1 MIMO Precodng Usng Rotatng Codebooks C Jang, M Wang, C Yang Abstract Next generaton wreless communcatons rely on multple nput multple output (MIMO) technques to acheve hgh data rates. eedback of channel nformaton can be used n MIMO precodng to fully actvate the strongest channel modes and mprove MIMO performance. Unfortunately, the bandwdth of the control channel va whch the feedback s conveyed s severely lmted. An mportant ssue s how to mprove the MIMO precodng performance wth mnmal feedback. In ths letter, we present a method that uses a rotatng codebook technque to effectvely mprove the precodng performance wthout the need of ncreasng feedback overhead. he basc dea of the rotatng codebook precodng s to expend the effectve precodng codebook sze va rotatng multple codebooks so that the number of feedback bts remans unchanged. Smulaton results are presented to show the performance gan of the proposed rotatng codebook precodng over the conventonal precodng. Index erms Multple-nput multple-output (MIMO), untary precodng, quantzed precodng. I. INRODUCION MUIPE nput multple output (MIMO) systems have become the most promsng canddates for the next generaton of hgh data rate wreless communcatons [1]. MIMO technology has been shown to provde sgnfcant system performance mprovement over conventonal systems by provdng communcaton lnks wth substantal dversty and capacty. hs s especally true when channel state nformaton (CSI) s avalable at the transmtter [2]. In fact, the closed loop capacty of a MIMO channel can be acheved by convertng the channel nto a set of parallel spatal layers va precodng and water-fllng power allocaton at the transmtter and a lnear MMSE flterng at the recever wheren the optmal precodng and MMSE flter are determned by the sngular value decomposton (SVD) of the MIMO channel matrx [3][4]. or tme dvson duplexng (DD) communcaton systems, where uplnk and downlnk channels share the same frequency band at dfferent tme, the down lnk channel nformaton can be estmated by the transmtter usng the uplnk plot (analogy feedback) due to the channel recprocty property. or frequency dvson duplexng (DD) systems, where dfferent frequency bands are allocated for the downlnk and the uplnk channels, the channels are not recprocal. herefore, a CSI feedback channel, va the uplnk control channel (dgtal feedback), s necessary to delver the estmated channel knowledge at the recever back to the transmtter. or a tme varyng channel, the channel knowledge at the transmtter should be updated regularly. he overhead, whch ncreases lnearly as the product of the number of antennas, the channel frequency selectvty and the feedback frequency, can be large. Hence, the MIMO technque utlzng full channel nformaton s mpractcal for the system wth lmted feedback channel capacty.
2 mted feedback MIMO sgnalng has been extensvely researched n the past []-[12]. In partcular, untary precodng [8]-[10] has shown to be an effectve form of lmted feedback MIMO that quantzes the precoder rather than the channel nformaton. he essental dea s that the transmt precoder s chosen from a fnte set of precodng matrces, called the codebook, known a pror to both the transmtter and recever (the transcevers). he recever chooses the optmal precoder (precodng matrx) from the codebook as a functon of the current channel state nformaton avalable at the recever and sends the ndex of ths matrx to the transmtter over a lmted bandwdth control channel. hs closed loop precodng technque provdes more flexblty than the drect channel quantzaton scheme n the feedback lmted channel and has been quckly adopted by the next generaton wreless standards to mprove MIMO performance n low moblty envronments [13][14][15]. Although the untary precodng technque has made the closed loop MIMO precodng more practcal, the feedback overhead can stll be sgnfcant dependng on the number of antennas, the number of sub-bands and the feedback frequency. herefore, the number of feedback bts for a practcal system s typcally lmted to 4 bts per sub-band [13][14] whch s far from suffcent to fully explot the MIMO precodng gan especally for systems wth large antenna arrays (e.g., eght transmsson antennas [18]) and systems that explot mult-user MIMO gans. In ths letter, a proposed technque, referred to as the rotatng codebook precodng, s amed at further explotng the MIMO precodng gan wthout ncurrng ncreased feedback. rst off, s a bref revew of the untary precodng n Secton II. Secondly, the rotatng codebook precodng technque and ts performance smulaton results are presented n Secton III, followed by a concluson n Secton IV. II. UNIARY PRECODING rst, let s consder a lnear precodng MIMO system where,,, modulaton symbols wth M matrx, whch defnes a mappng between the symbol vector s, producng a vector E s1 s2 s M s contans a vector of M ss I. he symbol vector s s then multpled by an M M ( M M precodng M ) transmt antennas and the modulaton x s of length M. Assumng a narrowband MIMO channel (e.g., a subcarrer of an ODM system) wth perfect synchronzaton, the baseband, dscrete-tme equvalent receved sgnal can be wrtten as y Hx w (1) where H s an M R M channel matrx wth ndependent entres dentcally dstrbuted accordng to CN (0,1), M R M s the number of receve antennas and w s a length M R nose vector. Agan, we assume that the entres of w are ndependent and dstrbuted accordng to CN (0, N0).
In closed-loop precodng, the recever chooses an M M precodng matrx. he transmtter precodes the symbol vector s wth, conveyed from the recever, before transmsson. he equvalent channel s H. hus, the capacty gven MIMO channel matrx H and the precodng matrx s C( ) logdet I H H logdet I H H (2) where s the rato of the transmt sgnal power to the recever nose power [1]. he best precoder that maxmzes the capacty s subject to the untary constrant, M ˆ arg max C( ) (3) * I. Based on the fact that functon extrema are obtaned when the matrx varables have the same egenvectors [17], the optmal beam drectons are gven by the egenvectors of 2 * HH UΛV UΛV VΛ V (4) 3 where H s decomposed va sngular value decomposton H UΛV. U and V are untary matrces and Λ s real dagonal (sngular values). he untary precodng matrx that maxmzes C s gven by ˆ V. (5) hat s, the optmal untary precodng matrx s the channel s egen matrx. hs structure s essentally a beamformer wth multple beams (equal powers) that maps or beamforms data symbols across the spatal channels n order to maxmze capacty or mnmze error rate. eedng back the actual ˆ from a recever to the transmtter s costly, and can be practcally prohbtve even f ˆ s lmted to a untary matrx. In the lmted feedback untary precodng system, as llustrated n g. 1, ˆ s further confned to a lmted number of choces,.e., ˆ, where s a fnte set of untary matrces of sze 2, 0 1 2 1,,,,, () called a codebook of precodng matrces. Obvously, the sze of the codebook s lmted by the number of bts per feedback,. Each precodng matrx s an M M untary matrx,.e., I M, 0 2 1. he 2 precodng matrces s ndependent of the current channel state and thereby can be predetermned and made avalable at both ends (.e., the transcevers) of the communcaton lnk. Based on the estmated channel knowledge, the recever selects a proper precodng matrx from the codebook based on a certan crteron, such as the one gven by (3),
ˆ arg max C( ), (7) and then conveys the correspondng code ndex,.e., the -bt preferred precodng matrx ndex î, to the transmtter va a control channel for symbol mappng characterzed by the preferred precodng matrx ˆ ˆ. g. 2 shows the closed-loop untary precodng performance wth varous codebook szes where =0 corresponds to the open loop scenaro where there s no feedback and precodng matrces are randomly chosen from a codebook at each transmsson at the transmtter. denotes that the actual (not quantzed) from (5) s fed back to the transmtter. he creaton of varable sze untary precodng codebooks can be found n [8]-[10]. he delay from the measurement of the channel to the applcaton of the precodng matrx s ~3 ms. ke any other feedback scheme, the closed-loop precodng gan dmnshes quckly as the channel moblty ncreases due to the feedback delay. It s evdent that the closed-loop precodng gan over open loop s dependent on the number of feedback bts or the sze of the codebook, especally n a low moblty envronment. ypcally, s lmted to 4 bts per subband per feedback to avod ncurrng excessve control channel overhead [13]. Clearly, there s a sgnfcant gap between 4 and for mprovement. he gap ncreases as the number of antennas ncreases. he goal of the proposed rotatng codebook scheme s to mnmze the gap wthout nvokng addtonal feedback bts. 4 III. ROAING CODEBOOS OR IMIED EEDBAC PRECODING It s known that the performance of the lmted feedback precodng system n a low moblty envronment depends on the sze of the codebook, determned by the number of feedback bts. Increasng the number of feedback bts no doubt enhances the performance. However, the ncrease n the number of feedback bts n turn requres a sgnfcant ncrease n the control channel bandwdth. he goal of our rotatng codebook scheme s to mprove the closed-loop precodng gan wthout ncreasng the number of feedback bts. rst, dfferent but equvalent codebooks of sze 2 are created: k,0 k 1. (8) Each codebook n ths set of codebooks s alternatvely used by the transcevers n a predetermned fashon. In general, the codebook used at tme t s determned by a predetermned sequence k (), t 0 ( t) 1, known to both the transmtter and the recever. he codebook used at feedback tme t s hence () t. In ths paper, the codebooks smply rotate n a sequental order, () t nmod, at each feedback tme, t n, where s the feedback nterval,.e.,
5,,,,,,,,,. (9) 0 1 2 1 0 1 2 hs rotatng codebook scheme s llustrated n g. 3 where the codebook used at feedback tme t s () t, n contrast to the conventonal closed-loop precodng (c.f., g. 1) where a same codebook s used at all tmes:,,,,,,,,,. (10) hs rotatng codebook mechansm gves the transcevers the chance to use more precodng matrces over tme than a sngle codebook could offer wthout usng addtonal feedback bts. hs approach offers the transcevers a tmes larger vrtual codebook, log, but stll usng feedback bts nstead of the usual +log bts for ths sze of codebook. However, snce only one codebook can be used at a partcular feedback tme t,.e., the matrx selecton process s performed on one codebook, () t, of sze 2 (other than 2 ), ˆ arg max C( ), (11) ( t) and snce each ndvdual codebook, () t s equvalent to the codebook n (7) wth the same sze 2, t the rotatng codebook scheme n ths form provdes no overall performance advantage over the conventonal precodng that uses a sngle codebook wth the same number of feedback bts. In order to beneft from the rotatng codebooks, the default precodng matrx scheme must be employed. 0,,2 1 for servng as an ndcaton to the he default precodng matrx scheme reserves an ndex transmtter that the default precodng matrx (.e., the precodng matrx used n the prevous transmsson whch s a matrx optmzed through the prevous precodng matrx selecton process) should be used for current transmsson. or each codebook k, 0 k 1, the code matrx wth ndex,.e., =, s therefore punctured for use as the default matrx. hat s, the orgnal th k n matrx k k s no longer avalable for selecton. he transcevers keep track of the precodng matrx that s used n the prevous transmsson. If the recever cannot fnd a precodng matrx, from the current codebook, that s better than the precodng matrx,.e., Cî C, the reserved ndex (.e., the default precodng matrx ndcator) s communcated to the transmtter. Otherwse, î s fed back to the transmtter. When the transmtter receves the reserved ndex, t wll contnue usng the prevous precodng matrx,, for the current transmsson. hs allows the optmzaton results from the prevous codebooks to be used n the optmzaton process wthn the current codebook. rom an equvalent pont of vew, the current codebook
s altered by replacng the orgnal th precodng matrx, () (), wth the prevously selected () t precodng matrx,.e., () (). he best precodng matrx s then selected from the altered codebook, t denoted as () t, based on current channel condtons Hence, (9) becomes ( t) t ˆ arg max C( ). (12) 0 1 2 1 0 1 2,,,,,,,,,. (13) Snce () t contans precodng matrces not just from the current codebook () t but also the optmal precodng matrx from the prevous codebooks, (12) can be equvalently wrtten as (gnorng the punctured matrx for now) ˆ arg max C( ) (14) ( ), tt, 1, Compared to (11), t s clear that ths mechansm practcally enables the transcevers to choose an optmal precodng matrx that s not just optmzed over the current codebook but s also optmzed over prevously rotated codebooks n a progressve or ncremental manner. hs ncremental optmzaton process n effect ncreases the sze of the codebook used at each feedback tme, creatng a vrtual codebook that s larger than the actual one used at any feedback nstance. It s thus clear that, n a statc envronment, the rotatng codebooks k,0k 1 wth feedback bts now perform closely to the conventonal sngle log codebook precodng that has a codebook wth log feedback bts, ˆ arg max C( ), (15) log n the sense that the rotatng codebooks k,0 k 1 eventually select the same optmal precodng log matrx as the sngle codebook as a result of the use of the default precodng matrx wth certan delay. hat s a gan of log bts wthout actually commttng log addtonal feedback bts. However, t requres more selecton tme for the rotatng codebook of feedback bts to select the optmal precodng matrx from log 2 2 matrces from codebooks of sze 2 than for the sngle codebook of log feedback bts to select the same optmal matrx from log 2 matrces from a sngle codebook. or the sngle codebook wth feedback bts of log, the selecton of the optmal matrx s nstantaneous snce wth log2 log feedback bts, all 2 matrces are mmedately accessble at each feedback tme. Whle for the rotatng codebook wth feedback bts, the log 2 matrces are spread over codebooks and
7 only 2 matrces are accessble at each feedback nstance. he rotatng codebook may (n the worst case) take a full rotaton perod ( feedback ntervals) to obtan the optmal precodng matrx provded by a sngle codebook log. Pror to that happenng, only a suboptmal matrx can be chosen from the rotated codebooks at each feedback. Wth each rotaton, more codebooks are traversed; a better suboptmal matrx s hence selected at each feedback. After rotatons ( feedback ntervals, ), all codebooks are searched and the optmal matrx s guaranteed to be selected. he average delay s 1. herefore, the rotatng 2 codebook precodng wth feedback bts and rotatng codebooks essentally performs the same as the conventonal precodng wth feedback bts of than 1 2, log when the channel coherence tme c s much longer 1 c. (1) 2 hat s, the effectve codebook sze of the rotatng codebooks wth feedback bts s 2 whch s tmes larger than the conventonal codebook wthout commttng log2 extra feedback bts. As the channel coherence tme becomes shorter, condton (1) no longer holds. he rotaton delay becomes a sgnfcant part of the coherence tme. Before the optmal precodng matrx s selected, only sub-optmal precodng matrces can be selected durng the delay perod. or example, after the frst codebook rotaton, the best precodng matrx from two codebooks (the current one and the prevous one) s selected for current precodng. After the second rotaton, the best precodng matrx s selected from three codebooks (the current one and the prevous two). After the -1th rotaton, the best precodng matrx can then be chosen from the codebooks. he effectve codebook sze 2 s hence larger than 2 but less than 2,.e., 1, correspondng to log2 effectve feedback bts. As the channel coherence tme becomes shorter than the rotaton perod,.e., c, only c codebooks can be rotated wthn the coherence tme. he effectve number of codebooks becomes 1. he correspondng effectve feedback bts s log2. In general, c 1 mn, (17).e., the effectve number of feedback bts log2 s log 2 (18) hat s, the rotatng codebook performance s upper bounded by (15) and lower bounded by (7). In other words, the rotatng codebook precodng performance wth feedback bts and rotatng codebooks s upper c
8 bounded by the conventonal precodng performance wth log feedback bts (when c ) and lower bounded by feedback bts (when c precodng wth the same feedback bts s thus always non-negatve. ). he rotatng codebook gan over the conventonal he codebooks used for rotaton s created by generatng ndependent but equvalent codebooks. One can also create such codebooks by smply splttng a larger codebook (mother codebook) nto 2 equvalent smaller codebooks (chld codebooks) k, 0 k 1 1 k and k0, each of sze 2, where. he remander of the paper wll utlze the latter approach for creatng rotatng codebooks. As a result of the use of the reserved ndex for ndcaton of the default precodng matrx, total of matrces from the mother codebook (one from each of the chld codebook) wll not be avalable for use by the transcevers. o avod the precodng matrx k, k 0,, 1, correspondng to the reserved ndex, from beng permanently punctured from the codebook k, the reserved ndex should be made varable over tme. hat s, nstead of fxng, s now made tme varant accordng to a predetermned pseudo random sequence 0 ( t) 2 1 known to both the transmtter and the recever. hat s, () t at tme t. g. 4 llustrates ths random puncturng approach. It s evdent that the precodng matrx beng punctured changes from tme to tme. In effect, the precodng matrx t punctured from codebook () t at () t () feedback tme t wll be avalable at feedback tme t when a dfferent matrx ( t) ( t ) s punctured. herefore, all the precodng matrces from the mother codebook are vrtually avalable over tme for the transcevers. IV. SIMUAION RESUS Smulatons were performed to verfy the rotatng codebook performance usng a MIMO-ODM smulator. g. 5 shows the performance gan of the rotatng codebooks wth 4 and number of feedback bts s 4 and the number of rotatng codebooks (each of sze 4 2 ) s 2 4 2 (.e., the ) over the conventonal precodng wth varous feedback bts. he method for creatng the codebooks s the same as n Secton II. It s clear that, at 3 km/h, the rotatng codebook scheme mproves the precodng performance from the conventonal precodng wth 4 (lower bound for 4, 2 ) to better than that of the conventonal precodng wth 8 feedback bts, correspondng to an effectve gan of log2 8 4 4 4 feedback bts (and effectvely 2 1 tmes larger codebook) convertng to 2 db gan n SNR. In fact,
9 for further lower moblty, 4 wth 2 can approach closer to the performance of 10 (upper bound for 4, 2 ) as shown n g. where the moble speed s 1 km/h. On the other hand, as the moblty ncreases, the rotatng codebook gan over the conventonal precodng s reduced accordngly. g. 7 shows the rotatng codebook precodng performance at 15 km/h wth decreased gan. he rotatng codebook gan over the conventonal precodng s less but stll sgnfcantly better than conventonal 4. g. 7 also shows that the rotatng codebook gan dmnshes at 0 km/h where the rotatng codebook precodng wth 4 and 2 performs the same as the conventonal precodng wth =4 (lower bound), although at hgh moblty the closed-loop precodng performance dfference among dfferent values of s already small. It s worth notng that, n theory, the statc (.e., c ) performance of the rotatng codebook precodng wth 4 approaches as. However, n a practcal moble envronment, the rotatng codebook gan s ultmately capped by the channel coherence tme. he choce of therefore depends on the lowest moblty that the applcaton s targeted at to reduce unnecessary memory for storage of codebooks. or envronment wth moblty hgher than 1 km/h, 2 4 s obvously more than suffcent. V. CONCUSION he full gan from MIMO precodng s acheved wth full CSI at the transmtter snce ths allows the transmtted sgnal to be shaped based on the egen-structure of the channel. eedback of CSI to the transmtter can thus enable the transmtter to better explot channel condtons to mprove the MIMO performance. However, the amount of channel nformaton fed back to the transmtter (.e., the sze of the codebook) s lmted by the often severely lmted feedback control channel bandwdth, thereby preventng the transmtter from obtanng full channel nformaton n order to fully explot MIMO precodng gan. In ths letter, presented s a rotatng codebook precodng approach that vrtually extends the sze of the codebook wthout the need of ncreasng the feedback bts. hat s, the rotatng codebook provdes the transcevers wth essentally a larger codebook, whle utlzng the same number of feedback bts of a smaller codebook. hs s made possble by the use of two key mechansms: 1) multple codebook rotaton that allows transcevers to see more precodng matrces wthout utlzng more feedback bts; and 2) default precodng matrx that enables the combnng of the optmzaton over prevous codebooks wth the current codebook. As a result, compared to the conventonal closed-loop precodng scheme, the transcevers have more precodng matrces for optmzaton, mprovng the closed-loop precodng performance. he actual gan of the rotatng codebook over the conventonal closed-loop precodng s determned by the effectve number of codebooks that the transcevers can traverse wthn the channel coherence tme. he rotatng codebook precodng hence
10 provdes more gan over the conventonal closed-loop precodng wth the same feedback bts n statonary and pedestran envronment (whch s typcally the scenaro that hgh rate data applcatons s targeted at) and reduces to the same performance as the conventonal closed-loop precodng at hgh moblty. hat s, the rotatng codebook precodng automatcally and fully explots the channel coherence provded by the envronment and converts t to the performance advantage, whch s not the case for the conventonal precodng. Although ths letter uses untary precodng as the applcaton paradgm, the rotatng codebook scheme s applcable to any codebook based precodng system wth lmted feedback. REERENCES [1] H. Bolcske, MIMO-ODM wreless systems: bascs, perspectves, and challenges, IEEE Wreless Commun, vol. 12, pp. 31-37, Aug. 200. [2] G. oschn and M. Gans, On lmts of wreless communcatons n a fadng envronment when usng multple antennas, Wreless Personal Commun., vol., pp.311-335, 1998. [3] A. Scglone, P. Stoca, s. Barbarossa, G. Gannaks, and H. Sampath, Optmal desgns for space-tme lnear precoders and decoders, IEEE rans. Sgnal Process., vol.50, no.5, pp.1051-104, May 2002. [4] H. Sampath, P. Stoca, and A. Paulraj, Generalzed lnear precoder and decoder desgn for MIMO channels usng the weghted MMSE crteron, IEEE rans. Commun., vol.49, pp.2198-220, Dec. 2001. [5] G. Care and S. Shama, On the achevable throughput of a mult-antenna Gaussan broadcast channel, IEEE rans. Inform. heory, vol. 49, no. 7, pp. 191-170, July 2003. [] D. ove, R. Heath Jr., W. Santpach, and M. Hong, What s the value of lmted feedback MIMO channels? IEEE Commun. Mag., vol. 42, pp. 54-59, Oct. 2004. [7] N. Jndal, MIMO broadcast channels wth fnte-rate feedback, IEEE rans. Inform. heory, vol. 52, no. 11, pp. 5045-500, Nov. 200. [8] D. ove, R. Heath, Jr., mted feedback untary precodng for spatal multplexng Systems, IEEE rans. Inf. heory, vol.51, Aug. 2005. [9] D. ove, R Heath, Jr., mted feedback untary precodng for orthogonal space-tme block codes, IEEE rans. Sgnal Processng, vol.53, January 2005. [10] I. m, S. Park, D. ove, S. m, Improved multuser MIMO untary precodng usng partal channel state nformaton and nsghts from the Remannan manfold, IEEE rans. Wreless Comm., vol. 8, pp. 4014-4023, August 2009. [11] S. J. m, H. m, C. S. Park, and. B. ee, On the performance of multuser MIMO systems n WCDMA/HSPA: Beamformng, feedback and user dversty, IEICE rans. Commun., vol. E98-B, pp. 211-219, Aug. 200. [12] M. hojastepour, X. Wang, M. Madhan, Desgn of multuser down lnear MIMO precodng systems wth quantzed feedback, IEEE rans. Vehcular echnology, vol. 58, pp. 4828-483, November 2009. [13]. han, E for 4G Moble Broadband: Ar Interface echnologes and Performance, Cambrdge Unversty Press, 2009. [14] M. Regel, A. Chndapol, D. roeselberg, Deployng Moble WMAX, Wley, 2010. [15] A. Greenspan, M. lerer, J. omck, R. Canch, J. Wlson, IEEE 802.20: Moble Broadband Wreless Access for the twenty-frst century, IEEE Communcatons Magazne, pp.5-3, July 2008. [1]. Cover, J. homas, Elements of Informaton heory, John Wley & Sons, 1991. [17] A. Marshall, I. Olkn, Inequaltes: heory of Majorzaton and ts Applcatons. Academc Press, 1979. [18] 3GPP R 3.912 V2.0.0, 3 rd Generaton Partnershp Project; echncal Specfcaton Group Rado Access Network; easblty Study for urther Advancements for E-URA (E-Advanced) (Release 9), Aug. 2009.
11 s U y î ˆ ( bts) ˆ g. 1. Block dagram of a lmted feedback untary precodng MIMO system. g. 2 Untary precodng performance for M 8, M 1 antenna confguraton. adng speeds = 3 km/h and 0 km/h R at 2GHz carrer frequency; Number of feedback bts = 0 (no feedback), 4,, 8, 10, and ; eedback nterval=3 msec.
12 s U y î J 0 ˆ ( bts) ˆ () t J 0 log g. 3. Schematc block dagram of the rotatng codebook precodng system where the codebook used at feedback tme t s () t. () t ( t) ( t1 ) ( t ) ( t ) 2 () t ( t ) t 1 eedback me t t+ t+(-1) t+ 2 g. 4. Illustraton of codebook random puncturng for rotatng codebook precodng, where denotes the precodng matrx that s punctured and replaced by the default precodng matrx.
13 g. 5 Rotatng codebook performance wth 4 and 2 ( M 8, M 1 R he number of feedback bts for the conventonal precodng s 4,, 8, and 10 ) at 3 km/h/2 GHz carrer frequency. ; eedback nterval = 3 msec. g.. Rotatng codebook performance wth 4 and 2 ( M 8, M 1 ) at 1 km/h at 2 GHz carrer R frequency. he number of feedback bts for the conventonal precodng s 4,, 8, and 10 ; eedback nterval = 3 msec.
14 g. 7 Rotatng codebook performance wth 4 and 2 at 15 and 0 km/h ( M 8, M 1). R