Performance Comparison of LCMV-based Space-time 2D Array and Ambiguity Problem

Transcription

1 Inernaional journal of cience Commerce and umaniie Volume No 2 No 3 April 204 Performance Comparion of LCMV-baed pace-ime 2D Arra and Ambigui Problem 2 o uan Chang and Jin hinghia Deparmen of Communicaion Engineering, I-hou Univeri, aiwan 2 Deparmen of Indurial echnolog Educaion, Naional Kaohiung Normal Univeri, aiwan Abrac- e dicu and compare hree differen pace-ime 2D arra which are he pace-ime join (J), he ime-pace cacade (C) and he pace-ime cacade (C) configuraion. he LCMV algorihm i applied for evaluaing he arra weighing funcion of hee 2D arra. he J can have beer performance han he oher wo, bu i ubjec o heav compuing load; while boh he C and C configuraion will encouner he ambigui problem under he muliple beam conrain iuaion. Keword: LCMV, J, C, C.. INODUCION In a wirele communicaion cenario a pace-ime 2D arra i applied o erac boh he angle of arrival (AOA) and he pecrum conen of a received ignal [-3]. he performance of a pace-ime 2D arra i deermined b how we combine he overall pace-ime 2D daa. In hi paper hree differen 2D configuraion, which are he pace-ime join (J), he ime-pace cacade (C) and he pace-ime cacade (C), are compared. here migh be no much novel he opic of a pace-ime 2D arra, bu we would menion he eiing of ambigui problem, defined in ecion IV, which hi problem can be eail ignored under he pace-ime ignal proceing cheme. hi paper i organized a follow. In ecion 2 we make definiion he rucure of J, C and C configuraion. In ecion 3 we appl he linear conrain minimum variance (LCMV) algorihm for evaluaing a pace-ime 2D arra weighing funcion [4, 5]. he ambigui problem encounered o boh he C and C configuraion i defined in ecion 4. imulaion reul are hown in ecion 5. Finall we make concluion. 2. CONFIGUAION OF PACE-IME 2D AAY Le u define a acked eering or ignal vecor a( f, f ) a ( f ) ( f ) a in a pace-ime domain, where j 2π f j 2π ( N ) f denoe he Kronecker produc beween he pace and ime eering vecor of a [ e... e ] and j 2π f j 2π ( N ) f a [ e... e ]. In hee wo vecor a and a, f and f are he normalized pace frequenc and Doppler frequenc wih repec o paricular pace angle and Doppler beam in he pace-ime domain and N and N are he degree of order in pace and ime domain, repecivel. Aume K differen received baeband ignal, { (), (),, ()}, wih parameer { f, f }, i=,, K. hee ignal ma include boh he deired 2 K i i ignal and inerference, which are independen among each oher. e ma epre he received ignal in vecor form a follow: () a( f, f ) () v() = D()+v(), () K i i i i In equaion (), D = [ a( f, f ) a( f, f )] i he eering mari, () [ () ()], and v() i he noie K K K vecor. For he convenience of epreion and anali, we ma dicard in equaion () and le N = n and N = 27

2 Inernaional journal of cience Commerce and umaniie Volume No 2 No 3 April 204 m in he following ecion. In hi paper, we define hree differen pace-ime configuraion in he following ubecion, which are he join combinaion of pace-ime (J), he cacade combinaion of pace and ime (C), and he cacade combinaion of ime and pace(c) configuraion. A. pace-ime join (J) configuraion In J configuraion, hown in Figure, he arra combine he received daa, afer muliplied wih he appropriae weighing funcion, a he final node of each channel and obain he arra' oupu, which i n m FJ j i ] m 2 2m n nm w Y, (2) where Y = [ i he acked daa vecor obained from aking ample from [] in equaion (). he arra weighing vecor = [w w w w w w ] i deigned o mee paricular m 2 2m n nm crieria. Boh Y and are wih he ame ize ( nm ). B. ime-pace cacade (C) configuraion he C configuraion, hown in Figure 2, i a wo-age combinaion proce rucure. A fir, all emporal ample in each channel are weighed and combined o obain channel oupu. In he ne ep, hee oupu from all emporal channel are combined paiall a he arra final node o form a ingle oupu. hi i a cacade of ime-pace ordered procedure. o make i clear, he oupu, j =,,n, in each channel, can be j epreed a: m j i w X, where j j j channel. he arra final oupu, b aking paial combinaion of j F n C j w j j X = [ j jm] and j i he emporal weighing vecor in each, i given b. (3) CY Daa vecor Y [ n ] and paial weighing vecor [ w w ] are all wih he ame ize C n (n ). e would like o epre equaion (3) a he ame form o equaion (2). Le [ 2 n ] which ha mn ize (m n). e can define a Khari-ao produc beween and, which i epreed a C [ w n wn ]. I i clear ha C i wih he ame ize (m n) o ha of. hen, we obain a acked ime-pace cacade weighing vecor vec( ) C. I i clear ha equaion (3) can be epreed a where Y =[ F n C j wj m 2 2m n nm] j = Y, (4) C CY i he ame a ha of Y in equaion (2). C. pace-ime cacade (C) configuraion In C cae we echange he proceing order of pace and ime, a revere order from ha of he C cae. A pace domain, he oupu a each napho,, i=,, m, i given b i n i j w i Xi, where X i = h [ i ni ] i a daa vecor wih ize (n ) and i i he pace domain weighing vecor a he i napho. he overall arra oupu i hen: F m C i w i i. (5) Y C In above queion, daa vecor Y [ ] and [ w w ] are wih ize m C m (m ). Le [ 2 m] and define nm [ w w ], all wih ize C m m (n m). e can obain a pace-ime cacade weighing vecor vec( ). he equaion (5) can be epreed a C 28

3 Inernaional journal of cience Commerce and umaniie Volume No 2 No 3 April 204 F m C i ] n 2 n2 m nm w i i Y = C CY, (6) where Y = [ i he rearrange of acked daa vecor of Y in equaion (2) and (4). Noe ha he oupu in equaion (2), (4) and (6) have he ame epreion --- Y, all wih he ame ize (nm ). owever, o updae he J weighing vecor one hould calculae a vecor wih nm dimenion once a ime, which he compuing load i proporional o he order of O ((nm) 3 ) ; while for he C and C cae he compuing load i wih order of O (m 3 ) and O (n 3 ) which i aboluel much maller han O ((nm) 3 ). 3. LCMV-BAED PACE-IME BEAMFOME he LCMV algorihm can be applied o adapive arra em o achieve among beam paern conrol, inerference uppreion, and capaci or performance improvemen purpoe. In hi ecion, we would appl he LCMV algorihm o hree differen pace-ime 2D arra configuraion for he comparion of he overall performance. A. he J cae he arra weighing under he LCMV algorihm i derived from olving he following conrain equaion: min( ) ubjec o D g. he oluion i = D(D D) g, (7) where i he covariance mari defined b = E( YY ), which i wih ize ( nm nm ), D i he eering mari decribed in equaion (), and g i he conrol vecor defined according o he deired repone of he conrain equaion. B. he C cae In he C cae, we have m j i w X and j j F n C j 29 w j j weighing vecor = C D (D D ) g and j = D (D D ) g j j j j j equaion min( ), ubjec o D g, and min( ), C C repecivel, where C. he C cae C and In he C cae, we have C j j j CY j are defined b E( Y Y ) and j E( XjXj). n i j w i Xi and FC w i m i i j j. Baed on LCMV, boh are evaluaed b olving ubjec o Y C D g, j. Baed on LCMV, boh weighing vecor = C D (D D ) g and i = D (D D ) g i i j i i i, where and i are defined b E( Y Y ) and i E( Xi Xi ), repecivel. 4. AMBIGUIY OF PACE-IME CACADE POCEING Boh of C and C will encouner he ambigui problem defined in hi ecion. Le u ake C for eample. In equaion (4), he overall weighing vecor of C i vec( ), C where [ w w ]. e ma e a imple cae for ea anali, which we would le C n n = [ w w and hu we have w w ]. Under hee aumpion, we can 2 n ] m [ n j j

4 Inernaional journal of cience Commerce and umaniie Volume No 2 No 3 April 204 derive a cloe form reul --- vec (. ) C In equaion (8), = C D (D D ) g. Le = D (D D ) g. e have C = D (D D ) g D (D D ) g D(D D) = ( g g ), (0) where and D D D. he difference beween equaion (0) and (7) come from he conrain vecor of g and g g. he ignificance of he difference beween equaion (0) and (7) i hown in Figure 4. In hi figure here i a wo-poin conrain in he pace-ime domain { ( f, f ) }={( F, ) (, )} 2 F F F. e can define wo eering vecor 2 aociaed wih hee wo poin in he pace-ime domain and hu he eering mari can be epreed a D= [ a ( F 2, F ) a( F, F 2)], a mari wih ( nm2 nm 2) ize. e can appl he J configuraion o ge he deired repone via he conrol vecor g. Alernaivel, when we appl eiher he C or C configuraion, we need wo-poin conrain equaion denoed b g and g, repecivel. he operaion of eiher g g or g g will enroll wo more poin under he conrain, locaed a {( F, F ) ( F 2, F 2)} in he pace-ime domain. hee wo bproduc conrain poin aboluel can no be avoided and we would call i ambigui, a hown in he Figure IMULAION EUL In hi ecion we compare he beam paern of he J and C configuraion. he arra parameer in hi ecion are n=2 and m=24. Figure 5 i he paern of J and Figure 6 i for he C cae, where he deired ignal i locaed a ( f, θ ) = (900, 60 ) and inerference i a ( f, θ ) =(880, 65 ). Becaue he deired ignal and inerference are raher cloe o each oher, hu in he C cae no onl he mainbeam ha diorion bu alo he idelobe level i much higher han ha of he J cae. hi how ha he J configuraion can have beer performance han boh he C and C, hough he C reul i no hown here. In Figure 7, J can eparae and preerve wo deired ignal uccefull b eing a 2-poin conrain a (900, 60 ) and (700, 80 ) on he angle v. frequenc domain. owever, if he C configuraion i applied, we hould e conrain o preerve boh 900 and 700 frequencie, a well a o preerve boh ignal from he AOA of 60 and 80 on he pace domain. hi will reul in a 4-poin conrain on he angle v. frequenc paern, hown in Figure CONCLUION In hi paper we appl he LCMV-baed arra proceing and how he eience of deecion ambigui problem encounered o boh he C and C pe arra configuraion under he muli-poin conrain. he J configuraion will have no ambigui problem and i performance i beer han boh he C and C cae. owever, he dimenion of J i much larger han he oher wo, which will involve he heav compuing load when he J weighing funcion i updaing. EFEENCE [] heau-hong Bor, and o-huan Chang, A implified linear conrain algorihm for mar anenna em, ICOM 99, Kaohiung, aiwan, pp , Nov [2] o-uan Chang, heau-hong Bor and Kaung-pin Chen LCMV-baed pace-ime 2-D beamformer deign for mar anenna em, Proceeding of he IAED inernaional conference on OC200, Banff, Canada, pp.58-63, June 27-29, 200. [3] o-huan Chang, heau-hong Bor, e al, Adapive beam paern anali of a pracical ring arra, ICOM 0, ainan, aiwan,

5 Inernaional journal of cience Commerce and umaniie Volume No 2 No 3 April 204 [4] Y. Ichikawa, K. omiuka,. Oboe, and K. Kagohima, Compuaional Complei educed MME Adapive Arra Anenna wih pace emporal Join Equalizaion, IEEE Communicaion magazine, pp , 200. [5] Linrang hang,. C. o, Li Ping, and Guiheng Liao, Adapive muliple-beamformer for recepion of coheren ignal wih known direcion in he preence of uncorrelaed inerference, ignal Proceing Vol. 84.pp , June X 2 X 2 X N N 2 N 2 N O J X N 2 N N N 2 N 2 N N O N 2 N N N Figure. he rucure of J configuraion Figure 2. he rucure of C configuraion X 2 2 X N N N 2 N N N 2 N 2 2 O Figure 3. he rucure of C configuraion Figure 6. Beam paern of C. 3

6 Inernaional journal of cience Commerce and umaniie Volume No 2 No 3 April 204 ambigui deired F deired ambigui F2 F F2 Figure7 Beam paern of J under wo-poin conrain. Figure 4. Ambigui conrain in pace-ime domain Figure 5. Beam paern of J. Figure 8. Beam paern of C under wo-poin conrain. 32