Beamforming in Short Time Fractional Fourier Domain (FRFD)

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Reynard Norris
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1 Rrch Jourl o Ald cc Egrg d Tchology 48: I: wll cc Orgzo ubd: y 8 Accd: Ju 8 Publhd: br 5 org hor T rcol ourr Do RD o Rh d H Gh Dr o Elcrcl d our Egrg Trb odr Uvry Thr Ir Abrc: Th udy o gr odl or h robl o borg rcol ourr do hrough whch grl oluo or obg wghd borg rcol ourr do or dr crr gh b dlvrd I ordr o gr h odl dcro o rcol dly coc rd hrough whch h rg vcor rcol ourr do clculd d odl or borg rcol ourr do grd whch lr o h ud or obg wghd borg do or rrow-bd gl To col our oluo hod or obg h l rcol do wll drco o rrvl l rcol ourr do Kyword: Arry rocg borg drco o rrvl rcol ourr do rcol ourr ror rg vcor ITRODUTIO org rry gl rocg chu ud or rry or drcol gl ro or rco Th chvd by cobg l h rry uch wy h gl rculr gl rc rucv rrc d whl ohr rc drucv rrc Vrou borg lgorh hv b rood ozgo d llr 98 Th covol borr d bc o h cod world wr d r lco o ourr-bd crl ly o o-orlly ld d Lr dv borr Gbrl 97 d clcl dly o chu wr ld o hc o' bly o rolv cloly cd gl ourc Th o o h dly o hod o or h o gl d h ld roluo o borg oghr wh crg ubr o ovl lco rwd r o rrchr cl gl rocg Howvr or rrc chu h uv oo o roluo orvl o d rc r Th ur ruld h rgc o h rr o roch cv rrch r or h r 3 roducd org RD d hor 3 h Zhou d To 8 colcd h lgorh To 4 I h hod org h b odl l lrg h ohr word hr wor w o O' wor Kuy l 997 o rry gl rocg Our udy d o olv h borg robl hrough rrc hod RD Th ovo bhd h rood hod h bly o h rcol ourr Tror rt o roc h chr gl br h h ordry ourr Tror T Thror rlcg h T wh h rt hould rov h rorc codrbly I h udy ovl org lgorh o wdbd L gl bd o h rt rood whch c b ld o lr rry Ilo o h rood lgorh r lo dvlod whch r bd o h crl rchg A h d o h udy urcl l d ulo r rd ollowd by dcuo o h rul ETHODOLOG rcol ourr ror: rcol ourr ror rr owrul hod gl rocg whch h b ud ovr wo dcd Th ror bg grlzo o ordry ourr ror dd ul o gl roo -rucy do gv gl 98 Lg b rbrry gl h-ordr rt dd : u [ ] K u d whr ; [] oror gl o RT K u rl uco: u K u ucc u u orrodg Auhor: o Rh Dr o Elcrcl d our Egrg Trb odr Uvry Thr Ir 35

2 Th RT lr ror h u clld h RT do ro h bov rloh obvou h h d rucy r h h d h RT do rcvly g h h RT ud -rucy ror d h RT c dor h chrcrc or h gl chgg ro h do o h rucy do hvg h ordr h rg ro crg o I rcc h Dcr RT DRT ud Th cooly ud lgorh o DRT O' lgorh Oz l 996 I h lgorh o h RT dgl couo h orgl gl u b orlzd I u h h orgl uou gl coc log h o d rucy h -do rro o h gl cod o h rvl [ ] d rucy-do rro cod o h rvl [ ] o h gl' bdwdh roduc whch lwy grr h uy bcu o h ucry rlo cu -do d rucy do hv dr do I ordr o cou h RT -do d rucy-do u b rord o h do who do I roduc h clg rr wh h do o d roduc cld coord: v R J Al c Eg Tchol 48: Th w coord v r clrly dol Th gl cod o h rvl [ ] d [ ] rcvly udr h w coord I ordr o hv h wo rvl ul o ch ohr choo o h h lgh o boh rvl r ul o h dol uy oh rvl r orlzd [ ] I h wly dd coord ccordg o h lg hor h gl c b rrd boh do wh l cd r Accordg o h lgorh h ror o h dgl gl DRT ordr dd blow: A cc 3 35 Th rt h oud y lco dgl gl rocg uch lrg Aldrz d Dur ; Erd l 999 gl roro Erd l 999 y yh Erd d Oz 998 uul y yh Erd l 996 y dcooo Kuy l 999; l u Wr lrg Kuy d Oz 998 g roro Kuy d Oz 998 d rcv roco l or col r o h rt d y o c d dgl gl rocg lco c b oud Oz l Th rt ugg ol rov y lco whr h ordry T ud c rovd r dgr o rdo corrodg o h choc o h ordr W c o rov h oluo o y robl h ulz h T by crryg h r rr hroughou h oluo d h zg ovr h rr Th rt o lly o rov h oluo o robl whr chr gl r volvd Th bcu chr gl or l h rucy l d hror hr ordr or whch uch gl coc Arry odl: L uor lr rry how g Th rry h or wh rrc or o h coord org o Th -h or coord loco - d = Au h wdbd gl ro r-ld g o h rry Thu h obrvd gl h -h or c b rd : r u c d c 4 whr h ddv wh Gu o Whch ud o b clly dd wh gl ourc ourc drco h h dly c wv vlocy d org rcol ourr do: I ordr or h odl o b rd rrclly r h coc o rcol dly hould b dd d h rlo' u b rcd or h dcr by u o whch h borg o wdbd gl h RD odld borg o rrowbd gl h ourr do

R J Al c Eg Tchol 48: 35-355 35 g : Uor lr rry rcol dly: DRT or h dly gl d ullco by ol gl c b rd blow: co co 5 co co 6 ow ob DRT h w g r o o h DRRT ob r o h cll rcol dly: co co co co 7 d h: cc 8

3 R J Al c Eg Tchol 48: g : Uor lr rry rcol dly: DRT or h dly gl d ullco by ol gl c b rd blow: co co 5 co co 6 ow ob DRT h w g r o o h DRRT ob r o h cll rcol dly: co co co co 7 d h: cc 8 odllg o borg RD: Accordg o h do rcol dly E 8 w ob rcol dly or ch or blow: Q Q 9 Th ccordg o h do DRT E 3 ob DRT E 9 d rc h odl or borg DRT: cc whr I ] [ h ouu o rry c b cocly rrd r or: A whr A 3 h rg vcor rd Th ouu o borr :

R J Al c Eg Tchol 48: 35-355 353 g : Ouu o covol borr RD ordry do or gl log o y 4 d h DrT o y c b ob: A 5 wh E 5 ld: 6 I h rd Orgl rry odl c b dgrd o: W A 7 IULATIO W dor h h rood hod yld rovd rul h

4 R J Al c Eg Tchol 48: g : Ouu o covol borr RD ordry do or gl log o y 4 d h DrT o y c b ob: A 5 wh E 5 ld: 6 I h rd Orgl rry odl c b dgrd o: W A 7 IULATIO W dor h h rood hod yld rovd rul h llr E Au ULA o = d h obrvd gl wdbd L coo blow: T ovol borr: Th rcvd gl o clud o o Th o c b odld col Gu drbud rdo ubr W u h 3 d R ch or gur how h ouu o covol borr T do d covol borr u rcol ourr do Th vlu o h R vr ro - d o 5d wh rvl o d A ch lvl o h R w ru 8 o rlo ulo d h RE obd how g 3 d or o h E or h ovol rt orr d ovol orr Adv orr VDR: I h rc o rog rrc h rg gl y b d by

5 R J Al c Eg Tchol 48: ur rror g 3: oro o h E or h covol rt borr d covol borr gud Ouu o ovol orr rcol ourr Do T Ouu o ovol orr 5 E v R Prood Orgl R d gud Ouu o VDR orr rcol ourr Do T g 5: Ouu o VDR borr rd or gl log o d rrc VDER rcol ourrdo VDR ourr Do ovol ourrdo ovol rcol ourrdo gud T g 6: oro o h E or h covol rt borr covol borr VDR d VDR RD g 4: Ouu o covol borr rd d ordry do or gl log o d rrc h rrc gl uch robl r vry chllgg or covol borr d hror dv borr r roducd o ddr h robl To ovrco h rrc robl w c u h VDR borr oulr dv borr Th VDR orr rrv h gl rrvg log drd drco whl ryg o ur gl cog ro ohr drco gur 4 how Ouu o ovol borr RD d Ordry Do or gl log o gur 5 how ouu o VDR b orr rd or gl log o d rrc Th vlu o h R vr ro - o 5 d wh rvl o d ch lvl o h R w ru 8 o rlo ulo d h RE or h ovol rt orr ovol orr VDR d VDR rd obd how g 6 I h gur c b clrly h h borg or wdbd L gl wh VDR lgorh bd o rt h br rorc cord o VDR lgorh bd o rucy do Eclly cv or obvou low R c OLUIO W hv rood w hod o borg ug h rt Th hod w how o b clly uul yldg llr rror or chr gl Th hod c b uul lo cv rdr d or 354

6 R J Al c Eg Tchol 48: wh w chr gl r rd o h rg Th ulo vrd h h w hod w br h h covol hod rucy do REEREE Aldrz L d Dur Adv rcol ourr do lrg J c Drc gl Proc 9: 88-9 Erd H Oz d D dlovc 996 yh o uul y drbuo ug h rcol ourr ror O ou 5: 88-3 Erd d H Oz 998 yh o grl lr y wh rd lrg cocuv rcol ourr do J O oc Ar A 5: Erd A Kuy d H Oz 999 Rd lrg cocuv rcol ourr do d lco o gl roro IEEE Tr gl Proc 47: Gbrl W 97 crl ly d dv rry urroluo chu Proc IEEE 6: Kuy A H Oz O Ar d L Ourl 997 Ol lrg rcol ourr do IEEE Trco o gl Proc 455: 9-43 Kuy A d H Oz 998 Ol g roro wh h rcol ourr ror J O oc Ar A 5: Kuy A H Oz d O Ar 999 Th rcol ourr do dcooo gl Proc 77: 5-9 ozgo RA d TW llr 98 Iroduco o Adv Arry Wly-Ircc w or : 555 V 98 Th rcol ordr ourr ror d lco o uu chc IA J Al h 53: 4-65 Oz H O Ar d A Kuy d Gözd 996 Dgl ouo o h rcol ourr Tror IEEE Tr gl Proc 44: 4-5 Oz H Z Zlvy d A Kuy Th rcol ourr Tror wh Alco Oc d gl Procg Wly w or : 53 I: To HW 4 LV org Algorh d o h rcol ourr Tror IA : I H Oz rh d L Ourl Prcv Proco h c- rucy Pl d rcol ourr ror J O oc Ar A 7: I A Kuy H Oz d H Oz b ouou d dcr rcol ourr do dcooo IEEE Irol orc o Acouc ch d gl Procg IAP ' D o Elcr Eg l Uv Ar : I d A hor 3 org ug rcol ourr ror IEEE Tr gl Proc 56: Zhou H d R To 8 rodbd org o L gl d o h rcol ourr Tror Irol orc o gl Procg Oc D o Elcro Eg g I o Tchol g :

Let's revisit conditional probability, where the event M is expressed in terms of the random variable. P Ax x x = =

Let's revisit conditional probability, where the event M is expressed in terms of the random variable. P Ax x x = = L's rvs codol rol whr h v M s rssd rs o h rdo vrl. L { M } rrr v such h { M } Assu. { } { A M} { A { } } M < { } { } A u { } { } { A} { A} ( A) ( A) { A} A A { A } hs llows us o cosdr h cs wh M { } [ (