research article rotating parabolic-reflector...

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2013 Article ID 583865 13 pageshttpdxdoiorg1011552013583865

Research ArticleRotating Parabolic-Reflector Antenna Target in SAR DataModel Characteristics and Parameter Estimation

Bin Deng Hong-Qiang Wang Yu-Liang Qin Sha Zhu and Xiang Li

College of Electronic Science and Engineering National University of Defense Technology China

Correspondence should be addressed to Bin Deng sagitdeng163com

Received 22 October 2012 Accepted 3 March 2013

Academic Editor Gui Gao

Copyright copy 2013 Bin Deng et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Parabolic-reflector antennas (PRAs) usually possessing rotation are a particular type of targets of potential interest to the syntheticaperture radar (SAR) community This paper is aimed to investigate PRArsquos scattering characteristics and then to extract PRArsquosparameters from SAR returns for supporting image interpretation and target recognition We at first obtain both closed-form andnumeric solutions to PRArsquos backscattering by geometrical optics (GO) physical optics and graphical electromagnetic computationrespectively Based on the GO solution a migratory scattering center model is at first presented for representing the movement ofthe specular point with aspect angle and then a hybrid model named the migratorymicromotion scattering center (MMSC)model is proposed for characterizing a rotating PRA in the SAR geometry which incorporates PRArsquos rotation into its migratoryscattering center model Additionally we in detail analyze PRArsquos radar characteristics on radar cross-section high-resolution rangeprofiles time-frequency distribution and 2D images which also confirm the models proposed A maximal likelihood estimator isdeveloped for jointly solving the MMSC model for PRArsquos multiple parameters by optimization By exploiting the aforementionedcharacteristics the coarse parameter estimation guarantees convergency upon global minima The signatures recovered can befavorably utilized for SAR image interpretation and target recognition

1 Introduction

Parabolic-reflector antennas (PRAs) due to their excellentbeam directionality are widely used in radar surveillancenavigation microwave communication and radio astron-omy Although they are antennas in themselves PRAs arealso present as a particular type of targets gaining growinginterest in some scenarios to the synthetic aperture radar(SAR) community [1] Their scattering and rotation charac-teristics which are obtainable with SAR may be of valuableaid for target recognition Therefore it is desirable to detectand recognize them from SAR returns or imagery Howeverthis work is difficult because of two reasons On one handthe backscattering of PRAs is so complicated that theycannot be characterized by point scattering centers and thedirectionality favorable as it is for PRAs as antennas resultsin aspect sensitivity and weak backscattering at most aspectangles when they are present as SAR targets On the otherhand their rotation due to mechanical scanning changesthe Doppler history and therefore makes their SAR images

blurred and enigmatic And unfortunately rotation cannotbe processed by the mature technique that is SAR groundmoving target indication (SARGMTI) though it is a validtool for uniformly moving targets [2]

Conventionally rotating PRAs (RPRAs) were investi-gated asmicromotion targets In 2009 the SARmicromotiontarget indication concept was presented to generalize theSARGMTI technique and to intensively examine the detec-tion and imaging problems for micromotion targets [3]Actually early in 2004 the picket fence shape was revealedto be representative of a rotating radar dish in the SARazimuth image [4] but the dish (viz PRA) was simplifiedas a combination of point scatterers which are not qualifiedfor characterizing PRArsquos complex scattering In 2005 and2010 wavelet decomposition and chirplet decompositionwere used to successfully separate the return of anRPRA fromthat of stationary parts The returns were collected by the USNavy APY-6 radar [5 6] After separation time-frequencydistribution (TFD) was utilized to analyze the phase modula-tion and autocorrelation to estimate the rotating frequency

2 International Journal of Antennas and Propagation

The RPRA detected was annotated in the SAR image [1]However the RPRArsquos signature blurred in SAR imagerywas not explained and the aforementioned approaches aresusceptible to range cell migration (RCM) TFD also usuallysuffers from cross-term interference In 2007 RPRA datacollected by amicrowave SAR system inUniversity of Zurichwere analyzed [7] A point-symmetrical image of the RPRAwas obtained and the time-frequency image showed a smallfraction of a sinusoidal phase modulation The conclusionshowever are only valid for slow rotating frequency and shortcoherent processing interval (CPI) PRArsquos radar characteris-tics have not been deeply studied either

In this paper we focus on characterization and non-TFD-based estimation of PRAs both stationary and rotating inspotlight SAR data There are different types of PRAs butwe restrict our scope to a rotation-symmetrical PRA witha single reflector Instead of processing returns in a singlerange cell we use the phase history in the wavenumberdomain that is the 2D Fourier transform (FT) of compleximagery to extract PRAs and their parameters We buildfor a PRA a new non-point-scattering center model able todescribe the aspect sensitivity and the sliding of scatterersand then incorporate the motion model into it to obtaina hybrid scatteringmotion model Radar characteristics ofPRAs are then analyzed Afterwards an estimation algorithmfor this model is proposed This idea fully exploits a prioriinformation on target models implements joint estimationof scattering and motion parameters and circumvents TFDand other tough issues such as clutter suppression detectionRCM correction and parameter estimation which have to beconducted separately and explicitly in SARGMTI

This paper is organized as follows In Section 2 geo-metrical optics (GO) physical optics (PO) and graphicalelectromagnetic computation (GRECO) are used to calculatethe backscattering or complex radar cross-section (RCS) ofa PRA Then based on the calculated results we propose anew scattering center model and a hybrid scatteringmotionmodel in Section 3 We then in Section 4 analyze the char-acteristics of PRAs A few interesting signatures of PRAsare revealed for example periodicity of high-resolutionrange profile (HRRP) sequences bowl- and bowknot-shapedimages Sections 5 and 6 provide a maximum likelihoodestimator and a few results and conclusions are reportedin Section 7 The derivation throughout the paper may be alittle difficult to follow and we will try our best to clarify theproblem

2 Complex RCS Calculation for PRA

Backscattering of an antenna consists of two parts that isstructured scattering whose scattering principle is the sameof as common objects and pattern scattering which is closelyrelated to antennarsquos loads Pattern scattering is weak fora PRA and hence not taken into account here We willcalculate PRArsquos complex RCS via GO PO and GRECO GOis simple and is particularly suitable for smooth surface POhas high accuracy near targetrsquos normal direction and canprovide closed-form solutions in some particular aspectsAnd GRECO as a tool and software incorporates many

Specularpoint

FrontBack

Radar

Migration of thespecular point

119863

minus119875

21205750

1198632

8119875

1205910

120578

119911

119909

119910

120578 = 0∘

120578 = arctan 1198632119875

S

O1O(F)

Figure 1 PRA geometry (119909-z slice)

electromagnetic methods and can efficiently provide high-precision numerical solutions

21 GO As Figure 1 shows let 119875 denote the focus parameterof a PRA 120591 and 120575 denote the radius and angle (relative to -zaxis) of any point on a PRA Other parameters encounteredare defined in Figure 1 where the focus is located at the origin119874-z is the symmetrical axis Then the amplitude of the PRArsquoscomplex RCS that is the square root of RCS can be obtainedfrom the GO theory [8]

1003816100381610038161003816radic120590GO

1003816100381610038161003816=

radic120587119875

cos2120578 (1)

Its phase can be easily derived from the position of thespecular point 119878 Combining the amplitude and phase wehave the complex RCS

radic120590GO =

radic120587119875

cos2120578exp(minus119895119870 119875

2 cos 120578) (2)

where 119870 = 4120587119891119888 is the wavenumber 119888 is the speed of lightand 119891 is the frequency When choosing PRArsquos bottom center1198741as the phase reference we need to multiply (2) with

exp(119895119870 sdot 1198752 sdot cos 120578) and then have

radic120590GO =

radic120587119875

cos2120578exp(minus119895119870

119875 sin21205782 cos 120578

) (3)

Equations (2) and (3) hold only when 120578 is chosen suchthat the specular point does not move outside the PRASpecifically when the incident field illuminates the PRA frontside 120578 should be subjected to

10038161003816100381610038161205781003816100381610038161003816le arctan 119863

2119875

(4)

22 PO To facilitate application of PO we use the Cartesiancoordinate system and the cylinder coordinate system Thenin Figure 1 for any point 119876 in the PRA suppose that its

International Journal of Antennas and Propagation 3

position in the cylindrical coordinate system is (120588 120593 119911) =

(120588 120593 1205882(2119875) minus 1198752) It can be vectorized in the Cartesian

coordinate system as

r1015840 = (119909 119910 119911)119879

= (120588 cos120593 120588 sin1205931205882

2119875

minus

119875

2

)

119879

(5)

The normal vector at 119876 is

n = (minus

120588 cos120593119875

minus

120588 sin120593119875

1)

119879

radic1 +

1205882

1198752

(6)

The backscattering direction can be expressed by

r = (sin 120578 0 cos 120578)119879 (7)

According to PO the complex RCS of the PRA can bewritten as

radic120590PO = minus119895

119896

radic120587

int

2120587

0

int

1198632

0

r sdot n exp (119895119870r sdot r1015840)radic1 + 1205882

1198752120588119889120588119889120593

(8)

Generally (8) can be calculated by using numericalmethods and its closed-form solution cannot be obtaineddue to its complexity However the closed-form solutionexists for some particular incident angles When 120578 = 0 forexample the complex RCS we derive is

radic120590PO1003816100381610038161003816120578=0

= minus1198952radic120587119875 sin(119870 1198632

16119875

) exp[minus119895119870(

119875

2

minus

1198632

16119875

)]

= radic120587119875[exp(minus119895119870 1198632

16119875

) minus exp(119895119870 1198632

16119875

)]

times exp[minus119895119870(

119875

2

minus

1198632

16119875

)]

(9)

23 GRECO GRECO is an effective tool for calculating(complex) RCS of electrically large objects [9] It is capable ofcalculating the reflected and diffracted fields with high pre-cision and efficiency by using graphic hardware accelerationand can also perform well for a wide range of aspect anglesTherefore we select it as a numeric method and benchmarkfor the GO and PO methods The result will be given inSection 6

3 Scattering Center Model and Hybrid Model

31 Migratory Scattering Model for a Stationary PRA Thescattering center models have developed from the pointscattering model damped exponentialProny model [10]through the geometric theory of diffraction (GTD) model[11] localized scattering model [12] and attributed scatteringmodel [13] to the polynomial model [14] and migratory

scattering model [15] Their difference lies in that they usedifferent models to characterize the dependency of complexRCS including amplitude andor phase on frequency andoraspect angle We can find from (2) that for a PRA thecomplex RCS amplitude depends on aspect angle 120579 (120579 = 120578

when aspect angle is relative to PRArsquos axis of symmetry)resulting in aspect sensitivity while the phase also dependson aspect angle but the dependency is not of the form ofsin 120579 or cos 120579 as ideal point scattering is (1 cos 120578 instead)which means that the scattering center will move when theaspect angle varies By (2) we have in effect proven that aPRA can be approximated by the migratory scattering modelcontaining a single scattering center and the approximationhas sound physical grounds However the amplitude term ofcomplex RCS in (2) enough as it is for coarse analysis is notan accurate model intended for parameter estimation sinceGO does not take into account more scattering componentsfor example edge diffraction Therefore we use a function119904(120578120579) to describe arbitrary aspect dependency Then from

(3) we obtain the migratory scattering model for a stationaryPRA

119892 (119870 120579) = 119904 (120579) exp[minus119895119870(119909 cos 120579 + 119910 sin 120579 +119875 sin2120578

120579

2 cos 120578120579

)]

(10)

where (119909 119910) are the coordinates of the PRArsquos bottom center1198741 and 120578

120579is the 120579-dependent incident angle

For instance When choosing PRArsquos geometrical center asthe phase reference (in order for agreement with the GRECOgeometry in Figure 3 and Section 4) we have

119892 (119870 1205791015840)

= 119904 (1205791015840) exp[minus119895119870(

sin 1205791015840

4

+

119875 sin2 (1205872 minus 1205791015840)2 cos (1205872 minus 1205791015840)

)]

= 119904 (1205791015840) exp[minus119895119870(

sin 1205791015840

4

+

119875cos21205791015840

2 sin 1205791015840)]

(11)

When 1205791015840 = 120596119905 is around 90∘ (PRArsquos front against radar)where 120596 is the angular frequency and 119905 is the slow timethe instantaneous Doppler can be obtained by taking thederivative of the phase in (11)

119891119889asymp

175

120582

cos (120596119905) asymp 875120587

120582

minus

175120596

120582

119905 (12)

Equation (12) indicates that the PRArsquos return in azimuthis a sinusoidal frequency-modulated (SFM) signal or approx-imately a linear frequency-modulated (LFM) signal when thePRA is illuminated broadside

32 Hybrid Model for a RPRA in the SAR Geometry It is wellknown that a PRA usually possesses rotation in operationThus we need to establish its motion model by findingthe range from SAR to the specular point 119878 For the SARgeometry shown in Figure 2 119874 is the center of the scenario


Slantplane view

119905 = 0

119884119888

119881119886

119883119888

119911998400

Spot

119909

119910

120579

119909

119910(119910998400)

120587 minus 120601119888119909998400

P119905

O O1O2

FO

S

H

Figure 2 SAR geometry with a RPRA

illuminated by SAR Assume that a PRA is rotating aroundits vertical axis and the pivot is the top of the vertical axisidentical to119874

1The angle between the119874-1199111015840 axis and the PRArsquos

axis of symmetry 120601119888 remains constant during the rotation

around 11987411198742 120579119888= minus arctan(119884

119888119883119888) is the slant angle When

119884119888= 0 SAR is side-looking and otherwise it operates in

squint-looking mode 120579 is the angle of light of sight (LOS) in119874-119909119910 (ie slant plane) The relationship between 120579 and 119905 is

119883119888tan 120579 = 119881

119886119905 minus 119884119888 (13)

When 120579119888= 0 and the extent of aspect angle 120579 is small

tan 120579 asymp 120579 and then (13) reduces to

120579 =

119881119886

119883119888

119905 (14)

Clearly 120579 also has the meaning of slow timeFor deriving the range model we at first make the

following definitionsIn 119874-119909119910 at slow time 119905

119875119905is at (minus119883

119888 119884119888minus 119881119886119905) (SARrsquos position)

1198741is at (119909 119910) (bottom center of the PRA)

In119874-119909101584011991010158401199111015840 at slow time 119905 the normalized vectors of 997888997888997888rarr1198741119865

and 997888997888rarrOH are

n1198741119865= (cos 120574 sin120601

119888 sin 120574 sin120601

119888 cos120601

119888)

nOH = (cos1205720sin1205730 sin120572

0sin1205730 cos120573

0)

where 120574 = 120596119905+1205740120596 = 2120587119891

119898is the rotating angular frequency

and 1205740is the initial phase both unknown and 120572

0as well as 120573

0

are known for parameter estimation in Section 5Then we can easily have

cos 120578120579= n1198741119865sdot n1198741119875119905

asymp n1198741119865sdot n119874119875119905

asymp n1198741119865sdot nOH

= sin120601119888sin1205730cos (120574 minus 120572

0) + cos120601

119888cos1205730

(15)

where the approximation is because 1198741119875119905is approximately

parallel to 119874119875119905and even to OH when the extent of 120579 is not

largeWe can also easily have

10038161003816100381610038161198751199051198741

1003816100381610038161003816= radic(119881

119886119905 minus 119884119888+ 119910)2

+ (119883119888+ 119909)2

(16)

From (3) or (10) we have

997888997888rarr1198741119878 sdot n1198751199051198741

=

119875 sin2120578120579

2 cos 120578120579

(17)

Then using the far-field plane wave assumption and (16)as well as (17) we can obtain the range from 119875

119905to 119878 at slow

time 119905

119877 (119905)

def= 119877120579=10038161003816100381610038161198751199051198741

1003816100381610038161003816+997888997888rarr1198741119878 sdot n1198751199051198741

= radic(119881119886119905 minus 119884119888+ 119910)2

+ (119883119888+ 119909)2

+

119875 sin2120578120579

2 cos 120578120579

asymp radic(119881119886119905 minus 119884119888)2

+ 1198832

119888+ 119909 cos 120579 + 119910 sin 120579 +

119875 sin2120578120579

2 cos 120578120579

(18)

where 120578120579can be calculated by (15)

After range compression and discarding insignificantfactors representing the beam pattern and window functionsSAR returns can be written in the frequency slow timedomain as

119866 (119870 120579) = 119904 (120579) exp [minus119895119870119877 (119905)] (19)

When processing spotlight SAR data with algorithmssuch as the polar format algorithm (PFA) one usuallycompensates the returns with the scenario centerrsquos phase thatis tomultiply (19)with exp(119895119870radic(119881

119886119905 minus 119884119888)2+ 1198832

119888) Hencewe

obtain the resulting returns


120579

2 cos 120578120579

)]

= 119904 (120579) exp [minus119895119870(119909 cos 120579 + 119910 sin 120579

+

119875

2

(

1

cos 120578120579

minus cos 120578120579))]

=

119868

sum

119894=0

119887 (119894) 120582119894(120579)

times exp [minus119895119870(119909 cos 120579 + 119910 sin 120579

+

119875

2

(

1

cos 120578120579

minus cos 120578120579))]

(20)

which is named the hybrid scatteringmotion model ormigratorymicromotion scattering center (MMSC) model


where 120582119894(120579) are orthogonal basis functions It is of the same

form as (10) derived based on a turntable but herein it isderived from the SAR geometry and its 120578

120579is determined

by (15) This model has incorporated motion parametersinto the scattering center model providing the rationale forjoint estimation of scattering-motion parameters It will beseen in next section that the model provides a very goodrepresentation of a PRA By the way the 2D inverse FT of(20) interpolation probably needed will produce an SARcomplex image

Assume that 120579 is minor and near 0∘ 120579119888= 0∘ and 120578

120579is

around 0∘ or 180∘ Then we have

1

cos 120578120579

asymp

2 minus cos 120578120579 when 120578

120579is around 0

∘

minus2 minus cos 120578120579 when 120578

120579is around 180

∘

(21)

Applying (14) (15) and (21) to (20) leads to

119892 (119870 120579)

asymp 119904 (120579) exp [minus119895119870 (119909 cos 120579 + 119910 sin 120579 minus 119875 cos 120578120579

+119875 sign (cos (120578120579)))]

asymp 119862 sdot 119904 (120579)

times exp [ minus 119895119870

times(

119881119886119910119905

119883119888

minus 119875 sin120601119888sin1205730cos (120596119905 + 120574

0minus 1205720))]

(22)

where 119862 is an insignificant unknown constant factor TheinstantaneousDoppler ormicro-Doppler can be obtained bytaking the derivative of the phase

119891119889asymp minus

2120596119875 sin120601119888sin1205730

120582

sin (120596119905 + 1205740minus 1205720) minus

2119881119886119910

120582119883119888

(23)

Both (22) and (23) indicate that the RPRArsquos azimuthreturn is an SFM signal but in effect it can be approximatedby an LFM signal due to broadside flash

4 Radar Characteristics of a PRA andValidation of the MMSC Model

The analysis proceeds with radar characteristics of a PRAboth stationary and rotating in terms of RCS HRRPs TFDand SAR images The results from GO PO and GRECO areall based on the following geometry and parameters

As shown in Figure 3 the origin is now set at its geometri-cal center 119874

2according to our GRECO softwarersquos definition

11987411198742= 11987421198743= 025m The PRArsquos depth 119874

11198743= 05m

119875 = 9m and 119863 = 6m The frequencies are from 98GHzto 102GHz stepped by 001 GHz with central frequency 119891

119888=

10GHz (therefore the bandwidth is 400MHz and the rangeresolution is 0375m) and the aspect angles are from minus180

∘

to 180∘ stepped by 01∘ Then we use GRECO to calculate thecomplex RCS of a PRA on a turntable (equivalent to radarrsquoscircling a stationary PRA at an angular frequency 120596 = 120579

1015840119905)

We now analyze the calculated results

180∘(minus180∘)

119863

119909

270∘(minus90∘)119910

90∘

120579998400

120579998400 = 0∘

O3O1 O2

A

B

Figure 3 PRA geometry for GO PO and GRECO

41 Stationary PRA

411 RCS Characteristics As shown by Figures 4(a) and 4(b)the PRArsquos RCS exhibits strong dependency on aspect angle1205791015840 Its scattering is intense (about 250m2 or 24 dB sdot m2 farlarger than the PRArsquos aperture area 120587(1198632)2 asymp 28m2) whenthe LOS deviates from its axis of symmetry by no morethan about 18∘ Otherwise its RCS abruptly declines to zeroThis phenomenon is generally called broadside flash [16]From (4) |120578| le arctan[6(2 times 9)] asymp 184

∘ which agreeswith GRECO results (solid line) and means that GO hascorrectly predicted the aspect supporting interval Figure 4(c)shows that GO can also give a coarse prediction for theRCS-1205791015840 function by frequency-free (1) For more accuratedescription 119904(120579) is therefore used in (10)

From Figure 4(b) we can also find that the RCS-1205791015840 curvesare almost identical for different frequencies except when1205791015840= plusmn 90

∘ indicating weak frequency dependency of RCSThis is confirmed by Figure 5(a) which shows the RCS-frequency functions are valued between 22 and 26 in dBm2 asmall range regardless of aspects (except 1205791015840 = plusmn 90

∘) There-fore ourMMSCmodel thoughnot consideringRCSrsquos depen-dency on frequency that is 119904(120579) does not contain 119891 in (20)is judged reasonable from this viewpoint However whenthe LOS is perpendicular to the PRArsquos bottom RCS takes onobvious oscillation for PO and GRECO (Figure 5(b)) Thereason may be stated in terms of multiple scatterers thatis there may be additional scattering centers besides thespecular point predicted by GO at this aspect

412 HRRP Characteristics Figure 6 depicts the HRRP seq-uence of the PRA derived from GRECO data Since HRRPsat some aspects are too weak to display in Figure 6(a) wemodify their amplitudes for every aspect into a normalizedvalue as shown in Figure 6(b) We will analyze its HRRPcharacteristics from the following points

(1) Overall strong HRRPs occur periodically when 1205791015840 isin[minus108

∘ minus72∘] cup [72

∘ 108∘] which is consistent with

our analysis of RCS The HPPRsrsquo intensity is almostidentical for front and back illumination


9060

30

0

330

300270

240

210

180

150

120

800

400 m 2

(a)

0 50 100 150 2000

500

1000

1500

0 100 200

0

50

minus200 minus150 minus100 minus50

minus100

minus50

minus100minus200

RCS

(m2)

Aspect angle (∘)

RCS

(dB

m2)

(b)

0 50 100 150 2000

100

200

300

400

500

600

700

800

900

1000

70 80 90 100 1100

200

400

600

minus200

Aspect angle (∘)minus100 minus50minus150

RCS

(m2)

GRECOGO

(c)

Figure 4 PRA RCSrsquos aspect dependency (a) GRECO at carrierfrequency 10GHz (b) GRECO at different frequencies (c) GRECOand GO at carrier frequency 10GHz

98 99 10 101 102

10

15

20

25

30

RCS

(dB

m2)

Frequency (GHz)

120579998400 = 90∘

(a)

98 99 10 101 102

200

400

600

800

1000

1200

GRECOPOGO

Frequency (GHz)

RCS

(m2)

(b)

Figure 5 RCSrsquos frequency dependency (a) GRECO at differentaspects (b) GRECO PO and GO 1205791015840 = 90

∘

(2) When illuminated broadside (1205791015840 = plusmn 90∘) the PRA

has two scattering centers at plusmn 025m with thesame amplitude (two peaks in Figure 6(a) only forthis aspect the bandwidth is 2GHz as opposed to400MHz to resolve them)This can be interpreted by(9) given by PO the second square-bracketed termof which clearly shows that the positions of the twoscattering centers are plusmn1198632(16119875) = plusmn 025m Andthe conclusion also confirms our conjecture on thereason of RCS oscillation in Figure 5(b) In effect thetwo centers correspond to the PRArsquos bottom centerand edge

(3) When 1205791015840 isin [minus108∘ minus72∘] cup [72∘ 108∘] and 1205791015840 = plusmn 90∘

the PRA assumes a single scattering center Whenthe PRArsquos front side is illuminated the center moves


0 50 100 150

0

2

4

6

75 80 85 90 95 100 108

0

05069

0 1 15 20005

01015

02025

03

Backilluminated

Frontilluminated

Farther from radar

Nearer

minus05

minus2

minus4

minus6

Rang

e (m

)

minus100 minus50minus150

Aspect angle (∘)

minus2 minus15 minus1 minus025 025Range (m)

(a)

0 50 100 150

0

2

4

6

minus2

minus4

minus6

Rang

e (m

)


Aspect angle (∘)

(b)

Figure 6 HRRP sequences of the PRA with (a) original and (b) normalized amplitude GRECO data are used

Nor

mal

ized

freq

uenc

y

70 75 80 85 90 95 100 105 110

0

01

02

03

04

minus01

minus02

minus03

minus04

minus05

Aspect angle (∘)

Spec

ular

pio

nt

(a)

Nor

mal

ized

freq

uenc

y

70 75 80 85 90 95 100 105 110

0

01

02

03

04

05

Aspect angle (∘)

minus01

minus02

minus03

minus04

minus05

(b)

Figure 7 TFD of the PRA with data from (a) GRECO and (b) our PRA scattering center model

further with deviation of the LOS from the PRArsquossymmetry axis The case is just the opposite when theback side is illuminated This conclusion also agreeswith our modelrsquos prediction (see (10) and (11)) Morespecifically in Figure 3 when 1205791015840 = 108

∘ the scatteringcenter should move to sin 12057910158404 + 119875cos21205791015840(2 sin 1205791015840) asymp069m which is in good agreement with the reality inthe top-right corner of Figure 6(a) Beyond this rangethe center will move outside the PRA and thereforedisappear

(4) When 1205791015840 notin [minus108∘ minus72∘] cup [72∘ 108∘] the HRRPs aretoo weak andwe have to resort to Figure 6(b) Clearlyfor these aspects the scattering is rather complicateddue to disappearing of the specular point and morescattering centers are present beyond the predictionof GO and our models Fortunately these HRRPs areso weak that they can be safely ignored

413 TFD Characteristics We also plot the TFD accordingto the PRArsquos complex RCS at the central frequency only foraspects at which the PRArsquos RCS is strong The PRA is placedon a ldquorotatablerdquo turntable and thus the aspect angle has thesense of slow time Figure 7(a) shows the TFD based onGRECO-derived data it is clear that the PRA contains threemain scattering centers One is the specular point relatedto the slant line in Figure 7(a) The slant line agrees wellwith (12) and its graphical presentation in Figure 7(b) Twois at the PRArsquos edge that is points 119860 and 119861 in Figure 3related to the top and bottom curves in Figure 7(a) Eachcurve is effectively a small fraction of a sine wave We nowtry to explain the vertical line in the middle of Figure 7(a)When 120579

1015840= 90

∘ different parts of the PRA have wide-range relative velocities from 0 corresponding 119874

1to the

maximum and minimum corresponding to point 119860 and 119861

in Figure 3 Therefore the instantaneous Doppler at 1205791015840 =


0 1 2 3 4

0

2

4

6

minus1minus2minus3minus4

Azimuth (m)

Rang

e (m

)

minus2

minus4

minus6

(a)

0 2 4

0

2

4

6

minus2minus4

Azimuth (m)

Rang

e (m

)

minus2

minus4

minus6

minus6 6

(b)

Figure 8 2D images of the PRA in a turntable by CBP using GRECO data (a) 1205791015840 isin [88∘ 92∘] (b) 1205791015840 isin [0∘ 180∘]

Table 1 SAR and RPRArsquos parameters

Bandwidth 400MHz Reference range119883119888

10000mCentral frequency 119891

11988810GHz PRArsquos focus parameter 119875 9m

Slant angle 120579119888

0 Diameter of the PRArsquos aperture119863 6mCPI 87 s PRA rotating frequency 119891

11989801sim03Hz

Airplane velocity 119881119886

200ms 120601119888

1205874

Angular extent 10∘ 1205740

120587

90∘ contains all the components throughout between the

maximal and minimal Doppler leading to a vertical linein Figure 7(a) In addition it is clear that though there areseveral components in Figure 7(a) the slant line has thestrongest intensity Thereby it is reasonable for our scatter-ing center models to contain only the specular scatteringcomponent

414 2119863 Image Characteristics The 2D images of the PRAare formulated by the convolution back-projection (CBP)algorithm displayed in Figure 8 for narrow and wide anglesrespectively From Figure 8(a) three scattering centers arepresent with themiddle bright one at the PRArsquos bottomcenterand the other darker two at its edgeThis is in accordancewithFigure 7(a) However in the wide-angle case (Figure 8(b))the PRA exhibits a bowl-shaped signature and the bowlrsquossize is equal to the true size of the PRA The two pointscorresponding to the PRArsquos edge are still visible at the bowlrsquosedge The bowl shape is due to the migration of the specularpoint within such a wide angle From Figure 6 it is wellknown at different aspects that the specular point is locatedat different positions and therefore they cannot be coherentlyintegrated into a single strong point in the resulting imageInstead the energy is spread over the bowl shape Moreoverthere is an other weak energy distributed at the left and rightedges of Figure 8(b) This energy results from other types ofscattering from the PRA for example creeping wave Againthe PRA when its back side faces radar will assume similar

signatures to Figure 8 but the bowlrsquos direction is the oppositeTo summarize 2D images provide a very good representationof PRArsquos structure and the specular scattering contributes toPRArsquos dominant energy

42 Rotating PRA When a PRA is rotating in the SARgeometry its radar characteristics will considerably vary dueto the rotation In what follows we proceed with the analysisof an RPRA based on the MMSC model since the real SARdata of an RPRA are unavailable for the moment Simulationparameters are shown in Table 1

421 RCS HRRP and TFD Characteristics Suppose 119891119898

=

03Hz which is also a typical value for real RPRAs InFigure 9(a) as is expected the RPRArsquos RCS sequence takes onperiodicity due to the rotation and the period is about 33 s ingood agreement with the true value 1119891

119898 Its TFD also shows

the periodicity (Figure 9(b)) In any segment the slantline-shaped TFD represents an LFM signal or a fraction of a SFMsignal as expected by (23) In effect a sine wave is impossibleto be displayed in the TF image owing to broadside flashNote also that the slant lines may be blurry due to shorttime and the TF-resolution limit This kind of periodicityis also characteristic of the HRRP sequence in Figure 9(c)However a curve similar to the left one in Figure 6(a) isnot present This is because that the back of the PRA hasno chance to be illuminated by SAR for the parameters inTable 1


0 1 2 3 40

50

100

150

200

250

RCS

(m2)


Slow time (s)

(a)

Nor

mal

ized

freq

uenc

y

0 2 4

0

05

minus05minus2minus4

Slow time (s)

(b)

0 1 2 3 4

0

2

4

6

minus2

minus4


Rang

e (m

)

minus4

Slow time (s)

(c)

Figure 9 RPRArsquos characteristics (a) RCS sequence (b) TFD (c) HRRP sequence

422 SAR Image Characteristics The presence of rotationchanges the Doppler history of the PRArsquos returns thusbringing out new signatures distinct from of the stationaryPRA When the PRA rotates at a small frequency it takeson blurred lines in the SAR image (Figure 10(a)) When itrotates more rapidly the image takes on a bowknot shapewith several intersectant lines (Figure 10(b)) The faster thePRA rotates the more lines there are When 119891

119898= 2Hz we

cannot distinguish these lines from each other As a resultthe image exhibits another kind of bowknots as shown inFigure 10(c) We also find that the angular extent of thebowknots is approximately equal to that of SAR that is10∘ in Figures 10(b) and 10(c) after the units of the two

axes are equalized These interesting signatures can providenew features for target recognition notwithstanding that theunderlying reason is enigmatic and remains to be explored inthe future

423 Effective Illumination Condition In Figure 9(c) wehave found that the back of the PRA is not illuminated

by SAR In effect even if it is illuminated there is notbackscattering strong enough if the incident angle is too largeIn such a case the illumination is not effective Then what isthe condition of effective illumination Clearly |120578

120579| le Δ or

120587minusΔ le |120578120579| le 120587+Δ needs and only needs to be satisfiedwhere

Δ is half the aspect supporting interval (eg 18∘) Substitutingit into (15) and after some simple manipulations we find thatthe sufficient and necessary conditions for effective front orback illumination are respectively

10038161003816100381610038161205730minus 120601119888

1003816100381610038161003816le Δ (24)

1205730+ 120601119888ge 120587 minus Δ (25)

If (24) and (25) are combined we readily obtain anecessary condition of both sides being illuminated

Δ ge

120587

3

(26)

However in general Δ lt 1205874 for PRAs Thereforeit is impossible for both sides of a PRA to be effectivelyilluminated


0 10 20 30 40-6

-4

-2

0

2

4

6


Azimuth (m)

Rang

e (m

)

(a)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4

minus6minus40 minus30 minus20 minus10

Azimuth (m)

(b)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)

(c)Figure 10 SAR images of an RPRA formulated by CBP when (a) 119891

119898= 02Hz (b) 119891

119898= 03Hz and (c) 119891

119898= 2Hz

The analyses made earlier have fully supported ourscattering centermodels and provided deep insight into PRArsquoscharacteristicsThese characteristics can be used as feature forSAR image interpretation and target detection or recognitionThey can also be used for parameterrsquos coarse estimation asshown in Section 5

5 MLE of Parameters in the MMSC ModelOnce theMMSCmodel is built the estimation for parametersin this model can be performed from measured data andthere are various ways available for example MLE [14] theregularization method [17] and the sparse representationmethod [15] though they were not intended for our problemherein Actually from a unified viewpoint all of them belongto optimization algorithms for solving inverse problems Inthis paper we develop anMLEmethod for theMMSCmodelwhich is equivalent to the nonlinear least squares estimation[14] Note that only one PRA target is assumed for simplicitybut it can be generalized for multiple targets by introducingRELAX or CLEAN techniques thereby circumventing theestimation of the model order

51 Maximal Likelihood Estimator In (20) 119904(120579) has beenparameterized in basis functions For notational expediency(20) can be further vectorized as

g = Φ (120599) b + v (27)

where g is the measured echo vector v is the noise vector120599

def= (119909 119910 119891

119898 1205740 120601119888 119875) is unknown b def

= [119887(0) 119887(119868)]119879 is

unknown and

119890119870120579

def= exp[minus119895119870(119909 cos 120579 + 119910 sin 120579 +

119875 sin2120578120579

2 cos 120578120579

)]

Φdef=

[

[

[

[

[

[

[

[

[

[

11989011987011205791

0

d

0 1198901198701120579119872

1198901198701198731205791

0

d

0 119890119870119873120579119872

]

]

]

]

]

]

]

]

]

]

[

[

[

1205820(1205791) 120582

119874(1205791)

1205820(120579119872) 120582

119874(120579119872)

]

]

]

(28)


0 20 40

0

2

4

6

Rang

e (m

)

minus2

minus4

minus6minus40 minus20

Azimuth (m)

Figure 11 PRArsquos image under SNR = minus3 dB by CBP

Table 2 Final estimated results

119909 (m) 119910 (m) 119891119898(Hz)

True value 0 0 03Coarse estimate 012 minus010 032Estimate 00013 00108 02995

1205740(rad) 120601

119888(rad) 119875 (m)

True value 1205872 1205874 9Coarse estimate 0 1205874 71205874 068 10Estimate 15715 07841 90179

where the derivation ofΦ is not difficult but tedious thereforenot given here

We try to solve for 120599 and b from (27) Clearly this is aninverse problem Using MLE we at first obtain 120599rsquos estimate

120599 = argmin

120599

100381710038171003817100381710038171003817

g minusΦ(Φ119867Φ)minus1

Φ119867g

100381710038171003817100381710038171003817

2

2

(29)

Afterwards by substituting 120599 into Φ we have the least-

squares estimate of b

b = (Φ119867Φ)minus1

Φ119867g (30)

Finally by use of b the reflectivity function can be foundas

(120579) =

119868

sum

119894=0

119887 (119894) 120582

119894(120579) (31)

52 Determination of the Initial Value of 120599 Equation (29) isan optimization function and it is easy to cause localminimaOne may either try to find efficient global optimization algo-rithms which have not been found to date or to determineinitial values with higher precision The latter line of thoughtentails a thorough understanding of the target characteristicswhich fortunately has been analyzed in detail in Section 4Therefore we give the coarse estimation method for 120599

0 2 40

5

10

15

20

25

Refle

ctiv

ity

Our methodTureConventional method

minus2minus4

Slow time (s)

Figure 12 Recovered reflectivity with respect to slow time

(119909 119910) From Figure 10 the center of an RPRArsquos blurredimage indicates the position of the scattering center So (119909 119910)can be estimated respectively by

119909 =

∬1199091015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

119910 =

∬1199101015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

(32)

where 119868(1199091015840 1199101015840) is the gray-level image119891119898 It can be estimated by finding the rotating period

from an HRRP sequence (see Figure 10) and then taking itsreciprocal

120601119888and 119875 We have not found any effective method

for estimating them separately However from (23) afterobtaining 119891

119898and 120596 we can find 119875 sin120601

119888as a whole by

estimating the Doppler slope 119875 can be set as a mean valuefrom a priori knowledge at the order of meters 120601

119888can then

be determined accordingly1205740 It is hard to be estimated We just take a set of values

of it as the coarse estimates so the objective function will beoptimized with different initial values resulting in multiplelocal minima the minimum of which is finally taken asthe global minimum We find through simulation that theoptimization is not sensitive to the choice of 120574

0rsquos initial values

6 Simulation Results

In this section we present some estimation results by theproposed method in Section 5 119891

119898= 03Hz 120572

0= 120587

1205730= 1205874 the signal-to-noise ratio (SNR) is minus3 dB and other

parameters are shown inTable 1TheDirac function is chosenas the basis function The optimization algorithm we use is


0 02 04 06 08 1 12 14 16 18 20

200

400

600

800

1000

1200O

bjec

tive f

unct

ion

119891119898 (Hz)

(a)

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

Obj

ectiv

e fun

ctio

n

119875 (m)

(b)

Figure 13 Objective function slices along (a) the rotating frequency and (b) the focus parameter

sequential quadratic programming (SQP) which has provensuccessful in solving our problem

The RPRArsquos SAR image is shown in Figure 11 which issimilar to Figure 10(b) except for being contaminated bynoise The coarse estimates for initial values are shown inTable 2 Using the estimator proposed we finally obtainthe estimates of the PRArsquos parameters which obviouslyagree well with their true values To our best knowledgethere is not any published work on extracting the overallinformation of an RPRA in SAR and so overall comparisoncannot be performed Once the six parameters are obtainedthe coefficients of the basis functions can be found andtherefore the reflectivity function (120579) results (Figure 12)Note that 120579 is proportional to slow time by (14) Figure 12indicates that the reflectivity is successfully recovered and theperiodicity implies the broadside flash signaturewhich can befavorably used in target recognition Also the square of thereflectivity is accordant with the RCS sequence in Figure 9(a)Although the conventional method also gives estimates thereflectivity by evaluating the mean of the returnsrsquo amplitudesequence it has poorer agreement with the reality andmore problematically this method needs finding range cellscontaining micromotion targets at first which is in itself atough job but is now circumvented by our approach throughbuilding the MMSC model in the phase domain

Again we can see through the slices of the objectivefunction shown by Figure 13 that improper initial values of119891119898

will lead to local minima but the coarsely estimatingprecision for it is able to guarantee the convexity of theobjective function within a necessary interval it is similarto other parameters except 119875 which tolerates larger error ofinitial estimates (Figure 13(b))

7 Conclusion

As a special type of targets in SAR RPRAs have a few interest-ing characteristics which can be exploited for understandingSAR images In this paper we have at first obtained the

closed-form and numeric solutions for PRArsquos backscatteringvia GO PO and GRECO respectively From its GO solutiona new scattering center model has been proposed for char-acterizing the migration of the specular point on the PRAand then the rotating parameters have been incorporatedinto the model to obtain a hybrid model or MMSC modelin SAR geometry Using the models and the scatteringdata calculated we have in detailed investigated the radarcharacteristics of a PRA and RPRA alike Their RCS HRRPsequences and TFD all show that a PRA exhibits broadsideflash as well as periodicity and contains a single migratoryscattering center for limited aspects whose energy is domi-nant The bowl-shaped image of a stationary PRA reflects itsstructure perfectly while the SAR image of an RPRA assumesdistinct signatures All these characteristics have confirmedthe MMSC model and they also provide novel informationon targetsrsquo structures and motions Then using this infor-mation for coarse estimation of PRArsquos parameters we havedeveloped amaximal likelihood estimator to solve theMMSCmodel Simulation results at minus3 dB of SNR demonstratethe estimatorrsquos performance The parameters estimated forexample the rotating frequency and focus parameter canfaithfully record the information on PRArsquos motion and sizeand therefore can be used advantageously as features fortarget recognition However we have concentrated only on aPRA in a homogeneous scenario which can be characterizedby white Gaussian noise and lacked consideration of cluttersstronger and more complex Additionally selection of basisfunctions has not been discussed either Notwithstandingits limitation this study provides a basis for interpretationof SAR images containing PRAs and the idea of hybridmodeling is also instrumental for other fields such asSARGMTI

Acknowledgment

The authors acknowledge the support from Provincial Sci-ence Foundation for Distinguished Young Scholars in Hunan


(no 11JJ1010) and National Science Fund of China (no61171133 no 61101182 and no 61025006)

References

[1] W J Miceli and L A Gross ldquoTest results from the ANAPY-6 SARGMTI surveillance tracking and targeting radarrdquo inProceedings of the IEEERadar Conference pp 13ndash17 Atlanta GaUSA May 2001

[2] X Li B Deng Y Qin H Wang and Y Li ldquoThe influence oftarget micromotion on SAR and GMTIrdquo IEEE Transactions onGeoscience and Remote Sensing vol 49 no 7 pp 2738ndash27512011

[3] B Deng G Wu Y Qin H Wang and X Li ldquoSARMMTI anextension to conventional SARGMTI and a combination ofSAR and micromotion techniquesrdquo in Proceedings of the IETInternational Radar Conference Guilin China April 2009

[4] S D Silverstein and C E Hawkins ldquoSynthetic aperture radarimage signatures of rotating objectsrdquo in Proceedings of theConference Record of the Thirty-Eighth Asilomar Conference onSignals Systems andComputers pp 1663ndash1667November 2004

[5] T Thayaparan S Abrol and S Qian ldquoMicro-doppler analysisof rotating target in SARrdquo Tech Rep TM-2005 DRDCOttawaCanada 2005

[6] T Thayaparan K Suresh S Qian K Venkataramaniah SSivaSankaraSai and K S Sridharan ldquoMicro-Doppler analysisof a rotating target in synthetic aperture radarrdquo IET SignalProcessing vol 4 no 3 pp 245ndash255 2010

[7] M Ruegg E Meier and D Nuesch ldquoVibration and rotationin millimeter-wave SARrdquo IEEE Transactions on Geoscience andRemote Sensing vol 45 no 2 pp 293ndash304 2007

[8] D C Jenn Radar and Laser Cross Section Engineering Ameri-can Institute of Aeronautics and Astronautics 2005

[9] 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

[10] M Hurst and R Mittra ldquoScattering center analysis via pronyrsquosmethodrdquo IEEE Transactions on Antennas and Propagation vol35 no 8 pp 986ndash988 1987

[11] L C Potter D M Chiang R Carriere and M J Gerry ldquoGTD-based parametricmodel for radar scatteringrdquo IEEETransactionson Antennas and Propagation vol 43 no 10 pp 1058ndash10671995

[12] L C Potter and R L Moses ldquoAttributed scattering centers forSAR ATRrdquo IEEE Transactions on Image Processing vol 6 no 1pp 79ndash91 1997

[13] M J Gerry L C Potter I J Gupta and A D Van Merwe ldquoAparametric model for synthetic aperture radar measurementsrdquoIEEE Transactions on Antennas and Propagation vol 47 no 7pp 1179ndash1188 1999

[14] R Jonsson A Genell D Losaus and F Dicander ldquoScatteringcenter parameter estimation using a polynomial model for theamplitude aspect dependencerdquo in Proceedings of the Algorithmsfor Synthetic Aperture Radar Imagery IX pp 46ndash57 OrlandoFla USA April 2002

[15] K R Varshney M Cetin J W Fisher and A SWillsky ldquoSparserepresentation in structured dictionaries with application tosynthetic aperture radarrdquo IEEE Transactions on Signal Process-ing vol 56 no 8 pp 3548ndash3561 2008

[16] R D Chaney A SWillsky and LM Novak ldquoCoherent aspect-dependent SAR image formationrdquo in Proceedings of the SPIE

Algorithms for Synthetic Aperture Radar Imagery vol 2230 pp256ndash274 1994

[17] B D Rigling ldquoUse of nonquadratic regularization in fourierimagingrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 45 no 1 pp 250ndash265 2009

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Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



The RPRA detected was annotated in the SAR image [1]However the RPRArsquos signature blurred in SAR imagerywas not explained and the aforementioned approaches aresusceptible to range cell migration (RCM) TFD also usuallysuffers from cross-term interference In 2007 RPRA datacollected by amicrowave SAR system inUniversity of Zurichwere analyzed [7] A point-symmetrical image of the RPRAwas obtained and the time-frequency image showed a smallfraction of a sinusoidal phase modulation The conclusionshowever are only valid for slow rotating frequency and shortcoherent processing interval (CPI) PRArsquos radar characteris-tics have not been deeply studied either

In this paper we focus on characterization and non-TFD-based estimation of PRAs both stationary and rotating inspotlight SAR data There are different types of PRAs butwe restrict our scope to a rotation-symmetrical PRA witha single reflector Instead of processing returns in a singlerange cell we use the phase history in the wavenumberdomain that is the 2D Fourier transform (FT) of compleximagery to extract PRAs and their parameters We buildfor a PRA a new non-point-scattering center model able todescribe the aspect sensitivity and the sliding of scatterersand then incorporate the motion model into it to obtaina hybrid scatteringmotion model Radar characteristics ofPRAs are then analyzed Afterwards an estimation algorithmfor this model is proposed This idea fully exploits a prioriinformation on target models implements joint estimationof scattering and motion parameters and circumvents TFDand other tough issues such as clutter suppression detectionRCM correction and parameter estimation which have to beconducted separately and explicitly in SARGMTI

This paper is organized as follows In Section 2 geo-metrical optics (GO) physical optics (PO) and graphicalelectromagnetic computation (GRECO) are used to calculatethe backscattering or complex radar cross-section (RCS) ofa PRA Then based on the calculated results we propose anew scattering center model and a hybrid scatteringmotionmodel in Section 3 We then in Section 4 analyze the char-acteristics of PRAs A few interesting signatures of PRAsare revealed for example periodicity of high-resolutionrange profile (HRRP) sequences bowl- and bowknot-shapedimages Sections 5 and 6 provide a maximum likelihoodestimator and a few results and conclusions are reportedin Section 7 The derivation throughout the paper may be alittle difficult to follow and we will try our best to clarify theproblem

2 Complex RCS Calculation for PRA

Backscattering of an antenna consists of two parts that isstructured scattering whose scattering principle is the sameof as common objects and pattern scattering which is closelyrelated to antennarsquos loads Pattern scattering is weak fora PRA and hence not taken into account here We willcalculate PRArsquos complex RCS via GO PO and GRECO GOis simple and is particularly suitable for smooth surface POhas high accuracy near targetrsquos normal direction and canprovide closed-form solutions in some particular aspectsAnd GRECO as a tool and software incorporates many

Specularpoint

FrontBack

Radar

Migration of thespecular point

119863

minus119875

21205750

1198632

8119875

1205910

120578

119911

119909

119910

120578 = 0∘

120578 = arctan 1198632119875

S

O1O(F)

Figure 1 PRA geometry (119909-z slice)

electromagnetic methods and can efficiently provide high-precision numerical solutions

21 GO As Figure 1 shows let 119875 denote the focus parameterof a PRA 120591 and 120575 denote the radius and angle (relative to -zaxis) of any point on a PRA Other parameters encounteredare defined in Figure 1 where the focus is located at the origin119874-z is the symmetrical axis Then the amplitude of the PRArsquoscomplex RCS that is the square root of RCS can be obtainedfrom the GO theory [8]

1003816100381610038161003816radic120590GO

1003816100381610038161003816=

radic120587119875

cos2120578 (1)

Its phase can be easily derived from the position of thespecular point 119878 Combining the amplitude and phase wehave the complex RCS

radic120590GO =

radic120587119875

cos2120578exp(minus119895119870 119875

2 cos 120578) (2)

where 119870 = 4120587119891119888 is the wavenumber 119888 is the speed of lightand 119891 is the frequency When choosing PRArsquos bottom center1198741as the phase reference we need to multiply (2) with

exp(119895119870 sdot 1198752 sdot cos 120578) and then have

radic120590GO =

radic120587119875

cos2120578exp(minus119895119870

119875 sin21205782 cos 120578

) (3)

Equations (2) and (3) hold only when 120578 is chosen suchthat the specular point does not move outside the PRASpecifically when the incident field illuminates the PRA frontside 120578 should be subjected to

10038161003816100381610038161205781003816100381610038161003816le arctan 119863

2119875

(4)

22 PO To facilitate application of PO we use the Cartesiancoordinate system and the cylinder coordinate system Thenin Figure 1 for any point 119876 in the PRA suppose that its





r1015840 = (119909 119910 119911)119879

= (120588 cos120593 120588 sin1205931205882

2119875

minus

119875

2

)

119879

(5)


n = (minus

120588 cos120593119875

minus

120588 sin120593119875

1)

119879

radic1 +

1205882

1198752

(6)


r = (sin 120578 0 cos 120578)119879 (7)



119896

radic120587

int

2120587

0

int

1198632

0


1198752120588119889120588119889120593

(8)


radic120590PO1003816100381610038161003816120578=0

= minus1198952radic120587119875 sin(119870 1198632

16119875

) exp[minus119895119870(

119875

2

minus

1198632

16119875

)]

= radic120587119875[exp(minus119895119870 1198632

16119875

) minus exp(119895119870 1198632

16119875

)]


119875

2

minus

1198632

16119875

)]

(9)








120579

2 cos 120578120579

)]

(10)




119892 (119870 1205791015840)

= 119904 (1205791015840) exp[minus119895119870(

sin 1205791015840

4

+


)]

= 119904 (1205791015840) exp[minus119895119870(

sin 1205791015840

4

+

119875cos21205791015840

2 sin 1205791015840)]

(11)


119891119889asymp

175

120582

cos (120596119905) asymp 875120587

120582

minus

175120596

120582

119905 (12)




Slantplane view

119905 = 0

119884119888

119881119886

119883119888

119911998400

Spot

119909

119910

120579

119909

119910(119910998400)

120587 minus 120601119888119909998400

P119905

O O1O2

FO

S

H









119883119888tan 120579 = 119881

119886119905 minus 119884119888 (13)



120579 =

119881119886

119883119888

119905 (14)



119875119905is at (minus119883





n1198741119865= (cos 120574 sin120601

119888 sin 120574 sin120601

119888 cos120601

119888)

nOH = (cos1205720sin1205730 sin120572

0sin1205730 cos120573

0)

where 120574 = 120596119905+1205740120596 = 2120587119891



0as well as 120573

0


cos 120578120579= n1198741119865sdot n1198741119875119905

asymp n1198741119865sdot n119874119875119905


= sin120601119888sin1205730cos (120574 minus 120572

0) + cos120601

119888cos1205730

(15)




10038161003816100381610038161198751199051198741

1003816100381610038161003816= radic(119881

119886119905 minus 119884119888+ 119910)2

+ (119883119888+ 119909)2

(16)


997888997888rarr1198741119878 sdot n1198751199051198741

=

119875 sin2120578120579

2 cos 120578120579

(17)


119905to 119878 at slow

time 119905

119877 (119905)

def= 119877120579=10038161003816100381610038161198751199051198741

1003816100381610038161003816+997888997888rarr1198741119878 sdot n1198751199051198741

= radic(119881119886119905 minus 119884119888+ 119910)2

+ (119883119888+ 119909)2

+

119875 sin2120578120579

2 cos 120578120579


+ 1198832

119888+ 119909 cos 120579 + 119910 sin 120579 +

119875 sin2120578120579

2 cos 120578120579

(18)



119866 (119870 120579) = 119904 (120579) exp [minus119895119870119877 (119905)] (19)


119886119905 minus 119884119888)2+ 1198832

119888) Hencewe



120579

2 cos 120578120579

)]

= 119904 (120579) exp [minus119895119870(119909 cos 120579 + 119910 sin 120579

+

119875

2

(

1

cos 120578120579

minus cos 120578120579))]

=

119868

sum

119894=0

119887 (119894) 120582119894(120579)


+

119875

2

(

1

cos 120578120579

minus cos 120578120579))]

(20)





120579is determined



120579is


1

cos 120578120579

asymp

2 minus cos 120578120579 when 120578

120579is around 0

∘


120579is around 180

∘

(21)


119892 (119870 120579)


+119875 sign (cos (120578120579)))]

asymp 119862 sdot 119904 (120579)


times(

119881119886119910119905

119883119888

minus 119875 sin120601119888sin1205730cos (120596119905 + 120574

0minus 1205720))]

(22)



2120596119875 sin120601119888sin1205730

120582

sin (120596119905 + 1205740minus 1205720) minus

2119881119886119910

120582119883119888

(23)







11198743= 05m


119888=


∘


1015840119905)


180∘(minus180∘)

119863

119909

270∘(minus90∘)119910

90∘

120579998400

120579998400 = 0∘

O3O1 O2

A

B


41 Stationary PRA












9060

30

0

330

300270

240

210

180

150

120

800

400 m 2

(a)

0 50 100 150 2000

500

1000

1500

0 100 200

0

50


minus100

minus50

minus100minus200

RCS

(m2)

Aspect angle (∘)

RCS

(dB

m2)

(b)

0 50 100 150 2000

100

200

300

400

500

600

700

800

900

1000

70 80 90 100 1100

200

400

600

minus200


RCS

(m2)

GRECOGO

(c)


98 99 10 101 102

10

15

20

25

30

RCS

(dB

m2)

Frequency (GHz)

120579998400 = 90∘

(a)

98 99 10 101 102

200

400

600

800

1000

1200

GRECOPOGO

Frequency (GHz)

RCS

(m2)

(b)


∘






0 50 100 150

0

2

4

6

75 80 85 90 95 100 108

0

05069

0 1 15 20005

01015

02025

03

Backilluminated

Frontilluminated

Farther from radar

Nearer

minus05

minus2

minus4

minus6

Rang

e (m

)


Aspect angle (∘)


(a)

0 50 100 150

0

2

4

6

minus2

minus4

minus6

Rang

e (m

)


Aspect angle (∘)

(b)


Nor

mal

ized

freq

uenc

y

70 75 80 85 90 95 100 105 110

0

01

02

03

04

minus01

minus02

minus03

minus04

minus05

Aspect angle (∘)

Spec

ular

pio

nt

(a)

Nor

mal

ized

freq

uenc

y

70 75 80 85 90 95 100 105 110

0

01

02

03

04

05

Aspect angle (∘)

minus01

minus02

minus03

minus04

minus05

(b)






1015840= 90


1to the




0 1 2 3 4

0

2

4

6


Azimuth (m)

Rang

e (m

)

minus2

minus4

minus6

(a)

0 2 4

0

2

4

6

minus2minus4

Azimuth (m)

Rang

e (m

)

minus2

minus4

minus6

minus6 6

(b)








11989801sim03Hz


200ms 120601119888

1205874


120587







=





0 1 2 3 40

50

100

150

200

250

RCS

(m2)


Slow time (s)

(a)

Nor

mal

ized

freq

uenc

y

0 2 4

0

05

minus05minus2minus4

Slow time (s)

(b)

0 1 2 3 4

0

2

4

6

minus2

minus4


Rang

e (m

)

minus4

Slow time (s)

(c)



119898= 2Hz we





120579| le Δ or



10038161003816100381610038161205730minus 120601119888

1003816100381610038161003816le Δ (24)

1205730+ 120601119888ge 120587 minus Δ (25)


Δ ge

120587

3

(26)



0 10 20 30 40-6

-4

-2

0

2

4

6


Azimuth (m)

Rang

e (m

)

(a)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)

(b)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)


119898= 02Hz (b) 119891

119898= 03Hz and (c) 119891

119898= 2Hz




g = Φ (120599) b + v (27)


def= (119909 119910 119891

119898 1205740 120601119888 119875) is unknown b def

= [119887(0) 119887(119868)]119879 is

unknown and

119890119870120579


119875 sin2120578120579

2 cos 120578120579

)]

Φdef=

[

[

[

[

[

[

[

[

[

[

11989011987011205791

0

d

0 1198901198701120579119872

1198901198701198731205791

0

d

0 119890119870119873120579119872

]

]

]

]

]

]

]

]

]

]

[

[

[

1205820(1205791) 120582

119874(1205791)

1205820(120579119872) 120582

119874(120579119872)

]

]

]

(28)


0 20 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)



119909 (m) 119910 (m) 119891119898(Hz)


1205740(rad) 120601

119888(rad) 119875 (m)




120599 = argmin

120599

100381710038171003817100381710038171003817


Φ119867g

100381710038171003817100381710038171003817

2

2

(29)




Φ119867g (30)


(120579) =

119868

sum

119894=0

119887 (119894) 120582

119894(120579) (31)


0 2 40

5

10

15

20

25

Refle

ctiv

ity


minus2minus4

Slow time (s)



119909 =

∬1199091015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

119910 =

∬1199101015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

(32)






119888as a whole by


119888can then






119898= 03Hz 120572

0= 120587




0 02 04 06 08 1 12 14 16 18 20

200

400

600

800

1000

1200O

bjec

tive f

unct

ion

119891119898 (Hz)

(a)

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

Obj

ectiv

e fun

ctio

n

119875 (m)

(b)






7 Conclusion



Acknowledgment




References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













r1015840 = (119909 119910 119911)119879

= (120588 cos120593 120588 sin1205931205882

2119875

minus

119875

2

)

119879

(5)


n = (minus

120588 cos120593119875

minus

120588 sin120593119875

1)

119879

radic1 +

1205882

1198752

(6)


r = (sin 120578 0 cos 120578)119879 (7)



119896

radic120587

int

2120587

0

int

1198632

0


1198752120588119889120588119889120593

(8)


radic120590PO1003816100381610038161003816120578=0

= minus1198952radic120587119875 sin(119870 1198632

16119875

) exp[minus119895119870(

119875

2

minus

1198632

16119875

)]

= radic120587119875[exp(minus119895119870 1198632

16119875

) minus exp(119895119870 1198632

16119875

)]


119875

2

minus

1198632

16119875

)]

(9)








120579

2 cos 120578120579

)]

(10)




119892 (119870 1205791015840)

= 119904 (1205791015840) exp[minus119895119870(

sin 1205791015840

4

+


)]

= 119904 (1205791015840) exp[minus119895119870(

sin 1205791015840

4

+

119875cos21205791015840

2 sin 1205791015840)]

(11)


119891119889asymp

175

120582

cos (120596119905) asymp 875120587

120582

minus

175120596

120582

119905 (12)




Slantplane view

119905 = 0

119884119888

119881119886

119883119888

119911998400

Spot

119909

119910

120579

119909

119910(119910998400)

120587 minus 120601119888119909998400

P119905

O O1O2

FO

S

H









119883119888tan 120579 = 119881

119886119905 minus 119884119888 (13)



120579 =

119881119886

119883119888

119905 (14)



119875119905is at (minus119883





n1198741119865= (cos 120574 sin120601

119888 sin 120574 sin120601

119888 cos120601

119888)

nOH = (cos1205720sin1205730 sin120572

0sin1205730 cos120573

0)

where 120574 = 120596119905+1205740120596 = 2120587119891



0as well as 120573

0


cos 120578120579= n1198741119865sdot n1198741119875119905

asymp n1198741119865sdot n119874119875119905


= sin120601119888sin1205730cos (120574 minus 120572

0) + cos120601

119888cos1205730

(15)




10038161003816100381610038161198751199051198741

1003816100381610038161003816= radic(119881

119886119905 minus 119884119888+ 119910)2

+ (119883119888+ 119909)2

(16)


997888997888rarr1198741119878 sdot n1198751199051198741

=

119875 sin2120578120579

2 cos 120578120579

(17)


119905to 119878 at slow

time 119905

119877 (119905)

def= 119877120579=10038161003816100381610038161198751199051198741

1003816100381610038161003816+997888997888rarr1198741119878 sdot n1198751199051198741

= radic(119881119886119905 minus 119884119888+ 119910)2

+ (119883119888+ 119909)2

+

119875 sin2120578120579

2 cos 120578120579


+ 1198832

119888+ 119909 cos 120579 + 119910 sin 120579 +

119875 sin2120578120579

2 cos 120578120579

(18)



119866 (119870 120579) = 119904 (120579) exp [minus119895119870119877 (119905)] (19)


119886119905 minus 119884119888)2+ 1198832

119888) Hencewe



120579

2 cos 120578120579

)]

= 119904 (120579) exp [minus119895119870(119909 cos 120579 + 119910 sin 120579

+

119875

2

(

1

cos 120578120579

minus cos 120578120579))]

=

119868

sum

119894=0

119887 (119894) 120582119894(120579)


+

119875

2

(

1

cos 120578120579

minus cos 120578120579))]

(20)





120579is determined



120579is


1

cos 120578120579

asymp

2 minus cos 120578120579 when 120578

120579is around 0

∘


120579is around 180

∘

(21)


119892 (119870 120579)


+119875 sign (cos (120578120579)))]

asymp 119862 sdot 119904 (120579)


times(

119881119886119910119905

119883119888

minus 119875 sin120601119888sin1205730cos (120596119905 + 120574

0minus 1205720))]

(22)



2120596119875 sin120601119888sin1205730

120582

sin (120596119905 + 1205740minus 1205720) minus

2119881119886119910

120582119883119888

(23)







11198743= 05m


119888=


∘


1015840119905)


180∘(minus180∘)

119863

119909

270∘(minus90∘)119910

90∘

120579998400

120579998400 = 0∘

O3O1 O2

A

B


41 Stationary PRA












9060

30

0

330

300270

240

210

180

150

120

800

400 m 2

(a)

0 50 100 150 2000

500

1000

1500

0 100 200

0

50


minus100

minus50

minus100minus200

RCS

(m2)

Aspect angle (∘)

RCS

(dB

m2)

(b)

0 50 100 150 2000

100

200

300

400

500

600

700

800

900

1000

70 80 90 100 1100

200

400

600

minus200


RCS

(m2)

GRECOGO

(c)


98 99 10 101 102

10

15

20

25

30

RCS

(dB

m2)

Frequency (GHz)

120579998400 = 90∘

(a)

98 99 10 101 102

200

400

600

800

1000

1200

GRECOPOGO

Frequency (GHz)

RCS

(m2)

(b)


∘






0 50 100 150

0

2

4

6

75 80 85 90 95 100 108

0

05069

0 1 15 20005

01015

02025

03

Backilluminated

Frontilluminated

Farther from radar

Nearer

minus05

minus2

minus4

minus6

Rang

e (m

)


Aspect angle (∘)


(a)

0 50 100 150

0

2

4

6

minus2

minus4

minus6

Rang

e (m

)


Aspect angle (∘)

(b)


Nor

mal

ized

freq

uenc

y

70 75 80 85 90 95 100 105 110

0

01

02

03

04

minus01

minus02

minus03

minus04

minus05

Aspect angle (∘)

Spec

ular

pio

nt

(a)

Nor

mal

ized

freq

uenc

y

70 75 80 85 90 95 100 105 110

0

01

02

03

04

05

Aspect angle (∘)

minus01

minus02

minus03

minus04

minus05

(b)






1015840= 90


1to the




0 1 2 3 4

0

2

4

6


Azimuth (m)

Rang

e (m

)

minus2

minus4

minus6

(a)

0 2 4

0

2

4

6

minus2minus4

Azimuth (m)

Rang

e (m

)

minus2

minus4

minus6

minus6 6

(b)








11989801sim03Hz


200ms 120601119888

1205874


120587







=





0 1 2 3 40

50

100

150

200

250

RCS

(m2)


Slow time (s)

(a)

Nor

mal

ized

freq

uenc

y

0 2 4

0

05

minus05minus2minus4

Slow time (s)

(b)

0 1 2 3 4

0

2

4

6

minus2

minus4


Rang

e (m

)

minus4

Slow time (s)

(c)



119898= 2Hz we





120579| le Δ or



10038161003816100381610038161205730minus 120601119888

1003816100381610038161003816le Δ (24)

1205730+ 120601119888ge 120587 minus Δ (25)


Δ ge

120587

3

(26)



0 10 20 30 40-6

-4

-2

0

2

4

6


Azimuth (m)

Rang

e (m

)

(a)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)

(b)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)


119898= 02Hz (b) 119891

119898= 03Hz and (c) 119891

119898= 2Hz




g = Φ (120599) b + v (27)


def= (119909 119910 119891

119898 1205740 120601119888 119875) is unknown b def

= [119887(0) 119887(119868)]119879 is

unknown and

119890119870120579


119875 sin2120578120579

2 cos 120578120579

)]

Φdef=

[

[

[

[

[

[

[

[

[

[

11989011987011205791

0

d

0 1198901198701120579119872

1198901198701198731205791

0

d

0 119890119870119873120579119872

]

]

]

]

]

]

]

]

]

]

[

[

[

1205820(1205791) 120582

119874(1205791)

1205820(120579119872) 120582

119874(120579119872)

]

]

]

(28)


0 20 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)



119909 (m) 119910 (m) 119891119898(Hz)


1205740(rad) 120601

119888(rad) 119875 (m)




120599 = argmin

120599

100381710038171003817100381710038171003817


Φ119867g

100381710038171003817100381710038171003817

2

2

(29)




Φ119867g (30)


(120579) =

119868

sum

119894=0

119887 (119894) 120582

119894(120579) (31)


0 2 40

5

10

15

20

25

Refle

ctiv

ity


minus2minus4

Slow time (s)



119909 =

∬1199091015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

119910 =

∬1199101015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

(32)






119888as a whole by


119888can then






119898= 03Hz 120572

0= 120587




0 02 04 06 08 1 12 14 16 18 20

200

400

600

800

1000

1200O

bjec

tive f

unct

ion

119891119898 (Hz)

(a)

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

Obj

ectiv

e fun

ctio

n

119875 (m)

(b)






7 Conclusion



Acknowledgment




References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Slantplane view

119905 = 0

119884119888

119881119886

119883119888

119911998400

Spot

119909

119910

120579

119909

119910(119910998400)

120587 minus 120601119888119909998400

P119905

O O1O2

FO

S

H









119883119888tan 120579 = 119881

119886119905 minus 119884119888 (13)



120579 =

119881119886

119883119888

119905 (14)



119875119905is at (minus119883





n1198741119865= (cos 120574 sin120601

119888 sin 120574 sin120601

119888 cos120601

119888)

nOH = (cos1205720sin1205730 sin120572

0sin1205730 cos120573

0)

where 120574 = 120596119905+1205740120596 = 2120587119891



0as well as 120573

0


cos 120578120579= n1198741119865sdot n1198741119875119905

asymp n1198741119865sdot n119874119875119905


= sin120601119888sin1205730cos (120574 minus 120572

0) + cos120601

119888cos1205730

(15)




10038161003816100381610038161198751199051198741

1003816100381610038161003816= radic(119881

119886119905 minus 119884119888+ 119910)2

+ (119883119888+ 119909)2

(16)


997888997888rarr1198741119878 sdot n1198751199051198741

=

119875 sin2120578120579

2 cos 120578120579

(17)


119905to 119878 at slow

time 119905

119877 (119905)

def= 119877120579=10038161003816100381610038161198751199051198741

1003816100381610038161003816+997888997888rarr1198741119878 sdot n1198751199051198741

= radic(119881119886119905 minus 119884119888+ 119910)2

+ (119883119888+ 119909)2

+

119875 sin2120578120579

2 cos 120578120579


+ 1198832

119888+ 119909 cos 120579 + 119910 sin 120579 +

119875 sin2120578120579

2 cos 120578120579

(18)



119866 (119870 120579) = 119904 (120579) exp [minus119895119870119877 (119905)] (19)


119886119905 minus 119884119888)2+ 1198832

119888) Hencewe



120579

2 cos 120578120579

)]

= 119904 (120579) exp [minus119895119870(119909 cos 120579 + 119910 sin 120579

+

119875

2

(

1

cos 120578120579

minus cos 120578120579))]

=

119868

sum

119894=0

119887 (119894) 120582119894(120579)


+

119875

2

(

1

cos 120578120579

minus cos 120578120579))]

(20)





120579is determined



120579is


1

cos 120578120579

asymp

2 minus cos 120578120579 when 120578

120579is around 0

∘


120579is around 180

∘

(21)


119892 (119870 120579)


+119875 sign (cos (120578120579)))]

asymp 119862 sdot 119904 (120579)


times(

119881119886119910119905

119883119888

minus 119875 sin120601119888sin1205730cos (120596119905 + 120574

0minus 1205720))]

(22)



2120596119875 sin120601119888sin1205730

120582

sin (120596119905 + 1205740minus 1205720) minus

2119881119886119910

120582119883119888

(23)







11198743= 05m


119888=


∘


1015840119905)


180∘(minus180∘)

119863

119909

270∘(minus90∘)119910

90∘

120579998400

120579998400 = 0∘

O3O1 O2

A

B


41 Stationary PRA












9060

30

0

330

300270

240

210

180

150

120

800

400 m 2

(a)

0 50 100 150 2000

500

1000

1500

0 100 200

0

50


minus100

minus50

minus100minus200

RCS

(m2)

Aspect angle (∘)

RCS

(dB

m2)

(b)

0 50 100 150 2000

100

200

300

400

500

600

700

800

900

1000

70 80 90 100 1100

200

400

600

minus200


RCS

(m2)

GRECOGO

(c)


98 99 10 101 102

10

15

20

25

30

RCS

(dB

m2)

Frequency (GHz)

120579998400 = 90∘

(a)

98 99 10 101 102

200

400

600

800

1000

1200

GRECOPOGO

Frequency (GHz)

RCS

(m2)

(b)


∘






0 50 100 150

0

2

4

6

75 80 85 90 95 100 108

0

05069

0 1 15 20005

01015

02025

03

Backilluminated

Frontilluminated

Farther from radar

Nearer

minus05

minus2

minus4

minus6

Rang

e (m

)


Aspect angle (∘)


(a)

0 50 100 150

0

2

4

6

minus2

minus4

minus6

Rang

e (m

)


Aspect angle (∘)

(b)


Nor

mal

ized

freq

uenc

y

70 75 80 85 90 95 100 105 110

0

01

02

03

04

minus01

minus02

minus03

minus04

minus05

Aspect angle (∘)

Spec

ular

pio

nt

(a)

Nor

mal

ized

freq

uenc

y

70 75 80 85 90 95 100 105 110

0

01

02

03

04

05

Aspect angle (∘)

minus01

minus02

minus03

minus04

minus05

(b)






1015840= 90


1to the




0 1 2 3 4

0

2

4

6


Azimuth (m)

Rang

e (m

)

minus2

minus4

minus6

(a)

0 2 4

0

2

4

6

minus2minus4

Azimuth (m)

Rang

e (m

)

minus2

minus4

minus6

minus6 6

(b)








11989801sim03Hz


200ms 120601119888

1205874


120587







=





0 1 2 3 40

50

100

150

200

250

RCS

(m2)


Slow time (s)

(a)

Nor

mal

ized

freq

uenc

y

0 2 4

0

05

minus05minus2minus4

Slow time (s)

(b)

0 1 2 3 4

0

2

4

6

minus2

minus4


Rang

e (m

)

minus4

Slow time (s)

(c)



119898= 2Hz we





120579| le Δ or



10038161003816100381610038161205730minus 120601119888

1003816100381610038161003816le Δ (24)

1205730+ 120601119888ge 120587 minus Δ (25)


Δ ge

120587

3

(26)



0 10 20 30 40-6

-4

-2

0

2

4

6


Azimuth (m)

Rang

e (m

)

(a)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)

(b)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)


119898= 02Hz (b) 119891

119898= 03Hz and (c) 119891

119898= 2Hz




g = Φ (120599) b + v (27)


def= (119909 119910 119891

119898 1205740 120601119888 119875) is unknown b def

= [119887(0) 119887(119868)]119879 is

unknown and

119890119870120579


119875 sin2120578120579

2 cos 120578120579

)]

Φdef=

[

[

[

[

[

[

[

[

[

[

11989011987011205791

0

d

0 1198901198701120579119872

1198901198701198731205791

0

d

0 119890119870119873120579119872

]

]

]

]

]

]

]

]

]

]

[

[

[

1205820(1205791) 120582

119874(1205791)

1205820(120579119872) 120582

119874(120579119872)

]

]

]

(28)


0 20 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)



119909 (m) 119910 (m) 119891119898(Hz)


1205740(rad) 120601

119888(rad) 119875 (m)




120599 = argmin

120599

100381710038171003817100381710038171003817


Φ119867g

100381710038171003817100381710038171003817

2

2

(29)




Φ119867g (30)


(120579) =

119868

sum

119894=0

119887 (119894) 120582

119894(120579) (31)


0 2 40

5

10

15

20

25

Refle

ctiv

ity


minus2minus4

Slow time (s)



119909 =

∬1199091015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

119910 =

∬1199101015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

(32)






119888as a whole by


119888can then






119898= 03Hz 120572

0= 120587




0 02 04 06 08 1 12 14 16 18 20

200

400

600

800

1000

1200O

bjec

tive f

unct

ion

119891119898 (Hz)

(a)

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

Obj

ectiv

e fun

ctio

n

119875 (m)

(b)






7 Conclusion



Acknowledgment




References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












120579is determined



120579is


1

cos 120578120579

asymp

2 minus cos 120578120579 when 120578

120579is around 0

∘


120579is around 180

∘

(21)


119892 (119870 120579)


+119875 sign (cos (120578120579)))]

asymp 119862 sdot 119904 (120579)


times(

119881119886119910119905

119883119888

minus 119875 sin120601119888sin1205730cos (120596119905 + 120574

0minus 1205720))]

(22)



2120596119875 sin120601119888sin1205730

120582

sin (120596119905 + 1205740minus 1205720) minus

2119881119886119910

120582119883119888

(23)







11198743= 05m


119888=


∘


1015840119905)


180∘(minus180∘)

119863

119909

270∘(minus90∘)119910

90∘

120579998400

120579998400 = 0∘

O3O1 O2

A

B


41 Stationary PRA












9060

30

0

330

300270

240

210

180

150

120

800

400 m 2

(a)

0 50 100 150 2000

500

1000

1500

0 100 200

0

50


minus100

minus50

minus100minus200

RCS

(m2)

Aspect angle (∘)

RCS

(dB

m2)

(b)

0 50 100 150 2000

100

200

300

400

500

600

700

800

900

1000

70 80 90 100 1100

200

400

600

minus200


RCS

(m2)

GRECOGO

(c)


98 99 10 101 102

10

15

20

25

30

RCS

(dB

m2)

Frequency (GHz)

120579998400 = 90∘

(a)

98 99 10 101 102

200

400

600

800

1000

1200

GRECOPOGO

Frequency (GHz)

RCS

(m2)

(b)


∘






0 50 100 150

0

2

4

6

75 80 85 90 95 100 108

0

05069

0 1 15 20005

01015

02025

03

Backilluminated

Frontilluminated

Farther from radar

Nearer

minus05

minus2

minus4

minus6

Rang

e (m

)


Aspect angle (∘)


(a)

0 50 100 150

0

2

4

6

minus2

minus4

minus6

Rang

e (m

)


Aspect angle (∘)

(b)


Nor

mal

ized

freq

uenc

y

70 75 80 85 90 95 100 105 110

0

01

02

03

04

minus01

minus02

minus03

minus04

minus05

Aspect angle (∘)

Spec

ular

pio

nt

(a)

Nor

mal

ized

freq

uenc

y

70 75 80 85 90 95 100 105 110

0

01

02

03

04

05

Aspect angle (∘)

minus01

minus02

minus03

minus04

minus05

(b)






1015840= 90


1to the




0 1 2 3 4

0

2

4

6


Azimuth (m)

Rang

e (m

)

minus2

minus4

minus6

(a)

0 2 4

0

2

4

6

minus2minus4

Azimuth (m)

Rang

e (m

)

minus2

minus4

minus6

minus6 6

(b)








11989801sim03Hz


200ms 120601119888

1205874


120587







=





0 1 2 3 40

50

100

150

200

250

RCS

(m2)


Slow time (s)

(a)

Nor

mal

ized

freq

uenc

y

0 2 4

0

05

minus05minus2minus4

Slow time (s)

(b)

0 1 2 3 4

0

2

4

6

minus2

minus4


Rang

e (m

)

minus4

Slow time (s)

(c)



119898= 2Hz we





120579| le Δ or



10038161003816100381610038161205730minus 120601119888

1003816100381610038161003816le Δ (24)

1205730+ 120601119888ge 120587 minus Δ (25)


Δ ge

120587

3

(26)



0 10 20 30 40-6

-4

-2

0

2

4

6


Azimuth (m)

Rang

e (m

)

(a)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)

(b)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)


119898= 02Hz (b) 119891

119898= 03Hz and (c) 119891

119898= 2Hz




g = Φ (120599) b + v (27)


def= (119909 119910 119891

119898 1205740 120601119888 119875) is unknown b def

= [119887(0) 119887(119868)]119879 is

unknown and

119890119870120579


119875 sin2120578120579

2 cos 120578120579

)]

Φdef=

[

[

[

[

[

[

[

[

[

[

11989011987011205791

0

d

0 1198901198701120579119872

1198901198701198731205791

0

d

0 119890119870119873120579119872

]

]

]

]

]

]

]

]

]

]

[

[

[

1205820(1205791) 120582

119874(1205791)

1205820(120579119872) 120582

119874(120579119872)

]

]

]

(28)


0 20 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)



119909 (m) 119910 (m) 119891119898(Hz)


1205740(rad) 120601

119888(rad) 119875 (m)




120599 = argmin

120599

100381710038171003817100381710038171003817


Φ119867g

100381710038171003817100381710038171003817

2

2

(29)




Φ119867g (30)


(120579) =

119868

sum

119894=0

119887 (119894) 120582

119894(120579) (31)


0 2 40

5

10

15

20

25

Refle

ctiv

ity


minus2minus4

Slow time (s)



119909 =

∬1199091015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

119910 =

∬1199101015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

(32)






119888as a whole by


119888can then






119898= 03Hz 120572

0= 120587




0 02 04 06 08 1 12 14 16 18 20

200

400

600

800

1000

1200O

bjec

tive f

unct

ion

119891119898 (Hz)

(a)

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

Obj

ectiv

e fun

ctio

n

119875 (m)

(b)






7 Conclusion



Acknowledgment




References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










9060

30

0

330

300270

240

210

180

150

120

800

400 m 2

(a)

0 50 100 150 2000

500

1000

1500

0 100 200

0

50


minus100

minus50

minus100minus200

RCS

(m2)

Aspect angle (∘)

RCS

(dB

m2)

(b)

0 50 100 150 2000

100

200

300

400

500

600

700

800

900

1000

70 80 90 100 1100

200

400

600

minus200


RCS

(m2)

GRECOGO

(c)


98 99 10 101 102

10

15

20

25

30

RCS

(dB

m2)

Frequency (GHz)

120579998400 = 90∘

(a)

98 99 10 101 102

200

400

600

800

1000

1200

GRECOPOGO

Frequency (GHz)

RCS

(m2)

(b)


∘






0 50 100 150

0

2

4

6

75 80 85 90 95 100 108

0

05069

0 1 15 20005

01015

02025

03

Backilluminated

Frontilluminated

Farther from radar

Nearer

minus05

minus2

minus4

minus6

Rang

e (m

)


Aspect angle (∘)


(a)

0 50 100 150

0

2

4

6

minus2

minus4

minus6

Rang

e (m

)


Aspect angle (∘)

(b)


Nor

mal

ized

freq

uenc

y

70 75 80 85 90 95 100 105 110

0

01

02

03

04

minus01

minus02

minus03

minus04

minus05

Aspect angle (∘)

Spec

ular

pio

nt

(a)

Nor

mal

ized

freq

uenc

y

70 75 80 85 90 95 100 105 110

0

01

02

03

04

05

Aspect angle (∘)

minus01

minus02

minus03

minus04

minus05

(b)






1015840= 90


1to the




0 1 2 3 4

0

2

4

6


Azimuth (m)

Rang

e (m

)

minus2

minus4

minus6

(a)

0 2 4

0

2

4

6

minus2minus4

Azimuth (m)

Rang

e (m

)

minus2

minus4

minus6

minus6 6

(b)








11989801sim03Hz


200ms 120601119888

1205874


120587







=





0 1 2 3 40

50

100

150

200

250

RCS

(m2)


Slow time (s)

(a)

Nor

mal

ized

freq

uenc

y

0 2 4

0

05

minus05minus2minus4

Slow time (s)

(b)

0 1 2 3 4

0

2

4

6

minus2

minus4


Rang

e (m

)

minus4

Slow time (s)

(c)



119898= 2Hz we





120579| le Δ or



10038161003816100381610038161205730minus 120601119888

1003816100381610038161003816le Δ (24)

1205730+ 120601119888ge 120587 minus Δ (25)


Δ ge

120587

3

(26)



0 10 20 30 40-6

-4

-2

0

2

4

6


Azimuth (m)

Rang

e (m

)

(a)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)

(b)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)


119898= 02Hz (b) 119891

119898= 03Hz and (c) 119891

119898= 2Hz




g = Φ (120599) b + v (27)


def= (119909 119910 119891

119898 1205740 120601119888 119875) is unknown b def

= [119887(0) 119887(119868)]119879 is

unknown and

119890119870120579


119875 sin2120578120579

2 cos 120578120579

)]

Φdef=

[

[

[

[

[

[

[

[

[

[

11989011987011205791

0

d

0 1198901198701120579119872

1198901198701198731205791

0

d

0 119890119870119873120579119872

]

]

]

]

]

]

]

]

]

]

[

[

[

1205820(1205791) 120582

119874(1205791)

1205820(120579119872) 120582

119874(120579119872)

]

]

]

(28)


0 20 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)



119909 (m) 119910 (m) 119891119898(Hz)


1205740(rad) 120601

119888(rad) 119875 (m)




120599 = argmin

120599

100381710038171003817100381710038171003817


Φ119867g

100381710038171003817100381710038171003817

2

2

(29)




Φ119867g (30)


(120579) =

119868

sum

119894=0

119887 (119894) 120582

119894(120579) (31)


0 2 40

5

10

15

20

25

Refle

ctiv

ity


minus2minus4

Slow time (s)



119909 =

∬1199091015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

119910 =

∬1199101015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

(32)






119888as a whole by


119888can then






119898= 03Hz 120572

0= 120587




0 02 04 06 08 1 12 14 16 18 20

200

400

600

800

1000

1200O

bjec

tive f

unct

ion

119891119898 (Hz)

(a)

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

Obj

ectiv

e fun

ctio

n

119875 (m)

(b)






7 Conclusion



Acknowledgment




References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 50 100 150

0

2

4

6

75 80 85 90 95 100 108

0

05069

0 1 15 20005

01015

02025

03

Backilluminated

Frontilluminated

Farther from radar

Nearer

minus05

minus2

minus4

minus6

Rang

e (m

)


Aspect angle (∘)


(a)

0 50 100 150

0

2

4

6

minus2

minus4

minus6

Rang

e (m

)


Aspect angle (∘)

(b)


Nor

mal

ized

freq

uenc

y

70 75 80 85 90 95 100 105 110

0

01

02

03

04

minus01

minus02

minus03

minus04

minus05

Aspect angle (∘)

Spec

ular

pio

nt

(a)

Nor

mal

ized

freq

uenc

y

70 75 80 85 90 95 100 105 110

0

01

02

03

04

05

Aspect angle (∘)

minus01

minus02

minus03

minus04

minus05

(b)






1015840= 90


1to the




0 1 2 3 4

0

2

4

6


Azimuth (m)

Rang

e (m

)

minus2

minus4

minus6

(a)

0 2 4

0

2

4

6

minus2minus4

Azimuth (m)

Rang

e (m

)

minus2

minus4

minus6

minus6 6

(b)








11989801sim03Hz


200ms 120601119888

1205874


120587







=





0 1 2 3 40

50

100

150

200

250

RCS

(m2)


Slow time (s)

(a)

Nor

mal

ized

freq

uenc

y

0 2 4

0

05

minus05minus2minus4

Slow time (s)

(b)

0 1 2 3 4

0

2

4

6

minus2

minus4


Rang

e (m

)

minus4

Slow time (s)

(c)



119898= 2Hz we





120579| le Δ or



10038161003816100381610038161205730minus 120601119888

1003816100381610038161003816le Δ (24)

1205730+ 120601119888ge 120587 minus Δ (25)


Δ ge

120587

3

(26)



0 10 20 30 40-6

-4

-2

0

2

4

6


Azimuth (m)

Rang

e (m

)

(a)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)

(b)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)


119898= 02Hz (b) 119891

119898= 03Hz and (c) 119891

119898= 2Hz




g = Φ (120599) b + v (27)


def= (119909 119910 119891

119898 1205740 120601119888 119875) is unknown b def

= [119887(0) 119887(119868)]119879 is

unknown and

119890119870120579


119875 sin2120578120579

2 cos 120578120579

)]

Φdef=

[

[

[

[

[

[

[

[

[

[

11989011987011205791

0

d

0 1198901198701120579119872

1198901198701198731205791

0

d

0 119890119870119873120579119872

]

]

]

]

]

]

]

]

]

]

[

[

[

1205820(1205791) 120582

119874(1205791)

1205820(120579119872) 120582

119874(120579119872)

]

]

]

(28)


0 20 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)



119909 (m) 119910 (m) 119891119898(Hz)


1205740(rad) 120601

119888(rad) 119875 (m)




120599 = argmin

120599

100381710038171003817100381710038171003817


Φ119867g

100381710038171003817100381710038171003817

2

2

(29)




Φ119867g (30)


(120579) =

119868

sum

119894=0

119887 (119894) 120582

119894(120579) (31)


0 2 40

5

10

15

20

25

Refle

ctiv

ity


minus2minus4

Slow time (s)



119909 =

∬1199091015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

119910 =

∬1199101015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

(32)






119888as a whole by


119888can then






119898= 03Hz 120572

0= 120587




0 02 04 06 08 1 12 14 16 18 20

200

400

600

800

1000

1200O

bjec

tive f

unct

ion

119891119898 (Hz)

(a)

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

Obj

ectiv

e fun

ctio

n

119875 (m)

(b)






7 Conclusion



Acknowledgment




References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 4

0

2

4

6


Azimuth (m)

Rang

e (m

)

minus2

minus4

minus6

(a)

0 2 4

0

2

4

6

minus2minus4

Azimuth (m)

Rang

e (m

)

minus2

minus4

minus6

minus6 6

(b)








11989801sim03Hz


200ms 120601119888

1205874


120587







=





0 1 2 3 40

50

100

150

200

250

RCS

(m2)


Slow time (s)

(a)

Nor

mal

ized

freq

uenc

y

0 2 4

0

05

minus05minus2minus4

Slow time (s)

(b)

0 1 2 3 4

0

2

4

6

minus2

minus4


Rang

e (m

)

minus4

Slow time (s)

(c)



119898= 2Hz we





120579| le Δ or



10038161003816100381610038161205730minus 120601119888

1003816100381610038161003816le Δ (24)

1205730+ 120601119888ge 120587 minus Δ (25)


Δ ge

120587

3

(26)



0 10 20 30 40-6

-4

-2

0

2

4

6


Azimuth (m)

Rang

e (m

)

(a)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)

(b)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)


119898= 02Hz (b) 119891

119898= 03Hz and (c) 119891

119898= 2Hz




g = Φ (120599) b + v (27)


def= (119909 119910 119891

119898 1205740 120601119888 119875) is unknown b def

= [119887(0) 119887(119868)]119879 is

unknown and

119890119870120579


119875 sin2120578120579

2 cos 120578120579

)]

Φdef=

[

[

[

[

[

[

[

[

[

[

11989011987011205791

0

d

0 1198901198701120579119872

1198901198701198731205791

0

d

0 119890119870119873120579119872

]

]

]

]

]

]

]

]

]

]

[

[

[

1205820(1205791) 120582

119874(1205791)

1205820(120579119872) 120582

119874(120579119872)

]

]

]

(28)


0 20 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)



119909 (m) 119910 (m) 119891119898(Hz)


1205740(rad) 120601

119888(rad) 119875 (m)




120599 = argmin

120599

100381710038171003817100381710038171003817


Φ119867g

100381710038171003817100381710038171003817

2

2

(29)




Φ119867g (30)


(120579) =

119868

sum

119894=0

119887 (119894) 120582

119894(120579) (31)


0 2 40

5

10

15

20

25

Refle

ctiv

ity


minus2minus4

Slow time (s)



119909 =

∬1199091015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

119910 =

∬1199101015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

(32)






119888as a whole by


119888can then






119898= 03Hz 120572

0= 120587




0 02 04 06 08 1 12 14 16 18 20

200

400

600

800

1000

1200O

bjec

tive f

unct

ion

119891119898 (Hz)

(a)

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

Obj

ectiv

e fun

ctio

n

119875 (m)

(b)






7 Conclusion



Acknowledgment




References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 40

50

100

150

200

250

RCS

(m2)


Slow time (s)

(a)

Nor

mal

ized

freq

uenc

y

0 2 4

0

05

minus05minus2minus4

Slow time (s)

(b)

0 1 2 3 4

0

2

4

6

minus2

minus4


Rang

e (m

)

minus4

Slow time (s)

(c)



119898= 2Hz we





120579| le Δ or



10038161003816100381610038161205730minus 120601119888

1003816100381610038161003816le Δ (24)

1205730+ 120601119888ge 120587 minus Δ (25)


Δ ge

120587

3

(26)



0 10 20 30 40-6

-4

-2

0

2

4

6


Azimuth (m)

Rang

e (m

)

(a)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)

(b)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)


119898= 02Hz (b) 119891

119898= 03Hz and (c) 119891

119898= 2Hz




g = Φ (120599) b + v (27)


def= (119909 119910 119891

119898 1205740 120601119888 119875) is unknown b def

= [119887(0) 119887(119868)]119879 is

unknown and

119890119870120579


119875 sin2120578120579

2 cos 120578120579

)]

Φdef=

[

[

[

[

[

[

[

[

[

[

11989011987011205791

0

d

0 1198901198701120579119872

1198901198701198731205791

0

d

0 119890119870119873120579119872

]

]

]

]

]

]

]

]

]

]

[

[

[

1205820(1205791) 120582

119874(1205791)

1205820(120579119872) 120582

119874(120579119872)

]

]

]

(28)


0 20 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)



119909 (m) 119910 (m) 119891119898(Hz)


1205740(rad) 120601

119888(rad) 119875 (m)




120599 = argmin

120599

100381710038171003817100381710038171003817


Φ119867g

100381710038171003817100381710038171003817

2

2

(29)




Φ119867g (30)


(120579) =

119868

sum

119894=0

119887 (119894) 120582

119894(120579) (31)


0 2 40

5

10

15

20

25

Refle

ctiv

ity


minus2minus4

Slow time (s)



119909 =

∬1199091015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

119910 =

∬1199101015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

(32)






119888as a whole by


119888can then






119898= 03Hz 120572

0= 120587




0 02 04 06 08 1 12 14 16 18 20

200

400

600

800

1000

1200O

bjec

tive f

unct

ion

119891119898 (Hz)

(a)

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

Obj

ectiv

e fun

ctio

n

119875 (m)

(b)






7 Conclusion



Acknowledgment




References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 10 20 30 40-6

-4

-2

0

2

4

6


Azimuth (m)

Rang

e (m

)

(a)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)

(b)

0 10 20 30 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)


119898= 02Hz (b) 119891

119898= 03Hz and (c) 119891

119898= 2Hz




g = Φ (120599) b + v (27)


def= (119909 119910 119891

119898 1205740 120601119888 119875) is unknown b def

= [119887(0) 119887(119868)]119879 is

unknown and

119890119870120579


119875 sin2120578120579

2 cos 120578120579

)]

Φdef=

[

[

[

[

[

[

[

[

[

[

11989011987011205791

0

d

0 1198901198701120579119872

1198901198701198731205791

0

d

0 119890119870119873120579119872

]

]

]

]

]

]

]

]

]

]

[

[

[

1205820(1205791) 120582

119874(1205791)

1205820(120579119872) 120582

119874(120579119872)

]

]

]

(28)


0 20 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)



119909 (m) 119910 (m) 119891119898(Hz)


1205740(rad) 120601

119888(rad) 119875 (m)




120599 = argmin

120599

100381710038171003817100381710038171003817


Φ119867g

100381710038171003817100381710038171003817

2

2

(29)




Φ119867g (30)


(120579) =

119868

sum

119894=0

119887 (119894) 120582

119894(120579) (31)


0 2 40

5

10

15

20

25

Refle

ctiv

ity


minus2minus4

Slow time (s)



119909 =

∬1199091015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

119910 =

∬1199101015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

(32)






119888as a whole by


119888can then






119898= 03Hz 120572

0= 120587




0 02 04 06 08 1 12 14 16 18 20

200

400

600

800

1000

1200O

bjec

tive f

unct

ion

119891119898 (Hz)

(a)

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

Obj

ectiv

e fun

ctio

n

119875 (m)

(b)






7 Conclusion



Acknowledgment




References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 20 40

0

2

4

6

Rang

e (m

)

minus2

minus4


Azimuth (m)



119909 (m) 119910 (m) 119891119898(Hz)


1205740(rad) 120601

119888(rad) 119875 (m)




120599 = argmin

120599

100381710038171003817100381710038171003817


Φ119867g

100381710038171003817100381710038171003817

2

2

(29)




Φ119867g (30)


(120579) =

119868

sum

119894=0

119887 (119894) 120582

119894(120579) (31)


0 2 40

5

10

15

20

25

Refle

ctiv

ity


minus2minus4

Slow time (s)



119909 =

∬1199091015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

119910 =

∬1199101015840sdot 119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

∬119868 (1199091015840 1199101015840) 11988911990910158401198891199101015840

(32)






119888as a whole by


119888can then






119898= 03Hz 120572

0= 120587




0 02 04 06 08 1 12 14 16 18 20

200

400

600

800

1000

1200O

bjec

tive f

unct

ion

119891119898 (Hz)

(a)

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

Obj

ectiv

e fun

ctio

n

119875 (m)

(b)






7 Conclusion



Acknowledgment




References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 02 04 06 08 1 12 14 16 18 20

200

400

600

800

1000

1200O

bjec

tive f

unct

ion

119891119898 (Hz)

(a)

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

Obj

ectiv

e fun

ctio

n

119875 (m)

(b)






7 Conclusion



Acknowledgment




References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article rotating parabolic-reflector...

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