Suppose that the collection $\{e_i\}_{i=1}^m$ forms a frame for $\R^k$, where each entry of the vector $e_i$ is a sub-Gaussian random variable. We consider expansions in such a frame, which are then quantized using a Sigma-Delta scheme. We show that an arbitrary signal in $\R^k$ can be recovered from its quantized frame coefficients up to an error which decays root-exponentially in the oversampling rate $m/k$. Here the quantization scheme is assumed to be chosen appropriately depending on the oversampling rate and the quantization alphabet can be coarse. The result holds with high probability on the draw of the frame uniformly for all signals. The crux of the argument is a bound on the extreme singular values of the product of a deterministic matrix and a sub-Gaussian frame. For fine quantization alphabets, we leverage this bound to show polynomial error decay in the context of compressed sensing. Our results extend previous results for structured deterministic frame expansions and Gaussian compressed sensing measurements.
In this paper we investigate encoding the bit-stream resulting from coarse Sigma-Delta quantization of finite frame expansions (i.e., overdetermined representations) of vectors. We show that for a wide range of finite-frames, including random frames and piecewise smooth frames, there exists a simple encoding algorithm ---acting only on the Sigma-Delta bit stream--- and an associated decoding algorithm that together yield an approximation error which decays exponentially in the number of bits used. The encoding strategy consists of applying a discrete random operator to the Sigma-Delta bit stre…
We construct high order low-bit Sigma-Delta $(ΣΔ)$ quantizers for the vector-valued setting of fusion frames. We prove that these $ΣΔ$ quantizers can be stably implemented to quantize fusion frame measurements on subspaces $W_n$ using $\log_2( {\rm dim}(W_n)+1)$ bits per measurement. Signal reconstruction is performed using a version of Sobolev duals for fusion frames, and numerical experiments are given to validate the overall performance.
Sampling above the Nyquist rate is at the heart of sigma-delta modulation, where the increase in sampling rate is translated to a reduction in the overall (mean-squared-error) reconstruction distortion. This is attained by using a feedback filter at the encoder, in conjunction with a low-pass filter at the decoder. The goal of this work is to characterize the optimal trade-off between the per-sample quantization rate and the resulting mean-squared-error distortion, under various restrictions on the feedback filter. To this end, we establish a duality relation between the performance of sigma-d…
We examine the uplink spectral efficiency of a massive MIMO base station employing a one-bit Sigma-Delta sampling scheme implemented in the spatial rather than the temporal domain. Using spatial rather than temporal oversampling, and feedback of the quantization error between adjacent antennas, the method shapes the spatial spectrum of the quantization noise away from an angular sector where the signals of interest are assumed to lie. It is shown that, while a direct Bussgang analysis of the Sigma-Delta approach is not suitable, an alternative equivalent linear model can be formulated to facil…
This paper focuses on channel estimation for mmWave MIMO systems with 1-bit spatial sigma-delta analog-to-digital converters (ADCs). The channel estimation performance with 1-bit spatial sigma-delta ADCs depends on the quantization noise modeling. Therefore, we present a new method for modeling the quantization noise by leveraging the deterministic input-output relation of the 1-bit spatial sigma-delta ADC. Using this new noise model, we propose an algorithm for channel estimation for a narrowband single-user mmWave line-of-sight MIMO system by determining the unknown angles and path attenuati…
This paper focuses on channel estimation in single-user and multi-user MIMO systems with multi-antenna base stations equipped with 1-bit spatial sigma-delta analog-to-digital converters (ADCs). A careful selection of the quantization voltage level and phase shift used in the feedback loop of 1-bit sigma-delta ADCs is critical to improve its effective resolution. We first develop a quantization noise model for 1-bit spatial sigma-delta ADCs. Using the developed noise model, we then present a two-step channel estimation algorithm to estimate a multipath channel parameterized by the gains, angles…
The one-bit spatial Sigma-Delta concept has recently been proposed as an approach for achieving low distortion and low power consumption for massive multi-input multi-output systems. The approach exploits users located in known angular sectors or spatial oversampling to shape the quantization noise away from desired directions of arrival. While reducing the antenna spacing alleviates the adverse impact of quantization noise, it can potentially deteriorate the performance of the massive array due to excessive mutual coupling. In this paper, we analyze the impact of mutual coupling on the uplink…
We address the connection between the multiple-description (MD) problem and Delta-Sigma quantization. The inherent redundancy due to oversampling in Delta-Sigma quantization, and the simple linear-additive noise model resulting from dithered lattice quantization, allow us to construct a symmetric and time-invariant MD coding scheme. We show that the use of a noise shaping filter makes it possible to trade off central distortion for side distortion. Asymptotically as the dimension of the lattice vector quantizer and order of the noise shaping filter approach infinity, the entropy rate of the di…
In Analog-to-digital (A/D) conversion, signal decimation has been proven to greatly improve the efficiency of data storage while maintaining high accuracy. When one couples signal decimation with the $ΣΔ$ quantization scheme, the reconstruction error decays exponentially with respect to the bit-rate. In this study, similar results have been proven for finite unitarily generated frames. We have devised a process called alternative decimation on finite frames that is compatible with $ΣΔ$ quantization up to the second order. In both cases, decimation results in exponential error decay with respec…
In this paper, we study the quantization errors of modulo sigma-delta modulated finite, asymptotically-infinite, infinite causal stable ARMA processes. We prove that the normalized quantization error can be taken as a uniformly distributed white noise for all the cases. Moreover, we find that this nice property is guaranteed by two different mechanisms: the high-enough quantization resolution \cite{Bennett1948}-\cite{WidrowKollar2008} and the asymptotic convergence of quantization errors for some quasi-stationary processes \cite{ChouGray1991}-\cite{LiChenLiZhang2009}, for different cases. But …
We consider channel estimation for an uplink massive multiple-input multiple-output (MIMO) system where the base station (BS) uses an array with low-resolution (1-2 bit) analog-to-digital converters and a spatial Sigma-Delta ($ΣΔ$) architecture to shape the quantization noise away from users in some angular sector. We develop a linear minimum mean squared error (LMMSE) channel estimator based on the Bussgang decomposition that reformulates the nonlinear quantizer model using an equivalent linear model plus quantization noise. We also analyze the uplink achievable rate with maximal ratio combin…
In Analog-to-digital (A/D) conversion, signal decimation has been proven to greatly improve the efficiency of data storage while maintaining high accuracy. When one couples signal decimation with the $ΣΔ$ quantization scheme, the reconstruction error decays exponentially with respect to the bit-rate. We build on our previous result, which extends signal decimation to finite frames, albeit only up to the second order. In this study, we introduce a new scheme called adapted decimation, which yields polynomial reconstruction error decay rate of arbitrary order with respect to the oversampling rat…
Quantization of compressed sensing measurements is typically justified by the robust recovery results of Candès, Romberg and Tao, and of Donoho. These results guarantee that if a uniform quantizer of step size $δ$ is used to quantize $m$ measurements $y = Φx$ of a $k$-sparse signal $x \in \R^N$, where $Φ$ satisfies the restricted isometry property, then the approximate recovery $x^#$ via $\ell_1$-minimization is within $O(δ)$ of $x$. The simplest and commonly assumed approach is to quantize each measurement independently. In this paper, we show that if instead an $r$th order $ΣΔ$ quantization …
In this paper, we study the problem of storing an archive of versioned data in a reliable and efficient manner in distributed storage systems. We propose a new storage technique called differential erasure coding (DEC) where the differences (deltas) between subsequent versions are stored rather than the whole objects, akin to a typical delta encoding technique. However, unlike delta encoding techniques, DEC opportunistically exploits the sparsity (i.e., when the differences between two successive versions have few non-zero entries) in the updates to store the deltas using compressed sensing te…
We propose the use of low bit-depth Sigma-Delta and distributed noise-shaping methods for quantizing the Random Fourier features (RFFs) associated with shift-invariant kernels. We prove that our quantized RFFs -- even in the case of $1$-bit quantization -- allow a high accuracy approximation of the underlying kernels, and the approximation error decays at least polynomially fast as the dimension of the RFFs increases. We also show that the quantized RFFs can be further compressed, yielding an excellent trade-off between memory use and accuracy. Namely, the approximation error now decays expone…
Sigma Delta quantization, a quantization method which first surfaced in the 1960s, has now been used widely in various digital products such as cameras, cell phones, radars, etc. The method samples an input signal at a rate higher than the Nyquist rate, thus achieves great robustness to quantization noise. Compressed Sensing (CS) is a frugal acquisition method that utilizes the possible sparsity of the signals to reduce the required number of samples for a lossless acquisition. One can deem the reduced number as an effective dimensionality of the set of sparse signals and accordingly, define a…
Sigma-Delta modulation is a popular method for analog-to-digital conversion of bandlimited signals that employs coarse quantization coupled with oversampling. The standard mathematical model for the error analysis of the method measures the performance of a given scheme by the rate at which the associated reconstruction error decays as a function of the oversampling ratio $λ$. It was recently shown that exponential accuracy of the form $O(2^{-rλ})$ can be achieved by appropriate one-bit Sigma-Delta modulation schemes. By general information-entropy arguments $r$ must be less than 1. The curren…
We show how cyclic $(f,σ,δ)$-codes over finite rings canonically induce a $\mathbb{Z}$-lattice in $\mathbb{R}^N$ by using certain quotients of orders in nonassociative division algebras defined using the skew polynomial $f$. This construction generalizes the one using certain $σ$-constacyclic codes by Ducoat and Oggier, which used quotients of orders in non-commutative associative division algebras defined by $f$, and can be viewed as a generalization of the classical Construction A for lattices from linear codes. It has the potential to be applied to coset coding, in particular to wire-tap co…
Visible light communications (VLC) are motivated by the radio-frequency (RF) spectrum crunch and fast-growing solid-state lighting technology. VLC relies on white light emitting diodes (LEDs) to provide communication and illumination simultaneously. Simple two-level on-off keying (OOK) and pulse-position modulation (PPM) are supported in IEEE standard due to their compatibility with existing constant current LED drivers, but their low spectral efficiency have limited the achievable data rates of VLC. Orthogonal frequency division multiplexing (OFDM) has been applied to VLC due to its high spec…
Compressed sensing posits that, within limits, one can undersample a sparse signal and yet reconstruct it accurately. Knowing the precise limits to such undersampling is important both for theory and practice. We present a formula that characterizes the allowed undersampling of generalized sparse objects. The formula applies to Approximate Message Passing (AMP) algorithms for compressed sensing, which are here generalized to employ denoising operators besides the traditional scalar soft thresholding denoiser. This paper gives several examples including scalar denoisers not derived from convex …
Coarsely quantized MIMO signalling methods have gained popularity in the recent developments of massive MIMO as they open up opportunities for massive MIMO implementation using cheap and power-efficient radio-frequency front-ends. This paper presents a new one-bit MIMO precoding approach using spatial Sigma-Delta ($ΣΔ$) modulation. In previous one-bit MIMO precoding research, one mainly focuses on using optimization to tackle the difficult binary signal optimization problem that arises from the precoding design. Our approach attempts a different route. Assuming angular MIMO channels, we apply …
Towards Digital Halftoning on Closed Manifolds--An Error Diffusion Scheme for the $2D$ Torus based on Sigma-Delta Quantization along the Rank-one Lattice
Technische Universität Darmstadt（达姆施塔特工业大学）；Technical University of Munich（慕尼黑工业大学）
In signal quantization, it is well-known that introducing adaptivity to quantization schemes can improve their stability and accuracy in quantizing bandlimited signals. However, adaptive quantization has only been designed for one-dimensional signals. The contribution of this paper is two-fold: i). we propose the first family of two-dimensional adaptive quantization schemes that maintain the same mathematical and practical merits as their one-dimensional counterparts, and ii). we show that both the traditional 1-dimensional and the new 2-dimensional quantization schemes can effectively quantiz…
We study the problem of approximately recovering signals on a manifold from one-bit linear measurements drawn from either a Gaussian ensemble, partial circulant ensemble, or bounded orthonormal ensemble and quantized using Sigma-Delta or distributed noise shaping schemes. We assume we are given a Geometric Multi-Resolution Analysis, which approximates the manifold, and we propose a convex optimization algorithm for signal recovery. We prove an upper bound on the recovery error which outperforms prior works that use memoryless scalar quantization, requires a simpler analysis, and extends the cl…
Compressed sensing is now established as an effective method for dimension reduction when the underlying signals are sparse or compressible with respect to some suitable basis or frame. One important, yet under-addressed problem regarding the compressive acquisition of analog signals is how to perform quantization. This is directly related to the important issues of how "compressed" compressed sensing is (in terms of the total number of bits one ends up using after acquiring the signal) and ultimately whether compressed sensing can be used to obtain compressed representations of suitable signa…
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 $\mathcal{X} \subset \mathbb{R}^n$, associated with the Euclidean metric, with points in the cube $\{\pm 1\}^m$ and we associate the cube with a pseudo-metric that approximates Euclidean distance among points in $\mathcal{X}$. 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…
