Parameter Optimization of Multi-element Synthetic Aperture Imaging Systems

Transcription

1 Paaee Opiizaio of Muli-elee Syheic Apeue Iagig Syses Vea Beha Isiue fo Paallel Pocessig Bulgaia Acadey of Scieces 5-A Acad. G. Bochev S., Sofia 1113, Bulgaia E-ail: Received: Jauay 19, 7 Acceped: Ma 16, 7 Published: Ma 7, 7 Absac: I coveioal ulasoud iagig syses wih phased aays, he fuhe ipovee of laeal esoluio equies elagig of he ube of aay elees ha i u iceases boh, he coplexiy ad he cos, of iagig syses. Muli-elee syheic apeue focusig (MSAF) syses ae a vey good aleaive o coveioal syses wih phased aays. The beefi of he syheic apeue is i educio of he syse coplexiy, cos ad acquisiio ie. I a MSAF syse cosideed i he pape, a goup of elees asi ad eceive sigals siulaeously, ad he asi bea is defocused o eulae a sigle elee espose. The echo eceived a each elee of a eceive sub-apeue is ecoded i he copue eoy. The pocess of asissio/ecepio is epeaed fo all posiios of a asi sub-apeue. All he daa ecodigs associaed wih each coespodig pai asi-eceive sub-apeue ae he focused syheically poducig a low-esoluio iage. The fial high-esoluio iage is foed by suig of he all low-esoluio iages associaed wih asi/eceive sub-apeues. A poble of paaee opiizaio of a MSAF syse is cosideed i his pape. The qualiy of iagig (laeal esoluio ad coas) is expessed i es of he bea chaaceisics bea widh ad side lobe level. The copaiso bewee he MSAF syse descibed i he pape ad a equivale coveioal phased aay syse shows ha he MSAF syse acquies iages of equivale qualiy uch fase usig oly a sall pa of he powe pe iage. Keywods: Ulasoud iagig, Syheic apeue desig, Paaee opiizaio, Siulaio aalysis. Ioducio Iages poduced by ulasoud iagig syses us be of sufficie qualiy i ode o povide accuae cliical iepeaio. The iage qualiy (laeal esoluio ad coas) is piaily deeied by he bea chaaceisics of a asduce used i a iagig syse. I coveioal ulasoud iagig syses wih phased aay (PA), all asduce elees asi sigals ad eceive he echoes, efleced fo he issue. Thus he ode coveioal PA syses poduce high-esoluio iages a high cos because he syse coplexiy ad hus he syse cos deped o he ube of asduce elees. The fuhe ipovee of laeal esoluio i a coveioal PA iagig syse equies elagig of he ube of asduce elees. I is ofe o possible because of physical cosais o oo high cos. The effec of high laeal esoluio ad coas ca be accoplished by usig vaious syheic apeue echiques. The beefi of he syheic apeue is i educio of he syse coplexiy ad cos. Moeove, i a syheic apeue iagig syse, he acquisiio ie ca be dasically educed ad he dyaical seeig ad focusig ca be applied i boh asi ad eceive. Thee ae diffee ehods fo syheic apeue iagig Syheic Receive Apeue (SRA) echique, Syheic Tasi Apeue (STA) echique, Syheic Apeue Focusig (SAF) echique, ad Muli-Elee Syheic Apeue Focusig (MSAF) echique. 64

2 I he PA iagig, he iage acquisiio ie is evaluaed as TPA = TREC, whee T REC is he ie eeded o acquie RF-sigals a a sigle sca diecio, is he ube of all sca diecios (ube of iage s). I he SRA iagig, a lage ube of asduce eceive elees is addessed wihou he sae ube of paallel eceive chaels [7, 8]. The SRA iagig echique ivolves asiig wih a full asi apeue ad eceivig wih seveal eceive sub-apeues. I eas ha seveal pulse asissios ae eeded o acquie a sigle RF. The iage acquisiio ie is evaluaed as TSRA = M TREC, whee M is he ube of asissios. I is iceased M ies copaed o he iage acquisiio ie of a PA iagig syse. I he STA iagig, a each ie oe aay elee asis a ulasoud pulse ad all asduce elees eceive he echoes [1, 9, 1]. The asduce elees ae fied cosequely, oe afe he ohe, ad he echo sigals eceived a each asduce elee ae soed i he copue eoy. Whe he sigals eceived fo each pai asieeceive have bee ecoded, a STA iage is ecosuced by he appopiae algoih. The ie eeded o acquie each STA iage is evaluaed aststa = TREC, whee is he ube of aay elees. The iage acquisiio ie is educed ( / ) ies copaed o a equivale PA iagig syse because as a ule >. The disadvaage is a huge daa eoy equied o soe RF-s i ode o poduce a STA iage. I he SAF iagig, each asduce elee acs as boh asie ad eceive [11, 1]. The full syheic apeue is syhesized afe all () aay elees have asied ad eceived he sigals. The iage acquisiio ie is evaluaed astsaf = TREC. The iage acquisiio ie is educed ( / ) ies copaed o a equivale PA iagig syse. The syse coplexiy is dasically educed because he cicuiy fo oly oe pai asie -eceive is equied. The Sigal-o-oise aio (SR), howeve, is educed ies copaed o a equivale PA iagig syse. I he MSAF iagig, i he asi ode, a goup of elees asi sigals siulaeously, ad he asi bea is defocused o eulae a sigle elee espose [, 3, 4, 5]. I he eceive ode, a goup of elees eceive he echo sigals. The acousic powe ad heefoe he SR ae iceased copaed o he SAF iagig whee a sigle asduce elee is used i asi. The iage acquisiio ie is evaluaed as TMSAF = M T REC, whee M is he ube of asissios. The iage acquisiio ie is educed ( / M ) ies copaed o a equivale PA iagig syse. Whe M <<, he educio i acquisiio ie ca be sufficiely lage. A vaia of he MSAF iagig, whee oly oe elee (L = 1) is dopped a he ex asissio, is cosideed i [5]. I his syse, a lage ube of pulse asissios (M >> 1) us be eployed i ode o obai a high-esoluio iage because he ube of elees used i a eceive sub-apeue is vey sall ( = 5). I geeal, he iage qualiy poduced by a MSAF iagig syse is piay deeied by he followig syse paaees he ube of asissios (M), he ube of elees i a asi sub-apeue ( ), he ube of elees i a eceive sub-apeue ( ), ad 65

3 he ube of elees (L) dopped a he ex asissio. I os edical applicaios, he iage acquisiio ie is equied o be iiized i ode o avoid he phase eos caused by issue oio duig he daa acquisiio. I ode o saisfy his equiee, oly a few pulse asissios should be used fo MSAF iage foaio (M = 3, 5, 7). Whe he ube of asissios is pese, i is vey ipoa o fid he opial cobiaio of he ohe syheic apeue paaees (L, ad ) i ode o guaaee he eeded qualiy of iagig. I his pape, he paaee opiizaio of a MSAF syse, i which he ube of asissios pe iage foaio is sall, is descibed. The syse paaees, M, ad L ae opiized usig he effecive apeue cocep fo bea pae aalysis. The iage qualiy (laeal esoluio ad coas) is expessed i es of he wo-way bea pae chaaceisics bea widh ad side lobe level. The ifluece of saplig ae of RF sigals o he iage qualiy is also ivesigaed. The copaiso aalysis bewee a 79-elee MSAF syse ad a equivale 79-elee PA syse shows ha he MSAF syse acquies iages of a equivale qualiy 4 ies fase usig oly 1/5 he powe pe iage. MSAF iagig Coside a MSAF syse wih a -elee viual a aay. A he fis sep, a a subaay wih elees asis a ulasoud pulse, ad a a sub-aay wih elees eceives he echo sigals. A he ex sep, he wo sub-apeues (asi ad eceive) ae oved by L elees ad he pocess of asissio/ecepio is epeaed. The piciple schee of daa acquisiio is illusaed i Fig. 1. TRASDUCER ARRAY T/Rec Seps 1 j i k 1 j1 i1 k1 L Viual asi elee M-1 L(M-1) jm-1 im-1 km-1 Fig. 1 Daa acquisiio i a MSAF iagig syse The ube of asissios eeded o ceae a syheic apeue equivale o a viual physical aay wih elees is: M = ( ) / L + 1 (1) We assue ha he sigal pocessig is ealized i he base bad fequecy doai, ad (M x ) s of he coplex apliude ae soed i he copue eoy afe he quadaue deecio. The iage foaio is caied ou i wo sages (Fig. ). 66

4 Sub-apeue 1 Sub-apeue Sub-apeue M RF RF RF D A T A A C Q U I S I T I O Iage 1 Iage Iage M Si Θ Si Θ Si Θ P A R T I A L B E A M F O R M I G Σ HIGH-RESOLUTIO IMAGE Fig. Iage foaio i a MSAF iagig syse Fisly, he coplex apliude of each low-esoluio iage associaed wih he coespodig sub-apeue pai asie-eceive is obaied as a paial beafoig su: S 1 = (, θ ) = wu, ( τ, ) exp( jφ, ), = 1,..., M () whee S (, θ ) is he coplex apliude focused a he poi (,θ), U, is he coplex apliude eceived a he h elee of he h eceive sub-apeue, τ, ad Φ, ae he ie delay ad phase applied o he h elee of he h eceive sub-apeue duig beafoig, w is he weighig coefficie applied o he h elee of a eceive subapeue. The phases Φ, used i beafoig ae evaluaed as: Φ, = πf τ, (3) The paaee f is he ceal fequecy of a asduce. The coplex apliude of he fial high-esoluio iage is foed as a su: S MSAF M 1 = (, θ ) = S (, θ ) (4) The fial B-iages ae obaied fo (4) afe he followig sigal pocessig evelope deecio, logaihic copessio, ad sca covesio. Paial beafoig I he paial beafoig, he daa ecodigs associaed wih a sub-apeue pai asie-eceive ae focused a a poi (, θ), whee θ ad ae he pola coodiaes. We assue ha he pola coodiae syse is oigiaed fo he syheic apeue cee as is show i Fig

5 -h eceive sub-apeue x x, -h asi sub-apeue, s θ -s (,θ) Syheic apeue cee. Fig. 3 The geoey of seeig o poi (Θ) The wo-way ie delay ha us be applied o he h elee of he h eceive sub-apeue is: τ + ec, = τ τ, whee τ, is he oe-way popagaio ie fo he h asi sub-apeue o he poi (, ec θ), ad τ, is he oe-way popagaio ie fo he poi (, θ) o he h elee of he h eceive sub-apeue. These ie delays ae evaluaed as: τ τ ec, = = + x x c + x, x c siθ, siθ The posiio of a asi sub-apeue ( x ) ad he posiio of a eceive sub-apeue elee ( ) abou he syheic apeue cee ae evaluaed as: x, M 1 Lλ 1 λ x = ( 1 ), x, = x + ( 1 ) (7) whee λ is he wave legh coespodig o he ceal fequecy f of a asduce. Spaial Saplig The spaial saplig of he bea space us be cosise wih he spaial yquis saplig cieia. Accodig o [6], he spaial fequecy badwidh ca be foud usig he effecive apeue fucio of a iagig syse. The effecive apeue fucio of a MSAF syse ca be expessed as: C = M = 1 A B, A = [,,... a( i )...,,], =,..., b,... b... b,..] (8) B [ + / / whee A is he full asi apeue duig he h fiig, B is he full eceive apeue duig he h fiig, a( i ) = 1 is he weighig coefficie applied o he h asi subapeue ad ( b /, b / 1,...,,..., / 1, / ) + b b+ b + ae weighig coefficies applied o he elees of he h eceive sub-apeue, ad is he covoluio opeao. The posiio of he h asi/eceive sub-apeue cees wihi a syhesized viual aay is (5) (6) 68

6 i = / + L( 1) + 1. The ube of o-zeo elees ( eff ) of he effecive apeue fucio (8) ca be valuaed as: [ + L( M 1)] E( / ) 1 (9) eff = whee E(.) is he iege ube value. I case of odd values of, he expessio (9) akes he fo: ( ) = L( M 1) + (1) eff MSAF I us be oed ha he effecive apeue exe of he h sub-apeue pai asieeceive is: ( ) = (11) eff LOW The Fouie asfo of he effecive apeue fucio gives he wo-way adiaio bea pae of a MSAF syse: eff 1 π W( k) = C( k)exp( j k) (1) = eff The spaial fequecy idex k i (1) aps io he bea agle θ by: k eff 1 si θ =, k =, ± 1,..., ± (13) eff Accodig o (13) he axial saplig ieval i θ fo boh, he low-esoluio iage ad he high-esoluio iage, is defied by he followig expessios: ( si θ ) = / ; ( si θ ) = / (14) high eff low Accodig o (14), he up saplig faco fo a low-esoluio iage is: K up =1 + D, whee D = L( M 1) / (15) The paaee D i (15) is he iepolaio faco of low-esoluio iages. Paaee opiizaio I he MSAF iagig, he iage qualiy depeds o he size of a eceive sub-apeue ( ), he ube of asissios (M) ad he sub-apeue spacig (L). The ube of elees () i a syheic apeue is deeied by hese paaees: = ( M 1) L + (16) I os edical applicaios he iage acquisiio ie is equied o be exeely sho o avoid pobles caused by he issue ovee duig he daa acquisiio. To saisfy such a equiee i is desiable o use oly a few asissios fo iage foaio (M < 1). I his secio, we assue ha M = 3, 5 ad 7. The ai of paaee opiizaio is o fid such a cobiaio of paaees M, ad L ha guaaees he sae high qualiy of iages as a equivale PA iagig syse wih asduce elees. The iage qualiy is expessed i es of he bea pae paaees of a syse - bea widh ( Θ), he fis side lobe peak (SLB-ea) ad he fa side lobe peak (SLB-fa). Fisly, he iage qualiy is aalyzed depedig o he paaee ses (M, L, ). Secodly, he ifluece of such facos as he ie 69

7 saplig ae ad he sigal-o-oise aio (SR) o he iage qualiy is sudied. I his secio fo he sake of siplificaio we assue ha a asi sub-apeue of a MSAF syse cosiss of oly oe elee, i.e. = 1. Bea pae aalysis. The effecive apeue fucio ad he coespodig wo-way adiaio bea pae of a MSAF syse is evaluaed fo paaee ses (, M, L) ha saisfy he followig equiee: ( M 1) L / (17) The equiee (17) is eeded o poduce he effecive apeue fucio ha is uiodal ad soohly filled. Such a effecive apeue fucio is foed by coveioal PA syses [6]. The effecive apeue ad he coespodig wo-way adiaio bea pae ca be copued by he expessios (8) ad (1), especively. I his sudy, fou diffee eceive sub-apeues wih 15, 33, 63 ad 19 elees ae used fo bea pae aalysis of a MSAF syse. Fo he case of Chebyshev weighig, he bea paaees of hese sub-apeues ae peseed i Table 1. This is a case of a sigle lowesoluio iage ceaed by a sigle sub-apeue pai asie-eceive, i.e. whe M = 1. I ca be see ha he bea widh ( Θ ) vaies i age fo (fo a 15-elee eceive sub-apeue) o 1.7 (fo a 19-elee sub-apeue). Fo he case whe M > 1, he bea paaees evaluaed fo vaious cobiaio of (M, L, ) ae peseed i Tables -5. Table 1. Bea paaees of a sigle eceive sub-apeue Chebyshev widow (5dB) Θ SLB-ea [db] SLB-fa [db] The bea paaees of a MSAF syse ha eploys a 15-elee eceive sub-apeue ae peseed i Table. These esuls show ha oly a sigle se of syse paaees ( = 15, M = 3, L = 1) eables o obai a high-coas MSAF iage povidig he side lobe level of - 6dB. The bea widh is educed fo fo a low-esoluio iage (M = 1) o 1.8 fo a MSAF iage, whe hee asissios ae used (M = 3) fo syheic apeue foaio. The syheic apeue of such a MSAF syse is equivale o a aay of 17 elees ( = 17). Table. Bea paaees of a MSAF syse (15-elee sub-apeue) Chebyshev widow - (5dB) M L Θ SLB-ea [db] SLB-fa [db]

8 The bea chaaceisics of a MSAF syse ha eploys a 33-elee eceive sub-apeue ae show i Table 3. Bea pae aalysis shows ha hee ae hee appopiae ses of syse paaees ha ake possible o obai he high-coas MSAF iages. These opial ses, ( = 33, M = 3, L = 3), ( = 33, M = 5, L = ) ad ( = 33, M = 7, L = 1), povide he side lobe level of -6dB, -5dB ad -7 db, especively. The secod vaia of syse paaees ( = 33, M = 5, L = ), howeve, is oe pefeable povidig he iial bea widh (4.95 ). This vaia of a MSAF syse ceaes a syheic apeue equivale o a a aay wih 41 elees ( = 41). Table 3. Bea paaees of a MSAF syse (33-elee sub-apeue) Chebyshev widow (5dB) M L Θ SLB-ea [db] SLB-fa [db] The bea chaaceisics of a MSAF syse, which eploys a 63-elee eceive subapeue ae show i Table 4. These esuls show ha hee ae hee opial paaee ses - ( = 63, M = 3, L = 6), ( = 63, M = 5, L = 4) ad ( = 63, M = 7, L = ) ha povide he elaively low level of side lobes (-5dB ad below). The secod paaee se ( = 63, M = 5, L = 4), howeve, is he os pefeable povidig he iial bea widh (.5 ). This se of paaees ceaes a syheic apeue equivale o a a aay wih 79 elees ( = 79). Fo ha vaia of a MSAF syse, he wo-way bea pae ad he coespodig effecive apeue fucio ae ploed i Fig. 4 ad Fig. 5, especively. I ca be see ha he side lobe level of -5dB is aiaied i he full 9 sca seco, ad he effecive apeue fucio is a liied covex fucio wih oly oe axiu. The bea chaaceisics of a MSAF syse ha eploys a 19-elee eceive sub-apeue ae show i Table 5. These ueical esuls show ha hee ae hee opial paaee ses ( = 19, M = 3, L = 13), ( = 19, M = 5, L = 8) ad ( = 19, M = 7, L = 6) ha poduce he elaively low level of side lobes (-5dB ad below). The secod paaee se ( = 19, M = 5, L = 8), howeve, is he os pefeable povidig he iial bea widh (1.6 ). This vaia of paaees ceaes a syheic apeue equivale o a a aay wih 161 elees ( = 161). 71

9 Table 4. Bea paaees of a MSAF syse (63-elee sub-apeue) Chebyshev widow (5dB) M L Θ SLB-ea [db] SLB-fa [db Bea pae (M-SAF-ehod); M=5 ;L=4 ;=63 ; Dela Q =.5 4 Effecive apeue (M-SAF-ehod); M=5 ;L=4 ;=63 ; eff = 95.5 Apliude [db] Bea Agle [degees] Fig. 4 Bea pae of a MSAF syse ( = 63; M = 5; L = 4) Fig. 5 Effecive apeue of a MSAF syse ( = 63; M = 5; L = 4) Table 6 copaes fou opial vaias of a MSAF syse wih a coespodig equivale PA syse wih he sae ube of phased aay elees (). I ay be see ha boh iagig syses, MSAF ad PA, poduce he ealy equivale iage qualiy bu he MSAF syses uilize he salle ube of acive asduce elees fo iage foaio ha he equivale coveioal PA syses ( << ). 7

10 Table 5. Bea paaees of a MSAF syse (19-elee sub-apeue) Chebyshev widow (5dB) M L Θ SLB-ea [db] SLB-fa [db] Table 6. Paaees of MSAF ad PA syses MSAF iagig syse PA iagig syse M L Θ SLB [db] Θ SLB [db] Saplig ae. A 9 seco sca of a poi age is siulaed i ode o sudy he effec of he saplig ae o he qualiy of MSAF iages. I siulaios, he sca agle is iceeed i seps of.45 fo -45 o 45. I case of a 33-elee eceive sub-apeue, he poi age was locaed a 7 fo he cee of a syheic apeue. I case of a 63- elee eceive sub-apeue, he disace o he poi age was. Fo each subapeue pai asie-eceive, all soed A-scas wee base-bad RF sigals sapled a ae of 5 MHz, 1 MHz ad 8 MHz. The bea widh ( Θ ) ad he side lobe peak (SLB) of a MSAF syse evaluaed fo hee vaias of saplig ae ae peseed i Table 7. Table 7. Bea paaees fo hee vaias of saplig ae Paaee Fs = 5 MHz Fs = 1 MHz Fs = 8 MHz M L Θ SLB [db] Θ SLB [db] Θ SLB [db] The followig paaees ae used i siulaios: cee fequecy 3.5 MHz, pulse duaio 1µs, SR 5dB, dyaic age 5dB, weighig i eceive Chebyshev (-5dB). The 73

11 ueical esuls show ha he saplig fequecy us be chose vey caefully i ode o guaaee he accepable level of side lobes. I ou case he saplig fequecy highe ha 8 MHz akes possible o esue he ea-opiu side lobe level of he MSAF bea pae. Resoluio of closely-spaced poi ages. Laeal esoluio of a MSAF syse is defied as a iial disace (i sca agle) bewee wo poi ages a which sepaae egisaio is jus disiguished o he display. A capabiliy of a MSAF syse o esolve wo closelyspaced poi ages wih diffee efleciviy is sudied by siulaio. Wih his ai i id, a seco sca of wo closely-spaced poi ages locaed a a disace of is siulaed i wo sages. The fis sage of siulaio fis he case whe boh poi ages have he ideical efleciviy. I his siulaio he SR is assued o be db. The ex sage of siulaio fis he case whe he poi ages ae o ideical i efleciviy, ad he diffeece i coas is 5dB. I boh cases i is assued ha he fis age is locaed a he sca agle of, he secod age a he sca agle of Θ, whee Θ is he bea widh of a MSAF syse. Fo a siulaed syse wih paaees = 63, M = 5 ad L = 4, he bea widh is Θ =.4. The iesiy of sigals efleced fo he closely-spaced ages is ploed i Fig. 6 fo ideical ages i coas ad Fig. 7 fo diffee ages i coas. I ca be see ha wo closely-spaced poi ages ae disiguishable oly if he diffeece i iesiy of hei echo sigals is isufficie. 1 Siulaed Bea pae (M-SAF ehod) Tage1 Tage 1 Siulaed Bea pae (M-SAF ehod).9.9 Tage Apliude [db] Apliude [db] Tage Bea Agle [degees] Fig. 6 Echoes fo closely-spaced poi ages (SR 1 = db; SR = db) Bea Agle [degees] Fig. 7 Echoes fo closely-spaced poi ages (SR 1 = db; SR = 15dB) Copaiso aalysis The effeciveess of a MSAF iagig syse is copaed wih ha of a equivale phased aay iagig syse. The copaiso aalysis is based o esiaio of he followig qualiy paaees sigal-o-oise aio, cosued powe pe iage, axial iesiy of a pulse, iage acquisiio ie, bea widh, ad side lobe level. The syse paaees used fo esiaio of he qualiy paaees ae show i Table 8. I he coveioal PA iagig, he iiu ie equied o fo a sigle high-esoluio iage is popoioal o he ube of sca s ad he ie eeded o ecod he eceived sigals duig he asissio, i.e. T = R c, ad c is he velociy of soud. Fo PA ax / = 1 ad R ax =, he iage acquisiio ie fo a PA iagig syse is T PA = 5 s. I eas ha a 1- sca seco ca be acquied a a ae of appoxiaely 19 iages pe secod. To icease he fae ae of a coveioal PA scae, he deph of view us be sacificed by deceasig he size of he iage, o he laeal esoluio us be sacificed by educig he ube of iage s. Howeve, he sae is o ue fo a MSAF iagig syse, i which he iage acquisiio ie is deeied by he ube of asi 74

12 sub-apeues ad equals T MSAF = MRax / c. I ha case, he pocess of iage acquisiio is speeded up ( TPA / TMSAF = / M ) ies. Fo M = 5, he iage acquisiio ie fo a MSAF syse is T MSAF = 1.3 s. I eas ha a 1- sca seco ca be acquied a a ae of 77 iages pe secod. I is 4 ies fase ha fo a PA syse. Table 8. Syse paaees used fo copaiso aalysis Syse paaees PA Syse MSAF Syse Sybol Value Sybol Value # of aay elees # of elees i a asi sub-apeue 79 1 # of elees i a eceive sub-apeue # of asissios - - M 5 Spacig bewee asi/eceive sub-apeues - - L 4 Exciaio pulse apliude 1 G 8 # of iage s 1 1 Maxial deph of iagig, [] R ax R ax Agula seco, [ ] Weighig fucio Chebyshev Chebyshev I a coveioal PA syse, he asi powe pe iage is popoioal o he ube of aay elees, he ube of sca s ad he aveage powe asied fo each elee, i.e. PPA = p. I a MSAF syse, a asi sub-apeue wih defocused elees siulaes a sigle viual asi elee i ode o icease he asied powe. The apliude of esulig asissio is iceased i popoio o. To icease he asi powe he exciaio apliude is aplified G ies (see Table 8). Theefoe, fo a MSAF syse, he asi powe pe iage is P G MSAF = M p. Copaed o a PA syse, he elaive asi powe fo a MSAF syse is: P / P = ( M G ) /( ) (18) MSAF PA Fo he syse paaees show i Table 8, he aio (18) is 1/5. I eas ha he asi eegy pe iage is educed 5 ies copaed o a equivale PA syse. Siilaly, he aio of he axiu iesiy i he asi ode ca be calculaed fo he wo iagig syses syse: I MSAF / I PA ( G ) / (19) Fo he syse paaees show i Table 8, he aio (19) is 1/8. I eas ha i he asi ode, he sigal iesiy is educed 8 ies copaed o a equivale PA syse. Accodig o [6], i a coveioal PA syse, he sigal is popoioal o he ube of aay elees, i.e. Sigal ~, while he oise is ivesely popoioal o he squae oo of he ube of aay elees, i.e. oise ~ 1/() 1/. Theefoe, fo a PA syse he sigal ooise aio is SR PAF ~ 3/. Siilaly, fo a MSAF syse we have: Sigal ~ G( ) 1/ ; 75

13 oise ~ 1/( M) 1/ ; SR MSAF ~ G( M ) 1/. The elaive SR fo a MSAF iagig syse is: SR 3 MSAF / SRPA = G M / () Fo he syse paaees show i Table 8, he aio () is.65. Theefoe, fo a MSAF syse, he losses i SR ae equal o 3.81 db, copaed o a PA syse. eveheless, he sigal-o-oise aio i a MSAF syse is wihi 6dB of a equivale PA syse. Table 9 suaizes he copaiso bewee he wo iagig syse. The copaiso aalysis shows ha he MSAF syse acquies iages of equivale iage qualiy 4 ies fase usig oly 1/5 he powe pe iage. Table 9. Qualiy paaees Syse Paaees PAF syse MSAF syse Relaive SR db -3.8dB Relaive powe/iage 1 1/5 Relaive axiu Iesiy 1 1/8 Relaive iage acquisiio Tie 1 1/4 Bea widh.79.5 Side lobe level -5dB -5dB Fially, i he MSAF syse, he coo ube of acive elees used i each asissio/ecepio is sufficiely less ( + = = 73) ha i a equivale PA syse ( + = = 158). Theefoe, he coplexiy of he MSAF iagig syse is educed wice copaed o a equivale PA iagig syse. Coclusios A echique fo paaee aalysis ad opiizaio of a MSAF syse is descibed. This echique ca be useful a he sage of desig of MSAF syses ha eploy oly a few asissios fo iage foaio. The esuls obaied show ha such syse paaees as he ube of asi/eceive sub-apeues, he ube elees i each asi/eceive apeue, he spacig bewee sub-apeues, ad he eeded saplig ae ca be opiized usig he effecive apeue appoach. The choice of he opial paaee se educes o a ade-off of he aowes bea agais he lowes side lobe level. The copaiso bewee he wo iagig syses deosaes ha he MSAF syse eployig oly a few (M = 3, 5, 7) pulse asissios pe iage ca poduce iages of equivale iage qualiy exeely fase ha a equivale coveioal PA syse. Fo os iagig applicaios, his acquisiio speed ca be fas eough o avoid pobles caused by issue ovee duig he iage acquisiio. Ackowledge This wok was fiacially suppoed by he Cee of Excellece BIS1++, Coac 16639, ad he Bulgaia aioal Sciece Fud, Ga MI-156/5. 76

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