Research ArticleA Compressive Sensing-Based Bistatic MIMO Radar ImagingMethod in the Presence of Array Errors

Zhigang Liu, Jun Li , Junqing Chang, and Yifan Guo

National Lab of Radar Signal Processing, Xidian University, Xi’an 710071, China

Correspondence should be addressed to Jun Li; junli01@mail.xidian.edu.cn

Received 27 June 2018; Accepted 28 August 2018; Published 25 November 2018

Academic Editor: Andrea Francesco Morabito

Copyright © 2018 Zhigang Liu 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.

A robust transmit-receive angle imagingmethod for bistatic MIMO radar based on compressed sensing is proposed. A new imagingmodel with array gain and phase error is established. The array gain error and phase error were modeled as a random interferencefor observation matrix by mathematical derivation. A constraint of observationmatrix error is constructed in optimization problemof sparse recovery to reduce the effect of the interference of observation matrix. Then, the iterative algorithm of the optimizationproblems is derived. The proposed recovery method is more robust than the existed method in small samples, especially in thecase of one snapshot. It is applicable in the case of relatively small array gain and phase errors. Simulation results confirm theeffectiveness of the proposed method.

1. Introduction

Bistatic MIMO radar has the potential advantages of bothbistatic radar and MIMO radar. Particularly, bistatic MIMOradar has the capability of obtaining the transmit angleinformation by processing the received data. As a new radarsystem, bistatic MIMO radar has been applied to targetlocalization [1–4], clutter cancellation [5, 6], and imaging[7, 8]. Compressed sensing (CS) is attracted technique usedin the field of radar imaging. Compressive sensing theorycan achieve high-resolution imaging in small samples.Introduction of compressive sensing to bistatic MIMO radarcan achieve more accurate imaging results. However, whenthe gain and phase errors exist in the array elements, thedetection accuracy and imaging quality of the radar willdeteriorate seriously. Many literatures have studied robustcompression sensing recovery methods [7, 9–12]. A sparserecovery-based transmit-receive angle imaging scheme isproposed for bistatic multiple-input multiple-output(MIMO) radar in [8]. The method is robust under largeerror conditions. However, the method does not take intoaccount the array phase error. Furthermore, it is not robustin the case of one snapshot. Therefore, it is necessary to

further study the robust recovery method of bistatic MIMOradar in the nonideal systems. Based on the above work, anew constraint is added to the sparse recovery method,and the corresponding algorithm is deduced. The simulationresults show that in the case of one snapshot, the methodproposed in [8] cannot accurately reconstruct the sparse sig-nal, whereas the proposed method can still recover thesparse signal robustly.

This paper is organized as follows. The sparse signalmodel for bistatic MIMO radar with array errors isderived in Section 2. In Section 3, a robust CS algorithmis proposed to achieve target imaging in the presence ofarray errors. The proposed method is tested via simula-tions and analysis, which appear in Section 4. Finally,Section 5 concludes the paper.

2. Signal Model of Bistatic MIMO Radar withArray Errors

The configuration of a bistatic MIMO radar is shown inFigure 1. Consider the radar has M-transmit and N-receiveuniform linear array. The transmitting signals are S ∈ CM×L,

HindawiInternational Journal of Antennas and PropagationVolume 2018, Article ID 9434360, 6 pageshttps://doi.org/10.1155/2018/9434360

where L is the length of transmitting signals. Assuming thatthe pth pixel point of the target is at the angle θtp, θrp . Thereceived signal can be written as

Yq =ArDqAtS + Eq q = 1, 2,… , Q, 1

where At = atp M×P and atp = 1 ej 2π/λ dt sin θtpej 2π/λ 2dt sin θtp

… ej 2π/λ M−1 dt sin θtp . Ar = arp N×P, where arp = 1ej 2π/λ dr sin θrpej 2π/λ 2dr sin θrp … ej 2π/λ N−1 dr sin θrp . At istransmit steering matrix and Ar is receive steering matrix ofP pixel points. λ represents the transmit signal wavelength.dt and dr are the ideal element space at the transmitter andreceiver. Dq = diag d1,… , dp denotes the scattering coeffi-cient of P pixel points of the target in the qth pulse period.The noise Eq is assumed to be independent, and zero-meancomplex Gaussian distribution withEq~Nc 0, σ2

nIN .As shown in Figure 1, the region of interest is divided into

two dimensional discrete set of angular positions Ω = θk,θl : k, l ∈ 1,… , G × 1,… , G . The target pixel distri-bution is assumed to be Xq ∈ℂG×G and the transmit signalis orthogonal waveform. After the matched filtering, thereceived signal can be expressed as Yq =ΑrXqΑt + Eq, thevector form of it is

yq = vec Yq = Αr ⊗Αt vec Xq + eq =Αxq + eq, 2

whereA =Αr ⊗Αt , xq = vec Xq , and eq = vec Eq . When thegain and phase errors exist for both the transmit and receivearray elements, the transmit and receive steering matrix canbe written as

Αt = Γt′Αt = I + Γt Αt =Αt + ΓtΑt =Αt + Εt,Αr = Γr′Αr = I + Γr Αr =Αr + ΓrΑr =Αr + Εr,

3

where Γt = diag ρt1,… , ρtM and Γr = diag ρr1,… , ρrN arethe diagonal matrices with array gain and phase errors atdiagonal elements. ρti = atie

jφti , where ati is gain error andφti is phase error in the transmit array elements. ρri = arie

jφri

, where ari is gain error and φri is phase error in the receivearray elements. The receiving data with the gain and phase

errors of the array can be written as Y~q =Α~

rXqΑ~t + Eq.

The vector form of it is

yq = vec Yq = Αr ⊗Αt vec Xq + eq= Αt + Εt ⊗ Αr + Εr xq + eq= Αt ⊗Αr +Αt ⊗ Εr + Εt ⊗Αr + Εt ⊗ Εr xq + eq= Α + Ε xq + eq,

4

where

Ε =Αt ⊗ Εr + Εt ⊗Αr + Εt ⊗ Εr 5

It can be observed from (4) and (5) that the array errorcan be modeled as an additive error matrix for ideal observa-tion matrix A. The next task is to construct the sparse recov-ery problem under this nonideal model.

3. Robust CS-Based Sparse Imaging Algorithm

Considering more general situation, the E matrix in (4) canbe relaxed as a random perturbation matrix which addedon the ideal observation matrix A. Inspired by the robustbeamforming method [12], the norm of the observationmatrix is constrained to reduce the effect of matrix E. A qua-dratic optimization problem with sparse constraint can beconstructed as

minB,x

y~q − Βx 22 + x 1s t B 2

2 =M B −A 22 ≤ ε 6

In this optimization problem, matrix Β is an unknownmatrix of the same size as the observation matrix A. Weuse it to approximate the actual observation matrix. Sincethe actual observation matrix is derived from the sum ofthe ideal observation matrixA and an unknown perturbationmatrix E, we obtain more accurate values of the actual obser-vation matrix by the above optimization methods.

The iterative method is derived to solve the optimizationproblem. In the kth iteration, the algorithm performs twosteps. In the first step, we solve the familiar convex optimiza-tion problem

x̂ k = arg minx

y~q − B k x 22 + x 1, 7

holding B matrix fixed. This is a convex optimization prob-lem that is known to yield unique sparse solutions. The nextstep is to hold the coefficients x fixed and find a better obser-vation matrix solution by using the Lagrange multipliermethod. The updating formula of the observation matrix is

B̂ k+1 = y x kH+ γA x k x k

H+ γ + μ I

−18

The above two steps are executed iteratively until an idealsolution is obtained.

Xq (k, l)

Receive array

Tran

smit

arra

y

1

1 2 3 N

23

M

Ω

𝜃r

𝜃t

Figure 1: Bistatic MIMO radar configuration.

2 International Journal of Antennas and Propagation

In the end, we introduce the derivation of the updatingformula for observation matrix. We construct the Lagrangefunction according to the optimization problem

f B, γ, μ = y − Bx 22 + x 1 + γ B −A 2

2 − ε

+ μ B 22 −M

9

By getting the partial derivative of the Lagrange functionto B and making it equal to zero, the optimal estimation ofthe actual observation matrix can be obtained.

∂f∂B = ∂ y − Bx 2

2∂B + γ

∂ B −A 22

∂B + μ∂ B 2

2∂B 10

The first item is

∂ y − Bx 22

∂B = ∂Tr y − Bx y − Bx H

∂B

= ∂Tr yyH∂B −

∂Tr yxHBH

∂B

−∂Tr BxyH

∂B + ∂Tr BxxHBH

∂B= 0 − yxH − xyH H + B xxH H + BxxH

= 2BxxH − 2yxH

11

The second item is

∂ B −A 22

∂B = ∂Tr B −A B −A H

∂B

= ∂Tr BBH

∂B −∂Tr BAH

∂B

−∂Tr ABH

∂B + ∂Tr AAH

∂B= 2B‐A‐A + 0 = 2B‐2A

12

The third item is

∂ B 22

∂B = 2B 13

Then, we get the estimation of the actual observationmatrix.

∂f∂B = 2BxxH‐2yxH + 2γ B‐A + 2μB = 0

⇒ B̂ = yxH + γA xxH + γ + μ I −114

The algorithm is summarized as Algorithm 1.For the error measurement parameter ε, we take the value

according to the empirical data. Usually we set ε as one tenthof the error value of the first iteration. That is to say, our algo-rithm stops when the estimation error of sparse signal to onetenth of the initial estimation error. This rule leads to relativegood results in practice. More appropriate selection methodsremain to be further studied.

Furthermore, the method we propose is a more generallyapplied method. As long as the signal model can be expressed

Input: y, A, λ, μOutput: x̂, B̂Step 1: Initialization: x0 = 0n×1, B0 =AStep 2: Fix B and solve the optimization problem

x̂ k = arg minx

y~q − B k x 22 + x 1

Step 3: Fix x and calculate the matrix BB̂ k+1 = y x k H + γA x k x k H + γ + μ I

−1

Step 4: Judge the condition of convergence

x k+1 − x k 22 ≤ ε

If the condition is satisfied, the iteration is stopped, and Step 2 is returned if it is not satisfied.

Algorithm 1: Algorithm Flow.

0 5 10 15 20 250.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

Iteration time

MSE

of t

he re

cove

ry re

sults

Figure 2: MSE curve with iteration time (M =N = 15, Q = 1, SNR = 10 dB, σt = σr = 0 2).

Table 1: The running time of each method.

Methods Method proposed in [8] Proposed method Direct CS

Runtime 0.902 s 11.205 s 0.751 s

3International Journal of Antennas and Propagation

as a form of random interference added to the observationmatrix, the proposed method can be used for sparse recon-struction. So algorithm is still effective in the case of manynonideal situations, such as array location uncertainties,and CS grid error.

4. Simulation Results

In this section, we simulate the method proposed in thisarticle and compare it with the method proposed in [8].It is assumed that the transmit array element and thereceive array element are uniform linear arrays with half-wavelength space between adjacent elements, and thenumber of transmit and receive array elements is 15. Bothof the transmit angular region and receive angular regionrange from 1° to 10°. Assume that there are two pixels of atarget at angle θt1, θr1 = 2°, 9° and θt2, θr2 = 9°, 2° . Weuse only one snapshot for the sparse recovery.

The form of the transmit and receive array gain andphase error is

Γt = diag exp N 0, σ2t ⊙ exp jN 0, σ2t ,

Γr = diag exp N 0, σ2r ⊙ exp jN 0, σ2

r ,15

where σt and σr are the parameters governing the arrayerrors.N 0, σ2

r denotes the Gaussian distribution. The singlesnapshot data is used in the simulations.

Figure 2 shows the MSE of the recovery results varyingwith iteration time. In the statistical sense, the algorithm con-verges at about 20 iteration times. The convergence rate ofthe algorithm is related to the size of the observation matrixand the selection of Lagrange multipliers.

Figure 3 shows the results of the image recovery using theproposed method compared with existing sparse methods. It

2 4 6 8 101

2

3

4

5

6

7

8

9

10

11

Transmit angle (deg)

Rece

ive a

ngle

(deg

)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a)

2 4 6 8 101

2

3

4

5

6

7

8

9

10

11

Transmit angle (deg)

Rece

ive a

ngle

(deg

)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b)

2 4 6 8 101

2

3

4

5

6

7

8

9

10

11

Transmit angle (deg)

Rece

ive a

ngle

(deg

)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(c)

2 4 6 8 101

2

3

4

5

6

7

8

9

10

11

Transmit angle (deg)

Rece

ive a

ngle

(deg

)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(d)

Figure 3: The performance of the proposed method compared with existing sparse methods (M =N = 15, θt1, θr1 = 2°, 9° , θt2, θr2= 9°, 2° , Q = 1, SNR = 10 dB, σt = σr = 0 3, γ = 0 01). (a) Original image. (b) Proposed method (Q = 1, MSE = 0 26). (c) Direct CSmethod (Q = 1, MSE = 0 45). (d) Method proposed in [8] (Q = 1, MSE = 1 69).

4 International Journal of Antennas and Propagation

can be observed that the image recovery of the direct CSmethod and the algorithm proposed in [8] have relativelyhigher sidelobes. It is obvious that the performance of theproposed method is prior to both of them.

The running time of the examples in Figure 3 byMATLAB in the same computer is listed in Table 1. It canbe observed that the computational complexity of the pro-posed method increases while it improves the performanceof the sparse recovery.

Figure 4 plots the mean square error of recoveryresults of the proposed method, the method in [8], andthe direct CS method. The governing parameters of thearray errors are equal for all the elements. The array error

parameter changes from 0 to 0.6 with the interval 0.01.For each simulation, 500 Monte Carlo trails are run. It isshown that the performance of the method in [8] is verypoor, even worse than the direct CS method, in the caseof one snapshot. The proposed method has the lowestMSE among three methods. The results confirm thatthe proposed method is robust for array errors and suit-able to one snapshot case. It can be observed fromFigure 4 that the performance of the proposed methodis degraded when the array gain and phase errors aregreater than 0.4. The result implies that the proposedmethod is applicable in the case of relatively small arraygain and phase errors.

0 0.002 0.004 0.006 0.008 0.010

50

100

150

200

250

300

Lagrange Multiplier/𝛾

Itera

tive t

imes

of a

lgor

ithm

conv

erge

nce

(a)

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.0116

18

20

22

24

26

28

30

32

34

36

Lagrange Multiplier/𝜇

Itera

tive t

imes

of a

lgor

ithm

conv

erge

nce

(b)

Figure 5: The influence of parameters on the convergence of the algorithm. (a) Convergent curve (μ = 0 001). (b) Convergent curve (γ = 0 01).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

101

100

10−1

10−2

Array gain and phase error

MSE

of t

he re

cove

ry re

sults

Method proposed in [12]Proposed methodDirect CS method

Figure 4: Comparison of the performance of sparse recovery methods.

5International Journal of Antennas and Propagation

Figure 5 (a) shows the effect of γ on the convergence ofthe algorithm. The parameter μ is set as 0.001 and γ increasesfrom 0.0001 to 0.01 with the interval 0.00005. For each γ, 100simulations are performed to get the average number of iter-ations required for convergence of the algorithm. The samesimulations are done to evaluate the effect of μ on the conver-gence of the algorithm in Figure 5 (b). It is shown that theiteration number will be small with the increase of γ and μ.However, too large γ and μ will result to the divergence ofthe algorithm.

5. Conclusions

A robust transmit-receive angle imaging method for bistaticMIMO radar based on CS had been proposed in this paper.The method can be used to enhance the performance of CSin the case of both gain and phase errors of the array. In addi-tion, only one snapshot is required in the proposed method.Simulation and analysis demonstrated the effectiveness of theproposed method.

Data Availability

The simulation data used to support the findings of this studyare included within the article.

Conflicts of Interest

The authors declare that they have no competing interests.

Acknowledgments

This study has been supported by the National NaturalScience Foundation of China under Contract no. 61431016.

References

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[3] J. Chen, H. Gu, and W. Su, “A new method for joint DOD andDOA estimation in bistatic MIMO radar,” Signal Processing,vol. 90, no. 2, pp. 714–718, 2010.

[4] J. F. Li, W. Y. Chen, and X. F. Zhang, “Angle estimation forbistatic MIMO radar with unknown mutual coupling basedon improved propagator method,” Applied Mechanics andMaterials, vol. 513-517, pp. 3029–3033, 2014.

[5] J. Li, Y. Guo, G. Liao, Q. Guo, and J. Xi, “Dimension reducedsparse recovery method for clutter suppression in bistaticMIMO radar,” in 2015 IEEE Radar Conference (RadarCon),Washington DC, USA, May 2015.

[6] G. Zheng, D. Zhang, Y. Zhang, J. Tang, C. Feng, and X. Hu,“Fast OMP algorithm for 3D parameters super-resolutionestimation in bistatic MIMO radar,” Electronics Letters,vol. 52, no. 13, pp. 1164–1166, 2016.

[7] N. Tong, Y. Zhang, G. Hu, X. He, and X. Hu, “Multiple-input–multiple-output radar super-resolution three-dimensionalimaging based on a dimension-reduction compressive

sensing,” IET Radar, Sonar & Navigation, vol. 10, no. 4,pp. 757–764, 2016.

[8] J. Li, S. Zhu, X. Chen, L. Lv, G. Liao, and M. Yi, “Sparserecovery for bistatic MIMO radar imaging in the presenceof array gain uncertainties,” International Journal of Anten-nas and Propagation, vol. 2014, no. 1, Article ID 807960, 6pages, 2014.

[9] Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensi-tivity to basis mismatch in compressed sensing,” IEEE Trans-actions on Signal Processing, vol. 59, no. 5, pp. 2182–2195,2011.

[10] G. Tang, B. N. Bhaskar, P. Shah, and B. Recht, “Compressedsensing off the grid,” IEEE Transactions on Information The-ory, vol. 59, no. 11, pp. 7465–7490, 2013.

[11] H. Zhu, G. Leus, and G. B. Giannakis, “Sparsity-cognizant totalleast-squares for perturbed compressive sampling,” IEEETransactions on Signal Processing, vol. 59, no. 5, pp. 2002–2016, 2011.

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