Iterative Target Localization in Distributed MIMO Radar ... - IEEE Xplore

IEEE SIGNAL PROCESSING LETTERS, VOL. 24, NO. 11, NOVEMBER 2017

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Iterative Target Localization in Distributed MIMO Radar From Bistatic Range Measurements Ali Noroozi, Amir Hosein Oveis, and Mohammad Ali Sebt

Abstract—This letter discusses the problem of determining the target position from bistatic range measurements in a multipleinput multiple-output radar system with widely separated antennas. By introducing nuisance parameters and taking the relationship between these parameters and the unknown target position into consideration, the problem is formulated as a constrained weighted least squares (CWLS) problem, which is a quadratically constrained quadratic programming problem. Since the constraints are nonconvex, finding the global solution is a difficult task. To solve this problem, an iterative CWLS method is developed in this letter. We iteratively approximate each nonconvex constraint as a linear constraint to convert the problem as a linearly constrained quadratic programming problem and then solve the problem. Numerical simulations are included to support and corroborate the theoretical developments. Index Terms—Bistatic range (BR), constrained weighted least squares (CWLS), iterative algorithm, target position.

I. INTRODUCTION ESEARCH into multiple-input multiple output (MIMO) radar systems has been growing over the past decade [1]–[3]. These systems have drawn considerable attention because of their remarkable advantages in enhancing detection and improving localization accuracy. MIMO radars are typically classified into two different categories, widely separated (distributed) [2] and colocated [3] antennas. The former makes use of spatial diversity, whereas the colocated MIMO radar takes advantage of the waveform diversity. In recent years, there have been several publications on target location estimation in the distributed MIMO radars [4]–[8], which indicates the importance of this issue. In general, the localization methods in these radars can be classified as either direct or indirect. The latter methods first measure the time delays using the cross-ambiguity function. Then, by multiplying these delays by the speed of signal propagation and performing some simple operations, the corresponding bistatic ranges (BRs), which is the sum of transmitter-target and target-receiver ranges, are calculated. These BRs form a set of elliptic equations through which the target position is obtained. The best linear

R

Manuscript received June 15, 2017; revised August 13, 2017; accepted August 28, 2017. Date of publication August 30, 2017; date of current version October 11, 2017. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ioannis D. Schizas. (Corresponding author: Mohammad Ali Sebt.) The authors are with the Department of Electrical Engineering, K. N. Toosi University of Technology, Tehran 1631714191, Iran (e-mail: ali_noroozi@ ee.kntu.ac.ir; [email protected]; [email protected]). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2017.2747479

unbiased estimator method is presented in [9], which linearizes the problem with a Taylor series expansion of the nonlinear equations. The major drawback of this approach is its dependency on the initial guess, which has to be sufficiently close to the target position. In [10], a closed-form least squares (LS) solution is presented in MIMO radar systems. In [11], a quadratically constrained quadratic programming (QCQP) problem is formed in order to linearize the ellipsoid equations and then a weighted least squares (WLS) algorithm is applied to realize it. The distributed approach [12] is a solution to find the target location. This method first divides the measurements into several groups based on the different transmitter elements or receiver elements, and then employs two-stage weighted least squares (TSWLS) estimator for each group to independently produce an estimate of target position. Next, the results from different groups are combined to form a composite estimate. In [13] and [14], a closed-form one-stage weighted least squares (OSWLS) method for target localization is presented in the general case of noise in two different conditions. Recently, two different solutions are presented in [15] and [16] to solve the problem, which can approximately attain the Cramer-Rao lower bound (CRLB) under low noise conditions. In this letter, we first formulate the target localization problem as a QCQP problem and then propose an iterative constrained weighted least squares (CWLS) algorithm to solve the problem. Unfortunately, the constraints are nonconvex and, therefore, the QCQP problem is nonconvex and NP-hard. To resolve this difficulty, we recursively approximate each nonconvex constraint as a linear constraint by substituting the previous estimate of the unknown parameter into the each of quadratic constraints. Indeed, we convert the QCQP problem to the linearly constrained quadratic programming (LCQP) problem by approximating the quadratic constraints by the linear constraints. The LCQP problem has the advantage of closed-form solution [17], [18], and thus the computation of the proposed method is quite efficient. In addition, the simulation results show that the proposed iterative CWLS method is able to converge to the optimal solution. The following notations are used throughout this letter. Bold lower- and upper-case letters denote column vectors and matrices, respectively. A diagonal matrix formed by the elements of a is represented by diag(a). The kth element of the vector a is denoted by [a]k , whereas the kth row of the matrix A is ˆ ˆ represented by A(k, :). A(or a) is the noisy version or estimated value of the matrix A (or a). ek denotes a zero vector of appropriate size except for the kth element which is one. 1 is a vector of all ones and 0 denotes a zero vector. I and O denote an identity matrix and a zero matrix, respectively. The size of the vectors/matrices can be understood from the context. The symbol ⊗ denotes the Kronecker product.

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II. PROBLEM FORMULATION

where

Consider a widely distributed MIMO radar system consisting of M transmit and N receive antennas in the threedimensional (3-D) space. The ith transmitter and the jth receiver are located at known positions xt,i = [ xt,i yt,i zt,i ]T and xr,j = [ xr,j yr,j zr,j ]T for i = 1 , ... , M and j = 1 , ... , N , respectively. The transmitters send out a set of orthogonal waveforms that are then reflected by an object at unknown position x0 = [ x0 y0 z0 ]T . Then, each receiver collects the direct and reflected signals from the transmitters and the target, respectively. The Euclidean distance between the target and the ith transmitter is rt,i = x0 − xt,i = (x0 − xt,i )2 + (y0 − yt,i )2 + (z0 − zt,i )2 .

1 2 rˆ + xTt,i xt,i − xTr,j xr,j 2 i,j

T ˆ 1,i (j, :) = xTt,i − xTr,j eTi ri,j ˆ1 = A ˆT ˆ T1,1 . . . A ,A A 1,M =

T θ = xT0 αT , α = [ α1 . . . αM ] .

(1)

(3)

1 2 ri,j + xTt,i xt,i − xTr,j xr,j . xTt,i − xTr,j x0 + ri,j αi = 2 (4)

Due to measurement errors and noise in the BRs, we express ri,j as rî,j − ni,j in (4), where rî,j denotes the BR measurement for the pair of the ith transmitter and the jth receiver and ni,j is a zero mean Gaussian noise added to the BR measurement of ith transmitter and the jth receiver. Then, (4) is rewritten as T xt,i − xTr,j x0 + (ˆ ri,j − ni,j ) αi 1 (ˆ ri,j − ni,j )2 + xTt,i xt,i − xTr,j xr,j . = 2

(5)

By ignoring the noise terms of second order in (5), the error term caused by the measurement noise becomes εi,j =

1 2 rî,j + xTt,i xt,i −xTr,j xr,j − xTt,i − xTr,j x0 −ˆ ri,j αi 2 (6)

where εi,j = (ˆ ri,j − αi )ni,j . Putting (6) into matrix form for all transmitters and receivers gives ˆ1 − A ˆ 1θ ε=b

(7)

(9)

where B = diag (ˆ r − α ⊗ 1) T T T ˆ r= ˆ ri = [ rî,1 . . . rî,N ] r1 . . . ˆ rTM , ˆ T T n = nT1 . . . nTM , ni = [ ni,1 . . . ni,N ] .

(10)

It is important to note that (7) represents a linear relationship with respect to θ. Therefore, the unknown vector θ can be determined by the WLS estimation. The WLS solution of (7), which minimizes the cost function εT Wε to θ, is given as −1 ˆ1 ˆ T1 WA ˆ T1 Wb ˆ1 θˆ = A A

By rearranging and then squaring both sides of (3) as well as defining αi = rt,i , after some algebraic manipulation, it follows that

(8)

ε = Bn

(2)

The BR defined as the sum of transmitter-target range rt,i and target-receiver range rr,j for the pair of the ith transmitter and the jth receiver is ri,j = rt,i + rr,j .

j

The error vector ε can also be expressed in terms of the noise vector n as

The Euclidean distance between the target and the jth receiver is given by rr,j = x0 − xr,j = (x0 − xr,j )2 + (y0 − yr,j )2 + (z0 − zr,j )2 .

T T ε = εT1 . . . εTM , εi = [ εi,1 . . . εi,N ]

T

ˆ 1,i ˆ1 = b ˆ T1,1 . . . b ˆT b , b 1,M

(11)

where W is a symmetric positive definite matrix given by −1 −1 = BQBT W = E ε εT

(12)

where Q = E[n nT ]−1 denotes the noise covariance matrix and B is given by (10). It is worth noting that the unknown vector θ consists of the unknown target position x0 and the nuisance parameters α. Since the unknown target position and each of the nuisance parameters, αi for i = 1, . . . , M, are not independent of each other, the WLS solution of (7) given by (11) is not the optimum minimizer of (7). To improve the localization accuracy, the relationship between the unknown parameter x0 and each of the nuisance parameters should be applied to the localization problem. As a result, by taking these relationships into account, the target localization problem can be formulated as follows [11]: min θ

s.t.

ˆ1 − A ˆ 1θ b

T

ˆ1 − A ˆ 1θ W b

αi2 = xT0 − xTt,i (x0 − xt,i ) ,

i = 1, . . . , M . (13)

The CWLS problem (13) is a nonconvex quadratic optimization problem with M quadratic equality constraints. We will discuss the solution of the target localization problem in the following section.

NOROOZI et al.: ITERATIVE TARGET LOCALIZATION IN DISTRIBUTED MIMO RADAR FROM BISTATIC RANGE MEASUREMENTS

III. ITERATIVE CWLS SOLUTION After performing some elementary manipulation, the CWLS target localization problem (13) can be reformulated as min θ

ˆT θ ˆ − 2b θ T Aθ

s.t.

T

θ −

sTi

Ci (θ − si ) = 0,

i = 1, . . . , M

(14)

where ˆ=A ˆ1 ˆ 1, b ˆ T1 Wb ˆ =A ˆ T1 WA A T . si = xTt,i 0T , Ci = diag 1T −eTi

(15)

Although the objective function of (14) is convex, the quadratic equality constraints are nonconvex. Thus, the CWLS problem (14) belongs to the class of nonconvex QCQP problems, which are NP-hard in general. In the discussion that follows, we develop an iterative approximation method to solve the CWLS problem. In the proposed algorithm, we replace one of the unknown vector θ with its estimation θˆ given by (11) and, therefore, each of nonconvex constraints becomes a linear equality constraint. Indeed, we convert the QCQP problem to the LCQP problem by approximating the quadratic constraints by the linear constraints. The main advantage of LCQP is that it has a closed-form solution. Consequently, the CWLS target localization problem becomes ˆT θ ˆ − 2b min θ T Aθ θ

s.t. where

cTi

θ = gi ,

i = 1, . . . , M

ci = Ci θˆ − si ,

(16)

gi = cTi si .

(17)

To derive the optimal solution of the CWLS problem (16), we put the linear constraints into the matrix form for easier manipulation and reformulate the equality constraints as Gθ = g

(18)

where T

G = [ c1 . . . c M ] ,

T

g = [ g1 . . . gM ] .

(19)

The general solution of (18) according to the generalized inverse theory of a matrix [19] is given by θ˜ = G† g + P⊥ ξ ⊥

(20) †

where ξ is an arbitrary vector, P = I − G G is a projection matrix onto the subspace orthogonal to that spanned by the rows of G. Note that G† represents the pseudo-inverse of G given as −1 . (21) G† = GT GGT Substituting (20) into the objective function of the problem (16) yields ˆ T θ = G † g + P⊥ ξ T A ˆ − 2b ˆ G † g + P⊥ ξ θ T Aθ ˆ T G † g + P⊥ ξ . (22) − 2b Minimizing the objective function of the problem (16) with respect to θ is equivalent to minimizing the right hand side of (22) with respect to ξ. Therefore, by taking the partial derivative of the right hand side of (22) with respect to ξ and setting the result equal to zero, the desired value of ξ is obtained. After some

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simple algebra, the minimum value of the objection function is obtained by taking the vector ξ as −1 ˆ − AG ˆ †g . ˆ ⊥ P⊥ b (23) ξ = P⊥ AP It is worth noting that the projection matrix P⊥ is symmetric of size (M + 3) × (M + 3) and idempotent. The projection matrix P⊥ is not full rank due to linearly dependent rows in ˆ ⊥ is singular. To obtain it, and as a result, the matrix P⊥ AP ˆ ⊥ , we can employ the singular value dethe inverse of P⊥ AP composition (SVD) technique. By assuming that the rank of the ˆ ⊥ is equal to m (where m < M + 3), the SVD matrix P⊥ AP form of this symmetric matrix is T V ˆ ⊥ = [ U U ] Σm O P⊥ AP (24) O O V T where U and VT are orthogonal matrices spanning the column ˆ ⊥ (when the matrix is symmetspace and row space of P⊥ AP ric, U = V), respectively, U and V T are orthogonal matrices ˆ ⊥ )T and of P⊥ AP ˆ ⊥ , respecspanning the null space of (P⊥ AP tively, and Σm denotes a diagonal matrix with m nonzero sinˆ ⊥ . In such a case, the inverse of P⊥ AP ˆ ⊥ gular values of P⊥ AP is computed as −1 −1 T T ˆ ⊥ P⊥ AP ≈ VΣ−1 (25) m U = UΣm U . The final solution in each iteration is then determined as a linear combination of the previous iteration estimate and the solution of the problem (16), given by (20). Thus, for the kth iteration, we have [20] θˆ(k ) = λθˆ(k −1) + (1 − λ) θ˜(k ) (26) where λ is a forgetting factor that lies in the range 0 < λ < 1 and its proper value leads to better convergence. Therefore, we iteratively solve the localization problem and update the solution and other parameters such as the weighting matrix in each iteration. The proposed iterative CWLS target localization algorithm is summarized as follows. 1) Specify initial values (k = 0) −1 ˆ 1, ˆ (0) = A ˆ T1 W(0) A W(0) = B(0) QBT(0) , A ˆ ˆ −1 θˆ(0) = A (0) b(0) .

ˆ1 , ˆ (0) = A ˆ T1 W(0) b b

2) Set k = k + 1 and formulate the approximate LCQP problem in the kth iteration ˆT ˆ (k−1) θ − 2b min θT A θ θ

(k−1)

cTi(k −1) θ = gi (k −1) , i = 1, . . . , M where ci (k −1) = Ci θˆ(k −1) −si , gi (k −1) = cTi(k −1) si. s.t.

(27) 3) Determine the solution of (27) given as −1 ⊥ ⊥ ˆ θ˜(k ) = G†(k −1) g(k −1) + P⊥ (k −1) P(k −1) A(k −1) P(k −1) ˆ (k −1) − A ˆ (k −1) G† b × P⊥ g (k −1) (k −1) (k −1) † where P⊥ (k −1) = I − G(k −1) G(k −1) .

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TABLE I POSITION (IN METERS) OF TRANSMITTERS AND RECEIVERS Tx no. i

xt , i

yt , i

zt , i

Rx no. j

xr, j

y r, j

zr, j

1 2 3 4 5 6 7 8 9

0 –300 –300 –200 –200 200 200 300 300

0 –200 200 –300 300 –300 300 –200 –200

150 150 100 200 100 100 80 120 –160

1 2 3 4 5 6 7 8 –

–450 –450 450 450 0 600 –600 0 –

–450 450 –450 450 600 0 0 –600 –

200 300 400 100 200 100 150 100 – Fig. 1. Performance comparison of different localization estimators in the first scenario.

4) Obtain the final solution of the localization problem in the kth iteration as θˆ(k ) = λθˆ(k −1) + (1 − λ) θ˜(k ) . 5) Check the convergence If θˆ(k ) − θˆ(k −1) /θˆ(k ) ≤ δ, where δ is the convergence terminate parameter, then the convergence is attained and the iteration is terminated, otherwise go to step (6). ˆ (k ) and then go ˆ (k ) , and b 6) Update the parameters W(k ) , A to step (2). The final point to note is that we find from extensive simulation runs that the proposed iterative CWLS method can converge in about 20 iterations. IV. SIMULATIONS In this section, two Monte Carlo simulation experiments are conducted to evaluate the proposed estimator to the problem of locating a single target in 3-D space. The performance of the proposed iterative CWLS method (referred to as “ICWLS”) is compared with that of the OSWLS method in [14], the TSWLS methods presented in [12], [15], and [16], and the CRLB derived in [14]. We consider a distributed MIMO radar arrangement such as [16] with M = 9 transmitters and N = 8 receivers. The position of all transmitters and receivers are given in Table I. In both simulations, the noise in the measured BR is modeled such as [16] as a zero-mean Gaussian random variable with a known variance, which is dependent only on the signal-to-noise ratio at each pair transmitter–receiver. As a consequence, the measurements of the BR were corrupted by additive Gaussian noise with the standard deviation σi,j = σ0 rt,i rr,j /R0 for i = 1, . . . , M and j = 1, . . . , N , where σ0 is a constant and R0 denotes the radius of the surveillance area (for further details, see [16]). The localization accuracy is evaluated according to the root mean squares error (RMSE) of the target position estimates, defined L as RMSE(x0 ) = x0, (l) − x0 2 /L, where x ˆ0,(l) del=1 ˆ notes the estimates of x0 at the lth ensemble run and L = 1000 is the number of ensemble runs. Two different scenarios similar to [16] are considered in this section. In the first, the target x0 is located at coordinates [ 100 400 200 ]T and the effect of increasing the noise level on the performance of the proposed method and other ones is examined. In the second, it is assumed that the target is flying parallel to the y-axis at x0 = 400 m and z0 = 200 m. The results of the first and second scenarios are shown in Figs. 1 and 2, respectively. Note that in both of them, the radius of the area of interest R0 is equal to 600 m.

Fig. 2. Performance comparison of different localization estimators in the second scenario.

Fig. 1 gives the RMSE of the different estimators as a function of σ0 , where σ0 varies from 1 to 1000. It is easy to find out that the iterative CWLS algorithm is more accurate and performs much better than other methods in the scenario considered here. Furthermore, from this figure, we can see a very good agreement between the proposed estimator and the CRLB accuracy over a wider range of noise levels (from σ0 = 1 to σ0 = 100). The RMSE of the different algorithms as the y-coordinate of the target varying from −R0 to R0 is also represented in Fig. 2. In the second scenario, the multiplier σ0 is set to 40. The result shows that the proposed method performs far better than any of the other algorithms for the case considered. We can also see that the proposed method is able to attain the CRLB accuracy.

V. CONCLUSION In this letter, the problem of locating a single target using BR measurements in a MIMO radar with widely separated antennas was investigated. An iterative CWLS algorithm was presented to determine the target position. By considering the quadratic relationship between the target position and each of nuisance parameters, the problem was formulated as a QCQP problem. Owing to the fact that our QCQP problem is nonconvex and NP-hard, we recursively approximated each quadratic constraint by linear constraint and then reformulated the problem as a LCQP problem, which has a closed-form solution in each iteration. Simulation results indicated that the proposed iterative CWLS method performs much better than the previous ones.

NOROOZI et al.: ITERATIVE TARGET LOCALIZATION IN DISTRIBUTED MIMO RADAR FROM BISTATIC RANGE MEASUREMENTS

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