research article a novel 3d imaging method for airborne downward-looking sparse array...

Research ArticleA Novel 3D Imaging Method for Airborne Downward-LookingSparse Array SAR Based on Special Squint Model

Xiaozhen Ren Yao Qin and Lihong Qiao

College of Information Science and Engineering Henan University of Technology Zhengzhou 450001 China

Correspondence should be addressed to Xiaozhen Ren rxz235163com

Received 30 June 2014 Revised 4 December 2014 Accepted 18 December 2014 Published 31 December 2014

Academic Editor Francisco Falcone

Copyright copy 2014 Xiaozhen Ren 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

Three-dimensional (3D) imaging technology based on antenna array is one of the most important 3D synthetic aperture radar(SAR) high resolution imaging modes In this paper a novel 3D imaging method is proposed for airborne down-looking sparsearray SAR based on the imaging geometry and the characteristic of echo signal The key point of the proposed algorithm is theintroduction of a special squint model in cross track processing to obtain accurate focusing In this special squint model pointtargets with different cross track positions have different squint angles at the same range resolution cell which is different fromthe conventional squint SAR However after theory analysis and formulation deduction the imaging procedure can be processedwith the uniform reference function and the phase compensation factors and algorithm realization procedure are demonstratedin detail As the method requires only Fourier transform and multiplications and thus avoids interpolations it is computationallyefficient Simulations with point scatterers are used to validate the method

1 Introduction

Traditional synthetic aperture radar (SAR) utilizes pulsecompression and synthetic aperture technique to form highresolution two-dimensional (2D) images of the observed areawithweather independence and all-day operation capabilities[1 2] However traditional 2D SAR works in side-lookingor squint mode and cannot obtain the 3D information ofthe scene Compared with 2D SAR 3D SAR has distinctadvantages in ground topography and forest height esti-mation and for solving layover effect in natural or urbanareasMultibaseline SAR tomography is an advanced 3D SARimaging mode which exploits the multibaseline nature toallow distinguishing multiple scatterers at different heightswithin the same azimuth-range resolution cell Therefore itallows measuring the scattering distribution in the 3D spaceUnfortunately for the current SAR tomography it is almostimpossible to avoid an uneven track distribution in repeat-pass data acquisition which is just the main reason for thestrong ambiguity in height [3 4]

Down-looking array SAR is an innovative imagingmodewhich was first introduced by Gierull in 1999 [5] In

down-looking array SAR 3D resolutions are obtained byapplying pulse compression technique in range directionvirtual aperture synthesis principle in azimuth direction andlinear array aperture synthesis in cross track direction [6ndash8]Down-looking array SAR can overcome restrictions of shad-ing and lay over effects in side-looking SAR and also avoidthe height ambiguity problem in SAR tomography causedby the uneven track distribution However down-lookingarray SAR requires more AD sampling devices and datacollection devices for echo recording than in traditional SARTherefore sparse linear array with multiple-input-multiple-output (MIMO) mode is often used in down-looking arraySAR which helps to reduce the complexity of the system[9 10]

Due to its advanced performance down-looking arraySAR has attracted wide attention Two airborne down-looking array SAR systems DRIVE [11] and ARTINO [6]are being developed at ONERA and FGANFHR respectivelyKlare [6] transformed the bistatic configuration to monos-tatic one based on the equivalent phase center principle andproposed a 3D imaging algorithm for cross track imagingusing beamforming operation However range cell migration

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 982014 10 pageshttpdxdoiorg1011552014982014

2 International Journal of Antennas and Propagation

is not considered Du et al [12] made an improvement ofthe method proposed in [6]They added the range migrationcorrection in the imaging processing to expand the imagingscene width in the cross track direction However the digitalbeamforming technology used in the paper requires the pointby point calculations and inserting the point spread functionfor focusing the elevation angle Therefore this methodsuffers from severe computational complexity Giret et al[13] made some simplifications to the dual square roots withFresnel approximation and proposed an algorithm derivedfrom that of monostatic configuration A shortcoming of thealgorithm is the range error caused by Fresnel approximationSubsequently an accurate 3D focused algorithm based onrange migration algorithm (RMA) was presented in [7] butneeds 3D stolt interpolation In a recent work somemethodsbased on compressive sensing (CS) were applied to arraySAR imaging [14 15] providing an improved resolution incross track direction However these methods suffered fromsevere computation loads and were hard to realize real-timeimaging

The main topic of this paper is to introduce an accurateand efficient 3D imaging algorithm for down-looking sparsearray SAR According to the analysis of the spatial geom-etry and the echo signal model a special squint model isintroduced for cross track focusingThe phase compensationfactors and algorithm realization procedure are demonstratedin detail

The rest of the paper is organized as follows Section 2presents the geometry and principle of airborne down-looking sparse array SAR system In Section 3 a novel3D imaging algorithm for down-looking sparse array SARis described in detail The performance of the method isinvestigated in Section 4 Finally Section 5 gives a brief con-clusion

2 Geometrical Model of AirborneDown-Looking Sparse Array SAR

Figure 1 shows the geometry of down-looking sparse arraySAR [15]119909denotes the azimuth direction119910denotes the crosstrack direction and 119911 denotes the height directionThe radarplatform flies at height119867 along the 119909-axis with velocity VThesparse linear array is mounted in the cross track directionalong the wings and is designed based on ARTINO arraydistributionThat is because that ARTINO array distributionis quite easy for hardware implementation [16] The sparselinear array composes119872 transmitting antenna elements and119873 receiving antenna elements and works in the time divisionmode Each time only one transmitting antenna elementtransmits signal and all the receiving antenna elementsreceive echo simultaneously [6] The transmitting antennaelements work sequentially and an aperture synthesis periodis acquired until all the transmitting antenna elements haveworked for once

According to the principle of equivalent phase center thesparse linear array formed by the ARTINO array distributioncan be equivalent to a virtual uniform linear array [6 10]

yx

z

H

v

Figure 1 Geometry of down-looking sparse array SAR

T1 T2 middot middot middot R1 R2 middot middot middot RNminus1 RN middot middot middot TMminus1 TM

TR1R2

RN

Delay time

P

Figure 2 Transmitting and receiving order of down-looking sparsearray SAR

The virtual antenna array is composed of 119872119873 virtual ele-ments and each virtual element transmits and receives byitself These virtual antenna elements are uniformly dis-tributed along the wings and centered at the 119910-axis Eachvirtual antenna element is located at the mean position of areal single transmitting element and a real single receivingelement Figure 2 shows the transmitting and the receivingorder of each antenna element for a down-looking sparsearray SAR 119875 is the target in the imaging scene 119879

119894with

119894 isin [1119872] denotes the transmitting antenna element and119877119895with 119895 isin [1119873] denotes the receiving antenna element

As shown in Figure 2 the transmitting antenna elementswork sequentially The antenna element 119879

1transmits signal

first and all the receiving antenna elements receive echosimultaneously Then the antenna element 119879

2transmits sig-

nal and the procedure loops until the entire transmittingantenna elements have worked for once and then an aperturesynthesis period is acquired

International Journal of Antennas and Propagation 3

3 Three-Dimensional Imaging Algorithm forAirborne Down-Looking Sparse Array SAR

Based on the principle of equivalent phase center the sparselinear array formed by the time division mode is equalto a virtual linear array and each virtual antenna elementtransmits and receives signal by itself

31 Equivalent Phase Error Compensation Consider the dataacquisition shown in Figure 1 At the slow time 120591

119898 the

positions of the transmitting and receiving antenna elementsare given by (119909 119910

119879119894 119867) and (119909 119910

119877119895 119867) respectivelyThen the

transmitting and receiving paths 119877119879119894

and 119877119877119895

of the pointscatterer 119875 positioned at (119909

119901 119910119901 119911119901) are given by

119877119879119894= radic(119909 minus 119909

119901)

2

+ (119910119879119894minus 119910119901)

2

+ (119867 minus 119911119901)

2

119877119877119895= radic(119909 minus 119909

119901)

2

+ (119910119877119895minus 119910119901)

2

+ (119867 minus 119911119901)

2

(1)

where 119909 = V120591119898denotes the azimuth position and 119910

119879119894and

119910119877119895

denote the cross track positions of the 119894th transmittingantenna element and the 119895th receiving antenna elementrespectively

Then the sum of the transmitting and receiving pathsof the point scatterer 119875 contains two square roots whichwill lead to complex computation for the following imagingprocess When the distance between the transmitting andreceiving antenna elements is far less than the observationdistance of the radar the above complete travelling path canbe equal to the dual echo paths from the virtual antennaelement located at 119910 = (119910

119879119894+ 119910119877119895)2 to point scatterer 119875

based on the principle of equivalent phase center [6] And theequivalent echo path can be written as

119877 = radic(119909 minus 119909119901)

2

+ (119910 minus 119910119901)

2

+ (119867 minus 119911119901)

2

(2)

Then the phase difference between the virtual antenna ele-ment and the reality antenna element should be compensatedby

Δ120593 =

2120587

120582

(119877119894119895minus 2119877) asymp

120587 (119910119879119894minus 119910119877119895)

2

2120582119877119875

(3)

where 120582 is the wavelength and 119877119875= radic(119909 minus 119909

119901)2+ (119867 minus 119911

119901)2

32MotionCompensation Furthermore the airborne down-looking sparse array SAR works in the time division modethe virtual antenna elements obtained from different pulserepetition period are not in a straight line with themovementof the platform Therefore in order to obtain a fully dis-tributed virtual uniform linear array the motion compensa-tion should be implemented The compensated phase caused

by the overtake or lag phases owing to the position differenceof the antenna elements is given by

Δ120593119897119896

=

4120587

120582

(radic(

119872VPRF

(119897 minus

119871

2

) + Δ119909119896minus 119909119901)

2

+ (119910 minus 119910119901)

2

+ (119867 minus 119911119901)

2

minusradic(

119872VPRF

(119897 minus

119871

2

) minus 119909119901)

2

+ (119910 minus 119910119901)

2

+ (119867 minus 119911119901)

2

)

119897 = 1 2 119871 119896 = 1 2 119872

(4)

where 119871 is the azimuth sample number and Δ119909119896is the move

spacing between the 119896th and the first transmitting antennaelement in the azimuth direction V is the velocity of theplatform and PRF is the pulse repetition frequency

33 Echo Signal After the process above the collected dataof airborne down-looking sparse array SAR can be regardedas received by the fully distributed virtual linear array Foran arbitrary point scatterer positioned at 119875 (119909

119901 119910119901 119911119901) the

distance between the 119898th virtual antenna element and thepoint scatterer is given by

119877 = radic(V120591 minus 119909119901)

2

+ (119910119898minus 119910119901)

2

+ (119867 minus 119911119901)

2

= radic(V120591 minus 119909119901)

2

+ 1198772

119861

(5)

where 120591 is the azimuth time and 119877119861

=

radic(119867 minus 119911119901)2+ (119910119898minus 119910119901)2 The echo signal received by

the119898th virtual antenna element can be written as

1199041( 120591 119910

119898) = 119886119903( minus

2119877

119888

) 119886119886(120591 minus 120591

119901)

times exp [119895120587120574 ( minus 2119877119888

)

2

minus 119895

4120587

120582

119877]

(6)

where denotes the fast time 119910119898

denotes the cross trackposition of the119898th virtual antenna element 120591

119901= 119909119901V is the

azimuth time of the point scatterer 119875 119888 is the light velocity 120574is the range frequency rate 119886

119903(sdot) is the rectangular window

function in range direction and 119886119886(sdot) is the rectangular

window function in azimuth direction

34 Azimuth and Range Compression As the signal receivedby each virtual antenna element in the airborne down-looking sparse array SAR can be treated as the echo datareceived in side-looking mode the classical 2D imagingmethods can be used to process the data received by eachvirtual antenna element which can be found in [17 18]


z

y

x

s1s2s3sMN middot middot middot

x0

Figure 3 2D focused images in cross track direction of down-looking sparse array SAR

Consequently a SAR image corresponding to the 119898thvirtual antenna element which is obtained after the azimuthand range direction focusing can be expressed as [17 18]

119904 ( 120591 119910119898) = 119860 sinc [119861

119903( minus

2119877119861

119888

)] sinc [119861119886(120591 minus 120591

119901)]

times exp(minus1198954120587119877119861120582

)

(7)

where119860 is the amplitude of the focused point target 119861119903is the

range bandwidth and 119861119886is the azimuth bandwidth

35 Cross Track Compression Suppose that the signalreceived by each virtual antenna element have been focusedby range-Doppler algorithm and then119872119873 2D SAR imagescan be obtained in cross track direction as shown in Figure 3Here 119872119873 is the number of the virtual antenna elementsIf all the SAR images have been coregistered first then inthe imaging area the azimuth positions of each point targetin all SAR images are invariable while only the distancesbetween the virtual antenna elements and the point targetvary with the positions of the virtual antenna elementsTherefore a range-cross track section corresponding to oneazimuth position can be focused at a time Then the 3Dimage of airborne down-looking array SAR can be obtainedby processing all the sections with the same procedure

Then the range-cross track section corresponding toazimuth time 120591 = 120591

0can be written as

119904 ( 119910119898 120591 = 120591

0) = 119860 sinc [119861

119903( minus

2119877119861

119888

)]

times sinc [119861119886(120591 minus 120591

119901)] exp(minus1198954120587119877119861

120582

)

= 1198601sinc [119861

119903( minus

2119877119861

119888

)] exp(minus1198954120587119877119861120582

)

(8)

where 1198601= 119860 sinc[119861

119886(1205910minus 120591119901)] As 119860

1has no effect on the

imaging process it will be ignored in the following analysisThe slant range between the119898th virtual antenna element and

z

y

H

ym

120593prp

zp P

yp

Figure 4 Geometry relations in cross track-height section

the point target positioned at (119909119901 119910119901 119911119901) in this section is

given by

119877119861= radic(119867 minus 119911

119901)

2

+ (119910119898minus 119910119901)

2

(9)

From (8) we can get that target responses take on sincfunction with positions of peak value distributing along therange trajectories in the cross track direction To simplifyfurther analysis we redraw the geometry relations of the crosstrack-height section in Figure 4 It can be seen that as thereal antenna array length is far less than the imaging sizein cross track direction for the point targets located in thesame height position the range trajectories and scanningangles are different from each other That is different fromthe characteristic of SAR working in squint mode as thepoint targets located in the same height position have thesame range trajectories and scanning angles for conventionalsquint SAR Therefore we introduce a special squint modelfor the cross track imaging in airborne down-looking arraySAR In this special squint model the distance between thepoint target and the center of the antenna array is definedas the equivalent squint range 119903

119901 and the corresponding

scanning angle is defined as equivalent squint angle 120593119901Then

from Figure 4 we can get that the point targets with differentcross track positions have different squint angles and squintranges at the same height in this special squint model Andthe slant range between the119898th virtual antenna element andthe point target positioned at (119910

119901 119911119901) can also be rewritten as

119877119861= radic(119867 minus 119911

119901)

2

+ (119910119898minus 119910119901)

2

= radic1199032

119901+ 1199102

119898minus 2119903119901119910119898sin120593119901

(10)

where

119903119901=

(119867 minus 119911119901)

cos120593119901

(11)

As the target responses distribute along the range trajec-tories in the cross track direction In order to concentrateenergy to realize focusing in the cross track direction work


only has to be done along this trajectory However the rangetrajectories of point targets at different heights are interlacedand the cross track focusing is difficult to realize in timedomain Transform the signal expressed in (8) into the range-cross track 2D frequency domain via the stationary phasemethod Then we have

119878 (119891119903 119891119910 120591 = 120591

0) = 119886119903(minus

119891119903

120574119890(119891119910 119877119861)

)

times exp (minus1198952120587119891119910119910119901) exp [120579 (119891

119903 119891119910)]

(12)

where 119891119903is the range frequency 119891

119910is the cross track fre-

quency and the range frequency rate is changed from 120574

to 120574119890(119891119910 119877119861) in cross track frequency domain which is

dependent on cross track frequency and on range

1

120574119890(119891119910 119877119861)

=

1

120574

minus

12058231198771198611198912

119910

21198882[1 minus (120582119891

1199102)

2

]

32 (13)

And the phase function 120579(119891119903 119891119910) is expressed as

120579 (119891119903 119891119910) = minus119895

4120587119903119901cos120593119901

120582

radic(1 +

119891119903

119891119888

)

2

minus (

120582119891119910

2

)

2

(14)

To aid further analysis 120579(119891119903 119891119910) is expanded into Taylorrsquos

series

120579 (119891119903 119891119910) = 120579 (119891

119903= 0 119891

119910) + 119891119903

119889120579 (119891119903 119891119910)

119889119891119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119891119903=0

+

1198912

119903

2

1198892120579 (119891119903 119891119910)

1198891198912

119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119891119903=0

+

1198913

119903

3

1198893120579 (119891119903 119891119910)

1198891198913

119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119891119903=0

+ sdot sdot sdot

= minus119895

4120587119903119901cos120593119901

120582

120572 (119891119910) minus 119895

4120587119903119901cos120593119901

119888120572 (119891119910)

119891119903

+ 119895

120587119903119901cos1205931199011198881198912

119910

21198913

1198881205723(119891119910)

1198912

119903minus 119895

120587119903119901cos1205931199011198881198912

119910

21198914

1198881205725(119891119910)

1198913

119903

+ sdot sdot sdot

(15)

where

120572 (119891119910) = radic1 minus (

120582119891119910

2

)

2

(16)

In (15) the first term represents the modulation in crosstrack direction determining the cross track compression thesecond term contains the information of range migrationreflecting the target position in the range direction the third

and fourth terms are the secondary range compression itemshowing the coupling of range and cross track in the signalspectrum the other higher order terms (power is greater thanthree) are very small and thus ignorable

From (15) we can get that the range migration and sec-ondary range compression items depend on the equivalentsquint range 119903

119901and squint angle 120593

119901of the point target and

the equivalent squint ranges and squint angles vary with thepositions of point targets However from Figure 4 we canget that when the targets are positioned at the same heightthe term 119903

119901cos120593119901is a constant which is irrelevant to the

target positions and is equal to the shortest distance from thetargets to the antenna array Therefore the range migrationand secondary range compression items can be compensateduniformly which can be constructed with target positionsindependence

From (15) the reference function for rangemigration cor-rection can be written as

119867RCM (119891119903 119891119910) = exp[1198954120587119888

(

1

120572 (119891119910)

minus 1) (119867 minus 119911119901) 119891119903]

(17)

And the reference function for secondary range compressioncan be written as

119867SRC (119891119903 119891119910)

= exp[

[

minus119895

120587119888 (119867 minus 119911119901) 1198912

119910

21198913

1198881205723(119891119910)

1198912

119903+ 119895

120587119888 (119867 minus 119911119901) 1198912

119910

21198914

1198881205725(119891119910)

1198913

119903]

]

(18)

From (17) and (18) we get that the rangemigration correctionterm and secondary range compression term are dependenton the height of the target In the real case we use theslant range 119877

119861to replace the range (119867 minus 119911

119901) in the imaging

processing when the scene width is relatively less than thedistance between the platform and the target

After the rangemigration correction and secondary rangecompression the range trajectory of each point target will becorrected to a line approximately Transform the signal intothe range time-cross track frequency domain and the rangecompression signal is given by

119904 ( 119891119910 120591 = 120591

0) = sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times exp (minus1198952120587119891119910119910119901)

times exp [minus1198954120587119903119901cos120593119901

120582

120572 (119891119910)]

(19)


Table 1 Parameters used for simulation

Parameter ValueCarrier frequency 375 GHzPulse bandwidth 300MHzPulse repetition frequency 1024HzChirp duration 10120583sRadar height 500mRadar velocity 50msNumber of transmitting antenna elements 4Number of receiving antenna elements 32Azimuth beam width 057∘

Cross track beam width 12∘

Table 2 The position parameters of the point targets

Point targets Azimuth (m) Range (m) Cross track (m)1 0 490 102 4 495 203 4 495 minus204 minus4 495 205 minus4 495 minus206 8 500 407 8 500 minus408 minus8 500 409 minus8 500 minus40

The next step is to proceed to the cross track focusingTransform the signal (19) into the range-cross track 2D timedomain

119904 ( 119910119898 120591 = 120591

0) = sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times exp[

[

minus119895

2120587 (119910119898minus 119910119901)

2

120582119903119901cos120593119901

]

]

times exp(minus1198954120587119903119901cos120593119901

120582

)

(20)

The first phase term in (20) represents a quadratic distortionwhich can be compensated by deramp processing And thequadratic phase reference function for cross track derampprocessing is

119867Deramp (119910119898) = exp(11989521205871199102

119898

120582119877119861

) (21)

Azi

mut

h (m

)

Range (m)485 490 495 500 505 510

0

5

10

02

04

06

08

minus5

minus10

Figure 5 Azimuth and range 2D imaging results of the center an-tenna element

050 0

10

480

490

500

510

Azimuth (m)

Cross track (m)

Rang

e (m

)

minus10

minus50

Figure 6 Final 3D image of down-looking sparse array SAR

Then a cross track direction Fourier transform is performedon each range gate to realize the cross track compressionand the focused range-cross track image can be expressedby

119904 ( 119891119910 120591 = 120591

0)

= sinc[119861119903( minus

2 (119867 minus 119911119901)

119888

)]

times sinc[119871119891(119891119910minus

2119910119901

120582119903119901cos120593119901

)]

times exp[minus11989521205871199102

119901

120582119903119901cos120593119901

] exp(minus1198954120587119903119901cos120593119901

120582

)

(22)

where 119871119891is the effective aperture in the cross track direction


Table 3 The image quality parameters of the selected point target

Image quality parameters Azimuth (dB) Range (dB) Cross track (dB)PSLR minus1320 minus1331 minus1341ISLR minus101069 minus112014 minus112879

Range (m)

Cros

s tra

ck (m

)

495 500 505

34

36

38

40

42

44

46

02

04

06

08

(a) Range-cross track section

02

04

06

08

Cross track (m)

Azi

mut

h (m

)35 40 45

4

6

8

10

12

(b) Azimuth-cross track section

02

04

06

08

Range (m)

Azi

mut

h (m

)

495 500 5054

6

8

10

12

(c) Azimuth-range section

Figure 7 2D contour plots of the extracted point target

From (22) it can be seen that the cross track resolution inthe frequency domain is

Δ119891119910=

1

119871119891

(23)

Moreover based on Doppler characteristic analysis of theecho signal in SAR [19] the relationship between the crosstrack frequency and the cross track position of the pointtarget can be written as

119891119910119901=

2

120582

sin120593119901asymp

2119910119901

120582 (119867 minus 119911119901)

(24)

Therefore from (23) and (24) the cross track resolutioncan be calculated as

Δ119910 asymp

120582 (119867 minus 119911119901)

2119871119891

(25)

From (25) we can get that the cross track resolution is nota constant which varies with the vertical distance betweenthe platform and the target

When all the range-cross track sections corresponding toeach azimuth position are processed following the procedurementioned above the 3D image of airborne down-lookingarray SAR can be achieved From (8) and (22) we get that


6 8 10minus30

minus20

minus13

minus3

0N

orm

aliz

ed am

plitu

de (d

B)

Azimuth (m)

(a) The profile of the azimuth compression

minus30

minus20

minus13

minus3

0

Nor

mal

ized

ampl

itude

(dB)

498 500 502Range (m)

(b) The profile of the range compression

minus30

minus20

minus13

minus3

0

Nor

mal

ized

ampl

itude

(dB)

36 38 40 42 44 46Cross track (m)

(c) The profile of the cross track compression

Figure 8 The profiles of the extracted point target

the 3D point spread function of airborne down-looking arraySAR can be denoted as

119901119904119891 (120591 119891119910)

sim sinc [119861119886(120591 minus 120591

119901)] sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times sinc[119871119891(119891119910minus

2119910119901

120582119903119901cos120593119901

)]

(26)

4 Simulation Results

In this section point target simulation is carried out to verifythe validity of the proposed imaging algorithm The mainparameters used for simulation are listed in Table 1

Here we suppose that there are nine point targets locatedat the sceneThe position parameters of the nine point targetsare shown in Table 2 Figure 5 shows the azimuth and range2D imaging results of the center antenna element Because thepoint targets located symmetrically to the flight direction areimaged in one resolution cell it can be seen that there are onlyfive point targets that are shown in Figure 5 after the azimuthand range focusing

With the 3D imaging processing by using the proposedalgorithm in this paper the surfaces of the final 3D imageare plotted at minus30 dB in Figure 6 As expected the image isreconstructed in 3D space and the positions of the nine pointtargets are consistent with the real situation in Table 2 FromFigure 6 we can get that the leftright ambiguity of the pointtargets caused by the symmetric range distances on both sidesof the flight direction can be resolved after cross track imagingprocessing


In order to reflect the detailed characteristics of theimaging results the point target located at (8 40 500) isextracted for individual analysis Figure 7 shows the contourplots of the imaging results of the extracted point targetFigure 7(a) shows the 2D image of the point target at therange-cross track section Figure 7(b) shows the 2D image ofthe point target at azimuth-cross track section Figure 7(c)shows the 2D image of the point target at azimuth-rangesection FromFigure 7 it is clear that the shape of the contourplots is regular main lobes and side lobes visibly dividedindicating that the point target is focused well

Then the image quality parameters that is the peak side-lobe ratio (PSLR) and the integrated sidelobe ratio (ISLR)are used to test the performance and focusing quality ofthe proposed method Taking the point target located at (840 500) as an example Figure 8 shows the profiles of theazimuth range and cross track compression From the abovesimulation results we can get that the compression resultsall present the shape of a sinc function Table 3 gives theperformance parameters of the point target Both the PSLRand the ISLR are close to the ideal values

5 Conclusions

Down-looking array SAR can reconstruct 3D images of theobserved area and overcome restrictions of shading and layover effects in side-looking SAR Therefore down-lookingarray SAR has challenging potential for 3D digital mapscomplex terrain mapping and so on In this paper a novel3D imaging algorithmcapable of focusing down-looking SARdata is proposed The principle behind the method is basedon a special squint model in cross track processing to obtainaccurate focusing And the phase compensation factors andalgorithm realization procedure are demonstrated in detailMoreover the method requires only Fourier transform andmultiplications making it suitable for practical applicationsRaw data of down-looking sparse array SAR is simulated andthe 3D image is achieved The results of the simulated dataconfirm the effectiveness of the proposed method

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 Sci-ence Foundation of China under Grant 61201390 andGrant 61201389 the Fundamental Research Funds for theHenan Provincial Colleges and Universities under Grant2014YWQQ10 and the Plan for Young Backbone Teacher ofHenan University of Technology

References

[1] D L Mensa High Resolution Radar Imaging Artech HouseDedham Mass USA 1981

[2] W G Carrara R S Goodman and R M Majewski SpotlightSynthetic Aperture Radar Signal Processing and AlgorithmsArtech House Boston Mass USA 1995

[3] B Wang Y P Wang W Hong W Tan and Y Wu ldquoStudiesonMB-SAR 3D imaging algorithm using Yule-WalkermethodrdquoScience China Information Sciences vol 53 no 9 pp 1848ndash18592010

[4] X X Zhu and R Bamler ldquoDemonstration of super-resolutionfor tomographic SAR imaging in urban environmentrdquo IEEETransactions on Geoscience and Remote Sensing vol 50 no 8pp 3150ndash3157 2012

[5] C H Gierull ldquoOn a concept for an airborne downward-looking imaging radarrdquo International Journal of Electronics andCommunications vol 53 no 6 pp 295ndash304 1999

[6] J Klare ldquoA new airborne radar for 3D imagingmdashsimulationstudy of ARTINOrdquo in Proceesings of the 6th European Con-ference on Synthetic Aperture Radar Dresden Germany May2006

[7] L Du Y P Wang W Hong W Tan and Y Wu ldquoA three-dimensional range migration algorithm for downward-looking3D-SAR with single-transmitting and multiple-receiving lineararray antennasrdquo EURASIP Journal on Advances in Signal Pro-cessing vol 2010 Article ID 957916 2010

[8] X-M Peng Y-P Wang W-X Tan W Hong and Y-R WuldquoFast wavenumber domain imaging algorithm for airbornedownward-looking array 3D-SAR based on region of interestpickrdquo Journal of Electronics and Information Technology vol 35no 7 pp 1525ndash1531 2013

[9] J Klare ldquoDigital beamforming for a 3D mimo sar-improve-ments through frequency andwaveform diversity (IGARSSrsquo08)rdquo in Proceedings of the IEEE International Geoscience andRemote Sensing Symposium pp 17ndash20 Boston Mass USA July2008

[10] Y N Hou Study of radar imaging technology based on sparsearray antenna [PhD dissertation] Institute of ElectronicsChinese Academy of Sciences 2010

[11] J F Nouvel H Jeuland G Bonin et al ldquoA Ka band imagingradar DRIVE on board ONERA motorgliderrdquo in Proceedingsof the International Geoscience and Remote Sensing Symposium(IGARSS rsquo06) pp 134ndash136 Denver Colo USA August 2006

[12] L Du Y P Wang and W Hong ldquoThree-dimensional imagingalgorithm for synthetic aperture radar with linear array anten-nas based on elevation angle compression principlerdquo Journal ofthe Graduate School of the Chinese Academy of Sciences vol 27no 6 pp 800ndash808 2010

[13] R Giret H Jeuland and P Enert ldquoA study of a 3D-SAR conceptfor a millimeter wave imaging radar onboard an UAVrdquo inProceedings of the 1st European Radar Conference (EURAD 04)pp 201ndash204 Amsterdam The Netherlands October 2004

[14] X Wen G Kuang J Hu R Zhan and J Zhang ldquoForward-looking imaging of scanning phased array radar based on thecompressed sensingrdquo Progress in Electromagnetics Research vol143 pp 575ndash604 2013

[15] X Z Ren L N Chen and J Yang ldquo3D imaging algo-rithm for down-looking MIMO array SAR based on Bayesiancompressive sensingrdquo International Journal of Antennas andPropagation vol 2014 Article ID 612326 9 pages 2014

[16] M Weiszlig J Ender O Peters et al ldquoAn airborne radar for threedimensional imaging and observationmdashtechnical realisationand status of ARTINOrdquo in Proceedings of the 6th EuropeanConference on Synthetic Aperture Radar Dresden GermanyMay 2006


[17] M Y Jin and C Wu ldquoA SAR correlation algorithm whichaccommodates large-range migrationrdquo IEEE Transactions onGeoscience and Remote Sensing vol 22 no 6 pp 592ndash597 1983

[18] M Soumekh Synthetic Aperture Radar Signal Processing withMATLAB Algorithms Wiley-Interscience New York NY USA1999

[19] G Lan and H G FrankDigital Processing of Synthetic ApertureRadar Data Algorithms and Implementation Artech House2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


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

Acoustics and VibrationAdvances in



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



is not considered Du et al [12] made an improvement ofthe method proposed in [6]They added the range migrationcorrection in the imaging processing to expand the imagingscene width in the cross track direction However the digitalbeamforming technology used in the paper requires the pointby point calculations and inserting the point spread functionfor focusing the elevation angle Therefore this methodsuffers from severe computational complexity Giret et al[13] made some simplifications to the dual square roots withFresnel approximation and proposed an algorithm derivedfrom that of monostatic configuration A shortcoming of thealgorithm is the range error caused by Fresnel approximationSubsequently an accurate 3D focused algorithm based onrange migration algorithm (RMA) was presented in [7] butneeds 3D stolt interpolation In a recent work somemethodsbased on compressive sensing (CS) were applied to arraySAR imaging [14 15] providing an improved resolution incross track direction However these methods suffered fromsevere computation loads and were hard to realize real-timeimaging

The main topic of this paper is to introduce an accurateand efficient 3D imaging algorithm for down-looking sparsearray SAR According to the analysis of the spatial geom-etry and the echo signal model a special squint model isintroduced for cross track focusingThe phase compensationfactors and algorithm realization procedure are demonstratedin detail

The rest of the paper is organized as follows Section 2presents the geometry and principle of airborne down-looking sparse array SAR system In Section 3 a novel3D imaging algorithm for down-looking sparse array SARis described in detail The performance of the method isinvestigated in Section 4 Finally Section 5 gives a brief con-clusion

2 Geometrical Model of AirborneDown-Looking Sparse Array SAR

Figure 1 shows the geometry of down-looking sparse arraySAR [15]119909denotes the azimuth direction119910denotes the crosstrack direction and 119911 denotes the height directionThe radarplatform flies at height119867 along the 119909-axis with velocity VThesparse linear array is mounted in the cross track directionalong the wings and is designed based on ARTINO arraydistributionThat is because that ARTINO array distributionis quite easy for hardware implementation [16] The sparselinear array composes119872 transmitting antenna elements and119873 receiving antenna elements and works in the time divisionmode Each time only one transmitting antenna elementtransmits signal and all the receiving antenna elementsreceive echo simultaneously [6] The transmitting antennaelements work sequentially and an aperture synthesis periodis acquired until all the transmitting antenna elements haveworked for once

According to the principle of equivalent phase center thesparse linear array formed by the ARTINO array distributioncan be equivalent to a virtual uniform linear array [6 10]

yx

z

H

v

Figure 1 Geometry of down-looking sparse array SAR

T1 T2 middot middot middot R1 R2 middot middot middot RNminus1 RN middot middot middot TMminus1 TM

TR1R2

RN

Delay time

P

Figure 2 Transmitting and receiving order of down-looking sparsearray SAR

The virtual antenna array is composed of 119872119873 virtual ele-ments and each virtual element transmits and receives byitself These virtual antenna elements are uniformly dis-tributed along the wings and centered at the 119910-axis Eachvirtual antenna element is located at the mean position of areal single transmitting element and a real single receivingelement Figure 2 shows the transmitting and the receivingorder of each antenna element for a down-looking sparsearray SAR 119875 is the target in the imaging scene 119879

119894with

119894 isin [1119872] denotes the transmitting antenna element and119877119895with 119895 isin [1119873] denotes the receiving antenna element

As shown in Figure 2 the transmitting antenna elementswork sequentially The antenna element 119879

1transmits signal

first and all the receiving antenna elements receive echosimultaneously Then the antenna element 119879

2transmits sig-

nal and the procedure loops until the entire transmittingantenna elements have worked for once and then an aperturesynthesis period is acquired





119898 the


119879119894 119867) and (119909 119910



and 119877119877119895


119901 119910119901 119911119901) are given by

119877119879119894= radic(119909 minus 119909

119901)

2

+ (119910119879119894minus 119910119901)

2

+ (119867 minus 119911119901)

2

119877119877119895= radic(119909 minus 119909

119901)

2

+ (119910119877119895minus 119910119901)

2

+ (119867 minus 119911119901)

2

(1)


119879119894and

119910119877119895





119877 = radic(119909 minus 119909119901)

2

+ (119910 minus 119910119901)

2

+ (119867 minus 119911119901)

2

(2)


Δ120593 =

2120587

120582

(119877119894119895minus 2119877) asymp

120587 (119910119879119894minus 119910119877119895)

2

2120582119877119875

(3)


119901)2+ (119867 minus 119911

119901)2



Δ120593119897119896

=

4120587

120582

(radic(

119872VPRF

(119897 minus

119871

2

) + Δ119909119896minus 119909119901)

2

+ (119910 minus 119910119901)

2

+ (119867 minus 119911119901)

2

minusradic(

119872VPRF

(119897 minus

119871

2

) minus 119909119901)

2

+ (119910 minus 119910119901)

2

+ (119867 minus 119911119901)

2

)

119897 = 1 2 119871 119896 = 1 2 119872

(4)




119901 119910119901 119911119901) the


119877 = radic(V120591 minus 119909119901)

2

+ (119910119898minus 119910119901)

2

+ (119867 minus 119911119901)

2

= radic(V120591 minus 119909119901)

2

+ 1198772

119861

(5)


=



1199041( 120591 119910

119898) = 119886119903( minus

2119877

119888

) 119886119886(120591 minus 120591

119901)

times exp [119895120587120574 ( minus 2119877119888

)

2

minus 119895

4120587

120582

119877]

(6)



119901= 119909119901V is the







z

y

x


x0



119904 ( 120591 119910119898) = 119860 sinc [119861

119903( minus

2119877119861

119888

)] sinc [119861119886(120591 minus 120591

119901)]

times exp(minus1198954120587119877119861120582

)

(7)





0can be written as

119904 ( 119910119898 120591 = 120591

0) = 119860 sinc [119861

119903( minus

2119877119861

119888

)]

times sinc [119861119886(120591 minus 120591

119901)] exp(minus1198954120587119877119861

120582

)

= 1198601sinc [119861

119903( minus

2119877119861

119888

)] exp(minus1198954120587119877119861120582

)

(8)

where 1198601= 119860 sinc[119861

119886(1205910minus 120591119901)] As 119860



z

y

H

ym

120593prp

zp P

yp



given by

119877119861= radic(119867 minus 119911

119901)

2

+ (119910119898minus 119910119901)

2

(9)






119877119861= radic(119867 minus 119911

119901)

2

+ (119910119898minus 119910119901)

2

= radic1199032

119901+ 1199102

119898minus 2119903119901119910119898sin120593119901

(10)

where

119903119901=

(119867 minus 119911119901)

cos120593119901

(11)




119878 (119891119903 119891119910 120591 = 120591

0) = 119886119903(minus

119891119903

120574119890(119891119910 119877119861)

)


119903 119891119910)]

(12)






1

120574119890(119891119910 119877119861)

=

1

120574

minus

12058231198771198611198912

119910

21198882[1 minus (120582119891

1199102)

2

]

32 (13)


120579 (119891119903 119891119910) = minus119895

4120587119903119901cos120593119901

120582

radic(1 +

119891119903

119891119888

)

2

minus (

120582119891119910

2

)

2

(14)


series

120579 (119891119903 119891119910) = 120579 (119891

119903= 0 119891

119910) + 119891119903

119889120579 (119891119903 119891119910)

119889119891119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119891119903=0

+

1198912

119903

2

1198892120579 (119891119903 119891119910)

1198891198912

119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119891119903=0

+

1198913

119903

3

1198893120579 (119891119903 119891119910)

1198891198913

119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119891119903=0

+ sdot sdot sdot

= minus119895

4120587119903119901cos120593119901

120582

120572 (119891119910) minus 119895

4120587119903119901cos120593119901

119888120572 (119891119910)

119891119903

+ 119895

120587119903119901cos1205931199011198881198912

119910

21198913

1198881205723(119891119910)

1198912

119903minus 119895

120587119903119901cos1205931199011198881198912

119910

21198914

1198881205725(119891119910)

1198913

119903

+ sdot sdot sdot

(15)

where

120572 (119891119910) = radic1 minus (

120582119891119910

2

)

2

(16)










119867RCM (119891119903 119891119910) = exp[1198954120587119888

(

1

120572 (119891119910)

minus 1) (119867 minus 119911119901) 119891119903]

(17)


119867SRC (119891119903 119891119910)

= exp[

[

minus119895

120587119888 (119867 minus 119911119901) 1198912

119910

21198913

1198881205723(119891119910)

1198912

119903+ 119895

120587119888 (119867 minus 119911119901) 1198912

119910

21198914

1198881205725(119891119910)

1198913

119903]

]

(18)






119904 ( 119891119910 120591 = 120591

0) = sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times exp (minus1198952120587119891119910119910119901)


120582

120572 (119891119910)]

(19)








119904 ( 119910119898 120591 = 120591

0) = sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times exp[

[

minus119895

2120587 (119910119898minus 119910119901)

2

120582119903119901cos120593119901

]

]


120582

)

(20)


119867Deramp (119910119898) = exp(11989521205871199102

119898

120582119877119861

) (21)

Azi

mut

h (m

)

Range (m)485 490 495 500 505 510

0

5

10

02

04

06

08

minus5

minus10


050 0

10

480

490

500

510

Azimuth (m)

Cross track (m)

Rang

e (m

)

minus10

minus50



119904 ( 119891119910 120591 = 120591

0)

= sinc[119861119903( minus

2 (119867 minus 119911119901)

119888

)]

times sinc[119871119891(119891119910minus

2119910119901

120582119903119901cos120593119901

)]

times exp[minus11989521205871199102

119901

120582119903119901cos120593119901

] exp(minus1198954120587119903119901cos120593119901

120582

)

(22)





Range (m)

Cros

s tra

ck (m

)

495 500 505

34

36

38

40

42

44

46

02

04

06

08


02

04

06

08

Cross track (m)

Azi

mut

h (m

)35 40 45

4

6

8

10

12


02

04

06

08

Range (m)

Azi

mut

h (m

)

495 500 5054

6

8

10

12




Δ119891119910=

1

119871119891

(23)


119891119910119901=

2

120582


2119910119901

120582 (119867 minus 119911119901)

(24)


Δ119910 asymp

120582 (119867 minus 119911119901)

2119871119891

(25)




6 8 10minus30

minus20

minus13

minus3

0N

orm

aliz

ed am

plitu

de (d

B)

Azimuth (m)


minus30

minus20

minus13

minus3

0

Nor

mal

ized

ampl

itude

(dB)

498 500 502Range (m)


minus30

minus20

minus13

minus3

0

Nor

mal

ized

ampl

itude

(dB)

36 38 40 42 44 46Cross track (m)




119901119904119891 (120591 119891119910)

sim sinc [119861119886(120591 minus 120591

119901)] sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times sinc[119871119891(119891119910minus

2119910119901

120582119903119901cos120593119901

)]

(26)








5 Conclusions




Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119898 the


119879119894 119867) and (119909 119910



and 119877119877119895


119901 119910119901 119911119901) are given by

119877119879119894= radic(119909 minus 119909

119901)

2

+ (119910119879119894minus 119910119901)

2

+ (119867 minus 119911119901)

2

119877119877119895= radic(119909 minus 119909

119901)

2

+ (119910119877119895minus 119910119901)

2

+ (119867 minus 119911119901)

2

(1)


119879119894and

119910119877119895





119877 = radic(119909 minus 119909119901)

2

+ (119910 minus 119910119901)

2

+ (119867 minus 119911119901)

2

(2)


Δ120593 =

2120587

120582

(119877119894119895minus 2119877) asymp

120587 (119910119879119894minus 119910119877119895)

2

2120582119877119875

(3)


119901)2+ (119867 minus 119911

119901)2



Δ120593119897119896

=

4120587

120582

(radic(

119872VPRF

(119897 minus

119871

2

) + Δ119909119896minus 119909119901)

2

+ (119910 minus 119910119901)

2

+ (119867 minus 119911119901)

2

minusradic(

119872VPRF

(119897 minus

119871

2

) minus 119909119901)

2

+ (119910 minus 119910119901)

2

+ (119867 minus 119911119901)

2

)

119897 = 1 2 119871 119896 = 1 2 119872

(4)




119901 119910119901 119911119901) the


119877 = radic(V120591 minus 119909119901)

2

+ (119910119898minus 119910119901)

2

+ (119867 minus 119911119901)

2

= radic(V120591 minus 119909119901)

2

+ 1198772

119861

(5)


=



1199041( 120591 119910

119898) = 119886119903( minus

2119877

119888

) 119886119886(120591 minus 120591

119901)

times exp [119895120587120574 ( minus 2119877119888

)

2

minus 119895

4120587

120582

119877]

(6)



119901= 119909119901V is the







z

y

x


x0



119904 ( 120591 119910119898) = 119860 sinc [119861

119903( minus

2119877119861

119888

)] sinc [119861119886(120591 minus 120591

119901)]

times exp(minus1198954120587119877119861120582

)

(7)





0can be written as

119904 ( 119910119898 120591 = 120591

0) = 119860 sinc [119861

119903( minus

2119877119861

119888

)]

times sinc [119861119886(120591 minus 120591

119901)] exp(minus1198954120587119877119861

120582

)

= 1198601sinc [119861

119903( minus

2119877119861

119888

)] exp(minus1198954120587119877119861120582

)

(8)

where 1198601= 119860 sinc[119861

119886(1205910minus 120591119901)] As 119860



z

y

H

ym

120593prp

zp P

yp



given by

119877119861= radic(119867 minus 119911

119901)

2

+ (119910119898minus 119910119901)

2

(9)






119877119861= radic(119867 minus 119911

119901)

2

+ (119910119898minus 119910119901)

2

= radic1199032

119901+ 1199102

119898minus 2119903119901119910119898sin120593119901

(10)

where

119903119901=

(119867 minus 119911119901)

cos120593119901

(11)




119878 (119891119903 119891119910 120591 = 120591

0) = 119886119903(minus

119891119903

120574119890(119891119910 119877119861)

)


119903 119891119910)]

(12)






1

120574119890(119891119910 119877119861)

=

1

120574

minus

12058231198771198611198912

119910

21198882[1 minus (120582119891

1199102)

2

]

32 (13)


120579 (119891119903 119891119910) = minus119895

4120587119903119901cos120593119901

120582

radic(1 +

119891119903

119891119888

)

2

minus (

120582119891119910

2

)

2

(14)


series

120579 (119891119903 119891119910) = 120579 (119891

119903= 0 119891

119910) + 119891119903

119889120579 (119891119903 119891119910)

119889119891119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119891119903=0

+

1198912

119903

2

1198892120579 (119891119903 119891119910)

1198891198912

119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119891119903=0

+

1198913

119903

3

1198893120579 (119891119903 119891119910)

1198891198913

119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119891119903=0

+ sdot sdot sdot

= minus119895

4120587119903119901cos120593119901

120582

120572 (119891119910) minus 119895

4120587119903119901cos120593119901

119888120572 (119891119910)

119891119903

+ 119895

120587119903119901cos1205931199011198881198912

119910

21198913

1198881205723(119891119910)

1198912

119903minus 119895

120587119903119901cos1205931199011198881198912

119910

21198914

1198881205725(119891119910)

1198913

119903

+ sdot sdot sdot

(15)

where

120572 (119891119910) = radic1 minus (

120582119891119910

2

)

2

(16)










119867RCM (119891119903 119891119910) = exp[1198954120587119888

(

1

120572 (119891119910)

minus 1) (119867 minus 119911119901) 119891119903]

(17)


119867SRC (119891119903 119891119910)

= exp[

[

minus119895

120587119888 (119867 minus 119911119901) 1198912

119910

21198913

1198881205723(119891119910)

1198912

119903+ 119895

120587119888 (119867 minus 119911119901) 1198912

119910

21198914

1198881205725(119891119910)

1198913

119903]

]

(18)






119904 ( 119891119910 120591 = 120591

0) = sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times exp (minus1198952120587119891119910119910119901)


120582

120572 (119891119910)]

(19)








119904 ( 119910119898 120591 = 120591

0) = sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times exp[

[

minus119895

2120587 (119910119898minus 119910119901)

2

120582119903119901cos120593119901

]

]


120582

)

(20)


119867Deramp (119910119898) = exp(11989521205871199102

119898

120582119877119861

) (21)

Azi

mut

h (m

)

Range (m)485 490 495 500 505 510

0

5

10

02

04

06

08

minus5

minus10


050 0

10

480

490

500

510

Azimuth (m)

Cross track (m)

Rang

e (m

)

minus10

minus50



119904 ( 119891119910 120591 = 120591

0)

= sinc[119861119903( minus

2 (119867 minus 119911119901)

119888

)]

times sinc[119871119891(119891119910minus

2119910119901

120582119903119901cos120593119901

)]

times exp[minus11989521205871199102

119901

120582119903119901cos120593119901

] exp(minus1198954120587119903119901cos120593119901

120582

)

(22)





Range (m)

Cros

s tra

ck (m

)

495 500 505

34

36

38

40

42

44

46

02

04

06

08


02

04

06

08

Cross track (m)

Azi

mut

h (m

)35 40 45

4

6

8

10

12


02

04

06

08

Range (m)

Azi

mut

h (m

)

495 500 5054

6

8

10

12




Δ119891119910=

1

119871119891

(23)


119891119910119901=

2

120582


2119910119901

120582 (119867 minus 119911119901)

(24)


Δ119910 asymp

120582 (119867 minus 119911119901)

2119871119891

(25)




6 8 10minus30

minus20

minus13

minus3

0N

orm

aliz

ed am

plitu

de (d

B)

Azimuth (m)


minus30

minus20

minus13

minus3

0

Nor

mal

ized

ampl

itude

(dB)

498 500 502Range (m)


minus30

minus20

minus13

minus3

0

Nor

mal

ized

ampl

itude

(dB)

36 38 40 42 44 46Cross track (m)




119901119904119891 (120591 119891119910)

sim sinc [119861119886(120591 minus 120591

119901)] sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times sinc[119871119891(119891119910minus

2119910119901

120582119903119901cos120593119901

)]

(26)








5 Conclusions




Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










z

y

x


x0



119904 ( 120591 119910119898) = 119860 sinc [119861

119903( minus

2119877119861

119888

)] sinc [119861119886(120591 minus 120591

119901)]

times exp(minus1198954120587119877119861120582

)

(7)





0can be written as

119904 ( 119910119898 120591 = 120591

0) = 119860 sinc [119861

119903( minus

2119877119861

119888

)]

times sinc [119861119886(120591 minus 120591

119901)] exp(minus1198954120587119877119861

120582

)

= 1198601sinc [119861

119903( minus

2119877119861

119888

)] exp(minus1198954120587119877119861120582

)

(8)

where 1198601= 119860 sinc[119861

119886(1205910minus 120591119901)] As 119860



z

y

H

ym

120593prp

zp P

yp



given by

119877119861= radic(119867 minus 119911

119901)

2

+ (119910119898minus 119910119901)

2

(9)






119877119861= radic(119867 minus 119911

119901)

2

+ (119910119898minus 119910119901)

2

= radic1199032

119901+ 1199102

119898minus 2119903119901119910119898sin120593119901

(10)

where

119903119901=

(119867 minus 119911119901)

cos120593119901

(11)




119878 (119891119903 119891119910 120591 = 120591

0) = 119886119903(minus

119891119903

120574119890(119891119910 119877119861)

)


119903 119891119910)]

(12)






1

120574119890(119891119910 119877119861)

=

1

120574

minus

12058231198771198611198912

119910

21198882[1 minus (120582119891

1199102)

2

]

32 (13)


120579 (119891119903 119891119910) = minus119895

4120587119903119901cos120593119901

120582

radic(1 +

119891119903

119891119888

)

2

minus (

120582119891119910

2

)

2

(14)


series

120579 (119891119903 119891119910) = 120579 (119891

119903= 0 119891

119910) + 119891119903

119889120579 (119891119903 119891119910)

119889119891119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119891119903=0

+

1198912

119903

2

1198892120579 (119891119903 119891119910)

1198891198912

119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119891119903=0

+

1198913

119903

3

1198893120579 (119891119903 119891119910)

1198891198913

119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119891119903=0

+ sdot sdot sdot

= minus119895

4120587119903119901cos120593119901

120582

120572 (119891119910) minus 119895

4120587119903119901cos120593119901

119888120572 (119891119910)

119891119903

+ 119895

120587119903119901cos1205931199011198881198912

119910

21198913

1198881205723(119891119910)

1198912

119903minus 119895

120587119903119901cos1205931199011198881198912

119910

21198914

1198881205725(119891119910)

1198913

119903

+ sdot sdot sdot

(15)

where

120572 (119891119910) = radic1 minus (

120582119891119910

2

)

2

(16)










119867RCM (119891119903 119891119910) = exp[1198954120587119888

(

1

120572 (119891119910)

minus 1) (119867 minus 119911119901) 119891119903]

(17)


119867SRC (119891119903 119891119910)

= exp[

[

minus119895

120587119888 (119867 minus 119911119901) 1198912

119910

21198913

1198881205723(119891119910)

1198912

119903+ 119895

120587119888 (119867 minus 119911119901) 1198912

119910

21198914

1198881205725(119891119910)

1198913

119903]

]

(18)






119904 ( 119891119910 120591 = 120591

0) = sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times exp (minus1198952120587119891119910119910119901)


120582

120572 (119891119910)]

(19)








119904 ( 119910119898 120591 = 120591

0) = sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times exp[

[

minus119895

2120587 (119910119898minus 119910119901)

2

120582119903119901cos120593119901

]

]


120582

)

(20)


119867Deramp (119910119898) = exp(11989521205871199102

119898

120582119877119861

) (21)

Azi

mut

h (m

)

Range (m)485 490 495 500 505 510

0

5

10

02

04

06

08

minus5

minus10


050 0

10

480

490

500

510

Azimuth (m)

Cross track (m)

Rang

e (m

)

minus10

minus50



119904 ( 119891119910 120591 = 120591

0)

= sinc[119861119903( minus

2 (119867 minus 119911119901)

119888

)]

times sinc[119871119891(119891119910minus

2119910119901

120582119903119901cos120593119901

)]

times exp[minus11989521205871199102

119901

120582119903119901cos120593119901

] exp(minus1198954120587119903119901cos120593119901

120582

)

(22)





Range (m)

Cros

s tra

ck (m

)

495 500 505

34

36

38

40

42

44

46

02

04

06

08


02

04

06

08

Cross track (m)

Azi

mut

h (m

)35 40 45

4

6

8

10

12


02

04

06

08

Range (m)

Azi

mut

h (m

)

495 500 5054

6

8

10

12




Δ119891119910=

1

119871119891

(23)


119891119910119901=

2

120582


2119910119901

120582 (119867 minus 119911119901)

(24)


Δ119910 asymp

120582 (119867 minus 119911119901)

2119871119891

(25)




6 8 10minus30

minus20

minus13

minus3

0N

orm

aliz

ed am

plitu

de (d

B)

Azimuth (m)


minus30

minus20

minus13

minus3

0

Nor

mal

ized

ampl

itude

(dB)

498 500 502Range (m)


minus30

minus20

minus13

minus3

0

Nor

mal

ized

ampl

itude

(dB)

36 38 40 42 44 46Cross track (m)




119901119904119891 (120591 119891119910)

sim sinc [119861119886(120591 minus 120591

119901)] sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times sinc[119871119891(119891119910minus

2119910119901

120582119903119901cos120593119901

)]

(26)








5 Conclusions




Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119878 (119891119903 119891119910 120591 = 120591

0) = 119886119903(minus

119891119903

120574119890(119891119910 119877119861)

)


119903 119891119910)]

(12)






1

120574119890(119891119910 119877119861)

=

1

120574

minus

12058231198771198611198912

119910

21198882[1 minus (120582119891

1199102)

2

]

32 (13)


120579 (119891119903 119891119910) = minus119895

4120587119903119901cos120593119901

120582

radic(1 +

119891119903

119891119888

)

2

minus (

120582119891119910

2

)

2

(14)


series

120579 (119891119903 119891119910) = 120579 (119891

119903= 0 119891

119910) + 119891119903

119889120579 (119891119903 119891119910)

119889119891119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119891119903=0

+

1198912

119903

2

1198892120579 (119891119903 119891119910)

1198891198912

119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119891119903=0

+

1198913

119903

3

1198893120579 (119891119903 119891119910)

1198891198913

119903

10038161003816100381610038161003816100381610038161003816100381610038161003816119891119903=0

+ sdot sdot sdot

= minus119895

4120587119903119901cos120593119901

120582

120572 (119891119910) minus 119895

4120587119903119901cos120593119901

119888120572 (119891119910)

119891119903

+ 119895

120587119903119901cos1205931199011198881198912

119910

21198913

1198881205723(119891119910)

1198912

119903minus 119895

120587119903119901cos1205931199011198881198912

119910

21198914

1198881205725(119891119910)

1198913

119903

+ sdot sdot sdot

(15)

where

120572 (119891119910) = radic1 minus (

120582119891119910

2

)

2

(16)










119867RCM (119891119903 119891119910) = exp[1198954120587119888

(

1

120572 (119891119910)

minus 1) (119867 minus 119911119901) 119891119903]

(17)


119867SRC (119891119903 119891119910)

= exp[

[

minus119895

120587119888 (119867 minus 119911119901) 1198912

119910

21198913

1198881205723(119891119910)

1198912

119903+ 119895

120587119888 (119867 minus 119911119901) 1198912

119910

21198914

1198881205725(119891119910)

1198913

119903]

]

(18)






119904 ( 119891119910 120591 = 120591

0) = sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times exp (minus1198952120587119891119910119910119901)


120582

120572 (119891119910)]

(19)








119904 ( 119910119898 120591 = 120591

0) = sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times exp[

[

minus119895

2120587 (119910119898minus 119910119901)

2

120582119903119901cos120593119901

]

]


120582

)

(20)


119867Deramp (119910119898) = exp(11989521205871199102

119898

120582119877119861

) (21)

Azi

mut

h (m

)

Range (m)485 490 495 500 505 510

0

5

10

02

04

06

08

minus5

minus10


050 0

10

480

490

500

510

Azimuth (m)

Cross track (m)

Rang

e (m

)

minus10

minus50



119904 ( 119891119910 120591 = 120591

0)

= sinc[119861119903( minus

2 (119867 minus 119911119901)

119888

)]

times sinc[119871119891(119891119910minus

2119910119901

120582119903119901cos120593119901

)]

times exp[minus11989521205871199102

119901

120582119903119901cos120593119901

] exp(minus1198954120587119903119901cos120593119901

120582

)

(22)





Range (m)

Cros

s tra

ck (m

)

495 500 505

34

36

38

40

42

44

46

02

04

06

08


02

04

06

08

Cross track (m)

Azi

mut

h (m

)35 40 45

4

6

8

10

12


02

04

06

08

Range (m)

Azi

mut

h (m

)

495 500 5054

6

8

10

12




Δ119891119910=

1

119871119891

(23)


119891119910119901=

2

120582


2119910119901

120582 (119867 minus 119911119901)

(24)


Δ119910 asymp

120582 (119867 minus 119911119901)

2119871119891

(25)




6 8 10minus30

minus20

minus13

minus3

0N

orm

aliz

ed am

plitu

de (d

B)

Azimuth (m)


minus30

minus20

minus13

minus3

0

Nor

mal

ized

ampl

itude

(dB)

498 500 502Range (m)


minus30

minus20

minus13

minus3

0

Nor

mal

ized

ampl

itude

(dB)

36 38 40 42 44 46Cross track (m)




119901119904119891 (120591 119891119910)

sim sinc [119861119886(120591 minus 120591

119901)] sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times sinc[119871119891(119891119910minus

2119910119901

120582119903119901cos120593119901

)]

(26)








5 Conclusions




Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















119904 ( 119910119898 120591 = 120591

0) = sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times exp[

[

minus119895

2120587 (119910119898minus 119910119901)

2

120582119903119901cos120593119901

]

]


120582

)

(20)


119867Deramp (119910119898) = exp(11989521205871199102

119898

120582119877119861

) (21)

Azi

mut

h (m

)

Range (m)485 490 495 500 505 510

0

5

10

02

04

06

08

minus5

minus10


050 0

10

480

490

500

510

Azimuth (m)

Cross track (m)

Rang

e (m

)

minus10

minus50



119904 ( 119891119910 120591 = 120591

0)

= sinc[119861119903( minus

2 (119867 minus 119911119901)

119888

)]

times sinc[119871119891(119891119910minus

2119910119901

120582119903119901cos120593119901

)]

times exp[minus11989521205871199102

119901

120582119903119901cos120593119901

] exp(minus1198954120587119903119901cos120593119901

120582

)

(22)





Range (m)

Cros

s tra

ck (m

)

495 500 505

34

36

38

40

42

44

46

02

04

06

08


02

04

06

08

Cross track (m)

Azi

mut

h (m

)35 40 45

4

6

8

10

12


02

04

06

08

Range (m)

Azi

mut

h (m

)

495 500 5054

6

8

10

12




Δ119891119910=

1

119871119891

(23)


119891119910119901=

2

120582


2119910119901

120582 (119867 minus 119911119901)

(24)


Δ119910 asymp

120582 (119867 minus 119911119901)

2119871119891

(25)




6 8 10minus30

minus20

minus13

minus3

0N

orm

aliz

ed am

plitu

de (d

B)

Azimuth (m)


minus30

minus20

minus13

minus3

0

Nor

mal

ized

ampl

itude

(dB)

498 500 502Range (m)


minus30

minus20

minus13

minus3

0

Nor

mal

ized

ampl

itude

(dB)

36 38 40 42 44 46Cross track (m)




119901119904119891 (120591 119891119910)

sim sinc [119861119886(120591 minus 120591

119901)] sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times sinc[119871119891(119891119910minus

2119910119901

120582119903119901cos120593119901

)]

(26)








5 Conclusions




Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Range (m)

Cros

s tra

ck (m

)

495 500 505

34

36

38

40

42

44

46

02

04

06

08


02

04

06

08

Cross track (m)

Azi

mut

h (m

)35 40 45

4

6

8

10

12


02

04

06

08

Range (m)

Azi

mut

h (m

)

495 500 5054

6

8

10

12




Δ119891119910=

1

119871119891

(23)


119891119910119901=

2

120582


2119910119901

120582 (119867 minus 119911119901)

(24)


Δ119910 asymp

120582 (119867 minus 119911119901)

2119871119891

(25)




6 8 10minus30

minus20

minus13

minus3

0N

orm

aliz

ed am

plitu

de (d

B)

Azimuth (m)


minus30

minus20

minus13

minus3

0

Nor

mal

ized

ampl

itude

(dB)

498 500 502Range (m)


minus30

minus20

minus13

minus3

0

Nor

mal

ized

ampl

itude

(dB)

36 38 40 42 44 46Cross track (m)




119901119904119891 (120591 119891119910)

sim sinc [119861119886(120591 minus 120591

119901)] sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times sinc[119871119891(119891119910minus

2119910119901

120582119903119901cos120593119901

)]

(26)








5 Conclusions




Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










6 8 10minus30

minus20

minus13

minus3

0N

orm

aliz

ed am

plitu

de (d

B)

Azimuth (m)


minus30

minus20

minus13

minus3

0

Nor

mal

ized

ampl

itude

(dB)

498 500 502Range (m)


minus30

minus20

minus13

minus3

0

Nor

mal

ized

ampl

itude

(dB)

36 38 40 42 44 46Cross track (m)




119901119904119891 (120591 119891119910)

sim sinc [119861119886(120591 minus 120591

119901)] sinc[119861

119903( minus

2 (119867 minus 119911119901)

119888

)]

times sinc[119871119891(119891119910minus

2119910119901

120582119903119901cos120593119901

)]

(26)








5 Conclusions




Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












5 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









research article a novel 3d imaging method for airborne downward-looking sparse array...

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