Algorithms for Compressive Sensing Signal Reconstruction with Applications
View this Special IssueResearch Article  Open Access
Ming Yin, Kai Yu, Zhi Wang, "Compressive Sensing Based Sampling and Reconstruction for Wireless Sensor Array Network", Mathematical Problems in Engineering, vol. 2016, Article ID 9641608, 11 pages, 2016. https://doi.org/10.1155/2016/9641608
Compressive Sensing Based Sampling and Reconstruction for Wireless Sensor Array Network
Abstract
For lowpower wireless systems, transmission data volume is a key property, which influences the energy cost and time delay of transmission. In this paper, we introduce compressive sensing to propose a compressed sampling and collaborative reconstruction framework, which enables realtime direction of arrival estimation for wireless sensor array network. In sampling part, random compressed sampling and 1bit sampling are utilized to reduce sample data volume while making little extra requirement for hardware. In reconstruction part, collaborative reconstruction method is proposed by exploiting similar sparsity structure of acoustic signal from nodes in the same array. Simulation results show that proposed framework can reach similar performances as conventional DoA methods while requiring less than 15% of transmission bandwidth. Also the proposed framework is compared with some data compression algorithms. While simulation results show framework’s superior performance, field experiment data from a prototype system is presented to validate the results.
1. Introduction
The monitoring of noncooperative targets using spatially distributed arrays has been one of the key problems in many applications such as biological acoustic studies [1], gunshot localization [2], and military target tracking [3]. Among all array signal processing methods, direction of arrival (DoA) estimation is an important one for source detection and localization.
In traditional array systems, arrays are connected by wires to a fusion center. In some other systems, like IAASNT [3], sensors in the same array are connected to a local processor. DoA estimation (or other array processing) is done there and only results are transmitted wirelessly to a remote fusion center. But the processors (like DSP) are usually too energyconsuming as well as expensive. These two weaknesses make largescale wireless deployment and maintenance difficult.
With advances in wireless communication, a few pioneer works [4, 5] deployed wireless nodes at spatially dispersed locations to form arrays. Then the AML [4] localization algorithm was used to locate acoustic sources, which validates the feasibility of wireless sensor array network (WSAN). However the energy and data volume for transmission were not considered. The large amount of raw sensor data needs to be communicated via broadband WiFi channels. Allen et al. [6] proposed the VoxNet system, which uses 802.11b wireless protocol to transmit data to the fusion center for processing. Even some strategies are used to reduce data volume; the lifetime is only about 8 hours. This huge energy consumption, along with a rather long transmission delay, would largely harm system performance. Aiming at longer independent detection, lowpower, lowerrate wireless protocol is indispensable. Thus many works focus on data reduction issue, like [7].
In this work, we also consider a WSAN configuration. Based on compressive sensing (CS) [8, 9], we propose a compressed sampling and collaborative reconstruction framework that can achieve obvious data reduction while keeping similar DoA performance. Instead of transmitting all sensor data to the fusion center, random sampling and deep quantization are performed to significantly reduce the amount of data that needs to be transmitted via wireless channel. Also, unlike traditional data compression algorithm, we place major computation burden at the fusion center side, requiring little computation at individual sensor node. Thus, this framework is very appealing to applications whose resources (energy, transmission, cost, etc.) are limited.
Our framework leverages sparsity in both frequency and space domain. We assume the acoustic signals of sources can be modeled as a sum of few dominant harmonic frequencies. As such, the time series at each sensor has a sparse spectral representation including few dominant spectral lines. Furthermore, sensors belonging to the same array often receive signal with the same frequency components, with only a phase shift due to time difference of arrival, because they are different in distance to the source. Leveraging these two sparsities with compressive sensing theory, we could reduce the data volume to our best.
We employed two methods to exploit the frequency sparsity in sampling the source signals: random sampling of the sensor data and deep quantization of sampled sensor data to the extent that only the sign bit is retained [10, 11]. To exploit the space domain sparsity, we only apply full reconstruction procedure on sampled data from the reference sensor node. Then the recovered harmonic frequencies’ structure will be used to reconstruct the sensor data of remaining sensor nodes using easier methods like least square approach with far fewer data samples than those sampled on the reference node. The procedure of the framework is depicted in Figure 1.
Previously, various works combined sparse reconstruction and array signal processing. Malioutov et al. [12] exploited the sparse distribution characteristic of sources in the angle domain and formulated array processing as a multiple measurement vector (MMV) problem. Baron et al. [13] proposed the joint sparsity model which can be used in sensor arrays. Also many works tried to introduce compressive sensing into wireless sensor network localization problem, like [14, 15]. However, the problem of data volume for transmission was not solved. Here in our work, our framework provides an acceptable solution to this problem for lowpower, lowcost WSAN systems. After successful reconstruction of the array data at the fusion center, either conventional DoA methods such as MUSIC [16] or the sparse representation based approach [12] may be applicable.
Although many differences exist, we compare our framework with modern data compression algorithms to show our data volume reduction ability, including the lossless weighted LZW coding [17], lossy JPEG [18], and MPEG1 [19]. Also a prototype microphone array platform has been implemented with offtheshelf devices. Experiment results are presented that convincingly validate our framework’s performance. The main contributions of this paper can be summarized as follows:(a)A novel compressed sampling collaborative reconstruction signal processing framework is proposed.(b)Random compressed sampling and a newly emerged 1bit compressed sampling method are employed. Thus sampling data volume is reduced, significantly cutting down the transmission burden.(c)Leveraging the high correlation of acoustic signals among array elements, a collaborative reconstruction method, works for both random compressed sampling and 1bit compressive sensing and is presented to drastically reduce computation cost and delay.(d)Numerical simulation and prototype experiment results are presented to validate the performance of the framework.
This paper is organized as follows: backgrounds of DoA estimation, compressive sensing, and 1bit compressive sensing are simply reviewed in Section 2. Our framework is presented in detail in Sections 3 and 4, about sampling and reconstruction, respectively. Simulation and experiment results are reported in Section 5 along with some analysis. And conclusion is presented in Section 6.
2. Background
2.1. Array Processing for DoA
Assume there are sources in the sensing field. The th source emits signal consisting of some harmonics: where and are the magnitude and the phase of the th harmonic at frequency and represents other less prominent components. For convenience, one assumes that each is an integral multiple of ().
The signal received at the th sensor is the summation of all signals from the sources. That is,where the reference sensor node receives from the th acoustic source. is the relative difference of propagation delay of th source’s signal from the reference node to the th sensor. is the additive Gaussian white noise at the th sensor with zeromean and variance .
Figure 2 shows the array model, in which is the reference orientation, is sensor position for the th sensor, and is target (source) position. In this work, we assume that the diameter of the sensor array is much smaller than the sensortotarget distance. That is,
As such, the acoustic wave emitted from the sources can be regarded as plane wave. All sensors in the same array should share the same incidence angle for signal from the same source. It is easy to verify that the relative time delay of the th source between the th sensor and the array centroid is
If we represent in the discrete Fourier basis , the corresponding Fourier coefficients can be given by
Consider all the sensors, the array data spectrum at the th frequency is given bywhere is defined as the steering vector corresponding to the th sources at the th frequency.
With traditional high resolution DoA spectral estimation, the sample covariance matrix of array data spectrum is computed. That is,Exploiting the low rank structure of the signal covariance matrix , numerous DoA methods, such as ESPRIT and MUSIC [16], have been proposed. Consider eigenvalue decomposition of the matrix :where and are diagonal matrices consisting of the eigenvalues of such that the eigenvalues in are larger than those in . Equation (8) implies that the steering vectors must lie within the span of the signal subspaces and perpendicular to the noise subspace spanned by columns of . With the MUSIC method, the DoA of targets are estimated as the peaks of the following function:
2.2. Compressive Sensing
Compressive sensing theory [8, 9] states that a sparse signal may be randomly sampled at subNyquist rate and then be reconstructed perfectly. Denote to be a sparse vector representing the sparse signal, to be the original signal, and to be the observed signal. According to compressive sensing theory, the observation matrix can be chosen to be a Gaussian random matrix or Bernoulli random matrix.
Given , the CS reconstruction problem is formulated as a constrained optimization problem:where is a preset threshold and is the norm of . The above problem formulation leads to various CS reconstruction algorithms, such as Compressive Sensing Matching Pursuit (CoSaMP), Orthogonal Matching Pursuit (OMP), magic [20], and Lasso.
2.3. 1Bit Compressive Sensing
Unlike conventional CS, 1bit compressive sensing only keeps the sign bit of measurements and discards the magnitude information [10]. In matrix form,
While traditional CS methods use equation constraint, 1bit CS reconstruction can be accomplished via optimization problem with an inequality constraint:where and the inequality is applied for each component of the vector. Thus, the 1bit CS reconstruction optimization problem can be expressed as follows:
The norm is used as a cost function that enforces the sparseness of under the sign constraints. Since the amplitude information is discarded in 1bit sampling, a unit sphere constraint is introduced to avoid getting the obvious wrong solution of .
Several algorithms have been proposed to solve the 1bit CS reconstruction problem. These include 1bit Linear Programming (1bit LP), Matching Sign Pursuit (MSP) algorithm, Binary Iterative Hard Thresholding (BIHT), RestrictedStep Shrinkage (RSS), and Fixed Point Continuation (FPC).
2.4. Multiple Measurement Vectors
For multiple sparse signal with the same sparse pattern, the multiple measurement vectors (MMV) [21] formulation utilizes the joint sparse in reconstruction procedure. The MMV problem can be formulated as solving multiple underdetermined systems of linear equations:where is a full rank matrix and is observation matrix consisting of consecutive observations of sparse vector represented by . It is further assumed that each column of the matrix has the same sparse structure.
Lots of algorithms have been proposed that exploit the joint sparsity in different ways. These include Simultaneous Orthogonal Matching Pursuit (SOMP), Reduce MMV and Boost (ReMBo), svd [12], and Multiple FOCal Underdetermined (MFOCUSS) algorithms.
3. Compressed Sampling
In this section, we provide two sampling methods designed for WSAN. We try to reduce the transmission data volume for array signal, which brings energy consumption and transmission delay issues. Compared with traditional data compression methods, CS performs dimension reduction with projection calculation. Further comparisons will be provided in later sections.
3.1. Random Compressed Sampling
Random projection is an important step in compressive sensing methods. But random number generation and projection calculation are required to perform random projection on nodes. These raise energy consumption and hardware requirement. Inspired by nonuniform sampling method, we combine the nonuniform sampling process with CS framework and model it by a measurement matrix. It is easy to implement and no extra computation is needed on wireless nodes.
So the measurement matrix is designed as follows: the measurement matrix represents the measurement process; each row has only one nonzero component 1 in the position. Following this matrix, the ADC take the samples at . A sequence is set with , andwhere is the random sampling interval between adjacent sample instances. Then the th element of this proposed random sampling matrix is given byAnd this is used as the measurement matrix in the random compressed sampling method. Although we call it random, this matrix (or sampling sequence) could be predefined and saved in nodes and fusion center. Nodes do not have to generate random number themselves.
3.2. 1Bit Compressed Sampling
1bit compressed sampling has its obvious advantage in reducing data volume. On condition that the sampling rate can be supported in sensor node, 1bit compressed sampling will be an appealing solution to WSAN monitoring applications. And there are two ways of implementation on nodes.(1)Hardware Implementation. Use random number generators and comparators on nodes. After random projection of raw signal, a comparator to zero is utilized to quantize the measurement. And, in fact, this kind of comparators is extremely cheap and fast.(2)Software Implementation. Perform random projection in software. The ADC samples at the Nyquist rate and only the signs of projected raw samples are kept. Such method has been successfully used in 1bit SigmaDelta converters, at the expense of very high sampling rate [22].
3.3. The Coherence of Compressed Sampling
To use CSbased methods, the incoherence of measurement matrix and sparsity matrix should be tested. Note that the structure of and the product amount to selecting some rows from the Discrete Fourier Transform (DFT) matrix . Hence their product is a partial Fourier transform matrix (submatrix of full DFT matrix). For the underlying random nonuniform sampling, the selection of these rows is determined by . Applying the Welch bound inequality [23], when is randomly selected from , satisfiesFor 1bit CS, binary stable embedding (BSE) is proposed to characterize 1bit CS system [24], and robust reconstruction is guaranteed when .
4. Collaborative Reconstruction
Conventional CS methods guarantee stable signal recovery with the prior information of sparsity. However, the sparsity is not the only prior information that helps to recover signal in this sensor array case. For spatially neighboring nodes in the same array, they receive highly correlated signals. The correlations are displayed in their similarity in spectrum, although different in phase shift. (They share the same support in frequency domain.) Thus independent reconstruction of individual sensor signal seems redundant.
4.1. Reconstruction for Random Compressed Sampling
Similar to the MMV formulation, the compressed measurements of all sensors can be formulated aswhere , , and . Note that each is a summation of different delay versions of sources signal; has the same sparse pattern. Based on such joint sparsity, the signal reconstruction problem can be described asUnder this joint reconstruction model, the theoretical measurement number of random compressed sampling is [25] while the normal CS requires [8]. The computational complexity of the above joint reconstruction is [21]; it sacrifices the realtime performance of signal reconstruction.
To solve this issue, we propose a collaborative signal reconstruction method. The main idea of this proposed framework is to choose one reference sensor node and reconstruct its signal. Then the support (nonzeros index of a sparse vector) of the reconstructed signal can be utilized to prune the CS problem to a least square problem.
We define to be the support of reference signal. Based on the known support , the joint reconstruction formulation can be simplified towhere and are the submatrices of and with column and row index .
It is obvious that and are and matrices (). In such case, the traditional illdetermined optimization problem is equivalent to solving equations with measurement and unknown variable. When , it has a least squares solution:
In conclusion, when given the support of signal, the signal reconstruction problem is simplified to a least square problem, which means we only need measurements to solve it. The pseudocode of collaborative reconstruction is presented as follows. (magic [20] is used as an example.)
Algorithm 1 (collaborative reconstruction algorithm).
Input Compressed sampling matrix ; Measurements .Output Ksparse approximation .(1)Solve by magic algorithm: (2)Estimate the support of :(3)Obtain the pruned reconstruction matrix:(4)Analytical solution of a least square problem:
4.2. Collaborative Reconstruction for 1Bit Sampling
For 1bit compressed sampling, whose measurements are the sign of projection for the original signal, the nonlinearity between the original signal and the measurements makes it hard to find an analytical solution. However, the obtained support can still help to simplify and boost the reconstruction process.
Similar to the reconstruction for random compressed sampling, the 1bit reconstruction problem can be modeled as where is the th column of .
For both sampling methods above, collaborative reconstruction algorithm simplifies the redundant reconstruction dictionary with the help of signal support. It avoids the redundancy in traditional CS signal reconstruction. Therefore, this algorithm can greatly reduce the computational complexity and increase the success rate of reconstruction while requiring less information. These merits have great value in achieving lowpower, realtime wireless sensing systems.
5. Performance Evaluation
In this section, extensive simulations are carried out to compare the proposed framework with traditional array processing and CS approaches. Consider the modern data compression approach and the wireless channel that has transmission error; our framework is compared with some local data compression approach under different transmission error cases. Finally, a hardware prototype is developed to validate the practical application of the framework through outdoor experiments.
5.1. Experiment and Simulation Settings
Theoretically, our framework works for any array formation when it satisfies the array spacing requirements. Here, a uniform linear array with 6 nodes and a to target space is chosen for both simulation and experiment. Considering that the acoustic signals of cars, trucks, or helicopters are usually dominated by a few harmonics, the source signal is assumed to be summation of harmonics at 500 Hz, 600 Hz, 700 Hz, and 800 Hz. Accordingly, a 0.2 m spacing is adopted to satisfy the halfwavelength requirements of array.
We choose the widely accepted MUSIC [16] algorithm for DoA estimation. The dimension of original signal (also reconstructed signal) in a snapshot is assumed to be 128, and the system sampling rate is set to be 2048 Hz. For expression simplicity in this section, we denote RCS as random compressed sampling, 1bS as 1bit compressed sampling, and CoR as collaborative reconstruction. The joint reconstruction method mentioned in Section 4 is used as benchmark while evaluating the proposed CoR method, denoted as JoR. To have fair comparison, we formulate the random compressed sampling based approach as an SOCP problem and 1bit CS base approach as an LP optimization. Both SOCP and LP are solved by Sedumi toolbox [26].
The first problem of CS in real application is the number of random measurements. Although some theoretic bounds have been proposed, it is still not clear how many samples will bring maximum benefit. Here the SNR of reconstructed signal is chosen as criteria. The SNR of raw signal is 10 dB and the reconstructed signals are compared with uncontaminated signal to obtain the same SNR. Figure 3 shows the relationship between the reconstructed SNR and the corresponding measurements . Table 1 shows the number of measurements necessary for a method to reach original 10 dB SNR.

As we can see here, CSbased methods obviously reduce the data volume, while our RCSCoR and 1bSCoR methods further reduce the number of measurements for nonreference nodes.
5.2. DoA Estimation Comparison
Figure 4 displays the DoA estimation result of three CSbased approaches using above empirical measurements numbers, and the RS (raw signal) denotes array processing with raw data, used as benchmark. First, we use settings from Table 1 for all methods. Results indicate that all methods present spatial spectrum clear enough to tell the source direction. RS has the best performance. For compressed sampling methods, reduction in sample amount caused information loss; therefore they all present worse performance but still clear directions.
(a)
(b)
Further numerical simulations are performed to statistically show the performances of these CSbased frameworks for DoA estimation purpose. The root mean square error (RMSE) and success detection rate of 500 independent trials are used as performance metrics. Here, if a DoA estimation result falls in range of true value, it is viewed as a success.
From the results in Figure 5, we could tell that using full raw signal simply provides the best performances. That makes sense for no information loss introduced. Therefore, the RS results may be considered as true value or benchmark of estimation performances. Other three CSbased methods are inferior to the RS approach, but still in a tolerable range. Among these three CSbased approaches, RCSJoR outperforms RCSCoR and 1bSCoR. This is because the RCSJoR approach is based on a joint optimization while others rely on the sparsity of single sensor.
(a)
(b)
In the above tests, we used settings from Table 1. These settings come from SNR analysis of reconstructed signal. The difference of these CSbased approaches in RMSE and success rate indicates that the criterion of reconstructed SNR may not be the most suitable one. Here, in Figure 6, the RMSE of DoA is chosen as criteria that evaluate different measurement number . To be fair, three times the RMSE value of RS approach is used as performance degradation threshold for each method and each SNR level (presented by dotted lines with corresponding color in Figure 6). This threshold may not be optimum but should provide credible results. Based on such threshold, we choose settings that could guarantee most SNR cases for each method. Then we have for RCSCoR, for RCSJoR, and for 1bSCoR. These settings are for nonreference nodes. Reference nodes still use Table 1 setting.
5.3. Compression Comparison
CSbased methods could also be considered as specialpurpose lossy compression methods. Ignoring algorithm complexity problem, other local data compression approaches can also reduce the data volume before wireless transmission. So here, we compare the performance of CSbased methods with local data compression methods. Three typical data compression methods are considered: the lossless weighted LZW coding [17], lossy JPEG [18], and MPEG audio coding layer 1 [19].
Referring to the technical manual of STM32 series MCU, the validated package load is 100 bytes and the PER (Package Error Rate) is with package of 20 bytes. Based on the provided parameter, the BER (Bit Error Rate) is set to be , , , and (corresponding to PER of , , , and , resp.). Here, we use the values mentioned in previous section, namely, 21 for RCSCoR, 15 for RCSJoR, and 120 bits for 1bSCoR. For methods other than 1bS, we normally use 16bit quantization in following computation.
Figure 7 shows DoA success rate of different BERs, and RMSE are presented as error bars. Obviously, CSbased methods display robust property under all the BERs, while not all existing compression methods can. The LZW has an obvious performance decline when the BER increases; the JPEG compression algorithm is heavily affected by the BERs; only MPEG1 displays a strong error tolerant capability, but not better than CSbased approaches. This is because the CSbased approaches focus on the sparse structure of data and the error of transmission can be regarded as noise and hence eliminated through reconstruction algorithms.
Table 2 lists the compression ratio of CSbased and traditional data compression approaches. The compression ratio for CSbased approaches is much high than that of the conventional compression approaches. Here, LZW has the smallest compression ratio; this may be because the repeatability of data is not good enough and extra data volume is occupied to construct coding dictionary. For different cases, CSbased methods could work with different parameter settings. This flexibility offers them better adaptability for various cases.

5.4. Field Experiment
To validate the proposed framework, an acoustic sensor array prototype system is developed to conduct some field experiments. As shown in Figure 8, this prototype mainly consists of a uniformly spaced linear array with six omnidirectional microphones and a multichannel recorder. Spacing between adjacent microphones is 0.2 m. Figure 9 shows the experiment setting. One source is placed at direction , few metres away. The sampling frequency is 2048 Hz and the DoA is estimated every 3 snapshots (each snapshot with 128 samples).
Figure 10 shows the spatial spectrum of CSbased approaches and also RS as benchmark. The DoA results are for 1bSCoR, for RCSCoR, for RCSJoR, and for RS. This result shows that our framework could work well on real offtheshelf devices.
5.5. Performance Summary
Till now, with simulation and experiment results, we can reach the following conclusions. (a) The proposed framework could significantly reduce sample data volume with a little performance loss; (b) RCSCoR could work with extremely simple nodes while 1bSCoR provide best DoA performance with least data. Which to use depends on the systems requirements.
6. Conclusion
In this paper, we propose a compressed sampling and collaborative reconstruction framework. By exploiting the sparse property of source signal, random compressed sampling and 1bit sampling require much less array data while still providing acceptable performance. By exploiting the high correlation among array signals, collaborative reconstruction effectively reconstructs array signal as well as further reducing the number of samples required for nonreference nodes.
Different methods have different advantages. RCSCoR method could work well with simple devices, and 1bSCoR provides the highest compression ratio. Both numerical simulation and experiments on offtheshelf device are presented to validate the usefulness of the proposed framework. Considering the convenience of implementation on hardware, this framework would be an excellent choice on lowcost, lowpower wireless sensor array network.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (NSFC61273079), in part by the National Natural Science Foundation of China Key Projects (U1509215), in part by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant no. XDA06020201), and in part by the Open Research Project of the State Key Laboratory of Industrial Control Technology, Zhejiang University (no. ICT1600208, no. ICT1600199, and no. ICT1600213).
References
 A. M. Ali, S. Asgari, T. C. Collier et al., “An empirical study of collaborative acoustic source localization,” Journal of Signal Processing Systems, vol. 57, no. 3, pp. 415–436, 2009. View at: Publisher Site  Google Scholar
 A. Ledeczi, A. Nadas, P. Volgyesi et al., “Countersniper system for urban warfare,” ACM Transactions on Sensor Networks, vol. 1, no. 2, pp. 153–177, 2005. View at: Publisher Site  Google Scholar
 Y. Li, Z. Wang, S. Zhuo et al., “The design and implement of acoustic array sensor network platform for online multitarget tracking,” in Proceedings of the 8th IEEE International Conference on Distributed Computing in Sensor Systems (DCOSS '12), pp. 323–328, IEEE, Hangzhou, China, May 2012. View at: Publisher Site  Google Scholar
 J. C. Chen, K. Yao, T. L. Tung, C. W. Reed, and D. Chen, “Source localization and tracking of a wideband source using a randomly distributed beamforming sensor array,” International Journal of High Performance Computing Applications, vol. 16, no. 3, pp. 259–272, 2002. View at: Publisher Site  Google Scholar
 J. C. Chen, L. Yip, J. Elson et al., “Coherent acoustic array processing and localization on wireless sensor networks,” Proceedings of the IEEE, vol. 91, no. 8, pp. 1154–1161, 2003. View at: Publisher Site  Google Scholar
 M. Allen, L. Girod, R. Newton, S. Madden, D. T. Blumstein, and D. Estrin, “VoxNet: an interactive, rapidlydeployable acoustic monitoring platform,” in Proceedings of the International Conference on Information Processing in Sensor Networks (IPSN '08), pp. 371–382, St. Louis, Mo, USA, April 2008. View at: Publisher Site  Google Scholar
 W. Hu, N. Bulusu, C. T. Chou, S. Jha, A. Taylor, and V. N. Tran, “Design and evaluation of a hybrid sensor network for cane toad monitoring,” ACM Transactions on Sensor Networks, vol. 5, no. 1, article 4, pp. 1–28, 2009. View at: Publisher Site  Google Scholar
 D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory, vol. 52, no. 4, pp. 1289–1306, 2006. View at: Publisher Site  Google Scholar  MathSciNet
 E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” Comptes Rendus Mathematique, vol. 346, no. 910, pp. 589–592, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 P. T. Boufounos and R. G. Baraniuk, “1Bit compressive sensing,” in Proceedings of the 42nd Annual Conference on Information Sciences and Systems (CISS '08), pp. 16–21, Princeton, NJ, USA, March 2008. View at: Publisher Site  Google Scholar
 P. T. Boufounos, “Reconstruction of sparse signals from distorted randomized measurements,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '10), pp. 3998–4001, IEEE, Dallas, Tex, USA, March 2010. View at: Publisher Site  Google Scholar
 D. Malioutov, M. Cetin, and A. S. Willsky, “A sparse signal reconstruction perspective for source localization with sensor arrays,” IEEE Transactions on Signal Processing, vol. 53, no. 8, pp. 3010–3022, 2005. View at: Publisher Site  Google Scholar  MathSciNet
 D. Baron, M. B. Wakin, M. F. Duarte, S. Sarvotham, and R. G. Baraniuk, “Distributed compressed sensing,” Tech. Rep., Department of Electrical and Computer Engineering, Rice University, Houston, Tex, USA, 2006. View at: Google Scholar
 Z. Xiahou and X. Zhang, “Adaptive localization in wireless sensor network through Bayesian compressive sensing,” International Journal of Distributed Sensor Networks, vol. 11, no. 8, Article ID 438638, 2015. View at: Publisher Site  Google Scholar
 T. L. N. Nguyen and Y. Shin, “Multiple target localization in wsns based on compressive sensing using deterministic sensing matrices,” International Journal of Distributed Sensor Networks, vol. 2015, Article ID 947016, 8 pages, 2015. View at: Publisher Site  Google Scholar
 R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Transactions on Antennas and Propagation, vol. 34, no. 3, pp. 276–280, 1986. View at: Google Scholar
 M. R. Nelson, “LZW data compression,” Dr. Dobb's Journal, vol. 14, no. 10, pp. 29–36, 1989. View at: Google Scholar
 G. K. Wallace, “The JPEG still picture compression standard,” Communications of the ACM, vol. 34, no. 4, pp. 30–44, 1991. View at: Publisher Site  Google Scholar
 K. Brandenburg and G. Stoll, “Iso/mpeg1 audio: a generic standard for coding of highquality digital audio,” Journal of the Audio Engineering Society, vol. 42, no. 10, pp. 780–792, 1994. View at: Google Scholar
 E. Candes and J. Romberg, “${l}_{1}$magic: recovery of sparse signals via convex programming,” 2005. View at: Google Scholar
 L. Sun, J. Liu, J. Chen, and J. Ye, “Efficient recovery of jointly sparse vectors,” in Proceedings of the 23rd Annual Conference on Neural Information Processing Systems (NIPS '09), pp. 1812–1820, December 2009. View at: Google Scholar
 P. M. Aziz, H. V. Sorensen, and J. Van der Spiegel, “An overview of sigmadelta converters,” IEEE Signal Processing Magazine, vol. 13, no. 1, pp. 61–84, 1996. View at: Publisher Site  Google Scholar
 P. Xia, S. Zhou, and G. B. Giannakis, “Achieving the welch bound with difference sets,” IEEE Transactions on Information Theory, vol. 51, no. 5, pp. 1900–1907, 2005. View at: Publisher Site  Google Scholar  MathSciNet
 L. Jacques, J. N. Laska, P. T. Boufounos, and R. G. Baraniuk, “Robust 1bit compressive sensing via binary stable embeddings of sparse vectors,” IEEE Transactions on Information Theory, vol. 59, no. 4, pp. 2082–2102, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 R. G. Baraniuk, V. Cevher, M. F. Duarte, and C. Hegde, “Modelbased compressive sensing,” IEEE Transactions on Information Theory, vol. 56, no. 4, pp. 1982–2001, 2010. View at: Publisher Site  Google Scholar  MathSciNet
 Sedumi, http://sedumi.ie.lehigh.edu/.
Copyright
Copyright © 2016 Ming Yin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.