Sigma Delta Quantization with Harmonic Frames and Partial Fourier Ensembles

Sigma Delta (\(\Sigma \Delta \)) quantization, a quantization method first surfaced in the 1960s, has now been widely adopted in various digital products such as cameras, cell phones, radars, etc. The method features a great robustness with respect to quantization noises through sampling an input signal at a Super-Nyquist rate. Compressed sensing (CS) is a frugal acquisition method that utilizes the sparsity structure of an objective signal to reduce the number of samples required for a lossless acquisition. By deeming this reduced number as an effective dimensionality of the set of sparse signals, one can define a relative oversampling/subsampling rate as the ratio between the actual sampling rate and the effective dimensionality. When recording these “compressed” analog measurements via Sigma Delta quantization, a natural question arises: will the signal reconstruction error previously shown to decay polynomially as the increase of the vanilla oversampling rate for the case of band-limited functions, now be decaying polynomially as that of the relative oversampling rate? Answering this question is one of the main goals in this direction. The study of quantization in CS has so far been limited to proving error convergence results for Gaussian and sub-Gaussian sensing matrices, as the number of bits and/or the number of samples grow to infinity. In this paper, we provide a first result for the more realistic Fourier sensing matrices. The main idea is to randomly permute the Fourier samples before feeding them into the quantizer. We show that the random permutation can effectively increase the low frequency power of the measurements, thus enhance the quality of \(\Sigma \Delta \) quantization.     

One‐bit sigma‐delta quantization with exponential accuracy

One‐bit quantization is a method of representing bandlimited signals by ±1 sequences that are computed from regularly spaced samples of these signals; as the sampling density λ → ∞, convolving these one‐bit sequences with appropriately chosen filters produces increasingly close approximations of the original signals. This method is widely used for analog‐to‐digital and digital‐to‐analog conversion, because it is less expensive and simpler to implement than the more familiar critical sampling followed by fine‐resolution quantization. However, unlike fine‐resolution quantization, the accuracy of one‐bit quantization is not well‐understood. A natural error lower bound that decreases like 2^?λ can easily be given using information theoretic arguments. Yet, no one‐bit quantization algorithm was known with an error decay estimate even close to exponential decay. In this paper, we construct an infinite family of one‐bit sigma‐delta quantization schemes that achieves this goal. In particular, using this family, we prove that the error signal for π‐bandlimited signals is at most O(2^?.07λ). © 2003 Wiley Periodicals, Inc.     

Event-triggered feedback stabilization of switched linear systems using dynamic quantized input

This paper is concerned with the co-design of event-triggered sampling, dynamic input quantization and constrained switching for a switched linear system. The mismatch between the plant and its corresponding controller is considered. This behavior is raised by switching within the event-triggered sampling interval. Accordingly, novel update laws of dynamic quantization parameter are designed separately for matched sampling intervals (without switching) and mismatched sampling intervals (with a switch). We technically transform the total variation (increment or decrement) of Lyapunov functions in one sampling interval into the discrete-time update of quantization parameter. Based on this transformation, a hybrid quantized control policy is developed. This policy, in conjunction with the average dwell-time switching law and the constructed event-triggered condition, can ensure the exponential stabilization of the switched system with finite-level quantized input. Besides, the event-triggered scheme is proved to be Zeno-free. The effectiveness of the developed method is verified by a simulation example.     

Sampling based on timing: Time encoding machines on shift-invariant subspaces

Sampling information using timing is an approach that has received renewed attention in sampling theory. The question is how to map amplitude information into the timing domain. One such encoder, called time encoding machine, was introduced by Lazar and Tóth (2004 [23]) for the special case of band-limited functions. In this paper, we extend their result to a general framework including shift-invariant subspaces. We prove that time encoding machines may be considered as non-uniform sampling devices, where time locations are unknown a priori. Using this fact, we show that perfect representation and reconstruction of a signal with a time encoding machine is possible whenever this device satisfies some density property. We prove that this method is robust under timing quantization, and therefore can lead to the design of simple and energy efficient sampling devices.     

Quantization of compressive samples with stable and robust recovery

In this paper we study the quantization stage that is implicit in any compressed sensing signal acquisition paradigm. We propose using Sigma–Delta (ΣΔ) quantization and a subsequent reconstruction scheme based on convex optimization. We prove that the reconstruction error due to quantization decays polynomially in the number of measurements. Our results apply to arbitrary signals, including compressible ones, and account for measurement noise. Additionally, they hold for sub-Gaussian (including Gaussian and Bernoulli) random compressed sensing measurements, as well as for both high bit-depth and coarse quantizers, and they extend to 1-bit quantization. In the noise-free case, when the signal is strictly sparse we prove that by optimizing the order of the quantization scheme one can obtain root-exponential decay in the reconstruction error due to quantization.     

Lattice quantization error for redundant representations

Redundant systems such as frames are often used to represent a signal for error correction, denoising and general robustness. In the digital domain quantization needs to be performed. Given the redundancy, the distribution of quantization errors can be rather complex. In this paper we study quantization error for a signal X in represented by a frame using a lattice quantizer. We completely characterize the asymptotic distribution of the quantization error as the cell size of the lattice goes to zero. We apply these results to get the necessary and sufficient conditions for the asymptotic form of the White Noise Hypothesis in the case of the pulse-code modulation scheme.     

具有量化误差的多步长非线性采样系统的镇定

该文研究一类非线性控制系统在采样器采样过程中产生量化误差的情况下多步长采样镇定问题. 运用近似DTD方法, 在非线性系统的近似离散时间模型上设计全局状态反馈镇定控制器. 当系统近似误差和采样量化误差被限制在一定的条件下, 可以得到含量化误差的多步长非线性采样系统是半全局实用渐近稳定. 最后, 仿真例子验证了所得结果的有效性.     

Smooth Frame-Path Termination for Higher Order Sigma-Delta Quantization

We study the performance of finite frames for the encoding of vectors by applying standard higher-order sigma-delta quantization to the frame coefficients. Our results are valid for any quantizer with accuracy ε > 0 operating in the no-overload regime. The frames under consideration are obtained from regular sampling of a path in a Hilbert space. In order to achieve error bounds that are comparable to results on higher-order sigma-delta for the quantization of oversampled bandlimited functions, we construct frame paths that terminate smoothly in the zero vector, that is, with an appropriate number of vanishing derivatives at the endpoint.     

Approximating a bandlimited function using very coarsely quantized data: Improved error estimates in sigma-delta modulation

Sigma-delta quantization is a method of representing bandlimited signals by sequences that are computed from regularly spaced samples of these signals; as the sampling density , convolving these one-bit sequences with appropriately chosen kernels produces increasingly close approximations of the original signals. This method is widely used for analog-to-digital and digital-to-analog conversion, because it is less expensive and simpler to implement than the more familiar critical sampling followed by fine-resolution quantization. We present examples of how tools from number theory and harmonic analysis are employed in sharpening the error estimates in sigma-delta quantization.

     

Frame Analysis of Irregular Periodic Sampling of Signals and Their Derivatives

Given a bandlimited signal, we consider the sampling of the signal and some of its derivatives in a periodic manner. The mathematical concept of frames is utilized in the analysis of the properties of the sequence of sampling functions. The frame operator of this sequence is expressed as a matrix-valued function multiplying a vector-valued function. An important property of this matrix is that the maximum and minimum eigenvalues are equal (in some sense) to the upper and lower frame bounds. We present a method for finding the dual frame and, thereby, a method for reconstructing the signal from its samples. Using the matrix approach we prove that the sequence of sampling functions is always complete in the cases of critical sampling and oversampling. A sufficient condition for the sequence of sampling functions to constitute a frame is derived. We show that if no sampling of the signal itself is involved, the sampling is not stable and cannot be stabilized by oversampling. Examples are considered, and the frame bounds in the case of sampling of the signal and its first derivative are calculated explicitly. Finally, the matrix approach can be similarly applied to other problems of signal representation.     

Local sampling set conditions in weighted shift-invariant signal spaces

How to represent a continuous signal in terms of a discrete sequence is a fundamental problem in sampling theory. Most of the known results concern global sampling in shift-invariant signal spaces. But in fact, the local reconstruction from local samples is one of the most desirable properties for many applications in signal processing, e.g. for implementing real-time reconstruction numerically. However, the local reconstruction problem has not been given much attention. In this article, we find conditions on a finite sampling set X such that at least in principle a continuous signal on a finite interval is uniquely and stably determined by their sampling value on the finite sampling set X in shift-invariant signal spaces.     

PDE-based signal quantization

We present a new method for signal restoration/quantization based on diffusion reaction model with memory term. We prove that the model is stable, with the existence and uniqueness results. We also propose a numerical approximation that we prove the convergence and present some experiments on noisy signals.     

论Whittaker-Shannon抽样定理及其一些推广

The aim of this paper is to present a survey of results concerning the Whittaker-Kotel'nikov-Raabe-Shannon-Someya sampling theorem and its various extensions obtained at Aachen since 1977. This theorem, basic in communication engineering, is often called the cardinal interpolation series theorem in mathematical circles. The interconnections of the sampling theorem (in the setting of Paley-Wiener space) with the theory of Fourier series and integrals are examined. Emphasis is placed upon error analysis, including the aliasing, round-off (or quantization), and time jitter errors. Some new error estimates are given, others are improved; many of the proofs are reduced to a common structure. Both deterministic and probabilistic methods are employed. whereas these results are worked out in detail, the paper also contains a brief discussion of some of the various generalizations.     

On the limit of sampled signal extrapolation with a wavelet signal model ∗

The extrapolation of sampled signals from a given interval using a wavelet model with various sampling rates is examined in this research. We present sufficient conditions on signals and wavelet bases so that the discrete-time extrapolated signal converges to its continuous-time counterpart when the sampling rate goes to infinity. Thus, this work provides a practical procedure to implement continuous-time signal extrapolation, which is important in wideband radar and sonar signal processing, with a discrete one via carefully choosing the sampling rate and the wavelet basis. A numerical example is given to illustrate our theoretical result.     

Error estimates for irregular sampling of band-limited functions on a locally compact Abelian group

Band-limited functions f can be recovered from their sampling values (f(x_i)) by means of iterative methods, if only the sampling density is high enough. We present an error analysis for these methods, treating the typical forms of errors, i.e., jitter error, truncation error, aliasing error, quantization error, and their combinations. The derived apply uniformly to whole families of spaces, e.g., to weighted L^p-spaces over some locally compact Abelian group with growth rate up to some given order. In contrast to earlier papers we do not make use of any (relative) separation condition on the sampling sets. Furthermore we discard the assumption on polynomial growth of the weights that has been used over Euclidean spaces. Consequently, even for the case of regular sampling, i.e., sampling along lattices in G, the results are new in the given generality.     

Approximation with Impulse Trains

The impulse train obtained by sampling an analog signal is studied directly. It is shown to represent the signal in a weak sense and to converge to the signal as the sampling interval converges to 0. The convergence is in the sense of Sobolev spaces and in the sense of another Hilbert space for which the shifted “delta functions” are orthogonal.     

An Asymptotic Sampling Recomposition Theorem for Gaussian Signals

For any Gaussian signal and every given sampling frequency we prove an asymptotic property of type Shannon’s sampling theorem, based on normalized cardinal sines, which keeps constant the sampling frequency. We generalize the Shannon’s sampling theorem for a class of non band–limited signals which plays a central role in the signal theory, the Gaussian map is the unique function which reachs the minimum of the product of the temporal and frecuential width. This solve a conjecture stated in [1] and suggested by [3].     

Local Sampling and Reconstruction in Shift-Invariant Spaces and Their Applications in Spline Subspaces

The local reconstruction from samples is one of the most desirable properties for many applications in signal processing. Local sampling is practically useful since we need only to consider a signal on a bounded interval and computer can only process finite samples. However, the local sampling and reconstruction problem has not been given as much attention. Most of known results concern global sampling and reconstruction. There are only a few results about local sampling and reconstruction in spline subspaces. In this article, we study local sampling and reconstruction in general shift-invariant spaces and multiple generated shift-invariant spaces with compactly supported generators. Then we give several applications in spline subspaces and multiple generated spline subspaces.     

A Noise-Shaping Theorem

Noise shaping is a process which aims to remove as much quantization noise as possible from the spectrum of a given band-limited signal when quantizing it (recall that the spectrum of a signal is the support of its Fourier transform). We provide a mathematical analysis of such a process, using methods of harmonic analysis on the unit disc. We are especially interested in conditions under which an analytic (but not necessary polynomial) function may be used as a transfer function of the process. Stability conditions will be given under terms of metric characteristics of the transfer function (stability means that the quantizer is never overloaded). Several kinds of quantizer transfer functions will be considered, especially midtread and midriser ones. We shall restrict our analysis to the case of deterministic (i.e., not random) signals. Some knowledge of Fourier analysis and signal processing will be presupposed.