Research Article The Fast Simulation of Scattering

Research ArticleThe Fast Simulation of Scattering Characteristics froma Simplified Time Varying Sea Surface

Yiwen Wei1 Lixin Guo12 and Xiao Meng1

1School of Physics and Optoelectronic Engineering Xidian University Xirsquoan 710071 China2State Key Laboratory of Integrated Services Networks Xidian University Xirsquoan 710071 China

Correspondence should be addressed to Yiwen Wei ywwei0910gmailcom

Received 29 December 2014 Revised 2 April 2015 Accepted 7 April 2015

Academic Editor Claudio Curcio

Copyright copy 2015 Yiwen Wei et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper aims at applying a simplified sea surface model into the physical optics (PO) method to accelerate the scatteringcalculation from 1D time varying sea surface To reduce the number of the segments and make further improvement on theefficiency of PO method a simplified sea surface is proposed In this simplified sea surface the geometry of long waves is locallyapproximated by tilted facets that are much longer than the electromagnetic wavelength The capillary waves are considered tobe sinusoidal line superimposing on the long waves The wavenumber of the sinusoidal waves is supposed to satisfy the resonantcondition of Braggwaves which is dominant in all the scattered short wave components Since the capillary wave is periodical withinone facet an analytical integration of the PO term can be performed The backscattering coefficient obtained from a simplified seasurface model agrees well with that obtained from a realistic sea surface The Doppler shifts and width also agree well with therealistic model since the capillary waves are taken into consideration The good agreements indicate that the simplified model isreasonable and valid in predicting both the scattering coefficients and the Doppler spectra

1 Introduction

The calculation of electromagnetic (EM) scattering froma time varying surface is important in many fields suchas radar surveillance target tracking and ocean remotesensing [1] Useful techniques have already been developedto provide realistic results They can be based on exactnumerical methods (MoM FEM FDTD and so on [2ndash5]) orapproximate approaches [6] Because numerical methods areunfortunately not efficient the approximate approaches arewidely used for the moment to calculate the scattered fieldfrom a large time varying surface Among them PO [7 8] ismost employed because it is simple and easy to implement

However the POmethod is still limited by the number ofunknowns when dealing with large time varying sea surfaceWhen dealing with the scattering problem form the sea itshould be noted that the sea surface should be divided intosmall segments whose length of each segment on the realisticsea surface should be 18sim110 wavelength of the incidentwave to accurately reflect the geometry characteristics ofthe sea Since each segment is that small the number of

the segments form the realistic sea surface will be very largeTo reduce the number of segments one can divide the seasurface into larger segments However the larger the segmentis the more inaccuracy will be shown in the scatteringcoefficient andDoppler spectrumThe inaccurate resultsmaybe caused by two reasons Firstly the phase difference onone segment will be neglected in this case This will lead tothe inaccuracy of the integration on each segment Secondlythe capillary waves superimposed on each segment are nottaken into consideration This will cause the fact that theDoppler value in some spectrum regions cannot be detectedby the radar Since the segment cannot be larger than 18sim110wavelength the number of the unknowns will be tremendousespecially when dealing with time varying large sea surfacewhich will limit the efficiency of PO method

In order to get accurate results with larger segments aswell as promote the efficiency of conventional PO methodwe add capillary waves [9] The capillary wave consideredto be sinusoidal line superimposing on each segment Thewavenumbers of the sinusoidal waves are supposed to satisfy

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 815913 8 pageshttpdxdoiorg1011552015815913

2 International Journal of Antennas and Propagation

x

yOne

segmentf(x)

119841i

vi

119844i

119844iz

120579i

120579998400i

119847loc

yloc119857loc

(a)

119841i

vi

119844i

119847loc

119844i

119844ref

x loc

119812inc

119815inc119815ref

119812ref

(b)

Figure 1 The illustration of the scattering problem from a sea surface (a) the global coordinate system and (b) the local coordinate system

the resonant condition of Bragg waves which are predom-inant in the scattered short wave component In this waywe substitute the simplified sea surface for the realistic seasurface approximately Since the capillary wave is periodicalwithin one facet the analytical expression of the inducedcurrents can be given after we get the currents on the firstperiod of the sinusoidal wave

This paper is organized as follows In Section 2 the POmethod is introduced and applied on the simplified seasurface model All the formulas are derived and all theexpressions are given in this part both horizontal (HH)polarization and vertical (VV) polarizations are consideredSeveral numerical simulations are exhibited in Section 3 toshow the validity and efficiency of the new model comparedwith the realistic sea surface Then this model is used toinvestigate the characteristic of the Doppler spectrum oftime varying sea surface Section 4 ends with a summaryof the new model and a proposition for further pertinentinvestigation

2 Formula

21 The Physical Optics Formulation The initial point ofphysical optics is the surface currents produced by an incom-ing electromagnetic wave (Einc

Hinc) Considering a 1D seasurface the induced electric currents J andmagnetic currentsM on each segment (the length of each segment is set as18sim110 wavelength of the incident wave to meet the divisioncriterion) are given by

J = n timesHM = E times n (1)

where n is the unit normal vector of the surface and rindicates the position of each segment E and H are respec-tively the total electric and magnetic fields at the surface ForTE case the horizontal polarization vector of the incidentwave h

119894is along y k

119894is the wavenumber of the incident

wave and the vertical polarization vector is k119894=

k119894times

h119894

The scattering problems in the global coordinate system andthe local coordinate system are shown in Figures 1(a) and1(b) respectively Given the slop of one segment the local

coordinate system can be built as xloc yloc zloc and eachcoordinate component can be expressed as

zloc = n

yloc = y = h119894

xloc = yloc times zloc

(2)

Let a plane wave illuminate on the rough sea surface thenthe incident field on the surface can be given as

Einc= y119864inc exp [119894

997888

k119894sdot

997888r119900]

Hinc=

k119894119864

inc exp [119894997888

k119894sdot

997888r119900]

120578

(3)

where 997888r119900is the position of each surface segment and 120578 is the

wave impedance in the free spaceInserting the TE reflection coefficient119877TE the electricEref

and magneticHref reflected fields are

Eref=

h119894119877TE119864

inc exp [119894997888

k119894sdot

997888r119900]

Href=

(

kref times h119894) 119864

inc exp [119894997888

k119894sdot

997888r119900]

120578

(4)

where kref =

k119894minus 2n(k

119894sdot n) Adding the contributions of

incident and reflected field the total electric and magneticfields can be written as a function of the incident electricfield the reflection coefficients and the intrinsic impedanceof the first medium Then going back to the boundary

International Journal of Antennas and Propagation 3

conditions (1) the equivalent currents can be reckoned fromthe total fields Consider

J = n timesH = n times (Hinc+Href

)

=

119864

inc exp [119894997888

k119894sdot

997888r119900]

120578

sdot [n times k119894+ 119877TEn times (kref times h

119894)]

=

119864

inc exp [119894997888

k119894sdot

997888r119900]

120578

sdot cos 120579incloc sdot (1 minus 119877TE) sdot yloc

(5)

M = E times n = (Einc+ Eref

) times n

= 119864

inc exp [119894997888

k119894sdot

997888r119900] (1 + 119877TE) sdot (yloc times n)

= 119864

inc exp [119894997888

k119894sdot

997888r119900] (1 + 119877TE) sdot xloc

(6)

The field scattered by the dielectric sea surface can readilybe determined as the radiation of J andM The scattered fieldcan be given by the well-known Stratton-Chu formula thatis

119864

sca= Esca

sdot

h119894

= 119894119896120578int

119904

119869119866 (r r1015840) 1198891199041015840 + int119904

119872(n1015840 sdot nabla119866 (r r1015840)) 1198891199041015840(7)

where n1015840 is the unit normal vector of each segment Theresulting expressions are written in terms of Greenrsquos function119866(r r1015840) and n1015840 sdot nabla119866(r r1015840) The far field approximation allowsexpressing these functions as [2] Consider

119866(r r1015840)1003816100381610038161003816

1003816|r|rarrinfin

=

119894

4

radic

2

120587119896

0119903

exp(minus1198941205874

)

sdot exp (1198941198960119903) exp [minus119894

997888

k119904sdot

997888r119900]

n1015840 sdot nabla1015840119866(r r1015840)1003816100381610038161003816

1003816|r|rarrinfin

=

119894

4

radic

2

120587119896

0119903

exp(minus1198941205874

)

sdot exp (1198941198960119903) (minus119894n sdot k

119904) exp [minus119894

997888

k119904sdot

997888r119900]

(8)

where997888

k119904is the scattering wavenumber vector

997888

k119904

=

119896

0(sin 120579119904x + cos 120579

119904y) Consequently the scattered fields are

119864

sca= minus

radic

119896

8120587119903

exp(minus1198941205874

) exp (1198941198960119903)

sdot int

119904

exp (minus119894997888

k119904sdot

997888r119900) [120578 sdot 119869 minus 119872 sdot (n sdot k

119904)] 119889119904

1015840

(9)

Similarly when the sea surface is illuminated by the inci-dent field Hinc

= y119867inc exp[119894997888

k119894sdot

997888r119900] (VV polarization) the

induced electric and magnetic currents can be written as

J = 119867

inc exp [119894997888

k119894sdot

997888r119900] sdot (1 + 119877TM) xloc

M = 119867

inc exp [119894997888

k119894sdot

997888r119900] sdot (1 minus 119877TM) sdot cos 120579

incloc sdot yloc

(10)

the scattered fields are

119867

sca= minus

radic

119896

8120587119903

exp (minus1198941205874

) exp (119894119896119903)

sdot int

119904

exp (minus119894997888

k119904sdot


119904)] 119889119904

1015840

(11)

The analytical expression of scattering coefficient can bedefined as

120590 = lim119903rarrinfin

2120587119903

1003816

1003816

1003816

1003816

119864

sca10038161003816

1003816

1003816

2

1003816

1003816

1003816

1003816

1003816

119864

inc100381610038161003816

1003816

1003816

2

119871

(12)

wherein 119871 is the length of the sea surfaceThe Doppler spectrum is a power spectral density of the

random time varying complex amplitude of the sea surfacescattering field to evaluate it we used a standard spectralestimation technique Thus the expression of the Dopplerspectrum is defined as

119878 (119891) =

1

119879

⟨

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

int

119879

0

119864

sca(119905) exp (1198942120587119891119905) 119889119905

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

2

⟩ (13)

where the angular bracket ⟨sdot⟩ stands for the averaging overall the surface realizations 119879 is the evolution time of the timevarying sea surface

22 The Simplified Sea Surface Model In this simplified seasurface model each segment on the sea surface is muchlonger than thewavelength of the incident wave All segmentsare locally approximated by tilted facet (In the followingpart of the paper the ldquolarge segmentsrdquo are all named asldquofacetrdquo) These facets are centered on the grid points that aregeometrically described by a discrete set of 119891

119892(119909 119905) 119891

119892(119909 119905)

can be regarded as the gravity wave The sinusoidal capillarywaves are added on the planar facetThe surface profilewithina planar facet can be represented as

119891 (119909 119905) = 119891

119892(119909

119900 119905) + 119891

1015840

119892(119909 minus 119909

119900) + 119861 (119896

119888)

sdot cos (119896119888119909 + 120596

119888119905 + 120593)

(14)

where 119891 is the height of the simplified sea surface 119861(sdot) repre-sents the amplitude of the superimposed capillary wave and119896

119888is the spatially short wavenumber which is supposed to

satisfy the resonant condition of Bragg wavesThe definitionsof 119861(119896

119888) and 119896

119888will be discussed in the following in detail

120596

119888represents the circle frequency 120596

119888

2= 119892119896

3

119888119896

3

119898 and 119892 =

98ms 119896119898asymp 370 radm 120593 is the random phase of the ripple


One facet

x

yf(x)

119844i

119844iz

120579i

119847loc

119857loc

(a)

One facet

x

yf(x)

119844i

119844iz

120579i

119847loc

119857loc

(b)

Figure 2 Configuration of sea surface (a) sea profile constructed by large facet simplified sea surface without capillary wave (b) thesimplified sea surface capillary waves are added

component 119891119892(119909

119900 119905) is the height of the gravity wave in the

middle point of the facet and 1198911015840119892is the slope of the facet

The configuration of the sea surface is depicted inFigure 2 Figure 2(a) shows the large-scale surface Each facetis flat in this model In Figure 2(b) the capillary waves areadded and each facet is modified by sinusoidal waves

Since the sinusoidal ripple is superimposed at micro-scopic level and is relevant for the Bragg resonant scatteringmechanism we assume that the echo waves are reflectedby the wave crest of the resonant sinusoidal ripples AsFigure 3(a) shows q = k

119904minus k119894 and q

119897is its projection on

the plane facet 120582119897is the wavelength of the resonant sinusoidal

ripples on the plane facet 120582119888is the corresponding wavelength

on the 119909-axis To meet the resonant condition 120582119897is set as

120582

119897=

120587

119896 sin 1205791015840119894

= 120582

119888radic

1 + (119891

1015840

119892)

2

(15)

The wavenumber of the sinusoidal ripple is 119896119888 which

can be written as 119896119888= 2120587120582

119888 According to (15) 119896

119888can be

determined as

119896

119888=

120587

119896 sin 1205791015840119894radic1 + (119891

1015840

119892)

2 (16)

and the amplitude of the ripple in (14) could be expressedby 119861(119896

119888) = 120587radic2119878(119896119888

)Δ119909

119892 119878(119896119888) is the capillary spectrum

located in the higher part of the sea spectrum and Δ119909119892is the

length of the facetFigure 3(b) shows that the facet is divided into 119873

119901

periods 119873119901= int(Δ119909

119892120582

119888) The currents in the first period

can be expressed as (5) (6) and (10) Each period is dividedinto small segments The lengths of the small segments areset as 18sim110 wavelength of the incident wave to meet thedivision criterion For the case ofHHpolarization bymovingthe exp(minus119894

997888

k119894sdot

997888r119900) term from (5) and (6) to (8) and moving

the n sdot k119904term from (8) to (5) and (6) 1120578 in (5) is removed

since it will counteract with the 120578 in (7) Then the currentscan be rewritten as

119869

119898119899= 119864

inc cos 120579incloc sdot (1 minus 119877TE) sdot 119889119897

119872

119898119899= 119864

inc(1 + 119877TE) sdot (n sdot k119904) sdot 119889119897

(17)

where the subscripts119898 and 119899 indicate the119898th segment in oneperiod and the 119899th period on one facet respectively Consider119889119897 = Δ119909

119888radic1 + (119891

1015840

119888)

2 1198911015840119888 which is the slope of each segment

is periodical on one facet 1198911015840119888is the differential of 119891(119909 119905) On

each facet 1198911015840119888can be expressed as

119891

1015840

119888= 119891

1015840(119909 119905) = 119891

1015840

119892minus 119861 (119896

119888) 119896

119888sin (119896119888119909 + 120596

119888119905) (18)

the scattered field from one facet can be expressed as

119864

sca= coef sdot

119873119904

sum

119898=1

119873119901

sum

119899=1

exp (minus119894997888q sdot

997888r119898119899) (119869

119898119899minus119872

119898119899)

(19)

where119873119904is the number of the segments in one period119873

119904=

int(120582119888Δ119909

119888) and

coef = minusradic

119896

8120587119903

exp(minus1198941205874

) exp (1198941198960119903)

(20)

Since 119869119898119899

and119872119898119899

vary periodically on one facet (19) can berewritten as

119864

sca= coef

sdot

119873119904

sum

119898=1

119873119901

sum

119899=1

(119869

119898119899minus119872

119898119899) exp (minus119894997888q sdot

997888r119898119899)

= coef

sdot

119873119904

sum

119898=1

[

[

(119869

1198981minus119872

1198981)

119873119901

sum

119899=1


997888r119898119899)

]

]

= coef119873119904

sum

119898=1

[

[


997888r1198981) (119869

1198981minus119872

1198981)

sdot

119873119901

sum

119899=1

exp [minus119894997888q (119899 minus 1) Δ

997888r ]]]

(21)


q119844i

119844s

120579998400i

119847loc

q l

120582l

120582c

(a)

119844i119844s

120579998400i

119847loc

Npth

period

middot middot middot2nd

period

1stperiod

(b)

Figure 3 The ripples on one facet

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

HH polarization

Simplified sea surfaceRealistic sea surfaceSimplified sea without capillary wave

X-bandf = 10GHzU = 5ms

120579i (deg)

120590(d

B)

(a)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

VV polarizationX-band


f = 10GHzU = 5ms

120590(d

B)

120579i (deg)

(b)

Figure 4 Backscattering coefficient from sea surface (a) HH and (b) VV

where Δ997888r is fixed on one facet and it can be expressed asΔ

997888r = 120582

119888x + 120582

119888119891

1015840

119892z Thus the scattered field from one facet

can be obtained by

119864

sca= coef

119873119904

sum

119898=1


997888r1198981) (119869

1198981minus119872

1198981)

sdot exp[minus119894 (119873119901minus 1) sdot

997888q sdot Δ

997888r2

]

sdot

sin (997888q sdot Δ

997888r sdot 119873

1199012)


997888r 2)

(22)

Summarizing the scattered field from all the facets we canobtain the total field from the sea surface

3 Numerical Simulations and Discussions

To validate the proposed simplified sea surface model sim-ulation results obtained from the simplified sea surface therealistic sea surface and the simplified sea without capillarywave will be shown and compared in the following

Figure 4 shows the comparison of the presented sim-plified model with the realistic sea surface for the angulardistribution of the backscattering coefficient (BSC) Thegeometry of observation is monostatic Both HH (a) and VV(b) polarization are taken into consideration The results are


minus100 minus80 minus60 minus40 minus20 0 20 40 60 80 10000

0102030405060708091011

HH polarizationX-band

Nor

mal

ized

Dop

pler

spec

trum

Frequency (Hz)


f = 10GHz120579i = 30∘

(a)



Nor

mal

ized

Dop

pler

spec

trum

Frequency (Hz)

HH polarizationX-bandf = 10GHz120579i = 30∘

(b)

Figure 5 Doppler spectra from time varying sea surface (a) HH and (b) VV

obtained by averaging 50 surface realizations The surfacesare generated using Monte Carlo [2] methods with 50 setsof Gaussian random numbers whose mean value is 0 andvariance is 1 The random numbers are called using the IMSLfunction library with 50 different seeds The parameters inthe simulation are as follows the frequency of the incidentwave is at the X-band (10GHz) the incident angle variesfrom 0∘ to 70∘ the length of the sea surface is 24576m thelength of each facet is set to 192m the wind speed at 10mabove the mean sea level is 119880

10= 5ms and the relative

permittivity of the sea water is calculated as 120576 = (5618 3442)

according to the Klein model [10] at 20∘C and 325 ofsalinity It is obvious that the BSCobtained from the proposedsea surface model and the realistic model are in fairly goodagreement for the whole backscattering region for bothpolarizations The BSC obtained by a simplified sea surfacewithout capillary wave have an obvious discrepancy withthe other two curves Thus the presented model is reliablefor predicting the backscattering coefficient We can alsoconclude that the resonant capillary wave plays an importantrole in the scattering result The simulation time obtainedfrom the simplified sea surface and realistic sea surface isalso presented in Table 1 The superiority of the simplifiedsea surface in terms of computation time is obvious Allresults are obtained on a computer with a 293GHz processor(Intel Core i3 CPU) 345GBmemory and Visual Fortran 65compiler All the comparisons of scattering results and thesimulating time indicate that the proposed sea surface modelis an accurate and efficient model

Subsequently the Doppler behaviors are observed withincident angle 120579

119894= 30

∘ and frequency 119891 = 100GHz Theresults obtained from the proposed simplified sea surface arecompared with the result from the realistic sea surface andfrom the simplified sea without capillary wave (Figure 5)These results are averaged over 50 samples and normalized

Table 1 Comparison of simulating time by two methods

Backscattering coefficient Doppler spectrumModel Time (s) Model Time (s)Simplified sea surface 18525 Simplified sea surface 478938Realistic sea surface 62323 Realistic sea surface 1089994

by their respective maximum values For the time-evolvingsimulations the time step Δ119905 is selected as 0005 s and 1024steps are selected for the sea surface realization Thus theevolution time 119879 is 512 s The other parameters are the sameas those given in Figure 4 The two curves obtained fromthe proposed sea surface model and the realistic model wellagree for both polarizations Doppler spectrum obtainedfrom the simplified sea without capillary waves cannot reflectthe Doppler characteristics of the sea surface It should benoticed that the Doppler shifts are almost the same for bothmethods whereas theDoppler width of the proposedmethodis a little smaller than the realistic sea surface It is mainlycaused by the fact that the other components of the capillarywaves are supposed to be absent After all we conclude thatthe simplified sea surface model is accurate enough andconvincible to describe the scattering characteristics of thesea

The simulation times of Doppler spectrum of both seasurface models are presented in Table 1 The simplified seasurface model takes much less time than the realistic seasurface especially in the Doppler spectrum simulation

In Figure 6 the Doppler spectra are illustrated for windspeed of 5ms at different incident angles The other param-eters in calculation are the same as those given in Figure 5With the incident angle increasing the widths of the Dopplerspectra first become broader and then shrink Figure 7 showsthe Doppler spectrum at different wind speed The incident


Frequency (Hz)minus100 minus80 minus60 minus40 minus20 0 20 40 60

00

01

02

03

04

05

06

07

08

09

10

11

Nor

mal

ized

Dop

pler

spec

trum

HH polarizationX-bandf = 10GHzU = 5ms

120579i = 10∘

120579i = 30∘120579i = 50∘

120579i = 60∘

Figure 6 Doppler spectra of different incident angle

Frequency (Hz)



Nor

mal

ized

Dop

pler

spec

trum

f = 10GHz120579i = 45∘

U = 3msU = 5msU = 7ms

Figure 7 Doppler spectra at different wind speed

angle is set as 120579119894= 45

∘ We can find that with the increasingwind speeds the velocities of the surface water and theorbital motion for the large-scale sea wave increase andthere has been a gradual increase for the Doppler spectrumfrequency shift which is coincident with themeasuring resultby Rozenberg et al [11] On the other hand the Dopplerspectrum broadens with the wind speed increasing in theroughness of the sea surface Thus the backscattering energywill be distributed over a wide region of the frequencydomain which has been discussed in [12 13] All these resultscan demonstrate that the simplified sea surface model is validto analyze the Doppler characteristic of the time varying seasurface

4 Conclusions

The PO method is combined with the simplified sea surfacemodel to calculate the backscattering coefficient and Dopplerspectrum from time varying sea surface With the simplifiedmodel the efficiency of POmethod is promotedThe charac-teristic of the Doppler spectrum of time varying sea surface isalso investigated by the proposedmodel Some shortcomingsstill exist in thismethod Since the coupling between differentsegments are not taken into consideration this method islimited to deal with the seawhose sea state is low ormoderateAnd this method will be inaccurate if the incident angle islargeThe 2D problem is not considered in this paperWe willpay more attention to these remaining problems in the futurestudy

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural ScienceFoundation for Distinguished Young Scholars of China(Grant no 61225002) the Specialized Research Fund forthe Doctoral Program of Higher Education (Grant no20100203110016) and the Fundamental Research Funds forthe Central Universities (Grant no K5051007001)

References

[1] F T Ulaby R K Moore and A K Fung Microwave RemoteSesing (Active and Passive) Addison Wesley New York NYUSA 1982

[2] L Tsang and J A Kong Scattering of Electromagnetic Wavesvol 2 John Wiley amp Sons 2001

[3] L Kuang and Y Q Jin ldquoBistatic scattering from a three-dimensional object over a randomly rough surface using theFDTD algorithmrdquo IEEE Transactions on Antennas and Propa-gation vol 55 no 8 pp 2302ndash2312 2007

[4] S H Lou L Tsang and C H Chan ldquoApplication of the finiteelement method to Monte Carlo simulations of scattering ofwaves by random rough surfaces penetrable caserdquo Waves inRandom Media vol 1 no 4 pp 287ndash307 1991

[5] Y W Wei L X Guo A Q Wang and Z S Wu ldquoApplicationof multiregion model to EM scattering from a dielectric roughsurface with or without a target above itrdquo IEEE Transactions onAntennas and Propagation vol 61 no 11 pp 5607ndash5620 2013

[6] T M Elfouhaily and C-A Guerin ldquoA critical survey of approx-imate scattering wave theories from random rough surfacesrdquoWaves in Random Media vol 14 no 4 pp R1ndashR40 2004

[7] W B Gordon ldquoHigh frequency approximation to the physicaloptics scattering integralrdquo IEEE Transactions on Antennas andPropagation vol 42 no 3 pp 427ndash432 1994

[8] J T Johnson ldquoOn the geometrical optics and physical opticsapproximations for scattering from exponentially correlatedsurfacesrdquo IEEE Transactions on Antennas and Propagation vol45 pp 2619ndash2629 1994


[9] M Zhang H Chen and H-C Yin ldquoFacet-based investigationon em scattering from electrically large sea surface with two-scale profiles theoretical modelrdquo IEEE Transactions on Geo-science and Remote Sensing vol 49 no 6 pp 1967ndash1975 2011

[10] L A Klein and C T Swift ldquoAn improved model for thedielectric constant of sea water at microwave frequenciesrdquo IEEETransactions on Antennas and Propagation vol AP-25 no 1 pp104ndash111 1977

[11] A D Rozenberg D C Quigley and W Kendall MelvilleldquoLaboratory study of polarized scattering by surface wavesat grazing incidence I wind wavesrdquo IEEE Transactions onGeoscience and Remote Sensing vol 33 no 4 pp 1037ndash10461995

[12] J T Johnson J V Toporkov and G S Brown ldquoA numeri-cal study of backscattering from time-evolving sea surfacescomparison of hydrodynamic modelsrdquo IEEE Transactions onGeoscience and Remote Sensing vol 39 no 11 pp 2411ndash24202001

[13] J V Toporkov and G S Brown ldquoNumerical study of theextended Kirchhoff approach and the lowest order small slopeapproximation for scattering from ocean-like surfaces doppleranalysisrdquo IEEE Transactions on Antennas and Propagation vol50 no 4 pp 417ndash425 2002

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



x

yOne

segmentf(x)

119841i

vi

119844i

119844iz

120579i

120579998400i

119847loc

yloc119857loc

(a)

119841i

vi

119844i

119847loc

119844i

119844ref

x loc

119812inc

119815inc119815ref

119812ref

(b)

Figure 1 The illustration of the scattering problem from a sea surface (a) the global coordinate system and (b) the local coordinate system

the resonant condition of Bragg waves which are predom-inant in the scattered short wave component In this waywe substitute the simplified sea surface for the realistic seasurface approximately Since the capillary wave is periodicalwithin one facet the analytical expression of the inducedcurrents can be given after we get the currents on the firstperiod of the sinusoidal wave

This paper is organized as follows In Section 2 the POmethod is introduced and applied on the simplified seasurface model All the formulas are derived and all theexpressions are given in this part both horizontal (HH)polarization and vertical (VV) polarizations are consideredSeveral numerical simulations are exhibited in Section 3 toshow the validity and efficiency of the new model comparedwith the realistic sea surface Then this model is used toinvestigate the characteristic of the Doppler spectrum oftime varying sea surface Section 4 ends with a summaryof the new model and a proposition for further pertinentinvestigation

2 Formula

21 The Physical Optics Formulation The initial point ofphysical optics is the surface currents produced by an incom-ing electromagnetic wave (Einc

Hinc) Considering a 1D seasurface the induced electric currents J andmagnetic currentsM on each segment (the length of each segment is set as18sim110 wavelength of the incident wave to meet the divisioncriterion) are given by

J = n timesHM = E times n (1)

where n is the unit normal vector of the surface and rindicates the position of each segment E and H are respec-tively the total electric and magnetic fields at the surface ForTE case the horizontal polarization vector of the incidentwave h

119894is along y k

119894is the wavenumber of the incident

wave and the vertical polarization vector is k119894=

k119894times

h119894

The scattering problems in the global coordinate system andthe local coordinate system are shown in Figures 1(a) and1(b) respectively Given the slop of one segment the local

coordinate system can be built as xloc yloc zloc and eachcoordinate component can be expressed as

zloc = n

yloc = y = h119894

xloc = yloc times zloc

(2)

Let a plane wave illuminate on the rough sea surface thenthe incident field on the surface can be given as

Einc= y119864inc exp [119894

997888

k119894sdot

997888r119900]

Hinc=

k119894119864

inc exp [119894997888

k119894sdot

997888r119900]

120578

(3)

where 997888r119900is the position of each surface segment and 120578 is the

wave impedance in the free spaceInserting the TE reflection coefficient119877TE the electricEref

and magneticHref reflected fields are

Eref=

h119894119877TE119864

inc exp [119894997888

k119894sdot

997888r119900]

Href=

(

kref times h119894) 119864

inc exp [119894997888

k119894sdot

997888r119900]

120578

(4)

where kref =

k119894minus 2n(k

119894sdot n) Adding the contributions of

incident and reflected field the total electric and magneticfields can be written as a function of the incident electricfield the reflection coefficients and the intrinsic impedanceof the first medium Then going back to the boundary




)

=

119864

inc exp [119894997888

k119894sdot

997888r119900]

120578


119894)]

=

119864

inc exp [119894997888

k119894sdot

997888r119900]

120578


(5)


) times n

= 119864

inc exp [119894997888

k119894sdot


= 119864

inc exp [119894997888

k119894sdot

997888r119900] (1 + 119877TE) sdot xloc

(6)


119864

sca= Esca

sdot

h119894

= 119894119896120578int

119904

119869119866 (r r1015840) 1198891199041015840 + int119904

119872(n1015840 sdot nabla119866 (r r1015840)) 1198891199041015840(7)


119866(r r1015840)1003816100381610038161003816

1003816|r|rarrinfin

=

119894

4

radic

2

120587119896

0119903

exp(minus1198941205874

)


997888

k119904sdot

997888r119900]

n1015840 sdot nabla1015840119866(r r1015840)1003816100381610038161003816

1003816|r|rarrinfin

=

119894

4

radic

2

120587119896

0119903

exp(minus1198941205874

)


119904) exp [minus119894

997888

k119904sdot

997888r119900]

(8)

where997888


997888

k119904

=

119896

0(sin 120579119904x + cos 120579


119864

sca= minus

radic

119896

8120587119903

exp(minus1198941205874

) exp (1198941198960119903)

sdot int

119904

exp (minus119894997888

k119904sdot


119904)] 119889119904

1015840

(9)


= y119867inc exp[119894997888

k119894sdot



J = 119867

inc exp [119894997888

k119894sdot

997888r119900] sdot (1 + 119877TM) xloc

M = 119867

inc exp [119894997888

k119894sdot


incloc sdot yloc

(10)


119867

sca= minus

radic

119896

8120587119903

exp (minus1198941205874

) exp (119894119896119903)

sdot int

119904

exp (minus119894997888

k119904sdot


119904)] 119889119904

1015840

(11)



2120587119903

1003816

1003816

1003816

1003816

119864

sca10038161003816

1003816

1003816

2

1003816

1003816

1003816

1003816

1003816

119864

inc100381610038161003816

1003816

1003816

2

119871

(12)



119878 (119891) =

1

119879

⟨

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

int

119879

0

119864

sca(119905) exp (1198942120587119891119905) 119889119905

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

2

⟩ (13)



119892(119909 119905) 119891

119892(119909 119905)


119891 (119909 119905) = 119891

119892(119909

119900 119905) + 119891

1015840

119892(119909 minus 119909

119900) + 119861 (119896

119888)

sdot cos (119896119888119909 + 120596

119888119905 + 120593)

(14)




119888) and 119896


120596


119888

2= 119892119896

3

119888119896

3

119898 and 119892 =



One facet

x

yf(x)

119844i

119844iz

120579i

119847loc

119857loc

(a)

One facet

x

yf(x)

119844i

119844iz

120579i

119847loc

119857loc

(b)


component 119891119892(119909










120582

119897=

120587

119896 sin 1205791015840119894

= 120582

119888radic

1 + (119891

1015840

119892)

2

(15)


can be written as 119896119888= 2120587120582

119888 According to (15) 119896

119888can be

determined as

119896

119888=

120587

119896 sin 1205791015840119894radic1 + (119891

1015840

119892)

2 (16)


119888) = 120587radic2119878(119896119888

)Δ119909




119901

periods 119873119901= int(Δ119909

119892120582



997888

k119894sdot




119869

119898119899= 119864


119872

119898119899= 119864


(17)


119888radic1 + (119891

1015840

119888)




119891

1015840

119888= 119891

1015840(119909 119905) = 119891

1015840

119892minus 119861 (119896

119888) 119896

119888sin (119896119888119909 + 120596

119888119905) (18)


119864

sca= coef sdot

119873119904

sum

119898=1

119873119901

sum

119899=1


997888r119898119899) (119869

119898119899minus119872

119898119899)

(19)


119904=

int(120582119888Δ119909

119888) and

coef = minusradic

119896

8120587119903

exp(minus1198941205874

) exp (1198941198960119903)

(20)

Since 119869119898119899

and119872119898119899


119864

sca= coef

sdot

119873119904

sum

119898=1

119873119901

sum

119899=1

(119869

119898119899minus119872

119898119899) exp (minus119894997888q sdot

997888r119898119899)

= coef

sdot

119873119904

sum

119898=1

[

[

(119869

1198981minus119872

1198981)

119873119901

sum

119899=1


997888r119898119899)

]

]

= coef119873119904

sum

119898=1

[

[


997888r1198981) (119869

1198981minus119872

1198981)

sdot

119873119901

sum

119899=1


997888r ]]]

(21)


q119844i

119844s

120579998400i

119847loc

q l

120582l

120582c

(a)

119844i119844s

120579998400i

119847loc

Npth

period


period

1stperiod

(b)


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

HH polarization



120579i (deg)

120590(d

B)

(a)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10



f = 10GHzU = 5ms

120590(d

B)

120579i (deg)

(b)



997888r = 120582

119888x + 120582

119888119891

1015840


can be obtained by

119864

sca= coef

119873119904

sum

119898=1


997888r1198981) (119869

1198981minus119872

1198981)


997888q sdot Δ

997888r2

]

sdot


997888r sdot 119873

1199012)


997888r 2)

(22)







0102030405060708091011


Nor

mal

ized

Dop

pler

spec

trum

Frequency (Hz)


f = 10GHz120579i = 30∘

(a)



Nor

mal

ized

Dop

pler

spec

trum

Frequency (Hz)


(b)







119894= 30









00

01

02

03

04

05

06

07

08

09

10

11

Nor

mal

ized

Dop

pler

spec

trum


120579i = 10∘

120579i = 30∘120579i = 50∘

120579i = 60∘


Frequency (Hz)



Nor

mal

ized

Dop

pler

spec

trum

f = 10GHz120579i = 45∘



angle is set as 120579119894= 45


4 Conclusions




Acknowledgments


References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












)

=

119864

inc exp [119894997888

k119894sdot

997888r119900]

120578


119894)]

=

119864

inc exp [119894997888

k119894sdot

997888r119900]

120578


(5)


) times n

= 119864

inc exp [119894997888

k119894sdot


= 119864

inc exp [119894997888

k119894sdot

997888r119900] (1 + 119877TE) sdot xloc

(6)


119864

sca= Esca

sdot

h119894

= 119894119896120578int

119904

119869119866 (r r1015840) 1198891199041015840 + int119904

119872(n1015840 sdot nabla119866 (r r1015840)) 1198891199041015840(7)


119866(r r1015840)1003816100381610038161003816

1003816|r|rarrinfin

=

119894

4

radic

2

120587119896

0119903

exp(minus1198941205874

)


997888

k119904sdot

997888r119900]

n1015840 sdot nabla1015840119866(r r1015840)1003816100381610038161003816

1003816|r|rarrinfin

=

119894

4

radic

2

120587119896

0119903

exp(minus1198941205874

)


119904) exp [minus119894

997888

k119904sdot

997888r119900]

(8)

where997888


997888

k119904

=

119896

0(sin 120579119904x + cos 120579


119864

sca= minus

radic

119896

8120587119903

exp(minus1198941205874

) exp (1198941198960119903)

sdot int

119904

exp (minus119894997888

k119904sdot


119904)] 119889119904

1015840

(9)


= y119867inc exp[119894997888

k119894sdot



J = 119867

inc exp [119894997888

k119894sdot

997888r119900] sdot (1 + 119877TM) xloc

M = 119867

inc exp [119894997888

k119894sdot


incloc sdot yloc

(10)


119867

sca= minus

radic

119896

8120587119903

exp (minus1198941205874

) exp (119894119896119903)

sdot int

119904

exp (minus119894997888

k119904sdot


119904)] 119889119904

1015840

(11)



2120587119903

1003816

1003816

1003816

1003816

119864

sca10038161003816

1003816

1003816

2

1003816

1003816

1003816

1003816

1003816

119864

inc100381610038161003816

1003816

1003816

2

119871

(12)



119878 (119891) =

1

119879

⟨

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

int

119879

0

119864

sca(119905) exp (1198942120587119891119905) 119889119905

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

2

⟩ (13)



119892(119909 119905) 119891

119892(119909 119905)


119891 (119909 119905) = 119891

119892(119909

119900 119905) + 119891

1015840

119892(119909 minus 119909

119900) + 119861 (119896

119888)

sdot cos (119896119888119909 + 120596

119888119905 + 120593)

(14)




119888) and 119896


120596


119888

2= 119892119896

3

119888119896

3

119898 and 119892 =



One facet

x

yf(x)

119844i

119844iz

120579i

119847loc

119857loc

(a)

One facet

x

yf(x)

119844i

119844iz

120579i

119847loc

119857loc

(b)


component 119891119892(119909










120582

119897=

120587

119896 sin 1205791015840119894

= 120582

119888radic

1 + (119891

1015840

119892)

2

(15)


can be written as 119896119888= 2120587120582

119888 According to (15) 119896

119888can be

determined as

119896

119888=

120587

119896 sin 1205791015840119894radic1 + (119891

1015840

119892)

2 (16)


119888) = 120587radic2119878(119896119888

)Δ119909




119901

periods 119873119901= int(Δ119909

119892120582



997888

k119894sdot




119869

119898119899= 119864


119872

119898119899= 119864


(17)


119888radic1 + (119891

1015840

119888)




119891

1015840

119888= 119891

1015840(119909 119905) = 119891

1015840

119892minus 119861 (119896

119888) 119896

119888sin (119896119888119909 + 120596

119888119905) (18)


119864

sca= coef sdot

119873119904

sum

119898=1

119873119901

sum

119899=1


997888r119898119899) (119869

119898119899minus119872

119898119899)

(19)


119904=

int(120582119888Δ119909

119888) and

coef = minusradic

119896

8120587119903

exp(minus1198941205874

) exp (1198941198960119903)

(20)

Since 119869119898119899

and119872119898119899


119864

sca= coef

sdot

119873119904

sum

119898=1

119873119901

sum

119899=1

(119869

119898119899minus119872

119898119899) exp (minus119894997888q sdot

997888r119898119899)

= coef

sdot

119873119904

sum

119898=1

[

[

(119869

1198981minus119872

1198981)

119873119901

sum

119899=1


997888r119898119899)

]

]

= coef119873119904

sum

119898=1

[

[


997888r1198981) (119869

1198981minus119872

1198981)

sdot

119873119901

sum

119899=1


997888r ]]]

(21)


q119844i

119844s

120579998400i

119847loc

q l

120582l

120582c

(a)

119844i119844s

120579998400i

119847loc

Npth

period


period

1stperiod

(b)


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

HH polarization



120579i (deg)

120590(d

B)

(a)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10



f = 10GHzU = 5ms

120590(d

B)

120579i (deg)

(b)



997888r = 120582

119888x + 120582

119888119891

1015840


can be obtained by

119864

sca= coef

119873119904

sum

119898=1


997888r1198981) (119869

1198981minus119872

1198981)


997888q sdot Δ

997888r2

]

sdot


997888r sdot 119873

1199012)


997888r 2)

(22)







0102030405060708091011


Nor

mal

ized

Dop

pler

spec

trum

Frequency (Hz)


f = 10GHz120579i = 30∘

(a)



Nor

mal

ized

Dop

pler

spec

trum

Frequency (Hz)


(b)







119894= 30









00

01

02

03

04

05

06

07

08

09

10

11

Nor

mal

ized

Dop

pler

spec

trum


120579i = 10∘

120579i = 30∘120579i = 50∘

120579i = 60∘


Frequency (Hz)



Nor

mal

ized

Dop

pler

spec

trum

f = 10GHz120579i = 45∘



angle is set as 120579119894= 45


4 Conclusions




Acknowledgments


References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










One facet

x

yf(x)

119844i

119844iz

120579i

119847loc

119857loc

(a)

One facet

x

yf(x)

119844i

119844iz

120579i

119847loc

119857loc

(b)


component 119891119892(119909










120582

119897=

120587

119896 sin 1205791015840119894

= 120582

119888radic

1 + (119891

1015840

119892)

2

(15)


can be written as 119896119888= 2120587120582

119888 According to (15) 119896

119888can be

determined as

119896

119888=

120587

119896 sin 1205791015840119894radic1 + (119891

1015840

119892)

2 (16)


119888) = 120587radic2119878(119896119888

)Δ119909




119901

periods 119873119901= int(Δ119909

119892120582



997888

k119894sdot




119869

119898119899= 119864


119872

119898119899= 119864


(17)


119888radic1 + (119891

1015840

119888)




119891

1015840

119888= 119891

1015840(119909 119905) = 119891

1015840

119892minus 119861 (119896

119888) 119896

119888sin (119896119888119909 + 120596

119888119905) (18)


119864

sca= coef sdot

119873119904

sum

119898=1

119873119901

sum

119899=1


997888r119898119899) (119869

119898119899minus119872

119898119899)

(19)


119904=

int(120582119888Δ119909

119888) and

coef = minusradic

119896

8120587119903

exp(minus1198941205874

) exp (1198941198960119903)

(20)

Since 119869119898119899

and119872119898119899


119864

sca= coef

sdot

119873119904

sum

119898=1

119873119901

sum

119899=1

(119869

119898119899minus119872

119898119899) exp (minus119894997888q sdot

997888r119898119899)

= coef

sdot

119873119904

sum

119898=1

[

[

(119869

1198981minus119872

1198981)

119873119901

sum

119899=1


997888r119898119899)

]

]

= coef119873119904

sum

119898=1

[

[


997888r1198981) (119869

1198981minus119872

1198981)

sdot

119873119901

sum

119899=1


997888r ]]]

(21)


q119844i

119844s

120579998400i

119847loc

q l

120582l

120582c

(a)

119844i119844s

120579998400i

119847loc

Npth

period


period

1stperiod

(b)


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

HH polarization



120579i (deg)

120590(d

B)

(a)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10



f = 10GHzU = 5ms

120590(d

B)

120579i (deg)

(b)



997888r = 120582

119888x + 120582

119888119891

1015840


can be obtained by

119864

sca= coef

119873119904

sum

119898=1


997888r1198981) (119869

1198981minus119872

1198981)


997888q sdot Δ

997888r2

]

sdot


997888r sdot 119873

1199012)


997888r 2)

(22)







0102030405060708091011


Nor

mal

ized

Dop

pler

spec

trum

Frequency (Hz)


f = 10GHz120579i = 30∘

(a)



Nor

mal

ized

Dop

pler

spec

trum

Frequency (Hz)


(b)







119894= 30









00

01

02

03

04

05

06

07

08

09

10

11

Nor

mal

ized

Dop

pler

spec

trum


120579i = 10∘

120579i = 30∘120579i = 50∘

120579i = 60∘


Frequency (Hz)



Nor

mal

ized

Dop

pler

spec

trum

f = 10GHz120579i = 45∘



angle is set as 120579119894= 45


4 Conclusions




Acknowledgments


References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










q119844i

119844s

120579998400i

119847loc

q l

120582l

120582c

(a)

119844i119844s

120579998400i

119847loc

Npth

period


period

1stperiod

(b)


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

HH polarization



120579i (deg)

120590(d

B)

(a)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10



f = 10GHzU = 5ms

120590(d

B)

120579i (deg)

(b)



997888r = 120582

119888x + 120582

119888119891

1015840


can be obtained by

119864

sca= coef

119873119904

sum

119898=1


997888r1198981) (119869

1198981minus119872

1198981)


997888q sdot Δ

997888r2

]

sdot


997888r sdot 119873

1199012)


997888r 2)

(22)







0102030405060708091011


Nor

mal

ized

Dop

pler

spec

trum

Frequency (Hz)


f = 10GHz120579i = 30∘

(a)



Nor

mal

ized

Dop

pler

spec

trum

Frequency (Hz)


(b)







119894= 30









00

01

02

03

04

05

06

07

08

09

10

11

Nor

mal

ized

Dop

pler

spec

trum


120579i = 10∘

120579i = 30∘120579i = 50∘

120579i = 60∘


Frequency (Hz)



Nor

mal

ized

Dop

pler

spec

trum

f = 10GHz120579i = 45∘



angle is set as 120579119894= 45


4 Conclusions




Acknowledgments


References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0102030405060708091011


Nor

mal

ized

Dop

pler

spec

trum

Frequency (Hz)


f = 10GHz120579i = 30∘

(a)



Nor

mal

ized

Dop

pler

spec

trum

Frequency (Hz)


(b)







119894= 30









00

01

02

03

04

05

06

07

08

09

10

11

Nor

mal

ized

Dop

pler

spec

trum


120579i = 10∘

120579i = 30∘120579i = 50∘

120579i = 60∘


Frequency (Hz)



Nor

mal

ized

Dop

pler

spec

trum

f = 10GHz120579i = 45∘



angle is set as 120579119894= 45


4 Conclusions




Acknowledgments


References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











00

01

02

03

04

05

06

07

08

09

10

11

Nor

mal

ized

Dop

pler

spec

trum


120579i = 10∘

120579i = 30∘120579i = 50∘

120579i = 60∘


Frequency (Hz)



Nor

mal

ized

Dop

pler

spec

trum

f = 10GHz120579i = 45∘



angle is set as 120579119894= 45


4 Conclusions




Acknowledgments


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











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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