Introduction

An important goal in source coding is to design quantizers that have both reasonable implementation complexity and performance close to the distortion-rate function of a source. Scalar quantizers have low implementation complexity, but their distortion performance is usually much worse than the distortion-rate function. Conversely, fixed blocklength constructive techniques for structured vector quantizers (VQ), such as the generalized Lloyd algorithm (GLA) [1] perform well, but their creation, storage, and encoding complexities each grow exponentially in both dimension and rate (also see [2,3]).

A number of complexity constrained VQs have been proposed in an attempt to improve upon scalar quantization while retaining low implementation complexity (e.g., see [4]). This paper makes use of two of these methods, lattice quantization and shape-gain quantization, together with wrapped spherical codes for channel coding [5]. Our proposed fixed rate quantizer does not have exponential complexity; in fact, the operating complexity grows linearly with the rate.

The quantizer presented in this paper is designed for a memoryless Gaussian source. One reason for studying the memoryless Gaussian source is that it naturally arises in numerous applications. For example, the prediction error signal in a DPCM (differential pulse code modulation) coder for moving pictures is well-modeled as Gaussian [6]. Also, discrete Fourier transform coefficients and holographic data can often be considered to be the output of a Gaussian source [7] (although some other aspects of images and speech are better modeled as Laplacian distributions [8,9]). Furthermore, a known filtering technique tends to make memoryless sources appear Gaussian, which makes the system insensitive to errors in modeling the input [10]. The Gaussian source is also easier mathematically to analyze compared to some other sources, because its distortion-rate function is known explicitly. In fact, the Gaussian is known to be the most difficult source to compress, in a rate vs. distortion sense [11]. Finally, the Gaussian source has provided a historical benchmark for measuring how close a practical quantizer can come to the theoretical performances predicted by Shannon [12,13,14,15,16,17,18,19,20,21,22,3,23,10,24,25,7,6,2,26,27,28].

Section II gives properties of Gaussian source coding and Section III describes the construction and performance analysis of the proposed wrapped shape-gain vector quantizer. It is shown how a fixed rate lattice quantizer can be transformed into a shape-gain quantizer. An asymptotic analysis gives the optimal high resolution tradeoff between allocating rate to the gain and shape quantizers, and the indexing problem is discussed. Section IV describes a specific implementation of the proposed Gaussian coder using the $24$-dimensional Leech lattice for the shape codebook. The performance is compared against other known quantizers and the computational complexity and confidence intervals are determined. For a memoryless Gaussian source, this shape-gain quantizer performs better than other quantizers in the literature at rates of three bits per sample or higher. Some extensions are given in Section V.