Guaranteed Stability of Sparse Recovery in Distributed Compressive Sensing MIMO Radar

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.


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][2][3][4][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.
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.

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 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 u    and u    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, ⟨k  , u    ⟩ is the velocity component from the th target and the th receiver, and ⟨k  , u    ⟩ 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.

Joint Sparse Representation.
We define the target state vector  = [  ,   , V  , V  ]; hence, the important properties of the target (position and velocity) are specified by .The whole target's state space is divided into  possible values {  , ∀ = 1, 2, . . ., }.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    (),  = 1, 2, . . ., .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    = 0. We arrange   into a   ×1 vector  = [  1 ,   2 , . . .,    ]  , where [⋅]  denotes the transpose of [⋅].We get joint sparse representation of the signal ensemble as In the expression of the measurement vector mentioned above, Ψ 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.

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 r.If the measurement matrix is represented by Φ, then the coherence between Φ and Ψ measures the largest correlation between them.Φ must be such that it has as little coherence with Ψ as possible.Since random matrices satisfy low coherence properties, we generate the entries of the (  ) × () dimensional measurement matrix Φ  from independent Gaussian distribution as the measurement matrix of the th receiver, where   ≪ .Considering the special structure of the joint sparse basis Ψ, we design the joint measurement matrix Φ 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 Φ without ignoring the independence of the receivers.
So, the new measurement vector in the presence of noise is where Γ = ΦΨ is defined as sensing matrix and e  is the measured noise.
In order to find the properties of targets, we need to recover the joint sparse vector  from the measurement y.It is an optimization problem with a noisy setting as follows: The recovery of joint sparse vector  from measurement y = Γ + e  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 Γ as a combination of subgroups Γ[]: 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 Γ = ΦΨ 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  (0) = 0 and the residual r (0) = y.In each subsequent iteration   , we project the residual vector r (  −1) onto all the subgroups of Γ and pick the Input: y-sampled measurement vector Φ-( CS   ) × (  ) measurement matrix Initialize All: reconstruct vector  (0) = 0; the residual r (0) = y; the sparsity ; sensing matrix Γ = ΦΨ; the index set Λ 0 = ⌀; Loop: set  = 1 and repeat until  >  (1) Arrange the   columns of Γ corresponding to the th cell into the matrix Γ () , get the product of each Γ () ( = 1, 2, . . ., ) and the residual  (0) , find the max product and get the corresponding index   = arg max =1,2,..., |⟨r −1 , Γ  ⟩| (2) Update the index set Λ  = Λ −1 ∪ {  } and update the set of reconstruct atoms group Γ (  ) 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 y.Then, we get the reconstructed sparse vector θ.The procedure of JOMP algorithm is shown as in Algorithm 1.

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 y obey precise equality y = Γ, where Γ ∈ R × .A more plausible scenario assumes sparse approximate representation: that there is an ideal noiseless signal x 0 = Γ 0 , but that we can observe only a noisy version y = x 0 + e, where ‖e‖ 2 ≤ .
The concept of coherence of the sensing matrix Γ is usually used as a criterion of the property of the sensing matrix.Assuming that the columns of Γ are normalized to unit  2 -norm, it is defined in terms of the Gram matrix G = Γ  Γ.With (, ) denoting entries of this matrix, the coherence is The theorem proposed in [18] said that if a noiseless sparse signal x 0 = Γ 0 satisfies the inequality  = ‖ 0 ‖ 0 < (1/ + 1)/4, the deviation of the ( 1, ) representation from The parameter  in Donoho's theorem indicates the worst case of the coherence between some columns of the sensing matrix Γ.Since the sensing matrix Γ 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 Γ, DCS-MIMO radar may still estimate targets' positions correctly by choosing the incoherent columns of Γ 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 Γ: where   is the mean of the nonzero absolute values of these off-diagonal elements in the Gram matrix G. 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 Γ.Then, the stability guarantee for DCS-MIMO radar system with   is proposed.
Theorem 1.Let the overcomplete sensing matrix Γ have average coherence   (Γ).If the sparse representation of the noiseless signal  0 = Γ 0 satisfies then the deviation of the ( 1, ) representation from  0 , assuming  ≥ , can be bounded by Proof.The stability bound can be posed as the solution to an optimization problem of the form max Put in words, we consider all representation vectors  0 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 V =  −  0 and similarly  = θ −  0 , we can rewrite the above problem as where  is the support of nonzero entries in  0 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 max where we denote Δ =  + .
The constraint ‖Γ‖ 2 ≤ Δ is not posed in terms of the absolute values in the vector , complicating the analysis; we now relax this constraint using incoherence of Γ.The Gram matrix of Γ is G = Γ  Γ, and the coherence used in this paper is the average off-diagonal amplitude   .Let 1 be the -by- matrix of all ones.Let G min denote the matrix whose offdiagonal elements are the smallest one of the off-diagonal entries of the Gram matrix G. Similarly, let G  denote the matrix whose off-diagonal elements are   .The constraint can be relaxed: Using this, (25) is bounded above by the value max 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  0 are concentrated in the initial slots of the vector; that is,  = 1, 2, . . ., .
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.
International Journal of Antennas and Propagation

Numerical Results
4.1.Targets' Parameters Estimation.In order to prove the effectiveness of the DCS-MIMO radar, we simulated a small scene with 2 transmitters and 2 receivers.We use the common Cartesian coordinate system.The transmitters are located at t 1 = [100, 0] m and t 2 = [200, 0] m, respectively.The receivers are located at r 1 = [0, 100] m and r 2 = [0, 200] m, respectively.The sample rate is   = 100 MHz.We choose the number of samples in each receiver to be  = 80; therefore, y has 160 entries.We divided the position space of the target into 11 × 13 grid points and the target's velocity space into 5 × 5 grid points.Therefore, the total number of possible target states  = 3575.We consider the presence of 2 targets.Hence, the sparse vector  formed by 7150 entries has only ×  = 4 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 SNR1 = 2 dB and SNR2 = 3 dB.
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.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.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.

Guaranteed Stability of
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.

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 10% 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.
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.

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.

4 )
Update the residual r  = y − Θ  ŝ ,  =  + 1 End loop Output: the index set Λ and the nonzero value ŝ Algorithm 1: The procedure of JOMP algorithm.

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

Figure 4 :
Figure4: Reconstruction probability for the system using joint sparse modeling and the system using CS in each receiver separately as a function of SNR.

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

Figure 6 :
Figure 6: Reconstruction probability for single target under different SNR condition.
With the noise of 2 dB
Note that if  is the minimizer of ‖ 0 + V‖ 1 under these constraints, then relaxing the constraints to all  satisfying ‖ 0 + ‖ 1 expands the feasible set.However, this is true only if  ≥  since otherwise V = 0 is not a feasible solution.
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