The history of the theory and practice of quantization dates to 1948, although similar ideas had appeared in the literature as long ago as 1898. The fundamental role of quantization in modulation and analog-to-digital conversion was first recognized during the early development of pulsecode modulation systems, especially in the 1948 paper of Oliver, Pierce, and Shannon. Also in 1948, Bennett published the first high-resolution analysis of quantization and an exact analysis of quantization noise for Gaussian processes, and Shannon published the beginnings of rate distortion theory, which would provide a theory for quantization as analog-to-digital conversion and as data compression. Beginning with these three papers of fifty years ago, we trace the history of quantization from its origins through this decade, and we survey the fundamentals of the theory and many of the popular and promising techniques for quantization.

In this paper, we study construction of structured regular quantizers for overcomplete expansions in . Our goal is to design structured quantizers which allow simple reconstruction algorithms with low complexity and which have good performance in terms of accuracy. Most related work to date in quantized redundant expansions has assumed that the same uniform scalar quantizer was used on all the expansion coefficients. Several approaches have been proposed to improve the reconstruction accuracy, with some of these methods having significant complexity. Instead, we consider the joint design of the overcomplete expansion and the scalar quantizers (allowing different step sizes) in such a way as to produce an equivalent vector quantizer (EVQ) with periodic structure. The construction of a periodic quantizer is based on lattices in and the concept of geometrically scaled-similar sublattices. The periodicity makes it possible to achieve good accuracy using simple reconstruction algorithms (e.g., linear reconstruction or a small lookup table).

Abstract — This paper proposes a distributed optimization framework for wireless multihop sensor networks base on a game theoretic approach. We show that the cross-layer optimization problem can be decomposed into two subproblems corresponding to two separate layers (the physical and the application layers) of the overall system. By modelling each subproblem as a noncooperative game, we aim to solve the noncovex application-layer rateallocation and physical-layer power-allocation subproblems in a distributed manner. Further, we prove the existence, uniqueness, and stability of the Nash equilibria for both games under certain sufficient conditions. Finally, we show that by using a set of dual variables as the market prices to coordinate the physical layer supply and the application layer demand, the overall optimization process strikes a right balance between the two layers in an overall cross-layer design. I.

A novel application and generalization of sigma--delta (61) modulation has emerged in three-phase power-electronic converters. A conventional modulator with scalar signals and binary quantizer is generalized to a modulator with vector signals and a hexagonal quantizer. Indeed, power-electronic switching states may be thought of as determining the quantizer outputs. The output spectrum is a key performance measure for both communications and power electronics. This paper analytically derives the output spectrum of the hexagonal modulator with a constant input using ergodic theory and Fourier series on the hexagon. The switching rate of the modulator is important for power-electronic design and formulas for the average switching rate are derived for constant and slowly varying sinusoidal inputs. Index Terms---Ergodic, power electronics, quantization, sigma--delta (61) modulation, spectral analysis.

A major focus of recent RF transceiver IC designs has been to increase both the integration and adaptability to multiple RF communication standards. Performing channel selection on chip at baseband allows the use of high-integration receiver architectures, and enhances programmability to different channel bandwidths and dynamic range requirements of multiple RF standards. A wideband, high-dynamic range sigma-delta modulator can be used to digitize both the desired signal and potentially stronger adjacent-channel interferers. In the digital domain, the decimation filter following the ADC can be easily made programmable. A 4th-order sigma-delta ADC which is capable of adapting to GSM (cellular) and DECT (cordless) communication standards is described. The ADC achieves 14 bits of resolution at 128x oversampling ratio (200kS/s Nyquist rate) for GSM, and 12 bits of i resolution at 32x oversampling ratio (1.4MS/s Nyquist rate) for DECT. Power reduction strategies are developed at both the sigma-delta architecture and circuit design levels. The experimental prototype, fabricated in a 0.35μm CMOS process, dissipates 70mW from a 3.3V supply.

Achievable distortion bounds are derived for the cascade of structured families of binary linear channel codes and binary lattice vector quantizers. It is known that for the cascade of asymptotically good channel codes and asymptotically good vector quantizers the end-to-end distortion decays to zero exponentially fast as a function of the overall transmission rate, and is achieved by choosing a channel code rate that is independent of the overall transmission rate. We show that for certain families of practical channel codes and binary lattice vector quantizers, the overall distortion can be made to decay to zero exponentially fast as a function of the square root of transmission rate. This is achieved by carefully choosing a channel code rate that decays to zero as the transmission rate grows. Explicit channel code rate schedules are obtained for several well-known families of channel codes.

Accuracy of oversampled analog-to-digital (A/D) conversion, the dependence of accuracy on the sampling interval and on the bit rate are characteristics fundamental to A/D conversion but not completely understood. These characteristics are studied in this paper for oversampled A/D conversion of band-limited signals in ( ). We show that the digital sequence obtained in the process of oversampled A/D conversion describes the corresponding analog signal with an error which tends to zero as in energy, provided that the quantization threshold crossings of the signal constitute a sequence of stable sampling in the respective space of band-limited functions. Further, we show that the sequence of quantized samples can be represented in a manner which requires only a logarithmic increase in the bit rate with the sampling frequency, = ( log ), and hence that the error of oversampled A/D conversion actually exhibits an exponential decay in the bit rate as the sampling interval tends to zero.

1) m-sequences: We use the notation fag to denote the sequence fa1;a2; 111g, and fakg to denote the shifted sequence fak+1;ak+2; 111g. We consider a binary m-sequence fag with period P. Some properties of m-sequences which will be used in our discussion in Section IV are [10] • A1: One period of an m-sequence contains exactly (P + 1)=2 ones and (P 0 1)=2 zeros. • A2: Shift and add: For any k (k 6 = 0modP), the sum of the m-sequence fag and its k shift fakg is another shift of the same m-sequence; i.e., (an 8 an+k) =an+l 8n where 8 denotes modulo two addition, and l is some other shift of the same sequence. • A3: Decimation: Consider the sequence fcg defined by cn = aJn; 8n. fcg is called the decimation by J of the sequence fag. The period of fcg is P=gcd (J; P). All m-sequences of period P can be constructed by decimations of fag. • A4: Periodic autocorrelation function: The periodic autocorrelation function of an m-sequence fag, is defined as and is given by Paa aa(l) aa = Paa aa(l) aa = P n=1

In this paper, we derive bounds on the structural similarity (SSIM) index as a function of quantization rate for fixed-rate uniform quantization of image discrete cosine transform (DCT) coefficients under the high-rate assumption. The space domain SSIM index is first expressed in terms of the DCT coefficients of the space domain vectors. The transform domain SSIM index is then used to derive bounds on the average SSIM index as a function of quantization rate for uniform, Gaussian, and Laplacian sources. As an illustrative example, uniform quantization of the DCT coefficients of natural images is considered. We show that the SSIM index between the reference and quantized images fall within the bounds for a large set of natural images. Further, we show using a simple example that the proposed bounds could be very useful for rate allocation problems in practical image and video coding applications.