Design of Forward Adaptive Piecewise Uniform Scalar Quantizer with Optimized Reproduction Level Distribution per Segments

摘要

The primary goal of the quantizer design is to determine the reproduction levels and the partition regions or cells such as to provide the minimum possible distortion for a fixed number of quantization levels N, or equivalently a fixed resolution R=log_2N [1, 3]. A quantizer support region can be divided into a variety of ways. If a quantizer support region consists of several segments, each of which contains several quantization cells and reproduction levels corresponding to a uniform quantizer, the quantizer is a piecewise uniform one [1, 2]. The piecewise uniform scalar quantizers (PUSQ) are widely used in practice due to their simple encoding procedure, which, in contrast to that of the nonuniform quantizer models, does not require the full search of the quantizer code book [1, 2]. The prevailing international standard for digital telephony, known as G.711 standard, defines a symmetric PUSQ by 8 bits of resolution and L=8 positive segments increased in length by a factor of 2 for each successive segments having 16 cells [1]. Particularly, the G.711 quantizers based on a piecewise uniform approximation to the A-law and μ-law compressor characteristics [1] divide the support region into a 2L=16 unequal segments, each of which has equal number of cells. Accordingly, the G.711 quantizer can be considered as a set of 2L uniform quantizers having an equal number of quantization levels, but the unequal support regions. Namely, each uniform quantizer, from the set of 2L quantizers composing the PUSQ, is designated to quantize the signals belonging to the corresponding segments so that, when the sample of the input signal to be quantized is within the i-th. segment, the corresponding z'-th uniform quantizer is then used. Along with the support region partition, according to the mentioned piecewise linear compressor characteristics, there are some novel propositions of the support region partition, i.e. of the piecewise uniform quantizer design. For instance, in the reference [4], the robustness conditions of the PUSQ based on a piecewise uniform approximation to the optimal compressor law are analysed. Additionally, in the reference [5], there is a proposition of the piecewise uniform vector quantizer that considers an unequal number of cells within the segments. However, in the reference [5], in the conclusion of the paper, the authors have highlighted that the main drawback of their method is that they do not have a manner for deciding how to determine the segments into which to divide the support region of a piecewise uniform quantizer for an arbitrary signal distribution. This comment has motivated us to propose an intuitively obtained solution, i.e. to research the performance of the PUSQ that defines the equidistant support region partion and the optimized distribution of the cells (or reproduction levels) within such defined segments. What is actually proposed in this paper is a PUSQ composed of 2L uniform quantizers having equal support regions, but possibly different number of quantization cells. Moreover, in order to provide an optimal manner of reproduction level distribution per segments, the granular distortion optimization is proposed, while such a constrained optimization problem is solved using the method of Lagrange multipliers [1].