Eurospeech 2001 - Scandinavia Wideband LSF Quantization by Generalized Voronoi Codes St´ ephane Ragot, Hassan Lahdili, and Roch Lefebvre Department of Electrical and Computer Engineering, University of Sherbrooke Sherbrooke, QC, J1K 2R1, Canada ragot,lahdili,lefebvre @gel.usherb.ca Abstract Presented a method for quantizing the wideband line spectrum frequencies (LSF) with a specific class of near-ellipsoidal lat- tice codes referred to as “generalized Voronoi codes”. Opti- mization procedures are described with respect to a weighted mean-square error (WMSE). The lattices , or are applied to quantize the LSF with no frequency splitting. Re- sults indicate that near-ellipsoidal lattice quantization allows to develop efficient one-stage algebraic wideband LSF quantiza- tion at competitive bit rates. 1. Introduction We address the problem of designing an efficient memoryless LSF quantization scheme for wideband linear predictive cod- ing. In practice this problem is typically “solved” by em- ploying constrained stochastic vector quantization (VQ) – e.g. split/multistage VQ. In this paper we propose an alternative technique using near-ellipsoidal lattice VQ. The proposed cod- ing system generalizes some unpublished results on narrowband LSF quantization [1] and it is similar in some respect to a nar- rowband technique described in [2]. The application of lattice VQ to wideband LSF quantization is mainly motivated by: the need in spectrum coding to optimize performance while minimizing both computational complexity and storage requirements [3, 4], the intrinsic scalability of lattice VQ which allows to trade bit rate and spectral quantization quality if variable bit rate allocation is allowed [4] – this is even more im- portant if a linear predictive coder is to be applied not only to speech but also to music signals, the possibility to share lattice quantization routines and tables with algebraic transform coding. A two-stage VQ-near-spherical lattice VQ configuration was proposed in [3, 4] with these motivations in mind. Although this technique is competitive in terms of performance/complexity trade-off, it resulted in quite inefficient codebooks because the LSF source is non-uniform and has a specific geometry due to LSF ordering. To understand its inherent structural constraints the two-stage coding scheme of [3] is sketched in Figure 1 (a) as fixed ball packing (i.e. duplicating the fixed near-spherical lattice stage to represent the source distribution). In particular the structural limitations are clear in the neighborhood of the LSF stability region and in sparse regions which are not ad- equately covered and where there may be “holes” in between the balls. These limitations could be mitigated by employing a This work was supported by the NSERC and VoiceAge Corp. LSF i+1 LSF i LSF i+1 LSF i (a) VQ-spherical lattice VQ (b) mixture of ellipoidal lattice VQ Figure 1: Ball packing as in [3, 4] vs packing of ellipsoids. pre-processing of the LSF source such as a temporal prediction prior to quantization, or by adding some flexibility to the near- spherical lattice VQ stage – e.g. a scale factor dependent on the codevectors selected in the VQ stage. In this paper we in- vestigate another alternative as illustrated in Figure 1 (b) : using near-ellipsoidal lattice VQ instead of fixed near-spherical lattice VQ. The paper is organized as follows. First we define clearly the proposed coding principle in Section 2. This principle is actually very similar to that of [2], except that we apply a dif- ferent system design and we use generalized Voronoi coding. The idea behind the algebraic structures used herein was origi- nally developed in [1] based on the lattices , and to quantize the narrowband LSF with one near-ellipsoidal lat- tice code ; for the sake of completeness we describe the defini- tion of generalized Voronoi VQ as well as the required indexing algorithms in Section 3, and present a fast (suboptimal) nearest- neighbor search algorithm taken from [5]. Source optimization for the proposed quantization scheme is considered in Section 4. This problem is only studied in the framework of source cod- ing based on parametric PDF estimation (as in [2, 6]). Results are presented in Section 5, before concluding in Section 6. 2. Coding Principle The proposed coding system is depicted in Figure 2. An ar- bitrary LSF vector is quantized by selecting the best recon- struction among candidates with re- spect to WMSE. In the general case, the candidates are found by a sequence of operations sketched in Figure 2 (a). This can be summarized as follows: (1) The are appropriate points in the LSF space. The trans- forms are unitary rotation matrices. The implement near- ellipsoidal lattice quantization. The scalar gains are