Multidimensional Lattice Quantization Of Information Sources

Abstract

In recent years vector quantization has emerged as a very efficient means for quantizing information sources. Traditionally, vector quantizers are based on a large codebook, consisting of an ensemble of representative waveforms of the signal to be quantized, which is stored at both the transmitter and the receiver 111. The process of quantization then involves a search through the codebook to find the waveform which best matches the signal to be quantized, and to transmit the corresponding index to the receiver. The required codebook is usually designed by means of a clustering algorithm, often referred to as the LBG-algorithm [2]. The main drawback of this approach is that both the memory requirements for the codebook and the search time for the best matching vector grow exponentially with the vector length N.These shortcomings can be eliminated by assigning the vectors of the quantizer to lie on a lattice in N-dimensional. The salient feature of this quantizer, termed a lattice quantizer, is that it is not necessary to store the codebook in memory but only a common set of algebraic rules is required both at the transmitter and receiver. Furthermore, it is not required to perform an exhaustive search to find the best codeword, but fast algorithms based on algebraic rules are available, which exploit the inherent structure of the lattices.The theoretical justification for lattice quantizers is given by alphabet-constrained rate-distortion theory [S][9], which indicates that performance close to the optimum promised by Shannon's rate-distortion theory [ 101 is achievable even if a finite reproduction alphabet, consisting of as little as 4 reproduction values, is used.An algorithm has been developed for the efficient calculation of alphabet-constrained reconstruction values [ 11). This algorithm can be applied to any source with arbitrary probability density function, and has used to calculate reconstruction values for the Gaussian, Uniform and Laplacian sources. By making use of these reconstruction values, an algorithm has been established for the design of lattice vector quantizers, based on the D , D 8 , E R and A lattices in 4,8 and 16 dimensions. The lattices make use of N-dimensional signal constellations which are constructed by concatenating constituent 2D-constellations [ 121. The principle of lattice vector quantization will be demonstrated for vector lengths N = 4,8 and 16, for the Gaussian, Laplacian and Uniform information source models respectively. It will be shown that excellent performance can be achieved at rates of 1 -3 bits/sample. Due to the inherent structure of the lattices considered, fast encoding of the information vectors is possible by means of the Viterbi algorithm [13].Furthermore, it will be demonstrated how lattice quantizers can be combined with binary trellis codes to form high performance sliding-window quantizers, which essentially amounts to application of the idea of multidimensional trellis-coded modulation proposed by Wei [ 141. Since these multidimensional trellis-coded qua...