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Optimizations of Multisite Radar System with MIMO Radars for Target Detection JIA XU, Member, IEEE Tsinghua University, Radar Academy of Airforce XI-ZENG DAI, Member, IEEE Tsinghua University XIANG-GEN XIA, Fellow, IEEE University of Delaware LI-BAO WANG National University of Defense Technology JI YU YING-NING PENG, Senior Member, IEEE Tsinghua University This paper proposes a novel multisite radar system (MSRS) with multiple-input and multiple-output (MIMO) radars, i.e., MIMO-MSRS system, to improve the detection performance of fluctuating targets. The proposed MIMO-MSRS system increases the local signal-to-noise ratio (SNR) by using digital beamforming (DBF) among all transmitting and receiving channels in a single site. Then it smoothes the target’s fluctuation via spatial diversity among the DBF outputs of different sites. For the MIMO-MSRS system, we derive the likelihood ratio test (LRT) detector at first based on the proposed signal model and spatial diversity conditions. Furthermore, with the derived statistics of the LRT detector in the fixed noise background, three optimization problems are discussed on the MIMO-MSRS system configurations, i.e., the numbers of sites and collocated channels in different sites. The first problem is to detect the lowest SNR target with a given probability of false alarm (PF ), probability of detection (PD ) and total system degrees of freedom (DOF). The second is to detect a target with the highest PD for a given PF , target SNR, and system DOF. The third is on the minimal system DOF to detect a target with a given PF , PD , and target SNR. For the uniform MIMO-MSRS system, both the standard optimal site number, i.e., the diversity DOF, and its closed-form approximation of the above three problems are obtained. Finally, some numerical results are also provided to demonstrate the effectiveness of the proposed MIMO-MSRS systems. Manuscript received June 3, 2008; revised November 3, 2009, and March 25, June 28, and September 7, 2010; released for publication November 7, 2010. IEEE Log No. T-AES/47/4/943056. Refereeing of this contribution was handled by Y. Abramovich. This research has been funded by China National Science Foundation under Grant 60971087 and 60902071, China Ministry Research Foundation under Grant 9140A07011810JW01 and 9140C130510DZ46, China Aerospace Innovation Fund under Grant CASC200904, and the Fund of Tsinghua National Laboratory for Information Science and Technology (TNList). X-G. Xia’s work was supported in part by a DEPSCoR Grant W911NF-07-1-0422. Authors’ addresses: J. Xu, X-Z. Dai, J. Yu, and Y-N. Peng, Dept. of Electronic Engineering, Tsinghua University, Beijing 100084, China, E-mail: ([email protected]); X-G. Xia, Dept. of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716; L-B. Wang, National University of Defense Technology, Changsha 410073, China. J. Xu is also with Radar Academy of Airforce, Wuhan 430010, China.

c 2011 IEEE 0018-9251/11/$26.00 °

I. INTRODUCTION It is known that the surroundings of a modern radar become more and more complicated [1—6]. Stealthy, far-range, high-altitude, and high-speed targets cause a serious challenge for effective radar detection, measuring, and tracking. For example, the radar cross sections (RCS) of a real target can fluctuate significantly even to 10 to 15 dB [1] with a small change of aspect angle. As a consequence, the target’s echoes of a single radar may possibly be generated from an aspect angle with an extremely small RCS and a target may easily be missed for detection. However, the multiple echoes of a multisite radar system (MSRS) [1, 6] may be assumed statistically independent as long as the spatial diversity condition is satisfied, and the fusion of multisite echoes may be helpful to detect the target by smoothing the RCS fluctuation. Therefore, MSRS systems have been widely discussed in radar fields to obtain the performance gain via spatial diversity. Besides, compared with the conventional monostatic radar, an MSRS system has superior abilities to cope with the threats of modern radar like strong jamming, low-altitude interception, antiradiation missiles, etc. Recently, multiple-input multiple-output (MIMO) radars have attracted increasing attention [1—24]. MIMO radars are characterized by using multiple antennas to simultaneously transmit orthogonal waveforms and by utilizing multiple antennas to receive the scattered echoes. Normally, the existing MIMO radars may be roughly divided into two classes [10]: one is with widely distributed transmitting and receiving (T/R) channels and the other is with closely collocated T/R channels. For the former, e.g., a statistical MIMO radar [8—10], the obvious radar aspect angle differences exist among separate T/R channels, which can improve fluctuating target detection via spatial diversity like the MSRS. For the latter, the signals among different T/R channels may be strongly correlated due to the closely collocated distribution. The latter may significantly improve radar performance [11—13] by capitalizing on the multiple waveforms with high coherency, e.g. the long dwell observation time [2—3], the improved parameter identifiability [10], the superior estimation performance [13—17] and the flexible digital beamforming (DBF) [14—15]. In fact, the MIMO radar with widely separated channels has been discussed in the MSRS with simultaneous multiple transmitters [6], while the MIMO radar with collated channels may add some different features for the design of the new radar system. In this paper we propose a novel MSRS system with MIMO radars, i.e., MIMO-MSRS system, to improve the detection performance of fluctuating targets. In each site of the MIMO-MSRS system, it coherently integrates all the T/R channels to generate

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the DBF output and increase the local signal-to-noise ratio (SNR). Notably, this spatial tightness among T/R channels in a specific site of the MIMO-MSRS system can be caused by the closely distributed transmitting elements, receiving elements, or both of them. Subsequently, the spatial diversity is applied to incoherently integrate DBF outputs of all sites and smooth the target RCS fluctuation. Furthermore, we call the MIMO-MSRS system with the uniform site sizes, i.e., the identical number of tightly T/R channels in different sites, as the uniform MIMO-MSRS system [23]. In this paper we derive the likelihood ratio test (LRT) detector for the MIMO-MSRS system at first based on the proposed signal model and spatial diversity conditions. Furthermore, the optimizations are discussed on the MIMO-MSRS system configurations to improve target detection performance. As for the specific system configurations, it is vital for the MIMO-MSRS system to determine the numbers of sites, i.e., the diversity degree of freedom (DOF), and the channels in each site. Therefore, in this paper three optimization problems of system configurations are investigated on the system configurations, but not on the physical multisite spatial optimization as conventional MSRS [6, 19]. Based on the derived LRT detector with constant probability of false alarm in the fixed noise background, three kinds of optimal configurations are discussed for the MIMO-MSRS system in this paper. The first is to detect the lowest SNR target with a given probability of false alarm (PF ), probability of detection (PD ) and system DOF. The second is to detect a target with the highest PD for a given PF , target SNR, and system DOF. Also, the third is on given PF , PD , and target SNR to detect a target with the minimal system DOF. Obviously, the first kind of optimization is normally needed for searching radar to detect targets as early as possible, or to enlarge the coverage diagram as large as possible. The second is normally needed for tracking radar to sustain the best detection performance for a specific target. The third problem is also important in real applications to reduce the system hardware cost. Furthermore, both the standard optimal DOF and its closed-form approximations of the above three problems are obtained for the uniform MIMO-MSRS systems in this paper. It is shown that the optimal spatial diversity DOF should be limited for the first and the third optimization problems, while it should be adjusted according to the target SNR and system DOF for the second one. Finally, numerical results are also provided to demonstrate the effectiveness of the proposed optimal uniform MIMO-MSRS system. This paper is organized as follows. In Section II, the definition and signal models, as well as the LRT detector are discussed for the MIMO-MSRS system. In Section III, the optimal diversity DOF and its 2330

Fig. 1. MIMO-MSRS system configuration.

closed-form approximations are given for three optimization problems of the uniform MIMO-MSRS system, respectively. In Section IV, numerical results are provided. Finally, some conclusions are given in Section V. II.

PROPOSED MIMO-MSRS SYSTEM AND ITS LRT DETECTOR

In the following, we give the signal model of our proposed MIMO-MSRS system based on the spatial diversity conditions. Then the LRT detector is further proposed for target detection. A. MIMO Radar Signal Model and Spatial Diversity Condition As shown in Fig. 1, M transmitting elements and N receiving elements are used by the MIMO-MSRS system for transmitting M orthogonal waveforms and receiving their echoes, simultaneously. Also, a fluctuating target, composed of numerous little scatterers, exists in the detection plane with a “centroid” located at (x0 , y0 ). The size of the fluctuating target is much larger than radar wavelength ¸ but is still irresolvable in a single range unit. Then the signal received by the nth receiving element may be given as r M Pt X ®nm sm (t ¡ ¿mn ) + nn (t), n = 1, : : : , N rn (t) = M m=1

(1) where Pt is the constant total transmitting power of radar, ¿mn = (Rm + Rn )=c denotes the time delay in the n ¡ mth channel caused by the propagating range Rm from the mth transmitting element to the target’s “centroid” and the range Rn from the target’s “centroid” to the nth receiving element, respectively, c is the light speed and sm (t) denotes the waveform emitted by the mth transmitting element, which

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Fig. 3. MIMO-MSRS system and its configuration.

Fig. 2. Signal processing flowchart of MIMO-MSRS system.

R satisfies sm (t)sl ¤ (t)dt = ±ml . In practice the above orthogonality condition may be difficult to be strictly satisfied among different waveforms and many works [4, 10, 18, 21—22] have been devoted to this field. Also, nn (t) represents the receiver thermal noise in the nth receiving element, and ®nm is given as s Gmt Gnr ¸2 ¾nm j'nm e (2) ®nm = 2 R2 (4¼)3 Rm n where Gmt and Gnr represent the antenna gains of the mth transmitting element and the nth receiving element, ¾nm and 'nm are a Rayleigh-distributed RCS and a uniform-distributed phase in [¡¼, ¼] among the multiple channel samplings, respectively. The fluctuating properties of ¾nm and 'nm are introduced by the joint scattering and phase modulations from the target’s numerous scatterers. That is, the fluctuating target satisfies Swerling II model [1] and ®nm can be regarded as a complex Gaussian-distributed variable with a variance ¾T2 . For the proposed MIMO-MSRS system, the matched-filter banks shown in Fig. 2 are used at first to generate N £ M channel outputs from N receiving elements. Because the time delays of different channels may be different, the joint space-time compensation is needed for the space-time coregistration among different channel samplings. Then, for the detection unit containing a target, the N £ M channel samplings may be approximated as r Et n = 1, : : : , N, m = 1, : : : , M ® + nnm , rnm = M nm (3) where Et = Pt Ts is the constant pulse transmitting energy of radar, Ts is the duration of transmitting

pulse, nnm is the white Gaussian-distributed noise samplings of the n ¡ mth channel with auto-correlation ¾n2 ±(˜t). Therefore, the SNR of each T/R channel sampling may be given as E ¾2 ½ = t T2 : (4) M¾n It is clearly shown that the SNR of a single T/R channel is reciprocal to the transmitting element number M with given ¾T2 , ¾n2 , and Et . Therefore, with the increase of M there is a tradeoff between the increased total number of T/R channels and the reduced SNR of a single T/R channel. In other words, the ultimate detection performance is reliant on the joint processing of multiple channels and may possibly be deteriorated due to the reduced SNR of each channel. The optimal M has been discussed in [26] and M is assumed fixed for simplicity in this paper. B. MIMO-MSRS System and its Signal Model It is obvious that total N £ M real T/R channels can be obtained for a MIMO radar as (3) with M transmitting elements and N receiving elements. These N £ M channel samplings can also be regarded as the outputs of N £ M virtual T/R elements [20] as in Fig. 3, and each element is composed of a real transmitting element and a real receiving element in Fig. 1. For example, there is the k ¡ jth T/R channel from the kth transmitting element to the jth receiving element and the l ¡ ith T/R channel from the lth transmitting element to the ith receiving element in Fig. 1. Then, the k ¡ jth and the l ¡ ith T/R channels can be represented as the kjth and the lith virtual T/R elements in Fig. 3, respectively. Furthermore, the aspect angle difference from the above two virtual T/R elements to target is denoted as μe in Fig. 3. To describe the statistical properties among

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different virtual T/R elements, the spatial diversity conditions have been discussed for the fluctuating target detection in MSRS in [6—7] and for statistical MIMO radar in [8]. According to the discussion in [6], we simply give the spatial diversity condition in this paper as ¸ μe > (5) 2LT where LT is the target size in the direction perpendicular to the bisector of μe . That is, ®nm for these two T/R elements are independently distributed from each other when (5) holds. Otherwise, when the strong complementary condition of (5), i.e., μe ¿

¸ 2LT

(6)

holds for two virtual T/R elements, the outputs of these two elements may be strongly correlated. For simplicity, ®nm are assumed identical for these two T/R elements in the following discussion. Furthermore, in accordance with the spatial diversity condition (5) or the coherent integration condition (6), these N £ M virtual T/R elements can be clustered into L separate sites as in Fig. 3 and each site includes a certain number of tightly distributed T/R channels. That is, the aspect angle differences among sites are satisfied with the spatial diversity condition as (5) while these differences among different channels meet the requirements of (6) in each site. Then the most natural strategy for the proposed MIMO-MSRS system is to increase local SNR via DBF among all the samplings in a specific site at first. Subsequently, the MIMO-MSRS system should smooth target RCS fluctuation via spatial diversity among all the DBF outputs. Because M transmitting elements and N receiving elements may totally form N £ M different T/R channels, the maximum system DOF of the MIMO-MSRS system is Ds = MN. Besides, M and N are all assumed to be larger than one for the proposed MIMO-MSRS system. That is, the proposed MIMO-MSRS system has at least one transmitting element and one receiving element. Furthermore, assuming there are L distributed sites for the MIMO-MSRS system and each site has ci , i = 1, : : : , L, closely collocated coherent T/R channels, we have L X Ds = ci : (7) i=1

In real applications, there may be some channels in a specific site that cannot satisfy well the condition (6), and these ci channels may not be very strictly correlated. The discussion of the de-correlation effect 2332

on the coherent integration may be found in [25] and is omitted for simplicity in this paper. After coherent integration among these channels, the DBF outputs of the different sites may be given as ai = vH i Ci ,

i = 1, : : : , L (8) p where Ci = Et =M[®i e¡jÁ1 , : : : , ®i e¡jÁci ]T and vi = [e¡jÁ1 , : : : , e¡jÁci ]T are ci £ 1-dimensional signal and steering vector for the ith site, respectively, ( )T and ( )H are matrix transpose operator and conjugate transpose operator, respectively, ®i , i = 1, : : : , L, are the L independently identically distributed (IID) complex Gaussian distributed scattering coefficients for different channels. Clearly, the MIMO-MSRS system is a generalized model for radar with multiple T/R channels. With different configurations of parameter vector  = [L, c1 , c2 , : : : , cL ], the MIMO-MSRS system can be degenerated into some special cases, e.g., distributed MIMO radar, single-input multiple-output (SIMO) radar, multiple-input single output (MISO), radar and single-input single-output (SISO) radar in [8]: 1) When L = Ds and c1 = c2 = ¢ ¢ ¢ = cDs = 1, the MIMO-MSRS system is degenerated into the so-called distributed MIMO radar. 2) When L = M, c1 = c2 = ¢ ¢ ¢ = cM = N and M sites are caused by M separate transmitting elements, the MIMO-MSRS system is degenerated into the MISO radar. 3) When L = N, c1 = c2 = ¢ ¢ ¢ = cN = M and N sites are caused by N separate receiving elements, the MIMO-MSRS system is degenerated into the SIMO radar. 4) When L = 1 and c1 = Ds , the MIMO-MSRS system is degenerated into the SISO radar. C.

Likelihood Ratio Test Detector of the MIMO-MSRS System

So, after the coherent integration via DBF in a specific site for all the channels’ samplings, the output SNR of the ith site may be given as ½i = ci

Et ¾T2 = ci ½, M¾n2

i = 1, : : : , L:

(9)

Furthermore, the DBF outputs are uncorrelated among different sites because the aspect angle difference among them satisfies the spatial diversity condition of (5). For the multisite radar application, the LRT has been derived as a weighted sum of square detector outputs [6, p. 125] for the multiple incoherent signals with mutually uncorrelated fluctuations. In this paper, we directly extract the output of the square detector of different sites as Xi = jai j2 ,

i = 1, : : : , L

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where the statistics of the detector Xi may be given as μ ¶ 8 1 Xi > > < f(Xi ) = c ¾ 2 exp ¡ c ¾ 2 i n i n μ ¶ > 1 Xi > : f(Xi ) = exp ¡ ci ¾n2 (1 + ½i ) ci ¾n2 (1 + ½i ) where ci ¾n2 denotes the local noise level of DBF output in the ith site. Therefore, the likelihood ratio (LR) of Xi is given as μ ¶ f(Xi j H1 ) 1 Xi ½i = exp ¤i (Xi ) = f(Xi j H0 ) (1 + ½i ) ci ¾n2 (1 + ½i ) ¶ μ ½i 1 (12) exp Ri = (1 + ½i ) (1 + ½i ) where the ratio detector Ri [28] is defined as Ri =

Xi , ci ¾n2

i = 1, 2, : : : , L:

(13)

Because Ri , i = 1, 2, : : : , L, are the site squared outputs normalized by the local noise level and they are independently distributed from each other, the logarithmic likelihood ratio (LLR) of the whole MIMO-MSRS system is a sum of the LLR of different sites as μ ¶ ¶ ¶ μ L μ X 1 ½i log(¤(X)) = log Ri : + (1 + ½i ) 1 + ½i i=1

(14) Therefore, the LRT detector of the MIMO-MSRS system may be given as [28] TMIMO-MSRS =

L X i=1

H1

wi Ri ? ´0

(15)

H0

where ´0 is a threshold preset according to a preset constant PF and ½i ci ½ wi = = : (16) 1 + ½i 1 + ci ½ Obviously, the LRT of the MIMO-MSRS system is a weighted sum of the ratio detector of all the sites Ri , i = 1, 2, : : : , L. Interestingly, it is easily found that (15) is just equivalent to the equation (5.54) in [6, p. 125]. For the implementation of the proposed LRT as (15), the local noise level ¾n2 should be estimated to generate the ratio detector Ri , i = 1, 2, : : : , L at first. Then, the SNRs of different sites are estimated to determine the weight as (16) with the T/R number ci . Finally, the weighted sum of the ratio detector is used for LRT. 1) Statistics of the MIMO-MSRS System LRT Detector at the H0 Hypothesis: When the hypothesis H0 holds for a certain detection unit, we have Ri »

Â22 ,

i = 1, 2, : : : , L

(17)

H0 ,

i = 1, 2, : : : , L

(11)

H1

where Â2» represents the chi-square distribution with DOF ». Furthermore, the LRT detector of the MIMO-MSRS system under hypothesis H0 can be rewritten as L X H1 TMIMO-MSRS jH0 = wi Ri ? ´0 (18) H0

i=1

where Ri , i = 1, 2, : : : , L, are the IID chi-square distributed variables with DOF of 2, i.e., the standard exponential distribution. It is known that the sum of independently identically exponentially distributed variables is distributed with the Erlang distribution [29, p. 11], i.e., a special case of Gamma distribution with integer scale parameter. Also, the sum of weighted independently identically exponential distributed variables may be well approximated as Gamma-distributed [30, eq. (3.1)], i.e., ³v ´ TMIMO-MSRS jH0 » g0 Â2v0 = ¡ 0 , 2g0 (19) 2 where ¡ (μ, ») represents the Gamma distribution with scale parameter μ and shape parameter », and μ ¶2 ´2 ³P PL ci ½ L 2 i=1 2 i=1 wi 1 + ci ½ v0 = PL = μ ¶2 (20a) 2 PL ci ½ i=1 wi i=1 1 + ci ½ PL wi2 = g0 = Pi=1 L i=1 wi

μ

¶2 ci ½ i=1 1 + ci ½ : PL ci ½ i=1 1 + ci ½

PL

(20b)

2) Statistics of the MIMO-MSRS System LRT Detector at the H1 Hypothesis: When the hypothesis H1 holds for a certain detection unit, we have Ri » (1 + ½i )Â22 ,

i = 1, 2, : : : , L:

(21)

From (14) the LRT detector of the MIMO-MSRS system in hypothesis H1 may be given as TMIMO-MSRS jH1 =

L X i=1

½i Ri = ½

L X

H1

ci Ri ? ´0 :

(22)

H0

i=1

Then, we have [30] TMIMO-MSRS jH1 » g1 Â2v1 = ¡

³v

1

2

, 2g1

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´

(23) 2333

where

´2 c ½ i=1 i

³P L

2 v1 = PL

i=1 (ci ½)

PL

i=1 g1 = P L

(ci ½)2

i=1 ci ½

2

2D 2 = PL s 2 i=1 ci

=

½ X 2 ci : Ds

(24)

L

(25)

i=1

3) Statistics of the MIMO-MSRS System Invariant LRT Detector: Furthermore, we can define an invariant detector [27] of TMIMO-MSRS as s 1 H1 TMSRS1 = TMIMO-MSRS ?´ (26) g0 H0 1 where ´1 is a preset threshold for invariant detector. Then, the statistics of TMSRS1 may be given as 8 ³v ´ 0 > H0 : ¡ v1 , k H1 2 where k = 2g1 =g0 . It should be pointed out that (26) is a generalized invariant detector for the MIMO-MSRS system with arbitrary configuration. In Section III, (26) is used to derive the LRT detector for the MIMO-MSRS system with uniform configuration. III. MIMO-MSRS SYSTEM WITH OPTIMAL CONFIGURATIONS A. Optimal Problem Overview for the MIMO-MSRS System Like the conventional radars, the MIMO-MSRS system can be used for many different applications. However, no matter what purpose is pursued, the rule of constant probability of false alarm is normally adopted for radar target detection. With the constraint of constant probability of false alarm in the fixed noise environment, different optimized system configurations can be adopted for MIMO-MSRS system. For example, detecting targets with the lowest SNR is normally needed for searching radar at a given PF and PD , which means that the largest coverage diagram may be obtained because the minimum detectable SNR is required for targets. Also, for given PF and target SNR, detecting a specific target with the highest PD is normally preferred for tracking radar. Thirdly, to detect target with a given PF and PD , the minimal system DOF may be helpful to reduce the system hardware cost. Therefore, the optimal configuration design of the MIMO-MSRS system can also be divided into three kinds of optimization problems as follows. Optimization problem 1: With a given Ds , constant PF and PD , obtain the minimum detectable SNR of the target, i.e., ½min , by optimizing the parameter vector 2334

 = [L, c1 , c2 , : : : , cL ] based on the LRT detector of the MIMO-MSRS system. Optimization problem 2: With a given system Ds , constant PF , and target SNR ½, obtain the maximum PD by optimizing the parameter vector  = [L, c1 , c2 , : : : , cL ] based on the LRT detector of the MIMO-MSRS system. Optimization problem 3: With a given constant PF and target SNR ½, obtain the minimum Ds by optimizing the parameter vector  = [L, c1 , c2 , : : : , cL ] based on the LRT detector of the MIMO-MSRS system. To detect the target with the minimum detectable SNR or to detect a certain target with the highest PD , we make a “tradeoff” by dividing the given Ds into two parts. One part is for coherent integration via DBF processing in each site, and the other is for incoherent integration via spatial diversity. The above three optimization problems are related to each other, but they may have different solutions. Obviously, the core problem of the above three problems can be solved by the high-dimensional optimization of parameter vector  = [L, c1 , c2 , : : : , cL ] with different rules. Normally, to determine the unknown Â, a two-step strategy can be used as follows. Step 1 Determine the range of parameter L, which can be varied in [1, : : : , Ds ]. Step 2 With a given L, to determine [c1 , c2 , : : : , cL ] P with the constraint of Ds = Li=1 ci . Normally, Step 2 is a multi-dimensional optimization problem which cannot be easily solved via analytical expressions. It should be realized by searching all possible combinations of  = [L, c1 , c2 , : : : , cL ]. An analytical solution of this problem may be too complicated and worth future studies. B. MIMO-MSRS System Optimal Configurations for Three Optimization Problems 1) Uniform MIMO-MSRS System: In fact, the above three optimization problems may be solved from (26) for the MIMO-MSRS system with an arbitrary configuration via searching all possible combinations. However, it is difficult to obtain closed-form expressions for arbitrary configurations. To reduce the complexity and obtain closed-form approximations, we assume that each site has the same channel number for coherent integration, i.e., ci =

Ds , L

i = 1, 2, : : : , L:

(28)

Then substituting (28) into (16), (20), (24), and (27), respectively, we have wi =

Ds ½ L + Ds ½

v0 = 2L

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v1 = 2L

(31)

2(L + Ds ½) : L

k=

Also, (27) may be rewritten as 8 < ¡ (L, 2) μ ¶ TMSRS1 » : ¡ L, 2(L + Ds ½) L

(32)

H0 :

H1

(33)

Then, the statistics of (33) may be rewritten as the chi-square distribution test as 8 2 H0 < Â2L TMSRS1 » L + D ½ : (34) s : H1 Â22L L With given PF , L, and ½, the probability of detection for a target may be given as 0 ¡1 1 LQÂ2 (PF ) 2L A (35) PD = QÂ2 @ 2L L + ½Ds where QÂ2 ( ) and 2L

Q¡1 2 () 2L

cumulative distribution functions of chi-square distribution with a DOF of 2L and its inverse function, respectively. Also, with given PF , PD , and L, the detectable SNR of the target may be given as ½MIMO-MSRS =

Ds Q¡1 2 (PD )

:

(36)

2) Solution for Problem 1 and its Approximation: Now the optimization problem 1 may be given as

Ds

= arg min @ L=1

¡1 L(Q¡1 2 (PF ) ¡ QÂ2 (PD )) 2L 2L

H1

(41)

where Q( ) and Q¡1 ( ) are the complementary cumulative distribution functions of standard Normal-distribution and its inverse function, respectively. Also, with given PF , PD , and L, the detectable SNR of the target may be given as ½MIMO-MSRS (L) =

L(Q¡1 (PF ) ¡ Q¡1 (PD )) ³ p ´ Ds Q¡1 (PD ) + L L(a ¡ b) ³ p ´ Ds b + L

(42)

where a = Q¡1 (PF ), b = Q¡1 (PD ). Applying the derivative operator on (42), we obtain à !0

(a ¡ b) ¡ p ¢ p ¢ + Ds b + L Ds b + L ¡ ¢ p (a ¡ b) Ds L + 2b LDs = : (43) p ¢2 p ¡ 2Ds2 L b + L

¡

1

Furthermore let ½0 (L¤SNR ) = 0, we have

Ds Q¡1 2 (PD ) 2L

= b8(erfc¡1 (2(1 ¡ PD )))2 c:

1

A : (37)

Because there is no explicit expression for Q¡1 2 ( ), this 2L

problem is normally realized via the numerical search. In this paper, to simplify this optimization, we provide an approximation of (37). When L is large enough, the statistics of TMSRS1 in (34) may be approximated as 8 H0 < N(2L, 4L) μ ¶ 2 TMSRS1 » : N 2(L + ½D ), 4(L + ½Ds ) H1 s L (38) where N(μ, ») represents the Gaussian distribution with mean μ and variance ». Define an invariant detector [27] of TMSRS1 as TMSRS2 =

(40)

L¤SNR = b4b2 c = b4(Q¡1 (PD ))2 c

Ds

L¤SNR = arg min(½MIMO-MSRS ) 0

H0

Now, with given PF , L, and ½, the probability of detection for a target may be approximated as à p ! Q¡1 (PF )L ¡ ½Ds L PD = Q L + ½Ds

½0MIMO-MSRS (L) = L(a ¡ b)

2L

L=1

8 μ D½ ¶ > N ¡ ps , 1 > > < L Ã μ TMSRS2 » ¶2 ! > ½D > s > N 0, 1 + : L

=

are the complementary

¡1 L(Q¡1 2 (PF ) ¡ QÂ2 (PD )) 2L 2L

We have

TMSRS1 ¡ 2(L + ½Ds ) p : 2 L

(39)

(44)

Clearly, the optimal site number is mainly determined by PD to obtain the minimum detectable SNR for the uniform MIMO-MSRS system. 3) Solution for Problem 2 and its Approximation Expression: Now the optimization problem 2 may also be given as 0 11 0 ¡1 Q 2 (PF ) D D  s s AA : L¤PD = arg max(PD ) = arg max @QÂ2 @ 2L 2L L + ½Ds L=1 L=1 (45) Because there is no explicit expression for Q¡1 2 ( ), this 2L

problem is also normally realized via the numerical search. To simplify this optimization, substituting (41) into (45) we can obtain the approximated optimal DOF as 0 0 11 Ds ½ ¡1 p Q (P ) ¡ F B B CC Ds B C ¶L C L¤PD = arg max B @Q @ μ AA : (46) ½Ds L=1 1+ L

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Let’s define

Ã

p ! aL ¡ ½Ds L PD (L) =Q L + ½Ds à p ! 1 1 aL ¡ ½Ds L = + erf ¡ p : 2 2 2(L + ½Ds )

(47)

p Because (d=dx)erf(x) = (2= ¼) exp(¡x2 ), we have à μ p ¶2 ! μ p ¶0 aL ¡ ½Ds L p 2(L + ½Ds )

1 PD0 (L) = ¡ p exp ¡ 2¼ = Z(L)

where

Ã

aL ¡ ½Ds L L + ½Ds

¡ p p ¢! ½Ds 2a + L ¡ ½Ds = L

(48)

2(L + ½Ds )2

0 Ã p !2 1 1 aL ¡ ½D s L A : (49) Z(L) = ¡ p exp @¡ p 2¼ 2(L + ½Ds )

Obviously, we have Z(L) < 0. If we let PD0 (L¤PD ) = 0 we then have 2a + and L¤PD

(50)

q q L¤PD ¡ ½Ds = L¤PD = 0

(51)

μ ¶2 q 2 = wopt ¡a + a + ½Ds μ ¶2 q = wopt ¡Q¡1 (PF ) + (Q¡1 (PF ))2 + ½Ds

(52) where wopt = 0:707 is an experiential weight, which is determined via a great number of numerical experiments. The necessity to adjust the optimal DOF is brought about by the Norm-distribution approximation as (38), which also causes the ultimate scaling of the optimal DOF. The accurate calculation of wopt may need some asymptotical and analytical methods for future studies. 4) Solution for Problem 3 and its Approximation Expression: To obtain the minimum system DOF with given PF and PD for a target with certain SNR, we can determine the needed system DOF from (36) as ¡1 L(Q¡1 2 (PF ) ¡ QÂ2 (PD )) 2L 2L Ds = : (53) ½Q¡1 2 (PD ) 2L

Now the optimization problem 3 may also be given as Ds

L¤Ds = arg min(Ds ) L=1 Ds

0

= arg min @ L=1

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¡1 L(Q¡1 2 (PF ) ¡ QÂ2 (PD )) 2L

2L

½Q¡1 2 (PD ) 2L

1

A:

(54)

Comparing (54) and (37), it can be easily found that the optimization problem 3 will have the identical optimal diversity DOF output as the optimization problem 1. In other words, these two problems are equivalent with the same optimal site number L for a uniform MIMO-MSRS system. IV. NUMERICAL EXPERIMENT RESULTS AND PERFORMANCE ANALYSIS To verify the concept of the MIMO-MSRS system and its optimal configurations, as well as the proposed closed-form approximations as (44) and (52), we design some numerical experiments in this section. In the following, the defaulted MIMO-MSRS system configuration parameters are M = 5 and N = 6. The SNR is defined as (4), i.e., ½ = Et ¾T2 =M¾n2 . A. The Numerical Experiments for Optimization Problem 1 In Fig. 4, with the fixed PD = 0:8 and the three different PF = 10¡6 , PF = 10¡8 , and PF = 10¡10 , three curves of the detectable SNR versus L are plotted according to (36), respectively. It is clear that there are optimal site numbers, i.e., system diversity DOFs, with respect to the minimum detectable SNR for the MIMO-MSRS system. When the real system diversity DOF L is larger than this small optimal system diversity DOF, the detectable SNR will be inversely increased with the increase of L. That is, to improve the ultimate system performance, more tightly located T/R channels should be allocated for DBF to increase the local SNR of a specific site. In Fig. 5 the optimal diversity DOF curves versus PD are calculated for the MIMO-MSRS system. The standard optimal DOF of the MIMO-MSRS system is obtained according to (37). Also the results with respect to PF = 10¡6 , PF = 10¡8 , and PF = 10¡10 are provided, respectively. In the above experiments, the range of PD is varied from 0.5 to 0.95 to cover the requirements of normal radar detectors. It is shown that only a small part of system DOF should be used for diversity DOF while the other should be used as the coherent integration DOF for DBF processing. B. The Numerical Experiments for Optimization Problem 2 In Fig. 6 with the fixed PF = 10¡6 and three different SNRs ½ = ¡3 dB, ½ = 0 dB, ½ = 3 dB, three curves of PD versus L are plotted according to (35), respectively. It is clear that the optimal site numbers, i.e., system diversity DOFs, exist with respect to the maximum PD for the MIMO-MSRS system. When the real system diversity DOF L is larger than the small optimal diversity DOF, the target PD will be inversely decreased, especially for the low SNR targets. Notably, the optimal value still exists when

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Fig. 4. Detectable SNR versus site number.

Fig. 5. Optimal system diversity DOF versus PD .

SNR=3dB though its peak is not as sharp as the other two cases. In Fig. 7 the optimal diversity DOF curves versus PF are calculated for the MIMO-MSRS system. The standard optimal DOF of the MIMO-MSRS system is plotted according to (45). Also the results with ½ = ¡3 dB, ½ = 0 dB, and ½ = 3 dB are all provided. In the above experiments, the range of PF is varied from 10¡5 to 10¡8 , which may cover the detection requirements of normal radar detectors. In Fig. 8 the optimal diversity DOF curves versus target SNR are calculated for the MIMO-MSRS system. The standard optimal DOF of the MIMO-MSRS system is given according to (45). Also

the results with respect to PF = 10¡6 , PF = 10¡7 , and PF = 10¡8 are all provided, which covers the detection requirements of normal radar detectors. Also, the SNR range is varied from ¡10 dB to 10 dB. It is clear that more system DOF should be used for spatial diversity with the increase of target SNR. C.

The Numerical Experiments for Optimization Problem 3

In Fig. 9 the optimal diversity DOF curves versus Ds are calculated for the MIMO-MSRS system. The standard optimal DOF of the MIMO-MSRS system is provided according to (45). Also, the results with respect to ½ = ¡3 dB, ½ = 0 dB, and ½ = 3 dB are

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Fig. 6. Probability of detection versus site number.

Fig. 7. Optimal system diversity DOF versus PF .

all provided. In these experiments, the system DOF is varied from 0 to 45. It is clear that the optimal diversity DOF is approximately linearly proportional to the system DOF. In Fig. 10 with the fixed PD = 0:8 and three different PF = 10¡6 , PF = 10¡8 , PF = 10¡10 , three curves of required Ds versus L are plotted according to (53), respectively. It is clear that the optimal site numbers, i.e., system diversity DOFs, should exist with respect to the minimum required Ds for the MIMO-MSRS system. Also, by comparing Fig. 10 and Fig. 4, it is also shown that the optimization problem 3 will have the identical optimal output as optimization problem 1, which verifies the conclusions in Section IIIB. 2338

D. General Discussion for the MIMO-MSRS System From the above results of Figs. 4—10, the following general conclusions can be drawn for the MIMO-MSRS system. 1) Because less DOF will be allocated for DBF in each site with the increase of L, the ultimate performance will be deteriorated for MIMO-MSRS system when L is larger than the optimal number. Therefore, there exists the optimal number L to obtain the optimal performance for the above three optimization problems as shown in Fig. 4, Fig. 6 and Fig. 10, respectively. 2) For both optimization problem 1 and the optimization problem 3, only a small optimal diversity DOF is needed as Fig. 4 and Fig. 10. Even for the

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Fig. 8. Optimal system diversity DOF versus SNR.

Fig. 9. Optimal system diversity DOF versus SNR with different SNR.

high-performance detection with PF = 10¡9 and PD = 0:95, the optimal DOF is only about 12. It is shown that a small optimal diversity DOF, as well as the limited number of orthogonal waveforms, should be needed for the proposed MIMO-MSRS system in practice. 3) For optimization problem 2, the MIMO-MSRS system should change its diversity DOF according to the target SNR, Ds , as well as the PF . Especially, the optimal diversity DOF is all monotonically increased with PF , target SNR, or Ds as Fig. 7, Fig. 8, Fig. 9, respectively. 4) In real applications, the proposed closed-form approximations (44) and (52) can be used to directly obtain a coarse estimation for above three optimization problems, respectively. Normally, if a

highly accurate estimation is needed, it is suggested to use numerical searching according to original definitions for the different optimization problems. V.

CONCLUSIONS

For fluctuating target detection, a novel MIMO-MSRS system, as well as its signal model and LRT detector, are proposed in this paper. Subsequently, three configuration optimization problems are discussed for the MIMO-MSRS system in the fixed-noise background. The first is to detect the lowest SNR target with a given PF , PD , and system DOF. The second is to detect the target with the highest PD under condition of a given PF , target SNR, and system DOF. The third is to use the minimal system DOF to detect a target with a given PF , PD ,

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Fig. 10. Required Ds versus site number.

and target SNR. For the uniform MIMO-MSRS system, both the standard optimal DOF and the closed-form approximations are obtained for the above three problems. It is shown that the optimal configurations of the MIMO-MSRS system should be the tradeoff between coherent integration via DBF and incoherent integration via spatial diversity. Finally, some numerical results are also provided to demonstrate the effectiveness of the proposed MIMO-MSRS systems. ACKNOWLEDGMENTS The authors would like to thank Professor Jian Yang, Professor Xiu-tan Wang, Professor Yong-Liang Wang, Mr. Bing-Qu Liu, Dr. Wen-Shu Xiao, Dr. Zuo Yu, Dr. Xia Bin, Dr. Shi-Bao Peng, and the anonymous reviewers for their useful discussions and suggestions.

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Dai, X. Z. Research on diversity detection and wideband synthesis based on MIMO radar. Ph.D. dissertation, Tsinghua University, China, 2008. Dai, X. Z., et al. A new method of improving the weak target detection performance based on the MIMO radar. In Proceedings of the 2006 CIE International Conference on Radar, Oct. 2006, 1—4. Yu, J., Xu, J., and Peng, Y. N. Upper bound of coherent integration loss for symmetrically distributed phase noise. IEEE Signal Processing Letters, 15 (2008), 661—664. Xu, J., et al. Optimal transmitting diversity degrees of freedom for statistical MIMO radar. In Proceedings of the IEEE International Radar Conference, May 2010. Guan, J., Peng, Y. N., and He, Y. Proof of CFAR by the use of the invariant test. IEEE Transactions on Aerospace and Electronic Systems, 36, 2 (Jan. 2000), 336—339. Guan, J., Peng, Y. N., and He, Y. Three types of distributed CFAR detection based on local test statistic. IEEE Transactions on Aerospace and Electronic Systems, 38, 1 (Jan. 2002), 278—288. Cox, D. R. Renewal Theory. London: Methuen Publishing, 1962. Box, G. E. P. Some theorems on quadratic forms applied in the study of analysis of variance problems, I. Effect of inequality of variance in the one-way classification. Annals of Mathematical Statistics, 25, 2 (1954), 290—302.

Jia Xu (M’05) was born in Anhui Province, P.R. China, in 1974. He received the B.S. and M.S. degree from Radar Academy of Air Force, Wuhan, China in 1995 and 1998, and the Ph.D. degree from Navy Engineering University, Wuhan, China, in 2001. Now he is an associated professor in the Department of Electronics Engineering, Tsinghua University, China. His current research interests include detection and estimation theory, SAR/ISAR imaging, target recognition, array signal processing and adaptive signal processing. Dr. Xu received the Outstanding Post-Doctor Honor of Tsinghua University in 2004. He has authored or coauthored more than 100 papers. He is a senior member of the Chinese Institute of Electronics. XU ET AL.: OPTIMIZATIONS OF MULTISITE RADAR SYSTEM WITH MIMO RADARS FOR TARGET DETECTION

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Xizeng Dai (M’07) was born in Liaoning Province, China, in 1978. He received the B.S. degree from Dalian Railway Institute, Dalian, Liaoning, China in 2001, M.S. degree from China Academy of Telecom Technology in 2004, and Ph.D. degree from Department of Electronic Engineering, Tsinghua University, Beijing, China in 2008. His current research interests are in the areas of array signal processing, MIMO radar and wireless communication.

Xiang-Gen Xia (M’97–SM’00–F’09) received his B.S. degree in mathematics from Nanjing Normal University, Nanjing, China, and his M.S. degree in mathematics from Nankai University, Tianjin, China, and his Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, in 1983, 1986, and 1992, respectively. He was a senior/research staff member at Hughes Research Laboratories, Malibu, CA, during 1995—1996. In September 1996, he joined the Department of Electrical and Computer Engineering, University of Delaware, Newark, where he is the Charles Black Evans Professor. He was a visiting professor at the Chinese University of Hong Kong during 2002—2003, where he is an adjunct professor. Before 1995, he held visiting positions in a few institutions. His current research interests include space-time coding, MIMO and OFDM systems, digital signal processing, and SAR and ISAR imaging. Dr. Xia has over 220 refereed journal articles published and accepted, and 7 U.S. patents awarded and is the author of the book, Modulated Coding for Intersymbol Interference Channels (Marcel Dekker, 2000). He received the National Science Foundation (NSF) Faculty Early Career Development (CAREER) Program Award in 1997, the Office of Naval Research (ONR) Young Investigator Award in 1998, and the Outstanding Overseas Young Investigator Award from the National Nature Science Foundation of China in 2001. He also received the Outstanding Junior Faculty Award of the Engineering School of the University of Delaware in 2001. He is currently an Associate Editor of the IEEE Transactions on Wireless Communications, IEEE Trasactions on Signal Processing, Signal Processing (EURASIP), and the Journal of Communications and Networks (JCN). He was a guest editor of Space-Time Coding and Its Applications in the EURASIP Journal of Applied Signal Processing in 2002. He served as an Associate Editor of the IEEE Transactions on Signal Processing during 1996—2003, the IEEE Transactions on Mobile Computing during 2001—2004, IEEE Transactions on Vehicular Technology during 2005—2008, the IEEE Signal Processing Letters during 2003—2007, and the EURASIP Journal of Applied Signal Processing during 2001—2004. Dr. Xia served as a Member of the Signal Processing for Communications Committee from 2000—2005 and is currently a Member of the Sensor Array and Multichannel (SAM) Technical Committee (since 2004) in the IEEE Signal Processing Society. He serves as IEEE Sensors Council Representative of IEEE Signal Processing Society (since 2002) and served as the Representative of IEEE Signal Processing Society to the Steering Committee for IEEE Transactions on Mobile Computing during 2005—2006. He was Technical Program Chair of the Signal Processing Symposium, Globecom 2007 in Washington, D.C. and the General Cochair of ICASSP 2005 in Philadelphia. 2342

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Li-Bao Wang was born in Hebeii Province, China, in 1980 He received the B.S. and M.S. degrees from Radar Academy of Air Force, Wuhan, China in 2003 and 2006. He is a Ph.D. candidate in Academy of Electronic Engineering and Science, National Defense University, ChangSha, HuNan, China. His current research interests are in the areas of radar imaging and moving target detection.

Ji Yu was born in Jiangxi Province, China, in 1982. He received the B.S. degree from Beijing Normal University, Beijing, China, in 2005. He is a Ph.D. candidate in Department of Electronic Engineering, Tsinghua University, Beijing, China. His current research interests are in the areas of moving target detection and tracking and array signal processing.

Ying-Ning Peng (M’93–SM’97) was born in Sichuan Province, P.R. China, in 1939. He received the B.S. and M.S. degrees from Tsinghua University, Beijing, China, in 1962 and 1965, respectively. Since 1993, he has been with the Department of Electronic Engineering, Tsinghua University, where he is now a Professor and Director of Institute of Signal Detection and Processing. He has worked with real-time signal processing for many years and has published more than 200 papers. His recent research interests include processing, parallel signal processing, and radar polarimetry. Professor Peng is a Fellow of Chinese Institute of Electronics. He has received many awards for his contributions to research and education in China. XU ET AL.: OPTIMIZATIONS OF MULTISITE RADAR SYSTEM WITH MIMO RADARS FOR TARGET DETECTION

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