A new family of linear dispersion code for fast sphere decoding. Creative Commons: Attribution 3.0 Hong Kong License

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1 tle A new famly of lnear dsperson code for fast sphere decodng Author(s) Da, XG; Cheung, SW; Yuk, I Ctaton he nd IEEE Canadan Conference on Electrcal and Computer Engneerng (CCECE 009), St. John's, L., 3-6 May 009. In Proceedngs of the nd CCECE, 009, p Issued Date 009 URL Rghts Creatve Commons: Attrbuton 3.0 Hong Kong Lcense

2 A EW FAMILY OF LIEAR DISPERSIO CODE FOR FAS SPHERE DECODIG X. G. Da, S. W. Cheung and. I. Yuk Department of Electrcal & Electronc Engneerng he Unversty of Hong Kong Hong Kong ABSRAC In ths paper, a new famly of Lnear Dsperson Codes (LDCs) that can be decoded usng a fast Sphere Decodng (SD) algorthm n MIMO systems s proposed. he basc prncple of ths structure s to make the LDC to have as many as possble the rows orthogonal n the dsperson matrces. Monte Carlo smulaton results show that the optmum LDCs wth ths orthogonal structure have nearly dentcal bt-error-rate (BER) performances as other optmal LDCs. We develop a smplfed Sphere Decodng (SD) algorthm that can sgnfcantly reduce the decodng complexty n decodng the new LDCs wth proposed orthogonal structure. Smulaton results show that the complexty reducton s more sgnfcant for MIMO system transmttng hgher level modulaton. For 4 MIMO systems transmttng 4 64QAM and 56QAM symbols n a block length of 4, the reductons are about 7-83% and 76-88%, respectvely. Index erms MIMO, Sphere decodng, orthogonal, complexty. IRODUCIO Lnear Dsperson Code (LDC) [] s well-known for ts advantages n provdng full-ergodc capacty to Multple- Input Multple-Output (MIMO) communcaton systems. However, the detecton complexty of LDCs has always been a problem n the desgn and mplementaton of hgh speed MIMO systems. he tremendous complexty of the Maxmum Lkelhood (ML) decodng process makes the mplementaton of hgh speed Space-tme block codes (SBCs) mpractcal. Lnear decodng algorthms lke Mnmum-Mean-Square Error (MMSE) [] and Zero- Forcng (ZF) [3] algorthms are much less complcated, but the achevable BER performances are not very satsfactory. Alamout [4] proposed a remarkable space-tme (S) code n 998 for MIMO systems wth two transmt antennas whch was later on extended to orthogonal space-tme block codes (OSBCs) [5]. he OSBCs and Alamout s code attracted much attenton because they allow the uses of very smple decodng algorthms to acheve the same BER performances as wth ML decodng. However, the man dsadvantage of the OSBCs s that they cannot acheve the full-transmsson rates for MIMO systems wth more than two transmt antennas [6]. In [7], Sphere Decodng (SD) was proposed to substantally reduce the complexty of ML decodng, yet havng the same BER performance. However, as Jalden et al. ponted out n [8], for a fxed sgnal-to-nose rato (SR), the complexty of SD ncreases exponentally wth the number of symbols jontly decoded. When data rate s hgh, SD s stll too complcated for practcal use. Improvements n dfferent aspects of SD have been proposed. In [9], a so-called Baba Pont method was used to set the ntal searched pont n SD. In [0], 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 search 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 []. In ths paper, we propose a new famly of fast-decodable full-dversty LDCs for MIMO systems and the LDCs can be desgned for arbtrary number of transmt and receve antennas. o reduce the decodng complexty, we also develop a smplfed SD for the codes to acheve the same BER performance as those usng 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. In Secton 3, our proposed new famly of LDCs together wth the correspondng smplfed SD algorthm s explaned. Monte Carlo smulaton results and dscussons are gven n Secton 4. Secton 5 concludes ths paper.. 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: /09/$ IEEE 34

3 R = HC+ W () 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 r codeword 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 vec( ) r,, r,,..., r,, r,,..., r,,..., r r r r, r = R = wth r,j beng the entry n the th row and j th column of matrx R and [ ] denotng matrx transposton, X = [ vec( M ), vec( M ),..., vec( M )], s = vec() s = s 0 = [ s,... s ] and w = vec( W ). K = I H, where r t K C, I s an dentty matrx and denotes the Kronecker product. Separatng the real and magnary parts of the elements n r, s and w of (3) and then vectorzng them gve the realvalued expresson: r = Gs + w where r = Re( ),Im( ) r r = [Re( s ),...,Re( s ),Im( s ),...,Im( s )] (4), s = [ s s ] Re( ),Im( ), and w = [Re( w ), Im( w )], wth Re(.) and Im(.) denotng the real and magnary parts, respectvely, of (.). In (4), G s a r real matrx gven by: Re( KX) Im( KX) G = (5) Im( KX) Re( KX) Here, we adopt an orderng scheme to construct our proposed code structure. Frst, we arrange the vector s n (4) to s = [Re( s), Im( s), Re( s), Im( s),..., Re( s ), Im( )] s (6) = [ s, s, s, s..., s ] 3 4 o keep the receved sgnal vector r n (4) unchanged, we need to arrange the columns of G correspondngly. hat s, for G = [ g,,, ], where, for =,,,, g g g denotes the th column of G, we arrange the columns of G to gve: G = [ g, g, g, g,..., g, g ] (7) + + Wth these arrangements, the receved sgnal vector r n (4) can be re-wrtten as: r = Gs + w (8) 3. PROPOSED LDC AD DECODIG ALGORIHM Proposed orthogonal row structure We propose a new famly of LDCs whch have the frst m dsperson matrces among the dsperson matrces { M } = n () satsfyng the followng condton: H MM j = 0 ( j;, j m) (9) where 0 s an t t matrx wth all elements beng zero and (.) H denotes the transpose conjugate of a matrx (.). he condton of (9) mples that among the frst m dsperson matrces, any row of one dsperson matrx s orthogonal to the rows of any other dsperson matrces. It can be easly proved that wth proper scalng, LDCs wth the dsperson matrces satsfyng (9) wll satsfy the full-capacty constrant and power constrant n [], whch are the basc requrements for full-ergodc capacty LDCs. Smplfed SD Sphere decodng attempts to obtan the soluton of [7]: ˆ s= arg mn r- Gs (0) s S 35

4 o do ths, we can frst conduct the QR decomposton of G : P G = Q () 0 ( r ) where P R s an upper trangular matrx, Q r r R s an orthonormal matrx, and r. hen (0) can be wrtten as: sˆ = arg mn y Ps s S () where y = Q r, Q s the frst orthonormal columns of Q. Based on (), the SD can now search only the ponts wthn a hypersphere centered at the receved sgnal wth radus d by solvng the followng nequalty teratvely [7]: y Ps d (3) o explan how the orthogonal row structure can be used to smplfy the SD process, we frst ntroduce the followng Lemma: Lemma : If a LDC has an orthogonal row structure and so satsfes (9), the elements p j, for =,,, m- and j = +, +,, m, n the upper trangular matrx P n () are all zeros. he proof s omtted here due to the page lmt. Accordng to Lemma, f a LDC wth an orthogonal row structure s used, there wll be a cluster of zeros n the upper trangular matrx P n (). In (3), there are a total of nequaltes, correspondng to the rows n the matrx equaton. o solve the nequalty teratvely, the SD starts from the bottom row of the matrx and works upwards from the th row. When the SD has completed the (m+) th row, the values of s, s,, s m+ wll have been determned. hen beneftng from the cluster of zeros n P, we do not need to use the values of s +, s +,, s m to determne the values of { s } for = to m. Instead, we 4. SIMULAIO RESULS Studes of ths new famly of LDCs have been carred out usng two 4 MIMO systems transmttng four 64QAM and 56QAM symbols n a block length of 4 (.e. t =, r = 4, = 4, = 4) over a block-fadng channel. In the constructon of these LDCs wth orthogonal row structure, we made the frst two dsperson matrces to satsfy the orthogonal condton of (9),.e. m=. hen we used random search wth the Rank & Determnant crteron to obtan the optmal LDC. For the LDCs wthout the orthogonal row structure, we also used random search wth the Rank & Determnant crteron to obtan the optmal LDC. o assess the BER performances of these optmum LDCs n the 4 MIMO system, Monte Carlo smulaton was used and the results are shown n Fgs. and for the 64QAM and 56QAM sgnals, respectvely. It can be seen that the BER performances of the optmum LDCs wth and wthout the orthogonal row structure are about the same. Accordng to [8], the complexty of SD can be measured by the number of nodes vsted n the tree search process. Monte Carlo smulaton was therefore used to examne the average numbers of vsted nodes to evaluate the complextes of the conventonal SD and smplfed SD for decodng the same optmum LDC wth the orthogonal row structure. he results on the complextes for the 4 MIMO systems transmttng four 64QAM and 56QAM symbols n a block length of 4 are shown n Fgs. 3 and 4, respectvely. In the 64QAM system, the smplfed SD reduces the complexty by 7-83%; whle n the 56QAM system, t reduces the complexty by 76-88%. Reducton s more sgnfcant for sgnals wth hgher-level modulaton. he reason s that for hgher-level modulaton, there are more branches n each node and so even more nodes n the lower levels. So for the same number of search levels reducton, more branches reducton and hence complexty reducton wll be acheved for hgher-level modulaton. can smply determne the value of s by hard decodng: s = ( y (4) p, ks k)/ p, k= m+ where a denotes the possble value of s 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. Fg. BER of optmal LDC usng 64QAM modulaton 36

5 nearly dentcal BER performances. However, the complexty of SD for LDCs wth our orthogonal row structure can be sgnfcantly reduced by usng a smplfed SD algorthm. he reducton s more sgnfcant for LDCs usng hgher level modulatons. 6. REFERECES [] Heath, R., and A. Paulraj, Lnear Dsperson Codes for MIMO Systems Based on Frame heory, IEEE rans. Sg. Proc., vol. 50, no. 0, pp , October, 00. Fg. BER of optmal LDC usng 56QAM modulaton [] arokh, V., et al., 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] Alamout, S. M., 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. on Informaton heory, vol. 45, no. 5, pp , July 999. Fg. 3 Complexty of conventonal SD and smplfed SD for 64QAM [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] 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. [8] Jalden J. and Ottersten B., On the complexty of sphere decodng n dgtal communcatons, IEEE rans. Sg., vol. 53, no. 4, pp , Aprl, 005. [9] 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. Fg. 4 Complexty of conventonal SD and smplfed SD for 56QAM 5. COCLUSIOS In ths paper the desgn of a new famly of fast-decodable full dversty LDC wth an orthogonal row structure n the dsperson matrces has been presented. Monte Carlo computer smulaton results have shown that the optmal LDCs wth and wthout our orthogonal row structure have [0] C. P. Schnorr and M. Euchner, Lattce bass reducton: Improved practcal algorthms and solvng subset sum problems, Math. Programmng, vol. 66, pp. 8 9, 994. [] 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. [] Ezo Bgler, Y Hong, Emanuele Vterbo, A Fast- Decodable, Quas-Orthogonal Space-me Block Code for 4x MIMO, Annual Allerton Conference on Communcaton, Control and Computng, Illnos, Sep. 6-8,