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Sigmadelta quantization and finite frames
 in Proc. Int. Conf. Acoustics, Speech and Signal Processing
, 2004
"... Abstract—Thelevel Sigma–Delta () scheme with step size is introduced as a technique for quantizing finite frame expansions for. Error estimates for various quantized frame expansions are derived, and, in particular, it is shown that quantization of a unitnorm finite frame expansion in achieves ap ..."
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Cited by 42 (6 self)
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Abstract—Thelevel Sigma–Delta () scheme with step size is introduced as a technique for quantizing finite frame expansions for. Error estimates for various quantized frame expansions are derived, and, in particular, it is shown that quantization of a unitnorm finite frame expansion in achieves approximation error 2 ( ( ) + 1) where is the frame size, and the frame variation ( ) is a quantity which reflects the dependence of the scheme on the frame. Here is thedimensional Euclidean 2norm. Lower bounds and refined upper bounds are derived for certain specific cases. As a direct consequence of these error bounds one is able to bound the mean squared error (MSE) by an order of 1 2. When dealing with sufficiently redundant frame expansions, this represents a significant improvement over classical pulsecode modulation (PCM) quantization, which only has MSE of order1 under certain nonrigorous statistical assumptions. also achieves the optimal MSE order for PCM with consistent reconstruction. Index Terms—Finite frames, Sigma–Delta quantization. I.
A CMOS Area Image Sensor With Pixel Level A/D Conversion
 IN ISSCC DIGEST OF TECHNICAL PAPERS
, 1995
"... A CMOS 64 x 64 pixel area image sensor chip using SigmaDelta modulation at each pixel for A/D conversion is described. The image data output is digital. The chip was fabricated using a 1.2µm two layer metal single layer poly nwell CMOS process. Each pixel block consists of a phototransistor and ..."
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Cited by 37 (7 self)
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A CMOS 64 x 64 pixel area image sensor chip using SigmaDelta modulation at each pixel for A/D conversion is described. The image data output is digital. The chip was fabricated using a 1.2µm two layer metal single layer poly nwell CMOS process. Each pixel block consists of a phototransistor and 22 MOS transistors. Test results demonstrate a dynamic range potentially greater than 93dB, a signal to noise ratio (SNR) of up to 61dB, and dissipation of less than 1mW with a 5V power supply.
Information Rates of Pre/Post Filtered Dithered Quantizers
 IEEE Trans. Information Theory
, 1997
"... We consider encoding of a source with a prespecified second order statistics, but otherwise arbitrary, by Entropy Coded Dithered (lattice) Quantization (ECDQ) incorporating linear preand postfilters. In the design and analysis of this scheme we utilize the equivalent additive noise channel model o ..."
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Cited by 26 (14 self)
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We consider encoding of a source with a prespecified second order statistics, but otherwise arbitrary, by Entropy Coded Dithered (lattice) Quantization (ECDQ) incorporating linear preand postfilters. In the design and analysis of this scheme we utilize the equivalent additive noise channel model of the ECDQ. For Gaussian sources and square error distortion measure, the coding performance of the pre/post filtered ECDQ approaches the ratedistortion function, as the dimension of the (optimal) lattice quantizer becomes large; actually, in this case the proposed coding scheme simulates the optimal forward channel realization of the ratedistortion function. For nonGaussian sources and finite dimensional lattice quantizers, the coding rate exceeds the ratedistortion function by at most the sum of two terms: the "information divergence of the source from Gaussianity" and the "information divergence of the quantization noise from Gaussianity". Additional bounds on the excess rate of the s...
Recursive Consistent Estimation with Bounded Noise
 IEEE TRANS. INFORM. TH
, 2001
"... Estimation problems with bounded, uniformly distributed noise arise naturally in reconstruction problems from over complete linear expansions with subtractive dithered quantization. We present a simple recursive algorithm for such boundednoise estimation problems. The meansquare error (MSE) of the ..."
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Cited by 25 (16 self)
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Estimation problems with bounded, uniformly distributed noise arise naturally in reconstruction problems from over complete linear expansions with subtractive dithered quantization. We present a simple recursive algorithm for such boundednoise estimation problems. The meansquare error (MSE) of the algorithm is "almost" (1/n²), where is the number of samples. This rate is faster than the (1/n) MSE obtained by standard recursive least squares estimation and is optimal to within a constant factor.
Alternative dual frames for digitaltoanalog conversion in sigmadelta quantization
 Adv. Comput. Math
"... Abstract. We design alternative dual frames for linearly reconstructing signals from SigmaDelta (Σ∆) quantized finite frame coefficients. In the setting of sampling expansions for bandlimited functions, it is known that a stable rth order SigmaDelta quantizer produces approximations where the app ..."
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Cited by 17 (3 self)
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Abstract. We design alternative dual frames for linearly reconstructing signals from SigmaDelta (Σ∆) quantized finite frame coefficients. In the setting of sampling expansions for bandlimited functions, it is known that a stable rth order SigmaDelta quantizer produces approximations where the approximation error is at most of order 1/λr, and λ> 1 is the oversampling ratio. We show that the counterpart of this result is not true for several families of redundant finite frames for Rd when the canonical dual frame is used in linear reconstruction. As a remedy, we construct alternative dual frame sequences which enable an rth order SigmaDelta quantizer to achieve approximation error of order 1/Nr for certain sequences of frames where N is the frame size. We also present several numerical examples regarding the constructions. 1.
Deterministic Analysis of Oversampled A/D Conversion and Sigma/Delta Modulation, and Decoding Improvements using Consistent Estimates
, 1993
"... Analogtodigital conversion (ADC) which consists in a double discretization of an analog signal in time and in amplitude is increasingly used in modern data acquisition. However, the conversion process always implies some loss of information due to amplitude quantization. Oversampling is the techni ..."
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Cited by 8 (0 self)
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Analogtodigital conversion (ADC) which consists in a double discretization of an analog signal in time and in amplitude is increasingly used in modern data acquisition. However, the conversion process always implies some loss of information due to amplitude quantization. Oversampling is the technique currently used to reduce this loss of accuracy. The error reduction can be performed by lowpass filtering the quantized signal, thus eliminating the high frequency components of the quantization error signal. This is the classical method used to reconstruct the analog signal from its oversampled and quantized version. This reconstruction scheme yields a mean squared error (MSE) inversely proportional to the oversampling ratio R. The fundamental question pursued in this thesis is the following: how much information is available in the oversampled and quantized version of a bandlimited signal for its reconstruction? In order to identify this information, it is essential to go back to the original description of quantization which is typically deterministic. We show that a reconstruction scheme fully takes this information into account
The effects of quantization noise and sensor nonideality on digitaldifferentiatorbased velocity measurement
 KAVANAGH: SHAFT ENCODER CHARACTERIZATION VIA THEORETICAL MODEL 801
, 1998
"... Abstract—This paper focuses on the nature of the rate error which arises when a firstorder digital differentiator is applied to the output of a uniform quantizer for the purpose of rate estimation. The quantizer input is assumed to be a constantrate signal which is subject to a uniformly distribut ..."
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Cited by 7 (2 self)
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Abstract—This paper focuses on the nature of the rate error which arises when a firstorder digital differentiator is applied to the output of a uniform quantizer for the purpose of rate estimation. The quantizer input is assumed to be a constantrate signal which is subject to a uniformly distributed noise source. New formulae are presented for the corresponding rms and spectral error characteristics. The results are applicable to the signal processing of sensor outputs, exemplified by the shaft encoderbased velocity estimation of an almost uniformly rotating mass. Both computergenerated finite data sets and experimental data derived from encoderbased shaft velocity measurements are utilized to verify the theoretical results. The results described are also applicable to a more general class of systems which involve the digital differentiation of quantized, noiseaffected signals, such as firstorder sigma–delta modulators with nominally constant input. Index Terms — Differentiation, digital measurements, optical transducers, optical velocity measurement, quantization,
Nonuniform sampling of periodic bandlimited signals
 IEEE TRANS. SIGNAL PROCESSING
, 2008
"... Digital processing techniques are based on representing a continuoustime signal by a discrete set of samples. This paper treats the problem of reconstructing a periodic bandlimited signal from a finite number of its nonuniform samples. In practical applications, only a finite number of values are ..."
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Cited by 6 (0 self)
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Digital processing techniques are based on representing a continuoustime signal by a discrete set of samples. This paper treats the problem of reconstructing a periodic bandlimited signal from a finite number of its nonuniform samples. In practical applications, only a finite number of values are given. Extending the samples periodically across the boundaries, and assuming that the underlying continuous time signal is bandlimited, provides a simple way to deal with reconstruction from finitely many samples. Two algorithms for reconstructing a periodic bandlimited signal from an even and an odd number of nonuniform samples are developed. In the first, the reconstruction functions constitute a basis while in the second, they form a frame so that there are more samples than needed for perfect reconstruction. The advantages and disadvantages of each method are analyzed. Specifically, it is shown that the first algorithm provides consistent reconstruction of the signal while the second is shown to be more stable in noisy environments. Next, we use the theory of finite dimensional frames to characterize the stability of our algorithms. We then consider two special distributions of sampling points: uniform and recurrent nonuniform, and show that for these cases, the reconstruction formulas as well as the stability analysis are simplified significantly. The advantage of our methods over conventional approaches is demonstrated by numerical experiments.