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29
Quantization
 IEEE TRANS. INFORM. THEORY
, 1998
"... 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 analogtodigital conversion was first recognized during the early development of pulsecode modula ..."
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Cited by 665 (11 self)
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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 analogtodigital 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 highresolution 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 analogtodigital 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.
Design of a lowlightlevel image sensor with an onchip sigmadelta analogtodigital conversion
 in CCDs and Outical Sensors 111, Proc. SPIE
"... The design of a lowlightlevel CMOS activepixelsensor (APS) with onchip, semiparallel analogtodigital (A/D) conversion is presented. The imager consists of a 128x128 array of active pixels at a 50 im pitch. Each column of pixels shares a 10bit A/D converter based on firstorder oversampled s ..."
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Cited by 18 (13 self)
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The design of a lowlightlevel CMOS activepixelsensor (APS) with onchip, semiparallel analogtodigital (A/D) conversion is presented. The imager consists of a 128x128 array of active pixels at a 50 im pitch. Each column of pixels shares a 10bit A/D converter based on firstorder oversampled sigmadelta (>z) modulation. The 10bit outputs of each converter are multiplexed and read out through a single set of outputs. A semiparallel architecture is chosen to achieve 30 frames/second operation even at low light levels. The sensor is designed for less than 10 e rms noise performance. A 28x28 activepixelsensor (APS) with 40x40p.tm2 pixels as well as individual elements of the sigmadelta modulator have been fabricated and tested using MOSIS * 2 m CMOS technology. 1.
A lowpower, lownoise, ultrawidedynamicrange CMOS imager with PixelParallel A/D conversion
 IEEE J. Solid State Circu. 2001
"... Abstract—A CMOS image sensor with pixelparallel analogtodigital (A/D) conversion fabricated with different array sizes and photodiode types in a threemetal 0.5 m process is presented. Nominal power dissipation is 40 nW per pixel at = 3 3 V. A/D conversion results from sampling a freerunning p ..."
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Cited by 17 (0 self)
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Abstract—A CMOS image sensor with pixelparallel analogtodigital (A/D) conversion fabricated with different array sizes and photodiode types in a threemetal 0.5 m process is presented. Nominal power dissipation is 40 nW per pixel at = 3 3 V. A/D conversion results from sampling a freerunning photocurrentcontrolled oscillator to give a firstorder – sequence. The sensor displays dynamic range capability of greater than 150 000: 1 and exhibits fixed pattern noise correctable to within 0.1 % of signal. Index Terms—Analog–digital conversion, image sensors, sigma–delta modulation, smart pixels. I.
Reconstruction of oversampled bandlimited signals from sigma delta encoded binary sequences
 IEEE Transactions on Signal Processing
, 1994
"... AbstractWe consider the application of EA modulators to analogtodigital conversion. We have previously shown that for constant input signals, optimal nonlinear decoding can achieve large gains in signaltonoise ratio (SNR) over linear decoding. In this paper we show a similar result for bandlimi ..."
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Cited by 11 (1 self)
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AbstractWe consider the application of EA modulators to analogtodigital conversion. We have previously shown that for constant input signals, optimal nonlinear decoding can achieve large gains in signaltonoise ratio (SNR) over linear decoding. In this paper we show a similar result for bandlimited input signals. The new nonlinear decoding algorithm is based on projections onto convex sets (POCS), and alternates between a timedomain operation and a band limitation to find a signal invariant under both. The timedomain operation results in a quadratic programming problem. The band limitation can be based on singular value decomposition of a certain matrix. We show simulation results for the SNR.performance of a POCSbased decoder and a linear decoder for the single loop, double loop and twostage CA modulators and for a specific fourthorder interpolative modulator. Depending on the modulator and the oversampling ratio, improvements in SNR of up to 1&20 dB can be achieved. I.
MLattice: A System For Signal Synthesis And Processing Based On ReactionDiffusion
 PROCESSING BASED ON REACTIONDIFFUSION. SCD THESIS, MIT
, 1994
"... This research begins with reactiondiffusion, first proposed by Alan Turing in 1952 to account for morphogenesis  the formation of hydranth tentacles, leopard spots, zebra stripes, etc. Reactiondiffusion systems have been researched primarily by biologists working on theories of natural pattern f ..."
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Cited by 5 (3 self)
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This research begins with reactiondiffusion, first proposed by Alan Turing in 1952 to account for morphogenesis  the formation of hydranth tentacles, leopard spots, zebra stripes, etc. Reactiondiffusion systems have been researched primarily by biologists working on theories of natural pattern formation and by chemists modeling dynamics of oscillating reactions. The past few years have seen a new interest in reactiondiffusion spring up within the computer graphics and image processing communities. However, reactiondiffusion systems are generally unbounded, making them impractical for many applications. In this thesis we introduce a bounded and more flexible nonlinear system, the "Mlattice", which preserves the natural patternformation properties of reactiondiffusion. On the theoretical front, we establish relationships between reactiondiffusion systems and paradigms in linear systems theory and certain types of artificial "neurallyinspired" systems. The Mlattice is closel...
On Incremental SigmaDelta Modulation with Optimal Filtering
 ACCEPTED FOR PUBLICATION TCASI
, 2005
"... The paper presents a quantizationtheoretic framework for studying incremental Σ∆ quantization systems. The framework makes it possible to efficiently compute the quantization intervals and hence the transfer function of the quantizer, and to determine the mean square error (MSE) and maximum error f ..."
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Cited by 4 (0 self)
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The paper presents a quantizationtheoretic framework for studying incremental Σ∆ quantization systems. The framework makes it possible to efficiently compute the quantization intervals and hence the transfer function of the quantizer, and to determine the mean square error (MSE) and maximum error for the optimal and conventional linear filters for first and second order incremental Σ∆ modulators. The results show that the optimal filter can significantly outperform conventional linear filters in terms of both MSE and maximum error. The performance of conventional Σ∆ quantizers is then compared to that of incremental Σ∆ with optimal filtering for bandlimited signals. It is shown that incremental Σ∆ can outperform the conventional approach in terms of signal to noise+distortion ratio (SNDR) and the characteristics of the power spectral density (PSD). The framework is also used to provide a simpler and more intuitive derivation of the Zoomer algorithm.
Improving the accuracy estimate for the first order sigmadelta modulator
 J. Amer. Math. Soc
"... Abstract. Based on experiments and numerical simulation, it has been widely believed that the average square error in the first order sigmadelta modulator with input of bandlimited signals decays like O(λ −3) as the sampling ratio λ goes to infinity. This conjecture remains as an open problem for m ..."
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Cited by 3 (0 self)
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Abstract. Based on experiments and numerical simulation, it has been widely believed that the average square error in the first order sigmadelta modulator with input of bandlimited signals decays like O(λ −3) as the sampling ratio λ goes to infinity. This conjecture remains as an open problem for many years. Combining tools from number theory, harmonic analysis, real analysis and complex analysis, this paper shows that the conjecture holds in some reasonable sense. 1.
Hexagonal SigmaDelta Modulation
, 2003
"... A novel application and generalization of sigmadelta (61) modulation has emerged in threephase powerelectronic converters. A conventional modulator with scalar signals and binary quantizer is generalized to a modulator with vector signals and a hexagonal quantizer. Indeed, powerelectronic ..."
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Cited by 3 (1 self)
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A novel application and generalization of sigmadelta (61) modulation has emerged in threephase powerelectronic converters. A conventional modulator with scalar signals and binary quantizer is generalized to a modulator with vector signals and a hexagonal quantizer. Indeed, powerelectronic 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 powerelectronic design and formulas for the average switching rate are derived for constant and slowly varying sinusoidal inputs. Index TermsErgodic, power electronics, quantization, sigmadelta (61) modulation, spectral analysis.
An Architecture for LowPower Real Time Image Analysis Using 3D Silicon Technology
 In Proc. SPIE AeroSense Symp
, 1998
"... The technology to build highly integrated 3dimensional computational image sensors by stacking and interconnecting layers of 2dimensional silicon ICs is being developed. Unlike multichip module (MCMV) packaging, in which interconnect lines are brought to the periphery of a chip stack to achieve ..."
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Cited by 2 (0 self)
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The technology to build highly integrated 3dimensional computational image sensors by stacking and interconnecting layers of 2dimensional silicon ICs is being developed. Unlike multichip module (MCMV) packaging, in which interconnect lines are brought to the periphery of a chip stack to achieve vertical integration, this new technology allows virtually unrestricted placement of vertical vias within the interior of the chip. The goal of this development is to enable high speed, high resolution image processing in compact low power wearable systems that would be coupled with a headmounted display (HMD). Potential applications for these systems include target tracking and image stabilization. In this talk we focus on the architecture of the 3D image sensor, which includes pixelparallel analogtodigital conversion and programmable digital processors for pixel and block operations. We show that 3D technology will allow at least an order of magnitude decrease in power dissipation over...
Power Analysis and Optimal Design of Opamps for Oversampled Converters
 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II
, 1999
"... In this paper, we address three issues related to the design of opamps for oversampled converters: the theoretical minimumpower bound for an ideal opamp, the best opamp choice in terms of power dissipation, and the best design strategy to reduce power dissipation. To be able to do so, we develop a ..."
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Cited by 2 (0 self)
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In this paper, we address three issues related to the design of opamps for oversampled converters: the theoretical minimumpower bound for an ideal opamp, the best opamp choice in terms of power dissipation, and the best design strategy to reduce power dissipation. To be able to do so, we develop a model that captures the dynamics of the integrator inside the modulator. Based on this model, power dissipation for various opamp topologies is computed and compared with each other. We show that classA opamps can consume one to six times the current consumed by an idealized classAB opamp, though the savings for practical classAB opamps is likely to be lower. For highresolution applications, classA topologies are most optimal, while for lower resolution lower power applications, classAB opamps may be more suitable. We further demonstrate a design strategy with the help of an example that can be used to minimize the power dissipation for classA opamps.