International Journal of Antennas and Propagation
Volume 2015 (2015), Article ID 421740, 10 pages
http://dx.doi.org/10.1155/2015/421740

Research Article

Guaranteed Stability of Sparse Recovery in Distributed Compressive Sensing MIMO Radar

Yu Tao, Gong Zhang, and Jindong Zhang

Key Laboratory of Radar Imaging and Microwave Photonics, Ministry of Education, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 27 May 2014; Revised 11 October 2014; Accepted 29 October 2014

Academic Editor: Matteo Pastorino

Copyright © 2015 Yu Tao 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.

Abstract.

Low SNR condition has been a big challenge in the face of distributed compressive sensing MIMO radar (DCS-MIMO radar) and noise in measurements would decrease performance of radar system. In this paper, we first devise the scheme of DCS-MIMO radar including the joint sparse basis and the joint measurement matrix. Joint orthogonal matching pursuit (JOMP) algorithm is proposed to recover sparse targets scene. We then derive a recovery stability guarantee by employing the average coherence of the sensing matrix, further reducing the least amount of measurements which are necessary for stable recovery of the sparse scene in the presence of noise. Numerical results show that this scheme of DCS-MIMO radar could estimate targets’ parameters accurately and demonstrate that the proposed stability guarantee could further reduce the amount of data to be transferred and processed. We also show the phase transitions diagram of the DCS-MIMO radar system in simulations, pointing out the problem to be further solved in our future work.

1. Introduction

Nowadays, detection of targets which are stealth or in strong interference has become an important requirement for radar system. Distributed placed antennas enable the system to view targets from multiple angles, providing spatial diversity and reducing the target radar cross sections (RCS) scintillations. Therefore, distributed MIMO radar can be employed to detect stealth targets. The difference of signals transmitted by each transmitter provides several information channels for distributed MIMO radar, enabling the MIMO radar system to achieve superior spatial resolution as compared to a traditional radar system [1–5]. But the amount of data from these information channels is always too huge to be processed, increasing the difficulty in hardware designing. Compressive sensing is used in MIMO radar to estimate the DOA in [6, 7] and it is shown that CS-MIMO radar system could estimate targets’ parameters accurately, using much fewer data than that needed in conventional MIMO radar.

Viewing targets from different angles with separated antennas, we can detect the stealth targets and estimate targets’ parameters as position and velocity [8]. The theory of distributed compressive sensing (DCS) was proposed in [9]. In a standard DCS scenario, signals measured by sensors are each individually sparse on some basis or all the signals share the locations of nonzero coefficients in the sparse vectors. Under the right conditions, a decoder at the collection point can jointly reconstruct all of the signals precisely. Such property of DCS happens to fit the distributed MIMO radar and in this paper we provide a practical scheme for DCS-MIMO radar system. The sparse targets scene of DCS-MIMO radar is shown in Figure 1.

Figure 1: Sparse targets scene of DCS-MIMO radar.

In the sparse targets scene of DCS-MIMO radar, as Figure 1 shows that the antennas are distributed and placed and they detect the region of interest from different angles. The pentagrams in Figure 1 represent the targets. Using distributed compressive sensing, distributed MIMO radar could precisely reconstruct the sparse scene with considerably lower amount of measurements than required by the Nyquist theorem. Many recovery algorithms have been proposed for compressive sensing in recent years. Algorithms inspired by MP include OMP [10], tree matching pursuit [11], stagewise OMP [12], CoSaMP [13], and IHT [14]. Different from algorithms of the match pursuit class, there are also FOCUSS [15] and sparse Bayesian learning (SBL) [16]. In DCS scenario, sparse recovery algorithms need to make the most of the common component and innovations of the received signals which are treated as a signal ensemble [9]. Many joint sparse recovery algorithms were employed to recover the sparse vector jointly, such as OSGA and SOMP [17]. In this paper, we propose a joint orthogonal matching pursuit (JOMP) algorithm to exploit the special structure of the joint sparse vector. It is demonstrated that using DCS with JOMP algorithm is more effective and more accurate than processing signals in each receiver with CS method separately.

Nonetheless, in order to ensure that the DCS-MIMO radar system could be realized in the application of engineering, many problems need to be discussed in depth. One of these problems is finding the least amount of measurements to ensure the stable recovery in low signal-to-noise ratio (SNR) condition. Article [18] establishes the possibility of stable recovery under a combination of sufficient sparsity and favorable structure of the overcomplete system. However, the theorem in [18] is too strict for radar application; hence, we can further reduce the amount of data. In this paper, we propose a stability guarantee for stable recovery in DCS-MIMO radar, using the average coherence of sensing matrix. We then find the least amount of measurements for our DCS-MIMO radar system using JOMP algorithm in low SNR condition.

This paper is organized as follows. In Section 2, we introduce DCS-MIMO radar system and give joint sparse model of the received signal ensemble and we process sparse reconstruction with JOMP algorithm. In Section 3, we propose the method to find the least amount of measurements to ensure stable recovery and we give the stability guarantee of DCS-MIMO radar. Numerical results are shown in Section 4 to demonstrate the effectiveness of the scheme of DCS-MIMO radar and the stability guarantee. Phase transitions diagram is shown to raise a new problem to be solved in future. Finally, Section 5 concludes this paper.

2. DCS-MIMO Radar

2.1. Signal Model

In this section, we describe DCS-MIMO radar system and give joint sparse representation of the received signal ensemble. We consider a system with transmitters and receivers and there are targets in the region of interest. We assume that all targets are moving in a two-dimensional plane. However, without loss of generality, we can also consider the three-dimensional occasion. We further assume that each of the targets contains multiple individual isotropic scatterers. We can express the collection of these scatterers as one point scatterer which represents the RCS center of gravity of these multiple scatterers [8]. The RCS center of the th target is located at with a velocity . The th transmitter and the th receiver are, respectively, located at and . Orthonormal waveforms are transmitted by each transmitter. Suppose the waveform transmitted by the th transmitter is . These signals travel in space and reflect off the surfaces of the targets and then are captured by the receivers. Further, we assume that the cross correlations between these waveforms are close to zero for different delays [8]. Let denote the attenuation corresponding to the th target between the th transmitter and the th receiver. The signal arriving at the th receiver can be expressed as

where is the delay and is the Doppler shift corresponding to the th target and denotes the additive noise in the th receiver:

where and denote the unit vector from the th transmitter to the th target and the unit vector from the th target to the th receiver, respectively. is the inner product operator. Hence, is the velocity component from the th target and the th receiver, and is the velocity component from the th transmitter to the th target. is the carrier frequency and is the speed of propagation of the wave in the medium.

2.2. Joint Sparse Representation

We define the target state vector ; hence, the important properties of the target (position and velocity) are specified by . The whole target’s state space is divided into possible values . Hence, each of the targets is associated with a state vector belonging to this grid. If the presence of a target at contributes to received signal in the th receiver, then we define

The sampled outputs of are given as . Then, we arrange into a matrix :

where is the sparse basis corresponding to the th receiver. Then, the joint sparse basis of all receivers can be expressed as

If the is the state of the th target, we define , where denote the attenuation value of the th target observed by the th receiver. Otherwise, the . We arrange into a vector , where denotes the transpose of . We get joint sparse representation of the signal ensemble as

In the expression of the measurement vector mentioned above, is known and only depends on the actual targets present in the illuminated area. The nonzero entries of represent the target attenuation values and the corresponding indices represent the positions and velocities. Further, the indices of nonzero entries of each are always the same; in other words, all signals in each receiver share the location of nonzero entries. So, we call this joint sparse modeling.

2.3. Joint Sparse Recovery

In the previous section, we get the joint sparse representation of the signal ensemble received by receivers. The theory of compressive sensing said that we can reconstruct the vector from far fewer samples than that contained in the vector . If the measurement matrix is represented by , then the coherence between and measures the largest correlation between them. must be such that it has as little coherence with as possible. Since random matrices satisfy low coherence properties, we generate the entries of the dimensional measurement matrix from independent Gaussian distribution as the measurement matrix of the th receiver, where . Considering the special structure of the joint sparse basis , we design the joint measurement matrix as

Employing this structure of joint measurement matrix, we can measure the signal ensemble simultaneously and design submeasurement matrix according to the different situation of each receiver such as the different SNR. On the other hand, it is much more convenient to optimize the joint measurement matrix without ignoring the independence of the receivers.

So, the new measurement vector in the presence of noise is

where is defined as sensing matrix and is the measured noise.

In order to find the properties of targets, we need to recover the joint sparse vector from the measurement . It is an optimization problem with a noisy setting as follows:

The recovery of joint sparse vector from measurement is one of the key points of this paper. Considering the special structure of the joint sparse vector , each has the same location of nonzero entries, which could be treated as joint sparsity. To define joint sparsity, we view as a combination of groups—assumed throughout the paper to be a length —with denoting the th group, that is,

where . Entries which are corresponding to the same target state vector are arranged into the group . Similarly, we can represent as a combination of subgroups :

We propose an extension of matching pursuit algorithm called joint orthogonal matching pursuit (JOMP) that exploits the knowledge of joint sparsity. Based on orthogonal matching pursuit (OMP) algorithm, we divided the columns of the sensing matrix into groups as shown in (11). Columns which are in the same group are corresponding to the same information cell (target state vector). In JOMP, we first initialize reconstructed vector and the residual . In each subsequent iteration , we project the residual vector onto all the subgroups of and pick the group that has the highest correlation with the residual. We update the estimated reconstructed vector:

We finally update the residual as

When the residual was finally updated by calculating the product of these groups and the received signal, we find the groups which are most correlated with the received signal . Then, we get the reconstructed sparse vector . The procedure of JOMP algorithm is shown as in Algorithm 1.

Input:     -sampled measurement vector

              - measurement matrix

Initialize All:  reconstruct vector ; the residual ; the sparsity ; sensing matrix ; the index set ;

Loop:  set and repeat until

(1) Arrange the columns of corresponding to the th cell into the matrix , get the product of each ()

      and the residual , find the max product and get the corresponding index

(2) Update the index set and update the set of reconstruct atoms

(3) Calculate

(4) Update the residual ,

End loop

Output: the index set and the nonzero value

Algorithm 1: The procedure of JOMP algorithm.

3. Stability Guarantee for DCS-MIMO Radar

Large amounts of data will bring great difficulties for the design of hardware system. So, it is important to find a lower limit of data amount to guarantee the stability of sparse recovery, providing benefits for implementation of the DCS-MIMO radar system. Hence, the stability guarantee for DCS-MIMO radar is the focus of this paper.

In most practical situations, it is not sensible to assume that the available data obey precise equality , where . A more plausible scenario assumes sparse approximate representation: that there is an ideal noiseless signal , but that we can observe only a noisy version , where .

The concept of coherence of the sensing matrix is usually used as a criterion of the property of the sensing matrix. Assuming that the columns of are normalized to unit -norm, it is defined in terms of the Gram matrix . With denoting entries of this matrix, the coherence is

The theorem proposed in [18] said that if a noiseless sparse signal satisfies the inequality , the deviation of the representation from , assuming , can be bounded by .

The parameter in Donoho’s theorem indicates the worst case of the coherence between some columns of the sensing matrix . Since the sensing matrix of DCS-MIMO radar is overcomplete and it could be regarded as a redundant dictionary, some columns would not be chosen during the process of sparse recovery; that is, even if is too large to guarantee the incoherentness of the sensing matrix , DCS-MIMO radar may still estimate targets’ positions correctly by choosing the incoherent columns of when the targets are in proper positions.

We simulated the distribution of the normalized cross correlation between the columns of the sensing matrix as Figure 2 shows. According to the definition, the coherence of the simulated sensing matrix is 0.8837. The value of is so large that the simulated sensing matrix seems not qualified according to the theorem in [18]. However, in practice, the sensing matrix is incoherent enough for DCS-MIMO radar since most of the cross correlation values are small. It is also demonstrated in a particular experiment as simulation 4.1 that DCS-MIMO radar is able to estimate the targets’ parameters accurately based on this sensing matrix. So, the parameter is too strict to assess the coherence of the sensing matrix of DCS-MIMO radar. A new evaluation criterion which could indicate the overall coherence of the redundant dictionary of DCS-MIMO radar is needed. Therefore, we use to denote the average value of the coherence of each pair of columns in the sensing matrix :

where is the mean of the nonzero absolute values of these off-diagonal elements in the Gram matrix . Compared to indicates the overall coherence of the redundant dictionary rather than the extreme conditions. By calculating the results in Figure 2, we get the average coherence of the simulated sensing matrix which is 0.1792. That is, even if the is large, small could still guarantee DCS-MIMO radar’s estimation accuracy. Hence, using to indicate the performance of DCS-MIMO radar is more practical than using . In this paper, we use the average coherence to assess the coherence of the sensing matrix . Then, the stability guarantee for DCS-MIMO radar system with is proposed.

Figure 2: The distribution of normalized cross correlation between the columns of .

Theorem 1. Let the overcomplete sensing matrix have average coherence . If the sparse representation of the noiseless signal satisfies

then the deviation of the representation from , assuming , can be bounded by

Proof. The stability bound can be posed as the solution to an optimization problem of the form

Put in words, we consider all representation vectors of bounded support, all possible realizations of bounded noise, and we ask for the largest error between the ideal sparse decomposition and its reconstruction from noisy data. Defining and similarly , we can rewrite the above problem as

Note that if is the minimizer of under these constraints, then relaxing the constraints to all satisfying expands the feasible set. However, this is true only if since otherwise is not a feasible solution. Thus, we consider

We now use and expand this set by exploiting the relation

where is the support of nonzero entries in with complement . Therefore, we get a further increase in value by replacing the feasible set in (20) with

Writing this out yields a new optimization problem with still larger value:

Then, we eliminate the noise vector , using

Expanding the feasible set of (21) using this observation gives

where we denote .
The constraint is not posed in terms of the absolute values in the vector , complicating the analysis; we now relax this constraint using incoherence of . The Gram matrix of is , and the coherence used in this paper is the average off-diagonal amplitude . Let be the -by- matrix of all ones. Let denote the matrix whose off-diagonal elements are the smallest one of the off-diagonal entries of the Gram matrix . Similarly, let denote the matrix whose off-diagonal elements are . The constraint

can be relaxed:

Using this, (25) is bounded above by the value

This problem is invariant under permutations of the entries in which preserve membership in and . It is also invariant under relabeling of coordinates. So, assume that all nonzero entries in are concentrated in the initial slots of the vector; that is, .
Putting where gives the first entries in and gives the remaining entries of , we obviously have

The norm on dominates the norm and is dominated by times the norm. Thus,

We define

Returning to (25) and using the definition above, we obtain a further reduction from an optimization problem on to an optimization problem on as

We further define , where , and rewrite (32) as

Then, we define . Then, over the region (33). Denoting , the first constraint defining that region takes the form

Since , sparsity requirement (9) leads to

Hence,

with the choice .
Requirement (35) puts a restriction on and , being free parameters of the problem. Using leads to the sparsity requirement in (16), since .

We can use (16) to find the least amount of measurements for stable recovery and confirm the guaranteed stability of DCS-MIMO radar. Compared to theorem proposed by Donoho et al. in [18], stability guarantee proposed in this paper further reduces the amount of data necessarily to be transferred and processed in DCS-MIMO radar for stable recovery of the sparse targets scene. This stability guarantee gives theoretical foundation for hardware implementation of DCS-MIMO radar and it also demonstrates the feasibility of targets’ parameters estimation in low SNR condition by DCS-MIMO radar.

4. Numerical Results

4.1. Targets’ Parameters Estimation

In order to prove the effectiveness of the DCS-MIMO radar, we simulated a small scene with transmitters and receivers. We use the common Cartesian coordinate system. The transmitters are located at and , respectively. The receivers are located at and , respectively. The sample rate is . We choose the number of samples in each receiver to be ; therefore, has entries. We divided the position space of the target into grid points and the target’s velocity space into grid points. Therefore, the total number of possible target states . We consider the presence of 2 targets. Hence, the sparse vector formed by 7150 entries has only nonzero entries corresponding to the targets.

The positions and the velocities of the targets are given as

The attenuations corresponding to the two targets are

We assume the SNR for the receivers are and .

From Figure 3, we can see that, using JOMP algorithm, the DCS-MIMO radar system is able to estimate the positions and velocities of the targets accurately in the presence of noise. Since it is not possible to plot the position and velocity on the same plot, we plotted the estimates of position and velocity separately.

(a) Position estimation

(b) Velocity estimation

Figure 3: Targets’ parameters estimation using JOMP.

In order to compare with the conventional CS method that uses CS to reconstruct the sparse vector in each receiver separately, we use 1000 independent Monte Carlo runs to generate these results. Here, we define that a successful estimation is that every receiver estimates the positions of targets accurately. Then, in Figure 4, we can see that reconstruction using joint sparse modeling has a higher reconstruction probability than the conventional CS approach. Therefore, we demonstrate the advantages of the reconstruction using joint sparse modeling and JOMP algorithm.

Figure 4: Reconstruction probability for the system using joint sparse modeling and the system using CS in each receiver separately as a function of SNR.

4.2. Guaranteed Stability of Joint Sparse Recovery

According to the theorem in Section 3, we simulated the Gram matrix of the sensing matrix of the DCS-MIMO radar system. After calculating the average off-diagonal amplitude , we show the relationship between and the number of measurements . For comparison, we also show the relationship between and the number of measurements according to Donoho’s theorem in [18].

Considering the joint sparsity of the received signal ensemble, we get the stability guarantee of the DCS-MIMO radar system based on the theorem proposed in Section 3. In Figure 5, when the value of the average coherence or the coherence is lower than the stability guarantee, stable recovery of sparse targets scene could be achieved. By comparison of Figures 5(a) and 5(b), we can find that the necessary number of measurements reduced from 120 to 80 in our simulation settings by the stability guarantee in this paper.

(a) Donoho’s stability guarantee

(b) Stability guarantee in this paper

Figure 5: Coherence and average coherence of the sensing matrix of different number of measurements with the different stability guarantee.

Figure 6 shows the probability of reconstruction with different SNR. We can see that when the measurements reach 80 which satisfy the stability guarantee proposed in this paper, the reconstruction probability is close to that which is corresponding to the 120 measurements. Therefore, it is demonstrated that our stability guarantee further reduces the necessary measurements for stable recovery of DCS-MIMO radar in low SNR condition, compared to Donoho’s stability guarantee.

Figure 6: Reconstruction probability for single target under different SNR condition.

4.3. Phase Transitions Diagram

In order to further study the impact of the joint sparsity and the amount of measurements on the performance of sparse reconstruction, we showed the phenomenon of phase transitions of joint sparse recovery in our DCS-MIMO radar system based on a large number of experiments. The phase transitions diagram would visually show the relationship between the joint sparsity, the amount of measurements, and the probability of sparse reconstruction.

The in Figure 7 denotes the ratio of the length of signal to the number of measurements . Success rates of 90%, 50%, and are indicated by the lower set of blue, green, and red curves, respectively. When the drops in the area under these curves, the DCS-MIMO radar system could reconstruct the sparse scene by the probability of the curve or higher probability.

(a) Noiseless

(b) With the noise of 2 dB

Figure 7: Phase transitions diagram.

We can find that when DCS-MIMO radar system is in the condition of low SNR, the performance falls rapidly while the joint sparsity increases. Hence, the new problem appears and we will somehow need to solve the joint sparse recovery with higher joint sparsity. This is the main content of our latter part of study.

5. Conclusion

In this paper, we have devised the scheme of the DCS-MIMO radar system and proposed the joint sparse modeling to get joint sparse representation of the received signal ensemble. This scheme provided us with the method for processing signal from different channels simultaneously and getting better performance in low SNR condition. We also proposed a modificatory stability guarantee for sparse recovery in the DCS-MIMO radar, employing the average coherence of the sensing matrix. This stability guarantee is demonstrated to be effective for the DCS-MIMO radar and could further reduce the necessary amount of measurements for stable recovery. On the other hand, since the stability guarantee has given us the least amount of measurements which is needed for stable recovery in low SNR condition, we will have theoretical foundation when designing the hardware of the DCS-MIMO radar. We have provided analytical results to show the feasibility of the DCS-MIMO radar system and the reliability of the proposed stability guarantee. At last, we raised a new problem with higher joint sparsity by analyzing the phase transitions diagram of DCS-MIMO radar, which points out the next research priorities in our future work.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China (61471191 and 61201367) and in part by Funding of Jiangsu Innovation Program for Graduate Education (CXZZ12_0155). Project is funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions and the Fundamental Research Funds for the Central Universities.

References

E. Fishler, A. Haimovich, R. Blum, D. Chizhik, L. Cimini, and R. Valenzuela, “MIMO radar: an idea whose time has come,” in Proceedings of the IEEE Radar Conference, pp. 71–78, April 2004.

J. Li and P. Stoica, “MIMO radar—diversity means superiority,” in Proceedings of the 14th Adaptive Sensor Array Processing Workshop (ASAP '06), pp. 1–6, Lincoln, Mass, USA, 2006.

X.-R. Li, Z. Zhang, W.-X. Mao, X.-M. Wang, J. Lu, and W.-S. Wang, “A study of frequency diversity MIMO radar beamforming,” in Proceedings of the IEEE 10th International Conference on Signal Processing (ICSP '10), pp. 352–356, Beijing, China, October 2010.

R. Sharma, “Analysis of MIMO radar ambiguity functions and implications on clear region,” in Proceedings of the IEEE International Radar Conference (RADAR '10), pp. 544–548, May 2010.

J. Li, G. Liao, and H. Griffiths, “Bistatic MIMO radar spacetime adaptive processing,” in Proceedings of the IEEE Radar Conference (RadarCon '11), pp. 498–501, Westin Crown Center in Kansas City, 2011.

Y. Yu, A. P. Petropulu, and H. V. Poor, “Compressive sensing for MIMO radar,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '09), pp. 3017–3020, Taipei, Taiwan, April 2009.

C.-Y. Chen and P. P. Vaidyanathan, “Compressed sensing in MIMO radar,” in Proceedings of the 42nd Asilomar Conference on Signals, Systems and Computers (ASILOMAR '08), pp. 41–44, Pacific Grove, Calif, USA, October 2008.

S. Gogineni and A. Nehorai, “Target estimation using sparse modeling for distributed {MIMO} radar,” IEEE Transactions on Signal Processing, vol. 59, no. 11, pp. 5315–5325, 2011.

D. Baron, M. F. Duarte, M. B. Wakin, S. Sarvotham, and R. G. Baraniuk, “Distributed compressive sensing,” Tech. Rep. TREE-0612, Rice University, Houston, Tex, USA, 2006.

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Transactions on Information Theory, vol. 53, no. 12, pp. 4655–4666, 2007.

M. F. Duarte, M. B. Wakin, and R. G. Baraniuk, “Fast reconstruction of piecewise smooth signals from random projections,” in Proceedings of the Signal Processing with Adaptative Sparse Structured Representations (SPARS '05), Rennes, France, November 2005.

D. L. Donoho, Y. Tsaig, I. Drori, and J. L. Starck, “Sparse solution of underdetermined systems of linear equations by stagewise orthogonal matching pursuit,” IEEE Transactions on Information Theory, vol. 58, no. 2, pp. 1094–1121, 2012.

D. Needell and J. A. Tropp, “CoSa{MP}: iterative signal recovery from incomplete and inaccurate samples,” Applied and Computational Harmonic Analysis. Time-Frequency and Time-Scale Analysis, Wavelets, Numerical Algorithms, and Applications, vol. 26, no. 3, pp. 301–321, 2009.

T. Blumensath, “Accelerated iterative hard thresholding,” Signal Processing, vol. 92, no. 3, pp. 752–756, 2012.

H. Jung, K. Sung, K. S. Nayak, E. Y. Kim, and J. C. Ye, “K-t FOCUSS: a general compressed sensing framework for high resolution dynamic MRI,” Magnetic Resonance in Medicine, vol. 61, no. 1, pp. 103–116, 2009.

Z. Zhang and B. D. Rao, “Sparse signal recovery with temporally correlated source vectors using sparse Bayesian learning,” IEEE Journal on Selected Topics in Signal Processing, vol. 5, no. 5, pp. 912–926, 2011.

M. F. Duarte, S. Sarvotham, D. Baron, M. B. Wakin, and R. G. Baraniuk, “Distributed compressed sensing of jointly sparse signals,” in Proceedings of the 39th Asilomar Conference on Signals, Systems and Computers, pp. 1537–1541, November 2005.

D. L. Donoho, M. Elad, and V. N. Temlyakov, “Stable recovery of sparse overcomplete representations in the presence of noise,” IEEE Transactions on Information Theory, vol. 52, no. 1, pp. 6–18, 2006.