In this semitutorial paper, a comprehensive survey of closest point search methods for lattices without a regular structure is presented. The existing search strategies are described in a unified framework, and differences between them are elucidated. An efficient closest point search algorithm, based on the Schnorr-Euchner (1995) variation of the Pohst (1981) method, is implemented. Given an arbitrary point x /spl isin/ /spl Ropf//sup m/ and a generator matrix for a lattice /spl Lambda/, the algorithm computes the point of /spl Lambda/ that is closest to x. The algorithm is shown to be substantially faster than other known methods, by means of a theoretical comparison with the Kannan (1983, 1987) algorithm and an experimental comparison with the Pohst (1981) algorithm and its variants, such as the Viterbo-Boutros (see ibid. vol.45, p.1639-42, 1999) decoder. Modifications of the algorithm are developed to solve a number of related search problems for lattices, such as finding a shortest vector, determining the kissing number, computing the Voronoi (1908)-relevant vectors, and finding a Korkine-Zolotareff (1873) reduced basis.

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Closest point search in lattices

Quantization is a process that maps a continous or discrete set of values into approximations that belong to a smaller set. Quantization is a lossy: some information about the original data is lost in the process. The key to a successful quantization is therefore the selection of an error criterion – such as entropy and signal-to-noise ratio – and the development of optimal quantizers for this criterion.

Quantization

Speech and audio compression has advanced rapidly in recent years spurred on by cost-effective digital technology and diverse commercial applications. Recent activity in speech compression is dominated by research and development of a family of techniques commonly described as code-excited linear prediction (CELP) coding. These algorithms exploit models of speech production and auditory perception and offer a quality versus bit rate tradeoff that significantly exceeds most prior compression techniques for rates in the range of 4 to 16 kb/s. Techniques have also been emerging in recent years that offer enhanced quality in the neighborhood of 2.4 kb/s over traditional vocoder methods. Wideband audio compression is generally aimed at a quality that is nearly indistinguishable from consumer compact-disc audio. Subband and transform coding methods combined with sophisticated perceptual coding techniques dominate in this arena with nearly transparent quality achieved at bit rates in the neighborhood of 128 kb/s per channel. >

Advances in speech and audio compression

Image subband and discrete cosine transform coefficients are modeled for efficient quantization and noiseless coding. Quantizers and codes are selected based on Laplacian, fixed generalized Gaussian, and adaptive generalized Gaussian models. The quantizers and codes based on the adaptive generalized Gaussian models are always superior in mean-squared error distortion performance but, generally, by no more than 0.08 bit/pixel, compared with the much simpler Laplacian model-based quantizers and noiseless codes. This provides strong motivation for the selection of pyramid codes for transform and subband image coding. >

On the modeling of DCT and subband image data for compression

We address the problem of frequency-selective channel estimation and symbol detection using superimposed training. The superimposed training consists of the sum of a known sequence and a data-dependent sequence that is unknown to the receiver. The data-dependent sequence cancels the effects of the unknown data on channel estimation. The performance of the proposed approach is shown to significantly outperform existing methods based on superimposed training (ST).

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Channel estimation and symbol detection for block transmission using data-dependent superimposed training

Lattice vector quantizer design procedures for nonuniform sources are presented. The procedures yield lattice vector quantizers with excellent performance and retaining the structure required for fast quantization. Analytical methods for truncating and scaling lattices to be used in vector quantizations are given, and their utility is demonstrated for independent and identically distributed (i.i.d.) Gaussian and Laplacian sources. An analytical technique for piecewise linear multidimensional compandor designs is evaluated for i.i.d. Gaussian and Laplacian sources by comparing its performance to that of the other vector quantizers. >

D.G. Jeong

Papers

Uniform and piecewise uniform lattice vector quantization for memoryless Gaussian and Laplacian sources