Approximating a bandlimited function using very coarsely quantized data: A family of stable sigma-delta modulators of arbitrary order

Daubechies, Ingrid; DeVore, Ronald A.

doi:10.4007/annals.2003.158.679

Cited by 171 publications

(252 citation statements)

References 12 publications

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“…Recently frames have been used to describe analog-to-digital (AD) conversion and specifically the quantization algorithm known as Σ∆. While there are numerous papers on Σ∆ in the engineering literature, [4,8,10,11,15,19,20] to which this list does not begin to do justice, a paper of Daubechies and DeVore [9] created great interest in the mathematical community [12][13][14]17,21]. In particular, Benedetto, Powell and Yilmaz [1,2] recently used finite frames to describe and examine first order Σ∆ schemes, followed by Bodmann and Paulsen [6].…”

Section: Introductionmentioning

confidence: 99%

On quantization of finite frame expansions: sigma-delta schemes of arbitrary order

2007

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In this note we will show that the so called Sobolev dual is the minimizer over all linear reconstructions using dual frames for stable r th order Σ∆ quantization schemes under the so called White Noise Hypothesis (WNH) design criteria. We compute some Sobolev duals for common frames and apply them to audio clips to test their performance against canonical duals and another alternate dual corresponding to the well known Blackman filter.

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Section: Introductionmentioning

confidence: 99%

On quantization of finite frame expansions: sigma-delta schemes of arbitrary order

2007

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“…Stability analysis of Σ∆ schemes can be quite complicated, especially for 1-bit higher order schemes, [10]. The following example shows a basic first order Σ∆ quantizer.…”

Section: Sigma-delta Quantization Of Finite Frame Expansionsmentioning

confidence: 99%

“…Sigma-Delta (Σ∆) schemes are a class of recursive algorithms that directly make use of dependencies among frame vectors in the quantization process. We give a brief description of Σ∆ schemes, but refer the reader to [10,23] for more detailed background. Suppose that {e n } N n=1 ⊂ R d is a frame for R d and that x ∈ R d has frame coefficients x n = x, e n .…”

Section: Sigma-delta Quantization Of Finite Frame Expansionsmentioning

confidence: 99%

“…The mathematical theory of Σ∆ quantization on the other hand is quite recent. After work by Gray, e.g., [14,16], Daubechies and DeVore [10] established an approximation theoretical framework and provided rigorous results on the relationship between robustness of Σ∆ schemes, redundancy of the associated representations, and the approximation error. The work in [10] gives an explicit construction of stable 1-bit Σ∆ schemes of general order r. Moreover, [10] that in the bandlimited setting the L ∞ approximation error associated with a stable rth order Σ∆ scheme behaves like 1/λ r where λ is the oversampling ratio.…”

Section: Introductionmentioning

confidence: 99%

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Sobolev Duals in Frame Theory and Sigma-Delta Quantization

Blum

Lammers

Powell

et al. 2009

J Fourier Anal Appl

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Abstract. A new class of alternative dual frames is introduced in the setting of finite frames for R d . These dual frames, called Sobolev duals, provide a high precision linear reconstruction procedure for Sigma-Delta (Σ∆) quantization of finite frames. The main result is summarized as follows: reconstruction with Sobolev duals enables stable rth order Sigma-Delta 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.

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“…An especially attractive statistical mechanics application is [24], and [9]. Now Shannon's view is motivated by signal processing, i.e., engineering of signals, see e.g., [12]: interpolation of functions (signals) on a continuum, determining band-limited functions defined on a continuum from their discrete samples. But in some cases outside the study of lattice graphs, one might only have available the particular given (countably discrete!)…”

Section: Signalsmentioning

confidence: 99%

A sampling theory for infinite weighted graphs

Jørgensen¹

2011

OpMath

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Abstract. We prove two sampling theorems for infinite (countable discrete) weighted graphs G; one example being "large grids of resistors" i.e., networks and systems of resistors. We show that there is natural ambient continuum X containing G, and there are Hilbert spaces of functions on X that allow interpolation by sampling values of the functions restricted only on the vertices in G. We sample functions on X from their discrete values picked in the vertex-subset G. We prove two theorems that allow for such realistic ambient spaces X for a fixed graph G, and for interpolation kernels in function Hilbert spaces on X, sampling only from points in the subset of vertices in G. A continuum is often not apparent at the outset from the given graph G. We will solve this problem with the use of ideas from stochastic integration.

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Approximating a bandlimited function using very coarsely quantized data: A family of stable sigma-delta modulators of arbitrary order

Cited by 171 publications

References 12 publications

On quantization of finite frame expansions: sigma-delta schemes of arbitrary order

On quantization of finite frame expansions: sigma-delta schemes of arbitrary order

Sobolev Duals in Frame Theory and Sigma-Delta Quantization

A sampling theory for infinite weighted graphs

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