetrical Scalar Quantizer with Extended Huffman Coding for Gaussian Source

Abstract

In this paper we propose a novel class of asymmetrical two-level scalar quantizers with extended Huffman coding that are designed to provide the required quality of the quantized signal, measured by SQNR (Signal to Quantization Noise Ratio), and for the average bit rate to approach the source entropy as close as possible. The only constraint in the design of the novel quantizer is that the value of SQNR decreases no more than 1 dB from the optimal SQNR Lloyd-Max's quantizer value. The two-level Lloyd-Max's quantizer with zero decision threshold is a special case of our quantizer. Unlike the two-level Lloyd-Max's quantizer having the decision threshold settled in zero, the novel quantizer with the same number of quantization levels proposes that the determination of the variable decision threshold is performed in a way that it has a non-negative value, which is designed depending on which SQNR has to be achieved. The basic idea is that, unlike to the Lloyd-Max's quantizer, the asymmetry of representation levels is assumed such that to provide an unequal probability of representation levels for the symmetric Gaussian probability density function (PDF). This in turn provides the proper basis for the further implementation of a lossless compression techniques. Output levels of a quantizer can be considered as a discrete source of symbols and can be coded using fixed-length codewords. However, a more effective manner of coding is by using entropy code with variable-length codewords. The bit rate of any lossless code is always higher than the entropy, where the aim is to approach the entropy as close as possible. To achieve this, symbols with large probabilities are coded with short codewords and less-probable symbols are coded with longer codewords. Among many lossless compression techniques, such as Huffman code, arithmetic code and Golomb-Rice code, the most suitable for utilization is the extended Huffman coding technique that achieves the lowest average length of code words. The procedure of Huffman coding includes determining the optimal length of code words for a given probability of symbols. It is sometimes beneficial to additionally reduce the bit rate by blocking more than one symbol together. Extended Huffman coding is the procedure of determining the optimal length of code words for blocks of two or more symbols. For that reason, we propose a quantizer that has only two representation levels and we apply extended Huffman coding on the output levels of this quantizer. As with Lloyd-Max's quantizer, these representation levels are determined from the centroid condition. It is shown that by using the extended Huffman coding technique and the set of quantizers with variable decision thresholds, approaching of the average bit rate to the source entropy can be achieved.