Mapping QR Decomposition Algorithm for MIMO Detection

Transcription

1 0 7th Internatonal ICST Conference on Communcatons and etworng n Chna (CIACO) appng QR Decomposton Algorthm for IO Detecton Xe, Xnng * ; Wang, Xngln # ; Tan, nqang #, Zhang, Feng # ; e, Zhqang * * Key Laboratory of Unversal Wreless Communcaton Bejng Unversty of Posts and Telecommuncatons Bejng, Chna * Xnng06806@yahoo.com.cn # Research and Technology Department oa Semens etwors Tech. (Bejng) Co. Ltd. Bejng, Chna # Xngln.wang@nsn.com Abstract In ths paper, a mappng QR decomposton (QRD) algorthm for IO detecton s proposed. Ths algorthm sgnfcantly reduces the overall complexty by elmnatng the square root and dvson operatons n tradtonal QR decomposton wthout any performance degradaton. Its combnaton wth lst sphere decodng (LSD) s re-derved to provde accurate soft nformaton. In addton, our dervaton shows that lnear nterpolaton-based QR decomposton s not feasble n practce as well. Keywords- multple-nput multple-output (IO); mappng QR decomposton(qrd); lst sphere decodng (LSD) I. ITRODUCTIO The multple-nput multple-output (IO) technque [] has been developed to ncrease data transmsson rates and ln qualty by utlzng multple antennas. The adopton of orthogonal frequency dvson multplexng (OFD) wth IO technology (IO-OFD) provdes hgh spectral effcency for the wdeband communcaton systems such as IEEE 80.n, IEEE 80.6m, and 3GPP-LTE. IO detector, that detects the transmtted data of each transmt antenna, plays an mportant role n the IO system. Several IO detecton algorthms []-[4] have been proposed to mprove the detecton performance and complexty, whch need the QR decomposton preprocessng to avod the complcated pseudo-nverse computaton of channel matrx [5]. owever, the tradtonal QR decomposton contans the dvson and square root operatons, leadng to a great deal of complexty n the actual hardware mplementaton. To address the above ssues, ths paper presents a mappng QR decomposton (QRD) algorthm n IO detecton system by mprovng the tradtonal Gram-Schmdt QR decomposton algorthm. The proposed algorthm greatly reduces the mplementaton complexty by elmnatng the square root and dvson operatons. Furthermore, the method to combne the QRD results wth lst sphere decodng (LSD) scheme s derved. Based on the change of the nose statstcal propertes, we develop a weghtng operaton n order to provde accurate soft nformaton for channel decoder. Theoretcal analyss and smulaton results ndcate that the proposed algorthm could dramatcally reduce the complexty wthout deteroratng the system bt error rate performance. The remander of ths paper s organzed as follows. Secton II descrbes the sgnal model and the tradtonal Gram-Schmdt QR decomposton [5]. In secton III, we ntroduce the mappng operator, and deduce the IO detecton algorthm based on mappng QR decomposton. Then, complexty analyss and numercal smulatons are shown n secton IV. Fnally, a bref concluson s gven n secton VI. II. SIGAL ODEL AD TRADITIOAL QR DECOPOSITIO A. Sgnal odel For the IO system wth transmttng antennas and recevng antennas, where. The transmttng antennas transmt the data streams x [ xx x ] T. Assume the IO fadng channel s, and the Addtve Whte Gaussan ose (AWG) n [ nn n ] T. ence, the recevng antennas receve the data streams y [ y y y ] T, and the IO sgnal model can be expressed as y=x+n () In the recever, the IO detector uses the receved sgnal y and the estmated channel response to detect the transmtted data. The QR decomposton s the basc preprocessng for most IO detecton algorthms. The channel matrx s decomposed nto Q and R matrces (=QR), where Q s an untary matrx and R s an upper trangular matrx. The upper trangular property of R matrx maes the subsequent detecton much smpler. Ths wor s supported by atonal Basc Research Program of Chna (973 Program 009CB3040), atonal Key Scentfc, Technologcal Project of Chna (0ZX ) and atonal atural Scence Foundaton of Chna (67099) //$ IEEE

2 B. Tradtonal QR decomposton Gram-Schmdt orthogonal factorzaton [5] s a tradtonal QR decomposton, and t s an teratve process whch can be expressed as: j j u = h q h q, u = h u q r q T T = = u () u u u = u uj 0 u u j (7) where =,,. The h and q are the th column vectors of and Q, respectvely, and r T s the th row vector of R. (.) and (.) T represents the ermtan and transpose of a matrx respectvely. III. IO DETECTIO BASED O APPIG QR DECOPOSITIO A. appng QR Decomposton A mappng functon :( QR, ) = ( QR, ) s ntroduced [6], whch s formulated by where s a dagnose matrx, and : Q = Q R=R (3) = r = r,,,,, j, j,, = r r = r r When s a square matrx = [ h h h ], and h, =, s the th column vector of. Then, accordng to (), the Q and R matrces become Q= [ q q q q] u u u u = [ ] u u u u T T T T T R = [ r r r r ] u q h q h q h u q h q h = u q h+ q h 0 u from (4) and (6), the matrx can be expressed by (4) (5) (6) Q = [ qq qq ] Then [ u u u u u u u ] = j j u q h q h3 q h 0 u u qh3 qh R uj q h+ q h 0 uj Besdes, the recurrence formula of accordng to () = (8) (9) u can be derved u = h h ( ) ( ) u j h h u j (0) u j and u h h From (8), (9) and (0), we deduce that all entres n Q and R matrces can be expressed as a seres of multplyadd of h and h, wthout square root and dvson operatons. In low-dmensonal case, when = =, that s, when s square matrx, we can rewrte the Q and R as Q = h ( h h) h ( h h) h () h h h h R = () 0 ( h h)( h h) ( h h)( h h) Furthermore, n hgh-dmensonal case, we ntroduce a matrx A and a functon F( d ) 36

3 A= h h h h h h hh hh hh = h h hh hh hh+ h h h h hh (3) then Q y = Rx+Q n () Then we can utlze the () for the tradtonal LSD detecton. Comparng the characterstc of Q, R and Q, R, we can fnd that R and R are both upper trangular matrces, but there s a lttle dfferent between the feature of Q and Q. Q Q=I,where I s an dentty matrx. owever, t F( d) = ( ) F( d \{ d}) atd (4) = and F({ φ}) = t where the set d{ d, d, dt}, d {,, }, =,, t ; a td represents the element n tth row and d th column of the matrx A; d \{ d } means d element s excluded n the set d. Then, for a channel matrx, we can derve the recurrence formulas for the entres of Q and R as q = F({,, } \ { }) h F({,, } \ { }) h + (5) + ( ) F({,, }\{}) h r, = F({,, }) (6) r = F({,, } \ {,, j }) (7) j () Q Q =(Q) (Q)=Q Q = Accordng to (4), the matrx s a dagonal matrx, whose dagonal elements are, = r,, = r, r, (3) The LSD detecton based on tradtonal QR decomposton manly uses the feature that Q Q s an dentty matrx and the upper trangular nature of R to do the follow-up IO detecton. In fact, f we relax the characterstc of Q Q, as long as Q Q s a dagonal matrx, we can also present the follow-up IO detecton. The process of dervaton as follows: Accordng to (3), (0) can be rewrtten as where, j, =,,, and < j Therefore, () and () can be rewrtten as [ ] [ F({ φ}) h F({}) h F({}) h ] Q = q q = (8) - - y=q (R)x + n = Q Rx + n ultplyng both sdes of the equaton wth get Q, we - Q y =(Q Q) Rx + Q n=rx+q n (4) F({}) F({}) R = 0 F({,}) (9) Smlar to the two-dmensonal case above, we can deduce the expressons of Q and R from (4)-(7) n the hgh-dmensonal case. The followng s the further nstructons about the tradtonal QR decomposton and mappng QR decomposton applcaton n LSD scheme [8], [9]. B. LSD Detecton For the tradtonal QR decomposton y=x+n=qrx+n (0) From (7), s a real dagonal matrx, then =, So (4) can be rewrtten as Q =(Q ) =(Q) =(Q) (Q) Let y =(Q) y, expressed as y = Rx + (Q) n (5) n=(q) n, then (5) can be y = Rx+n (6) 37

4 The above expresson expands to then E( nn ) = σ I (33) y r r r, x n y 0 r r, x n y3 = 0 0 r 3, x3 + n3 y 0 0 r, x n where (7) Because of the dagonal elements of E ( nn ) are not equal, we develop a weghtng operaton for (30) n order to provde accurate soft nformaton for the decoder. Then (30) s amended as ped = y r jx (34) = j + y r x n (8) Accordng to upper trangular characterstcs of the coeffcent matrx, we can wor out the transmtted sgnal components successvely from bottom to top usng the teratve method x = arg mn y r x (9) j j ped = y r x (30) Utlzng the tree search method [8][9], we choose a suffcent number of canddate vectors to do LLR calculaton: C. Interpolaton Feasblty Orgnally, mappng QR decomposton s used drectly for the nterpolaton of Q and R. Ths secton wll derve the feasblty of the nterpolaton algorthm. In [6], a mappng functon : Q = Q R=R was ntroduced to obtan the mapped LP matrces ( QR, ) = ( QR, ) ; consequently, the nterpolaton of Q and R matrces becomes applcable. In the 3GPP-LTE system, every sx subcarrers contan a plot. As shown n Fg., Q, R and Q 7, R 7 are derved from the channel responses of the plots and 7 utlzng mappng QR decomposton. Then, the Q and R of the data subcarrers are computed by nterpolatng Q, Q 7 and R, R 7. L( b / y ) max ped max ped (3) b B, + σ B = b, σ = where, x= map( b ), that s, b s the bt sequence correspondng to the symbol x, Gaussan Whte ose n, and σ s the varance of { / }, { / } B = b b =+ B = b b = (3), +, Thus, mappng QR decomposton can also be used n the detecton. owever, n (6), we fnd that the Covarance matrx of n s E( nn ) = E[( Q n)( Q n) ] = E( nn ) E( Q Q ) where E( nn ) = σ I, σ s the varance of Gaussan Whte ose n, and I represents the dentty matrx. Accordng to () and (3),, E( Q Q) = E( ) = Fgure. Drect lnear nterpolaton scheme. Reference [7] clamed that the, lnear nterpolated from and 7, corresponds to Q, R. owever, from () and (5), we fnd - - (Q) y = Rx + (Q) n = Rx + (Q) n - =Q Q( )Rx+(Q) n In order to ensure the low- complexty of LSD detecton, we should ensure that Q Q( ) R s an upper trangular matrx. We new that ( ) s a dagonal matrx, and R s an upper trangular matrx, so we need to mae sure that 38

5 Q Q s a dagonal matrx. Eq.() shows that Q meets ths requrement tself. owever, Q obtaned by lnear nterpolaton does not meet the requrement. The dervaton s as follows: We assume = [ m m ] and = [ ] 3 n n,where m and n are the th column of and 3,respectvely. After applyng the mappng QR decomposton for and 3, we obtan Q = [ a a] and Q 3 = b b, where a and b are the th column of Q and Q 3. Assumng the nterpolaton nterval s one, then we get Q = c c by applyng lnear nterpolaton between [ ] Q and Q 3. Then c = ( a + b ) c = ( a + b ) Q Q c c c c = c c c c That whether Q Q s a dagonal matrx depends on whether c c s a zero matrx. c c = ( a + b ) ( a + b ) = ( a b + b a) 4 Accordng to (4) and (5) Then = = ( ) ( ) a m a m m m m m m b n b n n n n n n = = ( ) ( ) [( )( ) ( )( c c = n n m n n n m n ) 4 + ( m m )( n m ) ( m m )( n m )] The above formula s not dentcally zero, so Q Q s not dentcally a dagonal matrx. Thus, Q obtaned by lnear nterpolaton of Q does not meet the requrement, whch wll greatly ncrease the complexty of LSD and worsen bt error rate performance. Thus, we conclude that t s not feasble to apply lnear nterpolaton on Q as shown n [6]. Even f there s an nterpolaton method wth excellent performance, t wll also be non-lnear nterpolaton, whch wll greatly ncrease the mplementaton complexty and delay. Thus, we propose a new nterpolaton scheme as llustrated n Fg.. We perform lnear nterpolaton for the channel responses of the plots, and then acqure Q and R of the data subcarrers by applyng the mappng QR decomposton for the nterpolaton results. IV Fgure. Proposed lnear nterpolaton scheme. COPLEXITY AALYSIS AD UERICAL SIULATIOS A. Complexty Analyss The computaton complextes of the tradtonal QR decomposton (QRD) and the mappng QR decomposton (QRD) for 4 4 IO are compared n Table I. TABLE I. COPLEXITY COPARISO OF QRD AD QRD QRD QRD Addton of real numbers ultplcaton of real numbers Dvson of real numbers 4 0 Square root of real numbers 4 0 Table I lsts the numbers of arthmetc operatons that are requred for QRD and QRD n the 4 4 IO system. It s shown that the square root and dvson operatons are not needed for QRD any more. The whole process of the mappng QR decomposton only exsts multplcaton and addton because the entres of Q and R can be expressed as a seres of multply-add of h and h. Compared wth tradtonal QR decomposton, the proposed algorthm could sgnfcantly reduce the complexty of the actual system mplementaton by elmnatng the square root and dvson operatons. B. umercal Smulatons To verfy the proposed IO detecton algorthm based on mappng QR decomposton, we carred out the followng smulaton based on LTE [0][][]. We assume that the channel between dfferent transmt and receve antennas s ndependent flat Raylegh fadng channel, and that the system has strct tmng synchronzaton and an deal channel estmaton. We evaluate the detecton performance by utlzng the turbo codng scheme. Specfc smulaton parameters are shown n Table II. 39

6 TABLE II. SIULATIO PARAETERS System parameters umercal value system bandwdth 0hz channel model flat Raylegh fadng channel modulaton 6QA, 64QA codng rate /3 mult-antenna transcever mode 4 4 IO detecton algorthm based on QRD, QRD In Fg. 3, t s shown that n the 4 4 IO system wthout channel codng, the bt error rate (BER) performance of the IO detecton algorthm based on mappng QR decomposton s same as that of the algorthm based on the tradtonal QR decomposton n both 6QA and 64QA cases. based on mappng QR decomposton yelds the same BER performance as the algorthm based on the tradtonal QR decomposton for the 4 4 IO case and 6QA or 64QA case and wth or wthout turbo channel codng case. V. COCLUSIO Ths paper proposed a mappng QR decomposton based IO detecton algorthm. We ntroduced a mappng and derved the general equatons functon ( QR, ) = ( QR, ) of Q and R for hgh IO dmensons. The avodance of square root and dvson operatons sgnfcantly reduced the complexty of the actual system mplementaton. Besdes, the method to combne the QRD results wth lst sphere decodng (LSD) scheme s re-derved. eanwhle, our dervaton shows that the lnear nterpolaton-based QR decomposton s not feasble n practce as well. Theoretcal analyss and smulaton results ndcate that the proposed algorthm could sgnfcantly reduce the complexty of the actual system mplementaton wthout ncreasng the system bt error rate, whch s a great beneft to the low-complexty IO-OFD system. Fgure 3. Performance comparson of IO detecton algorthm n hghorder modulaton. Fgure 4. Performance comparson of IO detecton algorthm n hghorder modulaton and turbo decodng. As llustrated n Fg. 4, when we combned channel codng wth the IO system, the system bt error rate (BER) rapdly reduce as the sgnal to ose Rato (SR) ncreases. In 6 QA and 64 QA cases, the BER curves of proposed IO detecton algorthm also overlap completely wth that of the algorthm based on the tradtonal QR decomposton. Thus, the IO detecton algorthm REFERECES [] G. J. Foschn, Layered space-tme archtecture for wreless communcaton n a fadng envronment when usng multple antennas, Bell Lab. Tech. J., vol., no., pp. 4 59, 996. [] R. Böhne, D. Wübben, V. Kühn, and K. D. Kammeyer, Reduced complexty SE detecton for BLAST archtectures, Proc. Globecom 03, vol. 4, pp. 58 6, Dec [3] K. W. Wong, C. Y. Tsu, R. S. K. Cheng, and W.. ow, A VLSI archtecture of a K-best lattce decodng algorthm for IO channels, n Proc. IEEE ISCAS 0, vol. 3, ay 00, pp [4] Y. C. Lang, D. K. Z. Kt, S. Attallah, W. S. Leon, and C. L. Xu, Dynamc QRD recevers for IO beamformng systems wth mperfect channel state nformaton, n Proc. IEEE ICICS 05, Dec. 005, pp [5] R. A. orn and C. R. Johnson, atrx Analyss. ew Yor: Cambrdge Press, 985. [6] D. Cescato,. Borgmann,. Bölcse, J. ansen, and A. Burg, Interpolaton-based QR decomposton n IO-OFD systems, n Proc. IEEE 6th Worshop on Sgnal Processng Advances n Wreless Communcatons, Jun. 005, pp [7] P. L. Chu, L. Z. uang, L. W. Cha, and Y.. uang, Interpolaton-Based QR Decomposton and Channel Estmaton Processor for IO-OFD System.pdf, IEEE TRASACTIOS O CIRCUITS AD SYSTES I: REGULAR PAPERS,0,pp.9-4 [8] Bertrand. ochwald, Stephan ten Brn, Achevng ear- Capacty on a ultple-antenna Channel, IEEE TRASACTIOS O COUICATIOS, VOL. 5, O. 3, ARC 003 [9] W.. Chn, QRD Based Tree Search Data Detecton for IO Communcaton Systems, Vehcular Technology Conference, 005. VTC 005-Sprng. 005 IEEE 6 st, pp [0] Evolved Unversal Terrestral Rado Access (E-UTRA); ultplexng and channel codng, 3GPP Techncal Specfcaton 36., Oct. 00 [] Evolved Unversal Terrestral Rado Access (E-UTRA); Physcal layer procedures, 3GPP Techncal Specfcaton 36.3, Oct. 00 [] Evolved Unversal Terrestral Rado Access (E-UTRA); Physcal channels and modulaton, 3GPP Techncal Specfcaton 36.. Oct