= (3) family of spreading sequences. The dimension of an Oppermann set with the code length of N is determined by

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Dominick Owen
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1 (EUSIPCO Europa Siga Procssig Cofrc, Gasgow, Aug. 9 Improv Corratio of Graiz DFT with oiar Phas for OFD a CDA Commuicatios Ai. Aasu a Haa Agirma-Tosu w Jrsy Istitut of Tchoogy Dpartmt of Ectrica & Computr Egirig Uivrsity Hights, war, J 7 USA Emai: Aasu@JIT.u ABSTT cty, Graiz Discrt Fourir Trasform ( has forwar as th thortica frwor to sig a varity of costat mouus orthogoa compx trasforms with o-iar phas. I this papr, w vauat th auto- a cross-corratio proprtis of svra soutios a compar thm with th popuar co fiis i Go, Wash a DFT. It is show that th rigs sigificat corratio prformac improvmts ovr ow co fiis. W prict that as OFD a CDA soutios wi offr prformac improvmts i muticarrir commuicatios systms of th futur. Ix Trms Graiz Discrt Fourir Trasform, Corratio Prformac, OFD, DT, CDA. I. ITODUCTIO Amog various iary spraig fiis, Go cos hav succssfuy us for asychroous commuicatios i DS/CDA systms u to thir owr cross-corratio faturs []. Wash, Go a svra othr ra spraig co sts ar sig to optimiz v corratio fuctios [-]. Howvr, th o corratios ar aso importat as much as v corratios. Thrfor, Fuumasa, Koho a Hii propos a w st of compx P squcs, ca Equa O a Ev (EOE squcs, with goo o a v corratios [7]. EOE squcs ar grat y usig o of th ra co sts,.g. Go a Wash. Spraig cos with o-iary ra chip vaus wr aso propos i th itratur i orr to improv thir auto- a cross-corratios. or rcty, rsarch has rfocus o costat pitu spraig cos u to th fficicy cocrs of o-iar F powr pifirs mpoy i wirss trascivrs. Hc, compx roots of uity wr propos as compx spraig cos y svra authors i th itratur. A cos of such a st ar pac o th uit circ of th compx pa. Fra-Zaoff, Chu a Opprma itrouc a varity of compx spraig cos [-]. orovr, Opprma has show that Fra- Zaoff a Chu Squcs ar th spcia cass of his fiy of spraig squcs. Th imsio of a Opprma st with th co gth of is trmi y Eur s totit fuctio. For th cas whr is a prim umr, th siz of th Opprrma co st is qua to -. This is o of th asic imitatios of Opprma Cos [,]. or rcty, Graiz Discrt Fourir Trasform ( was itrouc a it provis a thortica frwor whr may popuar costat mouus orthogoa fuctio sts icuig DFT a othrs show to th spcia soutios []. I cotrast to iar phas DFT, fiy xpors th phas spac i its tirty i orr to improv corratio proprtis of costat powr orthogoa spraig cos. W prst a its corratio proprtis with rspct to th w ow corratio mtrics aog with popuar fiis i Go a Wash i th foowig sctios of th papr. II. GEEALIZED DFT A th root of uity is a compx umr satisfyig th quatio [] z,,,... ( If z m hos Eq. ( ut z ;< m<, th z is fi as a primitiv th root of uity. Th compx umr ( π / z is th primitiv th root of uity with th smast positiv argumt. Th othr primitiv th roots of uity ar xprss as ( π / z,, 3,..., ( whr a ar co-prim. A primitiv th roots of uity satisfy th uiqu summatio proprty of a gomtric sris xprss as foows ( z, z,,,.., (3 z, >

2 ow, w fi a prioic, with th prio of, costat mouus, compx iscrt-tim squc ( as ( / ( ( π r z r,,,,..., - ( r r This compx squc ovr a fiit iscrt-tim itrva i a gomtric sris is xprss accorig to Eq. (3 as foows ( / ( π r r, r m, r m m, itgr This mathmatica proprty is utiiz with th factorizatio ito two orthogoa xpotia fuctios whr o fis th iscrt Fourir trasform (DFT st { ( } satisfyig, r m, r m m, itgr * ( / ( ( ( π Th otatio (* rprsts th compx cougat fuctio of a fuctio. ot that ω π / is th th root of uity o th uit circ a aso ca th futa frqucy fi i th uit of raias pr cyc. W ar goig to xpa th iar phas fuctios of Eq. ( i th fiitio of. Lt s graiz Eq. (5 y itroucig a prouct fuctio i th phas fi as ( ( ( a xprssig a costat pitu orthogoa st as foows, * ( ( r ( π / r (5 ( ( π / ( ( π / [ ( ( ], ( ( ( r m, ( ( ( r m m, itgr (7 Hc, th asis fuctios of th w orthogoa st ar fi as ( / ( ( ( π &,,..., - Th w orthogoa fuctio st i Eq. ( is ca th Graiz Discrt Fourir Trasform ( []. It is ot that thr ar ifiity may fuctio sts with costat powr ar avaia. As a xp, o might fi th iscrt tim ratioa fuctio ( i Eq. ( as th ratio of two poyomias, ( a ( D ( c Lt s assum that th omiator poyomia D ( is qua to o a th orr umrator poyomia i is fi as foows ( a a + a + a a 3 3 (9 ( I gra, { a, } cofficits ar ra umrs. ow, w wi fi svra corratio mtrics to compar various co sts. Ths corratio typs ar ow to ictat th prformac of a muticarrir commuicatios systm. III. COELATIO ETICS Sic thr ar ifiity may possi s with oiar a iar phas i th soutios spac, w fi a fw mtrics to compar th prformac of various co sts. Ths mtrics asicay p o th auto- a crosscorratio proprtis of th orthogoa sts. O th othr ha, th muti-usr itrfrc a sychroizatio iaiity of co sts ar thorticay show to p o thir auto- a cross-corratios proprtis. Thr ar svra iffrt typs of corratio fuctios fi i th itratur to charactriz co sts of iffrt fiis. I this stuy, w maiy focus o th aprioic corratio fuctio (F. F for two compx squcs (, ( of a -imsioa st is fi as []

3 m * ( ( + m, < m + m * ( ( (,, < m m m (, m Th foowig mtrics mpoy i th rut forc sarch procss to sig various xps prst i this papr. III.A aximum Vau of Auto- a Cross-Corratio Squcs: Th imum corratio vau of a st of squcs { ( };,,,,..., is cacuat as, ( whr is th imum vau of autocorratio squcs for th tir st otai from Eq. ( wh { ( ( };,,,..., as giv i th foowig quatio, ( m < m < (3 Simiary, is th imum vau of a possi crosscorratio squcs i a co st aso cacuat from Eq. ( a is xprss as ( m,, < m < ( I Eq. (3 a Eq. (, is th siz of th co st a is th co gth i th st. Sarwat show th ratioship tw th imum outof-phas auto-corratio a th imum crosscorratio as foows [3], ( + ( aig to th Wch ou for compx spraig squcs xprss as [], (5, ( I Ta, w ispay th achiva Wch ous for costat mouus compx spraig cos for various gths. Ta : Achiva Wch Bous for Various Spraig Co Lgths. ( III.B a Squar Vau of Auto- a Cross- Corratio Squcs: Th quatitativ masurs giv aov ar importat to highight th worst cas scarios. I cotrast, th avrag prformac couts mor i som appicatios. Thrfor, w ta ito accout th ma squar vau of cross-corratio squcs as aothr prformac mtric. Furthrmor, th avrag of ma squar auto-corratio squcs for ach co i th st,, a th avrag of ma squar cross-corratio squcs for a co pairs i th st, as foows [],, ( m m m, ar itrouc (7 m (, ( ( m III.C Th rit Factor (F : Co sychroizatio is crucia for th prformac of CDA systms a it is strogy rat to th auto-corratio proprtis of cos. I orr to icorporat this rquirmt i co vauatio aothr mtric ca th mrit factor (F was itrouc i []. Th mrit factor for th th co is th ratio of th rgy i th mai o of th autocorratio fuctio ovr th rgy i th si os a it is mathmaticay xprss as ( F ( m m (9 I CDA commuicatios systms, mrit factor is sir to as arg as possi i orr to improv th co sychroizatio a iaiity. xt, w ar goig to sarch

4 for soutios of a G matrix typ with rut forc sarch tchiqu with rspct to th corratio prformac mtrics fi aov i orr to a to sig matrics fi i Eq. ( ow. A IV. DESIG WITH DIAGOAL G ATIX I this sig xp w us a o-costat ( fuctio of Eq. ( i th phas for ach fuctio of th st xprss as ( a + a a ( + a ( Thrfor, th asis fuctios of th st ar fi accorig to Eq. ( as + ( π/ ( ( π/ ( ( π/ ( a ( (,,,..., - ot that th first xpotia trm of th ast quatio is mry th DFT r with iar phas whi th sco xpotia trm fis th G matrix a { ( } ar th row squcs of A matrix that is fi as foows A A G ( DFT I this form, y chagig th vaus of ra a a cofficits, o might otai may iffrt trasform sts with sira auto- a cross-corratio proprtis a oiar phas fuctios. I th prvious sctio, w fi svra mtrics for th vauatio of various spraig co sts. ow, w ispay th vaus of ths mtrics for optima A matrics otai i th soutios spac utiizig a rut-forc sarch whr th sarch rsoutio is fi y th iary vau a a cofficits with th corrspoig umr of its pr cofficit as fi aov. Ta tauats th optima vaus of th mtric aog with othr prformac mtrics for various sarch rsoutios fi as Δ, 5/ a whr is th sarch rsoutio its pr cofficit a < a, 5 for th co gth of. Ta : Vaus of Various trics wh Optima Dsig is Bas o th Prformac tric for th Co Lgth of. (its/c F (OPT Simiary, Ta 3 ispays th corratio mtrics for various ow cos aog with th optima A matrix otai through a sarch as o th sig mtric for th co gth of. Ta 3: Corratio Prformac trics for Various Popuar Spraig Co Fiis with th Co gth of. Co Wash [x] Wash-i [x], [3] DFT [x] / Go Opprma St, [,] (opt (m, p,.9, 7 A [x] (opt I Figurs.a a., w ispay th itr-pcs of th auto- a cross-corratio mtrics a, rspctivy, o oth of th sig partrs a a. For a muti-carrir commuicatios appicatio, pig o th systm ithr OFD as or CDA as, o ca choos optimum vaus of a a for th sir vaus of auto- a cross-corratio mtrics, a. I OFD systms, frqucy ocaizatio is mor importat a th optimizatio is prform o partrs whras i a DS-CDA systm, a oth ar F

5 quay importat. Th ow vaus of is sir for sychroizatio purposs of th systm. Simiary, th ow vaus of ar rquir to miimiz muti-usr itrfrc (UI that ictats th systm prformac.. COCLUSIOS.9..7 uti-carrir itrfrc a sychroizatio iaiity of orthogoa sts i a OFD or CDA as commuicatios systm ar thorticay show to p o thir auto- a cross-corratios proprtis. with oiar phas offrs a uifi frwor to sig costat mouus orthogoa sts. W propos i this papr a corratio-as optima orthogoa st sig mtho proviig prformac improvmts ovr th ow fiis. It is xpct that as orthogoa mutipxrs with improv prformac a sig fxiiitis wi aow us to ui ttr OFD a CDA as muticarrir commuicatios systms i th futur...5. a Figur.: Variatio of th cross-corratio mtric as a fuctio of th sig partrs a a. 5 A C 3 a Figur.a: Variatio of th auto-corratio mtric as a fuctio of th sig partrs a a. EFEECES []. Go, Optima Biary Squcs for Spra Spctrum utipxig, IEEE Tras. Iformatio Thory, pp. 9-, Oct. 97. [] K.G. Bauchp, Appicatios of Wash a at Fuctios. Acic Prss, 9. [3] A.. Aasu a. Pouri, Wash-i oiar Phas Orthogoa Trasforms for Dirct Squc CDA Commuicatios, IEEE Trasactios o Siga Procssig, pp. 3-3, Juy 7. [] D. Sarwat,. Pursy a W.Star, Error Proaiity for Dirct-Squc Spra-Spctrum utip-accss Commuicatios Part I: Uppr a Lowr Bous, IEEE Tras. o Commuicatios, vo. 3, pp , ay 9. [5] D. Sarwat a. Pursy, Prformac Evauatio for Phas-Co Spra-Spctrum utip-accss Commuicatio Part II: Co Squc Aaysis, IEEE Tras. o Commuicatios, vo. 5, pp. -3, Aug []. Goay, Th rit Factor of Log Low Autocorratio Biary Squcs, IEEE Tras. o Iformatio Thory, vo., pp , ay 9. [7] H. Fuumasa,. Koho a H. Imai, Dsig of Psuo ois Squcs with Goo O a Ev Corratio Proprtis for DS/CDA, IEEE Joura o Sct Aras i Commuicatios, vo., o. 5, Ju 99. [] I. Opprma a B.S. Vuctic, Compx Vau Spraig Squcs with a Wi ag of Corratio Proprtis, IEEE Trasactios o Commuicatios, pp , arch 997. [9] D.C. Chu, Poyphas Cos with Goo Prioic Corratio Proprtis, IEEE Trasactios o Iformatio Thory, pp. 7-7, Juy 97. [].L. Fra a S.A. Zaoff, "Phas Shift Pus Cos with Goo Prioic Corratio Proprtis," IE Tras. o Ifo. Thory, vo. IT-, pp. 3-3, 9. [] A.. Aasu a H. Agirma-Tosu, Graiz Discrt Fourir Trasforms: Thory a Dsig thos, IEEE Saroff Symposium, arch 9. [] Z. Guozh a L. Cog, Fiy Siz of Orthogoa Opprma Squcs, Ectroics Lttrs, vo. 37, pp. 33, ay. [3] D. Sarwat, Bous o Crosscorratio a Autocorratio of Squcs, IEEE Trasactios o Iformatio Thory, vo. 5, pp. 7-75, ovmr 979. [] L. Wch, Lowr Bous o th aximum Cross Corratio of Sigas, IEEE Trasactios o Iformatio Thory, vo., pp , ay 97. [5] B.J. Wysoci, Siga Formats for Co Divisio utip Accss Wirss twors, Ph.D. Thsis, Curti Uivrsity of Tchoogy, Austraia,.

Discrete Fourier Transform (DFT)

Discrete Fourier Transform (DFT) Discrt Fourir Trasorm DFT Major: All Egirig Majors Authors: Duc guy http://umricalmthods.g.us.du umrical Mthods or STEM udrgraduats 8/3/29 http://umricalmthods.g.us.du Discrt Fourir Trasorm Rcalld th xpotial