1 National Laboratory of Antennas and Microwave Technology Xidian University Xirsquoan 710071 China2 School of Electronic Engineering Beihang University Beijing 100191 China3 Science and Technology on Space Physics Laboratory Beijing 100076 China
Correspondence should be addressed to Mingyuan Man man-ming-yuan163com
Received 15 June 2014 Accepted 20 October 2014 Published 17 November 2014
Academic Editor Felipe Catedra
Copyright copy 2014 Mingyuan Man 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
A demonstrated hybrid method based on the combination of half-space physical optics method (PO) graphical-electromagneticcomputing (GRECO) and Monte Carlo method on echo signals from low-flying targets based on actual environment for airborneradar is presented in this paper The half-space physical optics method combined with the graphical-electromagnetic computing(GRECO) method to eliminate the shadow regions quickly and rebuild the target automatically is employed to calculate the radarcross section (RCS) of the conductive targets in half space fast and accurately The direct echo is computed based on the radarequationThe reflected paths from sea or ground surface cause multipath effects In order to accurately obtain the echo signals thephase factors are modified for fluctuations in multipath and the statistical average value of the echo signals is obtained using theMonte Carlo method A typical simulation is performed and the numerical results show the accuracy of the proposed method
1 Introduction
Due to Earth curvature terrain masking and clutter interfer-ence low altitude targets are difficult to detect by the groundradar [1ndash3] Hence the flying targets including militaryaircrafts and crushing missiles can make low or ultralowaltitude flying to avoid the radar detectionHowever airborneearly warning (AEW) radar plays a primary role in thedetection of low-flying aircraft which are out of the coverageof ground radars [4 5] The AEW system also plays animportant role in other aspects such as the coordination ofsearch and rescue and airborne rendezvous control Owing tovast cost in anAEW system experiment and other difficultiesthe simulation of the AEW system based on computers is aneffective way to evaluate its performance [6ndash10] An accurateestimate of the detection probability and other performancesin look-down mode for airborne radar can be obtained bya simulation code that reproduces the output of the signalprocessor in every practical operating condition To improvethe detection performance it is necessary to accurately obtainthe echo signals from low-flying targets for airborne radar
The problem has been studied by many other authors[1 2 11 12] The work finished in [1 2] has dealt with AEWradar system in detail and briefly introduced the theory ofAEW radar system on evaluating the electromagnetic charac-teristics of the interactions between the targets and the envi-ronmentThe tropospheric propagation channel backgroundis exactly described for low-altitude surveillance and wide-band frequency is utilized to mitigate and exploit specularmultipath channel characteristics in [11] The experimentaldata for a variety of terrain types is presented in [13] whichindicates that a statistical analysis is required for the signalstrength in practice However the experiments are carriedout just for one-way propagation The probability densityfunction (PDF) of targetrsquos echo in the presence of multipatheffects is discussed in [12] but only two ways of echoes areconsidered with the flat-earth geometry When the radarrsquosaltitude increases some important assumptions will not besatisfied
The statistical simulation of echo signals from low altitudetargets based on actual environment for airborne radar is pre-sented in this paper The low-flying target RCS is calculated
Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 416985 7 pageshttpdxdoiorg1011552014416985
2 International Journal of Antennas and Propagation
by using the quasistationary approximation The half-spacephysical optics integral equation is derived by introducingthe half-space Greenrsquos function into PO Combined with thegraphical-electromagnetic computing method the RCS ofconductive targets can be calculated in half space exactlyand efficiently Along with the description of the dynamicsituations the direct echo is computed based on the radarequation There are reflected paths from sea or groundsurface giving rise to multipath effects The phase factors aremodified using theMonte Carlomethod in order to eliminatethe fluctuations caused by multipath and the statisticalcharacteristics of echoes are obtained at last
The organization of this paper is shown as follows Inthe first part of Section 2 the relative incident direction isderived by coordinate transformation and the RCS of thelow-flying target is calculated by the half-space physic-opticsmethod combined with GRECO [14] The second part ofSection 2 describes multipath propagation modeling andintroduces the Monte Carlo method to modify the phasefactors of different paths for the statistical characteristics ofecho signals In Section 3 some simulations are given aswell as the results and comparisons among the simulationsConclusion is drawn in the final section
2 Theory
21 Dynamic RCS of Low-Fly Targets The echo signal poweris dependent on the target RCS As the low-altitude target isflying closely to the ground or sea surface its RCS is quitedifferent from the one in the free space The incident wavedirection of propagation is determined by the radar-targetengagement geometry and described by the azimuth angle 120593
119903
and the elevation angle 120579119903with respect to the target body-
fixed axes In order to calculate the target RCS of the targetilluminated by the radar 120593
119903and 120579
119903should be achieved by
coordinate transformation [15] with the given longitudeslatitudes heights and attitudes of the radar and the target(Table 1) When the incident wave direction of propagation isobtained the low-flying target RCS can be exactly calculatedusing half-space physic-optics method [16 17]
211 Half-Space PO for Radar Targets Considering an arbi-trarily shaped object illuminated by a plane wave in the halfspace surface 119878 is assumed to represent a closed surface oftarget The far zone scattering fields of the conductive targetin half space can be expressed as
119864119904(119903
1015840
) = minus119895120596119860 (1199031015840
) minus nablaΦ (1199031015840
) 1199031015840
isin 119878 (1)
where119860 andΦ represent the vector and scalar potentials dueto the surface current 119869(1199031015840) respectively
The half-space Greenrsquos function can be expressed by thevector or scalar potentials and the vector potential is notuniquely specified among literatures It is expressed in thispaper using the form mentioned in [18] as follows
119866119860= (119909119909 + 119910119910)119866
119909119909
119860+ 119909119866
119911119909
119860+ 119910119866
119911119910
119860+ 119866
119911119911
119860 (2)
in which
119866119910119910
119860= 119866
119909119909
119860 119866
119902119890
119910= 119866
119902119890
119909 (3)
where 119866119909119909
119860 119866119911119909
119860 119866119911119910
119860 and 119866119911119911
119860denote the spatial domain
half-space Greenrsquos function for the vector potentials and 119866119902119890
119909
119866119902119890
119910 and 119866119902119890
119911denote the spatial domain half-space Greenrsquos
function for the scalar potentials [19 20] According to theposition of targets the terrain below targets can be derivedfrom digital feature analysis data (DFAD) Then the groundcharacteristic is introduced into half-space Greenrsquos functionThe fluctuant ground is equivalent to the medium surface inlower half-space
As stated before the half-space Greenrsquos function has beenintroduced into PO to compute the electrically large targets inthe half space Combined with the graphical electromagneticcomputingmethod (GRECO) [14] the geometry informationof each illuminated plane is obtained by reading the colorsand depths of each pixel and the shadow regions are elim-inated by displaying lists technology of OpenGL to rebuildthe target Hidden surfaces of the image have been previouslyremoved by the hardware graphics acceleratorThenwemakeuse of the resolution to disperse the curve face into pixels thatsatisfy the requirement of the electromagnetic calculationmeanwhile the scene is rendered using the Phong localillumination model [21] For three light sources of purelygreen red and blue colors respectively located over each oneof the three coordinate axises the three color components forthis pixel are equal to the (119899
119909 119899
119910 119899
119911) components of the unit
normal to surface Meanwhile the depths of each pixel areobtained in the same way
The far zone scattering fields of the conductive target inhalf space is expressed as
119864119904(119903)
= minus119895120596∭V119866
119860sdot 119869 (119903
1015840
) 1198891199031015840
+119896 sdot 119904
120596
times∭V[119866
119902119890
119909
120597
120597119909119869 (119903
1015840
) + 119866119902119890
119910
120597
120597119910119869 (119903
1015840
) + 119866119902119890
119911
120597
120597119911119869 (119903
1015840
)] 1198891199031015840
(4)
Equation (4) is then calculated for every illuminated facetand the complex RCS due to scattering from every illumi-nated facet is calculated as [22]
radic120590 = lim119877rarrinfin
2radic120587119877119864119904sdot 119890
119903
119864119900
exp (119895119896119877) (5)
where exp(119895119896119877) is the phase term which is introduced intothe equation in order to account for the facet location withrespect to the global coordinate system and 119877 is the positionvector for the facetrsquos reference vertexwith respect to the globalcoordinate system
By using half-space PO to evaluate the scattering fromthe target in half-space note that the use of GRECO toremove shadowed surfaces is valid for the direct field inci-dence direction not for ground reflections This is valid onlyif the target is low flying because the incident andground-reflection directions are almost parallel In this case
International Journal of Antennas and Propagation 3
Table 1 Some primary parameters
Parameter Value Parameter ValueRadarrsquos longitude 123∘ Radarrsquos latitude 316∘
Targetrsquos latitude 319∘ Targetrsquos height 20mRadarrsquos emission power 1000 kW Frequency 04GHzEmission signal modality Linear frequency modulation Emission signal pulse width 13 120583sPulse compression ratio 13 120583s220 ns Aircraftrsquos velocity 400msAntenna maximal gain 32 dB Beam width 65∘
Polarization type Horizontal polarization Targetrsquos roll angle 5∘
20 40 60 80 100 120 140 160
0
10
20
30
Aspect angle
RCS
(dbs
m)
Ei
Ei
1205824x
z
y
8120582
minus10
minus20
minus30
Prediction of half spaceSimulation of free spaceSimulation of half space
Figure 1 Comparison of prediction and simulation of software foran 8120582 times 8120582 plane in half space (frequency = 10GHz)
the demonstrated half-space GRECO is appropriate to theanalysis of the scattering from the low-flying target
Figure 1 shows a perfectly conducting flat plane (8120582 times8120582) which is placed 1205824 above the soil 120582 represents thewavelength in the vacuumThe relative dielectric permittivityof the soil is 120576
119903= 40 and the relative magnetic permeability
is 120583119903= 10
Figure 1 shows that the half space RCS and the free spaceRCS matches very well at the backscatter angle Howeverat other incident angles there are noticeable differencesbetween them for 4-5 dbsm Hence the conclusion is drawnthat it is necessary to consider the effect of the half-spaceenvironment
22 Multipath Effects
221 Multipath Model In detecting low altitude targetsthe presence of reflections from sea or ground surface cancause multipath propagation In some literatures the implicit
assumption is that only one reflection path exists [12] How-ever in practice four different paths including direct-directdirect-reflected reflected-direct and reflected-reflected arecontributing to the radar received signals as shown analyt-ically in Figure 2 The total echo from a low-flying targetis thus a superposition of four ways of signals each witha different path Multipath propagation can cause signalcancellation in adaptive beamforming [23] and errors in low-angle radar tracking [24] A highly deterministic multipathmodel is constituted in [25]
The complex RCS can be written as radic120590119890119895120601119905 Based on theradar equation [1] the complex voltage coefficient of direct-direct echo signal received by the radar is expressed as
1198811(119905) = (
1198751199051205822
120590
(4120587)3
119871)
12
sdot119891 (120579
119889)
1198772119890119895(120601119903+120601119905)
(6)
where 119875119905is the radar transmit peak power 120582 is radar wave-
length 120590 is the RCS of target 119891(120579119889) is radar antenna gain
120579119889is the angle between the target and radar beam boresight
119871 is composite loss 119877 is the distance from the radar to thecenter of target and can be obtained through the coordinatesof radar and target and 120601
119903is the total phase brought by the
receiver systemSince the path differences of the direct and multipath
returns are usually less than the range resolution of the radarthe different returns are delivered into relevant range gatesresulting in fluctuations Thus the magnitude of the super-position of four signals can be expressed as
4 International Journal of Antennas and Propagation
Radar
Target
Path 1120579d
h1
h2
(a)
Radar
Target
Path 2120579d120579t
h1
h2
h998400
1
d1 d2
Ψ
Ψ
h998400
2
(b)
Radar
Target
Path 3120579d120579t
h1
h2
h998400
1
d1 d2
Ψ
Ψ
h998400
2
(c)
Radar
TargetPath 4
120579t
h1
h2
h998400
1
d1 d2
Ψ
Ψ
h998400
2
(d)
Figure 2 The radar-target geometry in the presence of multipath (a) Path 1 (b) Path 2 (c) Path 3 (d) Path 4
1205901120590
2120590
3 and120590
4are themagnitudes of complexRCS of differ-
ent ways 120588 is the reflection coefficientΔ1198711Δ119871
2 andΔ119871
3are
the path length differences of three reflected rays compared tothe direct 119877
1and 119877
2are the distances of radar-to-reflection
point and target-to-reflection point respectivelyMultipath interference from a rough surface generally
involves two components which are specular and dif-fuse reflections The specular reflection coefficient can beexpressed as
120588119904= 120588
0119877119904119863 (8)
where 1205880is Fresnel reflection coefficient 119877
119904is the specular
scattering factor and 119863 is divergence factor For horizontalpolarization 120588
0is expressed as [25]
1205880119867=sin120595 minus (120576
119888minus cos2120579)
12
sin120595 + (120576119888minus cos2120579)12
(9)
For vertical polarization it is expressed as
1205880119881=120576119888sin120595 minus (120576
119888minus cos2120579)
12
120576119888sin120595 + (120576
119888minus cos2120579)12
(10)
The divergence factor119863 is taken into account due to the cur-vature of the Earth and given as
119863 = [1 +2119877
11198772
119886 (1198771+ 119877
2) sin120595
]
12
(11)
The root mean square (RMS) value of the specular scatteringfactor is given as
119877119904= exp[minus2 (
2120587120590ℎsin120595120582
)
2
] (12)
Here 120590ℎis the RMS value of the wave height
The diffuse reflection coefficient is given as
120588119863= 120588
0119877119863 (13)
where 1205880is the Fresnel reflection coefficient defined in (9) and
(10) and 119877119863is the diffuse scattering coefficient [26]
The Rayleigh roughness criterion from optical theory iscommonly used to estimate the maximum surface irregular-ity that will not significantly lower the reflection coefficient[27 28]
Δℎ lt120582
8 sin120595 (14)
where Δℎ is the maximum peak-to-trough variations and 120595is the grazing angle of reflection point If (14) is satisfied thesurface can behave as an essentially smooth dielectric wherethe specular reflection will occur
222 Estimation Approaches Based on Monte Carlo MethodsThe one-way propagation experimental data for a varietyof terrain types are presented in [13] which indicate astatistical simulation is required for the signal strength inpractice References [1 2 11] treat the path length differencesΔ119871
1 Δ119871
2 and Δ119871
3as deterministic variables However the
errors caused by multipath geometry calculation are so largewith respect to the wavelength that they can confuse theinterference For a radar operating at 1 GHz an error of 015meter will cause a phase error of 180∘ Also due to airborneradar working at a large height the gracing angle of the targetis usually so large [1] that it causes a significant difference ofdirection between the direct and reflected ray which makesthe phases of scattering fields of the direct and reflected rayquite different from each other Hence the phase factors willbe treated as variables which satisfy some PDF The statisticpeculiarity of echo is calculated with Monte Carlo methods
International Journal of Antennas and Propagation 5
Monte Carlo methods are algorithm for solving variouskinds of computational problems by using random numbers[29] It is a means of treating mathematical problems byfinding a probabilistic analog and then obtaining approxi-mate answers to this analog by some experimental samplingprocedureThe solution of a problem by this method is closerin spirit to physical experiments than to classical numericaltechniques In our calculations the number of samples isincreased to achieve a fairly regular distribution As for theerror from the Monte Carlo sampling technique it can bemade negligible bymaking the number of samples sufficientlylarge
To simulate the instantaneous amplitude from echo sig-nals withMonte Carlomethod firstly the probabilistic modelaccording to the problem should be established From for-mula (7) the phase factors can be rewritten as
1205931= (
2120587Δ1198711
120582) mod (2120587)
1205932= (
2120587Δ1198712
120582) mod (2120587)
1205933= (
2120587Δ1198713
120582) mod (2120587)
(15)
where 1205931 120593
2 and 120593
3are the remainder of phase differences
divide 2120587 of three reflected rays compared to the direct Forairborne radar we know that the path length differences ofthree reflected rays compared to the direct can far exceedthe wave length so 120593
1 120593
2 and 120593
3can be supposed to three
sequences of random numbers from a uniform distributionin (0 2120587)
With the most widely used mixed congruence method[30] we generate three sequences of random numbers 119886
119894 119887
119894
and 119888119894(119894 = 1 2 119873) which is uniformly distributed in the
interval (0 1)
119886119894=[(119898
119886119886119894minus1+ 119899
119886) mod119872]
119872
119887119894=[(119898
119887119887119894minus1+ 119899
119887) mod119872]
119872
119888119894=[(119898
119888119888119894minus1+ 119899
119888) mod119872]
119872
119894 = 1 2 119873
(17)
0 2040
6080
100
0
50
1000
5
10
15
20
0533m
617m262m
Figure 3 The missile is placed 20m above the sea
where 119898119886 119898
119887 and 119898
119888are the multipliers 119899
119886 119899
119887 and 119899
119888are
the increment119872 is the modulusGiven120593
1= 2120587119886
119894120593
2= 2120587119887
119894 and120593
3= 2120587119888
119894in formula (16)
run the Monte Carlo simulation n times to estimate the echosignals from low altitude targets accurately based on actualenvironment for airborne radar
By using a sufficient number of samples a probability dis-tribution is obtained to describe echo signals The statisticalcharacteristics of the echo signals can be illustrated at last
3 Results and Discussion
Themissile is adopted in the simulation as shown in Figure 3with its RCS in Figure 4
The PDF of magnitude of echoes is shown in Figure 5when the target is flying at a height of 20m above the seaAccording to the radar equation the magnitude of direct rayis 64416 times 10minus7 V Figure 5 indicates that the magnitude ofechoes can be different from direct ray due to multipath
The magnitudes of direct echoes in the situations of dif-ferent targetrsquos heights are shown in Figure 6 and the PDFs ofmagnitude of echoes normalized to the direct ray in the samesituations are shown in Figure 7 As can be seen in Figure 6the magnitudes of direct echoes vary with targetrsquos heightsThe prime reason for that is the target RCSrsquos variation causedby the different azimuth angle 120593 and the elevation angle 120579with respect to the coordinate system of the target Figure 7shows the influences of multipath at different heights In theconditions of the target flying at low heights (about a fewhundred meters) the PDFs vary little with the heights whilethe normalized magnitude of echoes with respect to the peakof the PDF reduces a little As illustrated in Figure 7 when thetarget is at heights of below 2000m themagnitudes of echoesnormalized to the direct ray vary mainly between 0 and 4The peak probability densities are in the range of 0-1 and thecorresponding normalized magnitudes of echoes are in therange of 1-2 These results in Figure 7 are similar to the onesin Figures 14 and 15 of [13] which make the demonstratedalgorithm valid in this paper However the terrain in thispaper was produced from the digital feature analysis data
6 International Journal of Antennas and Propagation
Figure 4 The missilersquos RCS with frequency of 04GHz
Prob
abili
ty d
ensit
y
The magnitude of echoes (V) times10minus6
07
06
05
04
03
02
01
0
0 05 1 15 2 25 3
Figure 5 The PDF versus magnitude of echoes for the target at aheight of 20m above the sea
(DFAD) yet in [13] the desert terrain for simulations wasproduced from the defense mapping agency (DMA) data theexperiments weremade inNewMexico andNorthernMaineThe differences between the results in this paper and those in[13] are mainly caused by the differences in the terrains thefrequencies the geometries and attitudes of the targets theheights of the airborne radars and the targets the numberof samples and so forth As the target is flying higher theinfluences of multipath become less to the echoes until theindirect ways disappear In practice if the target height isat least moderately large the range resolution of the radaris less than the path differences of the direct and multipathreturns so the radar is able to resolve them As radar heightdecreases to a height of ground based radar the differencein angle between reflected and direct paths can be ignored
20100 500 1000 20000
1
2
3
4
5
6
7
8
The target height (m)
The m
agni
tude
of d
irect
ray
(V)
times10minus7
Figure 6 The magnitude of direct ray versus the target height
4Normalized magnitude of echoes
Prob
abili
ty d
ensit
y
0 05 1 15 2 25 3 35
0
05
1
15
2
25
3
h = 20mh = 100mh = 500m
h = 1000mh = 2000m
Figure 7 The PDF of magnitude of echoes at different targetrsquosheights
which causes the fluctuations in phase disappearing for low-flying targets and the PDF of magnitude of echoes is similarto Figure 6 of [12]
4 Conclusions
For airborne radar to detect low altitude targets a demon-strated hybrid method based on probability statistics andcomputational electromagnetics is proposed in this paperto give a statistical simulation of echo signals based onactual environmentThe half-space physical optics combinedwith the graphical-electromagnetic computing method isemployed to calculate the RCS of low-flying targets suffi-ciently accurately and efficiently in half space Considering
International Journal of Antennas and Propagation 7
the fluctuations due to the multipath effects the phasefactors are modified with theMonte Carlo methodWith thismethod the target echo signals from low altitude targets canbe obtained accurately for the radar simulation system andthe statistic characteristics of echo signals can be accuratelyand properly simulated The demonstrated method in thispaper is of great help for the radar researcher to evaluate thedetection probability and false alarm probability of specialtargets the signal-to-noise ratio (SNR) of emission pulsesand so on
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M Long Airborne Early Warning System Concepts ArtechBoston Mass USA 1992
[2] W Morchin Airborne Early Warning Radar Artech HouseNorwell Mass USA 1990
[3] L-XGuoA-QWang and JMa ldquoStudy onEMscattering from2-D target above 1-D large scale rough surface with low grazingincidence by parallel MOM based on PC Clustersrdquo Progress inElectromagnetics Research vol 89 pp 149ndash166 2009
[4] M I Skolnik Introduction to Radar SystemsMcGraw-Hill NewYork NY USA 3rd edition 2001
[5] Y Qu G S Liao S Q Zhu and X Y Liu ldquoPattern synthesisof planar antenna array via convex optimization for airborneforward looking radarrdquo Progress in Electromagnetics Researchvol 84 pp 1ndash10 2008
[6] S H Lim J H Han S Y Kim and N H Myung ldquoAzimuthbeam pattern synthesis for air-borne sar system optimizationrdquoProgress in Electromagnetics Research vol 106 pp 295ndash3092010
[7] Y-L Chang C-Y Chiang and K-S Chen ldquoSAR image sim-ulation with application to target recognitionrdquo Progress in Elec-tromagnetics Research vol 119 pp 35ndash57 2011
[8] M Zhang Y W Zhao H Chen and W Q Jiang ldquoSar imagingsimulation for composite model of ship on dynamic oceanscenerdquo Progress in Electromagnetics Research vol 113 pp 395ndash412 2011
[9] S Qiao Z G Shi K S Chen et al ldquoA new architecture ofUWB radar utilizing microwave chaotic signals and chaos syn-chronizationrdquo Progress in Electromagnetics Research vol 75 pp225ndash237 2007
[10] M-J Wang Z-S Wu Y-L Li and G Zhang ldquoHigh resolutionrange profile identifying simulation of laser radar based onpulse beam scattering characteristics of targetsrdquo Progress inElectromagnetics Research vol 96 pp 193ndash204 2009
[11] J G Teti Jr ldquoWide-band airborne radar operating considera-tions for low-altitude surveillance in the presence of specularmultipathrdquo IEEE Transactions on Antennas and Propagationvol 48 no 2 pp 176ndash191 2000
[12] S L Wilson and B D Carison ldquoRadar detection in multipathrdquoIEE Proceedings Radar Sonar and Navigation vol 146 no 1 pp45ndash52 1999
[13] L M Zurk ldquoExperimental observation and statistics of multi-path from terrain with application to overland height findingrdquo
IEEE Transactions on Antennas and Propagation vol 47 no 1pp 121ndash131 1999
[14] Y B Tao H Lin and H J Bao ldquoFrom CPU to GPU GPU-based electromagnetic computingrdquo Progress in ElectromagneticsResearch vol 81 pp 1ndash19 2008
[15] B Etkin andLD ReidDynamics of Flight Stability andControlJohn Wiley amp Sons New York NY USA 1996
[16] X F Li Y J Xie and R Yang ldquoBistatic RCS prediction forcomplex targets using modified current marching techniquerdquoProgress in Electromagnetics Research vol 93 pp 13ndash28 2009
[17] X F Li Y J Xie P Wang and T M Yang ldquoHigh-frequencymethod for scattering from electrically large conductive targetsin half-spacerdquo IEEE Antennas and Wireless Propagation Lettersvol 6 pp 259ndash262 2007
[18] K A Michalski and D Zheng ldquoElectromagnetic scattering andradiation by surfaces of arbitrary shape in layered media ITheoryrdquo IEEE Transactions on Antennas and Propagation vol38 no 3 pp 335ndash344 1990
[19] H Bagci A E Yilmaz V Lomakin and E Michielssen ldquoFastsolution of mixed-potential time-domain integral equations forhalf-space environmentsrdquo IEEE Transactions on Geoscience andRemote Sensing vol 43 no 2 pp 269ndash279 2005
[20] R C Acar and G Dural ldquoComplete set of closed-form Greenrsquosfunctions for cylindrically layered mediardquo in Proceedings of theIEEE Antennas and Propagation Society International Sympo-sium 2006 pp 2863ndash2866 2006
[21] J M Rius M Ferrando and L Jofre ldquoHigh-frequency RCSof complex radar targets in real-timerdquo IEEE Transactions onAntennas and Propagation vol 41 no 9 pp 1308ndash1319 1993
[22] E F Knott J F Shaeffer and M T Tuley Radar Cross SectionSciTech Publishing Raleigh NC USA 2004
[23] B Widrow K M Duvall R P Gooch and W C NewmanldquoSignal cancellation phenomena in adaptive antennas causesand curesrdquo IEEETransactions onAntennas and Propagation vol30 no 3 pp 469ndash478 1982
[24] WDWhite ldquoLow angle radar tracking in the presence ofmulti-pathrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 10 no 6 pp 335ndash352 1974
[25] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings Part F Radarand Signal Processing vol 138 no 2 pp 163ndash171 1991
[26] P Beckmann and A Spizzichino The Scattering of Electro-Magnetic Waves from Rough Surfaces Artech House NorwoodMass USA 1987
[27] C I Beard ldquoCoherent and incoherent scattering of microwavesfrom the oceanrdquo IEEE Transactions on Antenna and Propaga-tion vol 9 no 5 pp 470ndash483 1961
[28] N Pinel C Bourlier and J Saillard ldquoDegree of roughness ofrough layers extensions of the Rayleigh roughnessrdquo Progress inElectromagnetics Research B no 19 pp 41ndash63 2010
[29] M Mishra and N Gupta ldquoMonte Carlo integration techniquefor the analysis of electromagnetic scattering from conductingsurfacesrdquo Progress in Electromagnetics Research vol 79 pp 91ndash106 2008
[30] M A Herrador A G Asuero and A G Gonzalez ldquoEstimationof the uncertainty of indirect measurements from the propa-gation of distributions by using the Monte-Carlo method anoverviewrdquoChemometrics and Intelligent Laboratory Systems vol79 no 1-2 pp 115ndash122 2005
2 International Journal of Antennas and Propagation
by using the quasistationary approximation The half-spacephysical optics integral equation is derived by introducingthe half-space Greenrsquos function into PO Combined with thegraphical-electromagnetic computing method the RCS ofconductive targets can be calculated in half space exactlyand efficiently Along with the description of the dynamicsituations the direct echo is computed based on the radarequation There are reflected paths from sea or groundsurface giving rise to multipath effects The phase factors aremodified using theMonte Carlomethod in order to eliminatethe fluctuations caused by multipath and the statisticalcharacteristics of echoes are obtained at last
The organization of this paper is shown as follows Inthe first part of Section 2 the relative incident direction isderived by coordinate transformation and the RCS of thelow-flying target is calculated by the half-space physic-opticsmethod combined with GRECO [14] The second part ofSection 2 describes multipath propagation modeling andintroduces the Monte Carlo method to modify the phasefactors of different paths for the statistical characteristics ofecho signals In Section 3 some simulations are given aswell as the results and comparisons among the simulationsConclusion is drawn in the final section
2 Theory
21 Dynamic RCS of Low-Fly Targets The echo signal poweris dependent on the target RCS As the low-altitude target isflying closely to the ground or sea surface its RCS is quitedifferent from the one in the free space The incident wavedirection of propagation is determined by the radar-targetengagement geometry and described by the azimuth angle 120593
119903
and the elevation angle 120579119903with respect to the target body-
fixed axes In order to calculate the target RCS of the targetilluminated by the radar 120593
119903and 120579
119903should be achieved by
coordinate transformation [15] with the given longitudeslatitudes heights and attitudes of the radar and the target(Table 1) When the incident wave direction of propagation isobtained the low-flying target RCS can be exactly calculatedusing half-space physic-optics method [16 17]
211 Half-Space PO for Radar Targets Considering an arbi-trarily shaped object illuminated by a plane wave in the halfspace surface 119878 is assumed to represent a closed surface oftarget The far zone scattering fields of the conductive targetin half space can be expressed as
119864119904(119903
1015840
) = minus119895120596119860 (1199031015840
) minus nablaΦ (1199031015840
) 1199031015840
isin 119878 (1)
where119860 andΦ represent the vector and scalar potentials dueto the surface current 119869(1199031015840) respectively
The half-space Greenrsquos function can be expressed by thevector or scalar potentials and the vector potential is notuniquely specified among literatures It is expressed in thispaper using the form mentioned in [18] as follows
119866119860= (119909119909 + 119910119910)119866
119909119909
119860+ 119909119866
119911119909
119860+ 119910119866
119911119910
119860+ 119866
119911119911
119860 (2)
in which
119866119910119910
119860= 119866
119909119909
119860 119866
119902119890
119910= 119866
119902119890
119909 (3)
where 119866119909119909
119860 119866119911119909
119860 119866119911119910
119860 and 119866119911119911
119860denote the spatial domain
half-space Greenrsquos function for the vector potentials and 119866119902119890
119909
119866119902119890
119910 and 119866119902119890
119911denote the spatial domain half-space Greenrsquos
function for the scalar potentials [19 20] According to theposition of targets the terrain below targets can be derivedfrom digital feature analysis data (DFAD) Then the groundcharacteristic is introduced into half-space Greenrsquos functionThe fluctuant ground is equivalent to the medium surface inlower half-space
As stated before the half-space Greenrsquos function has beenintroduced into PO to compute the electrically large targets inthe half space Combined with the graphical electromagneticcomputingmethod (GRECO) [14] the geometry informationof each illuminated plane is obtained by reading the colorsand depths of each pixel and the shadow regions are elim-inated by displaying lists technology of OpenGL to rebuildthe target Hidden surfaces of the image have been previouslyremoved by the hardware graphics acceleratorThenwemakeuse of the resolution to disperse the curve face into pixels thatsatisfy the requirement of the electromagnetic calculationmeanwhile the scene is rendered using the Phong localillumination model [21] For three light sources of purelygreen red and blue colors respectively located over each oneof the three coordinate axises the three color components forthis pixel are equal to the (119899
119909 119899
119910 119899
119911) components of the unit
normal to surface Meanwhile the depths of each pixel areobtained in the same way
The far zone scattering fields of the conductive target inhalf space is expressed as
119864119904(119903)
= minus119895120596∭V119866
119860sdot 119869 (119903
1015840
) 1198891199031015840
+119896 sdot 119904
120596
times∭V[119866
119902119890
119909
120597
120597119909119869 (119903
1015840
) + 119866119902119890
119910
120597
120597119910119869 (119903
1015840
) + 119866119902119890
119911
120597
120597119911119869 (119903
1015840
)] 1198891199031015840
(4)
Equation (4) is then calculated for every illuminated facetand the complex RCS due to scattering from every illumi-nated facet is calculated as [22]
radic120590 = lim119877rarrinfin
2radic120587119877119864119904sdot 119890
119903
119864119900
exp (119895119896119877) (5)
where exp(119895119896119877) is the phase term which is introduced intothe equation in order to account for the facet location withrespect to the global coordinate system and 119877 is the positionvector for the facetrsquos reference vertexwith respect to the globalcoordinate system
By using half-space PO to evaluate the scattering fromthe target in half-space note that the use of GRECO toremove shadowed surfaces is valid for the direct field inci-dence direction not for ground reflections This is valid onlyif the target is low flying because the incident andground-reflection directions are almost parallel In this case
International Journal of Antennas and Propagation 3
Table 1 Some primary parameters
Parameter Value Parameter ValueRadarrsquos longitude 123∘ Radarrsquos latitude 316∘
Targetrsquos latitude 319∘ Targetrsquos height 20mRadarrsquos emission power 1000 kW Frequency 04GHzEmission signal modality Linear frequency modulation Emission signal pulse width 13 120583sPulse compression ratio 13 120583s220 ns Aircraftrsquos velocity 400msAntenna maximal gain 32 dB Beam width 65∘
Polarization type Horizontal polarization Targetrsquos roll angle 5∘
20 40 60 80 100 120 140 160
0
10
20
30
Aspect angle
RCS
(dbs
m)
Ei
Ei
1205824x
z
y
8120582
minus10
minus20
minus30
Prediction of half spaceSimulation of free spaceSimulation of half space
Figure 1 Comparison of prediction and simulation of software foran 8120582 times 8120582 plane in half space (frequency = 10GHz)
the demonstrated half-space GRECO is appropriate to theanalysis of the scattering from the low-flying target
Figure 1 shows a perfectly conducting flat plane (8120582 times8120582) which is placed 1205824 above the soil 120582 represents thewavelength in the vacuumThe relative dielectric permittivityof the soil is 120576
119903= 40 and the relative magnetic permeability
is 120583119903= 10
Figure 1 shows that the half space RCS and the free spaceRCS matches very well at the backscatter angle Howeverat other incident angles there are noticeable differencesbetween them for 4-5 dbsm Hence the conclusion is drawnthat it is necessary to consider the effect of the half-spaceenvironment
22 Multipath Effects
221 Multipath Model In detecting low altitude targetsthe presence of reflections from sea or ground surface cancause multipath propagation In some literatures the implicit
assumption is that only one reflection path exists [12] How-ever in practice four different paths including direct-directdirect-reflected reflected-direct and reflected-reflected arecontributing to the radar received signals as shown analyt-ically in Figure 2 The total echo from a low-flying targetis thus a superposition of four ways of signals each witha different path Multipath propagation can cause signalcancellation in adaptive beamforming [23] and errors in low-angle radar tracking [24] A highly deterministic multipathmodel is constituted in [25]
The complex RCS can be written as radic120590119890119895120601119905 Based on theradar equation [1] the complex voltage coefficient of direct-direct echo signal received by the radar is expressed as
1198811(119905) = (
1198751199051205822
120590
(4120587)3
119871)
12
sdot119891 (120579
119889)
1198772119890119895(120601119903+120601119905)
(6)
where 119875119905is the radar transmit peak power 120582 is radar wave-
length 120590 is the RCS of target 119891(120579119889) is radar antenna gain
120579119889is the angle between the target and radar beam boresight
119871 is composite loss 119877 is the distance from the radar to thecenter of target and can be obtained through the coordinatesof radar and target and 120601
119903is the total phase brought by the
receiver systemSince the path differences of the direct and multipath
returns are usually less than the range resolution of the radarthe different returns are delivered into relevant range gatesresulting in fluctuations Thus the magnitude of the super-position of four signals can be expressed as
4 International Journal of Antennas and Propagation
Radar
Target
Path 1120579d
h1
h2
(a)
Radar
Target
Path 2120579d120579t
h1
h2
h998400
1
d1 d2
Ψ
Ψ
h998400
2
(b)
Radar
Target
Path 3120579d120579t
h1
h2
h998400
1
d1 d2
Ψ
Ψ
h998400
2
(c)
Radar
TargetPath 4
120579t
h1
h2
h998400
1
d1 d2
Ψ
Ψ
h998400
2
(d)
Figure 2 The radar-target geometry in the presence of multipath (a) Path 1 (b) Path 2 (c) Path 3 (d) Path 4
1205901120590
2120590
3 and120590
4are themagnitudes of complexRCS of differ-
ent ways 120588 is the reflection coefficientΔ1198711Δ119871
2 andΔ119871
3are
the path length differences of three reflected rays compared tothe direct 119877
1and 119877
2are the distances of radar-to-reflection
point and target-to-reflection point respectivelyMultipath interference from a rough surface generally
involves two components which are specular and dif-fuse reflections The specular reflection coefficient can beexpressed as
120588119904= 120588
0119877119904119863 (8)
where 1205880is Fresnel reflection coefficient 119877
119904is the specular
scattering factor and 119863 is divergence factor For horizontalpolarization 120588
0is expressed as [25]
1205880119867=sin120595 minus (120576
119888minus cos2120579)
12
sin120595 + (120576119888minus cos2120579)12
(9)
For vertical polarization it is expressed as
1205880119881=120576119888sin120595 minus (120576
119888minus cos2120579)
12
120576119888sin120595 + (120576
119888minus cos2120579)12
(10)
The divergence factor119863 is taken into account due to the cur-vature of the Earth and given as
119863 = [1 +2119877
11198772
119886 (1198771+ 119877
2) sin120595
]
12
(11)
The root mean square (RMS) value of the specular scatteringfactor is given as
119877119904= exp[minus2 (
2120587120590ℎsin120595120582
)
2
] (12)
Here 120590ℎis the RMS value of the wave height
The diffuse reflection coefficient is given as
120588119863= 120588
0119877119863 (13)
where 1205880is the Fresnel reflection coefficient defined in (9) and
(10) and 119877119863is the diffuse scattering coefficient [26]
The Rayleigh roughness criterion from optical theory iscommonly used to estimate the maximum surface irregular-ity that will not significantly lower the reflection coefficient[27 28]
Δℎ lt120582
8 sin120595 (14)
where Δℎ is the maximum peak-to-trough variations and 120595is the grazing angle of reflection point If (14) is satisfied thesurface can behave as an essentially smooth dielectric wherethe specular reflection will occur
222 Estimation Approaches Based on Monte Carlo MethodsThe one-way propagation experimental data for a varietyof terrain types are presented in [13] which indicate astatistical simulation is required for the signal strength inpractice References [1 2 11] treat the path length differencesΔ119871
1 Δ119871
2 and Δ119871
3as deterministic variables However the
errors caused by multipath geometry calculation are so largewith respect to the wavelength that they can confuse theinterference For a radar operating at 1 GHz an error of 015meter will cause a phase error of 180∘ Also due to airborneradar working at a large height the gracing angle of the targetis usually so large [1] that it causes a significant difference ofdirection between the direct and reflected ray which makesthe phases of scattering fields of the direct and reflected rayquite different from each other Hence the phase factors willbe treated as variables which satisfy some PDF The statisticpeculiarity of echo is calculated with Monte Carlo methods
International Journal of Antennas and Propagation 5
Monte Carlo methods are algorithm for solving variouskinds of computational problems by using random numbers[29] It is a means of treating mathematical problems byfinding a probabilistic analog and then obtaining approxi-mate answers to this analog by some experimental samplingprocedureThe solution of a problem by this method is closerin spirit to physical experiments than to classical numericaltechniques In our calculations the number of samples isincreased to achieve a fairly regular distribution As for theerror from the Monte Carlo sampling technique it can bemade negligible bymaking the number of samples sufficientlylarge
To simulate the instantaneous amplitude from echo sig-nals withMonte Carlomethod firstly the probabilistic modelaccording to the problem should be established From for-mula (7) the phase factors can be rewritten as
1205931= (
2120587Δ1198711
120582) mod (2120587)
1205932= (
2120587Δ1198712
120582) mod (2120587)
1205933= (
2120587Δ1198713
120582) mod (2120587)
(15)
where 1205931 120593
2 and 120593
3are the remainder of phase differences
divide 2120587 of three reflected rays compared to the direct Forairborne radar we know that the path length differences ofthree reflected rays compared to the direct can far exceedthe wave length so 120593
1 120593
2 and 120593
3can be supposed to three
sequences of random numbers from a uniform distributionin (0 2120587)
With the most widely used mixed congruence method[30] we generate three sequences of random numbers 119886
119894 119887
119894
and 119888119894(119894 = 1 2 119873) which is uniformly distributed in the
interval (0 1)
119886119894=[(119898
119886119886119894minus1+ 119899
119886) mod119872]
119872
119887119894=[(119898
119887119887119894minus1+ 119899
119887) mod119872]
119872
119888119894=[(119898
119888119888119894minus1+ 119899
119888) mod119872]
119872
119894 = 1 2 119873
(17)
0 2040
6080
100
0
50
1000
5
10
15
20
0533m
617m262m
Figure 3 The missile is placed 20m above the sea
where 119898119886 119898
119887 and 119898
119888are the multipliers 119899
119886 119899
119887 and 119899
119888are
the increment119872 is the modulusGiven120593
1= 2120587119886
119894120593
2= 2120587119887
119894 and120593
3= 2120587119888
119894in formula (16)
run the Monte Carlo simulation n times to estimate the echosignals from low altitude targets accurately based on actualenvironment for airborne radar
By using a sufficient number of samples a probability dis-tribution is obtained to describe echo signals The statisticalcharacteristics of the echo signals can be illustrated at last
3 Results and Discussion
Themissile is adopted in the simulation as shown in Figure 3with its RCS in Figure 4
The PDF of magnitude of echoes is shown in Figure 5when the target is flying at a height of 20m above the seaAccording to the radar equation the magnitude of direct rayis 64416 times 10minus7 V Figure 5 indicates that the magnitude ofechoes can be different from direct ray due to multipath
The magnitudes of direct echoes in the situations of dif-ferent targetrsquos heights are shown in Figure 6 and the PDFs ofmagnitude of echoes normalized to the direct ray in the samesituations are shown in Figure 7 As can be seen in Figure 6the magnitudes of direct echoes vary with targetrsquos heightsThe prime reason for that is the target RCSrsquos variation causedby the different azimuth angle 120593 and the elevation angle 120579with respect to the coordinate system of the target Figure 7shows the influences of multipath at different heights In theconditions of the target flying at low heights (about a fewhundred meters) the PDFs vary little with the heights whilethe normalized magnitude of echoes with respect to the peakof the PDF reduces a little As illustrated in Figure 7 when thetarget is at heights of below 2000m themagnitudes of echoesnormalized to the direct ray vary mainly between 0 and 4The peak probability densities are in the range of 0-1 and thecorresponding normalized magnitudes of echoes are in therange of 1-2 These results in Figure 7 are similar to the onesin Figures 14 and 15 of [13] which make the demonstratedalgorithm valid in this paper However the terrain in thispaper was produced from the digital feature analysis data
6 International Journal of Antennas and Propagation
Figure 4 The missilersquos RCS with frequency of 04GHz
Prob
abili
ty d
ensit
y
The magnitude of echoes (V) times10minus6
07
06
05
04
03
02
01
0
0 05 1 15 2 25 3
Figure 5 The PDF versus magnitude of echoes for the target at aheight of 20m above the sea
(DFAD) yet in [13] the desert terrain for simulations wasproduced from the defense mapping agency (DMA) data theexperiments weremade inNewMexico andNorthernMaineThe differences between the results in this paper and those in[13] are mainly caused by the differences in the terrains thefrequencies the geometries and attitudes of the targets theheights of the airborne radars and the targets the numberof samples and so forth As the target is flying higher theinfluences of multipath become less to the echoes until theindirect ways disappear In practice if the target height isat least moderately large the range resolution of the radaris less than the path differences of the direct and multipathreturns so the radar is able to resolve them As radar heightdecreases to a height of ground based radar the differencein angle between reflected and direct paths can be ignored
20100 500 1000 20000
1
2
3
4
5
6
7
8
The target height (m)
The m
agni
tude
of d
irect
ray
(V)
times10minus7
Figure 6 The magnitude of direct ray versus the target height
4Normalized magnitude of echoes
Prob
abili
ty d
ensit
y
0 05 1 15 2 25 3 35
0
05
1
15
2
25
3
h = 20mh = 100mh = 500m
h = 1000mh = 2000m
Figure 7 The PDF of magnitude of echoes at different targetrsquosheights
which causes the fluctuations in phase disappearing for low-flying targets and the PDF of magnitude of echoes is similarto Figure 6 of [12]
4 Conclusions
For airborne radar to detect low altitude targets a demon-strated hybrid method based on probability statistics andcomputational electromagnetics is proposed in this paperto give a statistical simulation of echo signals based onactual environmentThe half-space physical optics combinedwith the graphical-electromagnetic computing method isemployed to calculate the RCS of low-flying targets suffi-ciently accurately and efficiently in half space Considering
International Journal of Antennas and Propagation 7
the fluctuations due to the multipath effects the phasefactors are modified with theMonte Carlo methodWith thismethod the target echo signals from low altitude targets canbe obtained accurately for the radar simulation system andthe statistic characteristics of echo signals can be accuratelyand properly simulated The demonstrated method in thispaper is of great help for the radar researcher to evaluate thedetection probability and false alarm probability of specialtargets the signal-to-noise ratio (SNR) of emission pulsesand so on
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M Long Airborne Early Warning System Concepts ArtechBoston Mass USA 1992
[2] W Morchin Airborne Early Warning Radar Artech HouseNorwell Mass USA 1990
[3] L-XGuoA-QWang and JMa ldquoStudy onEMscattering from2-D target above 1-D large scale rough surface with low grazingincidence by parallel MOM based on PC Clustersrdquo Progress inElectromagnetics Research vol 89 pp 149ndash166 2009
[4] M I Skolnik Introduction to Radar SystemsMcGraw-Hill NewYork NY USA 3rd edition 2001
[5] Y Qu G S Liao S Q Zhu and X Y Liu ldquoPattern synthesisof planar antenna array via convex optimization for airborneforward looking radarrdquo Progress in Electromagnetics Researchvol 84 pp 1ndash10 2008
[6] S H Lim J H Han S Y Kim and N H Myung ldquoAzimuthbeam pattern synthesis for air-borne sar system optimizationrdquoProgress in Electromagnetics Research vol 106 pp 295ndash3092010
[7] Y-L Chang C-Y Chiang and K-S Chen ldquoSAR image sim-ulation with application to target recognitionrdquo Progress in Elec-tromagnetics Research vol 119 pp 35ndash57 2011
[8] M Zhang Y W Zhao H Chen and W Q Jiang ldquoSar imagingsimulation for composite model of ship on dynamic oceanscenerdquo Progress in Electromagnetics Research vol 113 pp 395ndash412 2011
[9] S Qiao Z G Shi K S Chen et al ldquoA new architecture ofUWB radar utilizing microwave chaotic signals and chaos syn-chronizationrdquo Progress in Electromagnetics Research vol 75 pp225ndash237 2007
[10] M-J Wang Z-S Wu Y-L Li and G Zhang ldquoHigh resolutionrange profile identifying simulation of laser radar based onpulse beam scattering characteristics of targetsrdquo Progress inElectromagnetics Research vol 96 pp 193ndash204 2009
[11] J G Teti Jr ldquoWide-band airborne radar operating considera-tions for low-altitude surveillance in the presence of specularmultipathrdquo IEEE Transactions on Antennas and Propagationvol 48 no 2 pp 176ndash191 2000
[12] S L Wilson and B D Carison ldquoRadar detection in multipathrdquoIEE Proceedings Radar Sonar and Navigation vol 146 no 1 pp45ndash52 1999
[13] L M Zurk ldquoExperimental observation and statistics of multi-path from terrain with application to overland height findingrdquo
IEEE Transactions on Antennas and Propagation vol 47 no 1pp 121ndash131 1999
[14] Y B Tao H Lin and H J Bao ldquoFrom CPU to GPU GPU-based electromagnetic computingrdquo Progress in ElectromagneticsResearch vol 81 pp 1ndash19 2008
[15] B Etkin andLD ReidDynamics of Flight Stability andControlJohn Wiley amp Sons New York NY USA 1996
[16] X F Li Y J Xie and R Yang ldquoBistatic RCS prediction forcomplex targets using modified current marching techniquerdquoProgress in Electromagnetics Research vol 93 pp 13ndash28 2009
[17] X F Li Y J Xie P Wang and T M Yang ldquoHigh-frequencymethod for scattering from electrically large conductive targetsin half-spacerdquo IEEE Antennas and Wireless Propagation Lettersvol 6 pp 259ndash262 2007
[18] K A Michalski and D Zheng ldquoElectromagnetic scattering andradiation by surfaces of arbitrary shape in layered media ITheoryrdquo IEEE Transactions on Antennas and Propagation vol38 no 3 pp 335ndash344 1990
[19] H Bagci A E Yilmaz V Lomakin and E Michielssen ldquoFastsolution of mixed-potential time-domain integral equations forhalf-space environmentsrdquo IEEE Transactions on Geoscience andRemote Sensing vol 43 no 2 pp 269ndash279 2005
[20] R C Acar and G Dural ldquoComplete set of closed-form Greenrsquosfunctions for cylindrically layered mediardquo in Proceedings of theIEEE Antennas and Propagation Society International Sympo-sium 2006 pp 2863ndash2866 2006
[21] J M Rius M Ferrando and L Jofre ldquoHigh-frequency RCSof complex radar targets in real-timerdquo IEEE Transactions onAntennas and Propagation vol 41 no 9 pp 1308ndash1319 1993
[22] E F Knott J F Shaeffer and M T Tuley Radar Cross SectionSciTech Publishing Raleigh NC USA 2004
[23] B Widrow K M Duvall R P Gooch and W C NewmanldquoSignal cancellation phenomena in adaptive antennas causesand curesrdquo IEEETransactions onAntennas and Propagation vol30 no 3 pp 469ndash478 1982
[24] WDWhite ldquoLow angle radar tracking in the presence ofmulti-pathrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 10 no 6 pp 335ndash352 1974
[25] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings Part F Radarand Signal Processing vol 138 no 2 pp 163ndash171 1991
[26] P Beckmann and A Spizzichino The Scattering of Electro-Magnetic Waves from Rough Surfaces Artech House NorwoodMass USA 1987
[27] C I Beard ldquoCoherent and incoherent scattering of microwavesfrom the oceanrdquo IEEE Transactions on Antenna and Propaga-tion vol 9 no 5 pp 470ndash483 1961
[28] N Pinel C Bourlier and J Saillard ldquoDegree of roughness ofrough layers extensions of the Rayleigh roughnessrdquo Progress inElectromagnetics Research B no 19 pp 41ndash63 2010
[29] M Mishra and N Gupta ldquoMonte Carlo integration techniquefor the analysis of electromagnetic scattering from conductingsurfacesrdquo Progress in Electromagnetics Research vol 79 pp 91ndash106 2008
[30] M A Herrador A G Asuero and A G Gonzalez ldquoEstimationof the uncertainty of indirect measurements from the propa-gation of distributions by using the Monte-Carlo method anoverviewrdquoChemometrics and Intelligent Laboratory Systems vol79 no 1-2 pp 115ndash122 2005
Targetrsquos latitude 319∘ Targetrsquos height 20mRadarrsquos emission power 1000 kW Frequency 04GHzEmission signal modality Linear frequency modulation Emission signal pulse width 13 120583sPulse compression ratio 13 120583s220 ns Aircraftrsquos velocity 400msAntenna maximal gain 32 dB Beam width 65∘
Polarization type Horizontal polarization Targetrsquos roll angle 5∘
20 40 60 80 100 120 140 160
0
10
20
30
Aspect angle
RCS
(dbs
m)
Ei
Ei
1205824x
z
y
8120582
minus10
minus20
minus30
Prediction of half spaceSimulation of free spaceSimulation of half space
Figure 1 Comparison of prediction and simulation of software foran 8120582 times 8120582 plane in half space (frequency = 10GHz)
the demonstrated half-space GRECO is appropriate to theanalysis of the scattering from the low-flying target
Figure 1 shows a perfectly conducting flat plane (8120582 times8120582) which is placed 1205824 above the soil 120582 represents thewavelength in the vacuumThe relative dielectric permittivityof the soil is 120576
119903= 40 and the relative magnetic permeability
is 120583119903= 10
Figure 1 shows that the half space RCS and the free spaceRCS matches very well at the backscatter angle Howeverat other incident angles there are noticeable differencesbetween them for 4-5 dbsm Hence the conclusion is drawnthat it is necessary to consider the effect of the half-spaceenvironment
22 Multipath Effects
221 Multipath Model In detecting low altitude targetsthe presence of reflections from sea or ground surface cancause multipath propagation In some literatures the implicit
assumption is that only one reflection path exists [12] How-ever in practice four different paths including direct-directdirect-reflected reflected-direct and reflected-reflected arecontributing to the radar received signals as shown analyt-ically in Figure 2 The total echo from a low-flying targetis thus a superposition of four ways of signals each witha different path Multipath propagation can cause signalcancellation in adaptive beamforming [23] and errors in low-angle radar tracking [24] A highly deterministic multipathmodel is constituted in [25]
The complex RCS can be written as radic120590119890119895120601119905 Based on theradar equation [1] the complex voltage coefficient of direct-direct echo signal received by the radar is expressed as
1198811(119905) = (
1198751199051205822
120590
(4120587)3
119871)
12
sdot119891 (120579
119889)
1198772119890119895(120601119903+120601119905)
(6)
where 119875119905is the radar transmit peak power 120582 is radar wave-
length 120590 is the RCS of target 119891(120579119889) is radar antenna gain
120579119889is the angle between the target and radar beam boresight
119871 is composite loss 119877 is the distance from the radar to thecenter of target and can be obtained through the coordinatesof radar and target and 120601
119903is the total phase brought by the
receiver systemSince the path differences of the direct and multipath
returns are usually less than the range resolution of the radarthe different returns are delivered into relevant range gatesresulting in fluctuations Thus the magnitude of the super-position of four signals can be expressed as
4 International Journal of Antennas and Propagation
Radar
Target
Path 1120579d
h1
h2
(a)
Radar
Target
Path 2120579d120579t
h1
h2
h998400
1
d1 d2
Ψ
Ψ
h998400
2
(b)
Radar
Target
Path 3120579d120579t
h1
h2
h998400
1
d1 d2
Ψ
Ψ
h998400
2
(c)
Radar
TargetPath 4
120579t
h1
h2
h998400
1
d1 d2
Ψ
Ψ
h998400
2
(d)
Figure 2 The radar-target geometry in the presence of multipath (a) Path 1 (b) Path 2 (c) Path 3 (d) Path 4
1205901120590
2120590
3 and120590
4are themagnitudes of complexRCS of differ-
ent ways 120588 is the reflection coefficientΔ1198711Δ119871
2 andΔ119871
3are
the path length differences of three reflected rays compared tothe direct 119877
1and 119877
2are the distances of radar-to-reflection
point and target-to-reflection point respectivelyMultipath interference from a rough surface generally
involves two components which are specular and dif-fuse reflections The specular reflection coefficient can beexpressed as
120588119904= 120588
0119877119904119863 (8)
where 1205880is Fresnel reflection coefficient 119877
119904is the specular
scattering factor and 119863 is divergence factor For horizontalpolarization 120588
0is expressed as [25]
1205880119867=sin120595 minus (120576
119888minus cos2120579)
12
sin120595 + (120576119888minus cos2120579)12
(9)
For vertical polarization it is expressed as
1205880119881=120576119888sin120595 minus (120576
119888minus cos2120579)
12
120576119888sin120595 + (120576
119888minus cos2120579)12
(10)
The divergence factor119863 is taken into account due to the cur-vature of the Earth and given as
119863 = [1 +2119877
11198772
119886 (1198771+ 119877
2) sin120595
]
12
(11)
The root mean square (RMS) value of the specular scatteringfactor is given as
119877119904= exp[minus2 (
2120587120590ℎsin120595120582
)
2
] (12)
Here 120590ℎis the RMS value of the wave height
The diffuse reflection coefficient is given as
120588119863= 120588
0119877119863 (13)
where 1205880is the Fresnel reflection coefficient defined in (9) and
(10) and 119877119863is the diffuse scattering coefficient [26]
The Rayleigh roughness criterion from optical theory iscommonly used to estimate the maximum surface irregular-ity that will not significantly lower the reflection coefficient[27 28]
Δℎ lt120582
8 sin120595 (14)
where Δℎ is the maximum peak-to-trough variations and 120595is the grazing angle of reflection point If (14) is satisfied thesurface can behave as an essentially smooth dielectric wherethe specular reflection will occur
222 Estimation Approaches Based on Monte Carlo MethodsThe one-way propagation experimental data for a varietyof terrain types are presented in [13] which indicate astatistical simulation is required for the signal strength inpractice References [1 2 11] treat the path length differencesΔ119871
1 Δ119871
2 and Δ119871
3as deterministic variables However the
errors caused by multipath geometry calculation are so largewith respect to the wavelength that they can confuse theinterference For a radar operating at 1 GHz an error of 015meter will cause a phase error of 180∘ Also due to airborneradar working at a large height the gracing angle of the targetis usually so large [1] that it causes a significant difference ofdirection between the direct and reflected ray which makesthe phases of scattering fields of the direct and reflected rayquite different from each other Hence the phase factors willbe treated as variables which satisfy some PDF The statisticpeculiarity of echo is calculated with Monte Carlo methods
International Journal of Antennas and Propagation 5
Monte Carlo methods are algorithm for solving variouskinds of computational problems by using random numbers[29] It is a means of treating mathematical problems byfinding a probabilistic analog and then obtaining approxi-mate answers to this analog by some experimental samplingprocedureThe solution of a problem by this method is closerin spirit to physical experiments than to classical numericaltechniques In our calculations the number of samples isincreased to achieve a fairly regular distribution As for theerror from the Monte Carlo sampling technique it can bemade negligible bymaking the number of samples sufficientlylarge
To simulate the instantaneous amplitude from echo sig-nals withMonte Carlomethod firstly the probabilistic modelaccording to the problem should be established From for-mula (7) the phase factors can be rewritten as
1205931= (
2120587Δ1198711
120582) mod (2120587)
1205932= (
2120587Δ1198712
120582) mod (2120587)
1205933= (
2120587Δ1198713
120582) mod (2120587)
(15)
where 1205931 120593
2 and 120593
3are the remainder of phase differences
divide 2120587 of three reflected rays compared to the direct Forairborne radar we know that the path length differences ofthree reflected rays compared to the direct can far exceedthe wave length so 120593
1 120593
2 and 120593
3can be supposed to three
sequences of random numbers from a uniform distributionin (0 2120587)
With the most widely used mixed congruence method[30] we generate three sequences of random numbers 119886
119894 119887
119894
and 119888119894(119894 = 1 2 119873) which is uniformly distributed in the
interval (0 1)
119886119894=[(119898
119886119886119894minus1+ 119899
119886) mod119872]
119872
119887119894=[(119898
119887119887119894minus1+ 119899
119887) mod119872]
119872
119888119894=[(119898
119888119888119894minus1+ 119899
119888) mod119872]
119872
119894 = 1 2 119873
(17)
0 2040
6080
100
0
50
1000
5
10
15
20
0533m
617m262m
Figure 3 The missile is placed 20m above the sea
where 119898119886 119898
119887 and 119898
119888are the multipliers 119899
119886 119899
119887 and 119899
119888are
the increment119872 is the modulusGiven120593
1= 2120587119886
119894120593
2= 2120587119887
119894 and120593
3= 2120587119888
119894in formula (16)
run the Monte Carlo simulation n times to estimate the echosignals from low altitude targets accurately based on actualenvironment for airborne radar
By using a sufficient number of samples a probability dis-tribution is obtained to describe echo signals The statisticalcharacteristics of the echo signals can be illustrated at last
3 Results and Discussion
Themissile is adopted in the simulation as shown in Figure 3with its RCS in Figure 4
The PDF of magnitude of echoes is shown in Figure 5when the target is flying at a height of 20m above the seaAccording to the radar equation the magnitude of direct rayis 64416 times 10minus7 V Figure 5 indicates that the magnitude ofechoes can be different from direct ray due to multipath
The magnitudes of direct echoes in the situations of dif-ferent targetrsquos heights are shown in Figure 6 and the PDFs ofmagnitude of echoes normalized to the direct ray in the samesituations are shown in Figure 7 As can be seen in Figure 6the magnitudes of direct echoes vary with targetrsquos heightsThe prime reason for that is the target RCSrsquos variation causedby the different azimuth angle 120593 and the elevation angle 120579with respect to the coordinate system of the target Figure 7shows the influences of multipath at different heights In theconditions of the target flying at low heights (about a fewhundred meters) the PDFs vary little with the heights whilethe normalized magnitude of echoes with respect to the peakof the PDF reduces a little As illustrated in Figure 7 when thetarget is at heights of below 2000m themagnitudes of echoesnormalized to the direct ray vary mainly between 0 and 4The peak probability densities are in the range of 0-1 and thecorresponding normalized magnitudes of echoes are in therange of 1-2 These results in Figure 7 are similar to the onesin Figures 14 and 15 of [13] which make the demonstratedalgorithm valid in this paper However the terrain in thispaper was produced from the digital feature analysis data
6 International Journal of Antennas and Propagation
Figure 4 The missilersquos RCS with frequency of 04GHz
Prob
abili
ty d
ensit
y
The magnitude of echoes (V) times10minus6
07
06
05
04
03
02
01
0
0 05 1 15 2 25 3
Figure 5 The PDF versus magnitude of echoes for the target at aheight of 20m above the sea
(DFAD) yet in [13] the desert terrain for simulations wasproduced from the defense mapping agency (DMA) data theexperiments weremade inNewMexico andNorthernMaineThe differences between the results in this paper and those in[13] are mainly caused by the differences in the terrains thefrequencies the geometries and attitudes of the targets theheights of the airborne radars and the targets the numberof samples and so forth As the target is flying higher theinfluences of multipath become less to the echoes until theindirect ways disappear In practice if the target height isat least moderately large the range resolution of the radaris less than the path differences of the direct and multipathreturns so the radar is able to resolve them As radar heightdecreases to a height of ground based radar the differencein angle between reflected and direct paths can be ignored
20100 500 1000 20000
1
2
3
4
5
6
7
8
The target height (m)
The m
agni
tude
of d
irect
ray
(V)
times10minus7
Figure 6 The magnitude of direct ray versus the target height
4Normalized magnitude of echoes
Prob
abili
ty d
ensit
y
0 05 1 15 2 25 3 35
0
05
1
15
2
25
3
h = 20mh = 100mh = 500m
h = 1000mh = 2000m
Figure 7 The PDF of magnitude of echoes at different targetrsquosheights
which causes the fluctuations in phase disappearing for low-flying targets and the PDF of magnitude of echoes is similarto Figure 6 of [12]
4 Conclusions
For airborne radar to detect low altitude targets a demon-strated hybrid method based on probability statistics andcomputational electromagnetics is proposed in this paperto give a statistical simulation of echo signals based onactual environmentThe half-space physical optics combinedwith the graphical-electromagnetic computing method isemployed to calculate the RCS of low-flying targets suffi-ciently accurately and efficiently in half space Considering
International Journal of Antennas and Propagation 7
the fluctuations due to the multipath effects the phasefactors are modified with theMonte Carlo methodWith thismethod the target echo signals from low altitude targets canbe obtained accurately for the radar simulation system andthe statistic characteristics of echo signals can be accuratelyand properly simulated The demonstrated method in thispaper is of great help for the radar researcher to evaluate thedetection probability and false alarm probability of specialtargets the signal-to-noise ratio (SNR) of emission pulsesand so on
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M Long Airborne Early Warning System Concepts ArtechBoston Mass USA 1992
[2] W Morchin Airborne Early Warning Radar Artech HouseNorwell Mass USA 1990
[3] L-XGuoA-QWang and JMa ldquoStudy onEMscattering from2-D target above 1-D large scale rough surface with low grazingincidence by parallel MOM based on PC Clustersrdquo Progress inElectromagnetics Research vol 89 pp 149ndash166 2009
[4] M I Skolnik Introduction to Radar SystemsMcGraw-Hill NewYork NY USA 3rd edition 2001
[5] Y Qu G S Liao S Q Zhu and X Y Liu ldquoPattern synthesisof planar antenna array via convex optimization for airborneforward looking radarrdquo Progress in Electromagnetics Researchvol 84 pp 1ndash10 2008
[6] S H Lim J H Han S Y Kim and N H Myung ldquoAzimuthbeam pattern synthesis for air-borne sar system optimizationrdquoProgress in Electromagnetics Research vol 106 pp 295ndash3092010
[7] Y-L Chang C-Y Chiang and K-S Chen ldquoSAR image sim-ulation with application to target recognitionrdquo Progress in Elec-tromagnetics Research vol 119 pp 35ndash57 2011
[8] M Zhang Y W Zhao H Chen and W Q Jiang ldquoSar imagingsimulation for composite model of ship on dynamic oceanscenerdquo Progress in Electromagnetics Research vol 113 pp 395ndash412 2011
[9] S Qiao Z G Shi K S Chen et al ldquoA new architecture ofUWB radar utilizing microwave chaotic signals and chaos syn-chronizationrdquo Progress in Electromagnetics Research vol 75 pp225ndash237 2007
[10] M-J Wang Z-S Wu Y-L Li and G Zhang ldquoHigh resolutionrange profile identifying simulation of laser radar based onpulse beam scattering characteristics of targetsrdquo Progress inElectromagnetics Research vol 96 pp 193ndash204 2009
[11] J G Teti Jr ldquoWide-band airborne radar operating considera-tions for low-altitude surveillance in the presence of specularmultipathrdquo IEEE Transactions on Antennas and Propagationvol 48 no 2 pp 176ndash191 2000
[12] S L Wilson and B D Carison ldquoRadar detection in multipathrdquoIEE Proceedings Radar Sonar and Navigation vol 146 no 1 pp45ndash52 1999
[13] L M Zurk ldquoExperimental observation and statistics of multi-path from terrain with application to overland height findingrdquo
IEEE Transactions on Antennas and Propagation vol 47 no 1pp 121ndash131 1999
[14] Y B Tao H Lin and H J Bao ldquoFrom CPU to GPU GPU-based electromagnetic computingrdquo Progress in ElectromagneticsResearch vol 81 pp 1ndash19 2008
[15] B Etkin andLD ReidDynamics of Flight Stability andControlJohn Wiley amp Sons New York NY USA 1996
[16] X F Li Y J Xie and R Yang ldquoBistatic RCS prediction forcomplex targets using modified current marching techniquerdquoProgress in Electromagnetics Research vol 93 pp 13ndash28 2009
[17] X F Li Y J Xie P Wang and T M Yang ldquoHigh-frequencymethod for scattering from electrically large conductive targetsin half-spacerdquo IEEE Antennas and Wireless Propagation Lettersvol 6 pp 259ndash262 2007
[18] K A Michalski and D Zheng ldquoElectromagnetic scattering andradiation by surfaces of arbitrary shape in layered media ITheoryrdquo IEEE Transactions on Antennas and Propagation vol38 no 3 pp 335ndash344 1990
[19] H Bagci A E Yilmaz V Lomakin and E Michielssen ldquoFastsolution of mixed-potential time-domain integral equations forhalf-space environmentsrdquo IEEE Transactions on Geoscience andRemote Sensing vol 43 no 2 pp 269ndash279 2005
[20] R C Acar and G Dural ldquoComplete set of closed-form Greenrsquosfunctions for cylindrically layered mediardquo in Proceedings of theIEEE Antennas and Propagation Society International Sympo-sium 2006 pp 2863ndash2866 2006
[21] J M Rius M Ferrando and L Jofre ldquoHigh-frequency RCSof complex radar targets in real-timerdquo IEEE Transactions onAntennas and Propagation vol 41 no 9 pp 1308ndash1319 1993
[22] E F Knott J F Shaeffer and M T Tuley Radar Cross SectionSciTech Publishing Raleigh NC USA 2004
[23] B Widrow K M Duvall R P Gooch and W C NewmanldquoSignal cancellation phenomena in adaptive antennas causesand curesrdquo IEEETransactions onAntennas and Propagation vol30 no 3 pp 469ndash478 1982
[24] WDWhite ldquoLow angle radar tracking in the presence ofmulti-pathrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 10 no 6 pp 335ndash352 1974
[25] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings Part F Radarand Signal Processing vol 138 no 2 pp 163ndash171 1991
[26] P Beckmann and A Spizzichino The Scattering of Electro-Magnetic Waves from Rough Surfaces Artech House NorwoodMass USA 1987
[27] C I Beard ldquoCoherent and incoherent scattering of microwavesfrom the oceanrdquo IEEE Transactions on Antenna and Propaga-tion vol 9 no 5 pp 470ndash483 1961
[28] N Pinel C Bourlier and J Saillard ldquoDegree of roughness ofrough layers extensions of the Rayleigh roughnessrdquo Progress inElectromagnetics Research B no 19 pp 41ndash63 2010
[29] M Mishra and N Gupta ldquoMonte Carlo integration techniquefor the analysis of electromagnetic scattering from conductingsurfacesrdquo Progress in Electromagnetics Research vol 79 pp 91ndash106 2008
[30] M A Herrador A G Asuero and A G Gonzalez ldquoEstimationof the uncertainty of indirect measurements from the propa-gation of distributions by using the Monte-Carlo method anoverviewrdquoChemometrics and Intelligent Laboratory Systems vol79 no 1-2 pp 115ndash122 2005
4 International Journal of Antennas and Propagation
Radar
Target
Path 1120579d
h1
h2
(a)
Radar
Target
Path 2120579d120579t
h1
h2
h998400
1
d1 d2
Ψ
Ψ
h998400
2
(b)
Radar
Target
Path 3120579d120579t
h1
h2
h998400
1
d1 d2
Ψ
Ψ
h998400
2
(c)
Radar
TargetPath 4
120579t
h1
h2
h998400
1
d1 d2
Ψ
Ψ
h998400
2
(d)
Figure 2 The radar-target geometry in the presence of multipath (a) Path 1 (b) Path 2 (c) Path 3 (d) Path 4
1205901120590
2120590
3 and120590
4are themagnitudes of complexRCS of differ-
ent ways 120588 is the reflection coefficientΔ1198711Δ119871
2 andΔ119871
3are
the path length differences of three reflected rays compared tothe direct 119877
1and 119877
2are the distances of radar-to-reflection
point and target-to-reflection point respectivelyMultipath interference from a rough surface generally
involves two components which are specular and dif-fuse reflections The specular reflection coefficient can beexpressed as
120588119904= 120588
0119877119904119863 (8)
where 1205880is Fresnel reflection coefficient 119877
119904is the specular
scattering factor and 119863 is divergence factor For horizontalpolarization 120588
0is expressed as [25]
1205880119867=sin120595 minus (120576
119888minus cos2120579)
12
sin120595 + (120576119888minus cos2120579)12
(9)
For vertical polarization it is expressed as
1205880119881=120576119888sin120595 minus (120576
119888minus cos2120579)
12
120576119888sin120595 + (120576
119888minus cos2120579)12
(10)
The divergence factor119863 is taken into account due to the cur-vature of the Earth and given as
119863 = [1 +2119877
11198772
119886 (1198771+ 119877
2) sin120595
]
12
(11)
The root mean square (RMS) value of the specular scatteringfactor is given as
119877119904= exp[minus2 (
2120587120590ℎsin120595120582
)
2
] (12)
Here 120590ℎis the RMS value of the wave height
The diffuse reflection coefficient is given as
120588119863= 120588
0119877119863 (13)
where 1205880is the Fresnel reflection coefficient defined in (9) and
(10) and 119877119863is the diffuse scattering coefficient [26]
The Rayleigh roughness criterion from optical theory iscommonly used to estimate the maximum surface irregular-ity that will not significantly lower the reflection coefficient[27 28]
Δℎ lt120582
8 sin120595 (14)
where Δℎ is the maximum peak-to-trough variations and 120595is the grazing angle of reflection point If (14) is satisfied thesurface can behave as an essentially smooth dielectric wherethe specular reflection will occur
222 Estimation Approaches Based on Monte Carlo MethodsThe one-way propagation experimental data for a varietyof terrain types are presented in [13] which indicate astatistical simulation is required for the signal strength inpractice References [1 2 11] treat the path length differencesΔ119871
1 Δ119871
2 and Δ119871
3as deterministic variables However the
errors caused by multipath geometry calculation are so largewith respect to the wavelength that they can confuse theinterference For a radar operating at 1 GHz an error of 015meter will cause a phase error of 180∘ Also due to airborneradar working at a large height the gracing angle of the targetis usually so large [1] that it causes a significant difference ofdirection between the direct and reflected ray which makesthe phases of scattering fields of the direct and reflected rayquite different from each other Hence the phase factors willbe treated as variables which satisfy some PDF The statisticpeculiarity of echo is calculated with Monte Carlo methods
International Journal of Antennas and Propagation 5
Monte Carlo methods are algorithm for solving variouskinds of computational problems by using random numbers[29] It is a means of treating mathematical problems byfinding a probabilistic analog and then obtaining approxi-mate answers to this analog by some experimental samplingprocedureThe solution of a problem by this method is closerin spirit to physical experiments than to classical numericaltechniques In our calculations the number of samples isincreased to achieve a fairly regular distribution As for theerror from the Monte Carlo sampling technique it can bemade negligible bymaking the number of samples sufficientlylarge
To simulate the instantaneous amplitude from echo sig-nals withMonte Carlomethod firstly the probabilistic modelaccording to the problem should be established From for-mula (7) the phase factors can be rewritten as
1205931= (
2120587Δ1198711
120582) mod (2120587)
1205932= (
2120587Δ1198712
120582) mod (2120587)
1205933= (
2120587Δ1198713
120582) mod (2120587)
(15)
where 1205931 120593
2 and 120593
3are the remainder of phase differences
divide 2120587 of three reflected rays compared to the direct Forairborne radar we know that the path length differences ofthree reflected rays compared to the direct can far exceedthe wave length so 120593
1 120593
2 and 120593
3can be supposed to three
sequences of random numbers from a uniform distributionin (0 2120587)
With the most widely used mixed congruence method[30] we generate three sequences of random numbers 119886
119894 119887
119894
and 119888119894(119894 = 1 2 119873) which is uniformly distributed in the
interval (0 1)
119886119894=[(119898
119886119886119894minus1+ 119899
119886) mod119872]
119872
119887119894=[(119898
119887119887119894minus1+ 119899
119887) mod119872]
119872
119888119894=[(119898
119888119888119894minus1+ 119899
119888) mod119872]
119872
119894 = 1 2 119873
(17)
0 2040
6080
100
0
50
1000
5
10
15
20
0533m
617m262m
Figure 3 The missile is placed 20m above the sea
where 119898119886 119898
119887 and 119898
119888are the multipliers 119899
119886 119899
119887 and 119899
119888are
the increment119872 is the modulusGiven120593
1= 2120587119886
119894120593
2= 2120587119887
119894 and120593
3= 2120587119888
119894in formula (16)
run the Monte Carlo simulation n times to estimate the echosignals from low altitude targets accurately based on actualenvironment for airborne radar
By using a sufficient number of samples a probability dis-tribution is obtained to describe echo signals The statisticalcharacteristics of the echo signals can be illustrated at last
3 Results and Discussion
Themissile is adopted in the simulation as shown in Figure 3with its RCS in Figure 4
The PDF of magnitude of echoes is shown in Figure 5when the target is flying at a height of 20m above the seaAccording to the radar equation the magnitude of direct rayis 64416 times 10minus7 V Figure 5 indicates that the magnitude ofechoes can be different from direct ray due to multipath
The magnitudes of direct echoes in the situations of dif-ferent targetrsquos heights are shown in Figure 6 and the PDFs ofmagnitude of echoes normalized to the direct ray in the samesituations are shown in Figure 7 As can be seen in Figure 6the magnitudes of direct echoes vary with targetrsquos heightsThe prime reason for that is the target RCSrsquos variation causedby the different azimuth angle 120593 and the elevation angle 120579with respect to the coordinate system of the target Figure 7shows the influences of multipath at different heights In theconditions of the target flying at low heights (about a fewhundred meters) the PDFs vary little with the heights whilethe normalized magnitude of echoes with respect to the peakof the PDF reduces a little As illustrated in Figure 7 when thetarget is at heights of below 2000m themagnitudes of echoesnormalized to the direct ray vary mainly between 0 and 4The peak probability densities are in the range of 0-1 and thecorresponding normalized magnitudes of echoes are in therange of 1-2 These results in Figure 7 are similar to the onesin Figures 14 and 15 of [13] which make the demonstratedalgorithm valid in this paper However the terrain in thispaper was produced from the digital feature analysis data
6 International Journal of Antennas and Propagation
Figure 4 The missilersquos RCS with frequency of 04GHz
Prob
abili
ty d
ensit
y
The magnitude of echoes (V) times10minus6
07
06
05
04
03
02
01
0
0 05 1 15 2 25 3
Figure 5 The PDF versus magnitude of echoes for the target at aheight of 20m above the sea
(DFAD) yet in [13] the desert terrain for simulations wasproduced from the defense mapping agency (DMA) data theexperiments weremade inNewMexico andNorthernMaineThe differences between the results in this paper and those in[13] are mainly caused by the differences in the terrains thefrequencies the geometries and attitudes of the targets theheights of the airborne radars and the targets the numberof samples and so forth As the target is flying higher theinfluences of multipath become less to the echoes until theindirect ways disappear In practice if the target height isat least moderately large the range resolution of the radaris less than the path differences of the direct and multipathreturns so the radar is able to resolve them As radar heightdecreases to a height of ground based radar the differencein angle between reflected and direct paths can be ignored
20100 500 1000 20000
1
2
3
4
5
6
7
8
The target height (m)
The m
agni
tude
of d
irect
ray
(V)
times10minus7
Figure 6 The magnitude of direct ray versus the target height
4Normalized magnitude of echoes
Prob
abili
ty d
ensit
y
0 05 1 15 2 25 3 35
0
05
1
15
2
25
3
h = 20mh = 100mh = 500m
h = 1000mh = 2000m
Figure 7 The PDF of magnitude of echoes at different targetrsquosheights
which causes the fluctuations in phase disappearing for low-flying targets and the PDF of magnitude of echoes is similarto Figure 6 of [12]
4 Conclusions
For airborne radar to detect low altitude targets a demon-strated hybrid method based on probability statistics andcomputational electromagnetics is proposed in this paperto give a statistical simulation of echo signals based onactual environmentThe half-space physical optics combinedwith the graphical-electromagnetic computing method isemployed to calculate the RCS of low-flying targets suffi-ciently accurately and efficiently in half space Considering
International Journal of Antennas and Propagation 7
the fluctuations due to the multipath effects the phasefactors are modified with theMonte Carlo methodWith thismethod the target echo signals from low altitude targets canbe obtained accurately for the radar simulation system andthe statistic characteristics of echo signals can be accuratelyand properly simulated The demonstrated method in thispaper is of great help for the radar researcher to evaluate thedetection probability and false alarm probability of specialtargets the signal-to-noise ratio (SNR) of emission pulsesand so on
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M Long Airborne Early Warning System Concepts ArtechBoston Mass USA 1992
[2] W Morchin Airborne Early Warning Radar Artech HouseNorwell Mass USA 1990
[3] L-XGuoA-QWang and JMa ldquoStudy onEMscattering from2-D target above 1-D large scale rough surface with low grazingincidence by parallel MOM based on PC Clustersrdquo Progress inElectromagnetics Research vol 89 pp 149ndash166 2009
[4] M I Skolnik Introduction to Radar SystemsMcGraw-Hill NewYork NY USA 3rd edition 2001
[5] Y Qu G S Liao S Q Zhu and X Y Liu ldquoPattern synthesisof planar antenna array via convex optimization for airborneforward looking radarrdquo Progress in Electromagnetics Researchvol 84 pp 1ndash10 2008
[6] S H Lim J H Han S Y Kim and N H Myung ldquoAzimuthbeam pattern synthesis for air-borne sar system optimizationrdquoProgress in Electromagnetics Research vol 106 pp 295ndash3092010
[7] Y-L Chang C-Y Chiang and K-S Chen ldquoSAR image sim-ulation with application to target recognitionrdquo Progress in Elec-tromagnetics Research vol 119 pp 35ndash57 2011
[8] M Zhang Y W Zhao H Chen and W Q Jiang ldquoSar imagingsimulation for composite model of ship on dynamic oceanscenerdquo Progress in Electromagnetics Research vol 113 pp 395ndash412 2011
[9] S Qiao Z G Shi K S Chen et al ldquoA new architecture ofUWB radar utilizing microwave chaotic signals and chaos syn-chronizationrdquo Progress in Electromagnetics Research vol 75 pp225ndash237 2007
[10] M-J Wang Z-S Wu Y-L Li and G Zhang ldquoHigh resolutionrange profile identifying simulation of laser radar based onpulse beam scattering characteristics of targetsrdquo Progress inElectromagnetics Research vol 96 pp 193ndash204 2009
[11] J G Teti Jr ldquoWide-band airborne radar operating considera-tions for low-altitude surveillance in the presence of specularmultipathrdquo IEEE Transactions on Antennas and Propagationvol 48 no 2 pp 176ndash191 2000
[12] S L Wilson and B D Carison ldquoRadar detection in multipathrdquoIEE Proceedings Radar Sonar and Navigation vol 146 no 1 pp45ndash52 1999
[13] L M Zurk ldquoExperimental observation and statistics of multi-path from terrain with application to overland height findingrdquo
IEEE Transactions on Antennas and Propagation vol 47 no 1pp 121ndash131 1999
[14] Y B Tao H Lin and H J Bao ldquoFrom CPU to GPU GPU-based electromagnetic computingrdquo Progress in ElectromagneticsResearch vol 81 pp 1ndash19 2008
[15] B Etkin andLD ReidDynamics of Flight Stability andControlJohn Wiley amp Sons New York NY USA 1996
[16] X F Li Y J Xie and R Yang ldquoBistatic RCS prediction forcomplex targets using modified current marching techniquerdquoProgress in Electromagnetics Research vol 93 pp 13ndash28 2009
[17] X F Li Y J Xie P Wang and T M Yang ldquoHigh-frequencymethod for scattering from electrically large conductive targetsin half-spacerdquo IEEE Antennas and Wireless Propagation Lettersvol 6 pp 259ndash262 2007
[18] K A Michalski and D Zheng ldquoElectromagnetic scattering andradiation by surfaces of arbitrary shape in layered media ITheoryrdquo IEEE Transactions on Antennas and Propagation vol38 no 3 pp 335ndash344 1990
[19] H Bagci A E Yilmaz V Lomakin and E Michielssen ldquoFastsolution of mixed-potential time-domain integral equations forhalf-space environmentsrdquo IEEE Transactions on Geoscience andRemote Sensing vol 43 no 2 pp 269ndash279 2005
[20] R C Acar and G Dural ldquoComplete set of closed-form Greenrsquosfunctions for cylindrically layered mediardquo in Proceedings of theIEEE Antennas and Propagation Society International Sympo-sium 2006 pp 2863ndash2866 2006
[21] J M Rius M Ferrando and L Jofre ldquoHigh-frequency RCSof complex radar targets in real-timerdquo IEEE Transactions onAntennas and Propagation vol 41 no 9 pp 1308ndash1319 1993
[22] E F Knott J F Shaeffer and M T Tuley Radar Cross SectionSciTech Publishing Raleigh NC USA 2004
[23] B Widrow K M Duvall R P Gooch and W C NewmanldquoSignal cancellation phenomena in adaptive antennas causesand curesrdquo IEEETransactions onAntennas and Propagation vol30 no 3 pp 469ndash478 1982
[24] WDWhite ldquoLow angle radar tracking in the presence ofmulti-pathrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 10 no 6 pp 335ndash352 1974
[25] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings Part F Radarand Signal Processing vol 138 no 2 pp 163ndash171 1991
[26] P Beckmann and A Spizzichino The Scattering of Electro-Magnetic Waves from Rough Surfaces Artech House NorwoodMass USA 1987
[27] C I Beard ldquoCoherent and incoherent scattering of microwavesfrom the oceanrdquo IEEE Transactions on Antenna and Propaga-tion vol 9 no 5 pp 470ndash483 1961
[28] N Pinel C Bourlier and J Saillard ldquoDegree of roughness ofrough layers extensions of the Rayleigh roughnessrdquo Progress inElectromagnetics Research B no 19 pp 41ndash63 2010
[29] M Mishra and N Gupta ldquoMonte Carlo integration techniquefor the analysis of electromagnetic scattering from conductingsurfacesrdquo Progress in Electromagnetics Research vol 79 pp 91ndash106 2008
[30] M A Herrador A G Asuero and A G Gonzalez ldquoEstimationof the uncertainty of indirect measurements from the propa-gation of distributions by using the Monte-Carlo method anoverviewrdquoChemometrics and Intelligent Laboratory Systems vol79 no 1-2 pp 115ndash122 2005
International Journal of Antennas and Propagation 5
Monte Carlo methods are algorithm for solving variouskinds of computational problems by using random numbers[29] It is a means of treating mathematical problems byfinding a probabilistic analog and then obtaining approxi-mate answers to this analog by some experimental samplingprocedureThe solution of a problem by this method is closerin spirit to physical experiments than to classical numericaltechniques In our calculations the number of samples isincreased to achieve a fairly regular distribution As for theerror from the Monte Carlo sampling technique it can bemade negligible bymaking the number of samples sufficientlylarge
To simulate the instantaneous amplitude from echo sig-nals withMonte Carlomethod firstly the probabilistic modelaccording to the problem should be established From for-mula (7) the phase factors can be rewritten as
1205931= (
2120587Δ1198711
120582) mod (2120587)
1205932= (
2120587Δ1198712
120582) mod (2120587)
1205933= (
2120587Δ1198713
120582) mod (2120587)
(15)
where 1205931 120593
2 and 120593
3are the remainder of phase differences
divide 2120587 of three reflected rays compared to the direct Forairborne radar we know that the path length differences ofthree reflected rays compared to the direct can far exceedthe wave length so 120593
1 120593
2 and 120593
3can be supposed to three
sequences of random numbers from a uniform distributionin (0 2120587)
With the most widely used mixed congruence method[30] we generate three sequences of random numbers 119886
119894 119887
119894
and 119888119894(119894 = 1 2 119873) which is uniformly distributed in the
interval (0 1)
119886119894=[(119898
119886119886119894minus1+ 119899
119886) mod119872]
119872
119887119894=[(119898
119887119887119894minus1+ 119899
119887) mod119872]
119872
119888119894=[(119898
119888119888119894minus1+ 119899
119888) mod119872]
119872
119894 = 1 2 119873
(17)
0 2040
6080
100
0
50
1000
5
10
15
20
0533m
617m262m
Figure 3 The missile is placed 20m above the sea
where 119898119886 119898
119887 and 119898
119888are the multipliers 119899
119886 119899
119887 and 119899
119888are
the increment119872 is the modulusGiven120593
1= 2120587119886
119894120593
2= 2120587119887
119894 and120593
3= 2120587119888
119894in formula (16)
run the Monte Carlo simulation n times to estimate the echosignals from low altitude targets accurately based on actualenvironment for airborne radar
By using a sufficient number of samples a probability dis-tribution is obtained to describe echo signals The statisticalcharacteristics of the echo signals can be illustrated at last
3 Results and Discussion
Themissile is adopted in the simulation as shown in Figure 3with its RCS in Figure 4
The PDF of magnitude of echoes is shown in Figure 5when the target is flying at a height of 20m above the seaAccording to the radar equation the magnitude of direct rayis 64416 times 10minus7 V Figure 5 indicates that the magnitude ofechoes can be different from direct ray due to multipath
The magnitudes of direct echoes in the situations of dif-ferent targetrsquos heights are shown in Figure 6 and the PDFs ofmagnitude of echoes normalized to the direct ray in the samesituations are shown in Figure 7 As can be seen in Figure 6the magnitudes of direct echoes vary with targetrsquos heightsThe prime reason for that is the target RCSrsquos variation causedby the different azimuth angle 120593 and the elevation angle 120579with respect to the coordinate system of the target Figure 7shows the influences of multipath at different heights In theconditions of the target flying at low heights (about a fewhundred meters) the PDFs vary little with the heights whilethe normalized magnitude of echoes with respect to the peakof the PDF reduces a little As illustrated in Figure 7 when thetarget is at heights of below 2000m themagnitudes of echoesnormalized to the direct ray vary mainly between 0 and 4The peak probability densities are in the range of 0-1 and thecorresponding normalized magnitudes of echoes are in therange of 1-2 These results in Figure 7 are similar to the onesin Figures 14 and 15 of [13] which make the demonstratedalgorithm valid in this paper However the terrain in thispaper was produced from the digital feature analysis data
6 International Journal of Antennas and Propagation
Figure 4 The missilersquos RCS with frequency of 04GHz
Prob
abili
ty d
ensit
y
The magnitude of echoes (V) times10minus6
07
06
05
04
03
02
01
0
0 05 1 15 2 25 3
Figure 5 The PDF versus magnitude of echoes for the target at aheight of 20m above the sea
(DFAD) yet in [13] the desert terrain for simulations wasproduced from the defense mapping agency (DMA) data theexperiments weremade inNewMexico andNorthernMaineThe differences between the results in this paper and those in[13] are mainly caused by the differences in the terrains thefrequencies the geometries and attitudes of the targets theheights of the airborne radars and the targets the numberof samples and so forth As the target is flying higher theinfluences of multipath become less to the echoes until theindirect ways disappear In practice if the target height isat least moderately large the range resolution of the radaris less than the path differences of the direct and multipathreturns so the radar is able to resolve them As radar heightdecreases to a height of ground based radar the differencein angle between reflected and direct paths can be ignored
20100 500 1000 20000
1
2
3
4
5
6
7
8
The target height (m)
The m
agni
tude
of d
irect
ray
(V)
times10minus7
Figure 6 The magnitude of direct ray versus the target height
4Normalized magnitude of echoes
Prob
abili
ty d
ensit
y
0 05 1 15 2 25 3 35
0
05
1
15
2
25
3
h = 20mh = 100mh = 500m
h = 1000mh = 2000m
Figure 7 The PDF of magnitude of echoes at different targetrsquosheights
which causes the fluctuations in phase disappearing for low-flying targets and the PDF of magnitude of echoes is similarto Figure 6 of [12]
4 Conclusions
For airborne radar to detect low altitude targets a demon-strated hybrid method based on probability statistics andcomputational electromagnetics is proposed in this paperto give a statistical simulation of echo signals based onactual environmentThe half-space physical optics combinedwith the graphical-electromagnetic computing method isemployed to calculate the RCS of low-flying targets suffi-ciently accurately and efficiently in half space Considering
International Journal of Antennas and Propagation 7
the fluctuations due to the multipath effects the phasefactors are modified with theMonte Carlo methodWith thismethod the target echo signals from low altitude targets canbe obtained accurately for the radar simulation system andthe statistic characteristics of echo signals can be accuratelyand properly simulated The demonstrated method in thispaper is of great help for the radar researcher to evaluate thedetection probability and false alarm probability of specialtargets the signal-to-noise ratio (SNR) of emission pulsesand so on
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M Long Airborne Early Warning System Concepts ArtechBoston Mass USA 1992
[2] W Morchin Airborne Early Warning Radar Artech HouseNorwell Mass USA 1990
[3] L-XGuoA-QWang and JMa ldquoStudy onEMscattering from2-D target above 1-D large scale rough surface with low grazingincidence by parallel MOM based on PC Clustersrdquo Progress inElectromagnetics Research vol 89 pp 149ndash166 2009
[4] M I Skolnik Introduction to Radar SystemsMcGraw-Hill NewYork NY USA 3rd edition 2001
[5] Y Qu G S Liao S Q Zhu and X Y Liu ldquoPattern synthesisof planar antenna array via convex optimization for airborneforward looking radarrdquo Progress in Electromagnetics Researchvol 84 pp 1ndash10 2008
[6] S H Lim J H Han S Y Kim and N H Myung ldquoAzimuthbeam pattern synthesis for air-borne sar system optimizationrdquoProgress in Electromagnetics Research vol 106 pp 295ndash3092010
[7] Y-L Chang C-Y Chiang and K-S Chen ldquoSAR image sim-ulation with application to target recognitionrdquo Progress in Elec-tromagnetics Research vol 119 pp 35ndash57 2011
[8] M Zhang Y W Zhao H Chen and W Q Jiang ldquoSar imagingsimulation for composite model of ship on dynamic oceanscenerdquo Progress in Electromagnetics Research vol 113 pp 395ndash412 2011
[9] S Qiao Z G Shi K S Chen et al ldquoA new architecture ofUWB radar utilizing microwave chaotic signals and chaos syn-chronizationrdquo Progress in Electromagnetics Research vol 75 pp225ndash237 2007
[10] M-J Wang Z-S Wu Y-L Li and G Zhang ldquoHigh resolutionrange profile identifying simulation of laser radar based onpulse beam scattering characteristics of targetsrdquo Progress inElectromagnetics Research vol 96 pp 193ndash204 2009
[11] J G Teti Jr ldquoWide-band airborne radar operating considera-tions for low-altitude surveillance in the presence of specularmultipathrdquo IEEE Transactions on Antennas and Propagationvol 48 no 2 pp 176ndash191 2000
[12] S L Wilson and B D Carison ldquoRadar detection in multipathrdquoIEE Proceedings Radar Sonar and Navigation vol 146 no 1 pp45ndash52 1999
[13] L M Zurk ldquoExperimental observation and statistics of multi-path from terrain with application to overland height findingrdquo
IEEE Transactions on Antennas and Propagation vol 47 no 1pp 121ndash131 1999
[14] Y B Tao H Lin and H J Bao ldquoFrom CPU to GPU GPU-based electromagnetic computingrdquo Progress in ElectromagneticsResearch vol 81 pp 1ndash19 2008
[15] B Etkin andLD ReidDynamics of Flight Stability andControlJohn Wiley amp Sons New York NY USA 1996
[16] X F Li Y J Xie and R Yang ldquoBistatic RCS prediction forcomplex targets using modified current marching techniquerdquoProgress in Electromagnetics Research vol 93 pp 13ndash28 2009
[17] X F Li Y J Xie P Wang and T M Yang ldquoHigh-frequencymethod for scattering from electrically large conductive targetsin half-spacerdquo IEEE Antennas and Wireless Propagation Lettersvol 6 pp 259ndash262 2007
[18] K A Michalski and D Zheng ldquoElectromagnetic scattering andradiation by surfaces of arbitrary shape in layered media ITheoryrdquo IEEE Transactions on Antennas and Propagation vol38 no 3 pp 335ndash344 1990
[19] H Bagci A E Yilmaz V Lomakin and E Michielssen ldquoFastsolution of mixed-potential time-domain integral equations forhalf-space environmentsrdquo IEEE Transactions on Geoscience andRemote Sensing vol 43 no 2 pp 269ndash279 2005
[20] R C Acar and G Dural ldquoComplete set of closed-form Greenrsquosfunctions for cylindrically layered mediardquo in Proceedings of theIEEE Antennas and Propagation Society International Sympo-sium 2006 pp 2863ndash2866 2006
[21] J M Rius M Ferrando and L Jofre ldquoHigh-frequency RCSof complex radar targets in real-timerdquo IEEE Transactions onAntennas and Propagation vol 41 no 9 pp 1308ndash1319 1993
[22] E F Knott J F Shaeffer and M T Tuley Radar Cross SectionSciTech Publishing Raleigh NC USA 2004
[23] B Widrow K M Duvall R P Gooch and W C NewmanldquoSignal cancellation phenomena in adaptive antennas causesand curesrdquo IEEETransactions onAntennas and Propagation vol30 no 3 pp 469ndash478 1982
[24] WDWhite ldquoLow angle radar tracking in the presence ofmulti-pathrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 10 no 6 pp 335ndash352 1974
[25] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings Part F Radarand Signal Processing vol 138 no 2 pp 163ndash171 1991
[26] P Beckmann and A Spizzichino The Scattering of Electro-Magnetic Waves from Rough Surfaces Artech House NorwoodMass USA 1987
[27] C I Beard ldquoCoherent and incoherent scattering of microwavesfrom the oceanrdquo IEEE Transactions on Antenna and Propaga-tion vol 9 no 5 pp 470ndash483 1961
[28] N Pinel C Bourlier and J Saillard ldquoDegree of roughness ofrough layers extensions of the Rayleigh roughnessrdquo Progress inElectromagnetics Research B no 19 pp 41ndash63 2010
[29] M Mishra and N Gupta ldquoMonte Carlo integration techniquefor the analysis of electromagnetic scattering from conductingsurfacesrdquo Progress in Electromagnetics Research vol 79 pp 91ndash106 2008
[30] M A Herrador A G Asuero and A G Gonzalez ldquoEstimationof the uncertainty of indirect measurements from the propa-gation of distributions by using the Monte-Carlo method anoverviewrdquoChemometrics and Intelligent Laboratory Systems vol79 no 1-2 pp 115ndash122 2005
Figure 4 The missilersquos RCS with frequency of 04GHz
Prob
abili
ty d
ensit
y
The magnitude of echoes (V) times10minus6
07
06
05
04
03
02
01
0
0 05 1 15 2 25 3
Figure 5 The PDF versus magnitude of echoes for the target at aheight of 20m above the sea
(DFAD) yet in [13] the desert terrain for simulations wasproduced from the defense mapping agency (DMA) data theexperiments weremade inNewMexico andNorthernMaineThe differences between the results in this paper and those in[13] are mainly caused by the differences in the terrains thefrequencies the geometries and attitudes of the targets theheights of the airborne radars and the targets the numberof samples and so forth As the target is flying higher theinfluences of multipath become less to the echoes until theindirect ways disappear In practice if the target height isat least moderately large the range resolution of the radaris less than the path differences of the direct and multipathreturns so the radar is able to resolve them As radar heightdecreases to a height of ground based radar the differencein angle between reflected and direct paths can be ignored
20100 500 1000 20000
1
2
3
4
5
6
7
8
The target height (m)
The m
agni
tude
of d
irect
ray
(V)
times10minus7
Figure 6 The magnitude of direct ray versus the target height
4Normalized magnitude of echoes
Prob
abili
ty d
ensit
y
0 05 1 15 2 25 3 35
0
05
1
15
2
25
3
h = 20mh = 100mh = 500m
h = 1000mh = 2000m
Figure 7 The PDF of magnitude of echoes at different targetrsquosheights
which causes the fluctuations in phase disappearing for low-flying targets and the PDF of magnitude of echoes is similarto Figure 6 of [12]
4 Conclusions
For airborne radar to detect low altitude targets a demon-strated hybrid method based on probability statistics andcomputational electromagnetics is proposed in this paperto give a statistical simulation of echo signals based onactual environmentThe half-space physical optics combinedwith the graphical-electromagnetic computing method isemployed to calculate the RCS of low-flying targets suffi-ciently accurately and efficiently in half space Considering
International Journal of Antennas and Propagation 7
the fluctuations due to the multipath effects the phasefactors are modified with theMonte Carlo methodWith thismethod the target echo signals from low altitude targets canbe obtained accurately for the radar simulation system andthe statistic characteristics of echo signals can be accuratelyand properly simulated The demonstrated method in thispaper is of great help for the radar researcher to evaluate thedetection probability and false alarm probability of specialtargets the signal-to-noise ratio (SNR) of emission pulsesand so on
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M Long Airborne Early Warning System Concepts ArtechBoston Mass USA 1992
[2] W Morchin Airborne Early Warning Radar Artech HouseNorwell Mass USA 1990
[3] L-XGuoA-QWang and JMa ldquoStudy onEMscattering from2-D target above 1-D large scale rough surface with low grazingincidence by parallel MOM based on PC Clustersrdquo Progress inElectromagnetics Research vol 89 pp 149ndash166 2009
[4] M I Skolnik Introduction to Radar SystemsMcGraw-Hill NewYork NY USA 3rd edition 2001
[5] Y Qu G S Liao S Q Zhu and X Y Liu ldquoPattern synthesisof planar antenna array via convex optimization for airborneforward looking radarrdquo Progress in Electromagnetics Researchvol 84 pp 1ndash10 2008
[6] S H Lim J H Han S Y Kim and N H Myung ldquoAzimuthbeam pattern synthesis for air-borne sar system optimizationrdquoProgress in Electromagnetics Research vol 106 pp 295ndash3092010
[7] Y-L Chang C-Y Chiang and K-S Chen ldquoSAR image sim-ulation with application to target recognitionrdquo Progress in Elec-tromagnetics Research vol 119 pp 35ndash57 2011
[8] M Zhang Y W Zhao H Chen and W Q Jiang ldquoSar imagingsimulation for composite model of ship on dynamic oceanscenerdquo Progress in Electromagnetics Research vol 113 pp 395ndash412 2011
[9] S Qiao Z G Shi K S Chen et al ldquoA new architecture ofUWB radar utilizing microwave chaotic signals and chaos syn-chronizationrdquo Progress in Electromagnetics Research vol 75 pp225ndash237 2007
[10] M-J Wang Z-S Wu Y-L Li and G Zhang ldquoHigh resolutionrange profile identifying simulation of laser radar based onpulse beam scattering characteristics of targetsrdquo Progress inElectromagnetics Research vol 96 pp 193ndash204 2009
[11] J G Teti Jr ldquoWide-band airborne radar operating considera-tions for low-altitude surveillance in the presence of specularmultipathrdquo IEEE Transactions on Antennas and Propagationvol 48 no 2 pp 176ndash191 2000
[12] S L Wilson and B D Carison ldquoRadar detection in multipathrdquoIEE Proceedings Radar Sonar and Navigation vol 146 no 1 pp45ndash52 1999
[13] L M Zurk ldquoExperimental observation and statistics of multi-path from terrain with application to overland height findingrdquo
IEEE Transactions on Antennas and Propagation vol 47 no 1pp 121ndash131 1999
[14] Y B Tao H Lin and H J Bao ldquoFrom CPU to GPU GPU-based electromagnetic computingrdquo Progress in ElectromagneticsResearch vol 81 pp 1ndash19 2008
[15] B Etkin andLD ReidDynamics of Flight Stability andControlJohn Wiley amp Sons New York NY USA 1996
[16] X F Li Y J Xie and R Yang ldquoBistatic RCS prediction forcomplex targets using modified current marching techniquerdquoProgress in Electromagnetics Research vol 93 pp 13ndash28 2009
[17] X F Li Y J Xie P Wang and T M Yang ldquoHigh-frequencymethod for scattering from electrically large conductive targetsin half-spacerdquo IEEE Antennas and Wireless Propagation Lettersvol 6 pp 259ndash262 2007
[18] K A Michalski and D Zheng ldquoElectromagnetic scattering andradiation by surfaces of arbitrary shape in layered media ITheoryrdquo IEEE Transactions on Antennas and Propagation vol38 no 3 pp 335ndash344 1990
[19] H Bagci A E Yilmaz V Lomakin and E Michielssen ldquoFastsolution of mixed-potential time-domain integral equations forhalf-space environmentsrdquo IEEE Transactions on Geoscience andRemote Sensing vol 43 no 2 pp 269ndash279 2005
[20] R C Acar and G Dural ldquoComplete set of closed-form Greenrsquosfunctions for cylindrically layered mediardquo in Proceedings of theIEEE Antennas and Propagation Society International Sympo-sium 2006 pp 2863ndash2866 2006
[21] J M Rius M Ferrando and L Jofre ldquoHigh-frequency RCSof complex radar targets in real-timerdquo IEEE Transactions onAntennas and Propagation vol 41 no 9 pp 1308ndash1319 1993
[22] E F Knott J F Shaeffer and M T Tuley Radar Cross SectionSciTech Publishing Raleigh NC USA 2004
[23] B Widrow K M Duvall R P Gooch and W C NewmanldquoSignal cancellation phenomena in adaptive antennas causesand curesrdquo IEEETransactions onAntennas and Propagation vol30 no 3 pp 469ndash478 1982
[24] WDWhite ldquoLow angle radar tracking in the presence ofmulti-pathrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 10 no 6 pp 335ndash352 1974
[25] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings Part F Radarand Signal Processing vol 138 no 2 pp 163ndash171 1991
[26] P Beckmann and A Spizzichino The Scattering of Electro-Magnetic Waves from Rough Surfaces Artech House NorwoodMass USA 1987
[27] C I Beard ldquoCoherent and incoherent scattering of microwavesfrom the oceanrdquo IEEE Transactions on Antenna and Propaga-tion vol 9 no 5 pp 470ndash483 1961
[28] N Pinel C Bourlier and J Saillard ldquoDegree of roughness ofrough layers extensions of the Rayleigh roughnessrdquo Progress inElectromagnetics Research B no 19 pp 41ndash63 2010
[29] M Mishra and N Gupta ldquoMonte Carlo integration techniquefor the analysis of electromagnetic scattering from conductingsurfacesrdquo Progress in Electromagnetics Research vol 79 pp 91ndash106 2008
[30] M A Herrador A G Asuero and A G Gonzalez ldquoEstimationof the uncertainty of indirect measurements from the propa-gation of distributions by using the Monte-Carlo method anoverviewrdquoChemometrics and Intelligent Laboratory Systems vol79 no 1-2 pp 115ndash122 2005
International Journal of Antennas and Propagation 7
the fluctuations due to the multipath effects the phasefactors are modified with theMonte Carlo methodWith thismethod the target echo signals from low altitude targets canbe obtained accurately for the radar simulation system andthe statistic characteristics of echo signals can be accuratelyand properly simulated The demonstrated method in thispaper is of great help for the radar researcher to evaluate thedetection probability and false alarm probability of specialtargets the signal-to-noise ratio (SNR) of emission pulsesand so on
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] M Long Airborne Early Warning System Concepts ArtechBoston Mass USA 1992
[2] W Morchin Airborne Early Warning Radar Artech HouseNorwell Mass USA 1990
[3] L-XGuoA-QWang and JMa ldquoStudy onEMscattering from2-D target above 1-D large scale rough surface with low grazingincidence by parallel MOM based on PC Clustersrdquo Progress inElectromagnetics Research vol 89 pp 149ndash166 2009
[4] M I Skolnik Introduction to Radar SystemsMcGraw-Hill NewYork NY USA 3rd edition 2001
[5] Y Qu G S Liao S Q Zhu and X Y Liu ldquoPattern synthesisof planar antenna array via convex optimization for airborneforward looking radarrdquo Progress in Electromagnetics Researchvol 84 pp 1ndash10 2008
[6] S H Lim J H Han S Y Kim and N H Myung ldquoAzimuthbeam pattern synthesis for air-borne sar system optimizationrdquoProgress in Electromagnetics Research vol 106 pp 295ndash3092010
[7] Y-L Chang C-Y Chiang and K-S Chen ldquoSAR image sim-ulation with application to target recognitionrdquo Progress in Elec-tromagnetics Research vol 119 pp 35ndash57 2011
[8] M Zhang Y W Zhao H Chen and W Q Jiang ldquoSar imagingsimulation for composite model of ship on dynamic oceanscenerdquo Progress in Electromagnetics Research vol 113 pp 395ndash412 2011
[9] S Qiao Z G Shi K S Chen et al ldquoA new architecture ofUWB radar utilizing microwave chaotic signals and chaos syn-chronizationrdquo Progress in Electromagnetics Research vol 75 pp225ndash237 2007
[10] M-J Wang Z-S Wu Y-L Li and G Zhang ldquoHigh resolutionrange profile identifying simulation of laser radar based onpulse beam scattering characteristics of targetsrdquo Progress inElectromagnetics Research vol 96 pp 193ndash204 2009
[11] J G Teti Jr ldquoWide-band airborne radar operating considera-tions for low-altitude surveillance in the presence of specularmultipathrdquo IEEE Transactions on Antennas and Propagationvol 48 no 2 pp 176ndash191 2000
[12] S L Wilson and B D Carison ldquoRadar detection in multipathrdquoIEE Proceedings Radar Sonar and Navigation vol 146 no 1 pp45ndash52 1999
[13] L M Zurk ldquoExperimental observation and statistics of multi-path from terrain with application to overland height findingrdquo
IEEE Transactions on Antennas and Propagation vol 47 no 1pp 121ndash131 1999
[14] Y B Tao H Lin and H J Bao ldquoFrom CPU to GPU GPU-based electromagnetic computingrdquo Progress in ElectromagneticsResearch vol 81 pp 1ndash19 2008
[15] B Etkin andLD ReidDynamics of Flight Stability andControlJohn Wiley amp Sons New York NY USA 1996
[16] X F Li Y J Xie and R Yang ldquoBistatic RCS prediction forcomplex targets using modified current marching techniquerdquoProgress in Electromagnetics Research vol 93 pp 13ndash28 2009
[17] X F Li Y J Xie P Wang and T M Yang ldquoHigh-frequencymethod for scattering from electrically large conductive targetsin half-spacerdquo IEEE Antennas and Wireless Propagation Lettersvol 6 pp 259ndash262 2007
[18] K A Michalski and D Zheng ldquoElectromagnetic scattering andradiation by surfaces of arbitrary shape in layered media ITheoryrdquo IEEE Transactions on Antennas and Propagation vol38 no 3 pp 335ndash344 1990
[19] H Bagci A E Yilmaz V Lomakin and E Michielssen ldquoFastsolution of mixed-potential time-domain integral equations forhalf-space environmentsrdquo IEEE Transactions on Geoscience andRemote Sensing vol 43 no 2 pp 269ndash279 2005
[20] R C Acar and G Dural ldquoComplete set of closed-form Greenrsquosfunctions for cylindrically layered mediardquo in Proceedings of theIEEE Antennas and Propagation Society International Sympo-sium 2006 pp 2863ndash2866 2006
[21] J M Rius M Ferrando and L Jofre ldquoHigh-frequency RCSof complex radar targets in real-timerdquo IEEE Transactions onAntennas and Propagation vol 41 no 9 pp 1308ndash1319 1993
[22] E F Knott J F Shaeffer and M T Tuley Radar Cross SectionSciTech Publishing Raleigh NC USA 2004
[23] B Widrow K M Duvall R P Gooch and W C NewmanldquoSignal cancellation phenomena in adaptive antennas causesand curesrdquo IEEETransactions onAntennas and Propagation vol30 no 3 pp 469ndash478 1982
[24] WDWhite ldquoLow angle radar tracking in the presence ofmulti-pathrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 10 no 6 pp 335ndash352 1974
[25] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings Part F Radarand Signal Processing vol 138 no 2 pp 163ndash171 1991
[26] P Beckmann and A Spizzichino The Scattering of Electro-Magnetic Waves from Rough Surfaces Artech House NorwoodMass USA 1987
[27] C I Beard ldquoCoherent and incoherent scattering of microwavesfrom the oceanrdquo IEEE Transactions on Antenna and Propaga-tion vol 9 no 5 pp 470ndash483 1961
[28] N Pinel C Bourlier and J Saillard ldquoDegree of roughness ofrough layers extensions of the Rayleigh roughnessrdquo Progress inElectromagnetics Research B no 19 pp 41ndash63 2010
[29] M Mishra and N Gupta ldquoMonte Carlo integration techniquefor the analysis of electromagnetic scattering from conductingsurfacesrdquo Progress in Electromagnetics Research vol 79 pp 91ndash106 2008
[30] M A Herrador A G Asuero and A G Gonzalez ldquoEstimationof the uncertainty of indirect measurements from the propa-gation of distributions by using the Monte-Carlo method anoverviewrdquoChemometrics and Intelligent Laboratory Systems vol79 no 1-2 pp 115ndash122 2005
