Reduced Complexity MIMO MMSE-DFE

Transcription

1 Reduced Complexty MIMO MMSE-FE We-C Ka ad Gerald E. Sobelma epartmet o Electrcal ad Computer Egeerg Uersty o Mesota Meapols, MN USA ax8@um.edu, sobelma@um.edu Abstract We preset a reduced complexty MIMO MMSE-FE coecet computato algortm wc s based o a mproemet to te ast bloc algortm deeloped Re. []. We ae aceed a early 5% reducto complexty by elmatg te computato o uecessary terms. Te eects due to te precso artmetc o SNR are also cosdered ad a mmum xed-pot word legt o bts results o otceable perormace loss. Hardware mplemetato ssues deped o seeral actors ad tese are also dscussed. I. INTROUCTION Curret wreless systems would le to acee a Gb/s data rate. Howeer, sgle-put, sgle-output (SISO systems ca typcally oly prode a spectral ececy o ~4 bps/hz wc requres a 5 MHz badwdt order to deler a Gb/s trasmsso rate. Multple-put, multpleoutput (MIMO tecques are oe way to crease te spectral ececy ad a MIMO wreless system ca prode a Gb/s trasmsso usg oly a MHz bad, wc s more practcally aalable []. Res. [3]-[4] ae proposed desgs or te mplemetato o a MIMO wreless system te requecy-lat cael eromet. I te cael coerece badwdt s less ta te trasmtted sgal badwdt, te te cael s called requecy selecte or dsperse. Oterwse, t s called a requecy lat cael [6]. Te dsperse cael dstorts trasmtted sgals te orm o ter-symbol tererece (ISI, wc ot correctly compesated, ca lead to a g bt error rate. Equalzato as bee te soluto or te ISI problem. Tere exst seeral deret types o equalzato, amely, maxmum lelood sequece estmato (MSE, lear equalzato (E, ad decso eedbac equalzato (FE. Amog tese, FE as good perormace wt lower complexty ta MSE ad ece t as receed lots o researc atteto oer te last twety years. I te cael coerece tme s muc larger ta te symbol teral o te trasmtted sgal, te cael s called slow adg. Oterwse, t s called ast adg [6]. I Re. [5], two deret trasmsso scearos are preseted wc clearly RF ad Aalog rot ed Cael estmato Coecet computer FE Furter sgal processg Fgure. Receer bloc dagram or requecy selecte ad tme selecte caels. expla te requecy selecte cael wt slow adg ad wt ast adg. Fg. s a geeral bloc dagram o te receer dealg wt a requecy selecte cael wt ast adg (.e., seere tme selectty as Re. [5]. I a cael as requecy selectty ad seere tme selectty, a computato s eeded to costatly update te lter coecets order or te FE to wor eectely. A geerc bloc dagram o a FE structure s sow Fg.. I Re. [], two ast MIMO MMSE-FE coecet computato algortms (called SFBC-FE ad BFBC-FE are reported wc reduce te computatoal complexty o te tese spectral actorzatos o matrx ratoal spectra. I ts paper, a ew reduced complexty MIMO MMSE-FE coecet computato algortm (or RCFBC-FE or sort s preseted. Te arctecture s based o te ast computato algortm deeloped Re. []. Fte word legt eects are mportat ad ece a aalyss o te degradato SNR o te MIMO MMSE-FE we te lter coecets are computed a xed-pot maer s also cosdered. Te remader o ts paper s orgazed as ollows: I Secto II, te system model o te MIMO MMSE-FE ad ts SFBC-FE ad BFBC-FE algortms are dscussed. Te RCFBC-FE s preseted Secto III. Numercal examples, comparsos complexty betwee RCFBC-FE ad bloc FBC-FE ad te te precso eects o te xed-pot computer smulato are ge Secto IV. Hardware mplemetato ssues are dscussed Secto V. Fally, te coclusos are ge Secto VI. W N- y +N- W N- y +N- N: umber o Feedorward lter taps Nb: umber o Feedbac lter taps Δ: decso delay W y x Δ Nb Fgure. A geerc FE structure. + _ x B Nb x Δ I-B

2 Te otato used ts paper s summarzed as ollows:. E[ ] s te expected alue operator.. ( represets te complex cojugate trasposto. 3. ( m deotes a m-by- matrx. 4. ( [ : j, :] deotes te sub-matrx obtaed by extractg rows troug j. 5. e represets te t ut ector. II. SYSTEM MOE AN FAST AGORITHMS A. System Model Te system model used ts paper s based o a geeral lear, dsperse ad osy dgtal commucato system depcted Fg. 3 were te cael as puts ad o outputs ad (,j deotes te cael mpulse respose betwee t put ad te jt output ad (j represets AWGN o te jt output. I we urter assume tat te cael legt s deoted by (,j or cael (,j, te put ad output relato ca be wrtte as te orm, y = (, j ( = m= (, j m x ( m + ( j were s te tme stat. Furtermore, a oersamplg s assumed, say l samples per symbol perod, te output, mpulse respose ad ose would be l colum ectors. By groupg all te output samples rom te o cael to a l o colum ector, equato ca be wrtte a more compact orm, y = H x + x x m= m m (, (, y y were H m s a l o MIMO cael coecet at tme m, x -m s a colum ector at tme -m, ad s te maxmum legt o all o te o cael mpulse resposes. Oer a bloc o N symbol perods, equato ca be expressed a matrx orm as ollows, or more compactly, y+ N H H H y + N H H H = M M O O M y H H H x+ N x+ N M x + M + N + N + N : : :. + N + N (3 y = Hx + (4 B. Fast Algortms It s sow Re. [] tat te optmum eedbac ad eedorward lters are equatos (7 ad ( o Re. []. I order to compute tose lter coecets, te ast Colesy actorzato s utlzed o te matrx R R + H R H, xx = were R xx s te auto-correlato matrx o te put sgals, R s te auto-correlato matrx o te ose sources ad s te Colesy tragle [8]. Te auto-correlato uctos are deed as ollows. R R xx + N : + N : E[ x x ]. (5 + N : + N : E[ ]. (6 Two ast actorzatos, amely, stadard ad bloc actorzato, were deeloped Re. [] to perorm te ast Colesy actorzato wt a computatoal ececy o 3 3 O [ + ] compared to O [ + ] or te classcal Guaussa elmato metod. (, o ( o III. REUCE COMPEXITY MIMO MMSE-FE AN EFFECT OF FIXE-POINT IMPEMENTATION ( x Fgure 3. MIMO cael model ( o y A. Reduced Complexty Algortm We ow mae te ollowg obserato to urter mproe te computatoal ececy. Te optmum eedbac lter dered Re. [] (.e., ts equato (7 ca be urter smpled as ollows: opt [ B B ] B = (7 (( [ Δ+ : ( Δ+,:]

3 were B s computed by equato (9, Δ s te optmum delay, ad bot ad are parttos o szes (Δ+ (Δ+ ad (N +-Δ- (Δ+. (8 3 I oter words, we d tat te sub-bloc 3 s ot used so tat t s ot ecessary to compute t. B s computed a te ollowg equato, B were R3 e e =, e R3 e R =, we urter partto as, wt 3 o sze. 3 Sce te eedorward lter (.e., equato ( o Re. [] ad te eedbac lter are related troug te ollowg equato, W [ B ] R H R, opt = Δ opt te eedorward lter ca be easly smpled as te orm, W were opt = B (( [ Δ+ : ( Δ+,:] ( H [: ( Δ+,:] R are parttos (9 ( ( ( ( (ote tat = 3. (3 3 From equatos (7 ad (, we oly eed to compute te ad rater ta te etre matrx. Also, ca be computed by ertg. Tereore, te actual computatoal complexty s oly O[ ( Δ + ] compared to te preously reported O[ + ]. B. Perormace Aalyss Our perormace aalyss s based o te Artmetc SNR, wc s deed by: Rxx + ( Ree,m were R xx as (N + blocs o aoter dagoal matrx x o sze o ts dagoal. Ts perormace measure ca be smpled to: x. (5 R ee,m IV. NUMERICA EXAMPES Te umercal examples are ge below to sow tat we sae early 5% o te computatoal load o te bloc FBC-FE. Fte precso eects o te xed pot computato s also ge ere. A. Smulato Parameters Tree deret scearos were cosdered te smulatos o te precso eects. Scearo : Cael: trasmt ad recee ateas Cael legt: Feedorward taps: 3 Feedbac taps: ecso delay: Iput SNR:, 5, ad db Scearo : Cael: trasmt ad recee ateas Cael legt: 4 Feedorward taps: 3 Feedbac taps: 4 ecso delay: Iput SNR:, 5, ad db Scearo 3: Cael: 3 trasmt ad 3 recee ateas Cael legt: 4 Feedorward taps: 3 Feedbac taps: 4 ecso delay: Iput SNR:, 5, ad db B. Fte Precso Eects Based o te aboe tree smulato scearos, we coclude tat -bt xed-pot artmetc sould be used trougout te computato or a x cogurato order to ae o otceable perormace degradato compared to double-precso loatg pot. I te SNR at te receer put s low, e.g. db, te word legt ca be reduced to bts. For 3 x 3 coguratos, te mmum xed-pot word legt s also bts wtout ay otceable loss perormace. Fgs. 4 ~ 6 are te te precso eect or Scearo troug 3 respectely. C. Complexty Reductos Te ew RCFBC-FE algortm reduces te computatoal complexty by early 5% compared to te

4 Output : x wt c. legt = Number o Bts SNR= FP, SNR= SNR=5 FP, SNR=5 SNR= FP, SNR= Output : x wt c. legt = Number o Bts 8 6 SNR= FP, SNR= SNR=5 FP, SNR=5 SNR= FP, SNR= Fgure 4. Fte Precso Eects or Scearo. BFBC-FE. For example, or Scearo, we oly use multplcatos compared to 4456 multplcatos or te BFBC-FE. Also or Scearo, about 5% o te multplcatos are saed compared to te BFBC-FE (368 s Te ew algortm also saes more ta 5% o te umber o addtos requred te orgal algortm. Fgs. 7 ad 8 sow te umber o multplcatos ad addtos used by RCFBC ad BFBC, respectely. I bot gures, caels o legt ad legt 4 were cosdered. V. HARWARE IMPEMENTATION ISSUES Te MIMO MMSE-FE mplemetato depeds o seeral actors, e.g., ow ast te cael cages, system coguratos, ad te cael legt. Ge a 3G user wt a carrer requecy o MHz traelg at 55 mles per our, te user sees a cael coerece tme o about 5.7 ms ad we ca assume te cael stays costat or at most ms. Ts, tur, requres te FE to update ts coecets at a rate o tmes per secod. I we urter assume te cael ad system cogurato o Scearo, te te RCFBC-FE wll eed 368 -bt multplcatos ad 6 -bt addtos to update ts coecets eery ms. A TI TMSVC55x SP processor [8] ca perorm ts tas easly. Howeer, te system cogurato s 6 x 6 ad te cael legt s 4, te RCFBC wll requre early 8,353 -bt multplcatos ad 6,63 -bt addtos ms to update ts coecets. A TMSVC55x ca ot adle tat computato load. A dedcated -bt 9 Output : 3 x 3 wt c. legt = Number o Bts Fgure 6. Fte Precso Eect or Scearo 3. SNR= FP, SNR= SNR=5 FP, SNR=5 SNR= FP, SNR= Fgure 5. Fte Precso Eects or Scearo. multpler rug at a speed o at least 9 MHz could be used or suc a applcato [9]. VI. CONCUSIONS As wreless commucatos systems reac toward Gbps trasmsso, MIMO tecques ad wdebad trasmsso may be utlzed to acee ts goal. A gly moble user uder ts stuato aces a ast cagg, requecy selecte cael wc wll troduce tersymbol tererece to te data trasmsso. Re. [] preseted a MIMO MMSE-FE coecet computatoal algortm wc reduced te computato complexty to 3 3 O [ + ] rom O [ + ] or te classcal Guaussa elmato metod. We ae urter smpled tat algortm by elmatg te computato o uecessary terms. Our algortm lowers te computatoal complexty to O [ ( Δ + ]. Numercal smulatos sow tat a computatoal sags o early 5% ca be aceed. For good perormace comparable to tat o doubleprecso loatg-pot artmetc, te xed-pot word legt sould be bts. Curret g-speed, low-power multpler optos are aalable wc ca meet te requred computatoal load. Te actual ardware mplemetato Number o Multplcatos x 3 x 3 4 x 4 5 x 5 6 x 6 RCFBC c= RCFBC c=4 BFBC c= BFBC c=4 Fgure 7. Number o multplcatos used by RCFBC ad BFBC we te cael legt s ad 4.

5 Number o Addtos x 3 x 3 4 x 4 5 x 5 6 x 6 RCFBC c= RCFBC c=4 BFBC c= BFBC c=4 Fgure 8. Number o addtos used by RCFBC ad BFBC we te cael legt s ad 4. depeds o seeral actors, e.g. ow ast te cael cages, te cael legt ad te umber o trasmt ad recee ateas. I some cases, a dedcated ardware mplemetato wt a g speed -bt multpler ad adder wll be requred. Curret g-speed multplers wc ca ru at up to GHz mae ts coecet computato tecque easble. REFERENCES [] N. Al-ar ad A. H. Sayed, Te Fte-egt Mult-Iput Mult-Output MMSE-FE, IEEE Tras. O Sgal Processg, ol. 48, o., pp , Oct.. [] A. J. Paulraj,. A. Gore, ad R. U. Nabar, A Oerew o MIMO Commucatos-A Key to Ggabt Wreless, Proc. o te IEEE, ol. 9, o., pp. 98-8, Feb.. [3] Z. Guo ad P. Nlsso, A VSI Arctecture o te Scorr-Eucer ecoder or MIMO Systems, IEEE 6t CAS Symp. o Emergg Tec., Saga, Ca, ol., pp , May. [4] Z. Guo ad P. Nlsso, A VSI Implemetato o MIMO etecto or Future Wreless Commucatos, te t IEEE 3 Iteratoal Symp. o Persoal, Idoor ad Moble Rado Comm. Proc., ol. 3, pp , Sep. 3. [5] S. N. gga, N. Al-ar, A. Stamouls, ad A. R. Calderba, Great Expectatos: Te Value o Spatal ersty Wreless Networs, Proc. o te IEEE, ol. 9, o., pp. 9-7, Feb.. [6] J. W. Mar ad W. Zuag, Wreless Commucatos ad Networg, Upper Saddle Rer, NJ: Pearso Educato Ic., 3. [7] G. H. Golub ad C. F. Va oa, Matrx Computatos, 3 rd ed., Baltmore, M: Jo HopsUersty Press, 996. [8] Aalable: ttp://dspllage.t.com/ [9] H-C Cow; I-C Wey, A 3.3 V GHz g speed ppeled Boot multpler, IEEE Iteratoal Symposum o Crcuts ad Systems, ISCAS, ol., pp , May