Research ArticleLPI Optimization Framework for Target Tracking in RadarNetwork Architectures Using Information-Theoretic Criteria

Chenguang Shi1 Fei Wang1 Mathini Sellathurai2 and Jianjiang Zhou1

1 Key Laboratory of Radar Imaging and Microwave Photonics Ministry of EducationNanjing University of Aeronautics and Astronautics No 29 Yudao Street Qinhuai District Nanjing Jiangsu 210016 China

2 School of Engineering and Physical Sciences Heriot-Watt University Edinburgh UK

Correspondence should be addressed to Jianjiang Zhou zjjeenuaaeducn

Received 26 February 2014 Revised 29 May 2014 Accepted 17 June 2014 Published 6 July 2014

Academic Editor Shengqi Zhu

Copyright copy 2014 Chenguang Shi et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Widely distributed radar network architectures can provide significant performance improvement for target detection andlocalization For a fixed radar network the achievable target detection performance may go beyond a predetermined thresholdwith full transmitted power allocation which is extremely vulnerable in modern electronic warfare In this paper we study theproblem of low probability of intercept (LPI) design for radar network and propose two novel LPI optimization schemes based oninformation-theoretic criteria For a predefined threshold of target detection Schleher intercept factor is minimized by optimizingtransmission power allocation among netted radars in the network Due to the lack of analytical closed-form expression for receiveroperation characteristics (ROC) we employ two information-theoretic criteria namely Bhattacharyya distance and J-divergenceas the metrics for target detection performance The resulting nonconvex and nonlinear LPI optimization problems associatedwith different information-theoretic criteria are cast under a unified framework and the nonlinear programming based geneticalgorithm (NPGA) is used to tackle the optimization problems in the framework Numerical simulations demonstrate that ourproposed LPI strategies are effective in enhancing the LPI performance for radar network

1 Introduction

Radar network architecture which is often called as dis-tributed multiple-input multiple-output (MIMO) radar hasbeen recently put forward and is becoming an inevitabletrend for future radar system design [1ndash3] The perfor-mance of radar network heavily depends on optimal powerallocation and transmission waveform design so enhancedimprovements on target detection and information extrac-tion would be realized by spatial and signal diversities

Currently system design for target detection and infor-mation extraction performance improvement has been along-term research topic in the distributed radar networkliterature In [4] Fishler et al propose the distributedMIMOradar concept and analyze the target detection performancefor distributed MIMO radar Yang and Blum in [5] studythe target identification and classification for MIMO radaremploying mutual information (MI) and the minimummean-square error (MMSE) criteria The authors in [6]

investigate the problem of code design to improve the detec-tion performance of multistatic radar in the presence of clut-ter Niu et al propose localization and tracking approachesfor noncoherent MIMO radar which provides significantperformance enhancement over traditional phased arrayradar [7]

Power allocation problem in radar network architecturehas been attracting contentiously growing attention andsomeof the noteworthy publications include [8ndash14]Theworkof [8] investigates the scheduling and power allocation prob-lem in cognitive radar network for multiple-target trackingin which an optimization criterion is proposed to find asuitable subset of antennas and optimal transmitted powerallocation Godrich et al in [9ndash11] address the power alloca-tion strategies for target localization in distributed multiple-radar configurations and propose some performance drivenresource allocation schemes In [12] the authors investi-gate target threatening level based optimal power allocationfor LPI radar network where two effective algorithms are

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 654561 10 pageshttpdxdoiorg1011552014654561

2 International Journal of Antennas and Propagation

proposed to enhance the LPI performance for radar networkFurthermore in [13 14] several optimal power allocationalgorithms for distributed MIMO radars with heterogeneouspropagation losses are presented to enhance target detectionand information extraction performance However up tonow the low probability of intercept (LPI) optimization forradar network architecture is still an open problem which isplaying an increasingly important role in modern electronicwarfare [1 15ndash18]Therefore it is an urgent task to investigatethe LPI optimization problem in radar network

This paper will extend the results in [6] and propose twonovel LPI optimization algorithms based on information-theoretic criteria for radar network architecture Our pur-pose is to minimize Schleher intercept factor by optimizingtransmission power allocation among netted radars for apredefined threshold of target detection Due to the lackof analytical closed-form expression for receiver operationcharacteristics (ROC) we employ two information-theoreticcriteria including Bhattacharyya distance and J-divergenceas the metrics for target detection performance As demon-strated later the proposed algorithms can provide significantLPI performance improvement for radar network To thebest of the authorsrsquo knowledge no literature discussingthe information-theoretic criteria based LPI optimizationfor radar network architecture was conducted prior to thiswork

The remainder of this paper is organized as followsSection 2 provides the radar network system model andbinary hypothesis test We first derive Schleher interceptfactor for radar network in Section 3 and formulate theproblems of information-theoretic criteria based LPI opti-mization where the resulting nonconvex and nonlinear LPIoptimization problems associatedwith different information-theoretic criteria are cast under a unified framework andsolved through the nonlinear programming based geneticalgorithm (NPGA) Numerical examples are provided inSection 4 Finally conclusion remarks are drawn in Section 5

2 System Model and the Optimal Detector

21 Radar Network SNR Equation We consider a radarnetwork architecture with 119873

119905transmitters and 119873

119903receivers

which can be broken down into119873119905times119873119903transmitter-receiver

pairs each with a bistatic component contributing to theentirety of the radar network signal-to-noise ratio (SNR) [1]Depicted in Figure 1 is an example of 4 times 4 radar networkAll the radars have acquired and are tracking the targetwith their directional antenna beams The netted radars1198771198861198891198861199031 1198771198861198891198861199032 1198771198861198891198861199033 and 1198771198861198891198861199034 transmit orthogonalwaveforms (as solid lines) but receive and process all theseechoes that are reflected from the target (as dotted lines) andsend the estimates to one of the radars in the network for datafusion with data link

For the radar network here orthogonal polyphase codesare employed in the system which have a large main lobe-to-side lobe ratio These codes have a more complicatedsignal structuremaking it more difficult to be intercepted anddetected by a hostile intercept receiver

Target

Data link

Radar1

Radar4

Radar3

Radar2

Figure 1 Example of an LPI radar network

It is also supposed that the network system has a commonprecise knowledge of space and timeThe radar network SNRcan be calculated by summing up the SNR of each transmit-receive pair as [1] follows

SNRnet =

119873119905

sum

119894=1

119873119903

sum

119895=1

1198751199051198941198661199051198941198661199031198951205901199051198941198951205822

119894

(4120587)31198961198791199001198941198951198611199031198941198651199031198951198772

1199051198941198772

119903119895119871119894119895

(1)

where the 119875119905119894is the 119894th transmitter power 119866

119905119894is the 119894th

transmitting antenna gain 119866119903119895is the 119895th receiving antenna

gain 120590119905119894119895

is the radar cross-section (RCS) of the target forthe 119894th transmitter and 119895th receiver 120582

119894is the 119894th transmitted

wavelength 119896 is Boltzmannrsquos constant 119879119900119894119895

is the receivingsystem noise temperature at the 119895th receiver 119861

119903119894is the

bandwidth of the matched filter for the 119894th transmittedwaveform 119865

119903119895is the noise factor for the 119895th receiver 119871

119894119895is

the system loss between the 119894th transmitter and 119895th receiver119877119905119894is the distance from the 119894th transmitter to the target and

119877119903119895is the distance from the target to the 119895th receiver

22 Radar Network Signal Model According to the discus-sions in [14] the path gain contains the target reflectioncoefficient 119892

119894119895and the propagation loss factor 119901

119894119895 Based on

the central limit theorem 119892119894119895sim CN(0 119877

119892) where 119892

119894119895denotes

the target reflection gain between radar 119894 and radar 119895 Thepropagation loss factor 119901

119894119895is a function of radar antenna gain

and waveform propagation distance which is expressed asfollows

119901119894119895=

radic119866119905119894119866119903119895

119877119905119894119877119903119895

(2)

It is supposed that the transmitted waveform of the 119894thnetted radar is radic119875

119905119894119909119894(119905) and then the collected signals at

the 119895th receiver from a single point target can be written asfollows

119910119895(119905) =

119873119905

sum

119894=1

119901119894119895119892119894119895radic119875119905119894119909119894(119905 minus 120591119894119895) + 119899119895(119905) (3)

International Journal of Antennas and Propagation 3

where int |119909119895(119905)|2119889119905 = 1 120591

119894119895represents the time delay 119899

119895(119905)

denotes the noise at receiver 119895 and the Doppler effect isnegligible At the 119895th receiver the received signal is matchedfiltered by time response 119909lowast

119896(minus119905) and the output signal can be

expressed as follows

119910119895119896(119905) = int119910

119895(119905) sdot 119909

lowast

119896(120591 minus 119905) 119889120591

= 119901119895119896119892119895119896radic119875119905119896int119909119896(120591 minus 120591

119896119895) sdot 119909lowast

119896(120591 minus 119905) 119889120591 + 119899

119895119896(119905)

(4)

where 119899119895119896(119905) = int 119899

119895(120591)sdot119909lowast

119896(120591minus119905)119889120591 andint119909

119895(120591)sdot119909lowast

119896(120591+119905)119889120591 = 0

for 119896 = 119895The discrete time signal for the 119895th receiver can be

rewritten as follows

119903119895119896≜ 119910119895119896(120591119895119896) = 119901119895119896119892119895119896radic119875119905119896+ 120579119895119896 (5)

where 119903119896119895is the output of the matched filter at the receiver

119895 sampled at 120591119895119896 120579119895119896

= 119899119895119896(120591119895119896) and 120579

119895119896sim CN(0 119877

120579) As

mentioned before we assume that all the netted radars haveacquired and are tracking the target with their directionalradar beams and they transmit orthogonal waveforms whilereceiving and processing all these echoes that are reflectedfrom the target In this way we can obtain 120591

119895119896

23 Binary Hypothesis Test With all the received signals thetarget detection for radar network system leads to a binaryhypothesis testing problem

1198670 119903119894119895= 120579119894119895

1198671 119903119894119895= 119901119894119895119892119894119895radic119875119905119894+ 120579119894119895

(6)

where 1 le 119894 le 119873119905 1 le 119895 le 119873

119903 The likelihood ratio test can be

formulated as follows

1198670 119879 ≜

119872

prod

119894=1

119873

prod

119895=1

119891 (119903119894119895| 1198671)

119891 (119903119894119895| 1198670)

lt 120575

1198671 119879 ≜

119872

prod

119894=1

119873

prod

119895=1

119891 (119903119894119895| 1198671)

119891 (119903119894119895| 1198670)

gt 120575

(7)

As introduced in [14] the underlying detection problem canbe equivalently rewritten as follows

1198670 119903119894119895sim CN (0 119877

120579)

1198671 119903119894119895sim CN (0 119877

120579+ 1198751199051198941198771198921199012

119894119895)

(8)

Then we have the optimal detector as follows

1198670 119879 ≜

119873119905

sum

119894=1

119873119903

sum

119895=1

10038161003816100381610038161003816119903119894119895

10038161003816100381610038161003816

2 21198751199051198941199012

119894119895

119877120579+ 1198751199051198941198771198921199012

119894119895

lt 120575

1198671 119879 ≜

119873119905

sum

119894=1

119873119903

sum

119895=1

10038161003816100381610038161003816119903119894119895

10038161003816100381610038161003816

2 21198751199051198941199012

119894119895

119877120579+ 1198751199051198941198771198921199012

119894119895

gt 120575

(9)

where 120575 denotes the detection threshold

3 Problem Formulation

In this section we aim to obtain the optimal LPI performancefor radar network architecture by judiciously designing thetransmission power allocation among netted radars in thenetwork We first derive Schleher intercept factor for radarnetwork system and then formulate the LPI optimizationproblems based on information-theoretic criteria For apredefined threshold of target detection Schleher interceptfactor is minimized by optimizing transmission power allo-cation among netted radars It is indicated in [6] that theanalytical closed-form expression for ROC does not existAs such we resort to information-theoretic criteria namelyBhattacharyya distance and J-divergence In what followsthe corresponding LPI optimization problems associatedwith different information-theoretic criteria are cast under aunified framework and can be solved conveniently throughNPGA

31 Schleher Intercept Factor for Radar Network For radarnetwork it is supposed that all signals can be separatelydistinguished at every netted radar node Assuming thatevery transmitter-receiver combination in the network canbe the same and 119877

2

net ≜ 119877119905119894sdot 119877119903119895 in which case the radar

network SNR equation (1) can be rewritten as follows (seeAppendix A)

SNRnet = 119870rad119873119903119875119905

1198774net (10)

where

119870rad =1198661199051198661199031205901199051205822

(4120587)3119896119879119900119861119903119865119903119871 (11)

119875119905is the total transmitting power of radar network systemNote that when 119873

119905= 119873119903

= 1 we can obtain themonostatic case

SNRmon = 119870rad119875119905

1198774mon (12)

where 119877mon is the distance between the monostatic radar andthe target while for intercept receiver the SNR equation is

SNRint = 119870int119875119905

1198772

int (13)

4 International Journal of Antennas and Propagation

InterceptorTarget

Radar

Rint

Rrad

Figure 2 The geometry of radar target and interceptor

where

119870int =1198661199051198661198941205822

(4120587)2119896119879119900119861119894119865119894119871119894

(14)

SNRint is the SNR at the interceptor signal processor input1198661015840119905

is the gain of the radarrsquos transmitting antenna in the directionof the intercept receiver 119866

119894is the gain of the intercept

receiverrsquos antenna 119865119894is the interceptor noise factor 119861

119894is the

bandwidth of the interceptor 119877int is the range from radarnetwork to the intercept receiver and 119871

119894refers to the losses

from the radar antenna to the receiver For simplicity weassume that the intercept receiver is carried by the targetAs such the interceptor detects the radar emission from themain lobe that is 1198661015840

119905= 119866119905

Herein Schleher intercept factor is employed to evaluateLPI performance for radar network The definition of Schle-her intercept factor can be calculated as follows

120572 =119877int119877rad

(15)

where 119877rad is the detection range of radar and 119877int is theintercept range of intercept receiver as illustrated in Figure 2

Based on the definition of Schleher intercept factor if120572 gt 1 radar can be detected by the interceptor while if120572 le 1 radar can detect the target and the interceptor cannotdetect the radar Therefore radar can meet LPI performancewhen 120572 le 1 Moreover minimization of Schleher interceptfactor leads to better LPI performance for radar networkarchitecture

With the derivation of Schleher intercept factor inAppendix B it can be observed that for a predefined targetdetection performance the closer the distance between radarsystem and target is the less power the radar system needs totransmit on guarantee of target detection performance Forsimplicity the maximum intercept factor 120572max

mon is normalizedto be 1 when the monostatic radar transmits the maximalpower 119875max

tot and SNRnet = SNRmon Therefore when thetransmission power is 119875

119905 the intercept factor for radar

network system can be simplified as follows

120572net =120572mon

11987314

119903

= (119875119905

119875maxtot sdot 119873

119903

)

14

(16)

where 120572mon is the Schleher intercept factor for monostaticradar From (16) one can see that Schleher intercept factor120572net is reduced with the increase of the number of radarreceivers119873

119903and the decrease of the total transmission power

119875119905in the network system

32 Information-Theoretic Criteria Based LPI Optimization

321 Bhattacharyya Distance Based LPI OptimizationScheme It is introduced in [6] that Bhattacharyya distance119861(1199010 1199011) measures the distance between two probability

density functions (pdf) 1199010

and 1199011 The Bhattacharyya

distance provides an upper bound on the probability of falsealarm 119875fa and at the same time yields a lower bound on theprobability of detection 119875

119889

Consider two multivariate Gaussian distributions 1198750and

1198751 1198750sim CN(0 120590

0) and 119875

1sim CN(0 120590

1) the Bhattacharyya

distance 119861(1198750 1198751) can be obtained as [6]

119861 (1198750 1198751) = log

det [05 (1205900+ 1205901)]

radicdet (1205900) det (120590

1)

(17)

Let 119861[119891(119903 | 1198670) 119891(119903 | 119867

1)] represent Bhattacharyya distance

between 1198670and 119867

1 where 119891(119903 | 119867

0) and 119891(119903 | 119867

1) are

the pdfs of r under hypotheses 1198670and 119867

1 For the binary

hypothesis testing problem we have that

119861net ≜ 119861 [119891 (119903 | 1198670) 119891 (119903 | 119867

1)]

=

119873119905

sum

119894=1

119873119903

sum

119895=1

log(1 + 05120577

119894119895

radic1 + 120577119894119895

)

=

119873119905

sum

119894=1

119873119903

sum

119895=1

log[[

[

1 + 1198751199051198941198771198921199012

119894119895(2119877120579)minus1

radic1 + 1198751199051198941198771198921199012

119894119895(119877120579)minus1

]]

]

=

119873119905

sum

119894=1

119873119903

sum

119895=1

log[[[

[

1 + 119875119905119894119877119892119866119905119894119866119903119895(21198771205791198772

1199051198941198772

119903119895)minus1

radic1 + 119875119905119894119877119892119866119905119894119866119903119895(1198771205791198772

1199051198941198772

119903119895)minus1

]]]

]

(18)

Based on the discussion in [6] maximization of the Bhat-tacharyya distance minimizes the upper bound on 119875fa whileit maximizes the lower bound on 119875

119889 As expressed in (18)

the Bhattacharyya distance derived here can be applied toevaluate the target detection performance of radar networkas a function of different parameters such as the transmittingpower of each netted radar and the number of netted radarsin the network Intuitively the greater the Bhattacharyya dis-tance between the two distributions of the binary hypothesistesting problem the better the capability of radar networksystem to detect the target which would make the net-work system more vulnerable in modern electronic warfareTherefore the Bhattacharyya distance can provide guidanceto the problem of LPI optimization for radar networkarchitecture

Here we focus on the LPI optimization problem forradar network architecture where Schleher intercept factoris minimized by optimizing transmission power allocationamong netted radars in the network for a predetermined

International Journal of Antennas and Propagation 5

Bhattacharyya distance threshold such that the LPI perfor-mance is met on the guarantee of target detection perfor-mance Eventually the underlying LPI optimization problemcan be formulated as follows

min997888Pt

120572net

st 119861net ge 119861th

119873119905

sum

119894=1

119875119905119894le 119875

maxtot

0 le 119875119905119894le 119875

max119905119894

(forall119894)

(19)

where 997888Pt = [1198751199051 1198751199052 119875

119905119873119905

]119879 is the transmitting power of

radar network 119861th is the Bhattacharyya distance thresholdfor target detection 119875max

tot is the maximum total transmissionpower of radar network and 119875max

119905119894(for all 119894) is the maximum

transmission power of the corresponding netted radar node

322 J-Divergence Based LPI Optimization Scheme The J-divergence 119869(119901

0 1199011) is anothermetric tomeasure the distance

between two pdfs 1199010and 119901

1 It is defined as follows

119869 (1199010 1199011) ≜ 119863 (119901

0 1199011) + 119863 (119901

1 1199010) (20)

where 119863(sdot) is the Kullback-Leibler divergence It is shown in[19] that for any fixed value of 119875fa

119863[119891 (119903 | 1198670) 119891 (119903 | 119867

1)] = lim119873rarrinfin

[minus1

119873log (1 minus 119875

119889)]

(21)

and for any fixed value of 119875119889 we can obtain

119863[119891 (119903 | 1198671) 119891 (119903 | 119867

0)] = lim119873rarrinfin

[minus1

119873log (119875fa)] (22)

From (21) and (22) we can observe that for any fixed 119875fa themaximization of Kullback-Leibler divergence 119863[119891(119903 | 119867

0)

119891(119903 | 1198671)] results in an asymptotic maximization of 119875

119889

while for any fixed 119875119889the maximization of Kullback-Leibler

divergence119863[119891(119903 | 1198671) 119891(119903 | 119867

0)] results in an asymptotic

minimization of 119875faWith the derivation in [6] we have that

119869net ≜ 119869 [119891 (119903 | 1198670) 119891 (119903 | 119867

1)]

=

119873119905

sum

119894=1

119873119903

sum

119895=1

1205772

119894119895

1 + 120577119894119895

=

119873119905

sum

119894=1

119873119903

sum

119895=1

[1198751199051198941198771198921199012

119894119895(119877120579)minus1

]2

1 + 1198751199051198941198771198921199012

119894119895(119877120579)minus1

=

119873119905

sum

119894=1

119873119903

sum

119895=1

[119875119905119894119877119892119866119905119894119866119903119895(1198771205791198772

1199051198941198772

119903119895)minus1

]

2

1 + 119875119905119894119877119892119866119905119894119866119903119895(1198771205791198772

1199051198941198772

119903119895)minus1

(23)

Consequently the corresponding LPI optimization prob-lem can be expressed as follows

min997888119875119905

120572net

st 119869net ge 119869th

119873119905

sum

119894=1

119875119905119894le 119875

maxtot

0 le 119875119905119894le 119875

max119905119894

(forall119894)

(24)

where 119869th is the J-divergence threshold for target detection

33 The Unified Framework Based on NPGA In this subsec-tion we cast the LPI optimization problems based on variousinformation-theoretic criteria investigated earlier under aunified optimization framework Furthermore we formulatethe following general form of the optimization problems in(19) and (24)

min997888119875119905

120572net

st 120574net ge 120574th

119873119905

sum

119894=1

119875119905119894le 119875

maxtot

0 le 119875119905119894le 119875

max119905119894

(forall119894)

(25)

where 120574net isin 119861net 119869net and 120574th is the corresponding threshold

for target detectionIn this paper we utilize the nonlinear programming based

genetic algorithm (NPGA) to seek the optimal solutions tothe resulting nonconvex nonlinear and constrained problem(25)The NPGA has a good performance on the convergencespeed and it improves the searching performance of ordinarygenetic algorithm

The NPGA procedure is illustrated in Figure 3 wherethe population initialization module is utilized to initializethe population according to the resulting problem while thecalculating fitness value module is to calculate the fitnessvalues of individuals in the population Selection crossoverand mutation are employed to seek the optimal solutionwhere 119873 is a constant If the evolution is 119873rsquos multiples wecan use NP approach to accelerate the convergence speed

So far we have completed the derivation of Schle-her intercept factor for radar network architecture and

6 International Journal of Antennas and Propagation

Population initializationStart Calculate

fitness value Selection Crossover Mutation

Nonlinear YesYes

NoNo

Endmultiples

Isevolution Nrsquos the termination

conditions

Satisfy

Figure 3 Flow diagram of NPGA procedure

0 02 04 06 08 1 12 14 16 18 20

10

20

30

40

50

60

70

80

90

Schleher intercept factor

Bhat

tach

aryy

a dist

ance

2 times 2 radar network2 times 4 radar network4 times 2 radar network

4 times 4 radar network6 times 6 radar network

Figure 4 Bhattacharyya distance versus Schleher intercept factorfor different radar network architectures

the information-theoretic criteria based LPI optimizationschemes In what follows some numerical simulations areprovided to confirm the effectiveness of our presented LPIoptimization algorithms for radar network architecture

4 Numerical Simulations

In this section we provide several numerical simulations toexamine the performance of the proposed LPI optimizationalgorithms as (19) and (24) Throughout this section weassume that 119875max

tot = sum119873119905

119894=1119875119905119894= 24KW 119866

119905= 119866119903= 30 dB

119877120579= 10minus10 and 119877

119892= 1 The SNR is set to be 13 dB The

traditional monostatic radar can detect the target whose RCSis 005m2 in the distance 119877

119877MAX = 1061 km by transmittingthe maximum power 119875max

tot = 24KW where the interceptfactor is normalized to be 1 for simplicity

41 LPI Performance Analysis Figures 4 and 6 show theBhattacharyya distance and logarithmic J-divergence ver-sus Schleher intercept factor for different radar networkarchitectures respectively which are conducted 10

6 Monte

0 02 04 06 08 1 12 14 16 18 20

10

20

30

40

50

60

Schleher intercept factor

Bhat

tach

aryy

a dist

ance

4 times 4 radar network with Rg = 1

4 times 4 radar network with Rg = 5

4 times 4 radar network with Rg = 10

Figure 5 Bhattacharyya distance versus Schleher intercept factorfor different target scattering intensity with119873

119905= 119873119903= 4

Carlo trials We can observe in Figures 4 and 6 that asSchleher intercept factor increases from 120572net = 0 to 120572net =2 the achievable Bhattacharyya distance and logarithmic J-divergence are increased This is due to the fact that as theintercept factor increases more transmission power wouldbe allocated which makes the achievable Bhattacharyya dis-tance and logarithmic J-divergence increase correspondinglyas theoretically proved in (18) and (23) Furthermore it canbe seen from Figures 4 and 6 that with the same target detec-tion threshold Schleher intercept factor can be significantlyreduced as the number of transmitters and receivers in thenetwork system increases Therefore increasing the numberof netted radars can effectively improve the LPI performancefor radar networkThis confirms the LPI benefits of the radarnetwork architecture with more netted radars

As shown in Figures 5 and 7 we illustrate the Bhat-tacharyya distance and logarithmic J-divergence versusSchleher intercept factor for different target scattering inten-sity with 119873

119905= 119873119903= 4 respectively It is depicted that

as the target scattering intensity increases from 119877119892

= 1

to 119877119892

= 10 the achievable Bhattacharyya distance andlogarithmic J-divergence are significantly increased This is

International Journal of Antennas and Propagation 7

0 02 04 06 08 1 12 14 16 18 2

0

2

4

6

8

10

Schleher intercept factor

log(

J-di

verg

ence

)

minus10

minus8

minus6

minus4

minus2

2 times 2 radar network2 times 4 radar network4 times 2 radar network

4 times 4 radar network6 times 6 radar network

Figure 6 Logarithmic J-divergence versus Schleher intercept factorfor different radar network architectures

0 02 04 06 08 1 12 14 16 18 2

0

5

10

15

Schleher intercept factor

log(

J-di

verg

ence

)

minus10

minus5

4 times 4 radar network with Rg = 1

4 times 4 radar network with Rg = 5

4 times 4 radar network with Rg = 10

Figure 7 Logarithmic J-divergence versus Schleher intercept factorfor different target scattering intensity with119873

119905= 119873119903= 4

because the radar network system can detect the target withlarge scattering intensity easily with high 119875

119889and low 119875fa

42 Target Tracking with LPI Optimization In this subsec-tion we consider a 4 times 4 radar network system (119873

119905=

119873119903= 4) in the simulation and it is widely deployed in

modern battlefield The target detection threshold 120574th can

be calculated in the condition that the transmission powerof each radar is 6KW in the distance 150 km between theradar network and the target which is the minimum value ofthe basic performance requirement for target detection As

180 185 190 195 200 205 210 215 220180

185

190

195

200

205

210

215

220

Netted radar

Radar1Radar2

Radar3 Radar4

X (km)

Y(k

m)

Figure 8 The radar network system configuration in two dimen-sions

85 90 95 100 105 110 115 120 125 130100

120

140

160

180

200

220

240

260

280

X (km)

Y (k

m)

Real trajectoryTrack with PF

t = 1 s

t = 50 s

Figure 9 Target tracking scenario

mentioned before it is supposed that the intercept receiveris carried by the target It is depicted in Figure 8 that thenetted radars in the network are spatially distributed in thesurveillance area at the initial time 119905 = 0

We track a single target by utilizing particle filtering (PF)method where 5000 particles are used to estimate the targetstate Figure 9 shows one realization of the target trajectoryfor 50 s and the tracking interval is chosen to be 1 s Withthe radar network configuration in Figure 8 and the targettracking scenario in Figure 9 we can obtain the distanceschanging curve between the netted radars and the target inthe tracking process as depicted in Figure 10 Without loss ofgenerality we set1198771198861198891198861199031 as the distributed data fusion centerand capitalize the weighted average approach to obtain theestimated target state

8 International Journal of Antennas and Propagation

5 10 15 20 25 30 35 40 45 5080

85

90

95

100

105

110

115

120

125

130

Time (s)

Dist

ance

(km

)

Netted radar 1Netted radar 2

Netted radar 3Netted radar 4

Figure 10 The distances between the netted radars and the target

5 10 15 20 25 30 35 40 45 50Time (s)

B-netted radar 1B-netted radar 2

B-netted radar 3B-netted radar 4

Tran

smitt

ing

pow

er (k

W)

0

05

1

15

2

25

3

Figure 11 The transmitting power of netted radars utilizing Bhat-tacharyya distance based LPI optimization in the tracking process

To obtain the optimal transmission power allocation ofradar network we utilize NPGA to solve (19) and (24) Letthe population size be 100 let the crossover probability be06 and let the mutation probability be 001 The populationevolves 10 generations Figure 11 shows the transmittingpower of netted radars utilizing Bhattacharyya distance basedLPI optimization in the tracking process while Figure 12depicts the J-divergence based case Before 119905 = 36 s nettedradars 2 3 and 4 are selected to track the target which arethe ones closest to the target while netted radar 1 is selectedinstead of radar 2 after 119905 = 36 s which is because netted radars1 2 and 3 have the best channel conditions in the networkFrom Figures 11 and 12 we can see that the transmissionpower allocation is determined by the locations of single

Tran

smitt

ing

pow

er (k

W)

5 10 15 20 25 30 35 40 45 500

05

1

15

2

25

3

Time (s)

J-netted radar 1J-netted radar 2

J-netted radar 3J-netted radar 4

Figure 12 The transmitting power of netted radars utilizing J-divergence based LPI optimization in the tracking process

03

04

05

06

07

08

09

1

11Sc

hleh

er in

terc

ept f

acto

r

J-radar networkB-radar network

Ordinary radar networkMonostatic radar

5 10 15 20 25 30 35 40 45 50Time (s)

Figure 13 The normalized Schleher intercept factor comparison inthe tracking process

target relative to the netted radars and their propagationlosses To be specific in the LPI optimization process moretransmitting power is allocated to the radar nodes that arelocated closer to the target this is due to the fact that theysuffer less propagation losses

Figure 13 demonstrates the advantage of our proposedoptimization problems based on information-theoretic cri-teria The traditional monostatic radar transmits 24KWconstantly while the ordinary radar network has a constantsum of transmitted power 24KW and each radar nodetransmits uniform power One can see that Schleher interceptfactor for radar network employing the information-theoreticcriteria based LPI optimization strategies is strictly smaller

International Journal of Antennas and Propagation 9

than that of traditional monostatic radar and ordinary radarnetwork across the whole region which further shows theLPI enhancement by exploiting our presented LPI optimiza-tion schemes in radar network to defend against passiveintercept receiver Moreover it can be seen in Figure 13 thatin terms of the same system constraints and fundamen-tal quantity Bhattacharyya distance based LPI optimiza-tion is asymptotically equivalent to the J-divergence basedcase

43 Discussion According to Figures 4ndash13 we can deducethe following conclusions for radar network architecture

(1) From Figures 4 to 7 we can observe that as thepredefined threshold of target detection increasesmore transmission power would be allocated forradar network to meet the detection performancewhile the intercept factor is increased subsequentlywhich is vulnerable in electronic warfare In otherwords there exists a tradeoff between LPI and targetdetection performance in radar network system andthe LPI performance would be sacrificed with targetdetection consideration

(2) In the numerical simulations we observe that theproposed optimization schemes (19) and (24) canbe employed to enhance the LPI performance forradar network Based on the netted radarsrsquo spatialdistributionwith respect to the target we can improvethe LPI performance by optimizing transmissionpower allocation among netted radars As indicatedin Figures 11 and 12 netted radars with better channelconditions are favorable over others In additionit can be observed that exploiting our proposedalgorithms can effectively improve the LPI perfor-mance of radar network to defend against interceptreceiver and Bhattacharyya distance based LPI opti-mization algorithm is asymptotically equivalent tothe J-divergence based case under the same systemconstraints and fundamental quantity

5 Conclusions

In this paper we investigated the problem of LPI designin radar network architecture where two LPI optimizationschemes based on information-theoretic criteria have beenproposed The NPGA was employed to tackle the highlynonconvex and nonlinear optimization problems Simula-tions have demonstrated that our proposed strategies areeffective and valuable to improve the LPI performance forradar network and it is indicated that these two optimizationproblems are asymptotically equivalent to each other underthe same system constraints Note that only single target wasconsidered in this paper Nevertheless it is convenient to beextended to multiple targets scenario and the conclusionsobtained in this study suggest that similar LPI benefits wouldbe obtained for the multiple targets case Future work willlook into the adaptive threshold design of target detectionperformance in radar network architectures

Appendices

A

Assume that every transmitter-receiver combination in thenetwork can be the same and 1198772net ≜ 119877

119905119894sdot 119877119903119895 where the radar

network SNR (1) can be written as follows

SNRnet = 119870rad

119873119905

sum

119894=1

119873119903

sum

119895=1

119875119905119894

1198774net (A1)

where

119870rad =1198661199051198661199031205901199051205822

(4120587)3119896119879119900119861119903119865119903119871 (A2)

Assume that the sum of the effective radiated power (ERP)from all the radars in the network is equivalent to that ofmonostatic radar that is

ERP =

119873119905

sum

119894=1

119875119905119894119866119905119894= 119875119905119866119905 (A3)

where 119875119905and 119866

119905are the transmitting power and transmitting

antenna gain of the monostatic radar respectively For 119866119905119894=

119866119905(for all 119894) we can rewrite (A1) as follows

SNRnet = 119870rad119873119905119875119905

1198774net (A4)

B

According to (15) we can derive the intercept factor for radarnetwork as

120572net =119877int119877net

= (119875119905sdot 1198702

int sdot SNRnet

119870rad sdot 119873119903 sdot SNR2int)

14

(B1)

and the intercept factor for conventional monostatic radar as

120572mon =119877int119877mon

= (119875119905sdot 1198702

int sdot SNRmon

119870rad sdot SNR2int)

14

(B2)

When SNRnet = SNRmon we can readily obtain therelationship between the intercept factors for radar network120572net and for the monostatic case 120572mon

120572net =120572mon

11987314

119903

(B3)

Furthermore for monostatic radar we can assume that

SNRmon = 119870rad119875119905

1198774mon= 119870rad

119875maxtot

1198774

119877MAX (B4)

where 119875maxtot is the maximal power of the monostatic rada-

r and 119877119877MAX is the corresponding maximal detection range

Then we can obtain

119877mon119877119877MAX

= (119875119905

119875maxtot

)

14

(B5)

10 International Journal of Antennas and Propagation

Similarly for intercept receiver we have the following

SNRint = 119870int119875119905

1198772

int= 119870int

119875maxtot

1198772

119868119872119860119883

(B6)

119877int119877119868MAX

= (119875119905

119875maxtot

)

12

(B7)

where 119877119868MAX is the intercept range when the transmitting

power of radar is 119875maxtot

Using (15) (B5) and (B7) we can obtain the followingexpression

120572mon120572maxmon

=119877int119877mon

119877119868MAX119877119877MAX

= (119875119905

119875maxtot

)

14

(B8)

where 120572maxmon is Schleher intercept factor corresponding to the

maximal transmitting power 119875maxtot

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their comments that help to improve the quality of thispaper The support provided by the National Natural ScienceFoundation ofChina (Grant no 61371170) Funding of JiangsuInnovation Program for Graduate Education (CXLX13 154)the Fundamental Research Funds for the Central Universi-ties the Priority Academic Program Development of JiangsuHigher Education Institutions (PADA) and Key Laboratoryof Radar Imaging and Microwave Photonics Ministry ofEducation Nanjing University of Aeronautics and Astronau-tics Nanjing China is gratefully acknowledged

References

[1] E P Phillip Detecting and Classifying Low Probability ofIntercept Radar Artech House Boston Mass USA 2009

[2] AM Haimovich R S Blum and L J Cimini Jr ldquoMIMO radarwith widely separated antennasrdquo IEEE Signal Processing Maga-zine vol 25 no 1 pp 116ndash129 2008

[3] A L Hume and C J Baker ldquoNetted radar sensingrdquo in Proceed-ings of the 2001 CIE International Conference on Radar pp 23ndash26 2001

[4] E Fishler A Haimovich R S Blum L J Cimini Jr D Chizhikand R A Valenzuela ldquoSpatial diversity in radars-Models anddetection performancerdquo IEEE Transactions on Signal Processingvol 54 no 3 pp 823ndash838 2006

[5] Y Yang and R S Blum ldquoMIMO radar waveform designbased onmutual information andminimummean-square errorestimationrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 43 no 1 pp 330ndash343 2007

[6] MM Naghsh M HMahmoud S P ShahramM Soltanalianand P Stoica ldquoUnified optimization framework for multi-staticradar code design using information-theoretic criteriardquo IEEETransactions on Signal Processing vol 61 no 21 pp 5401ndash54162013

[7] R Niu R S Blum P K Varshney and A L Drozd ldquoTar-get localization and tracking in noncoherent multiple-inputmultiple-output radar systemsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 48 no 2 pp 1466ndash1489 2012

[8] P Chavali and A Nehorai ldquoScheduling and power allocationin a cognitive radar network for multiple-target trackingrdquo IEEETransactions on Signal Processing vol 60 no 2 pp 715ndash7292012

[9] H Godrich A Petropulu and H V Poor ldquoOptimal powerallocation in distributed multiple-radar configurationsrdquo inProceeding of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 11) pp 2492ndash2495Prague Czech Republic May 2011

[10] H Godrich A P Petropulu and H V Poor ldquoPower allocationstrategies for target localization in distributed multiple-radararchitecturesrdquo IEEE Transactions on Signal Processing vol 59no 7 pp 3226ndash3240 2011

[11] H Godrich A Petropulu and H V Poort ldquoEstimation per-formance and resource savings tradeoffs in multiple radarssystemsrdquo in Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo12) pp 5205ndash5208 Kyoto Japan March 2012

[12] C G Shi J J Zhou F Wang and J Chen ldquoTarget threateninglevel based optimal power allocation for LPI radar networkrdquoin Proceedings of the 2nd International Symposium on Instru-mentation and Measurement Sensor Network and Automation(IMSNA rsquo13) pp 634ndash637 Toronto Canada December 2013

[13] X Song P Willett and S Zhou ldquoOptimal power allocationfor MIMO radars with heterogeneous propagation lossesrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 12) pp 2465ndash2468Kyoto Japan March 2012

[14] X F Song P Willett S Zhou and J Glaz ldquoMIMO radar detec-tion with heterogeneous propagation lossesrdquo in Proceedings ofthe IEEE Statistical Signal Processing Workshop (SSP rsquo12) pp776ndash779 August 2012

[15] D C Schleher ldquoLPI radar fact or fictionrdquo IEEE Aerospace andElectronic Systems Magazine vol 21 no 5 pp 3ndash6 2006

[16] D Lynch Jr Introduction to RF Stealth Sci Tech Publishing2004

[17] A G Stove A L Hume and C J Baker ldquoLow probability ofintercept radar strategiesrdquo IEE Proceedings Radar Sonar andNavigation vol 151 no 5 pp 249ndash260 2004

[18] D E Lawrence ldquoLow probability of intercept antenna arraybeamformingrdquo IEEE Transactions on Antennas and Propaga-tion vol 58 no 9 pp 2858ndash2865 2010

[19] T Kailath ldquoThe divergence and Bhattacharyya distance mea-sures in signal detectionrdquo IEEE Transaction on CommunicationTechnology vol 15 no 2 pp 52ndash60 1967

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Shock and Vibration

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Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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International Journal of

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DistributedSensor Networks

International Journal of

2 International Journal of Antennas and Propagation

proposed to enhance the LPI performance for radar networkFurthermore in [13 14] several optimal power allocationalgorithms for distributed MIMO radars with heterogeneouspropagation losses are presented to enhance target detectionand information extraction performance However up tonow the low probability of intercept (LPI) optimization forradar network architecture is still an open problem which isplaying an increasingly important role in modern electronicwarfare [1 15ndash18]Therefore it is an urgent task to investigatethe LPI optimization problem in radar network

This paper will extend the results in [6] and propose twonovel LPI optimization algorithms based on information-theoretic criteria for radar network architecture Our pur-pose is to minimize Schleher intercept factor by optimizingtransmission power allocation among netted radars for apredefined threshold of target detection Due to the lackof analytical closed-form expression for receiver operationcharacteristics (ROC) we employ two information-theoreticcriteria including Bhattacharyya distance and J-divergenceas the metrics for target detection performance As demon-strated later the proposed algorithms can provide significantLPI performance improvement for radar network To thebest of the authorsrsquo knowledge no literature discussingthe information-theoretic criteria based LPI optimizationfor radar network architecture was conducted prior to thiswork

The remainder of this paper is organized as followsSection 2 provides the radar network system model andbinary hypothesis test We first derive Schleher interceptfactor for radar network in Section 3 and formulate theproblems of information-theoretic criteria based LPI opti-mization where the resulting nonconvex and nonlinear LPIoptimization problems associatedwith different information-theoretic criteria are cast under a unified framework andsolved through the nonlinear programming based geneticalgorithm (NPGA) Numerical examples are provided inSection 4 Finally conclusion remarks are drawn in Section 5

2 System Model and the Optimal Detector

21 Radar Network SNR Equation We consider a radarnetwork architecture with 119873

119905transmitters and 119873

119903receivers

which can be broken down into119873119905times119873119903transmitter-receiver

pairs each with a bistatic component contributing to theentirety of the radar network signal-to-noise ratio (SNR) [1]Depicted in Figure 1 is an example of 4 times 4 radar networkAll the radars have acquired and are tracking the targetwith their directional antenna beams The netted radars1198771198861198891198861199031 1198771198861198891198861199032 1198771198861198891198861199033 and 1198771198861198891198861199034 transmit orthogonalwaveforms (as solid lines) but receive and process all theseechoes that are reflected from the target (as dotted lines) andsend the estimates to one of the radars in the network for datafusion with data link

For the radar network here orthogonal polyphase codesare employed in the system which have a large main lobe-to-side lobe ratio These codes have a more complicatedsignal structuremaking it more difficult to be intercepted anddetected by a hostile intercept receiver

Target

Data link

Radar1

Radar4

Radar3

Radar2

Figure 1 Example of an LPI radar network

It is also supposed that the network system has a commonprecise knowledge of space and timeThe radar network SNRcan be calculated by summing up the SNR of each transmit-receive pair as [1] follows

SNRnet =

119873119905

sum

119894=1

119873119903

sum

119895=1

1198751199051198941198661199051198941198661199031198951205901199051198941198951205822

119894

(4120587)31198961198791199001198941198951198611199031198941198651199031198951198772

1199051198941198772

119903119895119871119894119895

(1)

where the 119875119905119894is the 119894th transmitter power 119866

119905119894is the 119894th

transmitting antenna gain 119866119903119895is the 119895th receiving antenna

gain 120590119905119894119895

is the radar cross-section (RCS) of the target forthe 119894th transmitter and 119895th receiver 120582

119894is the 119894th transmitted

wavelength 119896 is Boltzmannrsquos constant 119879119900119894119895

is the receivingsystem noise temperature at the 119895th receiver 119861

119903119894is the

bandwidth of the matched filter for the 119894th transmittedwaveform 119865

119903119895is the noise factor for the 119895th receiver 119871

119894119895is

the system loss between the 119894th transmitter and 119895th receiver119877119905119894is the distance from the 119894th transmitter to the target and

119877119903119895is the distance from the target to the 119895th receiver

22 Radar Network Signal Model According to the discus-sions in [14] the path gain contains the target reflectioncoefficient 119892

119894119895and the propagation loss factor 119901

119894119895 Based on

the central limit theorem 119892119894119895sim CN(0 119877

119892) where 119892

119894119895denotes

the target reflection gain between radar 119894 and radar 119895 Thepropagation loss factor 119901

119894119895is a function of radar antenna gain

and waveform propagation distance which is expressed asfollows

119901119894119895=

radic119866119905119894119866119903119895

119877119905119894119877119903119895

(2)

It is supposed that the transmitted waveform of the 119894thnetted radar is radic119875

119905119894119909119894(119905) and then the collected signals at

the 119895th receiver from a single point target can be written asfollows

119910119895(119905) =

119873119905

sum

119894=1

119901119894119895119892119894119895radic119875119905119894119909119894(119905 minus 120591119894119895) + 119899119895(119905) (3)

International Journal of Antennas and Propagation 3

where int |119909119895(119905)|2119889119905 = 1 120591

119894119895represents the time delay 119899

119895(119905)

denotes the noise at receiver 119895 and the Doppler effect isnegligible At the 119895th receiver the received signal is matchedfiltered by time response 119909lowast

119896(minus119905) and the output signal can be

expressed as follows

119910119895119896(119905) = int119910

119895(119905) sdot 119909

lowast

119896(120591 minus 119905) 119889120591

= 119901119895119896119892119895119896radic119875119905119896int119909119896(120591 minus 120591

119896119895) sdot 119909lowast

119896(120591 minus 119905) 119889120591 + 119899

119895119896(119905)

(4)

where 119899119895119896(119905) = int 119899

119895(120591)sdot119909lowast

119896(120591minus119905)119889120591 andint119909

119895(120591)sdot119909lowast

119896(120591+119905)119889120591 = 0

for 119896 = 119895The discrete time signal for the 119895th receiver can be

rewritten as follows

119903119895119896≜ 119910119895119896(120591119895119896) = 119901119895119896119892119895119896radic119875119905119896+ 120579119895119896 (5)

where 119903119896119895is the output of the matched filter at the receiver

119895 sampled at 120591119895119896 120579119895119896

= 119899119895119896(120591119895119896) and 120579

119895119896sim CN(0 119877

120579) As

mentioned before we assume that all the netted radars haveacquired and are tracking the target with their directionalradar beams and they transmit orthogonal waveforms whilereceiving and processing all these echoes that are reflectedfrom the target In this way we can obtain 120591

119895119896

23 Binary Hypothesis Test With all the received signals thetarget detection for radar network system leads to a binaryhypothesis testing problem

1198670 119903119894119895= 120579119894119895

1198671 119903119894119895= 119901119894119895119892119894119895radic119875119905119894+ 120579119894119895

(6)

where 1 le 119894 le 119873119905 1 le 119895 le 119873

119903 The likelihood ratio test can be

formulated as follows

1198670 119879 ≜

119872

prod

119894=1

119873

prod

119895=1

119891 (119903119894119895| 1198671)

119891 (119903119894119895| 1198670)

lt 120575

1198671 119879 ≜

119872

prod

119894=1

119873

prod

119895=1

119891 (119903119894119895| 1198671)

119891 (119903119894119895| 1198670)

gt 120575

(7)

As introduced in [14] the underlying detection problem canbe equivalently rewritten as follows

1198670 119903119894119895sim CN (0 119877

120579)

1198671 119903119894119895sim CN (0 119877

120579+ 1198751199051198941198771198921199012

119894119895)

(8)

Then we have the optimal detector as follows

1198670 119879 ≜

119873119905

sum

119894=1

119873119903

sum

119895=1

10038161003816100381610038161003816119903119894119895

10038161003816100381610038161003816

2 21198751199051198941199012

119894119895

119877120579+ 1198751199051198941198771198921199012

119894119895

lt 120575

1198671 119879 ≜

119873119905

sum

119894=1

119873119903

sum

119895=1

10038161003816100381610038161003816119903119894119895

10038161003816100381610038161003816

2 21198751199051198941199012

119894119895

119877120579+ 1198751199051198941198771198921199012

119894119895

gt 120575

(9)

where 120575 denotes the detection threshold

3 Problem Formulation

In this section we aim to obtain the optimal LPI performancefor radar network architecture by judiciously designing thetransmission power allocation among netted radars in thenetwork We first derive Schleher intercept factor for radarnetwork system and then formulate the LPI optimizationproblems based on information-theoretic criteria For apredefined threshold of target detection Schleher interceptfactor is minimized by optimizing transmission power allo-cation among netted radars It is indicated in [6] that theanalytical closed-form expression for ROC does not existAs such we resort to information-theoretic criteria namelyBhattacharyya distance and J-divergence In what followsthe corresponding LPI optimization problems associatedwith different information-theoretic criteria are cast under aunified framework and can be solved conveniently throughNPGA

31 Schleher Intercept Factor for Radar Network For radarnetwork it is supposed that all signals can be separatelydistinguished at every netted radar node Assuming thatevery transmitter-receiver combination in the network canbe the same and 119877

2

net ≜ 119877119905119894sdot 119877119903119895 in which case the radar

network SNR equation (1) can be rewritten as follows (seeAppendix A)

SNRnet = 119870rad119873119903119875119905

1198774net (10)

where

119870rad =1198661199051198661199031205901199051205822

(4120587)3119896119879119900119861119903119865119903119871 (11)

119875119905is the total transmitting power of radar network systemNote that when 119873

119905= 119873119903

= 1 we can obtain themonostatic case

SNRmon = 119870rad119875119905

1198774mon (12)

where 119877mon is the distance between the monostatic radar andthe target while for intercept receiver the SNR equation is

SNRint = 119870int119875119905

1198772

int (13)

4 International Journal of Antennas and Propagation

InterceptorTarget

Radar

Rint

Rrad

Figure 2 The geometry of radar target and interceptor

where

119870int =1198661199051198661198941205822

(4120587)2119896119879119900119861119894119865119894119871119894

(14)

SNRint is the SNR at the interceptor signal processor input1198661015840119905

is the gain of the radarrsquos transmitting antenna in the directionof the intercept receiver 119866

119894is the gain of the intercept

receiverrsquos antenna 119865119894is the interceptor noise factor 119861

119894is the

bandwidth of the interceptor 119877int is the range from radarnetwork to the intercept receiver and 119871

119894refers to the losses

from the radar antenna to the receiver For simplicity weassume that the intercept receiver is carried by the targetAs such the interceptor detects the radar emission from themain lobe that is 1198661015840

119905= 119866119905

Herein Schleher intercept factor is employed to evaluateLPI performance for radar network The definition of Schle-her intercept factor can be calculated as follows

120572 =119877int119877rad

(15)

where 119877rad is the detection range of radar and 119877int is theintercept range of intercept receiver as illustrated in Figure 2

Based on the definition of Schleher intercept factor if120572 gt 1 radar can be detected by the interceptor while if120572 le 1 radar can detect the target and the interceptor cannotdetect the radar Therefore radar can meet LPI performancewhen 120572 le 1 Moreover minimization of Schleher interceptfactor leads to better LPI performance for radar networkarchitecture

With the derivation of Schleher intercept factor inAppendix B it can be observed that for a predefined targetdetection performance the closer the distance between radarsystem and target is the less power the radar system needs totransmit on guarantee of target detection performance Forsimplicity the maximum intercept factor 120572max

mon is normalizedto be 1 when the monostatic radar transmits the maximalpower 119875max

tot and SNRnet = SNRmon Therefore when thetransmission power is 119875

119905 the intercept factor for radar

network system can be simplified as follows

120572net =120572mon

11987314

119903

= (119875119905

119875maxtot sdot 119873

119903

)

14

(16)

where 120572mon is the Schleher intercept factor for monostaticradar From (16) one can see that Schleher intercept factor120572net is reduced with the increase of the number of radarreceivers119873

119903and the decrease of the total transmission power

119875119905in the network system

32 Information-Theoretic Criteria Based LPI Optimization

321 Bhattacharyya Distance Based LPI OptimizationScheme It is introduced in [6] that Bhattacharyya distance119861(1199010 1199011) measures the distance between two probability

density functions (pdf) 1199010

and 1199011 The Bhattacharyya

distance provides an upper bound on the probability of falsealarm 119875fa and at the same time yields a lower bound on theprobability of detection 119875

119889

Consider two multivariate Gaussian distributions 1198750and

1198751 1198750sim CN(0 120590

0) and 119875

1sim CN(0 120590

1) the Bhattacharyya

distance 119861(1198750 1198751) can be obtained as [6]

119861 (1198750 1198751) = log

det [05 (1205900+ 1205901)]

radicdet (1205900) det (120590

1)

(17)

Let 119861[119891(119903 | 1198670) 119891(119903 | 119867

1)] represent Bhattacharyya distance

between 1198670and 119867

1 where 119891(119903 | 119867

0) and 119891(119903 | 119867

1) are

the pdfs of r under hypotheses 1198670and 119867

1 For the binary

hypothesis testing problem we have that

119861net ≜ 119861 [119891 (119903 | 1198670) 119891 (119903 | 119867

1)]

=

119873119905

sum

119894=1

119873119903

sum

119895=1

log(1 + 05120577

119894119895

radic1 + 120577119894119895

)

=

119873119905

sum

119894=1

119873119903

sum

119895=1

log[[

[

1 + 1198751199051198941198771198921199012

119894119895(2119877120579)minus1

radic1 + 1198751199051198941198771198921199012

119894119895(119877120579)minus1

]]

]

=

119873119905

sum

119894=1

119873119903

sum

119895=1

log[[[

[

1 + 119875119905119894119877119892119866119905119894119866119903119895(21198771205791198772

1199051198941198772

119903119895)minus1

radic1 + 119875119905119894119877119892119866119905119894119866119903119895(1198771205791198772

1199051198941198772

119903119895)minus1

]]]

]

(18)

Based on the discussion in [6] maximization of the Bhat-tacharyya distance minimizes the upper bound on 119875fa whileit maximizes the lower bound on 119875

119889 As expressed in (18)

the Bhattacharyya distance derived here can be applied toevaluate the target detection performance of radar networkas a function of different parameters such as the transmittingpower of each netted radar and the number of netted radarsin the network Intuitively the greater the Bhattacharyya dis-tance between the two distributions of the binary hypothesistesting problem the better the capability of radar networksystem to detect the target which would make the net-work system more vulnerable in modern electronic warfareTherefore the Bhattacharyya distance can provide guidanceto the problem of LPI optimization for radar networkarchitecture

Here we focus on the LPI optimization problem forradar network architecture where Schleher intercept factoris minimized by optimizing transmission power allocationamong netted radars in the network for a predetermined

International Journal of Antennas and Propagation 5

Bhattacharyya distance threshold such that the LPI perfor-mance is met on the guarantee of target detection perfor-mance Eventually the underlying LPI optimization problemcan be formulated as follows

min997888Pt

120572net

st 119861net ge 119861th

119873119905

sum

119894=1

119875119905119894le 119875

maxtot

0 le 119875119905119894le 119875

max119905119894

(forall119894)

(19)

where 997888Pt = [1198751199051 1198751199052 119875

119905119873119905

]119879 is the transmitting power of

radar network 119861th is the Bhattacharyya distance thresholdfor target detection 119875max

tot is the maximum total transmissionpower of radar network and 119875max

119905119894(for all 119894) is the maximum

transmission power of the corresponding netted radar node

322 J-Divergence Based LPI Optimization Scheme The J-divergence 119869(119901

0 1199011) is anothermetric tomeasure the distance

between two pdfs 1199010and 119901

1 It is defined as follows

119869 (1199010 1199011) ≜ 119863 (119901

0 1199011) + 119863 (119901

1 1199010) (20)

where 119863(sdot) is the Kullback-Leibler divergence It is shown in[19] that for any fixed value of 119875fa

119863[119891 (119903 | 1198670) 119891 (119903 | 119867

1)] = lim119873rarrinfin

[minus1

119873log (1 minus 119875

119889)]

(21)

and for any fixed value of 119875119889 we can obtain

119863[119891 (119903 | 1198671) 119891 (119903 | 119867

0)] = lim119873rarrinfin

[minus1

119873log (119875fa)] (22)

From (21) and (22) we can observe that for any fixed 119875fa themaximization of Kullback-Leibler divergence 119863[119891(119903 | 119867

0)

119891(119903 | 1198671)] results in an asymptotic maximization of 119875

119889

while for any fixed 119875119889the maximization of Kullback-Leibler

divergence119863[119891(119903 | 1198671) 119891(119903 | 119867

0)] results in an asymptotic

minimization of 119875faWith the derivation in [6] we have that

119869net ≜ 119869 [119891 (119903 | 1198670) 119891 (119903 | 119867

1)]

=

119873119905

sum

119894=1

119873119903

sum

119895=1

1205772

119894119895

1 + 120577119894119895

=

119873119905

sum

119894=1

119873119903

sum

119895=1

[1198751199051198941198771198921199012

119894119895(119877120579)minus1

]2

1 + 1198751199051198941198771198921199012

119894119895(119877120579)minus1

=

119873119905

sum

119894=1

119873119903

sum

119895=1

[119875119905119894119877119892119866119905119894119866119903119895(1198771205791198772

1199051198941198772

119903119895)minus1

]

2

1 + 119875119905119894119877119892119866119905119894119866119903119895(1198771205791198772

1199051198941198772

119903119895)minus1

(23)

Consequently the corresponding LPI optimization prob-lem can be expressed as follows

min997888119875119905

120572net

st 119869net ge 119869th

119873119905

sum

119894=1

119875119905119894le 119875

maxtot

0 le 119875119905119894le 119875

max119905119894

(forall119894)

(24)

where 119869th is the J-divergence threshold for target detection

33 The Unified Framework Based on NPGA In this subsec-tion we cast the LPI optimization problems based on variousinformation-theoretic criteria investigated earlier under aunified optimization framework Furthermore we formulatethe following general form of the optimization problems in(19) and (24)

min997888119875119905

120572net

st 120574net ge 120574th

119873119905

sum

119894=1

119875119905119894le 119875

maxtot

0 le 119875119905119894le 119875

max119905119894

(forall119894)

(25)

where 120574net isin 119861net 119869net and 120574th is the corresponding threshold

for target detectionIn this paper we utilize the nonlinear programming based

genetic algorithm (NPGA) to seek the optimal solutions tothe resulting nonconvex nonlinear and constrained problem(25)The NPGA has a good performance on the convergencespeed and it improves the searching performance of ordinarygenetic algorithm

The NPGA procedure is illustrated in Figure 3 wherethe population initialization module is utilized to initializethe population according to the resulting problem while thecalculating fitness value module is to calculate the fitnessvalues of individuals in the population Selection crossoverand mutation are employed to seek the optimal solutionwhere 119873 is a constant If the evolution is 119873rsquos multiples wecan use NP approach to accelerate the convergence speed

So far we have completed the derivation of Schle-her intercept factor for radar network architecture and

6 International Journal of Antennas and Propagation

Population initializationStart Calculate

fitness value Selection Crossover Mutation

Nonlinear YesYes

NoNo

Endmultiples

Isevolution Nrsquos the termination

conditions

Satisfy

Figure 3 Flow diagram of NPGA procedure

0 02 04 06 08 1 12 14 16 18 20

10

20

30

40

50

60

70

80

90

Schleher intercept factor

Bhat

tach

aryy

a dist

ance

2 times 2 radar network2 times 4 radar network4 times 2 radar network

4 times 4 radar network6 times 6 radar network

Figure 4 Bhattacharyya distance versus Schleher intercept factorfor different radar network architectures

the information-theoretic criteria based LPI optimizationschemes In what follows some numerical simulations areprovided to confirm the effectiveness of our presented LPIoptimization algorithms for radar network architecture

4 Numerical Simulations

In this section we provide several numerical simulations toexamine the performance of the proposed LPI optimizationalgorithms as (19) and (24) Throughout this section weassume that 119875max

tot = sum119873119905

119894=1119875119905119894= 24KW 119866

119905= 119866119903= 30 dB

119877120579= 10minus10 and 119877

119892= 1 The SNR is set to be 13 dB The

traditional monostatic radar can detect the target whose RCSis 005m2 in the distance 119877

119877MAX = 1061 km by transmittingthe maximum power 119875max

tot = 24KW where the interceptfactor is normalized to be 1 for simplicity

41 LPI Performance Analysis Figures 4 and 6 show theBhattacharyya distance and logarithmic J-divergence ver-sus Schleher intercept factor for different radar networkarchitectures respectively which are conducted 10

6 Monte

0 02 04 06 08 1 12 14 16 18 20

10

20

30

40

50

60

Schleher intercept factor

Bhat

tach

aryy

a dist

ance

4 times 4 radar network with Rg = 1

4 times 4 radar network with Rg = 5

4 times 4 radar network with Rg = 10

Figure 5 Bhattacharyya distance versus Schleher intercept factorfor different target scattering intensity with119873

119905= 119873119903= 4

Carlo trials We can observe in Figures 4 and 6 that asSchleher intercept factor increases from 120572net = 0 to 120572net =2 the achievable Bhattacharyya distance and logarithmic J-divergence are increased This is due to the fact that as theintercept factor increases more transmission power wouldbe allocated which makes the achievable Bhattacharyya dis-tance and logarithmic J-divergence increase correspondinglyas theoretically proved in (18) and (23) Furthermore it canbe seen from Figures 4 and 6 that with the same target detec-tion threshold Schleher intercept factor can be significantlyreduced as the number of transmitters and receivers in thenetwork system increases Therefore increasing the numberof netted radars can effectively improve the LPI performancefor radar networkThis confirms the LPI benefits of the radarnetwork architecture with more netted radars

As shown in Figures 5 and 7 we illustrate the Bhat-tacharyya distance and logarithmic J-divergence versusSchleher intercept factor for different target scattering inten-sity with 119873

119905= 119873119903= 4 respectively It is depicted that

as the target scattering intensity increases from 119877119892

= 1

to 119877119892

= 10 the achievable Bhattacharyya distance andlogarithmic J-divergence are significantly increased This is

International Journal of Antennas and Propagation 7

0 02 04 06 08 1 12 14 16 18 2

0

2

4

6

8

10

Schleher intercept factor

log(

J-di

verg

ence

)

minus10

minus8

minus6

minus4

minus2

2 times 2 radar network2 times 4 radar network4 times 2 radar network

4 times 4 radar network6 times 6 radar network

Figure 6 Logarithmic J-divergence versus Schleher intercept factorfor different radar network architectures

0 02 04 06 08 1 12 14 16 18 2

0

5

10

15

Schleher intercept factor

log(

J-di

verg

ence

)

minus10

minus5

4 times 4 radar network with Rg = 1

4 times 4 radar network with Rg = 5

4 times 4 radar network with Rg = 10

Figure 7 Logarithmic J-divergence versus Schleher intercept factorfor different target scattering intensity with119873

119905= 119873119903= 4

because the radar network system can detect the target withlarge scattering intensity easily with high 119875

119889and low 119875fa

42 Target Tracking with LPI Optimization In this subsec-tion we consider a 4 times 4 radar network system (119873

119905=

119873119903= 4) in the simulation and it is widely deployed in

modern battlefield The target detection threshold 120574th can

be calculated in the condition that the transmission powerof each radar is 6KW in the distance 150 km between theradar network and the target which is the minimum value ofthe basic performance requirement for target detection As

180 185 190 195 200 205 210 215 220180

185

190

195

200

205

210

215

220

Netted radar

Radar1Radar2

Radar3 Radar4

X (km)

Y(k

m)

Figure 8 The radar network system configuration in two dimen-sions

85 90 95 100 105 110 115 120 125 130100

120

140

160

180

200

220

240

260

280

X (km)

Y (k

m)

Real trajectoryTrack with PF

t = 1 s

t = 50 s

Figure 9 Target tracking scenario

mentioned before it is supposed that the intercept receiveris carried by the target It is depicted in Figure 8 that thenetted radars in the network are spatially distributed in thesurveillance area at the initial time 119905 = 0

We track a single target by utilizing particle filtering (PF)method where 5000 particles are used to estimate the targetstate Figure 9 shows one realization of the target trajectoryfor 50 s and the tracking interval is chosen to be 1 s Withthe radar network configuration in Figure 8 and the targettracking scenario in Figure 9 we can obtain the distanceschanging curve between the netted radars and the target inthe tracking process as depicted in Figure 10 Without loss ofgenerality we set1198771198861198891198861199031 as the distributed data fusion centerand capitalize the weighted average approach to obtain theestimated target state

8 International Journal of Antennas and Propagation

5 10 15 20 25 30 35 40 45 5080

85

90

95

100

105

110

115

120

125

130

Time (s)

Dist

ance

(km

)

Netted radar 1Netted radar 2

Netted radar 3Netted radar 4

Figure 10 The distances between the netted radars and the target

5 10 15 20 25 30 35 40 45 50Time (s)

B-netted radar 1B-netted radar 2

B-netted radar 3B-netted radar 4

Tran

smitt

ing

pow

er (k

W)

0

05

1

15

2

25

3

Figure 11 The transmitting power of netted radars utilizing Bhat-tacharyya distance based LPI optimization in the tracking process

To obtain the optimal transmission power allocation ofradar network we utilize NPGA to solve (19) and (24) Letthe population size be 100 let the crossover probability be06 and let the mutation probability be 001 The populationevolves 10 generations Figure 11 shows the transmittingpower of netted radars utilizing Bhattacharyya distance basedLPI optimization in the tracking process while Figure 12depicts the J-divergence based case Before 119905 = 36 s nettedradars 2 3 and 4 are selected to track the target which arethe ones closest to the target while netted radar 1 is selectedinstead of radar 2 after 119905 = 36 s which is because netted radars1 2 and 3 have the best channel conditions in the networkFrom Figures 11 and 12 we can see that the transmissionpower allocation is determined by the locations of single

Tran

smitt

ing

pow

er (k

W)

5 10 15 20 25 30 35 40 45 500

05

1

15

2

25

3

Time (s)

J-netted radar 1J-netted radar 2

J-netted radar 3J-netted radar 4

Figure 12 The transmitting power of netted radars utilizing J-divergence based LPI optimization in the tracking process

03

04

05

06

07

08

09

1

11Sc

hleh

er in

terc

ept f

acto

r

J-radar networkB-radar network

Ordinary radar networkMonostatic radar

5 10 15 20 25 30 35 40 45 50Time (s)

Figure 13 The normalized Schleher intercept factor comparison inthe tracking process

target relative to the netted radars and their propagationlosses To be specific in the LPI optimization process moretransmitting power is allocated to the radar nodes that arelocated closer to the target this is due to the fact that theysuffer less propagation losses

Figure 13 demonstrates the advantage of our proposedoptimization problems based on information-theoretic cri-teria The traditional monostatic radar transmits 24KWconstantly while the ordinary radar network has a constantsum of transmitted power 24KW and each radar nodetransmits uniform power One can see that Schleher interceptfactor for radar network employing the information-theoreticcriteria based LPI optimization strategies is strictly smaller

International Journal of Antennas and Propagation 9

than that of traditional monostatic radar and ordinary radarnetwork across the whole region which further shows theLPI enhancement by exploiting our presented LPI optimiza-tion schemes in radar network to defend against passiveintercept receiver Moreover it can be seen in Figure 13 thatin terms of the same system constraints and fundamen-tal quantity Bhattacharyya distance based LPI optimiza-tion is asymptotically equivalent to the J-divergence basedcase

43 Discussion According to Figures 4ndash13 we can deducethe following conclusions for radar network architecture

(1) From Figures 4 to 7 we can observe that as thepredefined threshold of target detection increasesmore transmission power would be allocated forradar network to meet the detection performancewhile the intercept factor is increased subsequentlywhich is vulnerable in electronic warfare In otherwords there exists a tradeoff between LPI and targetdetection performance in radar network system andthe LPI performance would be sacrificed with targetdetection consideration

(2) In the numerical simulations we observe that theproposed optimization schemes (19) and (24) canbe employed to enhance the LPI performance forradar network Based on the netted radarsrsquo spatialdistributionwith respect to the target we can improvethe LPI performance by optimizing transmissionpower allocation among netted radars As indicatedin Figures 11 and 12 netted radars with better channelconditions are favorable over others In additionit can be observed that exploiting our proposedalgorithms can effectively improve the LPI perfor-mance of radar network to defend against interceptreceiver and Bhattacharyya distance based LPI opti-mization algorithm is asymptotically equivalent tothe J-divergence based case under the same systemconstraints and fundamental quantity

5 Conclusions

In this paper we investigated the problem of LPI designin radar network architecture where two LPI optimizationschemes based on information-theoretic criteria have beenproposed The NPGA was employed to tackle the highlynonconvex and nonlinear optimization problems Simula-tions have demonstrated that our proposed strategies areeffective and valuable to improve the LPI performance forradar network and it is indicated that these two optimizationproblems are asymptotically equivalent to each other underthe same system constraints Note that only single target wasconsidered in this paper Nevertheless it is convenient to beextended to multiple targets scenario and the conclusionsobtained in this study suggest that similar LPI benefits wouldbe obtained for the multiple targets case Future work willlook into the adaptive threshold design of target detectionperformance in radar network architectures

Appendices

A

Assume that every transmitter-receiver combination in thenetwork can be the same and 1198772net ≜ 119877

119905119894sdot 119877119903119895 where the radar

network SNR (1) can be written as follows

SNRnet = 119870rad

119873119905

sum

119894=1

119873119903

sum

119895=1

119875119905119894

1198774net (A1)

where

119870rad =1198661199051198661199031205901199051205822

(4120587)3119896119879119900119861119903119865119903119871 (A2)

Assume that the sum of the effective radiated power (ERP)from all the radars in the network is equivalent to that ofmonostatic radar that is

ERP =

119873119905

sum

119894=1

119875119905119894119866119905119894= 119875119905119866119905 (A3)

where 119875119905and 119866

119905are the transmitting power and transmitting

antenna gain of the monostatic radar respectively For 119866119905119894=

119866119905(for all 119894) we can rewrite (A1) as follows

SNRnet = 119870rad119873119905119875119905

1198774net (A4)

B

According to (15) we can derive the intercept factor for radarnetwork as

120572net =119877int119877net

= (119875119905sdot 1198702

int sdot SNRnet

119870rad sdot 119873119903 sdot SNR2int)

14

(B1)

and the intercept factor for conventional monostatic radar as

120572mon =119877int119877mon

= (119875119905sdot 1198702

int sdot SNRmon

119870rad sdot SNR2int)

14

(B2)

When SNRnet = SNRmon we can readily obtain therelationship between the intercept factors for radar network120572net and for the monostatic case 120572mon

120572net =120572mon

11987314

119903

(B3)

Furthermore for monostatic radar we can assume that

SNRmon = 119870rad119875119905

1198774mon= 119870rad

119875maxtot

1198774

119877MAX (B4)

where 119875maxtot is the maximal power of the monostatic rada-

r and 119877119877MAX is the corresponding maximal detection range

Then we can obtain

119877mon119877119877MAX

= (119875119905

119875maxtot

)

14

(B5)

10 International Journal of Antennas and Propagation

Similarly for intercept receiver we have the following

SNRint = 119870int119875119905

1198772

int= 119870int

119875maxtot

1198772

119868119872119860119883

(B6)

119877int119877119868MAX

= (119875119905

119875maxtot

)

12

(B7)

where 119877119868MAX is the intercept range when the transmitting

power of radar is 119875maxtot

Using (15) (B5) and (B7) we can obtain the followingexpression

120572mon120572maxmon

=119877int119877mon

119877119868MAX119877119877MAX

= (119875119905

119875maxtot

)

14

(B8)

where 120572maxmon is Schleher intercept factor corresponding to the

maximal transmitting power 119875maxtot

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their comments that help to improve the quality of thispaper The support provided by the National Natural ScienceFoundation ofChina (Grant no 61371170) Funding of JiangsuInnovation Program for Graduate Education (CXLX13 154)the Fundamental Research Funds for the Central Universi-ties the Priority Academic Program Development of JiangsuHigher Education Institutions (PADA) and Key Laboratoryof Radar Imaging and Microwave Photonics Ministry ofEducation Nanjing University of Aeronautics and Astronau-tics Nanjing China is gratefully acknowledged

References

[1] E P Phillip Detecting and Classifying Low Probability ofIntercept Radar Artech House Boston Mass USA 2009

[2] AM Haimovich R S Blum and L J Cimini Jr ldquoMIMO radarwith widely separated antennasrdquo IEEE Signal Processing Maga-zine vol 25 no 1 pp 116ndash129 2008

[3] A L Hume and C J Baker ldquoNetted radar sensingrdquo in Proceed-ings of the 2001 CIE International Conference on Radar pp 23ndash26 2001

[4] E Fishler A Haimovich R S Blum L J Cimini Jr D Chizhikand R A Valenzuela ldquoSpatial diversity in radars-Models anddetection performancerdquo IEEE Transactions on Signal Processingvol 54 no 3 pp 823ndash838 2006

[5] Y Yang and R S Blum ldquoMIMO radar waveform designbased onmutual information andminimummean-square errorestimationrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 43 no 1 pp 330ndash343 2007

[6] MM Naghsh M HMahmoud S P ShahramM Soltanalianand P Stoica ldquoUnified optimization framework for multi-staticradar code design using information-theoretic criteriardquo IEEETransactions on Signal Processing vol 61 no 21 pp 5401ndash54162013

[7] R Niu R S Blum P K Varshney and A L Drozd ldquoTar-get localization and tracking in noncoherent multiple-inputmultiple-output radar systemsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 48 no 2 pp 1466ndash1489 2012

[8] P Chavali and A Nehorai ldquoScheduling and power allocationin a cognitive radar network for multiple-target trackingrdquo IEEETransactions on Signal Processing vol 60 no 2 pp 715ndash7292012

[9] H Godrich A Petropulu and H V Poor ldquoOptimal powerallocation in distributed multiple-radar configurationsrdquo inProceeding of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 11) pp 2492ndash2495Prague Czech Republic May 2011

[10] H Godrich A P Petropulu and H V Poor ldquoPower allocationstrategies for target localization in distributed multiple-radararchitecturesrdquo IEEE Transactions on Signal Processing vol 59no 7 pp 3226ndash3240 2011

[11] H Godrich A Petropulu and H V Poort ldquoEstimation per-formance and resource savings tradeoffs in multiple radarssystemsrdquo in Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo12) pp 5205ndash5208 Kyoto Japan March 2012

[12] C G Shi J J Zhou F Wang and J Chen ldquoTarget threateninglevel based optimal power allocation for LPI radar networkrdquoin Proceedings of the 2nd International Symposium on Instru-mentation and Measurement Sensor Network and Automation(IMSNA rsquo13) pp 634ndash637 Toronto Canada December 2013

[13] X Song P Willett and S Zhou ldquoOptimal power allocationfor MIMO radars with heterogeneous propagation lossesrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 12) pp 2465ndash2468Kyoto Japan March 2012

[14] X F Song P Willett S Zhou and J Glaz ldquoMIMO radar detec-tion with heterogeneous propagation lossesrdquo in Proceedings ofthe IEEE Statistical Signal Processing Workshop (SSP rsquo12) pp776ndash779 August 2012

[15] D C Schleher ldquoLPI radar fact or fictionrdquo IEEE Aerospace andElectronic Systems Magazine vol 21 no 5 pp 3ndash6 2006

[16] D Lynch Jr Introduction to RF Stealth Sci Tech Publishing2004

[17] A G Stove A L Hume and C J Baker ldquoLow probability ofintercept radar strategiesrdquo IEE Proceedings Radar Sonar andNavigation vol 151 no 5 pp 249ndash260 2004

[18] D E Lawrence ldquoLow probability of intercept antenna arraybeamformingrdquo IEEE Transactions on Antennas and Propaga-tion vol 58 no 9 pp 2858ndash2865 2010

[19] T Kailath ldquoThe divergence and Bhattacharyya distance mea-sures in signal detectionrdquo IEEE Transaction on CommunicationTechnology vol 15 no 2 pp 52ndash60 1967

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

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International Journal of

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DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 3

where int |119909119895(119905)|2119889119905 = 1 120591

119894119895represents the time delay 119899

119895(119905)

denotes the noise at receiver 119895 and the Doppler effect isnegligible At the 119895th receiver the received signal is matchedfiltered by time response 119909lowast

119896(minus119905) and the output signal can be

expressed as follows

119910119895119896(119905) = int119910

119895(119905) sdot 119909

lowast

119896(120591 minus 119905) 119889120591

= 119901119895119896119892119895119896radic119875119905119896int119909119896(120591 minus 120591

119896119895) sdot 119909lowast

119896(120591 minus 119905) 119889120591 + 119899

119895119896(119905)

(4)

where 119899119895119896(119905) = int 119899

119895(120591)sdot119909lowast

119896(120591minus119905)119889120591 andint119909

119895(120591)sdot119909lowast

119896(120591+119905)119889120591 = 0

for 119896 = 119895The discrete time signal for the 119895th receiver can be

rewritten as follows

119903119895119896≜ 119910119895119896(120591119895119896) = 119901119895119896119892119895119896radic119875119905119896+ 120579119895119896 (5)

where 119903119896119895is the output of the matched filter at the receiver

119895 sampled at 120591119895119896 120579119895119896

= 119899119895119896(120591119895119896) and 120579

119895119896sim CN(0 119877

120579) As

mentioned before we assume that all the netted radars haveacquired and are tracking the target with their directionalradar beams and they transmit orthogonal waveforms whilereceiving and processing all these echoes that are reflectedfrom the target In this way we can obtain 120591

119895119896

23 Binary Hypothesis Test With all the received signals thetarget detection for radar network system leads to a binaryhypothesis testing problem

1198670 119903119894119895= 120579119894119895

1198671 119903119894119895= 119901119894119895119892119894119895radic119875119905119894+ 120579119894119895

(6)

where 1 le 119894 le 119873119905 1 le 119895 le 119873

119903 The likelihood ratio test can be

formulated as follows

1198670 119879 ≜

119872

prod

119894=1

119873

prod

119895=1

119891 (119903119894119895| 1198671)

119891 (119903119894119895| 1198670)

lt 120575

1198671 119879 ≜

119872

prod

119894=1

119873

prod

119895=1

119891 (119903119894119895| 1198671)

119891 (119903119894119895| 1198670)

gt 120575

(7)

As introduced in [14] the underlying detection problem canbe equivalently rewritten as follows

1198670 119903119894119895sim CN (0 119877

120579)

1198671 119903119894119895sim CN (0 119877

120579+ 1198751199051198941198771198921199012

119894119895)

(8)

Then we have the optimal detector as follows

1198670 119879 ≜

119873119905

sum

119894=1

119873119903

sum

119895=1

10038161003816100381610038161003816119903119894119895

10038161003816100381610038161003816

2 21198751199051198941199012

119894119895

119877120579+ 1198751199051198941198771198921199012

119894119895

lt 120575

1198671 119879 ≜

119873119905

sum

119894=1

119873119903

sum

119895=1

10038161003816100381610038161003816119903119894119895

10038161003816100381610038161003816

2 21198751199051198941199012

119894119895

119877120579+ 1198751199051198941198771198921199012

119894119895

gt 120575

(9)

where 120575 denotes the detection threshold

3 Problem Formulation

In this section we aim to obtain the optimal LPI performancefor radar network architecture by judiciously designing thetransmission power allocation among netted radars in thenetwork We first derive Schleher intercept factor for radarnetwork system and then formulate the LPI optimizationproblems based on information-theoretic criteria For apredefined threshold of target detection Schleher interceptfactor is minimized by optimizing transmission power allo-cation among netted radars It is indicated in [6] that theanalytical closed-form expression for ROC does not existAs such we resort to information-theoretic criteria namelyBhattacharyya distance and J-divergence In what followsthe corresponding LPI optimization problems associatedwith different information-theoretic criteria are cast under aunified framework and can be solved conveniently throughNPGA

31 Schleher Intercept Factor for Radar Network For radarnetwork it is supposed that all signals can be separatelydistinguished at every netted radar node Assuming thatevery transmitter-receiver combination in the network canbe the same and 119877

2

net ≜ 119877119905119894sdot 119877119903119895 in which case the radar

network SNR equation (1) can be rewritten as follows (seeAppendix A)

SNRnet = 119870rad119873119903119875119905

1198774net (10)

where

119870rad =1198661199051198661199031205901199051205822

(4120587)3119896119879119900119861119903119865119903119871 (11)

119875119905is the total transmitting power of radar network systemNote that when 119873

119905= 119873119903

= 1 we can obtain themonostatic case

SNRmon = 119870rad119875119905

1198774mon (12)

where 119877mon is the distance between the monostatic radar andthe target while for intercept receiver the SNR equation is

SNRint = 119870int119875119905

1198772

int (13)

4 International Journal of Antennas and Propagation

InterceptorTarget

Radar

Rint

Rrad

Figure 2 The geometry of radar target and interceptor

where

119870int =1198661199051198661198941205822

(4120587)2119896119879119900119861119894119865119894119871119894

(14)

SNRint is the SNR at the interceptor signal processor input1198661015840119905

is the gain of the radarrsquos transmitting antenna in the directionof the intercept receiver 119866

119894is the gain of the intercept

receiverrsquos antenna 119865119894is the interceptor noise factor 119861

119894is the

bandwidth of the interceptor 119877int is the range from radarnetwork to the intercept receiver and 119871

119894refers to the losses

from the radar antenna to the receiver For simplicity weassume that the intercept receiver is carried by the targetAs such the interceptor detects the radar emission from themain lobe that is 1198661015840

119905= 119866119905

Herein Schleher intercept factor is employed to evaluateLPI performance for radar network The definition of Schle-her intercept factor can be calculated as follows

120572 =119877int119877rad

(15)

where 119877rad is the detection range of radar and 119877int is theintercept range of intercept receiver as illustrated in Figure 2

Based on the definition of Schleher intercept factor if120572 gt 1 radar can be detected by the interceptor while if120572 le 1 radar can detect the target and the interceptor cannotdetect the radar Therefore radar can meet LPI performancewhen 120572 le 1 Moreover minimization of Schleher interceptfactor leads to better LPI performance for radar networkarchitecture

With the derivation of Schleher intercept factor inAppendix B it can be observed that for a predefined targetdetection performance the closer the distance between radarsystem and target is the less power the radar system needs totransmit on guarantee of target detection performance Forsimplicity the maximum intercept factor 120572max

mon is normalizedto be 1 when the monostatic radar transmits the maximalpower 119875max

tot and SNRnet = SNRmon Therefore when thetransmission power is 119875

119905 the intercept factor for radar

network system can be simplified as follows

120572net =120572mon

11987314

119903

= (119875119905

119875maxtot sdot 119873

119903

)

14

(16)

where 120572mon is the Schleher intercept factor for monostaticradar From (16) one can see that Schleher intercept factor120572net is reduced with the increase of the number of radarreceivers119873

119903and the decrease of the total transmission power

119875119905in the network system

32 Information-Theoretic Criteria Based LPI Optimization

321 Bhattacharyya Distance Based LPI OptimizationScheme It is introduced in [6] that Bhattacharyya distance119861(1199010 1199011) measures the distance between two probability

density functions (pdf) 1199010

and 1199011 The Bhattacharyya

distance provides an upper bound on the probability of falsealarm 119875fa and at the same time yields a lower bound on theprobability of detection 119875

119889

Consider two multivariate Gaussian distributions 1198750and

1198751 1198750sim CN(0 120590

0) and 119875

1sim CN(0 120590

1) the Bhattacharyya

distance 119861(1198750 1198751) can be obtained as [6]

119861 (1198750 1198751) = log

det [05 (1205900+ 1205901)]

radicdet (1205900) det (120590

1)

(17)

Let 119861[119891(119903 | 1198670) 119891(119903 | 119867

1)] represent Bhattacharyya distance

between 1198670and 119867

1 where 119891(119903 | 119867

0) and 119891(119903 | 119867

1) are

the pdfs of r under hypotheses 1198670and 119867

1 For the binary

hypothesis testing problem we have that

119861net ≜ 119861 [119891 (119903 | 1198670) 119891 (119903 | 119867

1)]

=

119873119905

sum

119894=1

119873119903

sum

119895=1

log(1 + 05120577

119894119895

radic1 + 120577119894119895

)

=

119873119905

sum

119894=1

119873119903

sum

119895=1

log[[

[

1 + 1198751199051198941198771198921199012

119894119895(2119877120579)minus1

radic1 + 1198751199051198941198771198921199012

119894119895(119877120579)minus1

]]

]

=

119873119905

sum

119894=1

119873119903

sum

119895=1

log[[[

[

1 + 119875119905119894119877119892119866119905119894119866119903119895(21198771205791198772

1199051198941198772

119903119895)minus1

radic1 + 119875119905119894119877119892119866119905119894119866119903119895(1198771205791198772

1199051198941198772

119903119895)minus1

]]]

]

(18)

Based on the discussion in [6] maximization of the Bhat-tacharyya distance minimizes the upper bound on 119875fa whileit maximizes the lower bound on 119875

119889 As expressed in (18)

the Bhattacharyya distance derived here can be applied toevaluate the target detection performance of radar networkas a function of different parameters such as the transmittingpower of each netted radar and the number of netted radarsin the network Intuitively the greater the Bhattacharyya dis-tance between the two distributions of the binary hypothesistesting problem the better the capability of radar networksystem to detect the target which would make the net-work system more vulnerable in modern electronic warfareTherefore the Bhattacharyya distance can provide guidanceto the problem of LPI optimization for radar networkarchitecture

Here we focus on the LPI optimization problem forradar network architecture where Schleher intercept factoris minimized by optimizing transmission power allocationamong netted radars in the network for a predetermined

International Journal of Antennas and Propagation 5

Bhattacharyya distance threshold such that the LPI perfor-mance is met on the guarantee of target detection perfor-mance Eventually the underlying LPI optimization problemcan be formulated as follows

min997888Pt

120572net

st 119861net ge 119861th

119873119905

sum

119894=1

119875119905119894le 119875

maxtot

0 le 119875119905119894le 119875

max119905119894

(forall119894)

(19)

where 997888Pt = [1198751199051 1198751199052 119875

119905119873119905

]119879 is the transmitting power of

radar network 119861th is the Bhattacharyya distance thresholdfor target detection 119875max

tot is the maximum total transmissionpower of radar network and 119875max

119905119894(for all 119894) is the maximum

transmission power of the corresponding netted radar node

322 J-Divergence Based LPI Optimization Scheme The J-divergence 119869(119901

0 1199011) is anothermetric tomeasure the distance

between two pdfs 1199010and 119901

1 It is defined as follows

119869 (1199010 1199011) ≜ 119863 (119901

0 1199011) + 119863 (119901

1 1199010) (20)

where 119863(sdot) is the Kullback-Leibler divergence It is shown in[19] that for any fixed value of 119875fa

119863[119891 (119903 | 1198670) 119891 (119903 | 119867

1)] = lim119873rarrinfin

[minus1

119873log (1 minus 119875

119889)]

(21)

and for any fixed value of 119875119889 we can obtain

119863[119891 (119903 | 1198671) 119891 (119903 | 119867

0)] = lim119873rarrinfin

[minus1

119873log (119875fa)] (22)

From (21) and (22) we can observe that for any fixed 119875fa themaximization of Kullback-Leibler divergence 119863[119891(119903 | 119867

0)

119891(119903 | 1198671)] results in an asymptotic maximization of 119875

119889

while for any fixed 119875119889the maximization of Kullback-Leibler

divergence119863[119891(119903 | 1198671) 119891(119903 | 119867

0)] results in an asymptotic

minimization of 119875faWith the derivation in [6] we have that

119869net ≜ 119869 [119891 (119903 | 1198670) 119891 (119903 | 119867

1)]

=

119873119905

sum

119894=1

119873119903

sum

119895=1

1205772

119894119895

1 + 120577119894119895

=

119873119905

sum

119894=1

119873119903

sum

119895=1

[1198751199051198941198771198921199012

119894119895(119877120579)minus1

]2

1 + 1198751199051198941198771198921199012

119894119895(119877120579)minus1

=

119873119905

sum

119894=1

119873119903

sum

119895=1

[119875119905119894119877119892119866119905119894119866119903119895(1198771205791198772

1199051198941198772

119903119895)minus1

]

2

1 + 119875119905119894119877119892119866119905119894119866119903119895(1198771205791198772

1199051198941198772

119903119895)minus1

(23)

Consequently the corresponding LPI optimization prob-lem can be expressed as follows

min997888119875119905

120572net

st 119869net ge 119869th

119873119905

sum

119894=1

119875119905119894le 119875

maxtot

0 le 119875119905119894le 119875

max119905119894

(forall119894)

(24)

where 119869th is the J-divergence threshold for target detection

33 The Unified Framework Based on NPGA In this subsec-tion we cast the LPI optimization problems based on variousinformation-theoretic criteria investigated earlier under aunified optimization framework Furthermore we formulatethe following general form of the optimization problems in(19) and (24)

min997888119875119905

120572net

st 120574net ge 120574th

119873119905

sum

119894=1

119875119905119894le 119875

maxtot

0 le 119875119905119894le 119875

max119905119894

(forall119894)

(25)

where 120574net isin 119861net 119869net and 120574th is the corresponding threshold

for target detectionIn this paper we utilize the nonlinear programming based

genetic algorithm (NPGA) to seek the optimal solutions tothe resulting nonconvex nonlinear and constrained problem(25)The NPGA has a good performance on the convergencespeed and it improves the searching performance of ordinarygenetic algorithm

The NPGA procedure is illustrated in Figure 3 wherethe population initialization module is utilized to initializethe population according to the resulting problem while thecalculating fitness value module is to calculate the fitnessvalues of individuals in the population Selection crossoverand mutation are employed to seek the optimal solutionwhere 119873 is a constant If the evolution is 119873rsquos multiples wecan use NP approach to accelerate the convergence speed

So far we have completed the derivation of Schle-her intercept factor for radar network architecture and

6 International Journal of Antennas and Propagation

Population initializationStart Calculate

fitness value Selection Crossover Mutation

Nonlinear YesYes

NoNo

Endmultiples

Isevolution Nrsquos the termination

conditions

Satisfy

Figure 3 Flow diagram of NPGA procedure

0 02 04 06 08 1 12 14 16 18 20

10

20

30

40

50

60

70

80

90

Schleher intercept factor

Bhat

tach

aryy

a dist

ance

2 times 2 radar network2 times 4 radar network4 times 2 radar network

4 times 4 radar network6 times 6 radar network

Figure 4 Bhattacharyya distance versus Schleher intercept factorfor different radar network architectures

the information-theoretic criteria based LPI optimizationschemes In what follows some numerical simulations areprovided to confirm the effectiveness of our presented LPIoptimization algorithms for radar network architecture

4 Numerical Simulations

In this section we provide several numerical simulations toexamine the performance of the proposed LPI optimizationalgorithms as (19) and (24) Throughout this section weassume that 119875max

tot = sum119873119905

119894=1119875119905119894= 24KW 119866

119905= 119866119903= 30 dB

119877120579= 10minus10 and 119877

119892= 1 The SNR is set to be 13 dB The

traditional monostatic radar can detect the target whose RCSis 005m2 in the distance 119877

119877MAX = 1061 km by transmittingthe maximum power 119875max

tot = 24KW where the interceptfactor is normalized to be 1 for simplicity

41 LPI Performance Analysis Figures 4 and 6 show theBhattacharyya distance and logarithmic J-divergence ver-sus Schleher intercept factor for different radar networkarchitectures respectively which are conducted 10

6 Monte

0 02 04 06 08 1 12 14 16 18 20

10

20

30

40

50

60

Schleher intercept factor

Bhat

tach

aryy

a dist

ance

4 times 4 radar network with Rg = 1

4 times 4 radar network with Rg = 5

4 times 4 radar network with Rg = 10

Figure 5 Bhattacharyya distance versus Schleher intercept factorfor different target scattering intensity with119873

119905= 119873119903= 4

Carlo trials We can observe in Figures 4 and 6 that asSchleher intercept factor increases from 120572net = 0 to 120572net =2 the achievable Bhattacharyya distance and logarithmic J-divergence are increased This is due to the fact that as theintercept factor increases more transmission power wouldbe allocated which makes the achievable Bhattacharyya dis-tance and logarithmic J-divergence increase correspondinglyas theoretically proved in (18) and (23) Furthermore it canbe seen from Figures 4 and 6 that with the same target detec-tion threshold Schleher intercept factor can be significantlyreduced as the number of transmitters and receivers in thenetwork system increases Therefore increasing the numberof netted radars can effectively improve the LPI performancefor radar networkThis confirms the LPI benefits of the radarnetwork architecture with more netted radars

As shown in Figures 5 and 7 we illustrate the Bhat-tacharyya distance and logarithmic J-divergence versusSchleher intercept factor for different target scattering inten-sity with 119873

119905= 119873119903= 4 respectively It is depicted that

as the target scattering intensity increases from 119877119892

= 1

to 119877119892

= 10 the achievable Bhattacharyya distance andlogarithmic J-divergence are significantly increased This is

International Journal of Antennas and Propagation 7

0 02 04 06 08 1 12 14 16 18 2

0

2

4

6

8

10

Schleher intercept factor

log(

J-di

verg

ence

)

minus10

minus8

minus6

minus4

minus2

2 times 2 radar network2 times 4 radar network4 times 2 radar network

4 times 4 radar network6 times 6 radar network

Figure 6 Logarithmic J-divergence versus Schleher intercept factorfor different radar network architectures

0 02 04 06 08 1 12 14 16 18 2

0

5

10

15

Schleher intercept factor

log(

J-di

verg

ence

)

minus10

minus5

4 times 4 radar network with Rg = 1

4 times 4 radar network with Rg = 5

4 times 4 radar network with Rg = 10

Figure 7 Logarithmic J-divergence versus Schleher intercept factorfor different target scattering intensity with119873

119905= 119873119903= 4

because the radar network system can detect the target withlarge scattering intensity easily with high 119875

119889and low 119875fa

42 Target Tracking with LPI Optimization In this subsec-tion we consider a 4 times 4 radar network system (119873

119905=

119873119903= 4) in the simulation and it is widely deployed in

modern battlefield The target detection threshold 120574th can

be calculated in the condition that the transmission powerof each radar is 6KW in the distance 150 km between theradar network and the target which is the minimum value ofthe basic performance requirement for target detection As

180 185 190 195 200 205 210 215 220180

185

190

195

200

205

210

215

220

Netted radar

Radar1Radar2

Radar3 Radar4

X (km)

Y(k

m)

Figure 8 The radar network system configuration in two dimen-sions

85 90 95 100 105 110 115 120 125 130100

120

140

160

180

200

220

240

260

280

X (km)

Y (k

m)

Real trajectoryTrack with PF

t = 1 s

t = 50 s

Figure 9 Target tracking scenario

mentioned before it is supposed that the intercept receiveris carried by the target It is depicted in Figure 8 that thenetted radars in the network are spatially distributed in thesurveillance area at the initial time 119905 = 0

We track a single target by utilizing particle filtering (PF)method where 5000 particles are used to estimate the targetstate Figure 9 shows one realization of the target trajectoryfor 50 s and the tracking interval is chosen to be 1 s Withthe radar network configuration in Figure 8 and the targettracking scenario in Figure 9 we can obtain the distanceschanging curve between the netted radars and the target inthe tracking process as depicted in Figure 10 Without loss ofgenerality we set1198771198861198891198861199031 as the distributed data fusion centerand capitalize the weighted average approach to obtain theestimated target state

8 International Journal of Antennas and Propagation

5 10 15 20 25 30 35 40 45 5080

85

90

95

100

105

110

115

120

125

130

Time (s)

Dist

ance

(km

)

Netted radar 1Netted radar 2

Netted radar 3Netted radar 4

Figure 10 The distances between the netted radars and the target

5 10 15 20 25 30 35 40 45 50Time (s)

B-netted radar 1B-netted radar 2

B-netted radar 3B-netted radar 4

Tran

smitt

ing

pow

er (k

W)

0

05

1

15

2

25

3

Figure 11 The transmitting power of netted radars utilizing Bhat-tacharyya distance based LPI optimization in the tracking process

To obtain the optimal transmission power allocation ofradar network we utilize NPGA to solve (19) and (24) Letthe population size be 100 let the crossover probability be06 and let the mutation probability be 001 The populationevolves 10 generations Figure 11 shows the transmittingpower of netted radars utilizing Bhattacharyya distance basedLPI optimization in the tracking process while Figure 12depicts the J-divergence based case Before 119905 = 36 s nettedradars 2 3 and 4 are selected to track the target which arethe ones closest to the target while netted radar 1 is selectedinstead of radar 2 after 119905 = 36 s which is because netted radars1 2 and 3 have the best channel conditions in the networkFrom Figures 11 and 12 we can see that the transmissionpower allocation is determined by the locations of single

Tran

smitt

ing

pow

er (k

W)

5 10 15 20 25 30 35 40 45 500

05

1

15

2

25

3

Time (s)

J-netted radar 1J-netted radar 2

J-netted radar 3J-netted radar 4

Figure 12 The transmitting power of netted radars utilizing J-divergence based LPI optimization in the tracking process

03

04

05

06

07

08

09

1

11Sc

hleh

er in

terc

ept f

acto

r

J-radar networkB-radar network

Ordinary radar networkMonostatic radar

5 10 15 20 25 30 35 40 45 50Time (s)

Figure 13 The normalized Schleher intercept factor comparison inthe tracking process

target relative to the netted radars and their propagationlosses To be specific in the LPI optimization process moretransmitting power is allocated to the radar nodes that arelocated closer to the target this is due to the fact that theysuffer less propagation losses

Figure 13 demonstrates the advantage of our proposedoptimization problems based on information-theoretic cri-teria The traditional monostatic radar transmits 24KWconstantly while the ordinary radar network has a constantsum of transmitted power 24KW and each radar nodetransmits uniform power One can see that Schleher interceptfactor for radar network employing the information-theoreticcriteria based LPI optimization strategies is strictly smaller

International Journal of Antennas and Propagation 9

than that of traditional monostatic radar and ordinary radarnetwork across the whole region which further shows theLPI enhancement by exploiting our presented LPI optimiza-tion schemes in radar network to defend against passiveintercept receiver Moreover it can be seen in Figure 13 thatin terms of the same system constraints and fundamen-tal quantity Bhattacharyya distance based LPI optimiza-tion is asymptotically equivalent to the J-divergence basedcase

43 Discussion According to Figures 4ndash13 we can deducethe following conclusions for radar network architecture

(1) From Figures 4 to 7 we can observe that as thepredefined threshold of target detection increasesmore transmission power would be allocated forradar network to meet the detection performancewhile the intercept factor is increased subsequentlywhich is vulnerable in electronic warfare In otherwords there exists a tradeoff between LPI and targetdetection performance in radar network system andthe LPI performance would be sacrificed with targetdetection consideration

(2) In the numerical simulations we observe that theproposed optimization schemes (19) and (24) canbe employed to enhance the LPI performance forradar network Based on the netted radarsrsquo spatialdistributionwith respect to the target we can improvethe LPI performance by optimizing transmissionpower allocation among netted radars As indicatedin Figures 11 and 12 netted radars with better channelconditions are favorable over others In additionit can be observed that exploiting our proposedalgorithms can effectively improve the LPI perfor-mance of radar network to defend against interceptreceiver and Bhattacharyya distance based LPI opti-mization algorithm is asymptotically equivalent tothe J-divergence based case under the same systemconstraints and fundamental quantity

5 Conclusions

In this paper we investigated the problem of LPI designin radar network architecture where two LPI optimizationschemes based on information-theoretic criteria have beenproposed The NPGA was employed to tackle the highlynonconvex and nonlinear optimization problems Simula-tions have demonstrated that our proposed strategies areeffective and valuable to improve the LPI performance forradar network and it is indicated that these two optimizationproblems are asymptotically equivalent to each other underthe same system constraints Note that only single target wasconsidered in this paper Nevertheless it is convenient to beextended to multiple targets scenario and the conclusionsobtained in this study suggest that similar LPI benefits wouldbe obtained for the multiple targets case Future work willlook into the adaptive threshold design of target detectionperformance in radar network architectures

Appendices

A

Assume that every transmitter-receiver combination in thenetwork can be the same and 1198772net ≜ 119877

119905119894sdot 119877119903119895 where the radar

network SNR (1) can be written as follows

SNRnet = 119870rad

119873119905

sum

119894=1

119873119903

sum

119895=1

119875119905119894

1198774net (A1)

where

119870rad =1198661199051198661199031205901199051205822

(4120587)3119896119879119900119861119903119865119903119871 (A2)

Assume that the sum of the effective radiated power (ERP)from all the radars in the network is equivalent to that ofmonostatic radar that is

ERP =

119873119905

sum

119894=1

119875119905119894119866119905119894= 119875119905119866119905 (A3)

where 119875119905and 119866

119905are the transmitting power and transmitting

antenna gain of the monostatic radar respectively For 119866119905119894=

119866119905(for all 119894) we can rewrite (A1) as follows

SNRnet = 119870rad119873119905119875119905

1198774net (A4)

B

According to (15) we can derive the intercept factor for radarnetwork as

120572net =119877int119877net

= (119875119905sdot 1198702

int sdot SNRnet

119870rad sdot 119873119903 sdot SNR2int)

14

(B1)

and the intercept factor for conventional monostatic radar as

120572mon =119877int119877mon

= (119875119905sdot 1198702

int sdot SNRmon

119870rad sdot SNR2int)

14

(B2)

When SNRnet = SNRmon we can readily obtain therelationship between the intercept factors for radar network120572net and for the monostatic case 120572mon

120572net =120572mon

11987314

119903

(B3)

Furthermore for monostatic radar we can assume that

SNRmon = 119870rad119875119905

1198774mon= 119870rad

119875maxtot

1198774

119877MAX (B4)

where 119875maxtot is the maximal power of the monostatic rada-

r and 119877119877MAX is the corresponding maximal detection range

Then we can obtain

119877mon119877119877MAX

= (119875119905

119875maxtot

)

14

(B5)

10 International Journal of Antennas and Propagation

Similarly for intercept receiver we have the following

SNRint = 119870int119875119905

1198772

int= 119870int

119875maxtot

1198772

119868119872119860119883

(B6)

119877int119877119868MAX

= (119875119905

119875maxtot

)

12

(B7)

where 119877119868MAX is the intercept range when the transmitting

power of radar is 119875maxtot

Using (15) (B5) and (B7) we can obtain the followingexpression

120572mon120572maxmon

=119877int119877mon

119877119868MAX119877119877MAX

= (119875119905

119875maxtot

)

14

(B8)

where 120572maxmon is Schleher intercept factor corresponding to the

maximal transmitting power 119875maxtot

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their comments that help to improve the quality of thispaper The support provided by the National Natural ScienceFoundation ofChina (Grant no 61371170) Funding of JiangsuInnovation Program for Graduate Education (CXLX13 154)the Fundamental Research Funds for the Central Universi-ties the Priority Academic Program Development of JiangsuHigher Education Institutions (PADA) and Key Laboratoryof Radar Imaging and Microwave Photonics Ministry ofEducation Nanjing University of Aeronautics and Astronau-tics Nanjing China is gratefully acknowledged

References

[1] E P Phillip Detecting and Classifying Low Probability ofIntercept Radar Artech House Boston Mass USA 2009

[2] AM Haimovich R S Blum and L J Cimini Jr ldquoMIMO radarwith widely separated antennasrdquo IEEE Signal Processing Maga-zine vol 25 no 1 pp 116ndash129 2008

[3] A L Hume and C J Baker ldquoNetted radar sensingrdquo in Proceed-ings of the 2001 CIE International Conference on Radar pp 23ndash26 2001

[4] E Fishler A Haimovich R S Blum L J Cimini Jr D Chizhikand R A Valenzuela ldquoSpatial diversity in radars-Models anddetection performancerdquo IEEE Transactions on Signal Processingvol 54 no 3 pp 823ndash838 2006

[5] Y Yang and R S Blum ldquoMIMO radar waveform designbased onmutual information andminimummean-square errorestimationrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 43 no 1 pp 330ndash343 2007

[6] MM Naghsh M HMahmoud S P ShahramM Soltanalianand P Stoica ldquoUnified optimization framework for multi-staticradar code design using information-theoretic criteriardquo IEEETransactions on Signal Processing vol 61 no 21 pp 5401ndash54162013

[7] R Niu R S Blum P K Varshney and A L Drozd ldquoTar-get localization and tracking in noncoherent multiple-inputmultiple-output radar systemsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 48 no 2 pp 1466ndash1489 2012

[8] P Chavali and A Nehorai ldquoScheduling and power allocationin a cognitive radar network for multiple-target trackingrdquo IEEETransactions on Signal Processing vol 60 no 2 pp 715ndash7292012

[9] H Godrich A Petropulu and H V Poor ldquoOptimal powerallocation in distributed multiple-radar configurationsrdquo inProceeding of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 11) pp 2492ndash2495Prague Czech Republic May 2011

[10] H Godrich A P Petropulu and H V Poor ldquoPower allocationstrategies for target localization in distributed multiple-radararchitecturesrdquo IEEE Transactions on Signal Processing vol 59no 7 pp 3226ndash3240 2011

[11] H Godrich A Petropulu and H V Poort ldquoEstimation per-formance and resource savings tradeoffs in multiple radarssystemsrdquo in Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo12) pp 5205ndash5208 Kyoto Japan March 2012

[12] C G Shi J J Zhou F Wang and J Chen ldquoTarget threateninglevel based optimal power allocation for LPI radar networkrdquoin Proceedings of the 2nd International Symposium on Instru-mentation and Measurement Sensor Network and Automation(IMSNA rsquo13) pp 634ndash637 Toronto Canada December 2013

[13] X Song P Willett and S Zhou ldquoOptimal power allocationfor MIMO radars with heterogeneous propagation lossesrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 12) pp 2465ndash2468Kyoto Japan March 2012

[14] X F Song P Willett S Zhou and J Glaz ldquoMIMO radar detec-tion with heterogeneous propagation lossesrdquo in Proceedings ofthe IEEE Statistical Signal Processing Workshop (SSP rsquo12) pp776ndash779 August 2012

[15] D C Schleher ldquoLPI radar fact or fictionrdquo IEEE Aerospace andElectronic Systems Magazine vol 21 no 5 pp 3ndash6 2006

[16] D Lynch Jr Introduction to RF Stealth Sci Tech Publishing2004

[17] A G Stove A L Hume and C J Baker ldquoLow probability ofintercept radar strategiesrdquo IEE Proceedings Radar Sonar andNavigation vol 151 no 5 pp 249ndash260 2004

[18] D E Lawrence ldquoLow probability of intercept antenna arraybeamformingrdquo IEEE Transactions on Antennas and Propaga-tion vol 58 no 9 pp 2858ndash2865 2010

[19] T Kailath ldquoThe divergence and Bhattacharyya distance mea-sures in signal detectionrdquo IEEE Transaction on CommunicationTechnology vol 15 no 2 pp 52ndash60 1967

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Submit your manuscripts athttpwwwhindawicom

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Shock and Vibration

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Electrical and Computer Engineering

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Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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DistributedSensor Networks

International Journal of

4 International Journal of Antennas and Propagation

InterceptorTarget

Radar

Rint

Rrad

Figure 2 The geometry of radar target and interceptor

where

119870int =1198661199051198661198941205822

(4120587)2119896119879119900119861119894119865119894119871119894

(14)

SNRint is the SNR at the interceptor signal processor input1198661015840119905

is the gain of the radarrsquos transmitting antenna in the directionof the intercept receiver 119866

119894is the gain of the intercept

receiverrsquos antenna 119865119894is the interceptor noise factor 119861

119894is the

bandwidth of the interceptor 119877int is the range from radarnetwork to the intercept receiver and 119871

119894refers to the losses

from the radar antenna to the receiver For simplicity weassume that the intercept receiver is carried by the targetAs such the interceptor detects the radar emission from themain lobe that is 1198661015840

119905= 119866119905

Herein Schleher intercept factor is employed to evaluateLPI performance for radar network The definition of Schle-her intercept factor can be calculated as follows

120572 =119877int119877rad

(15)

where 119877rad is the detection range of radar and 119877int is theintercept range of intercept receiver as illustrated in Figure 2

Based on the definition of Schleher intercept factor if120572 gt 1 radar can be detected by the interceptor while if120572 le 1 radar can detect the target and the interceptor cannotdetect the radar Therefore radar can meet LPI performancewhen 120572 le 1 Moreover minimization of Schleher interceptfactor leads to better LPI performance for radar networkarchitecture

With the derivation of Schleher intercept factor inAppendix B it can be observed that for a predefined targetdetection performance the closer the distance between radarsystem and target is the less power the radar system needs totransmit on guarantee of target detection performance Forsimplicity the maximum intercept factor 120572max

mon is normalizedto be 1 when the monostatic radar transmits the maximalpower 119875max

tot and SNRnet = SNRmon Therefore when thetransmission power is 119875

119905 the intercept factor for radar

network system can be simplified as follows

120572net =120572mon

11987314

119903

= (119875119905

119875maxtot sdot 119873

119903

)

14

(16)

where 120572mon is the Schleher intercept factor for monostaticradar From (16) one can see that Schleher intercept factor120572net is reduced with the increase of the number of radarreceivers119873

119903and the decrease of the total transmission power

119875119905in the network system

32 Information-Theoretic Criteria Based LPI Optimization

321 Bhattacharyya Distance Based LPI OptimizationScheme It is introduced in [6] that Bhattacharyya distance119861(1199010 1199011) measures the distance between two probability

density functions (pdf) 1199010

and 1199011 The Bhattacharyya

distance provides an upper bound on the probability of falsealarm 119875fa and at the same time yields a lower bound on theprobability of detection 119875

119889

Consider two multivariate Gaussian distributions 1198750and

1198751 1198750sim CN(0 120590

0) and 119875

1sim CN(0 120590

1) the Bhattacharyya

distance 119861(1198750 1198751) can be obtained as [6]

119861 (1198750 1198751) = log

det [05 (1205900+ 1205901)]

radicdet (1205900) det (120590

1)

(17)

Let 119861[119891(119903 | 1198670) 119891(119903 | 119867

1)] represent Bhattacharyya distance

between 1198670and 119867

1 where 119891(119903 | 119867

0) and 119891(119903 | 119867

1) are

the pdfs of r under hypotheses 1198670and 119867

1 For the binary

hypothesis testing problem we have that

119861net ≜ 119861 [119891 (119903 | 1198670) 119891 (119903 | 119867

1)]

=

119873119905

sum

119894=1

119873119903

sum

119895=1

log(1 + 05120577

119894119895

radic1 + 120577119894119895

)

=

119873119905

sum

119894=1

119873119903

sum

119895=1

log[[

[

1 + 1198751199051198941198771198921199012

119894119895(2119877120579)minus1

radic1 + 1198751199051198941198771198921199012

119894119895(119877120579)minus1

]]

]

=

119873119905

sum

119894=1

119873119903

sum

119895=1

log[[[

[

1 + 119875119905119894119877119892119866119905119894119866119903119895(21198771205791198772

1199051198941198772

119903119895)minus1

radic1 + 119875119905119894119877119892119866119905119894119866119903119895(1198771205791198772

1199051198941198772

119903119895)minus1

]]]

]

(18)

Based on the discussion in [6] maximization of the Bhat-tacharyya distance minimizes the upper bound on 119875fa whileit maximizes the lower bound on 119875

119889 As expressed in (18)

the Bhattacharyya distance derived here can be applied toevaluate the target detection performance of radar networkas a function of different parameters such as the transmittingpower of each netted radar and the number of netted radarsin the network Intuitively the greater the Bhattacharyya dis-tance between the two distributions of the binary hypothesistesting problem the better the capability of radar networksystem to detect the target which would make the net-work system more vulnerable in modern electronic warfareTherefore the Bhattacharyya distance can provide guidanceto the problem of LPI optimization for radar networkarchitecture

Here we focus on the LPI optimization problem forradar network architecture where Schleher intercept factoris minimized by optimizing transmission power allocationamong netted radars in the network for a predetermined

International Journal of Antennas and Propagation 5

Bhattacharyya distance threshold such that the LPI perfor-mance is met on the guarantee of target detection perfor-mance Eventually the underlying LPI optimization problemcan be formulated as follows

min997888Pt

120572net

st 119861net ge 119861th

119873119905

sum

119894=1

119875119905119894le 119875

maxtot

0 le 119875119905119894le 119875

max119905119894

(forall119894)

(19)

where 997888Pt = [1198751199051 1198751199052 119875

119905119873119905

]119879 is the transmitting power of

radar network 119861th is the Bhattacharyya distance thresholdfor target detection 119875max

tot is the maximum total transmissionpower of radar network and 119875max

119905119894(for all 119894) is the maximum

transmission power of the corresponding netted radar node

322 J-Divergence Based LPI Optimization Scheme The J-divergence 119869(119901

0 1199011) is anothermetric tomeasure the distance

between two pdfs 1199010and 119901

1 It is defined as follows

119869 (1199010 1199011) ≜ 119863 (119901

0 1199011) + 119863 (119901

1 1199010) (20)

where 119863(sdot) is the Kullback-Leibler divergence It is shown in[19] that for any fixed value of 119875fa

119863[119891 (119903 | 1198670) 119891 (119903 | 119867

1)] = lim119873rarrinfin

[minus1

119873log (1 minus 119875

119889)]

(21)

and for any fixed value of 119875119889 we can obtain

119863[119891 (119903 | 1198671) 119891 (119903 | 119867

0)] = lim119873rarrinfin

[minus1

119873log (119875fa)] (22)

From (21) and (22) we can observe that for any fixed 119875fa themaximization of Kullback-Leibler divergence 119863[119891(119903 | 119867

0)

119891(119903 | 1198671)] results in an asymptotic maximization of 119875

119889

while for any fixed 119875119889the maximization of Kullback-Leibler

divergence119863[119891(119903 | 1198671) 119891(119903 | 119867

0)] results in an asymptotic

minimization of 119875faWith the derivation in [6] we have that

119869net ≜ 119869 [119891 (119903 | 1198670) 119891 (119903 | 119867

1)]

=

119873119905

sum

119894=1

119873119903

sum

119895=1

1205772

119894119895

1 + 120577119894119895

=

119873119905

sum

119894=1

119873119903

sum

119895=1

[1198751199051198941198771198921199012

119894119895(119877120579)minus1

]2

1 + 1198751199051198941198771198921199012

119894119895(119877120579)minus1

=

119873119905

sum

119894=1

119873119903

sum

119895=1

[119875119905119894119877119892119866119905119894119866119903119895(1198771205791198772

1199051198941198772

119903119895)minus1

]

2

1 + 119875119905119894119877119892119866119905119894119866119903119895(1198771205791198772

1199051198941198772

119903119895)minus1

(23)

Consequently the corresponding LPI optimization prob-lem can be expressed as follows

min997888119875119905

120572net

st 119869net ge 119869th

119873119905

sum

119894=1

119875119905119894le 119875

maxtot

0 le 119875119905119894le 119875

max119905119894

(forall119894)

(24)

where 119869th is the J-divergence threshold for target detection

33 The Unified Framework Based on NPGA In this subsec-tion we cast the LPI optimization problems based on variousinformation-theoretic criteria investigated earlier under aunified optimization framework Furthermore we formulatethe following general form of the optimization problems in(19) and (24)

min997888119875119905

120572net

st 120574net ge 120574th

119873119905

sum

119894=1

119875119905119894le 119875

maxtot

0 le 119875119905119894le 119875

max119905119894

(forall119894)

(25)

where 120574net isin 119861net 119869net and 120574th is the corresponding threshold

for target detectionIn this paper we utilize the nonlinear programming based

genetic algorithm (NPGA) to seek the optimal solutions tothe resulting nonconvex nonlinear and constrained problem(25)The NPGA has a good performance on the convergencespeed and it improves the searching performance of ordinarygenetic algorithm

The NPGA procedure is illustrated in Figure 3 wherethe population initialization module is utilized to initializethe population according to the resulting problem while thecalculating fitness value module is to calculate the fitnessvalues of individuals in the population Selection crossoverand mutation are employed to seek the optimal solutionwhere 119873 is a constant If the evolution is 119873rsquos multiples wecan use NP approach to accelerate the convergence speed

So far we have completed the derivation of Schle-her intercept factor for radar network architecture and

6 International Journal of Antennas and Propagation

Population initializationStart Calculate

fitness value Selection Crossover Mutation

Nonlinear YesYes

NoNo

Endmultiples

Isevolution Nrsquos the termination

conditions

Satisfy

Figure 3 Flow diagram of NPGA procedure

0 02 04 06 08 1 12 14 16 18 20

10

20

30

40

50

60

70

80

90

Schleher intercept factor

Bhat

tach

aryy

a dist

ance

2 times 2 radar network2 times 4 radar network4 times 2 radar network

4 times 4 radar network6 times 6 radar network

Figure 4 Bhattacharyya distance versus Schleher intercept factorfor different radar network architectures

the information-theoretic criteria based LPI optimizationschemes In what follows some numerical simulations areprovided to confirm the effectiveness of our presented LPIoptimization algorithms for radar network architecture

4 Numerical Simulations

In this section we provide several numerical simulations toexamine the performance of the proposed LPI optimizationalgorithms as (19) and (24) Throughout this section weassume that 119875max

tot = sum119873119905

119894=1119875119905119894= 24KW 119866

119905= 119866119903= 30 dB

119877120579= 10minus10 and 119877

119892= 1 The SNR is set to be 13 dB The

traditional monostatic radar can detect the target whose RCSis 005m2 in the distance 119877

119877MAX = 1061 km by transmittingthe maximum power 119875max

tot = 24KW where the interceptfactor is normalized to be 1 for simplicity

41 LPI Performance Analysis Figures 4 and 6 show theBhattacharyya distance and logarithmic J-divergence ver-sus Schleher intercept factor for different radar networkarchitectures respectively which are conducted 10

6 Monte

0 02 04 06 08 1 12 14 16 18 20

10

20

30

40

50

60

Schleher intercept factor

Bhat

tach

aryy

a dist

ance

4 times 4 radar network with Rg = 1

4 times 4 radar network with Rg = 5

4 times 4 radar network with Rg = 10

Figure 5 Bhattacharyya distance versus Schleher intercept factorfor different target scattering intensity with119873

119905= 119873119903= 4

Carlo trials We can observe in Figures 4 and 6 that asSchleher intercept factor increases from 120572net = 0 to 120572net =2 the achievable Bhattacharyya distance and logarithmic J-divergence are increased This is due to the fact that as theintercept factor increases more transmission power wouldbe allocated which makes the achievable Bhattacharyya dis-tance and logarithmic J-divergence increase correspondinglyas theoretically proved in (18) and (23) Furthermore it canbe seen from Figures 4 and 6 that with the same target detec-tion threshold Schleher intercept factor can be significantlyreduced as the number of transmitters and receivers in thenetwork system increases Therefore increasing the numberof netted radars can effectively improve the LPI performancefor radar networkThis confirms the LPI benefits of the radarnetwork architecture with more netted radars

As shown in Figures 5 and 7 we illustrate the Bhat-tacharyya distance and logarithmic J-divergence versusSchleher intercept factor for different target scattering inten-sity with 119873

119905= 119873119903= 4 respectively It is depicted that

as the target scattering intensity increases from 119877119892

= 1

to 119877119892

= 10 the achievable Bhattacharyya distance andlogarithmic J-divergence are significantly increased This is

International Journal of Antennas and Propagation 7

0 02 04 06 08 1 12 14 16 18 2

0

2

4

6

8

10

Schleher intercept factor

log(

J-di

verg

ence

)

minus10

minus8

minus6

minus4

minus2

2 times 2 radar network2 times 4 radar network4 times 2 radar network

4 times 4 radar network6 times 6 radar network

Figure 6 Logarithmic J-divergence versus Schleher intercept factorfor different radar network architectures

0 02 04 06 08 1 12 14 16 18 2

0

5

10

15

Schleher intercept factor

log(

J-di

verg

ence

)

minus10

minus5

4 times 4 radar network with Rg = 1

4 times 4 radar network with Rg = 5

4 times 4 radar network with Rg = 10

Figure 7 Logarithmic J-divergence versus Schleher intercept factorfor different target scattering intensity with119873

119905= 119873119903= 4

because the radar network system can detect the target withlarge scattering intensity easily with high 119875

119889and low 119875fa

42 Target Tracking with LPI Optimization In this subsec-tion we consider a 4 times 4 radar network system (119873

119905=

119873119903= 4) in the simulation and it is widely deployed in

modern battlefield The target detection threshold 120574th can

be calculated in the condition that the transmission powerof each radar is 6KW in the distance 150 km between theradar network and the target which is the minimum value ofthe basic performance requirement for target detection As

180 185 190 195 200 205 210 215 220180

185

190

195

200

205

210

215

220

Netted radar

Radar1Radar2

Radar3 Radar4

X (km)

Y(k

m)

Figure 8 The radar network system configuration in two dimen-sions

85 90 95 100 105 110 115 120 125 130100

120

140

160

180

200

220

240

260

280

X (km)

Y (k

m)

Real trajectoryTrack with PF

t = 1 s

t = 50 s

Figure 9 Target tracking scenario

mentioned before it is supposed that the intercept receiveris carried by the target It is depicted in Figure 8 that thenetted radars in the network are spatially distributed in thesurveillance area at the initial time 119905 = 0

We track a single target by utilizing particle filtering (PF)method where 5000 particles are used to estimate the targetstate Figure 9 shows one realization of the target trajectoryfor 50 s and the tracking interval is chosen to be 1 s Withthe radar network configuration in Figure 8 and the targettracking scenario in Figure 9 we can obtain the distanceschanging curve between the netted radars and the target inthe tracking process as depicted in Figure 10 Without loss ofgenerality we set1198771198861198891198861199031 as the distributed data fusion centerand capitalize the weighted average approach to obtain theestimated target state

8 International Journal of Antennas and Propagation

5 10 15 20 25 30 35 40 45 5080

85

90

95

100

105

110

115

120

125

130

Time (s)

Dist

ance

(km

)

Netted radar 1Netted radar 2

Netted radar 3Netted radar 4

Figure 10 The distances between the netted radars and the target

5 10 15 20 25 30 35 40 45 50Time (s)

B-netted radar 1B-netted radar 2

B-netted radar 3B-netted radar 4

Tran

smitt

ing

pow

er (k

W)

0

05

1

15

2

25

3

Figure 11 The transmitting power of netted radars utilizing Bhat-tacharyya distance based LPI optimization in the tracking process

To obtain the optimal transmission power allocation ofradar network we utilize NPGA to solve (19) and (24) Letthe population size be 100 let the crossover probability be06 and let the mutation probability be 001 The populationevolves 10 generations Figure 11 shows the transmittingpower of netted radars utilizing Bhattacharyya distance basedLPI optimization in the tracking process while Figure 12depicts the J-divergence based case Before 119905 = 36 s nettedradars 2 3 and 4 are selected to track the target which arethe ones closest to the target while netted radar 1 is selectedinstead of radar 2 after 119905 = 36 s which is because netted radars1 2 and 3 have the best channel conditions in the networkFrom Figures 11 and 12 we can see that the transmissionpower allocation is determined by the locations of single

Tran

smitt

ing

pow

er (k

W)

5 10 15 20 25 30 35 40 45 500

05

1

15

2

25

3

Time (s)

J-netted radar 1J-netted radar 2

J-netted radar 3J-netted radar 4

Figure 12 The transmitting power of netted radars utilizing J-divergence based LPI optimization in the tracking process

03

04

05

06

07

08

09

1

11Sc

hleh

er in

terc

ept f

acto

r

J-radar networkB-radar network

Ordinary radar networkMonostatic radar

5 10 15 20 25 30 35 40 45 50Time (s)

Figure 13 The normalized Schleher intercept factor comparison inthe tracking process

target relative to the netted radars and their propagationlosses To be specific in the LPI optimization process moretransmitting power is allocated to the radar nodes that arelocated closer to the target this is due to the fact that theysuffer less propagation losses

Figure 13 demonstrates the advantage of our proposedoptimization problems based on information-theoretic cri-teria The traditional monostatic radar transmits 24KWconstantly while the ordinary radar network has a constantsum of transmitted power 24KW and each radar nodetransmits uniform power One can see that Schleher interceptfactor for radar network employing the information-theoreticcriteria based LPI optimization strategies is strictly smaller

International Journal of Antennas and Propagation 9

than that of traditional monostatic radar and ordinary radarnetwork across the whole region which further shows theLPI enhancement by exploiting our presented LPI optimiza-tion schemes in radar network to defend against passiveintercept receiver Moreover it can be seen in Figure 13 thatin terms of the same system constraints and fundamen-tal quantity Bhattacharyya distance based LPI optimiza-tion is asymptotically equivalent to the J-divergence basedcase

43 Discussion According to Figures 4ndash13 we can deducethe following conclusions for radar network architecture

(1) From Figures 4 to 7 we can observe that as thepredefined threshold of target detection increasesmore transmission power would be allocated forradar network to meet the detection performancewhile the intercept factor is increased subsequentlywhich is vulnerable in electronic warfare In otherwords there exists a tradeoff between LPI and targetdetection performance in radar network system andthe LPI performance would be sacrificed with targetdetection consideration

(2) In the numerical simulations we observe that theproposed optimization schemes (19) and (24) canbe employed to enhance the LPI performance forradar network Based on the netted radarsrsquo spatialdistributionwith respect to the target we can improvethe LPI performance by optimizing transmissionpower allocation among netted radars As indicatedin Figures 11 and 12 netted radars with better channelconditions are favorable over others In additionit can be observed that exploiting our proposedalgorithms can effectively improve the LPI perfor-mance of radar network to defend against interceptreceiver and Bhattacharyya distance based LPI opti-mization algorithm is asymptotically equivalent tothe J-divergence based case under the same systemconstraints and fundamental quantity

5 Conclusions

In this paper we investigated the problem of LPI designin radar network architecture where two LPI optimizationschemes based on information-theoretic criteria have beenproposed The NPGA was employed to tackle the highlynonconvex and nonlinear optimization problems Simula-tions have demonstrated that our proposed strategies areeffective and valuable to improve the LPI performance forradar network and it is indicated that these two optimizationproblems are asymptotically equivalent to each other underthe same system constraints Note that only single target wasconsidered in this paper Nevertheless it is convenient to beextended to multiple targets scenario and the conclusionsobtained in this study suggest that similar LPI benefits wouldbe obtained for the multiple targets case Future work willlook into the adaptive threshold design of target detectionperformance in radar network architectures

Appendices

A

Assume that every transmitter-receiver combination in thenetwork can be the same and 1198772net ≜ 119877

119905119894sdot 119877119903119895 where the radar

network SNR (1) can be written as follows

SNRnet = 119870rad

119873119905

sum

119894=1

119873119903

sum

119895=1

119875119905119894

1198774net (A1)

where

119870rad =1198661199051198661199031205901199051205822

(4120587)3119896119879119900119861119903119865119903119871 (A2)

Assume that the sum of the effective radiated power (ERP)from all the radars in the network is equivalent to that ofmonostatic radar that is

ERP =

119873119905

sum

119894=1

119875119905119894119866119905119894= 119875119905119866119905 (A3)

where 119875119905and 119866

119905are the transmitting power and transmitting

antenna gain of the monostatic radar respectively For 119866119905119894=

119866119905(for all 119894) we can rewrite (A1) as follows

SNRnet = 119870rad119873119905119875119905

1198774net (A4)

B

According to (15) we can derive the intercept factor for radarnetwork as

120572net =119877int119877net

= (119875119905sdot 1198702

int sdot SNRnet

119870rad sdot 119873119903 sdot SNR2int)

14

(B1)

and the intercept factor for conventional monostatic radar as

120572mon =119877int119877mon

= (119875119905sdot 1198702

int sdot SNRmon

119870rad sdot SNR2int)

14

(B2)

When SNRnet = SNRmon we can readily obtain therelationship between the intercept factors for radar network120572net and for the monostatic case 120572mon

120572net =120572mon

11987314

119903

(B3)

Furthermore for monostatic radar we can assume that

SNRmon = 119870rad119875119905

1198774mon= 119870rad

119875maxtot

1198774

119877MAX (B4)

where 119875maxtot is the maximal power of the monostatic rada-

r and 119877119877MAX is the corresponding maximal detection range

Then we can obtain

119877mon119877119877MAX

= (119875119905

119875maxtot

)

14

(B5)

10 International Journal of Antennas and Propagation

Similarly for intercept receiver we have the following

SNRint = 119870int119875119905

1198772

int= 119870int

119875maxtot

1198772

119868119872119860119883

(B6)

119877int119877119868MAX

= (119875119905

119875maxtot

)

12

(B7)

where 119877119868MAX is the intercept range when the transmitting

power of radar is 119875maxtot

Using (15) (B5) and (B7) we can obtain the followingexpression

120572mon120572maxmon

=119877int119877mon

119877119868MAX119877119877MAX

= (119875119905

119875maxtot

)

14

(B8)

where 120572maxmon is Schleher intercept factor corresponding to the

maximal transmitting power 119875maxtot

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their comments that help to improve the quality of thispaper The support provided by the National Natural ScienceFoundation ofChina (Grant no 61371170) Funding of JiangsuInnovation Program for Graduate Education (CXLX13 154)the Fundamental Research Funds for the Central Universi-ties the Priority Academic Program Development of JiangsuHigher Education Institutions (PADA) and Key Laboratoryof Radar Imaging and Microwave Photonics Ministry ofEducation Nanjing University of Aeronautics and Astronau-tics Nanjing China is gratefully acknowledged

References

[1] E P Phillip Detecting and Classifying Low Probability ofIntercept Radar Artech House Boston Mass USA 2009

[2] AM Haimovich R S Blum and L J Cimini Jr ldquoMIMO radarwith widely separated antennasrdquo IEEE Signal Processing Maga-zine vol 25 no 1 pp 116ndash129 2008

[3] A L Hume and C J Baker ldquoNetted radar sensingrdquo in Proceed-ings of the 2001 CIE International Conference on Radar pp 23ndash26 2001

[4] E Fishler A Haimovich R S Blum L J Cimini Jr D Chizhikand R A Valenzuela ldquoSpatial diversity in radars-Models anddetection performancerdquo IEEE Transactions on Signal Processingvol 54 no 3 pp 823ndash838 2006

[5] Y Yang and R S Blum ldquoMIMO radar waveform designbased onmutual information andminimummean-square errorestimationrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 43 no 1 pp 330ndash343 2007

[6] MM Naghsh M HMahmoud S P ShahramM Soltanalianand P Stoica ldquoUnified optimization framework for multi-staticradar code design using information-theoretic criteriardquo IEEETransactions on Signal Processing vol 61 no 21 pp 5401ndash54162013

[7] R Niu R S Blum P K Varshney and A L Drozd ldquoTar-get localization and tracking in noncoherent multiple-inputmultiple-output radar systemsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 48 no 2 pp 1466ndash1489 2012

[8] P Chavali and A Nehorai ldquoScheduling and power allocationin a cognitive radar network for multiple-target trackingrdquo IEEETransactions on Signal Processing vol 60 no 2 pp 715ndash7292012

[9] H Godrich A Petropulu and H V Poor ldquoOptimal powerallocation in distributed multiple-radar configurationsrdquo inProceeding of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 11) pp 2492ndash2495Prague Czech Republic May 2011

[10] H Godrich A P Petropulu and H V Poor ldquoPower allocationstrategies for target localization in distributed multiple-radararchitecturesrdquo IEEE Transactions on Signal Processing vol 59no 7 pp 3226ndash3240 2011

[11] H Godrich A Petropulu and H V Poort ldquoEstimation per-formance and resource savings tradeoffs in multiple radarssystemsrdquo in Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo12) pp 5205ndash5208 Kyoto Japan March 2012

[12] C G Shi J J Zhou F Wang and J Chen ldquoTarget threateninglevel based optimal power allocation for LPI radar networkrdquoin Proceedings of the 2nd International Symposium on Instru-mentation and Measurement Sensor Network and Automation(IMSNA rsquo13) pp 634ndash637 Toronto Canada December 2013

[13] X Song P Willett and S Zhou ldquoOptimal power allocationfor MIMO radars with heterogeneous propagation lossesrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 12) pp 2465ndash2468Kyoto Japan March 2012

[14] X F Song P Willett S Zhou and J Glaz ldquoMIMO radar detec-tion with heterogeneous propagation lossesrdquo in Proceedings ofthe IEEE Statistical Signal Processing Workshop (SSP rsquo12) pp776ndash779 August 2012

[15] D C Schleher ldquoLPI radar fact or fictionrdquo IEEE Aerospace andElectronic Systems Magazine vol 21 no 5 pp 3ndash6 2006

[16] D Lynch Jr Introduction to RF Stealth Sci Tech Publishing2004

[17] A G Stove A L Hume and C J Baker ldquoLow probability ofintercept radar strategiesrdquo IEE Proceedings Radar Sonar andNavigation vol 151 no 5 pp 249ndash260 2004

[18] D E Lawrence ldquoLow probability of intercept antenna arraybeamformingrdquo IEEE Transactions on Antennas and Propaga-tion vol 58 no 9 pp 2858ndash2865 2010

[19] T Kailath ldquoThe divergence and Bhattacharyya distance mea-sures in signal detectionrdquo IEEE Transaction on CommunicationTechnology vol 15 no 2 pp 52ndash60 1967

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Shock and Vibration

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DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 5

Bhattacharyya distance threshold such that the LPI perfor-mance is met on the guarantee of target detection perfor-mance Eventually the underlying LPI optimization problemcan be formulated as follows

min997888Pt

120572net

st 119861net ge 119861th

119873119905

sum

119894=1

119875119905119894le 119875

maxtot

0 le 119875119905119894le 119875

max119905119894

(forall119894)

(19)

where 997888Pt = [1198751199051 1198751199052 119875

119905119873119905

]119879 is the transmitting power of

radar network 119861th is the Bhattacharyya distance thresholdfor target detection 119875max

tot is the maximum total transmissionpower of radar network and 119875max

119905119894(for all 119894) is the maximum

transmission power of the corresponding netted radar node

322 J-Divergence Based LPI Optimization Scheme The J-divergence 119869(119901

0 1199011) is anothermetric tomeasure the distance

between two pdfs 1199010and 119901

1 It is defined as follows

119869 (1199010 1199011) ≜ 119863 (119901

0 1199011) + 119863 (119901

1 1199010) (20)

where 119863(sdot) is the Kullback-Leibler divergence It is shown in[19] that for any fixed value of 119875fa

119863[119891 (119903 | 1198670) 119891 (119903 | 119867

1)] = lim119873rarrinfin

[minus1

119873log (1 minus 119875

119889)]

(21)

and for any fixed value of 119875119889 we can obtain

119863[119891 (119903 | 1198671) 119891 (119903 | 119867

0)] = lim119873rarrinfin

[minus1

119873log (119875fa)] (22)

From (21) and (22) we can observe that for any fixed 119875fa themaximization of Kullback-Leibler divergence 119863[119891(119903 | 119867

0)

119891(119903 | 1198671)] results in an asymptotic maximization of 119875

119889

while for any fixed 119875119889the maximization of Kullback-Leibler

divergence119863[119891(119903 | 1198671) 119891(119903 | 119867

0)] results in an asymptotic

minimization of 119875faWith the derivation in [6] we have that

119869net ≜ 119869 [119891 (119903 | 1198670) 119891 (119903 | 119867

1)]

=

119873119905

sum

119894=1

119873119903

sum

119895=1

1205772

119894119895

1 + 120577119894119895

=

119873119905

sum

119894=1

119873119903

sum

119895=1

[1198751199051198941198771198921199012

119894119895(119877120579)minus1

]2

1 + 1198751199051198941198771198921199012

119894119895(119877120579)minus1

=

119873119905

sum

119894=1

119873119903

sum

119895=1

[119875119905119894119877119892119866119905119894119866119903119895(1198771205791198772

1199051198941198772

119903119895)minus1

]

2

1 + 119875119905119894119877119892119866119905119894119866119903119895(1198771205791198772

1199051198941198772

119903119895)minus1

(23)

Consequently the corresponding LPI optimization prob-lem can be expressed as follows

min997888119875119905

120572net

st 119869net ge 119869th

119873119905

sum

119894=1

119875119905119894le 119875

maxtot

0 le 119875119905119894le 119875

max119905119894

(forall119894)

(24)

where 119869th is the J-divergence threshold for target detection

33 The Unified Framework Based on NPGA In this subsec-tion we cast the LPI optimization problems based on variousinformation-theoretic criteria investigated earlier under aunified optimization framework Furthermore we formulatethe following general form of the optimization problems in(19) and (24)

min997888119875119905

120572net

st 120574net ge 120574th

119873119905

sum

119894=1

119875119905119894le 119875

maxtot

0 le 119875119905119894le 119875

max119905119894

(forall119894)

(25)

where 120574net isin 119861net 119869net and 120574th is the corresponding threshold

for target detectionIn this paper we utilize the nonlinear programming based

genetic algorithm (NPGA) to seek the optimal solutions tothe resulting nonconvex nonlinear and constrained problem(25)The NPGA has a good performance on the convergencespeed and it improves the searching performance of ordinarygenetic algorithm

The NPGA procedure is illustrated in Figure 3 wherethe population initialization module is utilized to initializethe population according to the resulting problem while thecalculating fitness value module is to calculate the fitnessvalues of individuals in the population Selection crossoverand mutation are employed to seek the optimal solutionwhere 119873 is a constant If the evolution is 119873rsquos multiples wecan use NP approach to accelerate the convergence speed

So far we have completed the derivation of Schle-her intercept factor for radar network architecture and

6 International Journal of Antennas and Propagation

Population initializationStart Calculate

fitness value Selection Crossover Mutation

Nonlinear YesYes

NoNo

Endmultiples

Isevolution Nrsquos the termination

conditions

Satisfy

Figure 3 Flow diagram of NPGA procedure

0 02 04 06 08 1 12 14 16 18 20

10

20

30

40

50

60

70

80

90

Schleher intercept factor

Bhat

tach

aryy

a dist

ance

2 times 2 radar network2 times 4 radar network4 times 2 radar network

4 times 4 radar network6 times 6 radar network

Figure 4 Bhattacharyya distance versus Schleher intercept factorfor different radar network architectures

the information-theoretic criteria based LPI optimizationschemes In what follows some numerical simulations areprovided to confirm the effectiveness of our presented LPIoptimization algorithms for radar network architecture

4 Numerical Simulations

In this section we provide several numerical simulations toexamine the performance of the proposed LPI optimizationalgorithms as (19) and (24) Throughout this section weassume that 119875max

tot = sum119873119905

119894=1119875119905119894= 24KW 119866

119905= 119866119903= 30 dB

119877120579= 10minus10 and 119877

119892= 1 The SNR is set to be 13 dB The

traditional monostatic radar can detect the target whose RCSis 005m2 in the distance 119877

119877MAX = 1061 km by transmittingthe maximum power 119875max

tot = 24KW where the interceptfactor is normalized to be 1 for simplicity

41 LPI Performance Analysis Figures 4 and 6 show theBhattacharyya distance and logarithmic J-divergence ver-sus Schleher intercept factor for different radar networkarchitectures respectively which are conducted 10

6 Monte

0 02 04 06 08 1 12 14 16 18 20

10

20

30

40

50

60

Schleher intercept factor

Bhat

tach

aryy

a dist

ance

4 times 4 radar network with Rg = 1

4 times 4 radar network with Rg = 5

4 times 4 radar network with Rg = 10

Figure 5 Bhattacharyya distance versus Schleher intercept factorfor different target scattering intensity with119873

119905= 119873119903= 4

Carlo trials We can observe in Figures 4 and 6 that asSchleher intercept factor increases from 120572net = 0 to 120572net =2 the achievable Bhattacharyya distance and logarithmic J-divergence are increased This is due to the fact that as theintercept factor increases more transmission power wouldbe allocated which makes the achievable Bhattacharyya dis-tance and logarithmic J-divergence increase correspondinglyas theoretically proved in (18) and (23) Furthermore it canbe seen from Figures 4 and 6 that with the same target detec-tion threshold Schleher intercept factor can be significantlyreduced as the number of transmitters and receivers in thenetwork system increases Therefore increasing the numberof netted radars can effectively improve the LPI performancefor radar networkThis confirms the LPI benefits of the radarnetwork architecture with more netted radars

As shown in Figures 5 and 7 we illustrate the Bhat-tacharyya distance and logarithmic J-divergence versusSchleher intercept factor for different target scattering inten-sity with 119873

119905= 119873119903= 4 respectively It is depicted that

as the target scattering intensity increases from 119877119892

= 1

to 119877119892

= 10 the achievable Bhattacharyya distance andlogarithmic J-divergence are significantly increased This is

International Journal of Antennas and Propagation 7

0 02 04 06 08 1 12 14 16 18 2

0

2

4

6

8

10

Schleher intercept factor

log(

J-di

verg

ence

)

minus10

minus8

minus6

minus4

minus2

2 times 2 radar network2 times 4 radar network4 times 2 radar network

4 times 4 radar network6 times 6 radar network

Figure 6 Logarithmic J-divergence versus Schleher intercept factorfor different radar network architectures

0 02 04 06 08 1 12 14 16 18 2

0

5

10

15

Schleher intercept factor

log(

J-di

verg

ence

)

minus10

minus5

4 times 4 radar network with Rg = 1

4 times 4 radar network with Rg = 5

4 times 4 radar network with Rg = 10

Figure 7 Logarithmic J-divergence versus Schleher intercept factorfor different target scattering intensity with119873

119905= 119873119903= 4

because the radar network system can detect the target withlarge scattering intensity easily with high 119875

119889and low 119875fa

42 Target Tracking with LPI Optimization In this subsec-tion we consider a 4 times 4 radar network system (119873

119905=

119873119903= 4) in the simulation and it is widely deployed in

modern battlefield The target detection threshold 120574th can

be calculated in the condition that the transmission powerof each radar is 6KW in the distance 150 km between theradar network and the target which is the minimum value ofthe basic performance requirement for target detection As

180 185 190 195 200 205 210 215 220180

185

190

195

200

205

210

215

220

Netted radar

Radar1Radar2

Radar3 Radar4

X (km)

Y(k

m)

Figure 8 The radar network system configuration in two dimen-sions

85 90 95 100 105 110 115 120 125 130100

120

140

160

180

200

220

240

260

280

X (km)

Y (k

m)

Real trajectoryTrack with PF

t = 1 s

t = 50 s

Figure 9 Target tracking scenario

mentioned before it is supposed that the intercept receiveris carried by the target It is depicted in Figure 8 that thenetted radars in the network are spatially distributed in thesurveillance area at the initial time 119905 = 0

We track a single target by utilizing particle filtering (PF)method where 5000 particles are used to estimate the targetstate Figure 9 shows one realization of the target trajectoryfor 50 s and the tracking interval is chosen to be 1 s Withthe radar network configuration in Figure 8 and the targettracking scenario in Figure 9 we can obtain the distanceschanging curve between the netted radars and the target inthe tracking process as depicted in Figure 10 Without loss ofgenerality we set1198771198861198891198861199031 as the distributed data fusion centerand capitalize the weighted average approach to obtain theestimated target state

8 International Journal of Antennas and Propagation

5 10 15 20 25 30 35 40 45 5080

85

90

95

100

105

110

115

120

125

130

Time (s)

Dist

ance

(km

)

Netted radar 1Netted radar 2

Netted radar 3Netted radar 4

Figure 10 The distances between the netted radars and the target

5 10 15 20 25 30 35 40 45 50Time (s)

B-netted radar 1B-netted radar 2

B-netted radar 3B-netted radar 4

Tran

smitt

ing

pow

er (k

W)

0

05

1

15

2

25

3

Figure 11 The transmitting power of netted radars utilizing Bhat-tacharyya distance based LPI optimization in the tracking process

To obtain the optimal transmission power allocation ofradar network we utilize NPGA to solve (19) and (24) Letthe population size be 100 let the crossover probability be06 and let the mutation probability be 001 The populationevolves 10 generations Figure 11 shows the transmittingpower of netted radars utilizing Bhattacharyya distance basedLPI optimization in the tracking process while Figure 12depicts the J-divergence based case Before 119905 = 36 s nettedradars 2 3 and 4 are selected to track the target which arethe ones closest to the target while netted radar 1 is selectedinstead of radar 2 after 119905 = 36 s which is because netted radars1 2 and 3 have the best channel conditions in the networkFrom Figures 11 and 12 we can see that the transmissionpower allocation is determined by the locations of single

Tran

smitt

ing

pow

er (k

W)

5 10 15 20 25 30 35 40 45 500

05

1

15

2

25

3

Time (s)

J-netted radar 1J-netted radar 2

J-netted radar 3J-netted radar 4

Figure 12 The transmitting power of netted radars utilizing J-divergence based LPI optimization in the tracking process

03

04

05

06

07

08

09

1

11Sc

hleh

er in

terc

ept f

acto

r

J-radar networkB-radar network

Ordinary radar networkMonostatic radar

5 10 15 20 25 30 35 40 45 50Time (s)

Figure 13 The normalized Schleher intercept factor comparison inthe tracking process

target relative to the netted radars and their propagationlosses To be specific in the LPI optimization process moretransmitting power is allocated to the radar nodes that arelocated closer to the target this is due to the fact that theysuffer less propagation losses

Figure 13 demonstrates the advantage of our proposedoptimization problems based on information-theoretic cri-teria The traditional monostatic radar transmits 24KWconstantly while the ordinary radar network has a constantsum of transmitted power 24KW and each radar nodetransmits uniform power One can see that Schleher interceptfactor for radar network employing the information-theoreticcriteria based LPI optimization strategies is strictly smaller

International Journal of Antennas and Propagation 9

than that of traditional monostatic radar and ordinary radarnetwork across the whole region which further shows theLPI enhancement by exploiting our presented LPI optimiza-tion schemes in radar network to defend against passiveintercept receiver Moreover it can be seen in Figure 13 thatin terms of the same system constraints and fundamen-tal quantity Bhattacharyya distance based LPI optimiza-tion is asymptotically equivalent to the J-divergence basedcase

43 Discussion According to Figures 4ndash13 we can deducethe following conclusions for radar network architecture

(1) From Figures 4 to 7 we can observe that as thepredefined threshold of target detection increasesmore transmission power would be allocated forradar network to meet the detection performancewhile the intercept factor is increased subsequentlywhich is vulnerable in electronic warfare In otherwords there exists a tradeoff between LPI and targetdetection performance in radar network system andthe LPI performance would be sacrificed with targetdetection consideration

(2) In the numerical simulations we observe that theproposed optimization schemes (19) and (24) canbe employed to enhance the LPI performance forradar network Based on the netted radarsrsquo spatialdistributionwith respect to the target we can improvethe LPI performance by optimizing transmissionpower allocation among netted radars As indicatedin Figures 11 and 12 netted radars with better channelconditions are favorable over others In additionit can be observed that exploiting our proposedalgorithms can effectively improve the LPI perfor-mance of radar network to defend against interceptreceiver and Bhattacharyya distance based LPI opti-mization algorithm is asymptotically equivalent tothe J-divergence based case under the same systemconstraints and fundamental quantity

5 Conclusions

In this paper we investigated the problem of LPI designin radar network architecture where two LPI optimizationschemes based on information-theoretic criteria have beenproposed The NPGA was employed to tackle the highlynonconvex and nonlinear optimization problems Simula-tions have demonstrated that our proposed strategies areeffective and valuable to improve the LPI performance forradar network and it is indicated that these two optimizationproblems are asymptotically equivalent to each other underthe same system constraints Note that only single target wasconsidered in this paper Nevertheless it is convenient to beextended to multiple targets scenario and the conclusionsobtained in this study suggest that similar LPI benefits wouldbe obtained for the multiple targets case Future work willlook into the adaptive threshold design of target detectionperformance in radar network architectures

Appendices

A

Assume that every transmitter-receiver combination in thenetwork can be the same and 1198772net ≜ 119877

119905119894sdot 119877119903119895 where the radar

network SNR (1) can be written as follows

SNRnet = 119870rad

119873119905

sum

119894=1

119873119903

sum

119895=1

119875119905119894

1198774net (A1)

where

119870rad =1198661199051198661199031205901199051205822

(4120587)3119896119879119900119861119903119865119903119871 (A2)

Assume that the sum of the effective radiated power (ERP)from all the radars in the network is equivalent to that ofmonostatic radar that is

ERP =

119873119905

sum

119894=1

119875119905119894119866119905119894= 119875119905119866119905 (A3)

where 119875119905and 119866

119905are the transmitting power and transmitting

antenna gain of the monostatic radar respectively For 119866119905119894=

119866119905(for all 119894) we can rewrite (A1) as follows

SNRnet = 119870rad119873119905119875119905

1198774net (A4)

B

According to (15) we can derive the intercept factor for radarnetwork as

120572net =119877int119877net

= (119875119905sdot 1198702

int sdot SNRnet

119870rad sdot 119873119903 sdot SNR2int)

14

(B1)

and the intercept factor for conventional monostatic radar as

120572mon =119877int119877mon

= (119875119905sdot 1198702

int sdot SNRmon

119870rad sdot SNR2int)

14

(B2)

When SNRnet = SNRmon we can readily obtain therelationship between the intercept factors for radar network120572net and for the monostatic case 120572mon

120572net =120572mon

11987314

119903

(B3)

Furthermore for monostatic radar we can assume that

SNRmon = 119870rad119875119905

1198774mon= 119870rad

119875maxtot

1198774

119877MAX (B4)

where 119875maxtot is the maximal power of the monostatic rada-

r and 119877119877MAX is the corresponding maximal detection range

Then we can obtain

119877mon119877119877MAX

= (119875119905

119875maxtot

)

14

(B5)

10 International Journal of Antennas and Propagation

Similarly for intercept receiver we have the following

SNRint = 119870int119875119905

1198772

int= 119870int

119875maxtot

1198772

119868119872119860119883

(B6)

119877int119877119868MAX

= (119875119905

119875maxtot

)

12

(B7)

where 119877119868MAX is the intercept range when the transmitting

power of radar is 119875maxtot

Using (15) (B5) and (B7) we can obtain the followingexpression

120572mon120572maxmon

=119877int119877mon

119877119868MAX119877119877MAX

= (119875119905

119875maxtot

)

14

(B8)

where 120572maxmon is Schleher intercept factor corresponding to the

maximal transmitting power 119875maxtot

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their comments that help to improve the quality of thispaper The support provided by the National Natural ScienceFoundation ofChina (Grant no 61371170) Funding of JiangsuInnovation Program for Graduate Education (CXLX13 154)the Fundamental Research Funds for the Central Universi-ties the Priority Academic Program Development of JiangsuHigher Education Institutions (PADA) and Key Laboratoryof Radar Imaging and Microwave Photonics Ministry ofEducation Nanjing University of Aeronautics and Astronau-tics Nanjing China is gratefully acknowledged

References

[1] E P Phillip Detecting and Classifying Low Probability ofIntercept Radar Artech House Boston Mass USA 2009

[2] AM Haimovich R S Blum and L J Cimini Jr ldquoMIMO radarwith widely separated antennasrdquo IEEE Signal Processing Maga-zine vol 25 no 1 pp 116ndash129 2008

[3] A L Hume and C J Baker ldquoNetted radar sensingrdquo in Proceed-ings of the 2001 CIE International Conference on Radar pp 23ndash26 2001

[4] E Fishler A Haimovich R S Blum L J Cimini Jr D Chizhikand R A Valenzuela ldquoSpatial diversity in radars-Models anddetection performancerdquo IEEE Transactions on Signal Processingvol 54 no 3 pp 823ndash838 2006

[5] Y Yang and R S Blum ldquoMIMO radar waveform designbased onmutual information andminimummean-square errorestimationrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 43 no 1 pp 330ndash343 2007

[6] MM Naghsh M HMahmoud S P ShahramM Soltanalianand P Stoica ldquoUnified optimization framework for multi-staticradar code design using information-theoretic criteriardquo IEEETransactions on Signal Processing vol 61 no 21 pp 5401ndash54162013

[7] R Niu R S Blum P K Varshney and A L Drozd ldquoTar-get localization and tracking in noncoherent multiple-inputmultiple-output radar systemsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 48 no 2 pp 1466ndash1489 2012

[8] P Chavali and A Nehorai ldquoScheduling and power allocationin a cognitive radar network for multiple-target trackingrdquo IEEETransactions on Signal Processing vol 60 no 2 pp 715ndash7292012

[9] H Godrich A Petropulu and H V Poor ldquoOptimal powerallocation in distributed multiple-radar configurationsrdquo inProceeding of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 11) pp 2492ndash2495Prague Czech Republic May 2011

[10] H Godrich A P Petropulu and H V Poor ldquoPower allocationstrategies for target localization in distributed multiple-radararchitecturesrdquo IEEE Transactions on Signal Processing vol 59no 7 pp 3226ndash3240 2011

[11] H Godrich A Petropulu and H V Poort ldquoEstimation per-formance and resource savings tradeoffs in multiple radarssystemsrdquo in Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo12) pp 5205ndash5208 Kyoto Japan March 2012

[12] C G Shi J J Zhou F Wang and J Chen ldquoTarget threateninglevel based optimal power allocation for LPI radar networkrdquoin Proceedings of the 2nd International Symposium on Instru-mentation and Measurement Sensor Network and Automation(IMSNA rsquo13) pp 634ndash637 Toronto Canada December 2013

[13] X Song P Willett and S Zhou ldquoOptimal power allocationfor MIMO radars with heterogeneous propagation lossesrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 12) pp 2465ndash2468Kyoto Japan March 2012

[14] X F Song P Willett S Zhou and J Glaz ldquoMIMO radar detec-tion with heterogeneous propagation lossesrdquo in Proceedings ofthe IEEE Statistical Signal Processing Workshop (SSP rsquo12) pp776ndash779 August 2012

[15] D C Schleher ldquoLPI radar fact or fictionrdquo IEEE Aerospace andElectronic Systems Magazine vol 21 no 5 pp 3ndash6 2006

[16] D Lynch Jr Introduction to RF Stealth Sci Tech Publishing2004

[17] A G Stove A L Hume and C J Baker ldquoLow probability ofintercept radar strategiesrdquo IEE Proceedings Radar Sonar andNavigation vol 151 no 5 pp 249ndash260 2004

[18] D E Lawrence ldquoLow probability of intercept antenna arraybeamformingrdquo IEEE Transactions on Antennas and Propaga-tion vol 58 no 9 pp 2858ndash2865 2010

[19] T Kailath ldquoThe divergence and Bhattacharyya distance mea-sures in signal detectionrdquo IEEE Transaction on CommunicationTechnology vol 15 no 2 pp 52ndash60 1967

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 International Journal of Antennas and Propagation

Population initializationStart Calculate

fitness value Selection Crossover Mutation

Nonlinear YesYes

NoNo

Endmultiples

Isevolution Nrsquos the termination

conditions

Satisfy

Figure 3 Flow diagram of NPGA procedure

0 02 04 06 08 1 12 14 16 18 20

10

20

30

40

50

60

70

80

90

Schleher intercept factor

Bhat

tach

aryy

a dist

ance

2 times 2 radar network2 times 4 radar network4 times 2 radar network

4 times 4 radar network6 times 6 radar network

Figure 4 Bhattacharyya distance versus Schleher intercept factorfor different radar network architectures

the information-theoretic criteria based LPI optimizationschemes In what follows some numerical simulations areprovided to confirm the effectiveness of our presented LPIoptimization algorithms for radar network architecture

4 Numerical Simulations

In this section we provide several numerical simulations toexamine the performance of the proposed LPI optimizationalgorithms as (19) and (24) Throughout this section weassume that 119875max

tot = sum119873119905

119894=1119875119905119894= 24KW 119866

119905= 119866119903= 30 dB

119877120579= 10minus10 and 119877

119892= 1 The SNR is set to be 13 dB The

traditional monostatic radar can detect the target whose RCSis 005m2 in the distance 119877

119877MAX = 1061 km by transmittingthe maximum power 119875max

tot = 24KW where the interceptfactor is normalized to be 1 for simplicity

41 LPI Performance Analysis Figures 4 and 6 show theBhattacharyya distance and logarithmic J-divergence ver-sus Schleher intercept factor for different radar networkarchitectures respectively which are conducted 10

6 Monte

0 02 04 06 08 1 12 14 16 18 20

10

20

30

40

50

60

Schleher intercept factor

Bhat

tach

aryy

a dist

ance

4 times 4 radar network with Rg = 1

4 times 4 radar network with Rg = 5

4 times 4 radar network with Rg = 10

Figure 5 Bhattacharyya distance versus Schleher intercept factorfor different target scattering intensity with119873

119905= 119873119903= 4

Carlo trials We can observe in Figures 4 and 6 that asSchleher intercept factor increases from 120572net = 0 to 120572net =2 the achievable Bhattacharyya distance and logarithmic J-divergence are increased This is due to the fact that as theintercept factor increases more transmission power wouldbe allocated which makes the achievable Bhattacharyya dis-tance and logarithmic J-divergence increase correspondinglyas theoretically proved in (18) and (23) Furthermore it canbe seen from Figures 4 and 6 that with the same target detec-tion threshold Schleher intercept factor can be significantlyreduced as the number of transmitters and receivers in thenetwork system increases Therefore increasing the numberof netted radars can effectively improve the LPI performancefor radar networkThis confirms the LPI benefits of the radarnetwork architecture with more netted radars

As shown in Figures 5 and 7 we illustrate the Bhat-tacharyya distance and logarithmic J-divergence versusSchleher intercept factor for different target scattering inten-sity with 119873

119905= 119873119903= 4 respectively It is depicted that

as the target scattering intensity increases from 119877119892

= 1

to 119877119892

= 10 the achievable Bhattacharyya distance andlogarithmic J-divergence are significantly increased This is

International Journal of Antennas and Propagation 7

0 02 04 06 08 1 12 14 16 18 2

0

2

4

6

8

10

Schleher intercept factor

log(

J-di

verg

ence

)

minus10

minus8

minus6

minus4

minus2

2 times 2 radar network2 times 4 radar network4 times 2 radar network

4 times 4 radar network6 times 6 radar network

Figure 6 Logarithmic J-divergence versus Schleher intercept factorfor different radar network architectures

0 02 04 06 08 1 12 14 16 18 2

0

5

10

15

Schleher intercept factor

log(

J-di

verg

ence

)

minus10

minus5

4 times 4 radar network with Rg = 1

4 times 4 radar network with Rg = 5

4 times 4 radar network with Rg = 10

Figure 7 Logarithmic J-divergence versus Schleher intercept factorfor different target scattering intensity with119873

119905= 119873119903= 4

because the radar network system can detect the target withlarge scattering intensity easily with high 119875

119889and low 119875fa

42 Target Tracking with LPI Optimization In this subsec-tion we consider a 4 times 4 radar network system (119873

119905=

119873119903= 4) in the simulation and it is widely deployed in

modern battlefield The target detection threshold 120574th can

be calculated in the condition that the transmission powerof each radar is 6KW in the distance 150 km between theradar network and the target which is the minimum value ofthe basic performance requirement for target detection As

180 185 190 195 200 205 210 215 220180

185

190

195

200

205

210

215

220

Netted radar

Radar1Radar2

Radar3 Radar4

X (km)

Y(k

m)

Figure 8 The radar network system configuration in two dimen-sions

85 90 95 100 105 110 115 120 125 130100

120

140

160

180

200

220

240

260

280

X (km)

Y (k

m)

Real trajectoryTrack with PF

t = 1 s

t = 50 s

Figure 9 Target tracking scenario

mentioned before it is supposed that the intercept receiveris carried by the target It is depicted in Figure 8 that thenetted radars in the network are spatially distributed in thesurveillance area at the initial time 119905 = 0

We track a single target by utilizing particle filtering (PF)method where 5000 particles are used to estimate the targetstate Figure 9 shows one realization of the target trajectoryfor 50 s and the tracking interval is chosen to be 1 s Withthe radar network configuration in Figure 8 and the targettracking scenario in Figure 9 we can obtain the distanceschanging curve between the netted radars and the target inthe tracking process as depicted in Figure 10 Without loss ofgenerality we set1198771198861198891198861199031 as the distributed data fusion centerand capitalize the weighted average approach to obtain theestimated target state

8 International Journal of Antennas and Propagation

5 10 15 20 25 30 35 40 45 5080

85

90

95

100

105

110

115

120

125

130

Time (s)

Dist

ance

(km

)

Netted radar 1Netted radar 2

Netted radar 3Netted radar 4

Figure 10 The distances between the netted radars and the target

5 10 15 20 25 30 35 40 45 50Time (s)

B-netted radar 1B-netted radar 2

B-netted radar 3B-netted radar 4

Tran

smitt

ing

pow

er (k

W)

0

05

1

15

2

25

3

Figure 11 The transmitting power of netted radars utilizing Bhat-tacharyya distance based LPI optimization in the tracking process

To obtain the optimal transmission power allocation ofradar network we utilize NPGA to solve (19) and (24) Letthe population size be 100 let the crossover probability be06 and let the mutation probability be 001 The populationevolves 10 generations Figure 11 shows the transmittingpower of netted radars utilizing Bhattacharyya distance basedLPI optimization in the tracking process while Figure 12depicts the J-divergence based case Before 119905 = 36 s nettedradars 2 3 and 4 are selected to track the target which arethe ones closest to the target while netted radar 1 is selectedinstead of radar 2 after 119905 = 36 s which is because netted radars1 2 and 3 have the best channel conditions in the networkFrom Figures 11 and 12 we can see that the transmissionpower allocation is determined by the locations of single

Tran

smitt

ing

pow

er (k

W)

5 10 15 20 25 30 35 40 45 500

05

1

15

2

25

3

Time (s)

J-netted radar 1J-netted radar 2

J-netted radar 3J-netted radar 4

Figure 12 The transmitting power of netted radars utilizing J-divergence based LPI optimization in the tracking process

03

04

05

06

07

08

09

1

11Sc

hleh

er in

terc

ept f

acto

r

J-radar networkB-radar network

Ordinary radar networkMonostatic radar

5 10 15 20 25 30 35 40 45 50Time (s)

Figure 13 The normalized Schleher intercept factor comparison inthe tracking process

target relative to the netted radars and their propagationlosses To be specific in the LPI optimization process moretransmitting power is allocated to the radar nodes that arelocated closer to the target this is due to the fact that theysuffer less propagation losses

Figure 13 demonstrates the advantage of our proposedoptimization problems based on information-theoretic cri-teria The traditional monostatic radar transmits 24KWconstantly while the ordinary radar network has a constantsum of transmitted power 24KW and each radar nodetransmits uniform power One can see that Schleher interceptfactor for radar network employing the information-theoreticcriteria based LPI optimization strategies is strictly smaller

International Journal of Antennas and Propagation 9

than that of traditional monostatic radar and ordinary radarnetwork across the whole region which further shows theLPI enhancement by exploiting our presented LPI optimiza-tion schemes in radar network to defend against passiveintercept receiver Moreover it can be seen in Figure 13 thatin terms of the same system constraints and fundamen-tal quantity Bhattacharyya distance based LPI optimiza-tion is asymptotically equivalent to the J-divergence basedcase

43 Discussion According to Figures 4ndash13 we can deducethe following conclusions for radar network architecture

(1) From Figures 4 to 7 we can observe that as thepredefined threshold of target detection increasesmore transmission power would be allocated forradar network to meet the detection performancewhile the intercept factor is increased subsequentlywhich is vulnerable in electronic warfare In otherwords there exists a tradeoff between LPI and targetdetection performance in radar network system andthe LPI performance would be sacrificed with targetdetection consideration

(2) In the numerical simulations we observe that theproposed optimization schemes (19) and (24) canbe employed to enhance the LPI performance forradar network Based on the netted radarsrsquo spatialdistributionwith respect to the target we can improvethe LPI performance by optimizing transmissionpower allocation among netted radars As indicatedin Figures 11 and 12 netted radars with better channelconditions are favorable over others In additionit can be observed that exploiting our proposedalgorithms can effectively improve the LPI perfor-mance of radar network to defend against interceptreceiver and Bhattacharyya distance based LPI opti-mization algorithm is asymptotically equivalent tothe J-divergence based case under the same systemconstraints and fundamental quantity

5 Conclusions

In this paper we investigated the problem of LPI designin radar network architecture where two LPI optimizationschemes based on information-theoretic criteria have beenproposed The NPGA was employed to tackle the highlynonconvex and nonlinear optimization problems Simula-tions have demonstrated that our proposed strategies areeffective and valuable to improve the LPI performance forradar network and it is indicated that these two optimizationproblems are asymptotically equivalent to each other underthe same system constraints Note that only single target wasconsidered in this paper Nevertheless it is convenient to beextended to multiple targets scenario and the conclusionsobtained in this study suggest that similar LPI benefits wouldbe obtained for the multiple targets case Future work willlook into the adaptive threshold design of target detectionperformance in radar network architectures

Appendices

A

Assume that every transmitter-receiver combination in thenetwork can be the same and 1198772net ≜ 119877

119905119894sdot 119877119903119895 where the radar

network SNR (1) can be written as follows

SNRnet = 119870rad

119873119905

sum

119894=1

119873119903

sum

119895=1

119875119905119894

1198774net (A1)

where

119870rad =1198661199051198661199031205901199051205822

(4120587)3119896119879119900119861119903119865119903119871 (A2)

Assume that the sum of the effective radiated power (ERP)from all the radars in the network is equivalent to that ofmonostatic radar that is

ERP =

119873119905

sum

119894=1

119875119905119894119866119905119894= 119875119905119866119905 (A3)

where 119875119905and 119866

119905are the transmitting power and transmitting

antenna gain of the monostatic radar respectively For 119866119905119894=

119866119905(for all 119894) we can rewrite (A1) as follows

SNRnet = 119870rad119873119905119875119905

1198774net (A4)

B

According to (15) we can derive the intercept factor for radarnetwork as

120572net =119877int119877net

= (119875119905sdot 1198702

int sdot SNRnet

119870rad sdot 119873119903 sdot SNR2int)

14

(B1)

and the intercept factor for conventional monostatic radar as

120572mon =119877int119877mon

= (119875119905sdot 1198702

int sdot SNRmon

119870rad sdot SNR2int)

14

(B2)

When SNRnet = SNRmon we can readily obtain therelationship between the intercept factors for radar network120572net and for the monostatic case 120572mon

120572net =120572mon

11987314

119903

(B3)

Furthermore for monostatic radar we can assume that

SNRmon = 119870rad119875119905

1198774mon= 119870rad

119875maxtot

1198774

119877MAX (B4)

where 119875maxtot is the maximal power of the monostatic rada-

r and 119877119877MAX is the corresponding maximal detection range

Then we can obtain

119877mon119877119877MAX

= (119875119905

119875maxtot

)

14

(B5)

10 International Journal of Antennas and Propagation

Similarly for intercept receiver we have the following

SNRint = 119870int119875119905

1198772

int= 119870int

119875maxtot

1198772

119868119872119860119883

(B6)

119877int119877119868MAX

= (119875119905

119875maxtot

)

12

(B7)

where 119877119868MAX is the intercept range when the transmitting

power of radar is 119875maxtot

Using (15) (B5) and (B7) we can obtain the followingexpression

120572mon120572maxmon

=119877int119877mon

119877119868MAX119877119877MAX

= (119875119905

119875maxtot

)

14

(B8)

where 120572maxmon is Schleher intercept factor corresponding to the

maximal transmitting power 119875maxtot

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their comments that help to improve the quality of thispaper The support provided by the National Natural ScienceFoundation ofChina (Grant no 61371170) Funding of JiangsuInnovation Program for Graduate Education (CXLX13 154)the Fundamental Research Funds for the Central Universi-ties the Priority Academic Program Development of JiangsuHigher Education Institutions (PADA) and Key Laboratoryof Radar Imaging and Microwave Photonics Ministry ofEducation Nanjing University of Aeronautics and Astronau-tics Nanjing China is gratefully acknowledged

References

[1] E P Phillip Detecting and Classifying Low Probability ofIntercept Radar Artech House Boston Mass USA 2009

[2] AM Haimovich R S Blum and L J Cimini Jr ldquoMIMO radarwith widely separated antennasrdquo IEEE Signal Processing Maga-zine vol 25 no 1 pp 116ndash129 2008

[3] A L Hume and C J Baker ldquoNetted radar sensingrdquo in Proceed-ings of the 2001 CIE International Conference on Radar pp 23ndash26 2001

[4] E Fishler A Haimovich R S Blum L J Cimini Jr D Chizhikand R A Valenzuela ldquoSpatial diversity in radars-Models anddetection performancerdquo IEEE Transactions on Signal Processingvol 54 no 3 pp 823ndash838 2006

[5] Y Yang and R S Blum ldquoMIMO radar waveform designbased onmutual information andminimummean-square errorestimationrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 43 no 1 pp 330ndash343 2007

[6] MM Naghsh M HMahmoud S P ShahramM Soltanalianand P Stoica ldquoUnified optimization framework for multi-staticradar code design using information-theoretic criteriardquo IEEETransactions on Signal Processing vol 61 no 21 pp 5401ndash54162013

[7] R Niu R S Blum P K Varshney and A L Drozd ldquoTar-get localization and tracking in noncoherent multiple-inputmultiple-output radar systemsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 48 no 2 pp 1466ndash1489 2012

[8] P Chavali and A Nehorai ldquoScheduling and power allocationin a cognitive radar network for multiple-target trackingrdquo IEEETransactions on Signal Processing vol 60 no 2 pp 715ndash7292012

[9] H Godrich A Petropulu and H V Poor ldquoOptimal powerallocation in distributed multiple-radar configurationsrdquo inProceeding of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 11) pp 2492ndash2495Prague Czech Republic May 2011

[10] H Godrich A P Petropulu and H V Poor ldquoPower allocationstrategies for target localization in distributed multiple-radararchitecturesrdquo IEEE Transactions on Signal Processing vol 59no 7 pp 3226ndash3240 2011

[11] H Godrich A Petropulu and H V Poort ldquoEstimation per-formance and resource savings tradeoffs in multiple radarssystemsrdquo in Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo12) pp 5205ndash5208 Kyoto Japan March 2012

[12] C G Shi J J Zhou F Wang and J Chen ldquoTarget threateninglevel based optimal power allocation for LPI radar networkrdquoin Proceedings of the 2nd International Symposium on Instru-mentation and Measurement Sensor Network and Automation(IMSNA rsquo13) pp 634ndash637 Toronto Canada December 2013

[13] X Song P Willett and S Zhou ldquoOptimal power allocationfor MIMO radars with heterogeneous propagation lossesrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 12) pp 2465ndash2468Kyoto Japan March 2012

[14] X F Song P Willett S Zhou and J Glaz ldquoMIMO radar detec-tion with heterogeneous propagation lossesrdquo in Proceedings ofthe IEEE Statistical Signal Processing Workshop (SSP rsquo12) pp776ndash779 August 2012

[15] D C Schleher ldquoLPI radar fact or fictionrdquo IEEE Aerospace andElectronic Systems Magazine vol 21 no 5 pp 3ndash6 2006

[16] D Lynch Jr Introduction to RF Stealth Sci Tech Publishing2004

[17] A G Stove A L Hume and C J Baker ldquoLow probability ofintercept radar strategiesrdquo IEE Proceedings Radar Sonar andNavigation vol 151 no 5 pp 249ndash260 2004

[18] D E Lawrence ldquoLow probability of intercept antenna arraybeamformingrdquo IEEE Transactions on Antennas and Propaga-tion vol 58 no 9 pp 2858ndash2865 2010

[19] T Kailath ldquoThe divergence and Bhattacharyya distance mea-sures in signal detectionrdquo IEEE Transaction on CommunicationTechnology vol 15 no 2 pp 52ndash60 1967

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 7

0 02 04 06 08 1 12 14 16 18 2

0

2

4

6

8

10

Schleher intercept factor

log(

J-di

verg

ence

)

minus10

minus8

minus6

minus4

minus2

2 times 2 radar network2 times 4 radar network4 times 2 radar network

4 times 4 radar network6 times 6 radar network

Figure 6 Logarithmic J-divergence versus Schleher intercept factorfor different radar network architectures

0 02 04 06 08 1 12 14 16 18 2

0

5

10

15

Schleher intercept factor

log(

J-di

verg

ence

)

minus10

minus5

4 times 4 radar network with Rg = 1

4 times 4 radar network with Rg = 5

4 times 4 radar network with Rg = 10

Figure 7 Logarithmic J-divergence versus Schleher intercept factorfor different target scattering intensity with119873

119905= 119873119903= 4

because the radar network system can detect the target withlarge scattering intensity easily with high 119875

119889and low 119875fa

42 Target Tracking with LPI Optimization In this subsec-tion we consider a 4 times 4 radar network system (119873

119905=

119873119903= 4) in the simulation and it is widely deployed in

modern battlefield The target detection threshold 120574th can

be calculated in the condition that the transmission powerof each radar is 6KW in the distance 150 km between theradar network and the target which is the minimum value ofthe basic performance requirement for target detection As

180 185 190 195 200 205 210 215 220180

185

190

195

200

205

210

215

220

Netted radar

Radar1Radar2

Radar3 Radar4

X (km)

Y(k

m)

Figure 8 The radar network system configuration in two dimen-sions

85 90 95 100 105 110 115 120 125 130100

120

140

160

180

200

220

240

260

280

X (km)

Y (k

m)

Real trajectoryTrack with PF

t = 1 s

t = 50 s

Figure 9 Target tracking scenario

mentioned before it is supposed that the intercept receiveris carried by the target It is depicted in Figure 8 that thenetted radars in the network are spatially distributed in thesurveillance area at the initial time 119905 = 0

We track a single target by utilizing particle filtering (PF)method where 5000 particles are used to estimate the targetstate Figure 9 shows one realization of the target trajectoryfor 50 s and the tracking interval is chosen to be 1 s Withthe radar network configuration in Figure 8 and the targettracking scenario in Figure 9 we can obtain the distanceschanging curve between the netted radars and the target inthe tracking process as depicted in Figure 10 Without loss ofgenerality we set1198771198861198891198861199031 as the distributed data fusion centerand capitalize the weighted average approach to obtain theestimated target state

8 International Journal of Antennas and Propagation

5 10 15 20 25 30 35 40 45 5080

85

90

95

100

105

110

115

120

125

130

Time (s)

Dist

ance

(km

)

Netted radar 1Netted radar 2

Netted radar 3Netted radar 4

Figure 10 The distances between the netted radars and the target

5 10 15 20 25 30 35 40 45 50Time (s)

B-netted radar 1B-netted radar 2

B-netted radar 3B-netted radar 4

Tran

smitt

ing

pow

er (k

W)

0

05

1

15

2

25

3

Figure 11 The transmitting power of netted radars utilizing Bhat-tacharyya distance based LPI optimization in the tracking process

To obtain the optimal transmission power allocation ofradar network we utilize NPGA to solve (19) and (24) Letthe population size be 100 let the crossover probability be06 and let the mutation probability be 001 The populationevolves 10 generations Figure 11 shows the transmittingpower of netted radars utilizing Bhattacharyya distance basedLPI optimization in the tracking process while Figure 12depicts the J-divergence based case Before 119905 = 36 s nettedradars 2 3 and 4 are selected to track the target which arethe ones closest to the target while netted radar 1 is selectedinstead of radar 2 after 119905 = 36 s which is because netted radars1 2 and 3 have the best channel conditions in the networkFrom Figures 11 and 12 we can see that the transmissionpower allocation is determined by the locations of single

Tran

smitt

ing

pow

er (k

W)

5 10 15 20 25 30 35 40 45 500

05

1

15

2

25

3

Time (s)

J-netted radar 1J-netted radar 2

J-netted radar 3J-netted radar 4

Figure 12 The transmitting power of netted radars utilizing J-divergence based LPI optimization in the tracking process

03

04

05

06

07

08

09

1

11Sc

hleh

er in

terc

ept f

acto

r

J-radar networkB-radar network

Ordinary radar networkMonostatic radar

5 10 15 20 25 30 35 40 45 50Time (s)

Figure 13 The normalized Schleher intercept factor comparison inthe tracking process

target relative to the netted radars and their propagationlosses To be specific in the LPI optimization process moretransmitting power is allocated to the radar nodes that arelocated closer to the target this is due to the fact that theysuffer less propagation losses

Figure 13 demonstrates the advantage of our proposedoptimization problems based on information-theoretic cri-teria The traditional monostatic radar transmits 24KWconstantly while the ordinary radar network has a constantsum of transmitted power 24KW and each radar nodetransmits uniform power One can see that Schleher interceptfactor for radar network employing the information-theoreticcriteria based LPI optimization strategies is strictly smaller

International Journal of Antennas and Propagation 9

than that of traditional monostatic radar and ordinary radarnetwork across the whole region which further shows theLPI enhancement by exploiting our presented LPI optimiza-tion schemes in radar network to defend against passiveintercept receiver Moreover it can be seen in Figure 13 thatin terms of the same system constraints and fundamen-tal quantity Bhattacharyya distance based LPI optimiza-tion is asymptotically equivalent to the J-divergence basedcase

43 Discussion According to Figures 4ndash13 we can deducethe following conclusions for radar network architecture

(1) From Figures 4 to 7 we can observe that as thepredefined threshold of target detection increasesmore transmission power would be allocated forradar network to meet the detection performancewhile the intercept factor is increased subsequentlywhich is vulnerable in electronic warfare In otherwords there exists a tradeoff between LPI and targetdetection performance in radar network system andthe LPI performance would be sacrificed with targetdetection consideration

(2) In the numerical simulations we observe that theproposed optimization schemes (19) and (24) canbe employed to enhance the LPI performance forradar network Based on the netted radarsrsquo spatialdistributionwith respect to the target we can improvethe LPI performance by optimizing transmissionpower allocation among netted radars As indicatedin Figures 11 and 12 netted radars with better channelconditions are favorable over others In additionit can be observed that exploiting our proposedalgorithms can effectively improve the LPI perfor-mance of radar network to defend against interceptreceiver and Bhattacharyya distance based LPI opti-mization algorithm is asymptotically equivalent tothe J-divergence based case under the same systemconstraints and fundamental quantity

5 Conclusions

In this paper we investigated the problem of LPI designin radar network architecture where two LPI optimizationschemes based on information-theoretic criteria have beenproposed The NPGA was employed to tackle the highlynonconvex and nonlinear optimization problems Simula-tions have demonstrated that our proposed strategies areeffective and valuable to improve the LPI performance forradar network and it is indicated that these two optimizationproblems are asymptotically equivalent to each other underthe same system constraints Note that only single target wasconsidered in this paper Nevertheless it is convenient to beextended to multiple targets scenario and the conclusionsobtained in this study suggest that similar LPI benefits wouldbe obtained for the multiple targets case Future work willlook into the adaptive threshold design of target detectionperformance in radar network architectures

Appendices

A

Assume that every transmitter-receiver combination in thenetwork can be the same and 1198772net ≜ 119877

119905119894sdot 119877119903119895 where the radar

network SNR (1) can be written as follows

SNRnet = 119870rad

119873119905

sum

119894=1

119873119903

sum

119895=1

119875119905119894

1198774net (A1)

where

119870rad =1198661199051198661199031205901199051205822

(4120587)3119896119879119900119861119903119865119903119871 (A2)

Assume that the sum of the effective radiated power (ERP)from all the radars in the network is equivalent to that ofmonostatic radar that is

ERP =

119873119905

sum

119894=1

119875119905119894119866119905119894= 119875119905119866119905 (A3)

where 119875119905and 119866

119905are the transmitting power and transmitting

antenna gain of the monostatic radar respectively For 119866119905119894=

119866119905(for all 119894) we can rewrite (A1) as follows

SNRnet = 119870rad119873119905119875119905

1198774net (A4)

B

According to (15) we can derive the intercept factor for radarnetwork as

120572net =119877int119877net

= (119875119905sdot 1198702

int sdot SNRnet

119870rad sdot 119873119903 sdot SNR2int)

14

(B1)

and the intercept factor for conventional monostatic radar as

120572mon =119877int119877mon

= (119875119905sdot 1198702

int sdot SNRmon

119870rad sdot SNR2int)

14

(B2)

When SNRnet = SNRmon we can readily obtain therelationship between the intercept factors for radar network120572net and for the monostatic case 120572mon

120572net =120572mon

11987314

119903

(B3)

Furthermore for monostatic radar we can assume that

SNRmon = 119870rad119875119905

1198774mon= 119870rad

119875maxtot

1198774

119877MAX (B4)

where 119875maxtot is the maximal power of the monostatic rada-

r and 119877119877MAX is the corresponding maximal detection range

Then we can obtain

119877mon119877119877MAX

= (119875119905

119875maxtot

)

14

(B5)

10 International Journal of Antennas and Propagation

Similarly for intercept receiver we have the following

SNRint = 119870int119875119905

1198772

int= 119870int

119875maxtot

1198772

119868119872119860119883

(B6)

119877int119877119868MAX

= (119875119905

119875maxtot

)

12

(B7)

where 119877119868MAX is the intercept range when the transmitting

power of radar is 119875maxtot

Using (15) (B5) and (B7) we can obtain the followingexpression

120572mon120572maxmon

=119877int119877mon

119877119868MAX119877119877MAX

= (119875119905

119875maxtot

)

14

(B8)

where 120572maxmon is Schleher intercept factor corresponding to the

maximal transmitting power 119875maxtot

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their comments that help to improve the quality of thispaper The support provided by the National Natural ScienceFoundation ofChina (Grant no 61371170) Funding of JiangsuInnovation Program for Graduate Education (CXLX13 154)the Fundamental Research Funds for the Central Universi-ties the Priority Academic Program Development of JiangsuHigher Education Institutions (PADA) and Key Laboratoryof Radar Imaging and Microwave Photonics Ministry ofEducation Nanjing University of Aeronautics and Astronau-tics Nanjing China is gratefully acknowledged

References

[1] E P Phillip Detecting and Classifying Low Probability ofIntercept Radar Artech House Boston Mass USA 2009

[2] AM Haimovich R S Blum and L J Cimini Jr ldquoMIMO radarwith widely separated antennasrdquo IEEE Signal Processing Maga-zine vol 25 no 1 pp 116ndash129 2008

[3] A L Hume and C J Baker ldquoNetted radar sensingrdquo in Proceed-ings of the 2001 CIE International Conference on Radar pp 23ndash26 2001

[4] E Fishler A Haimovich R S Blum L J Cimini Jr D Chizhikand R A Valenzuela ldquoSpatial diversity in radars-Models anddetection performancerdquo IEEE Transactions on Signal Processingvol 54 no 3 pp 823ndash838 2006

[5] Y Yang and R S Blum ldquoMIMO radar waveform designbased onmutual information andminimummean-square errorestimationrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 43 no 1 pp 330ndash343 2007

[6] MM Naghsh M HMahmoud S P ShahramM Soltanalianand P Stoica ldquoUnified optimization framework for multi-staticradar code design using information-theoretic criteriardquo IEEETransactions on Signal Processing vol 61 no 21 pp 5401ndash54162013

[7] R Niu R S Blum P K Varshney and A L Drozd ldquoTar-get localization and tracking in noncoherent multiple-inputmultiple-output radar systemsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 48 no 2 pp 1466ndash1489 2012

[8] P Chavali and A Nehorai ldquoScheduling and power allocationin a cognitive radar network for multiple-target trackingrdquo IEEETransactions on Signal Processing vol 60 no 2 pp 715ndash7292012

[9] H Godrich A Petropulu and H V Poor ldquoOptimal powerallocation in distributed multiple-radar configurationsrdquo inProceeding of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 11) pp 2492ndash2495Prague Czech Republic May 2011

[10] H Godrich A P Petropulu and H V Poor ldquoPower allocationstrategies for target localization in distributed multiple-radararchitecturesrdquo IEEE Transactions on Signal Processing vol 59no 7 pp 3226ndash3240 2011

[11] H Godrich A Petropulu and H V Poort ldquoEstimation per-formance and resource savings tradeoffs in multiple radarssystemsrdquo in Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo12) pp 5205ndash5208 Kyoto Japan March 2012

[12] C G Shi J J Zhou F Wang and J Chen ldquoTarget threateninglevel based optimal power allocation for LPI radar networkrdquoin Proceedings of the 2nd International Symposium on Instru-mentation and Measurement Sensor Network and Automation(IMSNA rsquo13) pp 634ndash637 Toronto Canada December 2013

[13] X Song P Willett and S Zhou ldquoOptimal power allocationfor MIMO radars with heterogeneous propagation lossesrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 12) pp 2465ndash2468Kyoto Japan March 2012

[14] X F Song P Willett S Zhou and J Glaz ldquoMIMO radar detec-tion with heterogeneous propagation lossesrdquo in Proceedings ofthe IEEE Statistical Signal Processing Workshop (SSP rsquo12) pp776ndash779 August 2012

[15] D C Schleher ldquoLPI radar fact or fictionrdquo IEEE Aerospace andElectronic Systems Magazine vol 21 no 5 pp 3ndash6 2006

[16] D Lynch Jr Introduction to RF Stealth Sci Tech Publishing2004

[17] A G Stove A L Hume and C J Baker ldquoLow probability ofintercept radar strategiesrdquo IEE Proceedings Radar Sonar andNavigation vol 151 no 5 pp 249ndash260 2004

[18] D E Lawrence ldquoLow probability of intercept antenna arraybeamformingrdquo IEEE Transactions on Antennas and Propaga-tion vol 58 no 9 pp 2858ndash2865 2010

[19] T Kailath ldquoThe divergence and Bhattacharyya distance mea-sures in signal detectionrdquo IEEE Transaction on CommunicationTechnology vol 15 no 2 pp 52ndash60 1967

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 International Journal of Antennas and Propagation

5 10 15 20 25 30 35 40 45 5080

85

90

95

100

105

110

115

120

125

130

Time (s)

Dist

ance

(km

)

Netted radar 1Netted radar 2

Netted radar 3Netted radar 4

Figure 10 The distances between the netted radars and the target

5 10 15 20 25 30 35 40 45 50Time (s)

B-netted radar 1B-netted radar 2

B-netted radar 3B-netted radar 4

Tran

smitt

ing

pow

er (k

W)

0

05

1

15

2

25

3

Figure 11 The transmitting power of netted radars utilizing Bhat-tacharyya distance based LPI optimization in the tracking process

To obtain the optimal transmission power allocation ofradar network we utilize NPGA to solve (19) and (24) Letthe population size be 100 let the crossover probability be06 and let the mutation probability be 001 The populationevolves 10 generations Figure 11 shows the transmittingpower of netted radars utilizing Bhattacharyya distance basedLPI optimization in the tracking process while Figure 12depicts the J-divergence based case Before 119905 = 36 s nettedradars 2 3 and 4 are selected to track the target which arethe ones closest to the target while netted radar 1 is selectedinstead of radar 2 after 119905 = 36 s which is because netted radars1 2 and 3 have the best channel conditions in the networkFrom Figures 11 and 12 we can see that the transmissionpower allocation is determined by the locations of single

Tran

smitt

ing

pow

er (k

W)

5 10 15 20 25 30 35 40 45 500

05

1

15

2

25

3

Time (s)

J-netted radar 1J-netted radar 2

J-netted radar 3J-netted radar 4

Figure 12 The transmitting power of netted radars utilizing J-divergence based LPI optimization in the tracking process

03

04

05

06

07

08

09

1

11Sc

hleh

er in

terc

ept f

acto

r

J-radar networkB-radar network

Ordinary radar networkMonostatic radar

5 10 15 20 25 30 35 40 45 50Time (s)

Figure 13 The normalized Schleher intercept factor comparison inthe tracking process

target relative to the netted radars and their propagationlosses To be specific in the LPI optimization process moretransmitting power is allocated to the radar nodes that arelocated closer to the target this is due to the fact that theysuffer less propagation losses

Figure 13 demonstrates the advantage of our proposedoptimization problems based on information-theoretic cri-teria The traditional monostatic radar transmits 24KWconstantly while the ordinary radar network has a constantsum of transmitted power 24KW and each radar nodetransmits uniform power One can see that Schleher interceptfactor for radar network employing the information-theoreticcriteria based LPI optimization strategies is strictly smaller

International Journal of Antennas and Propagation 9

than that of traditional monostatic radar and ordinary radarnetwork across the whole region which further shows theLPI enhancement by exploiting our presented LPI optimiza-tion schemes in radar network to defend against passiveintercept receiver Moreover it can be seen in Figure 13 thatin terms of the same system constraints and fundamen-tal quantity Bhattacharyya distance based LPI optimiza-tion is asymptotically equivalent to the J-divergence basedcase

43 Discussion According to Figures 4ndash13 we can deducethe following conclusions for radar network architecture

(1) From Figures 4 to 7 we can observe that as thepredefined threshold of target detection increasesmore transmission power would be allocated forradar network to meet the detection performancewhile the intercept factor is increased subsequentlywhich is vulnerable in electronic warfare In otherwords there exists a tradeoff between LPI and targetdetection performance in radar network system andthe LPI performance would be sacrificed with targetdetection consideration

(2) In the numerical simulations we observe that theproposed optimization schemes (19) and (24) canbe employed to enhance the LPI performance forradar network Based on the netted radarsrsquo spatialdistributionwith respect to the target we can improvethe LPI performance by optimizing transmissionpower allocation among netted radars As indicatedin Figures 11 and 12 netted radars with better channelconditions are favorable over others In additionit can be observed that exploiting our proposedalgorithms can effectively improve the LPI perfor-mance of radar network to defend against interceptreceiver and Bhattacharyya distance based LPI opti-mization algorithm is asymptotically equivalent tothe J-divergence based case under the same systemconstraints and fundamental quantity

5 Conclusions

In this paper we investigated the problem of LPI designin radar network architecture where two LPI optimizationschemes based on information-theoretic criteria have beenproposed The NPGA was employed to tackle the highlynonconvex and nonlinear optimization problems Simula-tions have demonstrated that our proposed strategies areeffective and valuable to improve the LPI performance forradar network and it is indicated that these two optimizationproblems are asymptotically equivalent to each other underthe same system constraints Note that only single target wasconsidered in this paper Nevertheless it is convenient to beextended to multiple targets scenario and the conclusionsobtained in this study suggest that similar LPI benefits wouldbe obtained for the multiple targets case Future work willlook into the adaptive threshold design of target detectionperformance in radar network architectures

Appendices

A

Assume that every transmitter-receiver combination in thenetwork can be the same and 1198772net ≜ 119877

119905119894sdot 119877119903119895 where the radar

network SNR (1) can be written as follows

SNRnet = 119870rad

119873119905

sum

119894=1

119873119903

sum

119895=1

119875119905119894

1198774net (A1)

where

119870rad =1198661199051198661199031205901199051205822

(4120587)3119896119879119900119861119903119865119903119871 (A2)

Assume that the sum of the effective radiated power (ERP)from all the radars in the network is equivalent to that ofmonostatic radar that is

ERP =

119873119905

sum

119894=1

119875119905119894119866119905119894= 119875119905119866119905 (A3)

where 119875119905and 119866

119905are the transmitting power and transmitting

antenna gain of the monostatic radar respectively For 119866119905119894=

119866119905(for all 119894) we can rewrite (A1) as follows

SNRnet = 119870rad119873119905119875119905

1198774net (A4)

B

According to (15) we can derive the intercept factor for radarnetwork as

120572net =119877int119877net

= (119875119905sdot 1198702

int sdot SNRnet

119870rad sdot 119873119903 sdot SNR2int)

14

(B1)

and the intercept factor for conventional monostatic radar as

120572mon =119877int119877mon

= (119875119905sdot 1198702

int sdot SNRmon

119870rad sdot SNR2int)

14

(B2)

When SNRnet = SNRmon we can readily obtain therelationship between the intercept factors for radar network120572net and for the monostatic case 120572mon

120572net =120572mon

11987314

119903

(B3)

Furthermore for monostatic radar we can assume that

SNRmon = 119870rad119875119905

1198774mon= 119870rad

119875maxtot

1198774

119877MAX (B4)

where 119875maxtot is the maximal power of the monostatic rada-

r and 119877119877MAX is the corresponding maximal detection range

Then we can obtain

119877mon119877119877MAX

= (119875119905

119875maxtot

)

14

(B5)

10 International Journal of Antennas and Propagation

Similarly for intercept receiver we have the following

SNRint = 119870int119875119905

1198772

int= 119870int

119875maxtot

1198772

119868119872119860119883

(B6)

119877int119877119868MAX

= (119875119905

119875maxtot

)

12

(B7)

where 119877119868MAX is the intercept range when the transmitting

power of radar is 119875maxtot

Using (15) (B5) and (B7) we can obtain the followingexpression

120572mon120572maxmon

=119877int119877mon

119877119868MAX119877119877MAX

= (119875119905

119875maxtot

)

14

(B8)

where 120572maxmon is Schleher intercept factor corresponding to the

maximal transmitting power 119875maxtot

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their comments that help to improve the quality of thispaper The support provided by the National Natural ScienceFoundation ofChina (Grant no 61371170) Funding of JiangsuInnovation Program for Graduate Education (CXLX13 154)the Fundamental Research Funds for the Central Universi-ties the Priority Academic Program Development of JiangsuHigher Education Institutions (PADA) and Key Laboratoryof Radar Imaging and Microwave Photonics Ministry ofEducation Nanjing University of Aeronautics and Astronau-tics Nanjing China is gratefully acknowledged

References

[1] E P Phillip Detecting and Classifying Low Probability ofIntercept Radar Artech House Boston Mass USA 2009

[2] AM Haimovich R S Blum and L J Cimini Jr ldquoMIMO radarwith widely separated antennasrdquo IEEE Signal Processing Maga-zine vol 25 no 1 pp 116ndash129 2008

[3] A L Hume and C J Baker ldquoNetted radar sensingrdquo in Proceed-ings of the 2001 CIE International Conference on Radar pp 23ndash26 2001

[4] E Fishler A Haimovich R S Blum L J Cimini Jr D Chizhikand R A Valenzuela ldquoSpatial diversity in radars-Models anddetection performancerdquo IEEE Transactions on Signal Processingvol 54 no 3 pp 823ndash838 2006

[5] Y Yang and R S Blum ldquoMIMO radar waveform designbased onmutual information andminimummean-square errorestimationrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 43 no 1 pp 330ndash343 2007

[6] MM Naghsh M HMahmoud S P ShahramM Soltanalianand P Stoica ldquoUnified optimization framework for multi-staticradar code design using information-theoretic criteriardquo IEEETransactions on Signal Processing vol 61 no 21 pp 5401ndash54162013

[7] R Niu R S Blum P K Varshney and A L Drozd ldquoTar-get localization and tracking in noncoherent multiple-inputmultiple-output radar systemsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 48 no 2 pp 1466ndash1489 2012

[8] P Chavali and A Nehorai ldquoScheduling and power allocationin a cognitive radar network for multiple-target trackingrdquo IEEETransactions on Signal Processing vol 60 no 2 pp 715ndash7292012

[9] H Godrich A Petropulu and H V Poor ldquoOptimal powerallocation in distributed multiple-radar configurationsrdquo inProceeding of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 11) pp 2492ndash2495Prague Czech Republic May 2011

[10] H Godrich A P Petropulu and H V Poor ldquoPower allocationstrategies for target localization in distributed multiple-radararchitecturesrdquo IEEE Transactions on Signal Processing vol 59no 7 pp 3226ndash3240 2011

[11] H Godrich A Petropulu and H V Poort ldquoEstimation per-formance and resource savings tradeoffs in multiple radarssystemsrdquo in Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo12) pp 5205ndash5208 Kyoto Japan March 2012

[12] C G Shi J J Zhou F Wang and J Chen ldquoTarget threateninglevel based optimal power allocation for LPI radar networkrdquoin Proceedings of the 2nd International Symposium on Instru-mentation and Measurement Sensor Network and Automation(IMSNA rsquo13) pp 634ndash637 Toronto Canada December 2013

[13] X Song P Willett and S Zhou ldquoOptimal power allocationfor MIMO radars with heterogeneous propagation lossesrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 12) pp 2465ndash2468Kyoto Japan March 2012

[14] X F Song P Willett S Zhou and J Glaz ldquoMIMO radar detec-tion with heterogeneous propagation lossesrdquo in Proceedings ofthe IEEE Statistical Signal Processing Workshop (SSP rsquo12) pp776ndash779 August 2012

[15] D C Schleher ldquoLPI radar fact or fictionrdquo IEEE Aerospace andElectronic Systems Magazine vol 21 no 5 pp 3ndash6 2006

[16] D Lynch Jr Introduction to RF Stealth Sci Tech Publishing2004

[17] A G Stove A L Hume and C J Baker ldquoLow probability ofintercept radar strategiesrdquo IEE Proceedings Radar Sonar andNavigation vol 151 no 5 pp 249ndash260 2004

[18] D E Lawrence ldquoLow probability of intercept antenna arraybeamformingrdquo IEEE Transactions on Antennas and Propaga-tion vol 58 no 9 pp 2858ndash2865 2010

[19] T Kailath ldquoThe divergence and Bhattacharyya distance mea-sures in signal detectionrdquo IEEE Transaction on CommunicationTechnology vol 15 no 2 pp 52ndash60 1967

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 9

than that of traditional monostatic radar and ordinary radarnetwork across the whole region which further shows theLPI enhancement by exploiting our presented LPI optimiza-tion schemes in radar network to defend against passiveintercept receiver Moreover it can be seen in Figure 13 thatin terms of the same system constraints and fundamen-tal quantity Bhattacharyya distance based LPI optimiza-tion is asymptotically equivalent to the J-divergence basedcase

43 Discussion According to Figures 4ndash13 we can deducethe following conclusions for radar network architecture

(1) From Figures 4 to 7 we can observe that as thepredefined threshold of target detection increasesmore transmission power would be allocated forradar network to meet the detection performancewhile the intercept factor is increased subsequentlywhich is vulnerable in electronic warfare In otherwords there exists a tradeoff between LPI and targetdetection performance in radar network system andthe LPI performance would be sacrificed with targetdetection consideration

(2) In the numerical simulations we observe that theproposed optimization schemes (19) and (24) canbe employed to enhance the LPI performance forradar network Based on the netted radarsrsquo spatialdistributionwith respect to the target we can improvethe LPI performance by optimizing transmissionpower allocation among netted radars As indicatedin Figures 11 and 12 netted radars with better channelconditions are favorable over others In additionit can be observed that exploiting our proposedalgorithms can effectively improve the LPI perfor-mance of radar network to defend against interceptreceiver and Bhattacharyya distance based LPI opti-mization algorithm is asymptotically equivalent tothe J-divergence based case under the same systemconstraints and fundamental quantity

5 Conclusions

In this paper we investigated the problem of LPI designin radar network architecture where two LPI optimizationschemes based on information-theoretic criteria have beenproposed The NPGA was employed to tackle the highlynonconvex and nonlinear optimization problems Simula-tions have demonstrated that our proposed strategies areeffective and valuable to improve the LPI performance forradar network and it is indicated that these two optimizationproblems are asymptotically equivalent to each other underthe same system constraints Note that only single target wasconsidered in this paper Nevertheless it is convenient to beextended to multiple targets scenario and the conclusionsobtained in this study suggest that similar LPI benefits wouldbe obtained for the multiple targets case Future work willlook into the adaptive threshold design of target detectionperformance in radar network architectures

Appendices

A

Assume that every transmitter-receiver combination in thenetwork can be the same and 1198772net ≜ 119877

119905119894sdot 119877119903119895 where the radar

network SNR (1) can be written as follows

SNRnet = 119870rad

119873119905

sum

119894=1

119873119903

sum

119895=1

119875119905119894

1198774net (A1)

where

119870rad =1198661199051198661199031205901199051205822

(4120587)3119896119879119900119861119903119865119903119871 (A2)

Assume that the sum of the effective radiated power (ERP)from all the radars in the network is equivalent to that ofmonostatic radar that is

ERP =

119873119905

sum

119894=1

119875119905119894119866119905119894= 119875119905119866119905 (A3)

where 119875119905and 119866

119905are the transmitting power and transmitting

antenna gain of the monostatic radar respectively For 119866119905119894=

119866119905(for all 119894) we can rewrite (A1) as follows

SNRnet = 119870rad119873119905119875119905

1198774net (A4)

B

According to (15) we can derive the intercept factor for radarnetwork as

120572net =119877int119877net

= (119875119905sdot 1198702

int sdot SNRnet

119870rad sdot 119873119903 sdot SNR2int)

14

(B1)

and the intercept factor for conventional monostatic radar as

120572mon =119877int119877mon

= (119875119905sdot 1198702

int sdot SNRmon

119870rad sdot SNR2int)

14

(B2)

When SNRnet = SNRmon we can readily obtain therelationship between the intercept factors for radar network120572net and for the monostatic case 120572mon

120572net =120572mon

11987314

119903

(B3)

Furthermore for monostatic radar we can assume that

SNRmon = 119870rad119875119905

1198774mon= 119870rad

119875maxtot

1198774

119877MAX (B4)

where 119875maxtot is the maximal power of the monostatic rada-

r and 119877119877MAX is the corresponding maximal detection range

Then we can obtain

119877mon119877119877MAX

= (119875119905

119875maxtot

)

14

(B5)

10 International Journal of Antennas and Propagation

Similarly for intercept receiver we have the following

SNRint = 119870int119875119905

1198772

int= 119870int

119875maxtot

1198772

119868119872119860119883

(B6)

119877int119877119868MAX

= (119875119905

119875maxtot

)

12

(B7)

where 119877119868MAX is the intercept range when the transmitting

power of radar is 119875maxtot

Using (15) (B5) and (B7) we can obtain the followingexpression

120572mon120572maxmon

=119877int119877mon

119877119868MAX119877119877MAX

= (119875119905

119875maxtot

)

14

(B8)

where 120572maxmon is Schleher intercept factor corresponding to the

maximal transmitting power 119875maxtot

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their comments that help to improve the quality of thispaper The support provided by the National Natural ScienceFoundation ofChina (Grant no 61371170) Funding of JiangsuInnovation Program for Graduate Education (CXLX13 154)the Fundamental Research Funds for the Central Universi-ties the Priority Academic Program Development of JiangsuHigher Education Institutions (PADA) and Key Laboratoryof Radar Imaging and Microwave Photonics Ministry ofEducation Nanjing University of Aeronautics and Astronau-tics Nanjing China is gratefully acknowledged

References

[1] E P Phillip Detecting and Classifying Low Probability ofIntercept Radar Artech House Boston Mass USA 2009

[2] AM Haimovich R S Blum and L J Cimini Jr ldquoMIMO radarwith widely separated antennasrdquo IEEE Signal Processing Maga-zine vol 25 no 1 pp 116ndash129 2008

[3] A L Hume and C J Baker ldquoNetted radar sensingrdquo in Proceed-ings of the 2001 CIE International Conference on Radar pp 23ndash26 2001

[4] E Fishler A Haimovich R S Blum L J Cimini Jr D Chizhikand R A Valenzuela ldquoSpatial diversity in radars-Models anddetection performancerdquo IEEE Transactions on Signal Processingvol 54 no 3 pp 823ndash838 2006

[5] Y Yang and R S Blum ldquoMIMO radar waveform designbased onmutual information andminimummean-square errorestimationrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 43 no 1 pp 330ndash343 2007

[6] MM Naghsh M HMahmoud S P ShahramM Soltanalianand P Stoica ldquoUnified optimization framework for multi-staticradar code design using information-theoretic criteriardquo IEEETransactions on Signal Processing vol 61 no 21 pp 5401ndash54162013

[7] R Niu R S Blum P K Varshney and A L Drozd ldquoTar-get localization and tracking in noncoherent multiple-inputmultiple-output radar systemsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 48 no 2 pp 1466ndash1489 2012

[8] P Chavali and A Nehorai ldquoScheduling and power allocationin a cognitive radar network for multiple-target trackingrdquo IEEETransactions on Signal Processing vol 60 no 2 pp 715ndash7292012

[9] H Godrich A Petropulu and H V Poor ldquoOptimal powerallocation in distributed multiple-radar configurationsrdquo inProceeding of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 11) pp 2492ndash2495Prague Czech Republic May 2011

[10] H Godrich A P Petropulu and H V Poor ldquoPower allocationstrategies for target localization in distributed multiple-radararchitecturesrdquo IEEE Transactions on Signal Processing vol 59no 7 pp 3226ndash3240 2011

[11] H Godrich A Petropulu and H V Poort ldquoEstimation per-formance and resource savings tradeoffs in multiple radarssystemsrdquo in Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo12) pp 5205ndash5208 Kyoto Japan March 2012

[12] C G Shi J J Zhou F Wang and J Chen ldquoTarget threateninglevel based optimal power allocation for LPI radar networkrdquoin Proceedings of the 2nd International Symposium on Instru-mentation and Measurement Sensor Network and Automation(IMSNA rsquo13) pp 634ndash637 Toronto Canada December 2013

[13] X Song P Willett and S Zhou ldquoOptimal power allocationfor MIMO radars with heterogeneous propagation lossesrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 12) pp 2465ndash2468Kyoto Japan March 2012

[14] X F Song P Willett S Zhou and J Glaz ldquoMIMO radar detec-tion with heterogeneous propagation lossesrdquo in Proceedings ofthe IEEE Statistical Signal Processing Workshop (SSP rsquo12) pp776ndash779 August 2012

[15] D C Schleher ldquoLPI radar fact or fictionrdquo IEEE Aerospace andElectronic Systems Magazine vol 21 no 5 pp 3ndash6 2006

[16] D Lynch Jr Introduction to RF Stealth Sci Tech Publishing2004

[17] A G Stove A L Hume and C J Baker ldquoLow probability ofintercept radar strategiesrdquo IEE Proceedings Radar Sonar andNavigation vol 151 no 5 pp 249ndash260 2004

[18] D E Lawrence ldquoLow probability of intercept antenna arraybeamformingrdquo IEEE Transactions on Antennas and Propaga-tion vol 58 no 9 pp 2858ndash2865 2010

[19] T Kailath ldquoThe divergence and Bhattacharyya distance mea-sures in signal detectionrdquo IEEE Transaction on CommunicationTechnology vol 15 no 2 pp 52ndash60 1967

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 International Journal of Antennas and Propagation

Similarly for intercept receiver we have the following

SNRint = 119870int119875119905

1198772

int= 119870int

119875maxtot

1198772

119868119872119860119883

(B6)

119877int119877119868MAX

= (119875119905

119875maxtot

)

12

(B7)

where 119877119868MAX is the intercept range when the transmitting

power of radar is 119875maxtot

Using (15) (B5) and (B7) we can obtain the followingexpression

120572mon120572maxmon

=119877int119877mon

119877119868MAX119877119877MAX

= (119875119905

119875maxtot

)

14

(B8)

where 120572maxmon is Schleher intercept factor corresponding to the

maximal transmitting power 119875maxtot

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous reviewersfor their comments that help to improve the quality of thispaper The support provided by the National Natural ScienceFoundation ofChina (Grant no 61371170) Funding of JiangsuInnovation Program for Graduate Education (CXLX13 154)the Fundamental Research Funds for the Central Universi-ties the Priority Academic Program Development of JiangsuHigher Education Institutions (PADA) and Key Laboratoryof Radar Imaging and Microwave Photonics Ministry ofEducation Nanjing University of Aeronautics and Astronau-tics Nanjing China is gratefully acknowledged

References

[1] E P Phillip Detecting and Classifying Low Probability ofIntercept Radar Artech House Boston Mass USA 2009

[2] AM Haimovich R S Blum and L J Cimini Jr ldquoMIMO radarwith widely separated antennasrdquo IEEE Signal Processing Maga-zine vol 25 no 1 pp 116ndash129 2008

[3] A L Hume and C J Baker ldquoNetted radar sensingrdquo in Proceed-ings of the 2001 CIE International Conference on Radar pp 23ndash26 2001

[4] E Fishler A Haimovich R S Blum L J Cimini Jr D Chizhikand R A Valenzuela ldquoSpatial diversity in radars-Models anddetection performancerdquo IEEE Transactions on Signal Processingvol 54 no 3 pp 823ndash838 2006

[5] Y Yang and R S Blum ldquoMIMO radar waveform designbased onmutual information andminimummean-square errorestimationrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 43 no 1 pp 330ndash343 2007

[6] MM Naghsh M HMahmoud S P ShahramM Soltanalianand P Stoica ldquoUnified optimization framework for multi-staticradar code design using information-theoretic criteriardquo IEEETransactions on Signal Processing vol 61 no 21 pp 5401ndash54162013

[7] R Niu R S Blum P K Varshney and A L Drozd ldquoTar-get localization and tracking in noncoherent multiple-inputmultiple-output radar systemsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 48 no 2 pp 1466ndash1489 2012

[8] P Chavali and A Nehorai ldquoScheduling and power allocationin a cognitive radar network for multiple-target trackingrdquo IEEETransactions on Signal Processing vol 60 no 2 pp 715ndash7292012

[9] H Godrich A Petropulu and H V Poor ldquoOptimal powerallocation in distributed multiple-radar configurationsrdquo inProceeding of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 11) pp 2492ndash2495Prague Czech Republic May 2011

[10] H Godrich A P Petropulu and H V Poor ldquoPower allocationstrategies for target localization in distributed multiple-radararchitecturesrdquo IEEE Transactions on Signal Processing vol 59no 7 pp 3226ndash3240 2011

[11] H Godrich A Petropulu and H V Poort ldquoEstimation per-formance and resource savings tradeoffs in multiple radarssystemsrdquo in Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP rsquo12) pp 5205ndash5208 Kyoto Japan March 2012

[12] C G Shi J J Zhou F Wang and J Chen ldquoTarget threateninglevel based optimal power allocation for LPI radar networkrdquoin Proceedings of the 2nd International Symposium on Instru-mentation and Measurement Sensor Network and Automation(IMSNA rsquo13) pp 634ndash637 Toronto Canada December 2013

[13] X Song P Willett and S Zhou ldquoOptimal power allocationfor MIMO radars with heterogeneous propagation lossesrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP 12) pp 2465ndash2468Kyoto Japan March 2012

[14] X F Song P Willett S Zhou and J Glaz ldquoMIMO radar detec-tion with heterogeneous propagation lossesrdquo in Proceedings ofthe IEEE Statistical Signal Processing Workshop (SSP rsquo12) pp776ndash779 August 2012

[15] D C Schleher ldquoLPI radar fact or fictionrdquo IEEE Aerospace andElectronic Systems Magazine vol 21 no 5 pp 3ndash6 2006

[16] D Lynch Jr Introduction to RF Stealth Sci Tech Publishing2004

[17] A G Stove A L Hume and C J Baker ldquoLow probability ofintercept radar strategiesrdquo IEE Proceedings Radar Sonar andNavigation vol 151 no 5 pp 249ndash260 2004

[18] D E Lawrence ldquoLow probability of intercept antenna arraybeamformingrdquo IEEE Transactions on Antennas and Propaga-tion vol 58 no 9 pp 2858ndash2865 2010

[19] T Kailath ldquoThe divergence and Bhattacharyya distance mea-sures in signal detectionrdquo IEEE Transaction on CommunicationTechnology vol 15 no 2 pp 52ndash60 1967

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of