Using Geometrical Properties for Fast Indexation of Gaussian Vector Quantizers
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Research Article Using Geometrical Properties for Fast Indexation of Gaussian Vector Quantizers E. A. Vassilieva, D. Krob, and J. M. Steyaert Laboratoire d’Informatique de l’Ecole Polytechnique (LIX), Ecole Polytechnique, 91128 Palaiseau Cedex, France Received 2 November 2005; Revised 26 August 2006; Accepted 10 September 2006 Recommended by Satya Dharanipragada Vector quantization is a classical method used in mobile communications. Each sequence of d samples of the discretized vocal signal is associated to the closest d-dimensional codevector of a given set called codebook. Only the binary indices of these codevectors (the codewords) are transmitted over the channel. Since channels are generally noisy, the codewords received are often slightly different from the codewords sent. In order to minimize the distortion of the original signal due to this noisy transmission, codevectors indexed by one-bit different codewords should have a small mutual Euclidean distance. This paper is devoted to this problem of index assignment of binary codewords to the codevectors. When the vector quantizer has a Gaussian structure, we show that a fast index assignment algorithm based on simple geometrical and combinatorial considerations can improve the SNR at the receiver by 5dB with respect to a purely random assignment. We also show that in the Gaussian case this algorithm outperforms the classical combinatorial approach in the field. Copyright © 2007 E. A. Vassilieva et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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INTRODUCTION
Taking into account the constraints of the transmission channel between the base transceiver stations (BTS) and mobile stations (MS), voice is coded in mobile networks with the help of techniques allowing to minimize the quantity of information required for its good reconstitution. Among these techniques one finds vector quantization. This method consists of replacing the vector y from Rd , obtained by finite discretization of the input vocal signal, by the element ci , taken from a set C = {c0 , c1 , . . . , cN −1 } of vectors of reference, which is the closest to y. The set C is called a codebook and its elements the codevectors. Instead of transmitting the initial discretization y, one transmits a string of 0’s and 1’s which is the binary codeword b(ci ) associated with the codevector ci of the codebook C which is the closest to y. Because of some interfering noises on the transmission channel, the string s actually received can be different from b(ci ). The output signal is then c j such that b(c j ) = s (see Figure 1). In what follows, the mapping b that associates with each codevector ci a binary word b(ci ) representing a nonnegative integer will be called the indexation (or the index assignment) of the codebook. We will also refer to b(ci ) as the index associated with the codevector ci .
From the very start, vector quantizat
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