Non-Vanishing Space Time Block Code for Three Time Slots and Two Transmit Antennas

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1 Journal of Computer and Communcatons Publshed Onlne June 05 n Sces. Non-Vanshng Space me Block Code for hree me Slots and wo ransmt Antennas Al Azarbar Department of Electrcal and Computer Engneerng slamc Azad Unversty Parand Branch ehran ran Emal: azarbar@ee.sharf.edu eceved 0 March 05; accepted 0 May 05; publshed June 05 Copyrght 05 by author and Scentfc esearch Publshng nc. hs work s lcensed under the Creatve Commons Attrbuton nternatonal Lcense (CC BY). Abstract ecently space tme block codes (SBCs) are posed for mult-nput and mult-output (MMO) antenna systems. Desgnng an SBC wth both low decodng complexty and non-vanshng perty for the Long erm Evoluton Advanced (LE-A) remans an open ssue. n ths paper frst our prevously posed SBC s non-vanshng perty wll be completely descrbed. he posed SBC scheme has some nterestng pertes: ) the scheme can acheve full rate and full dversty; ) ts maxmum lkelhood (ML) decodng requres a jont detecton of three real symbols; ) the mnmum determnant values (MDVs) do not vansh by ncreasng sgnal constellaton szes; 4) compatble wth the sngle antenna transmsson mode. he sentence has been dropped. Second n order to mve BE performance we pose a varant of posed SBC. hs scheme further decreases the detecton complexty wth a rate reducton of %; moreover non-vanshng MDVs perty s preserved. he smulaton results show the second posed SBC has better BE performance compared wth other schemes. Keywords Space me Block Codes Maxmum Lkelhood Decodng Non-Vanshng Mnmum Determnant Value. ntroducton Space-tme block codes (SBCs) are known as well-suted technques that vde an effectve dversty method to mtgate the fadng n wreless channels. n these codes transmtted sgnals are repeated n dfferent tme slots by usng two or more transmt antennas. n order to vde dversty gan each replca of a sgnal must en- ow to cte ths paper: Azarbar A. (05) Non-Vanshng Space me Block Code for hree me Slots and wo ransmt Antennas. Journal of Computer and Communcatons

2 counter ndependent fadng. hus transmt (receve) antennas must be separated apprately. herefore f each replca of transmtted sgnals encounters ndependent fadng the bablty of occurrng deep fadng s very unlkely. Alamout code s the most popular SBC scheme for two-transmt antennas systems []. t acheves code rate one and full dversty transmsson usng two-tme slots for nformaton symbols. arokh and et al. generalzed Alamout code for systems wth an arbtrary number of transmt antennas whch are called orthogonal codes []. Although these codes vde full dversty for more than two transmt antennas wth lnear decodng complexty code rate s less than one. o ncrease code rate for more than two transmt antennas Quas-Orthogonal SBC (QSBC) scheme s ntroduced n [] and [4]. owever code rate s ncreased for QSBC scheme decodng complexty s hgher but not exponentally compared wth orthogonal codes. he hree Generaton Partnershp Project (GPP) started the next generaton wreless systems (4G) under the ject Long erm Evoluton Advanced (LE-A) n 008 [5]. n LE-A user equpment (UE) s mposed two transmt antennas. herefore SBC scheme can be the most popular canddate for the uplnk dversty gan [6]. Alamout SBC scheme sounds a sutable canddate n LE-A systems. Unfortunately n LE frame structure s dedcated -tme slots wth Alamout SBC scheme. hs has brought up an nterestng SBC desgn blem whch s compatble wth LE frame structure. ybrd SBC scheme as the frst scheme has been posed for -tme slots and two transmt antennas systems [7]. ts encodng matrx ncludes two tme slots Alamout scheme followed by one tme slot repetton transmsson. Although the ybrd scheme acheves code rate one and ts decodng complexty s lnear at recever does not acheve full dversty. o remove full dversty defect a class of QSBC scheme s posed by Le et al. n [8] [9]. he QSBC scheme acheves code rate one and full dversty. But there are two blems wth ths scheme. he frst blem s hghcomplexty of maxmum lkelhood (ML) decodng whch requres a jont detecton of two complex symbols (O(M )) where M s sze of the used symbol constellaton. he second blem s that the mnmum determnant values (MDVs) extremely vansh by ncreasng the symbol constellaton sze. ecently Fast-Group-Decodable SBC (Fast-GSBC) scheme has been posed n [0] []. As generc constructon method for odd-tme slot new GSBC scheme has been desgned for LE-A that acheves code rate one and full dversty wth symbol-wse decodng complexty (O(M )) []. GSBC scheme acheves code rate one and full dversty wth symbol-wse decodng complexty (O(M )). owever GSBC scheme reduces decodng complexty for -tme slots two transmt antennas ts MVDs vansh. Also ths scheme s not compatble wth sngle antenna. n [] a novel -tme slots SBC structure based on trace crteron has been desgned. o the best our knowledge ths s the frst -tme slots two transmt antennas SBC scheme whch has non-vanshng MDVs perty. n ths paper by usng ths SBC structure ) a -tme slots SBC scheme whch can acheve code rate one and full dversty wth decodng complexty of O(M.5 ) s posed ) a -tme slots code rate / SBC scheme whch can acheve full dversty wth symbol-wse decodng complexty s obtaned. Also we wll show that both schemes have non-vanshng MDVs perty. he smulaton results show that our frst scheme has the same bt error rate (BE) performance wth the GSBC scheme. owever BE for the second posed scheme s mved about 0. db compared wth the frst scheme. he rest of the paper s organzed as follows: Secton comprses two subsectons:.. Channel Model.. Code Desgn Crtera and.. evew of hree me Slots wo ransmt Antennas SBC Schemes. n Secton the Non-Vanshng MDVs Code ate One -me Slots SBC s ntroduced. hs secton ncludes four subsectons:.. Encodng matrx.. Parameter k Optmzaton.. Decodng Complexty and.4. Some Propertes. n Secton 4 Code ate / -tme slot -antenna SBC s ntroduced. n Secton 5 Smulaton esults and Dscusson s presented: 5.. Performance Comparson n ate One Scheme and 5.. Performance Comparson n ate / Scheme. Concluson s gven n Secton 6. Notatons: ereafter j = small letters bold letters and bold captal letters wll desgnate scalars vectors and matrces respectvely. f A s a matrx A A and tr(a) denote the conjugate-transpose transpose and trace of A respectvely; (.) (.) and (.) denote the real part the magnary part and the complex conjugate respectvely.. evew of -me Slots SBC for wo ransmt Antennas.. Cannel Model Consder a MMO system wth N t (N t = ) transmt antennas and N r receve antennas and wth quas-statc flat fadng of block length ( = ). t s assumed that the channel state nformaton (CS) to be known at the re- 75

3 cever but unknown at the transmtter. he nput-output relaton of ths system can be wrtten as where the normalzaton q ( ρ ) Y = q + Z () q = N t s to ensure that the ρ (SN at the recever) s ndependent of the number of the transmt antennas (N t ). s the Nt complex matrx of the transmtted symbols that are drawn constellaton. s the Nt Nr complex matrx that contans all the channel coeffcents wth zero mean and unt varance. Z s the Nr complex nose matrx and Y s the Nr complex matrx of the receved sgnals. he entres of Z are assumed to be..d. complex Gaussan random varables wth the bablty densty functon (pdf) CN ( 0 N 0 ) where CN (..) stands for the complex Gaussan pdf and N 0 for the nose varance... Code Desgn Crtera... ank and Determnant Crtera ecently SBC schemes manly rely on analyss of the par wse error bablty (PEP) P( ˆ ) whch s the bablty that ˆ s detected whle s transmtted. Chernoff bound Analyss of the PEP at hgh SN values results n ank crteron and Determnant crteron [4]. he SBC scheme has full dversty perty f ˆ s full rank for all codeword pars. he dversty gan at hgh SN values do- the dfference matrx mnates the steepness of the Bt Error ate (BE) curve. hus n SBC scheme desgn ensurng full dversty s mportant at hgh SN values. Afterwards codng gan should be maxmzed for gven average transmt power that leads to a good determnant crteron. he maxmum codng gan results n the mnmum PEP. Besdes maxmzng codng gan ths value should be constant for any symbol constellaton szes. hs perty s called non-vanshng MDV and has been establshed for several popular SBC schemes [5]-[8]. he non-vanshng MDV can beexploted through the use of the adaptve constellaton (adaptve rate) accordng to the wreless channel qualty.... race Crteron he trace crteron s less known but paramount for desgnng non-orthogonal SBC schemes [9] [0]. hs crteron states: to optmze performance of the BE SBC scheme say should be desgned so that the e- ˆ ˆ tr ˆ ˆ genvalues of the ( ) ( ) are as close as possble to each other and to and for whch the row-wse sum of the absolute values of the elements off the man dagonal s as small as tr ˆ ˆ plays the role of the Eucldean dstance between codewords. possble. Moreover the.. evew of hree me Slots wo ransmt Antennas SBC Schemes n ths secton the three tme slots two transmt antennas SBC schemes has been revewed. Also advantageous and dsadvantageous of the all schemes are ncluded.... ybrd SBC Scheme he hybrd scheme SBC s the frst -tme slot SBC scheme has been posed [7]. Encodng matrx of ths scheme s prescrbed n (). As can be seen n encodng matrx Alamout SBC s used at fst -tme slots and symbol s s repeated at the last tme slot over both antennas. Such SBC scheme has code rate one wth lnear decodng at recever and symbols are encoded by standard modulaton that has low hardware complexty. Because symbol s s not transmtted n dfferent tme slot full dversty does not acheve. * s s s SBC = * () s s s owever the scheme acheves code rate one and ts decodng complexty s lnear at recever does not acheve full dversty. 76

4 ... QSBC Scheme ncapactaton of ybrd SBC to acheve full dversty was good reason for author n [8] [9] to desgn a class of QSBC scheme QSBC compatble wth -tme slots systems. he encodng matrx of such scheme has a general form of QSBC where s = [ s s s ]. By defnng A = [ b b b ] s s s = * * * b s b s b s A yelds: π π j j 5 5 e e π π j j 5 5 A = e e (4) π π j j 5 5 e e QSBC has code rate one full dversty and backward compatble wth sngle antenna pertes. Also t s ven that ML decodng complexty of the QSBC s O(M ). here s two defects wth QSBC. Frst defect s hgh decodng complexty and second MDVs vansh by ncreasng symbol modulaton orders (.e. non-vanshng MDV perty does not preserve). Also the rotaton factors n matrx A may cause encoder and decoder accommodate three dfferent constellatons our smulatons show for 4-QAM modulaton. MDV s 7.8 whle for 6-QAM ths value s 0.. herefore n order to reduce the decodng complexty the code n subsecton.c.. s posed.... Group-Decodable SBC Scheme ecently Fast-Group-Decodable SBC (Fast-GSBC) scheme has been posed n [0] []. As generc constructon method for odd-tme slot new GSBC scheme wth arbtrary code dmenson ncludng odd tme slot has been desgned. Based on Fast-Group-Decodable SBC n [] a new SBC scheme for LE-A system was desgned as follows s+ s * s+ s s GSBC = (5) * * s + s s+ s s GSBC acheves code rate one and full dversty wth symbol-wse decodng complexty. Encoder/decoder only needs to accommodate one rotated constellaton that reduces hardware complexty. owever smlar to scheme n (4) GSBC cannot preserve non-vanshng MDVs perty. For 4-QAM and 6-QAM MDV s 6 but for 8-QAM the MVD s 6.8. Also ths scheme s not sutable for sngle antenna transmsson. So desgn of -tme slots two-transmt antennas SBC scheme wth non-vanshng MDVs s requred. n the next secton a novel SBC structure wth non-vanshng MDVs perty that has been posed n [] wll be presented.. Non-Vanshng MDVs Code ate One -me Slots SBC n ths secton ntally the encodng matrx s presented. hen parameter k s optmzed to maxmze the MDVs. Also we wll ve that our scheme acheves non-vanshng MDVs. Fnally the decodng complexty of the posed SBC scheme wth ML crteron s llustrated... Encodng Matrx n ths subsecton the blem s formulated. Fgure shows a schematc drawng of the posed SBC scheme. hree data symbols are transmtted consecutvely durng three tme slots. herefore full rate s acheved (an SBC rate s defned as the number of () 77

5 st antenna x x x nd antenna v v v Fgure. he posed SBC scheme. transmtted symbols durng tme slots). he st antenna transmts the three symbols x x and x durng three tme slots. he nd antenna transmts three symbols v v and v. Now the three symbols transmtted by the nd antenna wll be defned. One possble way to defne these symbols s to make vector v (v = [v v v ]) orthogonal to vector s (s = [x x x ]).e. s v = v s = 0. As wll be shown n the next subsecton ths causes the decodng complexty reduces from jont three complex symbols detecton to jont three real symbols; moreover results n sutable trace crtera [9] [0]. hen mve BE performance at hgh SN. o acheve ths goal three dfferent SBC schemes are posed as: x x x x x x x x x = * * = * * = * * x x 0 x 0 x 0 x x (6) n all of the above schemes the frst row s orthogonal to the second one. owever zero entry n the second row reduces dversty order. he followng soluton overcomes to ths defcency: v = v + v + v (7) * * * * * * where v ( = ) s the second row n matrx. herefore v yelds as v = x x x x x + x. n order to acheve power balance the symbols n the vector v are transmtted wth power. Now we can present the posed encodng matrx as: x x x * * * * * * = x x x x x + x (8) For full dversty symbols x x and x are selected from three dfferent symbol constellatons. Fgure represents these three symbol constellatons. From the represented constellatons x ( = ) are obtaned as []: Power ( k ) x = s + js x = + k s + jks x = + k ks + js. + n x and x guarantees ε x = ue of z. s and s are chosen from the standard QAM constellaton. Parameter k s unknown and must be optmzed to maxmze the MDVs. n the subsecton.b the parameter k wll be optmzed. Note that despte the frst row s orthogonal to the second one s nether orthogonal code nor quas-orthogonal. Consder K orthogonal code say orth then orth orth = α n whch = and s dentty matrx... Parameter K Optmzaton (9) ( = ) where ε [ z] s the excepted val- α s k = k n the prevous subsecton the three dfferent symbol constellatons were represented. he parameter k norma- 78

6 (.a) (a) (.b) (.c) (b) (c) Fgure. hree dfferent symbol constellatons. (a) he standard QAM constellaton that symbol x s chosen from; (b) he horzontally stretched QAM constellaton that symbol x s chosen from; (c) he vertcally stretched QAM constellaton that symbol x s chosen from. lzes symbols x and x. n ths subsecton the parameter k s optmzed. det det s s v s = 0 (0) s v v v Where ˆ = and and ˆ represent two dfferent codewords and det(.) denotes determnant value of (.). Accordng to v s = s v = 0 t s easy to show that s full rank. Note that when k = t s possble v = 0 although we consder 0< k < n.. hs vouches that determnant s always nonzero. hen ( * * * mndet = x x + xx xx + xx ) () = = where x ˆ = x x ( = ) and x represents the possble error n symbol x. Lemma : has non-vanshng-mdv and for d mn = ts value s 5.8. Prove: Consder M-ary QAM standard constellaton where the real and mage components of a symbol can be vewed as M -ary standard PAM and M -ary standard PAM symbols respectvely (e.g. 8-ary QAM constellaton can be consdered as 4-ary PAM and -ary PAM for real and mage components respectvely). hen for real components x = md mn x = m ( + k ) dmn and x ( = m k + k ) dmn d mn s the mnmum Eucldan dstance between the PAM constellaton ponts and here s consdered as. m ( = ) s an nteger such that m { [ M :: M+ ] 0}. Wth the above assumptons the optmzed MDV for 4-QAM (for traceablty constellaton s assumed 4- QAM) n () s acheved for m { 0 ± } observe that m { 0 ± } s compatble wth any QAM constellaton sze. As mentoned before when 0< k < the determnant n () s nonzero. o optmze MDV and show non-vanshng-mdv perty defne expresson D as follows: where z D ( z z z )( z z z zz zz zz) = m z = m ( + k ) and z ( m k k ) = () = +. n fact D s smlar the expresson n () 4 that normalzed by 6d mn and for traceablty x = x = x = 0 s consdered. Snce the frst term n () s always nonzero defne D = ( z+ z+ z+ zz zz+ zz) < D. Now suppose two followng cases: case: z = 0 ths case yelds: ( ) D = z + z + z z () Accordng to z and z and k the mnmum value s always obtaned for z = 0 and z 0. hen we defne D as follows mn m e m e m e 79

7 ( ) Note both z and z cannot be zero at the same tme. case : z 0 ths case yelds: { } D mn k = m k + k (4) ( ) ( ( )) ( ( )) mm k ( + k ) + mm ( + k ) D = m + m + k + m k + k + m m k + k m 0 ± s dfferent from m n (). t s clear for m = m = m 0 D has mnmum determnant value. hen D s defned as follows: mn ( ) D k = k + k k + k + k m (6) mn. n order to maxmze MDVs equate D ( k) and D ( k) mn : mn { } mn mn (5) D k = D k m m ±. (7) hs equalty yelds k = 0.66 and correspondng MDV s mn det = 5.8 when d mn =. therefore the obtaned mn det s the maxmum mn det for all k. m 0 ± s assumed ths assumpton s compatble wth any constellaton sze. Note that { }.. Decodng o llustrate the decodng complexty of the posed SBC scheme wth the ML crteron the decson metrc used for the ML decoder wll be derved. Consder a sngle antenna at recever N r = the ML decoder metrc s: y h (8) where h and y are represented n (). he objecton of the ML decoder s to obtan optmal between all of the possbltes whch mnmze (). After some manpulatons and and where ( ) ( ) y h = f x x x + f x x x (9) ( ) = { } + { } + { } + ( + + ) f x x x a x b x c x x x x h + x x x x + x x h = ( ) = { } + { } + { } + ( + + ) f x x x a x b x c x x x x h + xx xx + xx h a= rh r r h b= rh r r h c= rh r r h. * * * * * * = (0.) (0.) From (9) t s clear that f s ndependent of x x and x and f s ndependent of x x and x. he mnmzaton of (9) s equvalent to mnmze f and f ndependently. herefore the ML decodng requres a jont detecton of three real symbols (O(M.5 )). Compared wth QSBC scheme n [9] and GSBC scheme n [] the posed scheme for square symbol constellaton has lower decodng complexty than QSBC scheme and slghtly hgher decodng complexty than GSBC scheme. But for real constellatons such 80

8 as BPSK the posed scheme wll have hghest decodng complexty among the three tme slots two transmt antennas schemes..4. Some Propertes Smple but mportant pertes of the posed code are llustrated. full rate and full dversty t was mentoned that three nformaton symbol are transmtted from two antennas durng three tme slots. hs acheves full rate perty. Also when 0< k < the determnant of the dfference matrx ( ) s always nonzero. hs ensures full dversty perty. Non-vanshng MDVs t was ved n lemma always has nonzero determnant that guarantees acheves full dversty and ts MDVs do not vansh by ncreasng symbol constellaton szes. Non-vanshng MDVs perty dstngushes our scheme from other schemes have been posed for LE-A systems. Compatble wth sngle transmt antenna Our scheme has the perty that frst row s xxx n (8) such perty vde backward compatble for sngle antenna whch s desred n LE-A. QSBC encodng matrx n () s also compatble wth sngle antenna system whle has not such perty. GSBC 4. Code ate / -tme Slot -Antenna SBC t was shown has non-vanshng-mdv perty but ts decodng complexty slghtly s hgh. o reduce detecton complexty another SBC scheme s posed. has these pertes: ) detecton complexty reduces from order.5 to ; ) non-vanshng-mdv s preserved n ; ) code rate reduces from to /. Encodng Matrx. Consder detec- he structure of s same ton functon n (9) agan:. owever encodng of symbols s dfferent from ( ) = ( ) + ( + ) f s s s g s s s x x x x x x h At the above expresson the last term s called symbol nterference term. We can decrease nterference by omttng one of the nformaton symbols say x. n fact nstead of three symbols two symbols s s s transmtted. Note the structure of the encodng matrx s preserved for but encodng of symbols s changed as follows: and x = s x = s and x = js () x x x = * * * * * * x x x x x + x () where s and s are chosen from the M-QAM standard constellaton. Followng lemma shows ths scheme ensures non-vanshng MDVs perty too. Lemma : has non-vanshng-mdv and ts value s when d mn =. he determnant for s obtaned: (( ) ) = ( + ) + + ( ) det D s s s s s s. Lke lemma Consder M-ary QAM standard constellaton where the real and mage components of a symbol can be vewed as M -ary standard PAM and M -ary standard PAM symbols respectvely. d mn shows Eucldan dstance between the PAM constellaton ponts and here s consdered as. As regards the frst term n () () 8

9 s always greater than zero we defne D = s + s + ( s s ) < D. For traceablty ( s ) ( s ) hen D yelds: = = 0. D = s + s + s s (4) where s ˆ = s s = and as mentoned before s and s ˆ are chosen from M-QAM standard constellaton. hus s = md s mn = md mn where m ( = ) s an nteger such that m {[ M :: M ] 0} by 4d mn we obtan Assume followng two cases: Case : m = 0 or m = 0 hs case results n: When = or =. Case : m = m 0 hs case results n: (5) +. eplace (4) by (5) and normalze D = m + m + mm (6). D m = (7) m. D = (8) D s mnmum when m = ± or m = ± and D s mnmum when m = m =. herefore D = D =. hs means expresson n () s always nonzero for any symbol constellaton szes and has a mnmum. Accordng to (5) ths mnmum value s when d mn = s consdered. he mnmum value s non-vanshng and constant over all symbol constellaton szes. Compared wth Scheme cannot ensure full rate perty compared wth. But full dversty and non-vanshng pertes preserve n scheme smlar to scheme. Also t has hghest MDVs compared to the exstng three tme slots two transmt antennas SBCs [7] [9] [] and. t can be shown that ML decodng metrc can be calculated as the sum f + f where and { } { } * * * * f = rh r r h s + rh r r h s + ( s + s ) h + h + s s h = { } (? ) { } * * * * f = rh r r h s + rh r r h s + ( s + s ) h + h + ss h = (9.) (9.) snce f and f s just functon of s and s for = respectvely the mnmzaton of the ML metrc s equvalent to mnmzng the two metrcs f and f ndependently. hat mples has lower decodng complexty than and has the same decodng complexty compared to GSBC scheme for any symbol constellaton szes except for BPSK modulaton. n general able summarzes varous pertes of all schemes and gves a detaled comparson. 8

10 5. Smulaton esults and Dscusson n ths secton the smulaton results of the posed schemes and are shown for bts per channel use (bpcu) and 4bpcu throughputs. t s assumed that the ampltudes of fadng from each transmt antenna to the receve antenna are mutually uncorrelated aylegh-dstrbuted and the recever has perfect knowledge of the channel. 5.. Performance Comparson n ate One Scheme We frst gve performance comparson between and other -tme -antenna SBC schemes. Fgure shows the smulaton results for the posed SBC scheme for two transmt antennas and one receve antenna and compares BE performance wth SBC n [7] QSBC n [8] [9] and GSBC n [] scheme. Also Alamout scheme (two symbols are transmtted durng three tme slots) s consdered. he transmtted symbols are 4-QAM modulated for all of the schemes except Alamout scheme (8-QAM modulated) hence ther spectral effcences are bpcu. As can be observed from Fgure the posed SBC QSBC GSBC and Alamout schemes have the same dversty order but hybrd SBC scheme does not acheve full dversty. Also from Fgure t s clear that even wth lower decodng complexty the posed scheme outperforms the QSBC scheme. owever the posed scheme wth hgher decodng complexty than GSBC scheme has same smlar performance at hgh SN. For 4-QAM modulaton Note that the MDVs for the posed scheme QSBC and GSBC are and 6 respectvely. herefore t s expected that the both QSBC and GSBC schemes to have better BE performance upon the posed whch s n contrast wth smulaton results. But besde rank (dversty) and determnant crtera trace crteron must be consdered too. he trace crteron state: n uncorrelated aylegh fadng the lowest expected value for the unon bound to the parwse error event s obtaned when for all pars (transmtted matrx) and ˆ (detected matrx) the matrx ( = ˆ ) s dagonal wth all dagonal elements equal. Alternatvely the best suboptmal codes are those for whch the matrces are such that the man dagonal elements are as close as possble to each other and the row-wse sum of the absolute values of the elements off the man dagonal s as small as possble for each row. he matrx for the posed scheme s dagonal wth unequal man dagonal elements. owever both QSBC and GSBC schemes suffer from nonzero off the man dagonal elements that ncreases the absolute value of the off the man dagonal to man dagonal rato. Accordng to the trace crteron the posed scheme has better BE performance upon QSBC and GSBC schemes wth equal MVDs. n contrast accordng to the Alamout (8-QAM bpcu) ybrd (4-QAM bpcu) QSBC (4-QAM bpcu) GSBC (4-QAM bpcu) Proposed (4-QAM bpcu) QSBC (6-QAM 4bpcu) GSBC (6-QAM 4bpcu) QSBC (6-QAM 4bpcu) BE E b /N 0 (db) Fgure. BE curves of the frst posed GSBC QSBC hybrd and Alamout scheme. 8

11 determnant crteron the posed scheme should have worse BE performance than QSBC and GSBC scheme. herefore there s a tradeoff between lower determnant crteron and good trace crteron. hs tradeoff closes BE performance for all schemes at hgh SN for 4-QAM modulaton. Fgure shows BE performance of the posed SBC scheme and the QSBC scheme for 6-QAM modulaton (4 bpcu). t s clear that the QSBC scheme because of nonzero values off the man dagonal and lower MDV (for 6- QAM ts MDV s 0.) has poor BE performance compared wth other schemes e.g. at BE 0 4 both the GSBC and posed scheme about.5 db work better than the QSBC scheme. he MDVs of the posed scheme and GSBC scheme for 6-QAM modulaton are 6 and 5.8 respectvely. Nevertheless lke 4-QAM modulaton both schemes have same BE performance at hgh SN. 5.. Performance Comparson n ate / Scheme n ths subsecton BE curve for wth 4 bps throughput s presented and compared wth other schemes n Fgure 4. owever 64-QAM modulaton s used for and 6-QAM modulaton for other schemes smulaton results show the second posed scheme has lower BE than other schemes. Because the dagonal elements of the matrx n the second scheme are closer to each other can acheve trace crteron better than the frst scheme. n other sde has hgher codng gan than QSBC that result n good determnant crteron. herefore both trace and determnant crtera are domnant compared wth other schemes. he ft of ths can be seen n Fgure 4 that our second posed scheme outperforms 0. db n power effcency. Fgure 4. BE curves of the frst posed second posed GSBC QSBC schemes. able. Detaled comparson of varous pertes between schemes. Code ate Detecton Complexty Non-vanshng MDVs Compatble wth Sgnal Antenna Compatble wth LE-A Dversty Order Alamout [] 0.67 (Lnear Decoder) No SBC [7] (Lnear Decoder) No QSBC [9] (ML) No GSBC [] (ML) No No.5 (ML) / 0.67 (ML) No 84

12 6. Concluson n ths paper a novel SBC structure for three tme slots and two transmt antennas was posed. Based on ths structure two SBC schemes were posed. he frst scheme acheves full rate and full dversty pertes and has a jont three real symbols decodng complexty (O(M.5 )). Also the mnmum determnant value s constant for dfferent symbol constellaton szes. hen the posed scheme acheves non-vanshng-mdv perty. Also the posed scheme has the perty that frst row s xxx such perty vdes backward compatble for sngle antenna whch s desred n LE-A. owever the encoder (decoder) needs to accommodate dfferent constellatons whch may ncrease the hardware complexty at the encoder (decoder). he second scheme cannot acheve full rate. But full dversty and non-vanshng-mdv pertes are preserved. Moreover the second scheme reduces decodng complexty compared wth the frst one and s not compatble wth sngle antenna transmsson. Also ths scheme uses standard modulaton n encoder (decoder) whch s easly mplementable. Smulaton results show that our frst scheme has better BE performance than QSBC and smlar BE performance wth GSBC at hgh SN. Also the second scheme has the best BE performance compared wth other schemes. eferences [] Alamout S.M. (998) A Smple ransmtter Dversty Scheme for Wreless Communcaton. EEE Journal on Selected Areas n Communcatons ome [] arokh V. Jafarkhan. and Calderbank A.. (999) Space-me Block Codes from Orthogonal Desgns. EEE ransactons on nformaton heory [] Jafarkhan. (00) A Quas-Orthogonal Space-me Block Code. EEE ransactons on Communcatons [4] rkkonen O. Boaru A. and ottnen A. (000) Mnmal Nonorthogonalty ate Space-me Block Code for + x Antennas. 000 EEE 6th nternatonal Symposum on Spread Spectrum echnques and Applcatons [5] rd Generaton Partnershp Project (008) Evolved Unversal errestral ado Access (E-UA). Physcal Channels and Modulaton (elease 8) GPP S 6.. [6] Alcatel Shangha Bell Alcatel-Lucent (008) SBC- Scheme for Uplnk ransmt Dversty n LE-Advanced GPP SG AN WG Meetng #5 Bs. [7] Alcatel Shangha Bell Alcatel-Lucent (009) SBC- Scheme wth Nonpared Symbols for LE-Advanced Uplnk ransmt Dversty GPP SG AN WG Meetng #55 Bs. [8] Le Z.D. Yuen C. and Chn F. (00) hree-me-slot Quas-Orthogonal Space-me Block Code. 00 EEE nternatonal Conference on Communcatons (CC) Cape own -7 May [9] Le Z.D. Yuen C. and Chn F. (0) Quas-Orthogonal Space-me Block Codes for wo ransmt Antennas and hree me Slots. EEE ransactons on Wreless Communcatons [0] en.p. Guan Y.L. Yuen C. and Shen.J. (00) Fast-Group-Decodable Space-me Block Code. 00 EEE nformaton heory Workshop on nformaton heory (W 00 Caro) Caro 6-8 January [] en.p. Guan Y.L. Yuen C. Zhou Y. and Zhang E.Y. (0) Optmzaton of Fast-Decodable Full-ate SBC wth Non-Vanshng Determnants. EEE ransactons on Communcatons [] en.p. Yuen C. Guan Y.L. and Wang K.. (0) -me-slot Group-Decodable SBC wth Full ate and Full Dversty. EEE Communcatons Letters [] Ahmad M. and Vakly V.. (0) A Novel Space me Block Code Scheme for hree me Slots and wo ransmt Antennas. EEE Communcatons Letters [4] arokh V. Seshadr N. and Calderbank A.. (998) Space-me Codes for gh Data ate Wreless Communcaton: Performance Analyss and Code Constructon. EEE ransactons on nformaton heory [5] Yao. and Wornell G. (00) Achevng the Full MMO Dversty Multplexng Fronter wth otaton Based Space- me Codes. Proceedngs of the Allerton Conference on Communcaton Control and Computng Montcello - October 00 0 p. [6] avldar S. and Vswanath P. (006) Apxmately Unversal Codes over Slow Fadng Channels. EEE ransactons on nformaton heory

13 [7] Dayal P. and Varanas M.K. (00) An Optmal wo ransmt Antenna Spacetme Code and ts Stacked Extensons. Proceedngs of the Aslomar Conference on Sgnals Systems and Computers Pacfc Grove 8- November [8] Belfore J.-C. ekaya G. and Vterbo E. (005) he Golden Code: A Full-ate Space-me Code wth Non- Vanshng Determnants. EEE ransactons on nformaton heory [9] onescu D.M. (999) New esults on Space-me Code Desgn Crtera. Proceedngs of the 999 EEE Wreless Communcatons and Networkng Conference New Orleans -5 September [0] onescu D.M. (00) On Space-me Code Desgn. EEE ransactons on Wreless Communcatons 0-8. [] Snnokrot M.O. and Barry J.. (009) A Sngle-Symbol-Decodable Space me Block Code wth Full ate and Low Peak-to-Average Power ato. EEE ransactons on Wreless Communcatons