Performance Analysis of the V-BLAST Algorithm: An Analytical Approach
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1 Pefoance Analysis of te V-BLAST Algoit: An Analytical Appoac S. Loyka, F. Gagnon Abstact An analytical appoac to te pefoance analysis of te V-BLAST algoit is pesented in tis pape, wic is based on te analytical odel of te Ga-Scidt pocess. Closed-fo analytical expessions of te vecto signal at i-t pocessing step and its powe ae pesented. A igoous poof tat te ode at i-t step (witout optial odeing) is (n-+i) is given. It is sown tat te optial odeing is based on te least coelation citeion and tat te aftepocessing signal powe is deteined by te cannel coelation atices in a fasion siila to te cannel capacity. Index Tes MIMO, V-BLAST, ulti-antenna syste, fading I. INTODUCTION Infoation-teoetic consideations sow tat te ultipleinput ultiple-output (MIMO) counication acitectue is able to povide extaodinay ig spectal efficiencies in ic ultipat envionents, wic ae siply unattainable using conventional tecniques [-4]. Space-tie coding and/o a special signal pocessing algoit is to be ipleented at te eceive in ode to acieve at least pat of te MIMO cannel capacity. Diagonal Bell Labs Layeed Space-Tie (D-BLAST) algoit as been poposed by Foscini fo tis pupose, wic is capable of acieving a substantial pat of te MIMO capacity []. Howeve, a ig coplexity of te algoit ipleentation is its substantial dawback. A siplified vesion of te BLAST algoit is known as V-BLAST (vetical BLAST). It is capable of acieving ig spectal efficiency wile being elatively siple to ipleent [5]. Copeensive evaluation of te syste pefoance is equied because te atix wieless popagation cannel ay seveely degade te pefoance of tis algoit [6-8]. Soe peliinay studies including asyptotic analysis and nueical Monte-Calo siulations ave been epoted in [9]. Wile te nueical Monte-Calo appoac is useful fo any viewpoints, te analytical appoac povides deepe insigt and copeensive undestanding of te key points in te algoit opeation. In tis pape, we develop a unified analytical appoac to te analysis of te V-BLAST algoit opeation based on soe geneal geoetical ideas. Tis appoac is based on te closed-fo analytical odels of te key V-BLAST and Scool of Infoation Tecnology and Engineeing (SITE) Univesity of Ottawa, 6 Louis Pasteu, Ottawa, Ontaio, Canada KN 6N5 (e-ail: segey.loyka@ieee.og) Depatent of Electical Engineeing, Ecole de Tecnologie Supeieue 00, Note-Dae St. West, Monteal (Quebec), H3C K3, Canada associated syste coponents - intefeence nulling fo yet to be detected sybols (Ga-Scidt otogonalization pocess), intefeence subtaction fo aleady detected sybols, te optial odeing pocedue (based on te afte pocessing SN), optial (axiu atio o siila) cobining, and a statistical (coplex Gaussian) odel of te atix wieless popagation cannel. In paticula, we deive closed-fo analytical expessions fo te signal and noise vectos at eac pocessing step fo wieless cannel wit te geneal coelation atix. Based on tese esults, we give a igoous poof tat te ode at te i-t pocessing step is n-+i (wee n and ae te nube of x and Tx antennas coespondingly) fo uncoelated ayleig cannel and if no optial odeing is used. At te oent, we ae not able to analyze analytically te ode wen te optial odeing is ipleented, but nueical Monte-Calo analysis sows tat te effect of optial odeing is appoxiately to incease te afte pocessing SN by few dbs ate tan to incease te ode (as one would intuitively expect based on te selection cobining aguent). Note tat te analysis esults above allows one to find te outage cuves (i.e., fade level vesus outage pobability) and, ence, to estiate te aveage bit eo ate (BE), in soe cases analytically as well. II. V-BLAST ALGOITHM Te V-BLAST algoit as been discussed in details elsewee [5,9]. Hee we descibe its ain points fo copleteness and in ode to intoduce notations. Te ain idea of te BLAST acitectue is to split te infoation bit stea into seveal sub-steas and tansit te in paallel using a set of Tx antennas (te nube of Tx antennas equals te nube of sub-steas) at te sae tie and fequency. At te x side, eac x antennas sees all te tansitted signals, wic ae ixed due to te natue of te wieless popagation cannel. Using appopiate signal pocessing at te x side, tese signals can be unixed so tat te atix wieless cannel is tansfoed into a set of vitual paallel independent cannels (povided tat utltipat is ic enoug). Te following basic assuptions ae eployed: Te cannel is ando, quasistatic (i.e. fixed fo evey fae of infoation bits but vaying fo fae to fae), fequency independent (i.e., negligible delay spead) and wit coplex AWGN /0/$ IEEE. 5-
2 Te Tx signal vecto is copised of individual sybol sub-steas. No space-tie coding is eployed. Te noise vecto is copised of independent AWGN coponents wit equal vaiance. Te Tx signals, noise and cannel gains ae independent of eac ote Pefect cannel knowledge is assued to be available at te eceive. Tee is no pefoance degadation due to synconization and tiing eos. Te eceived signal vecto can be pesented in te following coplex baseband vecto fo [9]: wee [ ] T = Hq + n () q = q q is te tansitted sybol vecto, H is te cannel atix (i.e., te atix of coplex tansfe factos fo eac Tx to eac x antenna), and [ v v ] T n = n is te noise vecto. Pesenting te H, wee i is a colun vecto of tansfe factos fo i-t Tx antenna to all x antennas, te eceived signal can be pesented as: cannel atix in a colun-wise way, = [ ] = i iqi + n () Te V-BLAST pocessing begins wit te st Tx sybol and poceeds in sequence to te -t sybol. Wen te optial odeing pocedue is eployed, te Tx indexing is canged pio to te pocessing. Te ain steps of te V-BLAST pocessing (detection) algoit ae as follows [5,9]:. Te intefeence cancellation step: at te i-t pocessing step (i.e., wen te signal fo te i-t tansitte is detected) te intefeence fo te fist i- tansittes can be subtacted based on te estiations of te Tx sybols (wic ae actually assued to be eo-fee) and te knowledge of H.. Te intefeence nulling step: based on te knowledge of te cannel atix, te intefeence fo yet-to-bedetected sybols can be nulled out using te Ga- Scidt otogonalization pocess (applied to te colun vectos of H). 3. Te optial odeing pocedue: te ode of sybol pocessing is oganized accoding to tei afte-pocessing SNs in te deceasing ode (i.e., te sybol wit igest SN is detected fist). Te intefeence cancellation step can be expessed ateatically in a staigtfowad way [9]. Te eceived signal afte te cancellation at te i-t step is: wee sybols. i i' = ˆ j jq = j (3) q ˆ j ae te estiations of te aleady-detected { } i i i Figue. Geoetic illustation of te intefeence nulling out step: te eceived vecto (afte te intefeence cancellation) is decoposed into otogonal and paallel coponents wit espect to te space spanned by { } Te intefeence nulling step is based on te Ga- Scidt otogonalization pocedue, wic builds a set of otogonal vectos fo a set of linealy-independent vectos. At tis stage, we assue tat i ae linealy independent (otewise te V-BLAST algoit ust be odified taking into account all te linealy dependent colun vectos and deceasing te nube of independent bit sub-steas). Using te closed fo analytical expession fo te Ga-Scidt pocess [0, p.4] and afte soe ateatical developent (see Appendix fo details), we aive to te following expession of te eceived vecto afte intefeence nulling out and cancellation at te i-t step: q i i i i i i '' = wee i = j ji, eans deteinant wen n applied to a atix, i = i i, i j = η * k= kiηkj, * denotes coplex conjugate, and is te noalized cannel coelation atix built on[ ], (4) III. ANALYSIS OF THE V-BLAST ALGOITHM Fo te sake of notational siplicity, we fist descibe all te steps witout te noise contibution ( n = 0 ), wic is added to te analysis late. [, ] i i. + + = Te signal powe is siply expessed as: (5) 5-
3 i'' = qi i [, i] Fo tis esult and using (), it is staigtfowad to obtain a bit eo ate. It is instuctive to conside te case of =. At te fist pocessing step one obtains: ( ) (6) '' = q (7) wee =. Hence, te eceived powe and, consequently, SN, is deteined by te total eceived powe fo te st Tx antenna (tansitting te st bit stea), e.g. q, and by te noalized cannel coelation coefficient. We would like to epasize te siilaity of te esults above to te analogous esults on te MIMO cannel capacity [6,7], wic is also deteined by te cannel coelation atix (especially fo te case of x MIMO acitectue, i.e. []). One could intuitively conclude fo (7) tat te ode is n because of, wic actually eans n-t ode axiu-atio cobining (MC). Howeve, as we pove late, te effect of te last facto in (7) is suc tat te actual ode is n-. Let us now conside te optial odeing pocedue. To sepaate te effect of te tansitted sybol powe (i.e., q ) and of te noise powe fo te effect of te popagation cannel, we assue tat all q i ae equal (i.e., constant aplitude odulation) and all pe-banc noise powes ae also equal. If te -st Tx sybol is detected fist, ten te afte-pocessing powe is given by (7). If te -nd Tx sybol is detected fist, ten (7) sould be canged to: ( ) '' = q (8) Wen te noise powe is equal in all bances, te aftepocessing SN is popotional to te eceived sybol powe. Ten te optial odeing is to detect fist te sybol wit te igest i, i.e. te sae as fo te selection cobining. Howeve, as we sow late, tis does not esult in te incease of te ode (due to te last facto in (7) and(8)). Let us conside te optial odeing at te -st step fo abitay. Wen te i-t Tx sybol is detected fist, te signal powe afte intefeence nulling out is: Pi = qi i [] i wee is te full coelation atix (i.e., built [] i on[ ] ) and is te coelation atix built on all colun vectos except fo i. Unde te assuptions of equal q i and equal i (i.e., te sae eceived powe fo (6) evey tansit antenna), te optial odeing is to detect [] i fist te sybol wit te sallest. In fact, tis eans tat te oveall coelation aong [, i,, ] ust be igest and, consequently, te coelation between i and [, i,, ] ust be te lowest. Tus, te best odeing is to detect fist tat sybol wose colun popagation vecto as lowest coelation wit te ote vectos. Let us know conside te effect of te noise. Eq. 4 is genealized as follows: i'' = 0, i'' + n i'' (9) wee 0, i '' is given by (4) and n i '' = n (0) Using (0), te afte-pocessing noise powe at i-t step can be siply expessed as: wee ( ) Pn = n '' i i = n + i σ () σ = n j is pe-banc noise powe befoe pocessing, and is te expectation ove noise voltage. Note tat te afte-pocessing noise powe is less tan te total noise powe, wic is nσ. Tis is te consequence of te otogonal pojection pefoed by te Ga-Scidt pocess (see Fig. ). One also sould note tat te aftepocessing noise powe inceases wit i (step index), being te sallest in te st step and te sae as te total noise powe in te last step. Geoetical intepetation of te noise tansfoation duing te V-BLAST pocessing is te sae as in Fig.. IV. FADING OUTAGE CUVES AND DIVESITY ODE Based on te esults above, let us know analyze te signal fading in te V-BLAST syste. In paticula, we conside te outage pobabilities (i.e., te pobability tat te signal level is less tan te specified value) and ode (i.e., te asyptotic slope of te outage pobability cuve). We assue tat te cannel gains (i.e., te coponents of H) ae i.i.d. coplex Gaussians wit zeo ean and unit vaiance (i.e., we conside only te cannel vaiation due to ultipat and ignoe te absolute popagation loss and lagescale vaiation due to sadowing). Fist, we ignoe te 5-3
4 optial odeing pocedue and pove tat te ode at te i-t step is (n-+i). To deonstate te ain idea of te poof, let us conside fist te case of n==, i.e. H= [ ]. To be specific, we assue tat te -st Tx sybol is detected fist. Te intefeence nulling out can be expessed is a geneal atix fo: = Q () wee Q is an otogonal pojection atix, wic pojects to te diection otogonal to. Substituting () into (), one obtains (since we ae inteested in te eceived signal powe only, we ignoe noise in tis section): = qq (3) Tis eans tat te signal afte intefeence nulling out is popotional to tat pat of wic is otogonal to, see Fig., and te signal powe ~. e ψ Figue. Geoetical epesentation of intefeence nulling out: decoposition of into and. But te vecto agnitude is not affected by otation on an as a wole on angle ψ abitay angle. We otate [ ] so tat is paallel to e :, = 0. Tis can be expessed as: e i = A i (4) wee A is te otation atix, wic satisfies to (pesevation of lengt): + + A A = A A (5) wee + denotes conjugate tanspose. Using (4), one obtains: =,. It is staigtfowad to sow using (5) tat te coponents of as te sae distibution as te coponents of (note tat ψ is independent of ), i.e. i.i.d. coplex Gaussians wit unit vaiance. Hence, is ci-squaed ando vaiable wit two degees of feedo, ~ χ. Te sae is tue fo te signal powe. Tus, te ode in te -st step is one. Te siila consideation fo abitay n leads to te conclusion tat ~ χ( n ) (siply because as n- non-zeo coponents afte otation) and te ode is (n-). Te case of abitay is soewat oe coplex oweve staigtfowad to conside in te siila way. Fist, we otate te set [ ] as a wole so tat becoes paallel to e. In te second otation we keep fixed (i.e., a otation aound e axis) and position into [ e e ] plane. Te otations ae continued until is positioned into [ ee3 e ] ypeplane. Afte te otation, as (n-+) non-zeo coponents. Evey suc otation peseves te distibution of te coponents. Hence, ~ χ( n + ) and te ode is (n-+). Siila consideation fo te i-t step leads to te conclusion tat i ~ χ ( n + i ) and te ode is (n-+i). Note tat te lowest ode is at te -st step and te igest is at te last (i.e., n). Wen n=, no is obtained at te -st step. Unfotunately, we ae not able at te oent to analyze analytically te optial odeing pocedue due to ateatical coplications. Tus, we use nueical Monte- Calo siulations. Fist, te V-BLAST algoit outage cuves ave been siulated witout te optial odeing. No diffeence as been obseved between te analytical esults above and te Monte-Calo siulations (tus, te esults ae not sown ee), wic validates te analytical esults. Secondly, te V-BLAST outage cuves ave been siulated wit te optial odeing pocedue. Soe of te esults ae pesented in Fig Tey deonstate tat te effect of te optial odeing fo a odeate nube of antennas is to incease signal powe (and SN) ate tan to incease te ode. Howeve, as sown on Fig. 5, it is difficult to obseve tis tendency fo outage pobabilities ige tat appoxiately 0-4 wen n = 4. Te st and nd step cuves ae ixed in tis egion. V. CONCLUSIONS Using a closed-fo odel of te Ga-Scidt pocess, we ave developed an analytical appoac to te pefoance analysis of te V-BLAST algoit. In paticula, closed-fo analytical expessions ave been pesented fo te signal and noise vectos at i-t pocessing step. Te afte-pocessing signal powe is deteined by te cannel instantaneous coelation atices (in te sae fasion as te cannel capacity is). Te optial odeing is poved to be equivalent to te least coelation citeion. Pefoing te statistical analysis analytically fo ayleig uncoelated cannel, we ave poved tat te ode 5-4
5 Outage pobability st step MC: no nd step MC: nd ode Fade level, db Figue 3. Outage pobability cuves of te V-BLAST algoit fo n=. Outage pobability Fade level, db Figue 4. Outage pobability cuves of te V-BLAST algoit fo n=3. Outage pobability MC: nd ode st step st step MC: 3d ode nd step nd step MC: 3d ode MC: 4t ode Fade level, db Figue 5. Outage pobability cuves of te V-BLAST algoit fo n=4. at i-t pocessing step is (n-+i), povided tat no optial odeing is used. Nueical Monte-Calo siulations validate te analytical esults above and allow us to analyze te BLAST pefoance wit te optial odeing. Te effect of te optial odeing fo a odeate nube of antennas is appoxiately to incease te afte pocessing SN ate tan to incease te ode. Te sae is tue fo a lage nube of antennas and lowe outage pobabilities. Unfotunately, we ae not able at te oent to analyze analytically te outage pobability of te V-BLAST wit optial odeing due to te absence of an appopiate ateatical tecnique. Ode statistics ae usually eployed fo a cobining (o space-tie coding wit full ) analysis in a siila situation. Howeve, tis appoac does not apply diectly to te V-BLAST wit optial odeing due to ige diensionality of te poble. Hence, soe extension of te ode statistics appoac is equied to analyze analytically te optial odeing V-BLAST. In te pesent pape, we eployed a nueical Monte-Calo appoac to tis poble. VI. EFEENCES [] G.J. Foscini, M.J. Gans, On Liits of Wieless Counications in a Fading Envionent wen Using Multiple Antennas, Wieless Pesonal Counications, vol. 6, No. 3, pp , Mac 998. [] G.J Foscini, Layeed space-tie acitectue fo wieless counication in a fading envionent wen using ultiple antennas, Bell Lab. Tec. J., vol., N., pp. 4-59, 996. [3] I.E. Telata, "Capacity of Multi-Antenna Gaussian Cannels," AT&T Bell Lab. Intenal Tec. Meo., June 995 (Euopean Tans. Teleco., v.0, N.6, Dec.999) [4] G.G. ayleig, J.M. Cioffi, Spatio-Tepoal Coding fo Wieless Counications, IEEE Tans. Coun., v.46, N.3, pp , 998. [5] G.D. Golden, G.J. Foscini,.A. Valenzuela, P.W. Wolniansky, Detection Algoit and Initial Laboatoy esults Using V-BLAST Space-Tie Counication Acitectue, Electonics Lettes, vol. 35, No., pp.4-6, 7 t Januay 999. [6] S.L. Loyka, J.. Mosig, Cannel Capacity of N-Antenna BLAST Acitectue, Electonics Lettes, vol. 36, No.7, pp , Ma [7] S.L. Loyka, Cannel Capacity of MIMO Acitectue Using te Exponential Coelation Matix, IEEE Counicatons Lettes, v.5, N. 9, pp , Sept. 00 [8] S. Loyka, A. Kouki, Te Ipact of Coelation on Multi-Antenna Syste Pefoance: Coelation Matix Appoac, 00 IEEE Veicula Tecnology Confeence, Atlantic City, USA, Oct. 7-, pp [9] G.J Foscini et al, Siplified Pocessing fo Hig Spectal Efficiency Wieless Counication Eploying Multi-Eleent Aays, IEEE Jounal on Selected Aeas in Counications, v. 7, N., pp , Nov [0] F.. Gantae, Teoy of Matices, Nauka, Moscow, 988 (in ussian). [] S.L. Loyka, Cannel Capacity of Two-Antenna BLAST Acitectue, Electonics Lettes, vol. 35, No. 7, pp. 4-4, 9t Aug VII. APPENDIX A closed-fo analytical odel of te Ga-Scidt pocess is given in [0, pp. 3-]. Let us conside te i-st pocessing step. Assuing tat te fist (i-) sybols ae detected witout eos and te intefeence cancellation is accoplised, te eceived vecto at tis step is: 5-5
6 i i' = k kqk = k i kq = = k (A) Te otogonal pojection of i ' into te space spanned by { } is given by [0, p. 4]: = i, i i ' ' 0 Te coponent of i ' otogonal to { } is i, = i' i, = i' i' i' (A) (A3) Taking into account (A), we note tat te last ow in te nueato deteinant in (A3) includes coponents popotional to -st to (-i)-t ows, wic can be dopped out because tey do no affect te deteinant value. Tus, te only coponent of i ' tat gives contibution to te deteinant is i q i. Consequently, (A3) educes to: i, = q i i i i i (A4) elocating te last ow to te top position and te igt colun to te left position, one obtains: q i i i i i i, = Te signal powe is: (A5) i, = i, i (A6) Substituting (A6) into (A5), we obtain: ii i i qi i i i, = (A7) i We note tat te fist colun in te nueato deteinant in (A7) includes coponents popotional to -nd to ()-t coluns, wic can be dopped out. Consequently, (A7) educes to: i i qi i i i, = = (A8) i qi i [, i] 5-6
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