Quantization of compressive samples with stable and robust recovery

Saab, Rayan; Wang, Rongrong; Yılmaz, Özgür

doi:10.1016/j.acha.2016.04.005

Cited by 39 publications

(75 citation statements)

References 54 publications

(129 reference statements)

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“…In this subsection, we will show that for the first order Σ∆ scheme it is sufficient to choose a constant update δ = −N −1 u N −1 (30) in order to eliminate the boundary term of summation by parts in the error estimate (20). Namely, we will prove that this update causesũ N −1 = 0.…”

Section: A First Order σ∆ Schemementioning

confidence: 93%

“…Later, Σ∆ modulation schemes were extended to finite frame expansions in [14], [1], [2]. A number of works also study Σ∆ modulation in combination with compressed sensing [15], [9], [20]. For an overview of Σ∆ modulation in various settings and more general classes of noise shaping methods, we refer the reader to [4].…”

Section: B Sigma-delta Modulation In Mathematical Literaturementioning

confidence: 99%

“…Proof. For the first order quantizer, inequality (20) is obtained in (13). Now we focus on higher order quantizers with m ≥ 2.…”

Section: Error Analysis For σ∆ Schemes On a Circlementioning

confidence: 99%

“…The question to address in this section is how to improve the reconstruction error (20) and (21) by modifying the m-th order Σ∆ modulation scheme described in Section II.A.…”

Section: Modification Of the σ∆ Schemesmentioning

confidence: 99%

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Higher order 1-bit Sigma-Delta modulation on a circle

Graf

Krahmer

Krause-Solberg

2019

2019 13th International Conference on Sampling Theory and Applications (SampTA)

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Manifold models in data analysis and signal processing have become more prominent in recent years. In this paper, we will look at one of the main tasks of modern signal processing, namely, at analog-to-digital (A/D) conversion in connection with a simple manifold model -the circle. We will focus on Sigma-Delta modulation which is a popular method for A/D conversion of bandlimited signals that employs coarse quantization coupled with oversampling. Classical Sigma-Delta schemes provide mismatches and large errors at the initialization point if the signal to be converted is defined on the circle. In this paper, our goal is to get around these problems for Sigma-Delta schemes. Our results show how to design an update for the first and the second order schemes based on the reconstruction error analysis such that for the updated scheme the reconstruction error is improved.

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Section: A First Order σ∆ Schemementioning

confidence: 93%

Section: B Sigma-delta Modulation In Mathematical Literaturementioning

confidence: 99%

“…Proof. For the first order quantizer, inequality (20) is obtained in (13). Now we focus on higher order quantizers with m ≥ 2.…”

Section: Error Analysis For σ∆ Schemes On a Circlementioning

confidence: 99%

“…The question to address in this section is how to improve the reconstruction error (20) and (21) by modifying the m-th order Σ∆ modulation scheme described in Section II.A.…”

Section: Modification Of the σ∆ Schemesmentioning

confidence: 99%

See 2 more Smart Citations

Higher order 1-bit Sigma-Delta modulation on a circle

Graf

Krahmer

Krause-Solberg

2019

2019 13th International Conference on Sampling Theory and Applications (SampTA)

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“…Compressed sensing (CS) studies reconstruction of sparse signals from lower dimensional projections [17]. Distributed CS was studied in [18]- [22], sparse recovery from quantized projections was considered in [23]- [29], while [30], [31] proposed vector quantization schemes for bit-constrained distributed CS. Despite the similarity, there is a fundamental difference between distributed quantization of sparse signals and distributed CS with quantized observations: In the quantization framework, the measurements are the sparse signals, while in CS the observations are a linear projection of the signals.…”

Section: Introductionmentioning

confidence: 99%

Distributed Quantization for Sparse Time Sequences

Cohen

Shlezinger

Salamatian

et al. 2020

ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Analog signals processed in digital hardware are quantized into a discrete bit-constrained representation. Quantization is typically carried out using analog-to-digital converters (ADCs), operating in a serial scalar manner. In some applications, a set of analog signals are acquired individually and processed jointly. Such setups are referred to as distributed quantization. In this work we propose a distributed quantization scheme for representing a set of sparse time sequences acquired using conventional scalar ADCs. Our approach utilizes tools from secure group testing theory to exploit the sparse nature of the acquired analog signals, obtaining a compact and accurate representation while operating in a distributed fashion. We then show how our technique can be implemented when the quantized signals are transmitted over a multihop communication network providing a low-complexity network policy for routing and signal recovery. Our numerical evaluations demonstrate that the proposed scheme notably outperforms conventional methods based on the combination of quantization and compressed sensing tools.

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Fast Binary Embeddings and Quantized Compressed Sensing with Structured Matrices

Huynh

Saab

2019

Comm Pure Appl Math

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This paper deals with two related problems, namely distance‐preserving binary embeddings and quantization for compressed sensing. First, we propose fast methods to replace points from a subset Χ ⊂ ℝn, associated with the euclidean metric, with points in the cube {±1}m, and we associate the cube with a pseudometric that approximates euclidean distance among points in Χ. Our methods rely on quantizing fast Johnson‐Lindenstrauss embeddings based on bounded orthonormal systems and partial circulant ensembles, both of which admit fast transforms. Our quantization methods utilize noise shaping, and include sigma‐delta schemes and distributed noise‐shaping schemes. The resulting approximation errors decay polynomially and exponentially fast in m, depending on the embedding method. This dramatically outperforms the current decay rates associated with binary embeddings and Hamming distances. Additionally, it is the first such binary embedding result that applies to fast Johnson‐Lindenstrauss maps while preserving ℓ2 norms. Second, we again consider noise‐shaping schemes, albeit this time to quantize compressed sensing measurements arising from bounded orthonormal ensembles and partial circulant matrices. We show that these methods yield a reconstruction error that again decays with the number of measurements (and bits), when using convex optimization for reconstruction. Specifically, for sigma‐delta schemes, the error decays polynomially in the number of measurements, and it decays exponentially for distributed noise‐shaping schemes based on beta encoding. These results are near optimal and the first of their kind dealing with bounded orthonormal systems. © 2019 Wiley Periodicals, Inc.

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Quantization of compressive samples with stable and robust recovery

Cited by 39 publications

References 54 publications

Higher order 1-bit Sigma-Delta modulation on a circle

Higher order 1-bit Sigma-Delta modulation on a circle

Distributed Quantization for Sparse Time Sequences

Fast Binary Embeddings and Quantized Compressed Sensing with Structured Matrices

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