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Sparse target counting and localization in sensor networks based on compressive sensing
Abstract
In this paper, we propose a novel compressive sensing (CS) based approach for sparse target counting and positioning in wireless sensor networks. While this is not the first work on applying CS to count and localize targets, it is the first to rigorously justify the validity of the problem formulation. Moreover, we propose a novel greedy matching pursuit algorithm (GMP) that complements the well-known signal recovery algorithms in CS theory and prove that GMP can accurately recover a sparse signal with a high probability. We also propose a framework for counting and positioning targets from multiple categories, a novel problem that has never been addressed before. Finally, we perform a comprehensive set of simulations whose results demonstrate the superiority of our approach over the existing CS and non-CS based techniques.
... The importance of multi-target positioning technology has become increasingly prominent, which has attracted extensive attention of scholars. Sparse representation method provides a new and effective perspective to deal with multi-target location problems [4][5][6][7][8]. ...
... In [4], an original framework of multi-target localization based on sparse representation is proposed. Then, in the framework, Zhang et al. [5] proved that the location dictionary satisfies the restricted isometric property, and proposed a greedy matching pursuit algorithm for target location based on orthogonal matching pursuit (OMP) algorithm. Afterwards, several methods have been proposed to solve the multitarget positioning problem by transforming it into a sparse coding problem through the grid discretization of perceptive region [6][7][8]. ...
For large-scale high-dimensional positioning scenes, the massive number of grid points brings challenges to the multi-target positioning algorithms based on compressed sensing. To cope with the challenges, a fast multi-target localization method based on direction of arrival is proposed. A compressed sensing model is constructed for multi-target localization based on the DOA sequence measured by positioning nodes. Then, a two-stage fast matching pursuit algorithm is presented for sparse reconstruction, which consists of preliminary estimation and supports rectification. A process similar to orthogonal matching pursuit algorithm is adopted to get preliminary estimate result, but no nonlinear operations is employed for complexity reduction. Then another iterative process is carried out to rectify the chosen supports in preliminary result sequentially. Simulation results verify the effectiveness and accuracy of the proposed method for multi-target localization.
... , ψ G ] ∈ R G×G is a basis matrix, and each column ψ g ∈ R G×1 in Ψ denotes a basis vector. For a given signal x ∈ R G×1 , it can be expressed as [39] ...
... To simplify the notations, we first define κ = B/T p , and ∆F m = (m − 1) ∆f B, then the LFM signal (39) can be expressed as s m (t) = 1 T p e jπ(κt 2 +∆Fmt) (52) Substituting (52) into the first term of (40), we have ...
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Target localization is one of the most important research topics in the field of radar signal processing. In this paper, the problem of multi-target counting and localization in the distributed multiple-input multiple-output (MIMO) radar is investigated. We first analyze the theoretical bound of the multi-target localization accuracy in the discrete time signal model. It is determined by the Cramer-Rao lower bound (CRLB) at low signal-to-noise ratio (SNR) and the sampling lower bound (SLB) when the SNR is high. Furthermore, an innovative multi-target counting and localization scheme is developed, which is based on the energy modeling of the multiple transmitter-receiver paths and the compressive sensing theory. To solve the sparse vector recovery issue, we design a lightweight iterative greedy pursuit algorithm including the similarity evaluation strategy. The proposal utilizes the samples of the raw signals and belongs to the category of the direct localization. Nevertheless, it has significantly higher computational efficiency and lower communication burden than the conventional direct localization methods, while avoids the complex data association that encountered by the indirect localization methods. Finally, the simulation results validate the effectiveness and robustness of the proposed method.
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... This positioning technology relies solely on the sparsity of received signal strengths and receiver location information to locate multiple signal sources, enabling the positioning of several simultaneous cofrequency signal sources. One study [25] has rigorously demonstrated the efficacy of the sparse optimization problem formulation for positioning and proven that the sensing matrix satisfies the Restricted Isometry Property (RIP). ...
To address the threat posed by unknown signal sources within Mobile Crowd Sensing (MCS) systems to system stability and to realize effective localization of unknown sources in long-distance scenarios, this paper proposes a unilateral branch ratio decision algorithm (UBRD). This approach addresses the inadequacies of traditional sparse localization algorithms in long-distance positioning by introducing a time–frequency domain composite block sparse localization model. Given the complexity of localizing unknown sources, a unilateral branch ratio decision scheme is devised. This scheme derives decision thresholds through the statistical characteristics of branch residual ratios, enabling adaptive control over iterative processes and facilitating multisource localization under conditions of remote blind sparsity. Simulation results indicate that the proposed model and algorithm, compared to classic sparse localization schemes, are more suitable for long-distance localization scenarios, demonstrating robust performance in complex situations like blind sparsity, thereby offering broader scenario adaptability.
... The sparse signal reconstruction algorithm is based on the compressed sensing algorithm, which breaks through the Nyquist sampling limit, and realizes the sparse reconstruction of signals with fewer observations. A greedy matching pursuit method proposed by Zhang [12] is suitable for emitter localization using the orthogonal matching pursuit (OMP) algorithm [13,14], where the residual satisfies the orthogonal relationship with the selected column atoms in the over-complete dictionary [15]. On the basis of the block sparsity in the dictionary matrices, a block-OMP method, which improves the step of calculating the residual and column atoms in the OMP algorithm, was proposed by Eldar [16]. ...
The sparse direct position determination (DPD) method requires reconstructing the emitter position with prior knowledge. However, in non-cooperative localization scenarios, it is difficult to reconstruct the transmitted signal with the unknown signal form and propagation model. In this paper, a sparse DPD method based on time-difference-of-arrival (TDOA) information in correlation-domain is proposed. Different from the traditional sparse DPD method, the received signal is converted into correlation-domain, and the proposed dictionary matrix is generated by the quantized delay difference, which solves the pseudo-positioning problem. Compared to the conventional multi-signal classification (MUSIC) method, multi-frequency fusion (MFF) method, and two-step positioning algorithm, the proposed algorithm achieves higher positioning accuracy. The feasibility of the algorithm has been verified by both simulation and real-world measured tests.
... Compressed sensing (CS) has been widely used in many applications, such as channel estimation [1]- [3], data detection [4], [5], target localization [6], [7], etc. For a standard compressed sensing problem, a sparse signal x ∈ C N ×1 is to be recovered from measurements y ∈ C M ×1 (M < N ) under a linear observation model, ...
For many practical applications in wireless communications, we need to recover a structured sparse signal from a linear observation model with dynamic grid parameters in the sensing matrix. Conventional expectation maximization (EM)-based compressed sensing (CS) methods, such as turbo compressed sensing (Turbo-CS) and turbo variational Bayesian inference (Turbo-VBI), have double-loop iterations, where the inner loop (E-step) obtains a Bayesian estimation of sparse signals and the outer loop (M-step) obtains a point estimation of dynamic grid parameters. This leads to a slow convergence rate. Furthermore, each iteration of the E-step involves a complicated matrix inverse in general. To overcome these drawbacks, we first propose a successive linear approximation VBI (SLA-VBI) algorithm that can provide Bayesian estimation of both sparse signals and dynamic grid parameters. Besides, we simplify the matrix inverse operation based on the majorization-minimization (MM) algorithmic framework. In addition, we extend our proposed algorithm from an independent sparse prior to more complicated structured sparse priors, which can exploit structured sparsity in specific applications to further enhance the performance. Finally, we apply our proposed algorithm to solve two practical application problems in wireless communications and verify that the proposed algorithm can achieve faster convergence, lower complexity, and better performance compared to the state-of-the-art EM-based methods.
As a paradigm to recover the sparse signal from a small set of linear measurements, compressed sensing (CS) has stimulated a great deal of interest in recent years. In order to apply the CS techniques to wireless communication systems, there are a number of things to know and also several issues to be considered. However, it is not easy to come up with simple and easy answers to the issues raised while carrying out research on CS. The main purpose of this paper is to provide essential knowledge and useful tips that wireless communication researchers need to know when designing CS-based wireless systems. First, we present an overview of the CS technique, including basic setup, sparse recovery algorithm, and performance guarantee. Then, we describe three distinct subproblems of CS, viz., sparse estimation, support identification, and sparse detection, with various wireless communication applications. We also address main issues encountered in the design of CS-based wireless communication systems. These include potentials and limitations of CS techniques, useful tips that one should be aware of, subtle points that one should pay attention to, and some prior knowledge to achieve better performance. Our hope is that this article will be a useful guide for wireless communication researchers and even non-experts to grasp the gist of CS techniques.
Compressive Sensing, as an emerging technique in signal processing is reviewed in this paper together with its common applications. As an alternative to the traditional signal sampling, Compressive Sensing allows a new acquisition strategy with significantly reduced number of samples needed for accurate signal reconstruction. The basic ideas and motivation behind this approach are provided in the theoretical part of the paper. The commonly used algorithms for missing data reconstruction are presented. The Compressive Sensing applications have gained significant attention leading to an intensive growth of signal processing possibilities. Hence, some of the existing practical applications assuming different types of signals in real-world scenarios are described and analyzed as well.
The sparsity of the localization problem makes the Compression Sensing (CS) theory suitable for indoor localization in Wireless Local Area Networks (WLAN). However, in practice, we find that the location errors and computing complexity increase significantly as the dimensionality of the sparse vector and measurement matrix are high in a large environment, so most CS-based localization techniques are accompanied by coarse localization and AP selection stages. Therefore, in this paper, we first deduced the relationship between the number of Access Points (APs) and the dimensionality of the sparse vector theoretically to give the guideline that the number of sub-databases and APs should be obtained. Then an Adaptive Intuitionistic Fuzzy C-ordered Mean (AIFCOM) clustering is designed for the data with outliers in the environment with multipath effects. Finally, in the fine localization stage, we propose a Semi-tensor Product Compression Sensing (STP-CS) model to construct the measurement matrix, compared with the traditional CS model, our model not only remains more number of APs, but also decreases the dimensionality of measurement matrix, which can reduce the storage space and improve localization accuracy simultaneously.
Target localization is one of the most important research topics in the field of radar signal processing. In this article, the problem of multitarget enumeration and localization in the distributed multiple-input multiple-output radar with noncoherent processing mode is investigated. We first analyze the theoretical bound of the multitarget localization accuracy under the discrete time signal model and the Swerling 1 target model. It is determined by the Cram
r–Rao lower bound at a low signal-to-noise ratio (SNR) and the sampling lower bound when the SNR is high. Furthermore, an innovative multitarget enumeration and localization scheme is developed, which is based on the energy modeling of the multiple transmitter–receiver paths and the compressive sensing theory. To solve the sparse vector recovery issue, we design a lightweight iterative greedy pursuit algorithm including the similarity evaluation strategy. In addition, an iterative-based target position refinement process is designed to alleviate the off-grid problem caused by the spatial discretization. The proposal utilizes the samples of the raw signals and belongs to the category of the direct localization. Nevertheless, it has significantly higher computational efficiency and lower data communication burden than the conventional direct localization methods, while avoiding the complex data association encountered by the indirect localization methods. Finally, the simulation results validate the effectiveness and robustness of the proposed method.
This article proposes a sparsity-aware channel estimation scheme for reconfigurable intelligent surface (RIS)-assisted wireless communications. We present an angular domain-channel sparsity model in a closed-form mathematical expression. A comprehensive formulation of the RIS channel estimation problem based on the sparsity analysis and RIS reflection model is also presented in this manuscript. This work aims to achieve a practical RIS beamforming algorithm without requiring excessive training overhead or any sensor deployment. We achieve the goal by proposing two sparsity-aware RIS channel estimation schemes based on the compressive sensing (CS) algorithms such as the Dantzig selector (DS) and orthogonal matching pursuit (OMP). Differently from the existing works on the CS algorithms for RIS, we consider a fully passive RIS without any active sensor. We validate the theory and algorithm through both simulations and experiments. We have experimented orthogonal frequency division multiplexing (OFDM) communications on our 5.8 GHz 1-bit RIS testbed with QPSK, 16QAM, 64QAM, and 256QAM modulation schemes. By experiments, it is shown that the proposed scheme is able to adaptively form a beam toward the receiver and improves the quality of the wireless communication to a notable level. Thanks to the properties of the proposed scheme, the wireless channel can be estimated without excessive training time and complexity.
We introduce an algorithm, called matching pursuit , that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions. These waveforms are chosen in order to best match the signal structures. Matching pursuits are general procedures to compute adaptive signal representations. With a dictionary of Gabor functions a matching pursuit defines an adaptive time-frequency transform. We derive a signal energy distribution in the time-frequency plane, which does not include interference terms, unlike Wigner and Cohen class distributions. A matching pursuit isolates the signal structures that are coherent with respect to a given dictionary. An application to pattern extraction from noisy signals is described. We compare a matching pursuit decomposition with a signal expansion over an optimized wavepacket orthonormal basis, selected with the algorithm of Coifman and Wickerhauser.
We solve the problem of counting the total number of observable targets (e.g., persons, vehicles, landmarks) in a region using local counts performed by a dense field of sensors, each of which measures the number of targets nearby but not their identities nor any positional information. We formulate and solve several such problems based on the types of sensors and mobility of the targets. The main contribution of this paper is the adaptation of a topological integration theory — integration with respect to Euler characteristic — to yield complete solutions to these problems. 1. Topological enumeration. 1.1. Sensors. Sensor networks are poised to impact society in fundamental ways anal-ogous to the impact of the personal computer and the internet. The rapid development of small-scale sensor devices coupled with wireless ad hoc networking capability is giving birth to a wide variety of sensor networks for applications to agriculture, defense, environmental monitoring, and more. At present, there are severe limits on sizes and types of implementable networks. In partic-ular, there are trade-offs between power consumption, sensing complexity (how much data is gathered and processed on-board), sensor size, sensor range (how large a neighborhood of the sensor is scanned), and communication bandwidth. Current implementations of sen-sor networks in environmental and agricultural monitoring are limited to dozens or at most hundreds of sensors (see e.g., [20] for a typical deployment). This constraint will not persist.
In the letter we describe signal transmitted between two dual-polarized antennas using a quaternion notation. The channel can then be modeled by a single quaternion gain, instead of a matrix of four complex gains as it is done in a classical approach. The model is very useful as it allows for a simple accommodation of any polarization twist between the transmit and receive antennas, as well as, modeling random fading in both polarizations with or without cross-polar scattering
This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f∈CN and a randomly chosen set of frequencies Ω. Is it possible to reconstruct f from the partial knowledge of its Fourier coefficients on the set Ω? A typical result of this paper is as follows. Suppose that f is a superposition of |T| spikes f(t)=στ∈Tf(τ)δ(t-τ) obeying |T|≤CM·(log N)-1 · |Ω| for some constant CM>0. We do not know the locations of the spikes nor their amplitudes. Then with probability at least 1-O(N-M), f can be reconstructed exactly as the solution to the ℓ1 minimization problem. In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for CM which depend on the desired probability of success. Our result may be interpreted as a novel kind of nonlinear sampling theorem. In effect, it says that any signal made out of |T| spikes may be recovered by convex programming from almost every set of frequencies of size O(|T|·logN). Moreover, this is nearly optimal in the sense that any method succeeding with probability 1-O(N-M) would in general require a number of frequency samples at least proportional to |T|·logN. The methodology extends to a variety of other situations and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one- or two-dimensional) object from incomplete frequency samples - provided that the number of jumps (discontinuities) obeys the condition above - by minimizing other convex functionals such as the total variation of f.
An empirical formula for propagation loss is derived from Okumura's report in order to put his propagation prediction method to computational use. The propagation loss in an urban area is presented in a simple form: A + B log10R, where A and B are frequency and antenna height functions and R is the distance. The introduced formula is applicable to system designs for UHF and VHF land mobile radio services, with a small formulation error, under the following conditions: frequency range 100-1500 MHz, distance 1-20 km, base station antenna height 30-200 m, and vehicular antenna height 1-10 m.
In this paper we consider a mobile cooperative network that is tasked with building a map of the spatial variations of a parameter of interest, such as an obstacle map or an aerial map. We propose a new framework that allows the nodes to build a map of the parameter of interest with a small number of measurements. By using the recent results in the area of compressive sensing, we show how the nodes can exploit the sparse representation of the parameter of interest in the transform domain in order to build a map with minimal sensing. The proposed work allows the nodes to efficiently map the areas that are not sensed directly. To illustrate the performance of the proposed framework, we show how the nodes can build an aerial map or a map of obstacles with sparse sensing. We furthermore show how our proposed framework enables a novel non-invasive approach to mapping obstacles by using wireless channel measurements.
We demonstrate a simple greedy algorithm that can reliably recover a vector v ?? ??d from incomplete and inaccurate measurements x = ?? v + e . Here, ?? is a N x d measurement matrix with N <<d, and e is an error vector. Our algorithm, Regularized Orthogonal Matching Pursuit (ROMP), seeks to provide the benefits of the two major approaches to sparse recovery. It combines the speed and ease of implementation of the greedy methods with the strong guarantees of the convex programming methods. For any measurement matrix ?? that satisfies a quantitative restricted isometry principle, ROMP recovers a signal v with O ( n ) nonzeros from its inaccurate measurements x in at most n iterations, where each iteration amounts to solving a least squares problem. The noise level of the recovery is proportional to ??{log n } || e ||2. In particular, if the error term e vanishes the reconstruction is exact.
Compressive sampling offers a new paradigm for acquiring signals that are compressible with respect to an orthonormal basis. The major algorithmic challenge in compressive sampling is to approximate a compressible signal from noisy samples. This paper describes a new iterative recovery algorithm called CoSaMP that delivers the same guarantees as the best optimization-based approaches. Moreover, this algorithm offers rigorous bounds on computational cost and storage. It is likely to be extremely efficient for practical problems because it requires only matrix–vector multiplies with the sampling matrix. For compressible signals, the running time is just O(Nlog2N), where N is the length of the signal.