cs radar- new applications

7/23/2019 CS Radar- New applications

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Compressed Sensing Radar New Concepts ofIncoherent Continuous Wave Transmissions

Filippo Biondi Italian Ministry of Defence Military Radio Frequency Agengy (MiRFA)

Piazza Renato Villoresi RMA !iopippoo"g#ail$co#Abstract The most of the conventional SAR, ISAR or otherkind of high resolution radars are today based on LinearFrequency odulation !LF" transmissions, #here the rangeresolution is directly $ro$ortional to the used radio%frequencyband and the $ulse com$ression $rocedure is based on theatched Filter !F" theory& This traditional a$$roach can behighly limited to the transmitted signal band#idth, todaye'tremely de$endent on the electromagnetic s$ectrumavailability& This $a$er demonstrates that incoherenttransmissions can be focused using the (S reconstructiontheory, erasing the need of a $ulse com$ression stage, based onF, and a required receiver analog%to%digital conversionband#idth reduction& These kind of transmissions can be

useful for anti%)amming transmissions, needing lo#er $o#er, ifcom$ared bet#een conventional based high resolution radars&

Inde' Terms Synthetic aperture radar (SAR) CompressedSensing (CS) Interior !oint "ethods (I!") #igita$Wave$et Transform (#WT)%

I% INTR'CTI&NAR is designed to otain high reso$ution images of ani$$uminated scene% A conventiona$ SAR radar systemtransmits an e$ectromagnetic wideand pu$se that in

the most of the cases can e a *+" or a pseudorandomcoded noise signa$% The received echo is stored in memoryand post corre$ated using a "+ ,-.% /efore storing the range

history the physica$ signa$ is samp$ed at the Ny0uist Rate(NR) and a great amount of data can premature$y saturatethe onoard sensor storage memory% In ,1. a new approachto radar imaging ased on the concept of CS has eenintroduced% In CS an inco%erent $inear pro2ection is used toac0uire an efficient representation of a compressi$e signa$direct$y using 2ust a few measurements ,3.% The signa$ isthen reconstructed y so$ving an inverse pro$em ased on

the minimi4ing the - norm% This wor5 can e performedthrough *inear !rogramming (*!) Second &rder Cone!rogramming (S&C!) or maye using greedy a$gorithmsased on "atching !ursuit ("!)% Random pro2ection of ane$ectromagnetic physica$ signa$ can e performed sending tothe ground a Continuous Wave (CW) pseudonoise signa$and co$$ecting the scattered echoes having an energyaccording y the radar e0uation% The pro$em is that thediscreti4ed timefre0uency N y N p$ane grid has to econstituted y a radar scattering environment e6isting in asparse configuration% As indicated in ,37. can e assumedthat a signa$ is dense in the oservation domain and a sparseversion its resu$t can e estimated y pro2ecting the signa$

on the Ndimensiona$ space =[- &&'] thatprovides the fo$$owing 8sparse representation9

x=i=-

i=(

n

i

n

i

% (-)

Where the {ni}i{-& '} are the inde6 of each n ieing one of the e$ements of the sparsity inducting asis andthe parameters {i} are the associated coefficients% Theaove descried mode$ can e represented in the fo$$owingcompact matri6 representation9

x= (1)

The CS theory shows that aout 8random pro2ectionscontains enough information to reconstruct piece smoothsigna$s or mi6ed dense of spi5es and smooth signa$s thatcan e rea$istica$$y associated to the radar range profi$e% The

CS framewor5 measures the pro2ections i= x & Vi of thesigna$ in to an a$ternative set of asis functions {Vi}i &ca$$ed the measurement asis% In the present case this asisdoes not provide sparse representation of the e$ements{

i}i and an incoherence of two asis are needed in

order to recover a $arge set of i from an a$ternative si4ed

of measurements {yi }i $ This incoherence proprietycommon$y ho$ds many change of asis and inc$udes a$so the:AAR#WT (:#WT) case that is treated in this paper ,;.%

The mode$ recovers a consistent set of coefficients yperforming the CS optimi4ation procedure% Summari4ing

the pro$em given a set of yM

measurements

yi=x &V i with the " numer of random measurementsare concrete$y $ess than N dimensiona$ity of the origina$range vector% It is possi$e to recover the origina$ signa$having the sparsest transform {i } that agrees with theoserved coefficients yi 9

=argmin- s%t% y=where =V (3)

and V=[ V- &&VM* ]

*is the matri6 representation of the

measurement asis the so ca$$ed sensing matri6% The new

matri6 =V=[i &&'] is the ho$ographic ase%+or this wor5 the matrices that has een used to imp$ementthe ho$ographic mode$ are9 < constituted y random vectorsand constituted a matri6 that performs the #WT of thesmooth signa$ and the two matrices wor5s independent$y%The origina$ set of coefficients {i} are recovered y CSusing a *og/arrier A$gorithm where detai$s are reportedin ,7. where the C& pro$em is so$ved y a S&C! So$ution

S

2015 3rd International Workshop on Compressed Sensing Theory and its Applications to Radar, Sonar, and Remote Sensing(CoSeRa)

978-1-4799-7420-7/15/$31.00 2015 IEEE 204


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(S&C!s) using a generic $ogarrier a$gorithm trough asuccessfu$$y computationa$ ott$enec5 Newton ca$cu$ation%The idea is to so$ve tomographic pro$ems y interiorpointmethods for so$ving conve6 optimi4ation a$gorithms thatinc$udes a generic optimi4ation pro$em with ine0ua$ityconstraints9 Interior!oint"ethods (I!") so$ves the genericoptimi4ation pro$em y app$ying the Newton=s method inorder to convert an ine0ua$ity constrained pro$em versus a

se0uence of e0ua$ity constrained pro$ems% This paperconsiders a particu$ar I!" a$gorithm ca$$ed the $ogarriermethod% The goa$ is to appro6imate$y formu$ate constrainedpro$em to which Newton=s method can e app$ied% To so$vethe pro$em has een imp$emented a >matri6 free? so$verased on con2ugate gradients where detai$s are e6p$ained in,7.% The optimi4ed vertica$ ref$ectivity function + is estimatein a varia$e Newton steps and numer of $ogarrieriterations% The new theory of CS ena$es so thereconstruction of sparse signa$s using far fewer samp$es ormeasurements than NR% This advantage can e usefu$ inorder to e$iminate the need for the "+ in the radar receiverto reduce the samp$ing fre0uency e$ow the NR to save theon oard storage memory and to save of the e$ectromagnetic

and pu$se signa$ occupancy ,@ .% This wor5 is structuredaccording to the fo$$owing scheme9 Section II is descriesthe recovery mode$ structure formed y a random comp$e6matri6 2ointed y an :AAR wave$et transformation matri6

having the function to sparsify synthetic targets that e6istsin a smooth scattering configuration% Section III isdedicated to the e6perimenta$ resu$ts where it isdemonstrated the feasii$ity to reconstruct radar rangeprofi$es according to three different typo$ogies9 pointscattererB distriuted scatterer and hyrid scattererconstituted y the presence of point and distriuted

scatterers% The resu$ts are estimated app$ying the "+ andthe CS theory% It is demonstrated the e6istence of a greatperformances enhancement estimated from the CS resu$ts%

II% T: RC&


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Isometry !ropriety (RI!) and can e suita$e to reconstructradar range profi$es constituted y iso$ated point targetsdistriuted targets and range profi$es constituted y the othof them% The generated hyrid mode$ wi$$ e the transmittedcoded signa$% The impact of the e$ectromagnetic energyagainst the targets wi$$ generate the oservation vector

according to the radar e0uation% The fo$$owing paragraphwi$$ provide some theory descriing the #WT used in thismode$ in order to generate sparsity%

III% G!RI"NTA* RS'*TSThis section reports a$$ the e6perimenta$ resu$ts performedon three different simu$ated datasets immersed in noise%The first one is a range synthetic stac5 formed y -F pointscatterers with different energy pea5s% The second dataset isa range synthetic stac5 having a comp$ete$y smooth energyvariation the "AT*A/ function :eaviSine has eenchosen in order to satisfy this case of study% The $ast datasethas the same range synthetic data vector configurationdescried in the previous$y case ut some point scatterers

having high Radar Cross Section (RCS) on the smoothorographic surface has een added in order to simu$ate ahyrid and more comp$icated environment composed of asmooth energy variation signa$ mi6ed with strong energyradar echoes% The first case can e assimi$ates as an ISARradar ac0uisition ecause the range is composed y pointscatterers e6isting in a deep $ac5 and $ow energy

ac5ground% The second and the $ast studycase are morecomp$iant to SAR ac0uisitions where two environmentsconstituted y smooth and regu$ar$y distriutedac5scattered energy a$$ocated at different range reso$utionce$$s has een designed% The resu$ts proposed in this paperare estimated using a "+ co$$ecting signa$s received y a

c$assica$ chirp radar and using the - norm

minimi4ation method so transmitting noise signa$saccording to the scheme in +ig%- and 1 depicted% Thesimu$ated c$assica$ radar has an *+" ased transmittedetween 1HH":4 of and and in a centra$ fre0uency set inthe Gand%

,$- Point .catterers Range Profile (I.AR /ase)The case of study treated in this section is referred to atypica$$y range stac5 ac0uired y an ISAR ecause thecompressed range profi$e is formed y severa$ pointscatterers positioned on a deep $ow energy ac5ground% Thesynthetic environment is in +ig%3 (Top) depicted where theassociated compressed range profi$e is in +ig%3 (/ottom)depicted% +or this case a compressor imp$emented y a "+

has een used% In +ig%; (Top) the synthetic environment iscompared to the - norm minimi4ation resu$t representedin +ig%; (/ottom)% The CS resu$t is estimated providing a$soan undersamp$ing factor of H%7% In +ig%7 the error e6istingetween the synthetic data and the estimated matchedfi$terresu$t is reported% In +ig%@ the the CS error is reported%Comparing the $ast two resu$ts it is possi$e to appreciate

Fig&.9 Reconstruction error estimated y considering theresu$t in +ig%3 depicted% The matchedfi$ter methodgenerates great errors ecause of the presence energyside$oes%

Fig&/9 Reconstruction error estimated y considering theresu$t in +ig%; depicted% The CS method generates $essenergy error if compared to the error $eve$ estimated y the"+%

Fig&09 Radar range profi$e estimated y compressing thechirp signa$ response using the "+% +ig% (Top)9 Synthetictarget% +ig% (/ottom)9 stimated target%

Fig&1 Radar range profi$e estimated y $-normminimi4ation% +ig%F (Top)9 Synthetic target% +ig%F (/ottom)9stimated target%


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that the CS error is far $ess than the c$assica$ one estimatedy the matchedfi$ter%

,$, Distri!uted .catterers Range Profile (Regularrograp%ic Variation& First .AR /ase)

The case of study treated in this section is referredprocessing a typica$$y range stac5 ac0uisition performed y

a SAR where the compressed range profi$e is constituted ysevera$ point scatterer positioned a$ong a regu$ar$y andsmooth$y energy ac5scattering range profi$e for each rangereso$ution ce$$% In order to simu$ate this 5ind of signa$ the"AT*A/ :eaviSine function has een generated% Thesynthetic environment is in +ig% (Top) depicted where theassociated compressed range profi$e is in +ig% (/ottom)depicted% +or this case a compressor imp$emented y a "+has een used% In +ig%F (Top) the synthetic environment is

compared to the - norm minimi4ation resu$trepresented in +ig%F (/ottom)% The CS resu$t is estimatedproviding a$so an undersamp$ing factor of H%7% In +ig% theerror e6isting etween the synthetic data and the estimatedmatchedfi$ter resu$t is reported% In +ig%-H the CS error is

reported% Comparing the $ast two resu$ts it is possi$e toappreciate that the CS error is far $ess than the c$assica$ oneestimated y the matchedfi$ter%

,$0 Distri!uted and Point .catterers Range Profile(Irregular rograp%ic Variation& .econd .AR /ase)

The case of study treated in this section is referred to amodified version of the previous studycase ecause somepoint scatterers with right RCS has een inserted on the:eaviSene "AT*A/ function% These targets simu$atescorner ref$ectors and they are regu$ar$y spaced one to eachother% The synthetic range profi$e is in +ig% -- (Top)depicted oserving the picture it is possi$e to view the

scattering echoes of two corner ref$ectors situated at therange point near @7H% In +ig%-- (/ottom) it is possi$e tooserve the resu$t estimated y the "+% In +ig%-1 (/ottom)the CS compressed range profi$e is reported% &serving theresu$t it is possi$e to appreciate that the range profi$e iscorrect$y reconstructed a$so for the present hyrid caseconstituted y point and distriuted scatterers% The error$eve$s are reported in +ig%-3 and +ig%-; where thematchedfi$ter and the CS resu$ts are respective$y depicted%The error attriuted to the CS processing is orders $ess thatthe error attriuted to the c$assica$ radar where amatchedfi$ter is emp$oyed% In +ig -7 is reported aparticu$ar of the range estimated vector starting from therange point numer @HH up to the range point numer HH

of the resu$ts depicted in +ig%-- (Top and /ottom) and+ig%-1 (Top and /ottom)% The resu$ts are the one estimatedfrom the hyrid case composed y the point and continuousscatterers e6isting a$ong the range $ine% The resu$t of +ig% -7(*eft) is the synthetic target the resu$t of +ig%-7 (Center) isthe one estimated y the matchedfi$ter and the resu$testimated in +ig%-7 (Right) is the resu$t estimated y the

Fig&29 Reconstruction error estimated y consideringthe resu$t in +ig% depicted% The matchedfi$ter methodgenerates great energy error amount%

Fig&--9 Radar range profi$e estimated y $-normminimi4ation% +ig%-- (Top)9 Synthetic target% +ig%--(/ottom)9 stimated target%

Fig&-*9 Radar range profi$e estimated y $-normminimi4ation% +ig%-1 (Top)9 Synthetic target% +ig%-1(/ottom)9 stimated target%

Fig&-39 Reconstruction error estimated y considering theresu$t in +ig%F depicted% The $-minimi4ation methodgenerates a very $ow error $eve$ if compared to the onemeasured from the matchedfi$ter data%


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CS% +or the CS compression method no ripp$e andside$oes are detected%

IApprova4ione de$!iano Na4iona$e di Riparti4ione #e$$e +re0uen4e? (M' SerieMenera$e n% 13 in data 1---1HHF Supp$% &rd% n% 177)%

Fig&-+9 Reconstruction error estimated y considering theresu$t in +ig%-- depicted% The matchedfi$ter methodgenerates a great error $eve$ if compared to the onemeasured from the CS resu$t estimated y $-minimi4ationoptimi4ation process%

Fig&-9 Reconstruction error estimated y considering theresu$t in +ig%-1 depicted% The CS generates a very $ow error$eve$ if compared to the one measured from thematchedfi$ter data%

Fig&-.9 !articu$ar starting from the range point numer@HH up to the range point numer HH of the resu$tsdepicted in +ig%-- (Top and /ottom) and +ig%-1 (Top and/ottom)% The resu$ts are the one estimated from the hyridcase composed y the point and continuous scattererse6isting a$ong the range $ine% +ig%-7 (*eft)9 Synthetic data+ig%-7 (Center)9 "atchedfi$ter resu$t +ig%-7 (right)9 CSresu$t estimated y the CS%


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