Linear dispersion code with an orthogonal row structure for simplifying sphere decoding

Transcription

1 tle Lnear dsperson code wth an orthogonal row structure for smplfyng sphere decodng Author(s) Da XG; Cheung SW; Yuk I Ctaton he 0th IEEE Internatonal Symposum On Personal Indoor and Moble Rado Communcatons (PIMRC 009) okyo Japan 3-6 September 009. In Proceedngs of the 0th PIMRC 009 p Issued Date 009 URL Rghts IEEE Internatonal Symposum on Personal Indoor and Moble Rado Communcatons. Copyrght IEEE.

2 Lnear Dsperson Code wth an Orthogonal Row Structure for Smplfyng Sphere Decodng X. G. Da S. W. Cheung and. I. Yuk Department of Electrcal & Electronc Engneerng he Unversty of Hong Kong Hong Kong Abstract In ths paper we propose a famly of full dversty fast decodable Lnear Dsperson Codes (LDCs) for use n MIMO systems. hese codes have as many as possble the orthogonal rows n the dsperson matrces of the LDC. One of the man advantages of ths famly of LDCs s that t can be used n MIMO systems wth arbtrary numbers of transmtted and receved antennas and arbtrary transmsson rates. We also develop a smplfed Sphere Decodng (SD) algorthm to sgnfcantly reduce the decodng complexty for ths famly of LDCs. Monte Carlo smulaton shows that the optmum LDCs wth or wthout ths orthogonal structure have nearly dentcal bt-error rate (BER) performances but the complexty can be sgnfcantly reduced. For a 4 MIMO system transmttng 8 QPSK symbols n a block length of 4 the reducton s about 8-5% and for a 3 3 MIMO system transmttng 0 QPSK symbols n a block length of 6 the reducton s about 8-49%. Keywords- MIMO Sphere decodng orthogonal complexty I. IRODUCIO Lnear Dsperson Code (LDC) [] s well-known for ts advantage n provdng full-ergodc capacty and good error probablty performance to Multple-Input Multple-Output (MIMO) communcaton systems. However to acheve the optmum bt-error rate (BER) performance the tremendous complexty of the optmum Maxmum Lkelhood (ML) decodng process makes the mplementaton of hgh speed LDCs mpractcal. Mnmum-Mean-Square Error (MMSE) [] and Zero-Forcng (ZF) [3] algorthms are much less complcated but the achevable BER performances are not very satsfactory. Researchers have been workng hard to develop smple decodng algorthms to acheve optmum BERs for LDCs. Alamout [4] proposed a remarkable space-tme (S) code n 998 for MIMO systems wth two transmt antennas. Later on n [5] orthogonal space-tme block codes (OSBCs) were proposed wth the same propertes as Alamout s code.e. allowng the use of a very smple ML decodng algorthm. However the man dsadvantage of the OSBCs s that they cannot acheve full-transmsson rates for MIMO systems wth more than two transmt antennas [6]. Quas-orthogonal SBCs [7] were then proposed to support full-transmsson rates but at a cost of lower dversty gans. In [8] Sphere Decodng (SD) was proposed to substantally reduce the complexty of ML decodng yet havng the same BER performance. SD can be appled to all lnear SBCs. However as Jalden et al. ponted out [9] for a fxed sgnal-tonose rato (SR) the complexty of SD ncreases exponentally wth the number of symbols jontly decoded and ths makes SD stll too complcated for hgh-data-rate transmssons. Improvements n dfferent aspects of SD have been proposed. In [0] a so-called Baba Pont method was used to set the ntal searched pont n SD. In [] a Schnorr Euchner (SE) enumeraton method was proposed to refne the search strategy of SD. In [] Paredes et al. reduced the complexty of SD by reducng the number of searched levels n the tree search process. By combnng the advantages of OSBCs and SD they constructed a famly of fast decodable full-rate full dversty codes for a MIMO system. Bgler et al. extended ths concept to a 4 MIMO system and developed a famly of quas-orthogonal structured codes leadng to a reducton of 4 levels n the tree search for SD [3]. A new famly of fast-decodable full-rank flexble-rate lnear dsperson codes (LDCs) for MIMO systems wth arbtrary numbers of transmt and receve antennas was proposed n [4]. hs new famly of LDCs has as many as possble the orthogonal rows n the dsperson matrces. he advantages of the orthogonal structure for dfferent modulatons were studed n [5]. In ths paper we study ths famly of LDCs for dfferent numbers of transmt and receve antennas. We also use a smplfed SD for the codes to acheve the same BER performance as conventonal SD or ML decodng but wth much less decodng complexty. he rest of ths paper s organzed as follows. he system model used for the study s defned n Secton. he prncple of SD s descrbed n Secton 3. In Secton 4 the famly of LDCs together wth the correspondng smplfed SD algorthm s descrbed. Monte Carlo smulaton results and dscussons are gven n Secton 5. Secton 6 concludes ths paper. II. SYSEM MODEL he system model used for the study s an t r MIMO system wth t transmt antennas and r receve antennas over a quas-statc Raylegh fadng channel. he r receved sgnal matrx R s gven by: R = HC+ W () /09/ $ IEEE 76

3 r t where the entres of H C represent the channel coeffcents whch are assumed to be perfectly known at the t recever but not at the transmtter C C s the codeword r matrx wth block length and W C represents the complex addtve whte Gaussan nose (AWG) matrx wth elements beng ndependently and dentcally dstrbuted (d) and followng the normal dstrbuton C(0 0 ). For LDCs the codeword C can be expressed as []: C= M s () = where { M } = are the dsperson matrces of the LDC { s } = are the transmtted symbols takng values from some complex constellaton n a fnte set S and s the number of symbols n one codeword. All elements n R H C and W of () are complex varables. he transmtted symbols n () can be expressed n vector form as s = [ s s s3 s4... s ]. hen substtutng () nto () and takng vectorzaton on both sdes yelds [] r = KXs + w (3) where r = vec( R) = r r... r r... r... r r r r t wth r j beng the entry n the th row and j th column of matrx R and [ ] denotng matrx transposton X = [ vec( M0) vec( M)... vec( M )] s = vec() s = s = [ s... s ] and w = vec( W ). r t In (3) K = I H where K C I s an dentty matrx and denotes the Kronecker product. III. SPHERE DECODIG o facltate the mplementaton of SD decodng we need to transform the model usng complex varables n (3) to an equvalent model usng real varables as follows. Separatng the real and magnary parts of the elements n r s and w of (3) and then takng vectorzaton on both sdes gve the realvalued expresson: r = Gs + w (4) [ ] where r = Re( )Im( ) r r = Re( s) Im( s) = [Re( s)... Re( s ) Im( )...Im( )] s s w = [Re( w ) Im( w )] and Re(.) and Im(.) denote the real and magnary parts of (.) respectvely. In (4) G s a r real matrx gven by: Re( KX) Im( KX) G = Im( KX) Re( KX) Here we arrange the vector to = [Re( s )Im( s )Re( s )Im( s )...Re( s )Im( s )] (5) = [ s s s s... s ] (6) 3 4 o keep the receved sgnal vector r n (4) unchanged we need to arrange the columns of G correspondngly. We denote the columns of G as [ col ( G ) col ( G ) col ( G ) ] where col ( G ) for = s the th column of G and we re-arrange the columns of G to gve: G = [ col G ( G ) col+ ( G ) col( G )... col( G ) col( )] (7) Wth these arrangements the receved sgnal vector r n (4) can be re-wrtten as: r = Gs + w (8) SD attempts to fnd the soluton to the least square problem [8] ˆ s = arg mn r- Gs (9) S Usng the QR decomposton on matrx G gves P G = Q (0) 0 ( r ) where P R s an upper trangular matrx Q = [Q r r Q ] R s an orthonormal matrx and here r needs to be satsfed. Matrces Q and Q are the frst and last r - orthonormal columns of Q respectvely. Due to the nvarance of the Frobenus norm to orthogonal transforms (9) can be re-wrtten as [9] sˆ = arg mn y Ps () S where y = Q r. SD searches for ŝ only over those ponts that satsfy the constrant y P d () Denotng a= a( ) = y P as a functon of () can be wrtten as a d (3) = where a s the th element of a. Because of the upper trangular structure of P a s a functon of only where j denotes the j th element of. o begn the search we frst consder the nequalty ( ) a s d (4) where a( ) ndcates that a s a functon of. (4) s a necessary condton of (3) because a s non-negatve. Solvng ths nequalty gves the upper and lower bounds of the 763

4 possble values of. For each choce of nequalty a s s a s d ( ) + ( ) the followng (5) agan a necessary condton of (3) gves the upper and lower bounds of the possble values of. Gong on n ths way the possble values of all the elements of s can be found. Here we denote (4) the th level nequalty (5) the (-) th level and so on. Durng the searchng process when an nequalty cannot be satsfed the algorthm goes back to the prevous level selects another possble value of and then contnues. After a possble vector s s found the value of d s reduced and the search starts all over agan. he algorthm stops when (3) s volated for all possble values of and the last found s s taken as the result. IV. EW LDC AD DECODIG ALGORIHM From the descrpton n Secton 3 we can see that the process of SD can be consdered as a tree search [9]. In ths secton we llustrate how we can reduce the number of levels n the tree search by usng LDC wth orthogonal row structure hence to reduce the complexty of the SD [4]. A. Orthogonal row structure hs famly of LDCs has the frst m dsperson matrces among the dsperson matrces { M } = n () satsfyng the followng condton: <M (p:) M j (q:)> =0 (6) for j; j m; p t q t where M (k:) denotes the k th -row vector of M and <a b> denotes the nner product of vectors a and b. hs means that among the frst m dsperson matrces the rows of one dsperson matrx are orthogonal to the rows of any other dsperson matrx. It can be easly proved that LDCs wth the dsperson matrces satsfyng (6) wll satsfy the full-capacty constrant []. It can also be rgorously shown that for the LDCs satsfyng (6) the elements p j for = m- and j = + + m of the upper trangular matrx P n (0) are all zeros. For smplcty we omt the proof n ths paper. B. Smplfed SD ote that a = y P. hanks to the zeros n P a (for = m) s a functon of only and s s... s m rather than.... As a result (for = m) appears only n the th level. So to make a decson on (for = m) we do not need the knowledge of the (+) th level to the m th level. Furthermore snce (for = m) does not affect any other level to decde the value of we only need to choose the value of that mnmzes a rather than calculatng the upper and lower bounds of. hus the decodng for these symbols s smply hard decodng: k k)/ k= m+ s = ( y p s p for = m (7) where a means the possble value of closest to a. It should be noted that compared to the complexty of tree search the complexty of hard decodng can be neglected and our proposed code structure can reduce the tree search by m levels n SD wthout causng any degradaton n BER performance. V. SIMULAIO RESULS he advantages of the orthogonal structure for dfferent modulatons were studed n [5]. Here we nvestgate the LDCs for dfferent numbers of transmt and receve antennas. Studes of ths famly of LDCs have been carred out usng a 4 MIMO system transmttng eght QPSK symbols n a block length of 4 (.e. t = r = 4 = 4 = 8) and a 3 3 MIMO system transmttng ten QPSK symbols n a block length of 6 (.e. t = 3 r = 3 = 6 = 0) over a blockfadng channel. In the constructon of these LDCs wth orthogonal row structure we frst made the frst two dsperson matrces to satsfy the orthogonal condton n (6).e. m=. hen we used random search wth the Rank & Determnant crteron to obtan the optmal LDCs. For the LDCs wthout the orthogonal row structure we smply use random search wth the Rank & Determnant crteron to obtan the optmal LDC. o assess the BER performances of these optmum LDCs Monte Carlo smulaton was used and the results are shown n Fg.. It can be seen that the BER performance of the optmum LDCs wth and wthout the orthogonal row structure are about the same. Here the complexty of SD s measured by the number of nodes vsted n the tree search process []. In the 4 MIMO systems transmttng eght QPSK symbols n a block length of 4 the SD needs 6 levels n decodng. However the smplfed SD algorthm can reduce the decodng to a -level SD and 4 hard decsons reducng 5% of the search levels. he complexty reducton s not lnearly proportonal to the number of search levels because the search process depends on many random factors so t s dffcult to estmate the actual complexty reducton. Monte Carlo smulaton agan was used to evaluate the complextes of the conventonal SD and smplfed SD for decodng the optmum LDCs wth the orthogonal row structure. he results on the complextes for the 4 MIMO system transmttng eght QPSK symbols n a block length of 4 and the 3 3 system transmttng ten QPSK symbols n a block length of 6 are shown n Fg. and Fg. 3 respectvely. In the 4 system the smplfed SD reduces the complexty by 8-5%; whle n the 3 3 system the smplfed SD reduces the complexty by 8-49%. 764

5 VI. COCLUSIOS In ths paper we have presented the desgn of a famly of fast-decodable full dversty LDC wth an orthogonal row structure and nvestgated these LDCs for dfferent numbers of transmt and receve antennas. Monte Carlo computer smulatons have shown that the optmal LDCs wth and wthout the orthogonal row structure for dfferent numbers of transmt and receve antennas have nearly dentcal BER performances. However the complexty of SD for LDCs wth orthogonal row structure can be sgnfcantly reduced by usng a smplfed SD algorthm. For a 4 MIMO system transmttng eght QPSK symbols n a block length of 4 and a 3 3 MIMO system transmttng ten QPSK symbols n a block length of 6 the respectve reductons are about 8-5% and 8-49%. Fgure. BERs of optmal LDCs n 4 and 3 3 MIMO systems. Fgure. Complextes of conventonal and smplfed SDs n 4 MIMO system. Fgure 3. Complextes of conventonal and smplfed SDs n 3 3 MIMO system. REFERECES [] R. Heath and A. Paulraj Lnear Dsperson Codes for MIMO Systems Based on Frame heory IEEE rans. Sg. Proc. vol. 50 no. 0 pp October 00. [] V. arokh. Seshadr and A. R. Calderbank Space-me Codes for Hgh Data Rate Wreless Communcaton: Performance Crtera n the Presence of Channel Estmaton Errors Moblty and Multple Paths IEEE rans. Commun. vol. 47 no. pp February 999. [3] G. D. Golden G. J. Foschn R. A. Valenzuela and P. W. Wolnansky Detecton algorthm and ntal laboratory results usng the V-BLAS space-tme communcaton archtecture Electroncs Letters vol. 35 no. pp. 4 5 January 999. [4] S. M. Alamout A Smple ransmt Dversty echnque for Wreless Communcatons IEEE Journal Select. Areas Commun. vol. 6 no. 8 pp October 998. [5] V. arokh H. Jafarkhan and A.R. Calderbank Space-tme block codes from orthogonal desgns IEEE rans. Inf. heory vol. 45 no. 5 pp July 999. [6] O. rkkonen and A. Hottnen Square-matrx embeddable space tme block codes for complex sgnal constellatons IEEE rans. Inf. heory vol. 48 no. pp February 00. [7] H. Jafarkhan A quas-orthogonal space tme block code IEEE Commun. Letters vol. 49 no. pp. 4 January 00. [8] U. Fncke and M. Pohst Improved methods for calculatng vectors of short length n lattce ncludng a complexty analyss Math. Comput. vol. 44 no. 70 pp Aprl 985. [9] J. Jalden and B. Ottersten On the complexty of sphere decodng n dgtal communcatons IEEE rans. Sg. Proc. vol. 53 no. 4 pp Aprl 005. [0] E. Agrell. Erksson A. Vardy and K. Zeger Closest pont search n lattces IEEE rans. Inf. heory vol. 48 no. 8 pp. 0 4 August 00. [] C. P. Schnorr and M. Euchner Lattce bass reducton: Improved practcal algorthms and solvng subset sum problems Math. Programmng vol. 66 pp [] J. Paredes A.B. Gershman and M. G. Alkhanar A space-tme code wth non-vanshng determnants and fast maxmum lkelhood decodng Proc. ICASSP. vol. pp Aprl 007. [3] E. Bgler Y. Hong and E. Vterbo "On fast decodable space-tme block codes" IEEE rans. Inf. heory Feb 009. [4] X.G. Da S. W. Cheung and. I. Yuk A Fast-Decodable Code Structure for Lnear Dsperson Codes IEEE rans. on Wreless Commun. Aug 009. [5] X.G. Da S. W. Cheung and. I. Yuk. "A ew Famly of Lnear Dsperson Code for Fast Sphere Decodng" Canadan Conference on Electrcal and Computer Engneerng 09 pp May