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34
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 782 (12 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.
Distributed crosslayer optimization of wireless sensor networks: A game theoretic approach
 Proc. of the GlobeCom 2006. San Francisco: IEEE Communications Society
, 2006
"... Abstract — This paper proposes a distributed optimization framework for wireless multihop sensor networks base on a game theoretic approach. We show that the crosslayer optimization problem can be decomposed into two subproblems corresponding to two separate layers (the physical and the application ..."
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Cited by 23 (4 self)
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Abstract — This paper proposes a distributed optimization framework for wireless multihop sensor networks base on a game theoretic approach. We show that the crosslayer 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 applicationlayer rateallocation and physicallayer powerallocation 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 crosslayer design. I.
Efficient Quantization for Overcomplete Expansions in R^N
 IEEE TRANS. INFORM. THEORY
, 2003
"... 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 quant ..."
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Cited by 15 (4 self)
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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 scaledsimilar sublattices. The periodicity makes it possible to achieve good accuracy using simple reconstruction algorithms (e.g., linear reconstruction or a small lookup table).
Rate Bounds on SSIM Index of Quantized Images
, 2008
"... In this paper, we derive bounds on the structural similarity (SSIM) index as a function of quantization rate for fixedrate uniform quantization of image discrete cosine transform (DCT) coefficients under the highrate assumption. The space domain SSIM index is first expressed in terms of the DCT co ..."
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Cited by 14 (1 self)
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In this paper, we derive bounds on the structural similarity (SSIM) index as a function of quantization rate for fixedrate uniform quantization of image discrete cosine transform (DCT) coefficients under the highrate 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.
Sobolev duals in frame theory and SigmaDelta quantization
"... Abstract. A new class of alternative dual frames is introduced in the setting of finite frames for Rd. These dual frames, called Sobolev duals, provide a high precision linear reconstruction procedure for SigmaDelta (Σ∆) quantization of finite frames. The main result is summarized as follows: recon ..."
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Cited by 12 (4 self)
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Abstract. A new class of alternative dual frames is introduced in the setting of finite frames for Rd. These dual frames, called Sobolev duals, provide a high precision linear reconstruction procedure for SigmaDelta (Σ∆) quantization of finite frames. The main result is summarized as follows: reconstruction with Sobolev duals enables stable rth order SigmaDelta schemes to achieve deterministic approximation error of order O(N−r) for a wide class of finite frames of size N. This asymptotic order is generally not achievable with canonical dual frames. Moreover, Sobolev dual reconstruction leads to minimal mean squared error under the classical white noise assumption. 1.
Efficient bitrate scalability for weighted squared error optimization in audio coding
 IEEE Trans. Audio, Speech, Lang. Process
, 2006
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Distributed Functional Scalar Quantization Simplified
, 2013
"... Distributed functional scalar quantization (DFSQ) theory provides optimality conditions and predicts performance of data acquisition systems in which a computation on acquired data is desired. We address two limitations of previous works: prohibitively expensive decoder design and a restriction to ..."
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Cited by 5 (5 self)
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Distributed functional scalar quantization (DFSQ) theory provides optimality conditions and predicts performance of data acquisition systems in which a computation on acquired data is desired. We address two limitations of previous works: prohibitively expensive decoder design and a restriction to source distributions with bounded support. We show that a much simpler decoder has equivalent asymptotic performance to the conditional expectation estimator studied previously, thus reducing decoder design complexity. The simpler decoder features decoupled communication and computation blocks. Moreover, we extend the DFSQ framework with the simpler decoder to source distributions with unbounded support. Finally, through simulation results, we demonstrate that performance at moderate coding rates is well predicted by the asymptotic analysis, and we give new insight on the rate of convergence.
Extended frequencybanddecomposition sigmadelta A/D converter
 Analog Integrated Circuits and Signal Processing
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Joint Source Coding, Routing and Power Allocation in Wireless Sensor Networks
"... Abstract—This paper proposes a crosslayer optimization framework for the wireless sensor networks. In a wireless sensor network, each sensor makes a local observation of the underlying physical phenomenon and sends a quantized version of the observation to a central location via wireless links. As ..."
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Cited by 3 (1 self)
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Abstract—This paper proposes a crosslayer optimization framework for the wireless sensor networks. In a wireless sensor network, each sensor makes a local observation of the underlying physical phenomenon and sends a quantized version of the observation to a central location via wireless links. As the sensor observations are often partial and correlated, the network performance is a complicated and nonseparable function of individual data rates at each sensor. In addition, due to the shared nature of wireless medium, nearby transmissions often interfere with each other. Thus, the traditional “bitpipe ” model for network link capacity no longer holds. This paper deals with the joint optimization of source quantization, routing, and power control in a wireless sensor network. We follow a separate source and channel coding approach and show that the overall network optimization problem can be naturally decomposed into a source coding subproblem at the application layer and a wireless power control subproblem at the physical layer. The interfaces between the layers are precisely the dual optimization variables. In addition, we introduce a novel source coding model at the application layer, which allows the efficient design of practical source quantization schemes at each sensor. Finally, we propose a dual algorithm for the overall network optimization problem. The dual algorithm, when combined with a columngeneration method, allows an efficient solution for the overall network optimization problem. Index Terms—Channel capacity region, convex optimization, dual decomposition, power control, ratedistortion region, routing, sensor networks, source coding. I.
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.