CHAPTER 3 DETECTION TECHNIQUES FOR MIMO SYSTEMS

Transcription

1 4 CAPTER 3 DETECTION TECNIQUES FOR MIMO SYSTEMS 3. INTRODUCTION The man challenge n he paccal ealzaon of MIMO weless sysems les n he effcen mplemenaon of he deeco whch needs o sepaae he spaally mulplexed daa seams. Seveal algohms offeng vaous ade-offs beween pefomance and compuaonal complexy have been dscussed n leaue. Lnea deecon schemes povde nfeo eo pefomance wh much educed complexy whle Maxmum Lkelhood Deeco (MLD) algohm povdes opmum pefomance wh hgh complexy. In hs chape, a vaey of hese echnques ae evaluaed usng dffeen pedeemned pefomance and complexy cea. MIMO deecon echnques ae caegozed no hee man caegoes; lnea schemes, successve nefeence cancellaon, and ee-seach echnques. Lnea schemes ae easy o mplemen bu leads o hgh degadaon n pefomance. Successve nefeence cancellaon schemes exac he ansmed symbol accodng o a cean pemuaon dependng on channel max. 3. MAXIMUM LIKELIOOD DETECTOR Maxmum Lkelhood Deeco (MLD) s consdeed as he opmum deeco fo a MIMO sysem gven by Equaon (3.). The

2 43 ansmed sgnal could be effecvely ecoveed a he eceve based on he followng mnmum dsance ceon, k,,, N k x ag mn x (3.) x x x x whee x s he esmaed symbol veco. Usng he above ceon, MLD compaes he eceved sgnal wh all possble ansmed sgnal veco whch s modfed by channel max and esmaes ansm symbol veco x. Alhough MLD acheves he bes pefomance and dvesy ode, eques a bue-foce seach whch has an exponenal complexy n he numbe of ansm anennas and consellaon sze. Fo example, f he modulaon scheme s 64-QAM and 4 ansm anenna, a oal of 64 4 = compasons pe symbol ae equed o be pefomed fo each ansmed symbol. Thus, fo hgh modulaon ode and hgh ansm anenna, becomes nfeasble. N MLD 3.3 LINEAR DETECTION TECNIQUES The dea behnd lnea deecon echnques s o lnealy fle eceved sgnals usng fle maces, as depced n Fgue 3.. Ths caegoy ncludes Zeo-Focng (ZF) and Mnmum Mean Squae Eo (MMSE) echnques. Alhough lnea deecon schemes ae easy o mplemen, hey lead o hgh degadaon n he acheved dvesy ode and eo pefomance due o he lnea fleng. Fgue 3. MIMO SM wh lnea eceve

3 Zeo-Focng Zeo-Focng (ZF) echnque s he smples MIMO deecon echnque, whch was poposed by Foschn (996), whee fleng max s consuced usng he ZF pefomance-based ceon. ZF can be mplemened by usng he nvese of he channel max o poduce he esmae of ansmed veco x. x x x (3.) whee denoes he pseudo-nvese. Consdeng he nose em, he pospocessng sgnal s gven by: x R x n x n (3.3) x consss of he decoded veco x plus a combnaon of he nveed channel max and he unknown nose veco. Because he pseudo-nvese of he channel max may have hgh powe when he channel max s llcondoned, he nose vaance s consequenly nceased and he pefomance s degaded. To allevae fo he nose enhancemen noduced by he ZF deeco, he MMSE deeco was poposed, whee he nose vaance s consdeed n he consucon of he fleng max Mnmum Mean Squae Eo Mnmum Mean Squae Eo (MMSE) appoach allevaes he nose enhancemen poblem by akng no consdeaon he nose powe

4 45 when consucng he fleng max usng he MMSE pefomance-based ceon. The veco esmaes poduced by an MMSE fleng max becomes x I (3.4) whee s he nose vaance. The em (/SNR = ) offes a ade-off beween he esdual nefeence and he nose enhancemen. As he SNR gows lage, he MMSE deeco conveges o he ZF deeco, bu a low SNR pevens he wos Egen values fom beng nveed (Wubben e al 00). 3.4 SUCCESSIVE INTERFERENCE CANCELLATION SCEMES 3.4. Maxmum A poseo Pobably (MAP) The MAP decson ceon s based on selecng he symbol coespondng o he maxmum of he se of poseo pobables P xm y. The poseo pobably s defned as { P( xm y)} P(sgnal beng ansmed y ) (3.5) whee y s he eceved veco. Alhough MAP ule offes opmal eo pefomance, suffes fom complexy ssues Vecal Bell-Labs Layeed Space Tme (V-BLAST) V-BLAST (Vecal Bell-Labs Layeed Space-Tme) eceve ulzes a layeed achecue and apples successve cancellaon by splng he channel vecally (Foschn and Gans 998). Alhough lnea deecon echnques ae easy o mplemen, hey lead o hgh degadaon n he

5 46 acheved dvesy ode due o he lnea fleng. Fgue 3. shows he basc achecue of a MIMO sysem wh V-BLAST a he deeco. The V-BLAST deecon algohm s a ecusve pocedue ha exacs he componens of he ansmed veco, x, accodng o a cean odeng k, k,, k of he ndces of he elemens of x. The odeng s a N pemuaon of,,, N and hs pemuaon depends on channel max,. Fgue 3. VBLAST achecue The symbol deecon algohm fo MIMO sysems whch combnes feaues of ZF, MMSE, MAP and V-BLAST ules ae mplemened. Combnaons lke VBLAST/ZF, VBLAST/MMSE and VBLAST/MMSE/MAP ae also mplemened. VBLAST/MMSE/MAP algohm offes supeo eo pefomance han schemes avalable n he leaue (Wubben e al 00). Though he complexy s slghly nceased, s sll lowe han he convenonal Maxmum Lkelhood (ML) mehod V-BLAST/ZF Deecon Algohm The V-BLAST/ZF deecon Algohm s a combnaon of ZF and V-BLAST. The Pseudo code of he deecon algohm s gven below.

6 47 Inalzaon W Recuson k ag mn W foj k k j y W k k a ˆk Q y k a ˆ k k W k denoes he Mooe-Penose pseudo nvese of. Ths deecon scheme has he advanage of pefomng sgnfcanly bee han ZF and MMSE echnques, bu no as good as MLD echnque (Golden e al 999) V-BLAST/MMSE Deecon Algohm V-BLAST s combned wh MMSE and he pseudo code fo he deecon algohm s gven below. The man dawback of hs scheme les n he compuaonal complexy, because mulple calculaons of he pseudo nvese of he channel max ae equed (Wubben e al 00). Inalzaon W I whee / SNR N

7 48 Recuson k ag mn W foj k k j y W k k a ˆk Q y k a ˆ k k W k V-BLAST/MMSE/MAP Algohm The MAP decson based on selecng he symbol wh maxmum poseo pobably s combned wh BLAST/MMSE. The pseudo code fo a V-BLAST/MMSE/MAP algohm s as follows. Inalzaon W I N Recuson y W. s Q( y ) p f y x f y s, j { k, k,.., k } j j j j s A j j k ag max{ p }, j { k, k,.., k } j

8 49 xˆ s k k x ˆ k k N k k k W I Each elemen of x belongs o a common modulaon alphabe A, x A, {,,... N }. The vecos y ae he esmae of he ansmed sgnal x and Q s a quanze ha maps s agumen o he neaes sgnal pon usng Eucldean dsance. The densy funcon fj s gven as f y x exp y s (3.6) j j j j j j j whee j o j N W ; j anges ove {,,... N }. Ths echnque s found o be he mos advanageous as gves almos equal eo pefomance as MLD echnque wh educed complexy. 3.5 COMPLEXITY ANALYSIS Complexy analyss allows o esmae he poenal cos of communcaon sysems and o denfy possble bolenecks fo he hadwae mplemenaon. The complexy of he deecos wll only be compued n ems of flops, whch s anohe measue popoonal o he unnng me. Even hough hee s no eal consensus n he dgal communcaons communy on how exacly o nepe he concep of complexy, s geneally defned as he numbe of floang pon opeaons (addons, mulplcaons ec.) whch ae equed o compue he esmae of he

9 50 ansmed veco x o he unnng me of he algohm when mplemened on some specfc plafom. Thee s also ypcally a adeoff beween he complexy of a deeco and s pefomance n ems of eo pobably. The opmal, ML, deeco whch povdes he mnmum pobably of eo s ofen pohbvely complex whle he compuaonally smples deecos wll have a poo pefomance n ems of eo pobably. Fo example, consde a max mulplcaon beween wo maces A, B of dmensons C D and D E, complexy s found o be as a oal of C( D ) E addons and CDE mulplcaons. To we complex addons and complex mulplcaons n ems of eal addons and eal mulplcaons, s easly vefed ha one complex addon consss of wo eal addons - he eal and he magnay pa of he wo complex numbes ae added. The complexy of he algohm s gven n ems of complex floang pon opeaons (flops). A complex mulplcaon / dvson eque 3 flops, and complex addon eques one flop Complexy Analyss of ZF Calculaon of he pseudo nvese of channel max s he man pocess n ZF, whee he pseudo nvese s defned as W (3.7) Dmensons of he maces W,, ae N N, N N, and N N especvely. In ode o fnd he complexy n calculang pseudo nvese of he channel max, he complexy of need o be found. Accodng o he ule of max mulplcaon, he complexy of he poduc s

10 5 N ( N ) Ac and N NM c. Ac efes o complex addon and M c efes o complex mulplcaon. Inveng Fnally he nvese of s mulpled by s N ( N ) N A and c s gven n Equaon (3.8); has a complexy of 3 N Ac and 3 c N M. and he complexy nvolved N NM c. Complexy nvolved n he peamble phase C ( flops) 7N 7N N N (3.7) (3.8) 3 ZF pe Now fo he sep; x (3.9) The complexy of hs poduc s N ( N ) complex addons and N N complex mulplcaons. The complexy of slcng consellaon pons equals N log ( ) M eal addons. N M-ay ence he complexy of he ZF algohm s gven as C N N N N N N N log M (3.0) 3 ZF (flops) (/ ) ( ) 3.5. Complexy Analyss of MMSE The complexy of he MMSE algohm s almos equal o he complexy of he ZF mehod. In he peamble pocessng sage, he max W needs o be deemned. W I N (3.) The calculaon of hs max has almos he same compuaonal complexy as he deemnaon of pseudo-nvese n ZF.

11 5 Snce addonal complexy nvolved s s eal, and s added wh he dagonal elemens of N, he eal addons. The oal complexy 3 nvolved n peamble s 4 N N (8N ) N N N eal addons and 4N 8N N eal mulplcaons. 3 mulplcaons s The oal complexy of MMSE whch nvolves eal addons and C N N N N N N N log M (3.) 3 MMSE (flops) (/ ) ( ) Complexy Analyss of VBLAST/ZF The pocessng of he ZF-VBLAST algohm can be dvded no wo pas: he pocessng dung he peamble and pocessng of he payload. Based on he assumpon ha he MIMO channel s sac dung a veco ansmsson, he odeng and he wegh vecos can be deemned dung he peamble pocessng. Dung he payload pocessng he acual deecon and SIC s pefomed. In ode o fnd he weghng vecos, an eave algohm ha consss of wo seps has o be pefomed. Sep : Compue he pseudo-nvese of Sep : Fnd he mnmum squaed lengh ow of. Ths wegh veco s modfed o be he las ow and pemue he columns of accodngly. Calculaon of he pseudo nvese of N N nvolves 3 4 (8 ) N N N N N eal addons and 4N 8N N eal 3 mulplcaons. Then he mnmum squaed lengh s obaned hough he compuaon of he pseudo nvese of and he mnmum value. Fndng he mnmum of N values has a complexy of N eal addons. Snce he

12 53 algohm s an eave algohm, he complexy n ems of floang pon opeaons s gven as C N N N N N N N N N VBLAST / ZF ( flops) 7 N N log ( M ) (3.3) Complexy Analyss of VBLAST/MMSE The complexy of MMSE wh BLAST can be deemned n he same way as ZF wh BLAST, bu wh a slgh modfcaon n he deemnaon of wegh vecos. Compaed o ZF/BLAST, he addon of esuls n 3 4 (8 ) N eal addons. The complexy nvolved s N N N N N N eal addons and 4N 8N N eal 3 mulplcaons. The complexy s slghly hghe han VBLAST/ZF because of he addon of I o. The oal complexy n ems of floang pon opeaons s deved as N N N N N N N NN N N log ( M ) (3.4) Complexy Analyss of MAP/VBLAST/MMSE The complexy of VBLAST/MMSE/MAP nvolves he same opeaon as VBLAST/MMSE and he selecon of maxmum aposeo pobably. The MAP decson ceon s based on selecng he symbol coespondng o he maxmum of he se of poseo pobables and hs nvolves a complexy of M N flops leadng o a compuaonal complexy of N N N N N N N NN N N log ( M ) (3.5)

13 54 The complexy of ohe deecon algohms n ems of he numbe of ansm anennas N, numbe of eceve anennas consellaon sze M s abulaed n Table 3.. N and Table 3. Compason of he deecon schemes Deecon Algohm ZF MMSE VBLAST/ZF VBLAST/MMSE VBLAST/MMSE/MAP MLD Complexy (flops), C 3 7N 7N N N 4NN N log ( M ) 3 7N 7N N N 4NN N log ( M ) N N N N N N N N N N N log ( M ) N N N N N N N N N N N log ( M ) N N N N N N N N N N N log ( M ) MN N M Fom he above analyss, s clealy undesood ha he ecusve algohms (VBLAST) ae compuaonally easy o be mplemened wh an eo pefomance almos compaable o MLD echnque. The genealzed expessons gven n Table 3. ae valdaed n Table 3. fo a 4 4 MIMO sysem employng 64-QAM modulaon wh vaous deecon schemes a he eceve.

14 55 Table 3. Complexy analyss fo 64-QAM, 4 4sysem Deecon Scheme Complexy (FLOPS) ZF 964 MMSE 968 V-BLAST/ZF 435 V-BLAST/MMSE 30 V-BLAST/MMSE/MAP 486 MLD (Ideal eo pefomance) Fgue 3.3 gves he eo pefomance compason of deecon schemes lke ML, ZF, MMSE and VBLAST fo a (4 4) MIMO spaal mulplexng sysem. MLD gves he bes eo pefomance. A an eo ae of 0, MMSE povdes db SNR gan ove ZF. Ths s because he Mnmum Mean Squae Eo (MMSE) appoach allevaes he nose enhancemen poblem by akng no consdeaon he nose powe when consucng he fleng max. Fgue 3.3 Eo pefomance of a 4 4 MIMO SM wh lnea eceves

15 56 Fgue 3.4 Eo pefomance of a 4 4 MIMO V-BLAST eceves Fgue 3.4 shows he compason of V-BLAST combned wh ZF, MMSE and MAP deecon schemes. A lowe SNR, he BER pefomance s smla fo all combos bu fo hghe SNR's (.e) fo an SNR of above 6 db, he MAP algohm domnaes VBLAST/ZF and VBLAST/MMSE. A an eo ae of 0, he MAP algohm has an mpovemen of aound.5 db as compaed o ohe schemes. Even hough boh VBLAST/MAP/ZF and VBLAST/MAP/MMSE povde he same pefomance as MLD, he scheme wh lesse complexy s chosen fo fuhe dscussons. Fom he above analyss, VBLAST/MMSE/MAP can be chosen as he bes deecon scheme wh much educed complexy and a bee eo pefomance. 3.6 CONCLUSION Lnea deecon schemes lke ZF and MMSE ae compaed wh he opmal Maxmum Lkelhood Deeco (MLD) algohm. MLD povdes opmum pefomance wh hgh complexy. Lnea schemes ae easy o

16 57 mplemen bu leads o hgh degadaon n pefomance. Successve nefeence cancellaon schemes exac he ansmed symbol accodng o a cean pemuaon dependng on channel max. The lnea deecon schemes ae combned wh successve nefeence cancellaon and a vaey of hese echnques ae evaluaed usng dffeen pedeemned pefomance and complexy cea. Thus, he compaave sudy of vaous deecon schemes n ems of pefomance and complexy analyss ae analysed fo dffeen anenna confguaons. Fom he obaned esuls can be concluded ha VBLAST/ MMSE/MAP acheves supeo B Eo Raes whle eanng he lowcomplexy naue of he V-BLAST.