 freely available
 reusable
Remote Sensing 2013, 5(2), 631647; doi:10.3390/rs5020631
Published: 4 February 2013
Abstract
: The frequency diverse multipleinputmultipleoutput (FDMIMO) radar synthesizes a wideband waveform by transmitting and receiving multiple frequency signals simultaneously. For FDMIMO radar imaging, conventional imaging methods based on Matched Filter (MF) cannot enjoy good imaging performance owing to the few and incomplete wavenumberdomain coverage. Higher resolution and better imaging performance can be obtained by exploiting the sparsity of the target. However, good sparse recovery performance is based on the assumption that the scatterers of the target are positioned at the prediscretized grid locations; otherwise, the performance would significantly degrade. Here, we propose a novel approach of sparse adaptive calibration recovery via iterative maximum a posteriori (SACRiMAP) for the general offgrid FDMIMO radar imaging. SACRiMAP contains three loop stages: sparse recovery, offgrid errors calibration and parameter update. The convergence and the initialization of the method are also discussed. Numerical simulations are carried out to verify the effectiveness of the proposed method.1. Introduction
The multipleinputmultipleoutput (MIMO) radar system has attracted much attention recently, due to the additional degrees of freedom and the higher spatial resolution [1]. Through space diversity, the whole array aperture is extended by virtual sensors, which are equivalently constructed by different combinations of transmitters and receivers. The MIMO radar system with frequency diversity (FDMIMO) has been researched widely in the recent years [2–4], since its range resolution can be further improved by simultaneously fusing the echoes coming from different transmitters to form a wideband signal [5]. Hence, here we focus our research on the FDMIMO radar imaging.
For a restricted number of transmitters and receivers, the wavenumberdomain coverage is incomplete, so the traditional imaging methods based on matched filter (MF) fail to achieve good performance. Nonetheless, in most radar imaging applications, the scatterers of the target are often distributed sparsely, i.e., the number of actual scatterers is much smaller than that of the potential scatterers. Much existing research has been dedicated to sparse recovery techniques for MIMO radar imaging [6–8], such as orthogonal matching pursuit (OMP) and basis pursuit (BP). Based on compressive sensing (CS), higher resolution and better imaging performance can be obtained by exploiting the sparsity of the scatterers.
However, most existing sparse recovery techniques require the scatterers to be located exactly on the prediscretized grid. When offgrid scatterers exist, their performance would be severely affected. In MIMO radar imaging [6,9], since the scatterers are distributed in a continuous scene, the offgrid problem usually emerges, even if the discretized grid is dense, which would lead to the mismatch of the sensing matrix. Using a denser grid may alleviate the mismatch level; however, it is still not an advisable remedy for the offgrid target imaging, since a denser grid may dramatically enhance the coherence between the column of the sensing matrix, which causes the violation of the restricted isometry property (RIP) condition for reliable sparse recovery. Therefore, in this paper, we consider more practical situations with offgrid scatterers for the FDMIMO radar imaging.
For linear measurement equation y = ϕx + e, the perturbed CS problem, where the sensing matrix ϕ is unknown or subject to an unknown perturbation, has been a hot research topic. Gleichman and Eldar [10] introduce a concept named blind CS, where the sensing matrix is assumed unknown. Chi et al.[11] analyze the performance of CS methods when the mismatch of the sensing matrix exists and verify that the sparse recovery performance is likely to suffer from a large error when the mismatch is large. However, the research in [10,11] is limited to theoretical analysis of the performance loss and does not devise any modified approaches. Hereafter, some algorithms have been proposed to deal with the mismatch of the sensing matrix and have some practical application to offgrid direction of arrival (DOA) estimation. Zhu et al.[12] propose a feasible approach, named sparse total least squares (STLS), to alleviate the effect of mismatch, which, however, is inefficient and timeconsuming. Han et al.[13] introduce faster and more robust algorithms, total leastsquares Focal Underdetermined System Solver (TLSFOCUSS) and Synchronous Descending Focal Underdetermined System Solver (SDFOCUSS), then apply SDFOCUSS to the DOA multiple measurement vectors (MMV) model. From the Bayesian perspective, STLS in [12] and TLSFOCUSS, SDFOCUSS in [13] are all equivalent to obtaining a maximum a posteriori (MAP) solution by assuming that the mismatch errors are white Gaussian distributed. However, since we do not have enough priori information about the scatterer distribution of the target, a uniform distribution of the offgrid error conforms more to a practical physical situation than a Gaussian distribution when we make arbitrary discretization of the imaging area. Yang et al.[14] formulate the offgrid DOA estimation problem using sparse Bayesian inference (SBI) and recover the source signal and the matrix mismatch through expectationmaximization (EM) iteration. In SBI, the unknown solutions are assumed to be timevarying; however, during the short imaging time, the reflection coefficients of the scatterers to be solved are timeinvariant. Hence, a new algorithm needs to be developed to take into account the timeinvariant unknowns.
In this paper, we propose an approach of sparse adaptive calibration recovery via iterative maximum a posteriori (SACRiMAP) for the offgrid FDMIMO radar imaging. As a matter of fact, SACRiMAP is within the framework of Bayesian CS [15]. The offgrid error (the distance from the actual scatterer to the nearest assumed grid point) that lies within a bounded interval is assumed to be uniformly distributed rather than Gaussian distributed, as in [12,13]. SACRiMAP adaptively calibrates the offgrid errors and, meanwhile, seeks the optimal target reconstruction results. Through iterative MAP, it turns the nonconvex optimization problem of the offgrid sparse recovery to three main stages: sparse recovery, offgrid errors calibration and parameter update. The offgrid errors and the power of noise are dynamically and adaptively calibrated, which enhance the robustness of the proposed algorithm. In [16–18], the authors also linearize the offgrid errors using the first order Taylor approximation and establish an alternating process to obtain the optimized recovery results and offgrid errors estimation. However, since they have not deduced the imaging problem from the Bayesian maximum a posteriori aspect, the estimation of the power of noise is unavailable. Moreover, the proposed algorithm can reconstruct scatterers accurately even under a coarsely discretized grid.
The outline of this paper is as follows. In Section 2, the sparse recovery of the offgrid target for the FDMIMO imaging problem is formulated. In Section 3, a novel algorithm, named SACRiMAP, for the offgrid sparse recovery problem is proposed. In Section 4, extensive numerical simulations are presented to verify the proposed algorithm. Finally, in Section 5, the conclusions are drawn.
Notations used in this paper are as follows. Boldcase letters are reserved for vectors and matrices. ‖x‖_{p} denote the l_{p} norm of a vector x. ‖A‖_{2} is the spectral norm of the matrix A. diag(x) is a diagonal matrix with its diagonal entries being the entries of a vector x. ⊙ is the Hadamard (elementwise) product. (·)^{T}, (·)^{T} and vec (·) denote the transpose, the conjugate transpose operation and the vectorization operation, respectively.
2. Problem Formulation
2.1. FDMIMO Imaging Problem
Consider the scenario of a MIMO radar system composed by M transmitters and N receivers, as illustrated in Figure 1. Here, we adopt a narrowband FDMIMO radar system, which is proposed in [4]. Considering that the linear frequency modulated (LFM) signal has long been used in radar systems, because of its implementation simplicity, constant modulus and high range resolution [5], here, we focus our derivation on the LFMbased FDMIMO radar imaging. The different transmitter elements transmit LFM signals within different bands, respectively.
Assuming that all of the M waveforms have an equal chirp rate, and letting s_{m}(t) (for m = 1,2,...,M) be the transmitted LFM waveform for the mth transmitter, therefore, we have:
Suppose the imaging scene contains U radial range bins and V angle bins. Setting K = UV, we define σ_{k}, R_{k}, θ_{k} (for k= 1,2,...,K) as the complex reflection coefficient, the radial range and the impinging angle (relative to the array normal), respectively. By defining s(t) = [s_{1}(t), s_{2}(t), …, s_{M}(t)]^{T}, the echoes of N receivers z(t) = [z_{1}(t), z_{2}(t), …, z_{N}(t)]^{T} can be written in the following form:
After orthogonal separation, we can obtain the received signal of the mnth virtual sensor corresponding to the mth transmitter and the nth receiver as:
Supposing that the reference range is R_{0}, for LFM signal, we have:
Here, we consider uniform linear arrays (ULA) for the transmitters and receivers with d_{t} = Nd and d_{r} = d, where d_{t} and d_{r} refer to the distances between adjacent transmitting and receiving antennas, respectively. So, we can get d_{tm} + d_{rn} = ((m − 1)N + n − 1)d. What is more, the corresponding derivation of ULA in this paper can be directly generalized to other configurations of the transceiver array.
Define z = vec(z_{mn}(q)) with its size MNQ × 1, e = vec(e_{mn}(q)) with the same size, then set:
Moreover, we let:
Defining σ = [σ_{1},…,σ_{K}], then the echoes can be redefined as:
The available observations in the wavenumber domain point are (2(f_{m} + γt′_{q})/c, ((m − 1)N + n − 1)d/λ) (m = 1,…,M, n = 1,…,N, q = 1,…,Q) [20], which are too few to achieve good recovery performance by traditional imaging methods based on matched filter (MF). However, for most targets, especially those in the air, the number of actual scatterers is much smaller than that of potential scatterers. Therefore, the sparsity of the target is an appropriate priority by which more accurate imaging performance can be expected. Herein, we consider utilizing the sparsity of the scatterers to realize high resolution imaging.
2.2. Sparse Recovery of the OffGrid Target for FDMIMO Imaging Problem
In most cases of radar imaging, the scatterers are distributed in a continuous scene. The sparsity is more of a kind of quantitative description. However, if we use the sparsity prior directly into Equation (12), the offgrid problem would emerge, which means that, no matter how densely we discretize the imaging scene, the scatterers could be offgrid. To alleviate the effect of the offgrid problem on the performance of the sparse recovery, we must make some amendments to Equation (12). Considering the offgrid problem, we have:
Substituting Equations (2) and (13) into Equation (9), the echoes can be redefined as:
Since the offgrid errors are restricted in the region of one radial range bin or angle bin, we can assume that the offgrid errors are significantly small; then, we can make the following approximations by the Taylor expansion:
Since {Δr_{k}, Δθ_{k}} are small, we ignore the crossterm about Δr_{k}, Δθ_{k}. Thus, we can approximate the sensing matrix as a linear combination of the errors {Δr_{k}, Δθ_{k}}:
Then, we can get the matrixvector form corresponding to Equation (14):
Thus, our pursuit is to find out the optimized σ′ and Λ; then, we can get the corresponding reflection coefficients σ. We cast both σ′ and Λ as unknown parameters and search for the sparsest solution and simultaneously estimate the offgrid errors.
3. SACRiMAP Algorithm
3.1. Basic Idea of the Proposed Algorithm
Inheriting the Bayesian idea, we can estimate σ′ and Λ via the MAP method. Suppose that the vector form of reflection coefficient of the scatterers σ is sparse, then the prior distribution of σ′ satisfies [21]:
Further, suppose that e satisfies the complex normal distribution with mean zero and covariance matrix ξI (ξ is the power of the noise and I denotes the identity matrix). Assume Λr and Λθ as independent, identically uniformly distributions as follows:
Taking the negative logarithm of Equation (21), we turn the MAP problem into the following optimization problem:
The minimization of F with respect to σ′, Λ and ξ is a complex nonlinear optimization problem; therefore, we adopt an alternatively iterative method. Based on the iterative MAP idea, we propose a novel sparse recovery algorithm for more generalized offgrid FDMIMO radar imaging, named sparse adaptive calibration recovery via iterative maximum a posteriori (SACRiMAP).
3.2. Algorithm Description
SACRiMAP includes mainly three steps. Defining l as the counter of iteration and assuming the lth estimations are obtained, we alternatively optimize σ′^{(}^{l}^{+1)}, the offgrid errors Λ^{(}^{l}^{+1)} and the parameter ξ^{(}^{l}^{+1)} in the (l+1)th iteration. The detailed algorithm is stated as follows:
(1) Sparse Recovery
Assuming that Λ^{(}^{l}^{)} and ξ^{(}^{l}^{)} are obtained, we seek for the optimal σ′^{(}^{l}^{+1)} to minimize the following equivalent cost function:
Letting ∂F_{1}/∂σ′^{(}^{l}^{+1)} = 0, we have:
From Equation (24), defining A = H + ĤΛ^{(}^{l}^{)} and α = pξ^{(}^{l}^{)}/2, we can get that:
Moreover, considering MNQ < K, with the aid of the matrix inversion formula [23], we can obtain:
(2) OffGrid Errors Calibration
In this stage, we use σ′^{(}^{l}^{+1)} and ξ^{(}^{l}^{)} to estimate {Δr_{k}^{(}^{l}^{+1)}, Δθ_{k}^{(}^{l}^{+1)}}, for k = 1,2,...,K, which are corresponding to minimizing the following equivalent cost function:
Observe that if σ′_{k}^{(}^{l}^{+1)} = 0, then Δr_{k} = 0 and Δθ_{k} = 0, which suggests that we only need to calculate those {Δr_{k}^{(}^{l}^{+1)}, Δθ_{k}^{(}^{l}^{+1)}} that correspond to σ′_{k}^{(}^{l}^{+1)} ≠ 0, while setting other errors to zero. Let Δ = [Δr_{1},...,K, Δθ_{1},...,K]^{T}, and:
Equation (30) is a convex, constrained least square optimization problem, hence its optimal solution can be efficiently obtained.
(3) Parameter Update
To alleviate the effect of parameter estimation on the performance of SACRiMAP, here, we embed a dynamically and adaptively parameter update process in SACRiMAP. Similarly, setting ∂F/∂ξ^{(}^{l}^{+1)} = 0 leads to:
Then, set l → l + 1 and repeat the three stages above until SACRiMAP shows no obvious improvement.
Remark (1) The Initialization of SACRiMAP
Here, we initialize Λ^{(0)} = 0, then, we get σ′^{(0)} by matched filter (MF). The jth item of σ′^{(0)} satisfies:
Then, we have the initial estimation of the noise power as:
Remark(2) The Convergence of SACRiMAP
We can conclude that the cost function F of SACRiMAP decreases with the iteration index l. The detailed proof of this conclusion is given in the Appendix—Proof of Remark (2).
Remark(3) The Applicability and Limitation of SACRiMAP
Though the algorithm of SACRiMAP is deduced in the case of the FDMIMO radar imaging scene, it is also applicable for the other imaging scenarios that can be described by Equation (18). In addition, we should also notice that Equation (18) is only approximately satisfied when the higher order terms of Taylor expansion are ignorable. When the offgrid errors turn bigger, the approximation of Equation (18) is violated, then the performance of SACRiMAP would largely deteriorate.
Based on the above idea, SACRiMAP can be described, as shown in Table 1.
4. Numerical Simulations
In this section, we present several numerical simulation results to illustrate the performance of the proposed algorithm. The SACRiMAP is implemented in Matlab, and Equation (30) is solved using CVX [24]. SACRiMAP is terminated as $\frac{{\Vert {{\sigma}^{\prime}}^{(l+1)}{{\sigma}^{\prime}}^{(l)}\Vert}_{2}}{{\Vert {{\sigma}^{\prime}}^{(l)}\Vert}_{2}}\le 1\times {10}^{6}$ or the maximum number of iterations, set to 50, is reached.
4.1. Verification of SACRiMAP
The simulation conditions are given in Table 2. From Table 2, we know that the wavenumberdomain coverage is (2MB/c, (MN − 1)d/λ) = (1,99), so the limit of the range and angle resolution achieved by MF is Res_{r} = 1 m, Res_{θ} ≈ 0.01 rad [20]. We set the signaltonoise ratio (SNR) to 20 dB.
The original scatterer distribution of the target is shown in Figure 2. There are U = 40 radial range bins and V = 40 angle bins whose adjacent intervals are 0.5 m and 0.005 rad, respectively. And there are 10 scatterers with the unit reflection coefficient in the scene of interest. From Figure 2, we know that the scatterers of the scene are off the grid.
In the following simulations, besides SACRiMAP, other algorithms will be involved: MF, OMP, FOCUSS, STLS [12] and TLSFOCUSS [13]. STLS seeks to solve the following optimization problem:
From the Bayesian perspective, STLS is equivalent to searching for the MAP solutions by assuming that the added noise is white Gaussian, σ′ is Laplacian and Λ is Gaussian.
Similarly, TLSFOCUSS is equivalent to searching for the MAP solutions by assuming that the added noise is white Gaussian, σ′ is l_{p}term forced and Λ is Gaussian.
Figure 3(a–d) shows the imaging results by MF, OMP, FOCUSS, STLS, TLSFOCUSS and our proposed method (SACRiMAP), where the blue circles represent the true scatterers. As stated above, the imaging results by MF totally fail, because of the incomplete wavenumberdomain coverage. The OMP and FOCUSS methods are likely to fail, since the offgrid scatterers exist. Comparing with OMP and FOCUSS, STLS and TLSFOCUSS can get improved recovery results. However, the offgrid errors cannot be calibrated accurately, so the imaging performance is not good. In contrast, our proposed method can deal with the offgrid scatterers, and thus, the imaging performance is good. Furthermore, as shown in Figure 4, during the whole imaging process, the cost function F keeps decreasing after each iteration, as proved in Remarks (2).
4.2. NMSE of the Imaging Results under Various SNR Conditions
In this subsection, we study the imaging error with respect to the SNR level. The target recovery errors, as well as the offgrid recovery errors, are averaged over 30 trials. In each trial, the offgrid errors are uniformly distributed in the region of one radial range bin or angle bin. We use the same data as Section 4.1, and expect that the SNR varies from 0 dB to 40 dB with interval 5 dB.
Assuming that the offgrid errors are known a priori, then it can obtain the best imaging result, since it exploits the exact, oracle information of the offgrid errors. We name it as oracle sparse recovery (OSR) [25] for comparison with other approaches. The normalized mean square error (NMSE) of the target recovery results by OMP, FOCUSS, STLS, TLSFOCUSS, SACRiMAP and OSR versus SNR are shown in Figure 5. From Figure 5, it can be seen that by OMP and FOCUSS, the target recovery errors always maintain a high level, even when the SNR increases. Using STLS and TLSFOCUSS, we can obtain improved recovery results to some extent. However, since the offgrid errors are uniformly distributed, the assumption about Λ in STLS and TLSFOCUSS could not capture the property of Λ, so the target recovery and the offgrid error recovery results by STLS and TLSFOCUSS are not good. However, SACRiMAP can get a better target recovery and also offgrid recovery result when the SNR is greater than 15 dB. Expect the ideal case of OSR and SACRiMAP has the smallest recovery error.
4.3. NMSE of the Imaging Results under Various Discretized Grid Intervals
In this subsection, we assume that the size of the imaging area and the actual scatterer distribution of the target are invariant, then we change the discretized grid interval and, consequently, the number of the discretized radial range bins and angle bins. We assign SNR = 20 dB, and the target recovery errors, as well as the offgrid recovery errors, are averaged over 30 trials.
Figure 6 presents the corresponding simulation results. When the grid interval is less than 0.4 (normalized by Res_{r} and Res_{θ}), the dense grid enhances the coherence between the column of the sensing matrix, which leads to the violation of the RIP condition for reliable sparse recovery. From Figure 6, when the grid interval is 0.2, OMP, FOCUSS, STLS, TLSFOCUSS, SACRiMAP and OSR, all have different levels of performance deterioration. When the interval is greater than 0.4 (normalized by Res_{r} and Res_{θ}), a nearly constant NMSE is obtained using OSR, since the offgrid errors are assumed to be known exactly. The errors of OMP, FOCUSS, STLS and TLSFOCUSS always maintain a high level with the discretized grid interval. Besides, the error of SACRiMAP also increases with the discretized grid interval; however, the level of NMSE is lower than those of OMP and FOCUSS and apparently higher than that of OSR. When the interval is 1 (normalized by Res_{r} and Res_{θ}), the SACRiMAP fails to improve the recovery performance, because a larger interval leads to higher approximation errors due to high order terms of Taylor expansion. Such a behavior is consistent with our analysis.
5. Conclusions
This paper presents the SACRiMAP method to realize high resolution imaging of sparse, but offgrid, targets for the FDMIMO radar system. Unlike traditional sparse recovery methods, SACRiMAP adaptively adjusts the offgrid errors, meanwhile seeking the optimal target reconstruction result. Through iterative MAP, it turns the nonconvex optimization problem of offgrid sparse recovery to three main stages: sparse recovery, offgrid errors calibration and parameter update. Benefited from adaptively adjusting and updating, SACRiMAP has some merits, e.g., there is no need for accurate initialization and improved robustness to noise. The derivations and numerical simulations illustrate the effectiveness of the new method, which shows the potential for the method to be applied in practical systems.
In this paper, we only consider the case of FDMIMO radar imaging; however, the framework in this paper can be extended to other imaging radar systems, such as generalized MIMO radar imaging and passive radar imaging, so SACRiMAP will have wider applications. Here, we only consider the first order Taylor approximation of the offgrid errors. Higher order approximations can be adopted in the future to further reduce the modeling error and, hence, to achieve higher precision. Moreover, in the offgrid problem formulation, the scatterers and offgrid errors are jointly sparse. Inspired by the recent works on block and structured sparsity [26–28], our future work is to fully excavate the joint sparsity to accelerate the convergence of SACRiMAP and, meanwhile, improve the recovery performance.
This work was supported by the HiTech Research and Development Program of China under Grant Project No. 2011AA120103.
References
 Fishler, E.; Haimovich, A.; Blum, R.; Chizhik, D.; Cimini, L.; Valenzuela, R. MIMO Radar: An Idea Whose Time Has Come. Proceedings of IEEE Radar Conference, Boston, MA, USA, 17– 20 April 2007; pp. 71–78.
 Dai, X.Z.; Xu, J.; Peng, Y.N. High Resolution Frequency MIMO Radar. Proceedings of IEEE Radar Conference, Boston, MA, USA, 17– 20 April 2007; pp. 693–698.
 Zhang, J.J.; Suppappola, A.P. MIMO Radar with Frequency Diversity. Proceedings of International Waveform Diversity Design (WDD), Orlando, FL, USA, 8–13 February 2009; pp. 208–212.
 Liu, C.C.; Chen, W.D. Sparse Frequency Diverse MIMO Radar Imaging. Proceedings of the 46th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, USA, 4–7 November 2012.
 Wang, W.Q. Spacetime coding MIMOOFDM SAR for highresolution imaging. IEEE Trans. Geosci. Remote Sens 2011, 49, 1–11, doi:10.1109/TGRS.2010.2093589.
 Chen, C.Y.; Vaidyanathan, P.P. Compressive Sensing in MIMO Radar. Proceedings of 42nd Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, USA, 26–29 October 2008; pp. 41–44.
 Yu, Y.; Petropulu, A.P.; Poor, H.V. CSSF MIMO radar: Lowcomplexity compressive sensing based MIMO radar that uses step frequency. IEEE Trans. Aerosp. Electron. Syst 2012, 48, 1490–1504, doi:10.1109/TAES.2012.6178074.
 Tan, X.; Roberts, W.; Li, J.; Stoica, P. Sparse learning via iterative minimization with application to MIMO radar imaging. IEEE Trans. Signal Process 2011, 59, 1088–1101, doi:10.1109/TSP.2010.2096218.
 Zhang, L.; Xing, M.; Qiu, C.; Li, J.; Bao, Z. Achieving higher resolution ISAR imaging with limited pulses via compressive sampling. IEEE Geosci. Remote Sens. Lett 2009, 6, 567–571, doi:10.1109/LGRS.2009.2021584.
 Gleichman, S.; Eldar, Y. Blind compressed sensing. IEEE Trans. Inform. Theory 2011, 57, 6958–6975, doi:10.1109/TIT.2011.2165821.
 Chi, Y.; Scharf, L.; Pezeshki, A.; Calderbank, A. Sensitivity to basis mismatch in compressed sensing. IEEE Trans. Signal Process 2011, 59, 2182–2195, doi:10.1109/TSP.2011.2112650.
 Zhu, H.; Leus, G.; Giannakis, G. Sparsitycognizant total leastsquares for perturbed compressive sampling. IEEE Trans. Signal Process 2011, 59, 2002–2016, doi:10.1109/TSP.2011.2109956.
 Han, X.B.; Zhang, H.; Li, G. Fast Algorithms for Sparse Recovery with Perturbed Dictionary, Available online: http://arxiv.org/abs/1111.6237 (accessed on 1 May 2012).
 Yang, Z.; Xie, L.; Zhang, C. OffGrid Direction of Arrival Estimation Using Sparse Bayesian Inference, Available online: http://arxiv.org/abs/1108.5838 (accessed on 17 September 2012).
 Ji, S.; Xue, Y.; Carin, L. Bayesian compressive sensing. IEEE Trans. Signal Process 2008, 56, 2346–2356, doi:10.1109/TSP.2007.914345.
 Huang, T.; Liu, Y.; Meng, H.; Wang, X. Adaptive matching pursuit with constrained total least squares. EURASIP J. Adv. Signal Process 2012, doi:10.1186/16876180201276.
 Tang, G.; Bhaskar, B.N.; Shah, P.; Recht, B. Compressed Sensing off the Grid, Available online: http://arxiv.org/abs/1207.6053 (accessed on 25 July 2012).
 Fannjiang, A.; Tseng, H. Compressive Radar with OffGrid and Extended Targets, Available online: http://arxiv.org/abs/1209.6399 (accessed on 28 September 2012).
 Carrara, W.G.; Goodman, R.S.; Majewski, R.M. Spotlight Synthetic Aperture RadarSignal Processing Algorithm; Artech House: Norwood, MA, USA, 1995.
 Liu, C.C.; Xu, H.; He, X.Z.; Chen, W.D. The Distributed Passive Radar 3D Imaging and Analysis in Wavenumber Domain. Proceedings of IEEE International Conference on Signal Processing (ICSP), Beijing, China, 24– 28 October 2010; pp. 2051–2054.
 Wipf, D.P.; Rao, B.D. Sparse bayesian learning for basis selection. IEEE Trans. Signal Process 2004, 52, 2153–2164, doi:10.1109/TSP.2004.831016.
 Gorodnitsky, I.F.; Rao, B.D. Sparse signal reconstructions from limited data using FOCUSS: A reweighted minimum norm algorithm. IEEE Trans. Signal Process 1997, 45, 600–616, doi:10.1109/78.558475.
 Franklin, J.N. Matrix Theory; Dover Publications: Dover, UK, 2000.
 CVX Research Inc. CVX: Matlab Software for Disciplined Convex Programming, Version 2.0 Beta. Available online: http://cvxr.com/cvx (accessed on 4 September 2012).
 Yang, Z.; Zhang, C.; Xie, L. Robustly Stable Signal Recovery in Compressed Sensing with Structured Matrix Perturbation, Available online: http://arxiv.org/abs/1112.0071 (accessed on 14 March 2012).
 Zheng, J.; Kaveh, M. DirectionofArrival Estimation Using A Sparse Spatial Spectrum Model with Uncertainty. Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Prague, Czech Republic, 22– 27 May 2011; pp. 2848–2551.
 Eldar, Y.; Kuppinger, P.; Bolcskei, H. Blocksparse signals: Uncertainty relations and efficient recovery. IEEE Trans. Signal Process 2010, 58, 3042–3054, doi:10.1109/TSP.2010.2044837.
 Bach, F.; Jenatton, R.; Mairal, J.; Obozinski, G. Structured Sparsity through Convex Optimization, Available online: http://arxiv.org/abs/1109.2397 (accessed on 20 April 2012).
Appendix
Proof of Remark (2)
Proof: It is equivalent to prove that the cost function F of SACRiMAP decreases in each of the three stages: sparse recovery, offgrid errors calibration and parameter update. Firstly, in the sparse recovery stage, Λ^{(}^{l}^{)} and ξ^{(}^{l}^{)} are kept unchanged. We write F = F_{1} + R_{1}, where R_{1} is independent of σ′. The difference between F_{1}(σ′^{(}^{l}^{+1)}) and F_{1}(σ′^{(}^{l}^{)}) totally reflects the change of F. Furthermore, since there exists an internal iteration, we have to consider F_{1}(σ′^{(}^{l}^{,}^{s}^{+1)}) in the (s+1)th internal step, i.e., we have to compare between F_{1}(σ′^{(}^{l}^{,}^{s}^{+1)}) and F_{1}(σ′^{(}^{l}^{,}^{s}^{)}). It is showed in [21] that F_{1}(σ′^{(}^{l}^{,}^{s}^{+1)}) is smaller than F_{1}(σ′^{(}^{l}^{,}^{s}^{)}) based on the fact that f(y) = y^{q} (y > 0, 0 < q < 1) have the concave property. Therefore, the cost function keeps decreasing till the internal iteration converges, yielding σ′^{(}^{l}^{+1)}. So, there is no doubt that F_{1}(σ′^{(}^{l}^{+1)}) < F_{1}(σ′^{(}^{l}^{)}) as σ′^{(}^{l}^{)} and σ′^{(}^{l}^{+1)} represent the initial value and the converged value of the internal iteration, respectively. Therefore, we prove that F_{1}, i.e., F decreases after the sparse recovery stage.
Secondly, in the offgrid errors calibration stage, similarly, we rewrite F = F_{2} + R_{2}, where R_{2} is independent of Λ. Equation (28) is equivalent to Equation (30), so we have F_{2}(Λ^{(}^{l}^{+1)}) = F_{2}(Δ^{(}^{l}^{+1)}) and F_{2}(Λ^{(}^{l}^{)}) = F_{2}(Δ^{(}^{l}^{)}). Furthermore, we can deduce that:
Finally, since the parameter ξ^{(}^{l}^{+1)} is computed according to ∂F/∂ξ^{(}^{l}^{+1)} = 0, it would definitely result in the decrease of the cost function F after the parameter update stage.
Based on all the analysis above, we know that the cost function F of SACRiMAP decreases in each stage of the (l+1)th iteration. Therefore, It can be proved that F decreases with the iteration index l.
Table 1. The main steps of sparse adaptive calibration recovery via iterative maximum a posteriori (SACRiMAP). 
Sparse Adaptive Calibration Recovery via Iterative MAP (SACRiMAP) 

Input: z, H 
Initialization: Set ${{\mathit{\sigma}}^{\prime}}_{j}^{(0)}={\mathit{h}}_{j}^{H}\mathit{z}/\left({\mathit{h}}_{j}^{H}{\mathit{h}}_{j}\right),{\mathit{\Lambda}}^{(0)}=0,{\xi}^{(0)}={\Vert \mathit{z}\mathit{H}{{\mathit{\sigma}}^{\prime}}^{(0)}\Vert}_{2}^{2}/\left(\mathit{MNQ}\right)$ 
Iteration (denote l as the counter of iteration):

Output: σ′ = σ′^{(}^{l}^{+1)}, Λ = Λ^{(}^{l}^{+1)}. 
Table 2. Simulation conditions. 
Parameter  Value 

Number of transmitters M  10 
Number of receivers N  10 
Bandwidth of each transmitted signal B  15 MHz 
Waveform pulse duration T  2 μs 
Number of snapshots Q  10 
Carrier frequency of the first transmitter f_{c}  10 GHz 
Interelement spacing of the transmitters dt  0.3 m 
Interelement spacing of the receivers dr  0.03 m 