Optimal Power Allocation in Distributed Multiple-Radar ... - IEEE Xplore
Optimal Power Allocation in Distributed Multiple-Radar Configurations Hana Godrich† , Athina Petropulu , and H. Vincent Poor† Rutgers University, Piscataway, NJ 08854 † Princeton University, Princeton, NJ 08544 Abstract—A performance driven power allocation scheme is proposed for target localization in widely distributed multiple-radar architectures. For a total transmitted power goal, power may be uniformly allocated among all transmit stations. This will result with a specific target location estimation mean-square error (MSE) that may be evaluated using the Cramer-Rao bound (CRB). However, in the case of target tracking, where previous knowledge of the system exists, uniform allocation might not be the most energy efficient method. In this paper, the following optimization problem is considered: find an optimal power allocation among the transmit radar stations, such that the total transmitted energy is minimized for a given performance objective. To address this problem, the Karush-Kuhn-Tucker (KKT) conditions for the resulting nonconvex optimization problem are formulated and a set of parallel, nonoverlapping, optimization problems is derived. This approach supports distributed processing of the optimal power allocation. Additionally, the analytical expressions provide imperative understanding of the relation between the system characteristics and the manner in which power is allocated. It is shown that uniform power allocation is not in general optimal and that considerable power savings may be attained through power adaptation. Index Terms—MIMO radar, Multistatic radar, CRB, nonconvex optimization, target localization.
I. I NTRODUCTION The notion of resource-aware design is of critical importance when it comes to radar applications that include mobile deployment of stations, such as surveillance radars. Furthermore, power management is an essential part of military operations in hostile environments, where low-probability-of-intercept (LPI) operation may be required [1]. Radar architectures employing multiple, widely distributed stations, such as multiple-input multiple-output (MIMO) radar systems with widely spread antennas [2] and multistatic/multisite radar systems [3], have been recently introduced. These systems have been shown to offer significant advantages in terms of enhanced target localization capabilities, by exploiting increased spatial spread [4]. For a given total transmitted power, uniform power allocation is commonly used at the transmitters [2]. Target localization estimation mean-square error (MSE) attainable through the use of this allocation defines a performance threshold. Hence, when prior estimation of the target radar cross section (RCS) is available, which is typical in target tracking or over several estimation cycles, is uniform allocation the best strategy? I. e., one use less energy and still provide the same estimation MSE? In this paper we capitalize this question and propose an appropriate power allocation scheme that minimizes the overall energy needs. The resulting optimization problem is a nonconvex, nonlinear, optimization problem. One way to deal with this type of problem is to choose an appropriate convex relaxation of the original problem and use the optimal solution to find a local minimum [5]. Another is through the use of Lagrange multipliers and the Karush-Kuhn-Tucker (KKT) conditions and decomposition techniques [6]- [7]. In our previous 0 The research was supported by the Office of Naval Research under Grant N00014-09-1-0342.
978-1-4577-0539-7/11/$26.00 ©2011 IEEE
study [8], relaxation methods have been used, demonstrating that uniform power allocation is not necessarily optimal. The proposed solution was reliant on central processing and standard optimization tools, such as CVX [9]. In this paper, we capitalizing on the intuition that optimal solutions that minimize power utilization should be found on the boundaries, to generate an analytical solution to the optimization problem. Using domain decomposition methods, the closed-form expressions are obtained and formulated as a set of parallel subproblems. The derived expressions facilitate distributed processing at the multiple transmit and receive stations. The analytical solution supports evaluation of the relation between the optimal power allocation and the system parameters. As the target location estimation MSE performance is tightly lower bounded by the Cramer-Rao bound (CRB) at high signal-to-noise ratio (SNR) [10], the latter may be used to evaluate the MSE for a given power allocation. The paper is organized as follows: The system model and the CRB are introduced in Section II. The total transmitted energy for a given MSE threshold is minimized in Section III using the KKT conditions and domain decomposition techniques. Numerical analysis of the proposed power allocation scheme is provided in Section IV. Finally, Section V concludes the paper. II. S YSTEM M ODEL AND P RELIMINARIES We consider a distributed multiple radar system with P transmit and Q receive radars, forming an P × Q distributed multiple radar system. An extended target, with a center of mass located at position ({> |), is assumed. The variation in the location of the targets’ center of mass, as viewed by the set of radars, is assumed to be small with respect to the system resolution capabilities. The system is tracking the target’s location and has available estimates for unknown parameters, such as the target RCS, from previous cycles. The search cell is confined to ({f ± nf@> |f ± nf@). The transmit and receive radars are located in a two dimensional plane. The P transmit radars are arbitrarily located at coordinates ({pW { > |pW { ), p = 1> = = = > P , and the Q receiver radars are arbitrarily located at coordinates ({qU{ > |qU{ ), q = 1> = = = > Q. A set of orthogonal waveforms is transmitted, each U with a lowpass equivalent vp (w), where Tp |vp (w)|2 gw = 1, and Tp is the duration of the p-th transmitted signal. The waveform effective bandwidth is denoted by p and is defined in [1]. The waveforms’ transmitted powers spw{ are constrained by maximal values pw{max = [s1w{ max > s2w{ max > ===> sPw{ max ]W . Let p>q ({> |) denote the propagation time of a signal transmitted by radar p, reflected by the target, and received by radar q: UpW { + UqU{ > (1) f where UpW { is the range from transmitter p to the target, the range from the target to receiver q is denoted by UqU{ , and f is the speed of light. The baseband representation for the signal transmitted from
2492
p>q ({> |) =
ICASSP 2011
radar p received at radar q is I up>q (w) = p>q spw{ kp>q vp (w 3 p>q ) +zp>q (w)=
(2)
The term p>q x U2 1U2 represents the variation in the signal pW { qU{ strength due to path loss effects. The target RCS kp>q is modeled as being deterministic and complex, and is assumed to be unknown. The term zp>q (w) represents circularly symmetric, zero-mean, complex Gaussian noise, spatially and temporally white with autocorrelation function 2z ( ). W We define a vector of unknown parameters u = {> |> hW , where h = [k1>1 > k1>2 > ===> kP>Q ]W . The following vector notation is defined for later use: pw{ = [s1w{ > s2w{ > ===> sPw{ ]W , and = [ 1>1 > 1>2 > ===> P>Q ]W . The CRB matrix, C{>| , on the target localization MSE has been derived in [8], resulting in + P ,31 [ jdp jfp C{>| (u> pw{ ) = spw{ > (3) jfp jep p=1
where the elements jdp , jep , and jfp , are defined as jdp jep
2 {pW { 3 { {qU{ 3 { + > (4) UpW { UqU{ q=1 2 Q [ |pW { 3 | |qU{ 3 | p p>q |kp>q |2 + > (5) UpW { UqU{ q=1
=
p
=
and jfp
=
Q [
p
p>q |kp>q |2
Q [
p>q |kp>q |2
q=1
×
{pW { 3 { {qU{ 3 { + UpW { UqU{ 3| >
(6)
|pW { 3 | |qU{ + UpW { UqU{
8 2 2
where p = 2 f2p . The trace of the matrix C{>| represents a lower z bound on the sum of the MSEs for the target location estimation, i.e., tr (C{>| ) $ 2{ + 2| , where 2{ and 2| are the target’s { and | location estimation MSEs, respectively. Following some additional matrix manipulations, the trace of the CRB matrix C{>| can be expressed as 2{>| (pw{ ) = tr (C{>| ) =
bW pw{ > pWw{ Apw{
(7)
where b = (gd + ge ), A = gd geW 3gf gfW , where gd , ge , and gf are defined by the elements in (4), (5), and (6). For the case of uniform power allocation, pw{u = su [1> 1> ===> 1]WP ×1 , the trace of the CRB matrix is W1 2{>| (pw{u ) = s1u 1bW A1 . As the CRB is known to be asymptotically tight to the maximum likelihood estimator (MLE) MSE at high SNR [10], it is used to represent the localization MSE as a function of the power allocation MSE in the power allocation schemes provided next. III. P OWER A LLOCATION S TRATEGY For a given total transmitted power, denoted by suW rwdo = su W P, the acheivable MSE may be calculated using 2{>| (pw{u ) = P bW 1 . This value is set as the target localization MSE sxW rwdo 1W A1 threshold, denoted by max . Total radiated energy may be minimized through appropriate allocation of power among the transmit radars,
such that the MSE performance goal, max , is achieved with minimal power requirements. This can be formulated as the following optimization problem: minimize
1W pw{ >
v=w=
pw{ 3 pw{max $ 0> pw{min 3 pw{ $ 0> max pWw{ Ah pw{ 3 bWh pw{ = 0>
pw{
(8)
where the vector bh = b (˜ u) and u) are lW Ah = A (˜ k the matrix W ˜ calculated based on the vector u ˜= { ˜> |˜> h of prior estimates of the target RCS, obtained in previous cycles, and the search cell center coordinates, ({f > |f ), are used for the estimated target location (˜ {> |˜). The optimization problem in (8) is nonconvex due to the third equality constraint [5]. A method based on the KKT conditions and domain decomposition is derived next, offering a parallel non-overlapping set of optimization problems, generating feasible solutions for the master problem defined in (8). For the power optimization problem given in (8) the Lagrangian function is of the form L(pw{ > > > ) = 1W pw{ + max pWw{ Ah pw{ 3 bWh pw{ (9) +W (pw{ 3 pw{max ) + W (pw{min 3 pw{ (10) )>
and the appropriate KKT conditions [6] are Qpw{ L =1 + W max Ah + AWh pWw{ 3 bh W W + 3 = 0> W W W max pWW w{ Ah pw{ 3 bh p w{ = 0> Wp sWpw{ 3spw{ max = 0> Wp (spw{ min 3 sWpw{ ) = 0> Wp D 0, Wp D 0=
(O1) (O2) (O3) (O4) (O5)
(11)
Linear Independence Constraint Qualifications (LICQs) [11] hold for this problem as it can be shown that Qpw{ (pWw{ 3 pw{max )W u = 1W u =0 and Qpw{ (pw{max 3 pWw{ )W u = 31W u =0 have only the trivial solution of u = 0 and the LICQ on the equality constraint may be rewritten using the first KKT condition in (11), resulting in 1 W ( W 3 W 3 1) u = 0> (12) W which has only the trivial solution of u = 0, as the Lagrange multiplier W is assumed to satisfy W 6= 0. The LICQ holds even for cases where W = W = 0. A solution to (8) may be obtained by choosing the first and second inequality constraints in (8) to be inactive and therefore, Wp = Wp = 0, p = 1> ==> P , reducing the set of KKT conditions in (11) to (13) 1 + W max AhW h pWw{ 3 bh = 0> and
W W W max pWW w{ AhW h pw{ 3 bh pw{ = 0>
(14)
AWh .
From the KKT condition in (13), the where AhW h = Ah + optimal power allocation vector, pWw{ , may be calculated as B 1 W pw{ = (15) bh 3 W 1 > max
and from the KKT condition in (14), the optimal value of the Lagrange multiplier, W , may be derived, resulting in v 1W BAh B1 W 1>2 = ± > (16) W bh (BAh B 3 B) bh where B = A31 hW h . The solution in (15) reveals a Lagrangian leveling mechanism in the form of 1 1, for the total power minimization
2493
scheme. The leveling Lagrange factors, 1 , defines an adaptation of vector bh values, shifting the origin of its elements such that transmitters with better channel conditions, corresponding to higher weights in the relevant row in the matrix B and vector bh , are assigned higher transmit powers compared with transmitters with weaker propagation paths. The use of the term channel in this context refers to the overall attenuation over a given propagation path between transmit radar p, the target, and receive radar q, defined as channel (p> q). The channel conditions incorporate the effects of propagation path fading, target reflectivity, and geometric position of the transmitreceive pair with respect to the target. The solution in (15) is obtained while ignoring the upper and lower bounds on the values of pw{ . Thus the resulting solution pWw{ may be outside these boundary constraints. As the objective function is to minimize the total transmitted power, a reasonable place to look for feasible solutions is on the boundaries. In the following, domain decomposition methods are used to generate a set of P + 1 optimization problems. For the decomposition, the transmit powers are regarded as P vertices, divided into two groups: one includes interior points and the other includes points located on the boundaries. A global set of vertices is defined as P := {s1w{ > s2w{ > = = = > sPw{ }. We define a set BN as one that includes all possible vectors pw{2N M RN{1 , 0$N$ P, generated by choosing N different elements out of P, and the set IN as a set of complementary vectors pw{1N M R(P3N)×1 , such that IN := {pw{1N : pw{1N = pw{ \pw{2N }. Vectors pw{1N in the set IN include interior vertices and vectors pw{2N in the set BN include vertices that lay on the boundaries. We W W regroup the vector pw{ as pw{N = pWw{1N > p , defining a w{2N lW k set UN := pw{N : (pw{ \pw{2N )W > pWw{2N . Rearranging the matrix Ah and vector bh to conform with the partitioning of the vector pw{N results in A11(PN)×(PN) A12(PN)×N AhN = > (17) A21N×(PN) A22N×N
and bhN = 0
b1(PN)×1 = b2N×1
(18)
0
Define vector pNErxqg M YN , as the possible boundary values assigned to pw{2N . The set of possible q 0 r values of pNErxqg is YN := 0 pNErxqg : 1W2 pNErxqg $ suW rwdo . The latter condition on the set YN reduces the search to sets that minimize the total transmit power when compared with the simple solution of uniform power allocation. The master optimization problem in (8) is decomposed to P + 1 optimization problems of the form minimize
1W pw{N >
v=w=
max pWw{N AhN pw{ 3 bWhN pw{N = 0>
pw{N MUN
and the appropriate KKT conditions for this subproblem are k lW W 3 bhN = 0> 1 + W max AhW hN pWW w{1N > pNErxqg
(19)
(20)
and
0
=
k l k lW W WW W max pWW (21) w{1N > pNErxqg AhN pw{1N > pNErxqg k lW W WW 3bhN pw{1N > pNErxqg = (22)
Each problem performs domain boundary search by enforcing pNErxqg on the power allocation vector pw{N . The resulting optimal
2494
power allocation for the internal point vector, pWw{1N , is of the form 1 1 (B11 b1 + B12 b2 ) 3 W (B11 11 + B12 12 ) > pWw{1N = max (23) and the optimal Lagrange multiplier value, W , is obtained by solving the quadratic equation 2
d (W ) + e W + f = 0>
(24)
where the expressions for the coefficients d , e , and f are provided in Appendix A. Evaluating all possible solutions pWw{1N and W in (23) and (24) for pNErxqg M YN , for all pw{2N M BN , results in a set of feasible solutions pWw{N . The ones that follow the master problem constraints spw{ min $ sWpw{ $ spw{ max > p = 1> = = = > P , are added to a local 0 feasible set FN . Finally an optimal set is found for subproblem N through the evaluation of , + W W W FN := pw{N : arg min 0 1 pw{N = (25) p w{
N
MFN
The search for potential feasible solutions is enhanced by introducing a modification to the assumed boundary conditions pw{max , in which a modified upper bound pw{mod (N) A pw{max is set and used in the evaluation of pWw{1N and W in (23) and (24). The initial value for 1W pw{mod (N) is set to half of 1W pWw{N obtained in the first search cycle. At each iteration, pw{mod (N) is updated to follow the direction that minimizes the objective function 1W pWw{N . The resulting solutions that follow the constraints spw{ min $ sWpw{ $ spw{ max > p = 1> = = = > P are included in the local feasible set F. A universal set of optimal points is created as FDoo = F1 F2 === FP +1 . The set of optimal values for the master problem is then obtained by pWw{ = arg min ir = 1W pw{ = (26) pw{ MF Doo
The following process supports parallel processing where each subproblems may be calculated at a different receiver or centrally, using parallel computing. A local processing center for the N-th subproblem needs to get the values of matrix Ah and vectors , h, and bh . Information of the local subset of optimal points FN is transferred to the central processor. The universal set FDoo is used in the solution of the master problem, as expressed by (26). IV. N UMERICAL A NALYSIS
The proposed power allocation method dilutes the error variation through adequate distribution of the transmit power. To evaluate the effect of the radars’ spread, four different angular spreads of radars with respect to the target are chosen, each with 5 transmit radars and 7 receive radars, as illustrated in Figure 1 in Case 1 through Case 4. The ranges from the transmit and receive radars to the target are set to be all equal. Two target RCS models are used: the first mimics a scenario of two transmitters with high reflectivity conditions (transmitters 1 and 5), while one exhibits significant loss (transmitter 2). The second demonstrates a scenario of three transmitters with high reflectivity conditions (transmitters 2, 3, and 4), while one exhibits significant loss (transmitter 1). The given MSE threshold is set to max = 10m2 . In Table 1, numerical analysis is presented for radar spreads given in Case 1 through Case 4 with RCS model 1. For each scenario, the value of the total transmit power that would have been allocated given a choice of uniform power allocation is calculated, denoted by sxW rwdo . An optimal power allocation vector, pWw{min , is obtained through the use of the proposed domain decomposition algorithm.
Case 1
Case 2
4000
T5
T2
R2
2000 1000 0 −1000
3000
T3 T1 R1
R4
R7
R6
−3000
V. C ONCLUSIONS
2000
4000
−1000 −1000
0
1000
TX radar RX radar
2000
3000
4000
Case 4
4000
T5
T4
3000
T3
1000
T1 0 −2000 R7 1000
R6
R5
2000
T4
2000
T2
0
T3 R4 R3 T2 R2 T1 R1
Target
0
2000
−4000 −1000
R5
0
Case 3 4000
T4
2000
T4
−2000
R7R6
1000
T5
R5
−2000
−4000 −4000
obtained by using the proposed power allocation algorithm, compared with uniform power allocation. The level of savings is reliant on the geometric spread and the RCS values.
4000
R3
3000
0
R1 R2 R3 R4
−1000 −2000
4000
T2 T1 R1
R7
−3000
3000
T3
T5
−4000 −4000
R6
R2 R5 −2000
R4 0
R3 2000
4000
In this paper, a power allocation scheme that minimizes the total radiating power required to accomplish a predetermined localization MSE threshold has been derived. The resulting power allocation nonconvex optimization problem has been solved through domain decomposition methods. The resulting parallel set of optimization problems supports distributed processing. The power allocation expressions reveal a Lagrangian leveling mechanism that weights the overall system conditions for a given transmitter and allocates power appropriately. It has been shown that uniform power allocation is not necessarily the best choice, and significant power savings can be obtained through the proposed strategy. Optimal distribution of power is shown to be reliant on the radars’ angular spread with respect to the target location and the overall channels conditions.
Fig. 1. Multiple radar layouts with equal ranges between transmit and receive radars and the target; Cases 1, 2, 3, and 4.
The resulting minimal total transmitted power is denoted by sWw{W rwdo . The resulting transmit power vector, pWw{min , allocates most of the energy to the two transmitters with no propagation loss (transmitters 1 and 5). Table 1: Minimize power with RCS 1 and max = 10m2 . case 1 case 2 case 3 case 4 sxW rwdo 162 189 332 312 sWw{W rwdo 90 89 151 145 sWw{W rwdo @sxW rwdo 0=56 5 6 5 6 0=47 5 6 0=46 5 6 0=47 80 68 36 37 91: 91: 91: 91: : : 9 : 9 : 9 9 91: 91: 91: 91: pWw{min 9 : 9 : 9 : 9 : 718 718 718 718 62 80 50 50
The analysis for the second RCS is presented in Table 2. In the resulting optimal power allocation vector, pWw{min , it is observed that most of the energy is split between only two radars out of the three that experience the best channels. Table 2: Minimize power with RCS 2 and max case 1 case 2 case 3 sxW rwdo 116 281 442 sWw{W rwdo 77 161 279 sWw{W rwdo @sxW rwdo 0=66 6 5 5 6 0=63 5 6 0=75 1 1 1 9150: 980: 934: : 9 9 : 9 : 9 1 : 91: 940: pWw{min : 9 9 : 9 : 71268 7788 718 1 1 1
= 10m2 . case 4 369 208 0=56 6 5 1 9100: : 9 9 99 : : 9 7 7 8 1
For Case 1 and Case 4 in Table 2, most of the power is allocated to transmitters 2 and 3 while for Case 2 and Case 3 it is split between transmitters 2 and 4. The difference between the two is related to the position of the three transmitters in each of these cases, as can be seen from Figure 1. In Case 2 and Case 3, transmitters 2 and 4 have a wider angular spread compared with transmitters 2 and 3, while in Case 1 and Case 4, transmitters 2 and 3 have a wider angular spread. Over all, savings of up to 46% in the required energy are
A PPENDIX A Q UADRATIC E QUATION (24).
COEFFICIENTS FOR THE
The values of the coefficients for the quadratic equation given in (24) are as follows: coefficient d is k d = (B11 b1 + B12 b2 )W AW11 3 bW1 l + max pWNErxqg A12 + AW21
× (B11 b1 + B12 b2 ) 3 max bW2 pNErxqg + 2max pWNErxqg A22 pNErxqg > (27) coefficient e is k (28) e = bW1 3 (B11 b1 + B12 b2 )W A11 + AW11 l 3 max pWNErxqg A12 + AW21 × (B11 11 + B11 12 ) >
and coefficient f is
f = 3 (B11 11 + B11 12 )W A11 (B11 11 + B11 12 ) =
(29)
R EFERENCES
[1] M. Skolnik, Introduction to Radar Systems, New York: McGraw-Hill, 3rd, 2002. [2] H. Godrich, A. M. Haimovich and R. S. Blum, "Concepts and applications of a MIMO radar system with widely separated antennas," in MIMO Radar Signal Processing, edited by J. Li and P. Stoica, New York: John Wiley, 2008. [3] V. S. Chernyak, Fundamental of Multisite Radar Systems: Multistatic Radars and Multi Radar Systems, London: CRC Press, 1998. [4] H. Godrich, A. M. Haimovich, and R. S. Blum, "Target localization accuracy gain in MIMO radar based system,” IEEE Trans. Inf. Theory, vol. 56, no. 6, pp. 2783 - 2803, June, 2010. [5] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge, UK: Cambridge University Press, 2004. [6] M. S. Bazaraa and C. M. Shetty, Nonlinear Programming, New York: John Wiley, 1979. [7] M. S. Ingber, J. A. Tanski, and P. Alsing, "A domain decomposition tool for boundary element methods," Engineering Analysis with Boundary Elements, vol. 31, no. 11, pp. 890-896, November 2007. [8] H. Godrich, A. Petropulu, and H. V. Poor, “Resource allocation schemes for target localization in distributed multiple radar architectures,” in Proc. Euro. Signal Procss. Conf., Aalborg, Denmark, 2010. [9] M. Grant and S. Boyd, "CVX: Matlab software for disciplined convex programming," October 2008. [Online]. Available: http://stanford.edu/~boyd/cvx. [10] H. V. Poor, An Introduction to Signal Detection and Estimation, New York; Springer, 2nd ed, 1994 [11] J. Gauvin, "A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming," Mathematical Programming, Vol. 12, pp. 136–138, 1977.
2495
I. I NTRODUCTION The notion of resource-aware design is of critical importance when it comes to radar applications that include mobile deployment of stations, such as surveillance radars. Furthermore, power management is an essential part of military operations in hostile environments, where low-probability-of-intercept (LPI) operation may be required [1]. Radar architectures employing multiple, widely distributed stations, such as multiple-input multiple-output (MIMO) radar systems with widely spread antennas [2] and multistatic/multisite radar systems [3], have been recently introduced. These systems have been shown to offer significant advantages in terms of enhanced target localization capabilities, by exploiting increased spatial spread [4]. For a given total transmitted power, uniform power allocation is commonly used at the transmitters [2]. Target localization estimation mean-square error (MSE) attainable through the use of this allocation defines a performance threshold. Hence, when prior estimation of the target radar cross section (RCS) is available, which is typical in target tracking or over several estimation cycles, is uniform allocation the best strategy? I. e., one use less energy and still provide the same estimation MSE? In this paper we capitalize this question and propose an appropriate power allocation scheme that minimizes the overall energy needs. The resulting optimization problem is a nonconvex, nonlinear, optimization problem. One way to deal with this type of problem is to choose an appropriate convex relaxation of the original problem and use the optimal solution to find a local minimum [5]. Another is through the use of Lagrange multipliers and the Karush-Kuhn-Tucker (KKT) conditions and decomposition techniques [6]- [7]. In our previous 0 The research was supported by the Office of Naval Research under Grant N00014-09-1-0342.
978-1-4577-0539-7/11/$26.00 ©2011 IEEE
study [8], relaxation methods have been used, demonstrating that uniform power allocation is not necessarily optimal. The proposed solution was reliant on central processing and standard optimization tools, such as CVX [9]. In this paper, we capitalizing on the intuition that optimal solutions that minimize power utilization should be found on the boundaries, to generate an analytical solution to the optimization problem. Using domain decomposition methods, the closed-form expressions are obtained and formulated as a set of parallel subproblems. The derived expressions facilitate distributed processing at the multiple transmit and receive stations. The analytical solution supports evaluation of the relation between the optimal power allocation and the system parameters. As the target location estimation MSE performance is tightly lower bounded by the Cramer-Rao bound (CRB) at high signal-to-noise ratio (SNR) [10], the latter may be used to evaluate the MSE for a given power allocation. The paper is organized as follows: The system model and the CRB are introduced in Section II. The total transmitted energy for a given MSE threshold is minimized in Section III using the KKT conditions and domain decomposition techniques. Numerical analysis of the proposed power allocation scheme is provided in Section IV. Finally, Section V concludes the paper. II. S YSTEM M ODEL AND P RELIMINARIES We consider a distributed multiple radar system with P transmit and Q receive radars, forming an P × Q distributed multiple radar system. An extended target, with a center of mass located at position ({> |), is assumed. The variation in the location of the targets’ center of mass, as viewed by the set of radars, is assumed to be small with respect to the system resolution capabilities. The system is tracking the target’s location and has available estimates for unknown parameters, such as the target RCS, from previous cycles. The search cell is confined to ({f ± nf@> |f ± nf@). The transmit and receive radars are located in a two dimensional plane. The P transmit radars are arbitrarily located at coordinates ({pW { > |pW { ), p = 1> = = = > P , and the Q receiver radars are arbitrarily located at coordinates ({qU{ > |qU{ ), q = 1> = = = > Q. A set of orthogonal waveforms is transmitted, each U with a lowpass equivalent vp (w), where Tp |vp (w)|2 gw = 1, and Tp is the duration of the p-th transmitted signal. The waveform effective bandwidth is denoted by p and is defined in [1]. The waveforms’ transmitted powers spw{ are constrained by maximal values pw{max = [s1w{ max > s2w{ max > ===> sPw{ max ]W . Let p>q ({> |) denote the propagation time of a signal transmitted by radar p, reflected by the target, and received by radar q: UpW { + UqU{ > (1) f where UpW { is the range from transmitter p to the target, the range from the target to receiver q is denoted by UqU{ , and f is the speed of light. The baseband representation for the signal transmitted from
2492
p>q ({> |) =
ICASSP 2011
radar p received at radar q is I up>q (w) = p>q spw{ kp>q vp (w 3 p>q ) +zp>q (w)=
(2)
The term p>q x U2 1U2 represents the variation in the signal pW { qU{ strength due to path loss effects. The target RCS kp>q is modeled as being deterministic and complex, and is assumed to be unknown. The term zp>q (w) represents circularly symmetric, zero-mean, complex Gaussian noise, spatially and temporally white with autocorrelation function 2z ( ). W We define a vector of unknown parameters u = {> |> hW , where h = [k1>1 > k1>2 > ===> kP>Q ]W . The following vector notation is defined for later use: pw{ = [s1w{ > s2w{ > ===> sPw{ ]W , and = [ 1>1 > 1>2 > ===> P>Q ]W . The CRB matrix, C{>| , on the target localization MSE has been derived in [8], resulting in + P ,31 [ jdp jfp C{>| (u> pw{ ) = spw{ > (3) jfp jep p=1
where the elements jdp , jep , and jfp , are defined as jdp jep
2 {pW { 3 { {qU{ 3 { + > (4) UpW { UqU{ q=1 2 Q [ |pW { 3 | |qU{ 3 | p p>q |kp>q |2 + > (5) UpW { UqU{ q=1
=
p
=
and jfp
=
Q [
p
p>q |kp>q |2
Q [
p>q |kp>q |2
q=1
×
{pW { 3 { {qU{ 3 { + UpW { UqU{ 3| >
(6)
|pW { 3 | |qU{ + UpW { UqU{
8 2 2
where p = 2 f2p . The trace of the matrix C{>| represents a lower z bound on the sum of the MSEs for the target location estimation, i.e., tr (C{>| ) $ 2{ + 2| , where 2{ and 2| are the target’s { and | location estimation MSEs, respectively. Following some additional matrix manipulations, the trace of the CRB matrix C{>| can be expressed as 2{>| (pw{ ) = tr (C{>| ) =
bW pw{ > pWw{ Apw{
(7)
where b = (gd + ge ), A = gd geW 3gf gfW , where gd , ge , and gf are defined by the elements in (4), (5), and (6). For the case of uniform power allocation, pw{u = su [1> 1> ===> 1]WP ×1 , the trace of the CRB matrix is W1 2{>| (pw{u ) = s1u 1bW A1 . As the CRB is known to be asymptotically tight to the maximum likelihood estimator (MLE) MSE at high SNR [10], it is used to represent the localization MSE as a function of the power allocation MSE in the power allocation schemes provided next. III. P OWER A LLOCATION S TRATEGY For a given total transmitted power, denoted by suW rwdo = su W P, the acheivable MSE may be calculated using 2{>| (pw{u ) = P bW 1 . This value is set as the target localization MSE sxW rwdo 1W A1 threshold, denoted by max . Total radiated energy may be minimized through appropriate allocation of power among the transmit radars,
such that the MSE performance goal, max , is achieved with minimal power requirements. This can be formulated as the following optimization problem: minimize
1W pw{ >
v=w=
pw{ 3 pw{max $ 0> pw{min 3 pw{ $ 0> max pWw{ Ah pw{ 3 bWh pw{ = 0>
pw{
(8)
where the vector bh = b (˜ u) and u) are lW Ah = A (˜ k the matrix W ˜ calculated based on the vector u ˜= { ˜> |˜> h of prior estimates of the target RCS, obtained in previous cycles, and the search cell center coordinates, ({f > |f ), are used for the estimated target location (˜ {> |˜). The optimization problem in (8) is nonconvex due to the third equality constraint [5]. A method based on the KKT conditions and domain decomposition is derived next, offering a parallel non-overlapping set of optimization problems, generating feasible solutions for the master problem defined in (8). For the power optimization problem given in (8) the Lagrangian function is of the form L(pw{ > > > ) = 1W pw{ + max pWw{ Ah pw{ 3 bWh pw{ (9) +W (pw{ 3 pw{max ) + W (pw{min 3 pw{ (10) )>
and the appropriate KKT conditions [6] are Qpw{ L =1 + W max Ah + AWh pWw{ 3 bh W W + 3 = 0> W W W max pWW w{ Ah pw{ 3 bh p w{ = 0> Wp sWpw{ 3spw{ max = 0> Wp (spw{ min 3 sWpw{ ) = 0> Wp D 0, Wp D 0=
(O1) (O2) (O3) (O4) (O5)
(11)
Linear Independence Constraint Qualifications (LICQs) [11] hold for this problem as it can be shown that Qpw{ (pWw{ 3 pw{max )W u = 1W u =0 and Qpw{ (pw{max 3 pWw{ )W u = 31W u =0 have only the trivial solution of u = 0 and the LICQ on the equality constraint may be rewritten using the first KKT condition in (11), resulting in 1 W ( W 3 W 3 1) u = 0> (12) W which has only the trivial solution of u = 0, as the Lagrange multiplier W is assumed to satisfy W 6= 0. The LICQ holds even for cases where W = W = 0. A solution to (8) may be obtained by choosing the first and second inequality constraints in (8) to be inactive and therefore, Wp = Wp = 0, p = 1> ==> P , reducing the set of KKT conditions in (11) to (13) 1 + W max AhW h pWw{ 3 bh = 0> and
W W W max pWW w{ AhW h pw{ 3 bh pw{ = 0>
(14)
AWh .
From the KKT condition in (13), the where AhW h = Ah + optimal power allocation vector, pWw{ , may be calculated as B 1 W pw{ = (15) bh 3 W 1 > max
and from the KKT condition in (14), the optimal value of the Lagrange multiplier, W , may be derived, resulting in v 1W BAh B1 W 1>2 = ± > (16) W bh (BAh B 3 B) bh where B = A31 hW h . The solution in (15) reveals a Lagrangian leveling mechanism in the form of 1 1, for the total power minimization
2493
scheme. The leveling Lagrange factors, 1 , defines an adaptation of vector bh values, shifting the origin of its elements such that transmitters with better channel conditions, corresponding to higher weights in the relevant row in the matrix B and vector bh , are assigned higher transmit powers compared with transmitters with weaker propagation paths. The use of the term channel in this context refers to the overall attenuation over a given propagation path between transmit radar p, the target, and receive radar q, defined as channel (p> q). The channel conditions incorporate the effects of propagation path fading, target reflectivity, and geometric position of the transmitreceive pair with respect to the target. The solution in (15) is obtained while ignoring the upper and lower bounds on the values of pw{ . Thus the resulting solution pWw{ may be outside these boundary constraints. As the objective function is to minimize the total transmitted power, a reasonable place to look for feasible solutions is on the boundaries. In the following, domain decomposition methods are used to generate a set of P + 1 optimization problems. For the decomposition, the transmit powers are regarded as P vertices, divided into two groups: one includes interior points and the other includes points located on the boundaries. A global set of vertices is defined as P := {s1w{ > s2w{ > = = = > sPw{ }. We define a set BN as one that includes all possible vectors pw{2N M RN{1 , 0$N$ P, generated by choosing N different elements out of P, and the set IN as a set of complementary vectors pw{1N M R(P3N)×1 , such that IN := {pw{1N : pw{1N = pw{ \pw{2N }. Vectors pw{1N in the set IN include interior vertices and vectors pw{2N in the set BN include vertices that lay on the boundaries. We W W regroup the vector pw{ as pw{N = pWw{1N > p , defining a w{2N lW k set UN := pw{N : (pw{ \pw{2N )W > pWw{2N . Rearranging the matrix Ah and vector bh to conform with the partitioning of the vector pw{N results in A11(PN)×(PN) A12(PN)×N AhN = > (17) A21N×(PN) A22N×N
and bhN = 0
b1(PN)×1 = b2N×1
(18)
0
Define vector pNErxqg M YN , as the possible boundary values assigned to pw{2N . The set of possible q 0 r values of pNErxqg is YN := 0 pNErxqg : 1W2 pNErxqg $ suW rwdo . The latter condition on the set YN reduces the search to sets that minimize the total transmit power when compared with the simple solution of uniform power allocation. The master optimization problem in (8) is decomposed to P + 1 optimization problems of the form minimize
1W pw{N >
v=w=
max pWw{N AhN pw{ 3 bWhN pw{N = 0>
pw{N MUN
and the appropriate KKT conditions for this subproblem are k lW W 3 bhN = 0> 1 + W max AhW hN pWW w{1N > pNErxqg
(19)
(20)
and
0
=
k l k lW W WW W max pWW (21) w{1N > pNErxqg AhN pw{1N > pNErxqg k lW W WW 3bhN pw{1N > pNErxqg = (22)
Each problem performs domain boundary search by enforcing pNErxqg on the power allocation vector pw{N . The resulting optimal
2494
power allocation for the internal point vector, pWw{1N , is of the form 1 1 (B11 b1 + B12 b2 ) 3 W (B11 11 + B12 12 ) > pWw{1N = max (23) and the optimal Lagrange multiplier value, W , is obtained by solving the quadratic equation 2
d (W ) + e W + f = 0>
(24)
where the expressions for the coefficients d , e , and f are provided in Appendix A. Evaluating all possible solutions pWw{1N and W in (23) and (24) for pNErxqg M YN , for all pw{2N M BN , results in a set of feasible solutions pWw{N . The ones that follow the master problem constraints spw{ min $ sWpw{ $ spw{ max > p = 1> = = = > P , are added to a local 0 feasible set FN . Finally an optimal set is found for subproblem N through the evaluation of , + W W W FN := pw{N : arg min 0 1 pw{N = (25) p w{
N
MFN
The search for potential feasible solutions is enhanced by introducing a modification to the assumed boundary conditions pw{max , in which a modified upper bound pw{mod (N) A pw{max is set and used in the evaluation of pWw{1N and W in (23) and (24). The initial value for 1W pw{mod (N) is set to half of 1W pWw{N obtained in the first search cycle. At each iteration, pw{mod (N) is updated to follow the direction that minimizes the objective function 1W pWw{N . The resulting solutions that follow the constraints spw{ min $ sWpw{ $ spw{ max > p = 1> = = = > P are included in the local feasible set F. A universal set of optimal points is created as FDoo = F1 F2 === FP +1 . The set of optimal values for the master problem is then obtained by pWw{ = arg min ir = 1W pw{ = (26) pw{ MF Doo
The following process supports parallel processing where each subproblems may be calculated at a different receiver or centrally, using parallel computing. A local processing center for the N-th subproblem needs to get the values of matrix Ah and vectors , h, and bh . Information of the local subset of optimal points FN is transferred to the central processor. The universal set FDoo is used in the solution of the master problem, as expressed by (26). IV. N UMERICAL A NALYSIS
The proposed power allocation method dilutes the error variation through adequate distribution of the transmit power. To evaluate the effect of the radars’ spread, four different angular spreads of radars with respect to the target are chosen, each with 5 transmit radars and 7 receive radars, as illustrated in Figure 1 in Case 1 through Case 4. The ranges from the transmit and receive radars to the target are set to be all equal. Two target RCS models are used: the first mimics a scenario of two transmitters with high reflectivity conditions (transmitters 1 and 5), while one exhibits significant loss (transmitter 2). The second demonstrates a scenario of three transmitters with high reflectivity conditions (transmitters 2, 3, and 4), while one exhibits significant loss (transmitter 1). The given MSE threshold is set to max = 10m2 . In Table 1, numerical analysis is presented for radar spreads given in Case 1 through Case 4 with RCS model 1. For each scenario, the value of the total transmit power that would have been allocated given a choice of uniform power allocation is calculated, denoted by sxW rwdo . An optimal power allocation vector, pWw{min , is obtained through the use of the proposed domain decomposition algorithm.
Case 1
Case 2
4000
T5
T2
R2
2000 1000 0 −1000
3000
T3 T1 R1
R4
R7
R6
−3000
V. C ONCLUSIONS
2000
4000
−1000 −1000
0
1000
TX radar RX radar
2000
3000
4000
Case 4
4000
T5
T4
3000
T3
1000
T1 0 −2000 R7 1000
R6
R5
2000
T4
2000
T2
0
T3 R4 R3 T2 R2 T1 R1
Target
0
2000
−4000 −1000
R5
0
Case 3 4000
T4
2000
T4
−2000
R7R6
1000
T5
R5
−2000
−4000 −4000
obtained by using the proposed power allocation algorithm, compared with uniform power allocation. The level of savings is reliant on the geometric spread and the RCS values.
4000
R3
3000
0
R1 R2 R3 R4
−1000 −2000
4000
T2 T1 R1
R7
−3000
3000
T3
T5
−4000 −4000
R6
R2 R5 −2000
R4 0
R3 2000
4000
In this paper, a power allocation scheme that minimizes the total radiating power required to accomplish a predetermined localization MSE threshold has been derived. The resulting power allocation nonconvex optimization problem has been solved through domain decomposition methods. The resulting parallel set of optimization problems supports distributed processing. The power allocation expressions reveal a Lagrangian leveling mechanism that weights the overall system conditions for a given transmitter and allocates power appropriately. It has been shown that uniform power allocation is not necessarily the best choice, and significant power savings can be obtained through the proposed strategy. Optimal distribution of power is shown to be reliant on the radars’ angular spread with respect to the target location and the overall channels conditions.
Fig. 1. Multiple radar layouts with equal ranges between transmit and receive radars and the target; Cases 1, 2, 3, and 4.
The resulting minimal total transmitted power is denoted by sWw{W rwdo . The resulting transmit power vector, pWw{min , allocates most of the energy to the two transmitters with no propagation loss (transmitters 1 and 5). Table 1: Minimize power with RCS 1 and max = 10m2 . case 1 case 2 case 3 case 4 sxW rwdo 162 189 332 312 sWw{W rwdo 90 89 151 145 sWw{W rwdo @sxW rwdo 0=56 5 6 5 6 0=47 5 6 0=46 5 6 0=47 80 68 36 37 91: 91: 91: 91: : : 9 : 9 : 9 9 91: 91: 91: 91: pWw{min 9 : 9 : 9 : 9 : 718 718 718 718 62 80 50 50
The analysis for the second RCS is presented in Table 2. In the resulting optimal power allocation vector, pWw{min , it is observed that most of the energy is split between only two radars out of the three that experience the best channels. Table 2: Minimize power with RCS 2 and max case 1 case 2 case 3 sxW rwdo 116 281 442 sWw{W rwdo 77 161 279 sWw{W rwdo @sxW rwdo 0=66 6 5 5 6 0=63 5 6 0=75 1 1 1 9150: 980: 934: : 9 9 : 9 : 9 1 : 91: 940: pWw{min : 9 9 : 9 : 71268 7788 718 1 1 1
= 10m2 . case 4 369 208 0=56 6 5 1 9100: : 9 9 99 : : 9 7 7 8 1
For Case 1 and Case 4 in Table 2, most of the power is allocated to transmitters 2 and 3 while for Case 2 and Case 3 it is split between transmitters 2 and 4. The difference between the two is related to the position of the three transmitters in each of these cases, as can be seen from Figure 1. In Case 2 and Case 3, transmitters 2 and 4 have a wider angular spread compared with transmitters 2 and 3, while in Case 1 and Case 4, transmitters 2 and 3 have a wider angular spread. Over all, savings of up to 46% in the required energy are
A PPENDIX A Q UADRATIC E QUATION (24).
COEFFICIENTS FOR THE
The values of the coefficients for the quadratic equation given in (24) are as follows: coefficient d is k d = (B11 b1 + B12 b2 )W AW11 3 bW1 l + max pWNErxqg A12 + AW21
× (B11 b1 + B12 b2 ) 3 max bW2 pNErxqg + 2max pWNErxqg A22 pNErxqg > (27) coefficient e is k (28) e = bW1 3 (B11 b1 + B12 b2 )W A11 + AW11 l 3 max pWNErxqg A12 + AW21 × (B11 11 + B11 12 ) >
and coefficient f is
f = 3 (B11 11 + B11 12 )W A11 (B11 11 + B11 12 ) =
(29)
R EFERENCES
[1] M. Skolnik, Introduction to Radar Systems, New York: McGraw-Hill, 3rd, 2002. [2] H. Godrich, A. M. Haimovich and R. S. Blum, "Concepts and applications of a MIMO radar system with widely separated antennas," in MIMO Radar Signal Processing, edited by J. Li and P. Stoica, New York: John Wiley, 2008. [3] V. S. Chernyak, Fundamental of Multisite Radar Systems: Multistatic Radars and Multi Radar Systems, London: CRC Press, 1998. [4] H. Godrich, A. M. Haimovich, and R. S. Blum, "Target localization accuracy gain in MIMO radar based system,” IEEE Trans. Inf. Theory, vol. 56, no. 6, pp. 2783 - 2803, June, 2010. [5] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge, UK: Cambridge University Press, 2004. [6] M. S. Bazaraa and C. M. Shetty, Nonlinear Programming, New York: John Wiley, 1979. [7] M. S. Ingber, J. A. Tanski, and P. Alsing, "A domain decomposition tool for boundary element methods," Engineering Analysis with Boundary Elements, vol. 31, no. 11, pp. 890-896, November 2007. [8] H. Godrich, A. Petropulu, and H. V. Poor, “Resource allocation schemes for target localization in distributed multiple radar architectures,” in Proc. Euro. Signal Procss. Conf., Aalborg, Denmark, 2010. [9] M. Grant and S. Boyd, "CVX: Matlab software for disciplined convex programming," October 2008. [Online]. Available: http://stanford.edu/~boyd/cvx. [10] H. V. Poor, An Introduction to Signal Detection and Estimation, New York; Springer, 2nd ed, 1994 [11] J. Gauvin, "A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming," Mathematical Programming, Vol. 12, pp. 136–138, 1977.
2495
