A note on the singular value decomposition of inverse-orthogonal circulant jacket matrices

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1 Jourl of Egrg Thology Volum 6 Spl Issu o Thology Iovos d Applos O. 7 PP A o o h sgulr vlu domposo of vrs-orhogol rul jk mrs K. M. Cho Md. Abdul Lf Srkr N. Ykov Moo Ho L *. Dprm of Elros d Iformo Egrg Cobuk Nol Uvrsy Joju-s 5496 Kor. Th Fuly of Mhms d Iforms Shum Uvrsy Shum 97 Bulgr. *Corrspodg uhor: Moo Ho L Absr. Ths sudy vsgs h sgulr vlu domposo (SVD) of vrs orhogol rul Jk (CJ) mrs whh r bl o b osrud by vrs orhogol Toplz mrs. Ths CJ mrs r ppld o vry of flds of grg d mhml dspls suh s sgl prossg wrlss ommuos d so forh. Th SVD of vrs orhogol CJ mrs s bld o ompu smll mdum d lrg sz mrs wh hghr ury. Kywords: Toplz d Jk mrs Crul Jk (CJ) mrs Eg vlu domposo (EVD) Sgulr vlu domposo (SVD).. Iroduo Th mrs suh s Toplz [] rul [ 3] Hdmrd [4] d Jk mrs [5 6 7] hv mpor rol umrl lyss d sgl prossg. Th prl vlus of Hdmrd rsformo suh s r wghd Hdmrd rsformo rprs hr orrspodg sgls d mgs. Th lss of Jk mrs lso os h lss of rl d omplx Hdmrd mrs. Jk mrs grd by vrs orhogol Toplz mrs r f rul Jk (CJ) mrs. W prs vry smpl domposo of CJ mrs whh provd ff gvlu d sgulr-vlu domposo srgs hs ppr. Ths gvlu d sgulr-vlu domposo hqus r xsvly usd sgl prossg d wrlss ommuos whh r brodly usd pr hks o hr ffly ompubl vrs mrs. Th Jk mrs r grlzo of omplx Hdmrd mrs. A squr mrx M ( m j ) s lld Jk mrx [5] f s vrs ssfs - [ M ] j m j.. h vrs mrx b obd by kg lm-ws vrs d rspos up o glgbl os for. Equvlly hs mrs ssfy h followg rlos: whr s h Krokr dl - fuo of wo vrbls usully grs j m k j j () k mjk f j j. f j () 39

2 Jourl of Egrg Thology Volum 6 Spl Issu o Thology Iovos d Applos O. 7 PP If C s ozro os h h dfo of Jk mrx b rwr s follows. A squr m m mrx j j j m j j j m Jm jm jm j m m s lld Jk mrx f s ormlzd lm-ws vrs d rspos ssfs J J I. m m m J m / j / j / j / j / j / j C / j / j / j m m m m m m A Jk mrx whh h modulus of h ry s uy s lld omplx Hdmrd mrx [4]. I s sy o s h f K s jk mrx h for vry prmuo mrs P P d for vry vrbl dgol mrx D D h mrx H PD KD P s Jk mrx s wll. Jk mrs rld hs fsho r lld quvl. Fdg ll jk mrs up o quvl os urs ou o b hllgg problm d hs b solvd oly up o ordrs 5 []. Th propry of Jk mrs s h for y wo dffr rows j s j ssry o hv h Jk odo T d s. (3) s js Rqurm of Eq. () sd of usul r produ s gv s bd lgbr proprs bou Jk mrs - for xmpl mulplo of wo Jk mrs grl s o Jk mrx. Howvr Jk mrs hv som rsg omborl proprs. For xmpl f w mulply som rows or olums by y o-zro lm h h rsulg mrx rms Jk mrx. Ths yp of quvl opro spls h sp lrg lsss of mrs. I orhogol s w mulply oly by. W olud hs so wh som xmpls of Jk mrs. Exmpls of Jk mrs r h followg mrs of ordr d 3 rspvly: A A3 3 whr.. so lrv oo s 3 3 A Numrous xmpls b foud [6]. Som of h Jk mrs hv prmrs. For xmpl for vry ozro h followg mrx 393

3 Jourl of Egrg Thology Volum 6 Spl Issu o Thology Iovos d Applos O. 7 PP A4 s o oly Jk mrx bu lso symmr mrx. Also s sy o hk h h Vdrmod mrx of -h roos of uy x x x x x W x x x x x x s Jk mrx oo [7]. Ths ppr s orgzd s follows. Frs w vsg h sysm modl So. I So 3 w show h rlo bw h rul d vrs orhogol rul Jk (CJ) mrs. I So 4 w lul h gvor d gvlus of vrs CJ mrs. I So 5 w ompu h SVD of vrs CJ mrs whh b ppld o wrlss MIMO ommuos show So 6. Flly h olusos r drw So 7.. Sysm Modl A Toplz mrx s mrx suh h T { k j; k j } whr k j k for som lms j. I ohr words h Toplz mrxs hv dgol-os lms d hv h form T. (4) Suh mrs hv b foud my pplos. For xmpl suppos h x ( x x x ) T s olum vor dog pu d h k s zro for k. Th h vor wh rs k k k y T x z (5) y x rprss h oupu of h dsr m usl m-vr flr h wh mpuls rspos k. Equvlly hs s mrx d vor formulo of dsr-m ovoluo of dsr m pu wh dsr m flr. As ohr xmpl suppos h { X } s dsr m rdom pross wh m fuo gv by h xpos m E( X ) d ovr fuo gv by h xpos K ( k j) E[( X m )( X m )]. k k X k k j j Th sgl prossg hory suh s prdo smo do lssfo rgrsso d 394

4 Jourl of Egrg Thology Volum 6 Spl Issu o Thology Iovos d Applos O. 7 PP ommuos d formo hory r mos horoughly dvlopd udr h ssumpo h h m s os d h h ovr s Toplz.. K ( k j) K ( k j) whh s h pross s sd o X b wkly sory. (Th rms ovr sory d sod ordr sory lso r usd wh h ovr s ssumd o b Toplz.) I hs s h ovr mrs K [ K ( k j); k j ] r Toplz mrs. My hors of wkly sory prosss volv X pplos of Toplz mrs. Toplz mrs lso gv hd wh soluos o dffrl d grl quos spl fuos problms d mhods physs mhms sss d sgl prossg. 3. Crul d Ivrs Orhogol Crul Jk Mrs A ommo spl s of Toplz mrs whh wll rsul sgf smplfo d ply fudml rol dvlopg mor grl rsuls wh vry row of h mrx s rgh yl shf of h row bov so h k ( k) for k. I hs s h rsulg mrx boms k X C. (6) A mrx of hs form s lld rul mrx d usully s dod by r( ). Crul mrs hv b md good us of vrous grg rs suh s h dsr Fourr rsform (DFT) d yl ods for rror orro. I [9] hr s xmpl of vrs-orhogol Toplz or Toplz Jk (TJ) mrs of ordr 4 TJ b 4 b b b b ( ). b b b b W k dvg of h dgol mrs h b usd o rpl h mrx wh h rul Jk mrx. O rsg hg bou h formod mrx hs hr dpd prmrs whh r o mor h hos of rul mrs wh h sm sz. Ths xr prmr howvr s gvg rs du o h followg o-rvl symmry of Toplz mrs. W us h oo Dg o rfr o h... dgol mrx wh dgol rs =. Thrfor squr vrs-orhogol Toplz mrx s lld CJ mrx whr s rbrry omplx umbr d b s ozro omplx umbr. Th w rrv Lmm s follows: Lmm. L T b Toplz mrx of ordr d l b rbrry omplx umbr whl l b b ozro omplx umbr. Th h followg mrx s Toplz mrx s wll: ' T Dg b b b T Dg b b b. () (7) 395

5 Jourl of Egrg Thology Volum 6 Spl Issu o Thology Iovos d Applos O. 7 PP Proof: I s sy o s h h ( j)-h ry of ' T lds o j j b [9]. Now w show sruurl rsul w.r.. Toplz-Jk mrs Proposo. Suppos h T s Toplz Jk mrx. Th wh h oos from Eq. (4) w hv l l l l... (9) l Proof: No h h Eq. (9) holds rvlly for l. W bg h proof by showg h holds for l. W ssum h 3. Cosdrg h Eq. (3) wh h frs wo rows of T.. for j w rh Nx h Eq. (3) for h pr of rows j 3 Addg () rsuls 3 o boh sds d by ms of Eq. () s sy o prov h h Eq. (9) holds for l. I ddo wh l w g o l l l l... () Now by usg mhml duo w prov h h Eq. () holds for ll l. W ssum h 4 bus ws lrdy prov h h Eq. () holds for l 3. Th w osdr h Eq. (3) for h pr of rows l3 l : d rwr usg h Eq. () s l lk k l k l k. l k k Addg l / l o boh sds d by ms of Eq. () s sy o prov h h Eq. () holds for l l. W ompl h proof by usg h Eq. () for h osuv rms l d l h. l l l l l l l l l l l l Now w mov o Thorm bou Toplz-Jk TJ d rul-jk CJ mrx. Thorm. Evry Toplz-Jk mrx s quvl o rul-jk mrx. Proof Thorm. L TJ b Toplz-Jk mrx s h Eq. (4) d ssum x / whr h opror dos h prpl h roo. By Lmm h mrx 396

6 Jourl of Egrg Thology Volum 6 Spl Issu o Thology Iovos d Applos O. 7 PP s Toplz mrx quvl o CJ dg x x TJ dg x x () TJ. Ths CJ s lld rul-jk mrx. To s hs s j ough o show h j j for vry j.... I s lr h j j x. O h ohr hd by usg Proposo w hv x x x. j j j j j j j Usg h Eq. (7) h Eq. () f =b== d =4 h h [CJ] 4 s gv by CJ 4 C C. (3) C C C C Proposo. L CJ b blok mrx of ordr =p. If [C ] p d [C ] p r p p C C Jk mrs h [CJ] s blok rul Jk mrx f d oly f RT RT C C C C whr RT s h rprol rspos. (4) Proof. S C d C r Jk mrs w hv C C RT blok rul Jk mrx f d oly f CJ CJ RT I RT pi d C C pi p p. No h [CJ] s. Th [CJ] s Jk mrx f d oly f CJ CJ I H [CJ] s Jk mrx f d oly f RT RT RT RT C C C C p I CC C p C C RT RT C C C C C C C pi p C C C C. RT RT By usg Proposo w my osru my blok rul mrs. O xmpl of blok rul jk mrs: L S C C C RT I d C C I RT Ad w rh C C d C r Jk mrs of ordr. Morovr C C C C CJ RT RT C C C C (5) (6). (7) () 397

7 Jourl of Egrg Thology Volum 6 Spl Issu o Thology Iovos d Applos O. 7 PP Thrfor [CJ] s blok rul vrs orhogol Jk mrx. Crul submrs of sz r suh h h submrx hs h propry h ll rows r yl shfs of h frs row d ll olums r yl shfs of h frs olum. W shll rfr o hs submrs s rul bloks. Also w df usr s quvl lsss s h ss of usr groups whos orrspodg olums of Hdmrd mrs form rul bloks. A rul mrx s fully spfd by o vor whh pprs s s frs olum. Thrfor hs mrs r mpor bus hy r dgolzd by h DFT d h hr lr quos b qukly solvd usg h ury fs Fourr rsform. 4. Egvor d gvlu domposo (EVD) Bus by Thorm vry Toplz Jk mrx s quvl o rul mrs [9] s url o look o dls how h gvlus gvors d gvlu domposo (EVD) of rul mrx b obd. Also o h ors o h usul symmr mrx S (h hv h propry S S T ) h Toplz mrs d rul mrs prulr r prsymmr [3]. Ths ms h rl mrx B b b b b b b B b b b s prsymmr f s symmr bou s orhs-souhws dgol.. for ll j b. j b j Ths s quvl o rqurg B EB T E whr T s h rspos d E s h xhg mrx E. Lmm. If T s Toplz (rul) mrx h h followg sms r ssfd: ) T s prsymmr mrx; b) If T s osgulr h s vrs T s lso prsymmr. I ors o dgol mrs whos gvlus r hr dgol lms [] h umrl drmo of h gvlus of prsymmr mrs rqurs dous lulo. Thr gdomposo dpds o whhr hr dmso s odd or v. W wll dlv o h EVD of rul mrs followg []. L b h uvrsl rul mrx gv by r( ). I s sy o prov h mrx C s rul f d oly f CΠ=ΠC (9) 39

8 Jourl of Egrg Thology Volum 6 Spl Issu o Thology Iovos d Applos O. 7 PP Dfo. L b fxd gr d s h mgry u. L os s. Do ( ) dg( ). () Thorm. F F whr * s h ojug rspos d F s h ormlzd mrx of h ury dsr Fourr Trsform F ( ) ( ) jk/ ( ). ( ) ( ) ( )( ) Usg h uvrsl rul mrx Π d Thorm w hv Thorm 3. If C s rul s dgolzd by F. Mor prsly C F F whr dg( p () p ( ) p ( )) for p () x x x j. If r h -h roos of uy h gvlus of C r j T d h gvors v ( ) j.... j j j () j j j j j... As osqu of h xpl formul for h gvlus bov h drm of rul mrx b ompud s: j j j () j d( C) ( ). 5. Th Sgulr-vlu Domposo Usg h Eq. (7) [9 Thorm.] w g o h dgol mrs h b usd o rpl h mrx wh rul Jk mrs. W hv CJ 4 4 T ( b ) b b b b b b b b. (3) 399

9 Jourl of Egrg Thology Volum 6 Spl Issu o Thology Iovos d Applos O. 7 PP C C Th Eq. (3) s bld o hv Jk mrs h pr of whr C C Th hrrs polyoml of C s so h gvlus r {()()( )( )}. C b b d C b b. 3 6 b 6 b 4b 3 4 d( C xi ) x 4 x x 4x x 4b 4b x ( x 4 4 ) x ( x ) b b d h gvors r S w do o hv y smpl sprum of h mrx C w prod wh h sgulr vlu domposo (SVD): CJ 4 b. b T (4) Aohr xmpl of h Toplz Jk 6 6 mrx w w w 3w w 9 3w TJ w 7 9 3w 3 9 w 7 9 3w 3 43 w 7 9 3w b rpld wh h rul Jk mrx usg x=3 s (5) 4

10 Jourl of Egrg Thology Volum 6 Spl Issu o Thology Iovos d Applos O. 7 PP CJ dg( x x x ) TJ dg( x x x ) r( ) C C C C (6) whr C C 3 3 d rspvly. If = = d b= h rul vor x(b)=x() s osrud s follows x xp b xp xp Thrfor hs vor looks lk s 7 7 x xp xp...xp. Thrfor s of CJ mrx [CJ] (b) s gv by whr C 7 7 CJ b D TJ b D b C C 7 7 C C C (7) h dgol mrx D dg h vrs of hs dgol mrx D dg / / 3 / 4 / 5 / 6 / 7 mrx d Toplz-Jk 4

11 Jourl of Egrg Thology Volum 6 Spl Issu o Thology Iovos d Applos O. 7 PP TJ b b b b b b b b b 7 7 b b b b b b b b 7 7 b b b b b b b b 7 7 b b b b b b b b 3 b b b 7 7 b b b b b b b b 7 7 b b b b b b b b 7 7 b b b b b b b b 7 b b b 7 b 4 3 b W lso o h from Eq. (7) h gvlu of CJ mrs s whos mulply s s h of DFT mrs.. 6. Wrlss MIMO Applos W osdr H I T I I J I Σ I J JΣJ b J Σ J whr J s h Jk mrx h ssfs JJ - =I J J= s vrs Jk mrx d Σ s h dgol mrx gv by Th wll-kow py formul s gv by J = J Σ J T b J Σ J J Σ. () H C log d I H H bps / Hz (9) 4

12 Jourl of Egrg Thology Volum 6 Spl Issu o Thology Iovos d Applos O. 7 PP whr γ = ρ / N s h sgl-o-os ro (SNR) ρ s h rgy of h rsmd sgls d N s h powr sprl dsy of h ddv os whr QQ H z. Th SVD of HHH s gv by H H H H H H H H H JΣJ JΣ J JΣΣ J QΛQ (3) I d Λ s h dgol mrx wh s lms gv s Howvr h sysm py b wr s C whr r = m( m ) = m dos h rk of H d f... m. (3) f m... r m log bps / Hz (3) s h gvlu of h mrx HH H. I Fgur w ompr h vrs orhogol Jk hl gs h ovol hl [-]. Usg MATLAB sofwr w prform h Mo Crlo smulo γ = [dbs] d gr by hl rlzos h smulos. W obsrv h Fgur h hl py ovr h vrs orhogol hl s slghly rsd hgh SNR rgm omprd o Toplz d Jk hls whh b bg hlp for mg xpolly growg dmd for hghr py ommuo sysms. Fgur. Comprso of MIMO hl py mployg Eq. (7) Eq. (3). 7. Colusos To sum up w hv sudd h CJ mrs h hv smlr sruur of spl kd of Toplz suh s rdol rul mrs hs ppr. Th grs dvgs of hs CJ mrs r h hy fl vry fs d sous ompuo of vrs CJ mrs smmg from h f h vrs CJ mrs r dgosbl by fs Fourr rsform mrs d h hy qu wh h sgulr vlu domposo hqu. Thr r my pplos of CJ mrs suh s h Krokr MIMO hl [6] d blok rul Jk []. Akowldgm Ths work ws suppord by MEST 5RAAA5977 NRF Kor d hks o Fr Szöllős h hs drvd Eq. (7). 43

13 Jourl of Egrg Thology Volum 6 Spl Issu o Thology Iovos d Applos O. 7 PP Rfrs [] Gry M. R Toplz d Crul Mrs: A rvw Sford hp://.sford.du/~gry/oplz.pdf. [] Dvs P.J Crul Mrs Amr Mhml So [3] G H. Golub F. Chrls Lod V. Mrx Compuos Th Jo Hopks Uvrsy Prss 3d 996. [4] Hordm K. J. Hdmrd Mrs d Thr Applos Pro Uvrsy Prss 7. [5] L M. H. A w rvrs Jk rsform d s fs lgorhm IEEE Trsos o rus d sysms II ; 47: [6] L M. H. Jk Mrs: Cosruos d Is Applos for Fs Cooprv Wrlss Sgl Prossg LAP LAMBERT Publshg Grmy. [7] L M. H. Borssov L. Y. O Jk Trsform Ovr F Flds IEEE ISIT Soul Kor Ju - July 3 9. [] D P. O mhod for osruo of vrs orhogol mrs Rom. Jour. Phys.9; 54: [9] L M. H. d Fr Szöllős A No o Ivrs Orhogol Toplz Mrs Elro Jourl of Lr Algbr 3; 6: [] Shd J. Eglyss of dgols AURORA Thl Rpor TR- Isu for Appld Mhms d Numrl Alyss V Uvrsy of Thology. [] L M.H. H H. d Zhg X.D. MIMO Commuo Mhod d Sysm usg h Blok Crul Jk Mrx USA P 6; 4/6.54 3/3. [] L M. H. Mlgh M.M. Sog W. Fs Mhod for Prodg d Dodg of Dsrbuv Mulplpu d Mulpl-oupu Chls Rly bsd Dod-d-Forwrd Cooprv Wrlss Nworks Th IET ommuo ; 4():

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