PAPER On the Capacity of MIMO Wireless Channels

Transcription

1 IEICE TRANS. COMMUN., VOL.E87 B, NO.3 MARCH PAPER On he Capaciy of MIMO Wieless Channels Hyundong SHIN a, Suden Membe and Jae Hong LEE, Nonmembe SUMMARY In his pape, we pesen a new closed-fom fomula fo he egodic capaciy of muliple-inpu muliple-oupu MIMO wieless channels. Assuming independen and idenically disibued i.i.d. Rayleigh fla-fading beween anenna pais and equal powe allocaion o each of he ansmi anennas, he egodic capaciy of such channels is expessed in closed fom as finie sums of he exponenial inegals which ae he special cases of he complemenay incomplee gamma funcion. Using he asympoic capaciy ae of MIMO channels, which is defined as he asympoic gowh ae of he egodic capaciy, we also give simple appoximae expessions fo he MIMO capaciy. Numeical esuls show ha he appoximaions ae quie accuae fo he enie ange of aveage signal-o-noise aios. key wods: channel capaciy, muliple anennas, MIMO channels 1. Inoducion In eseach aeas on wieless communicaions, mulipleinpu muliple-oupu MIMO sysems equipped wih muliple anennas a boh ansmi and eceive ends have ecenly dawn consideable aenion in esponse o he inceasing equiemens on daa ae and qualiy in adio links [1] [9]. Mulipah signal popagaion has long been hough as an impaimen limiing he sysem capaciy and eliable communicaion in wieless channels, ogehe wih he consain of powe and bandwidh. In MIMO sysems, his mulipah popagaion due o he ich scaeing in wieless channels is used o impove achievable daa ae and link qualiy. Recen seminal wok in [1] and [2] has shown ha he use of muliple anennas a boh he ansmie and he eceive significanly inceases he infomaion-heoeic capaciy fa beyond ha of single-anenna sysems. As he numbe of anennas a boh he ansmie and he eceive ges lage, he capaciy inceases linealy wih he minimum of he numbe of ansmi and eceive anennas fo a given fixed signal-o-noise aio SNR [1] [3], even if he fades beween anenna pais ae coelaed [4]. In paicula, Telaa [1] deived he analyic expession fo he egodic capaciy of independen and idenically disibued i.i.d. Rayleigh fla-fading MIMO channels in inegal fom involving he Laguee polynomials and povided a look-up able obained by numeically evaluaing he inegals o find he egodic capaciy fo diffeen numbes of ansmi and eceive anennas. To dae, and o he bes knowledge of he Manuscip eceived Augus 26, Manuscip evised July 28, The auhos ae wih he School of Elecical Engineeing, Seoul Naional Univesiy, Seoul , Koea. a shd71@sun.ac.k auhos, no closed-fom expessions fo he egodic capaciy of he MIMO channel ae available fo a finie numbe of anennas alhough hose fo he channel wih muliple anennas a only one end of link have been epoed in pevious wok [10] [12]. The main objecive of his pape is o exend he analysis in [1] o obain closed-fom expessions fo he egodic capaciy of i.i.d. Rayleigh fla-fading MIMO channels. We also inoduce he capaciy ae whichisdefinedashe gowh ae of egodic capaciy wih espec o he numbe of anennas. In andom maix heoy RMT [16] [19], i is well known ha he eigenvalues of a lage class of andom maix ensembles have fewe andom flucuaions as he maix dimension ges lage, i.e., he andom disibuion of he eigenvalues conveges o a deeminisic disibuion in he limi of infinie maix sie. Fom hese esuls of he RMT, pevious sudies [1], [4] have shown ha he asympoic capaciy ae of MIMO channels conveges o a noneo consan deemined by he aveage SNR and he limiing aio beween he numbes of ansmi and eceive anennas. Using his asympoic capaciy ae of MIMO channels, we pesen vey accuae appoximaion fomulas fo he MIMO egodic capaciy. We also show ha he capaciy aes of single-inpu muliple-oupu SIMO and muliple-inpu single-oupu SIMO channels appoach eo asympoically as he numbe of anennas ends o infiniy, in conas o MIMO channels. The emainde of his pape is oganied as follows. The nex secion povides a bief eview on he egodic capaciy fo MIMO channels. In Sec. 3, we deive he closedfom fomula fo he MIMO egodic capaciy. In Sec. 4, we define he capaciy aes of MIMO, MISO, and SIMO channels and analye hei asympoic behavio as he numbe of anennas ends o infiniy. In Sec. 5 we pesen appoximae fomulas fo he MIMO egodic capaciy and conclusions ae pesened in Sec Capaciy of MIMO Channels In his secion, we biefly eview he capaciy fomula fo MIMO channels. Conside a poin-o-poin communicaion link wih ansmi and eceive anennas. Thoughou he pape we efe o α = min, } and β = max, }, and esic ou analysis o he fequency-fla fading case. The oal powe of he complex ansmied signal veco x C is consained o P egadless of he numbe of anennas, namely

2 672 IEICE TRANS. COMMUN., VOL.E87 B, NO.3 MARCH 2004 E [ x x ] P 1 whee he supescip denoes he anspose conjugae. A each symbol ineval, he eceived signal veco y C is given by y = Hx + n 2 whee H C is a andom channel maix and n is a complex dimensional addiive whie Gaussian noise AWGN veco wih i.i.d. ciculaly symmeic Gaussian componens, E [ nn ] = σ 2 ni. The enies H ij, i = 1, 2,, and j = 1, 2,,, ofh ae he complex channel gains beween ansmi anenna j and eceive anenna i, which ae modeled as i.i.d. complex Gaussian andom vaiables wih eo mean and uni vaiance, i.e., H ij CN0, 1. In his case, he aveage SNR a each eceive anenna is equal o = P/σ 2 n. Fuhemoe, we assume ha he channel is pefecly known o he eceive bu unknown o he ansmie. When he ansmied signal veco x is composed of saisically independen equal powe componens each wih a ciculaly symmeic complex Gaussian disibuion, he channel capaciy unde ansmi powe consain P is given by [1], [2] C = log 2 de I + HH bis/s/h. 3 The egodic mean capaciy of he andom MIMO channel, which is he Shannon capaciy obained by assuming i is possible o code ove many independen channel ealiaions, is evaluaed by aveaging C wih espec o he andom maix channel H, namely [1] C, = E [ log 2 de I + HH ]. 4 Fo some specific models of he channel maix, C, can be evaluaed by saisical simulaions. Howeve, hese numeical maix calculaions may be vey lenghy, especially when he numbe of anennas is vey lage. Define HH W = < H H, hen, he andom maix W has he cenal Wisha disibuion wih paamees α and β, and is andom eigenvalues ae of gea inees in mulivaiae saisics see [16] [19], and efeences heein. Using he singula value decomposiion heoem and he geneal esuls fom he RMT, Telaa [1] deived he analyic fomula fo he egodic capaciy of MIMO channels in 4 as follows: C, = α 0 5 log λ/ p λ λ dλ 6 whee p λ λ is he disibuion of a andomly seleced eigenvalue of W. The disibuion p λ λ isgivenby[1] p λ λ = 1 α k=0 k!λ β α e λ [ L β α β α + k! k λ ] 2,λ 0 7 whee L m k is he Laguee polynomial of ode k defined as [20, Eq ] L m k = 1 k! e m dk e k+m} d k k k + m = 1 i i k i i!. 8 i=0 whee n k = n!/k!n k! is he binomial coefficien. 3. Closed-Fom Fomulas fo he Capaciy We now deive a closed-fom expession fo he capaciy of MIMO channels. Theoem 1: The egodic capaciy in bis/s/h of an i.i.d. Rayleigh fading MIMO channel wih ansmi and eceive anennas unde oal ansmi powe consain and equal powe allocaion is given by 1 C, = e / i 2l! log 2 e 2 2k i l!i! β α + i! 2k 2l 2β 2α + 2l β α + l! k l 2l i β α+i } E j+1 whee E n is he exponenial inegal of ode n defined by [20, p.xxxv] E n = 1 9 e u u n du, n = 0, 1,, Re[] > Poof: Fom 8 and he ideniy of [20, Eq ] [ L m k ] 2 Γk + m + i k 2l! 2k 2l k l = 2 2k k! l!γm + l + 1 L2m 2l 2 11 whee Γ is he gamma funcion, he disibuion p λ λ can be ewien as p λ λ = 1 α k=0 l=0 k!λ β α e λ [ L β α β α + k! k λ ] 2 λ β α e λ = 1 k 2l! 2k 2l k l α 2 2k l!β α + l! L2β 2α 2l 2λ k=0 l=0 = 1 1 i 2l!λ β α+i e λ α 2 2k i l!i!β α + l! 2k 2l 2β 2α + 2l k l 2l i },λ Subsiuing 12 ino 6, he egodic capaciy is wien as I n epesens he n n ideniy maix.

3 SHIN and LEE: ON THE CAPACITY OF MIMO WIRELESS CHANNELS 1 i 2l! C, = log 2 e 2 2k i l!i!β α + l! 2k 2l 2β 2α + 2l k l 2l i } ln1 + λ/λ β α+i e λ dλ 0 } } I. 13 To evaluae he inegal I in 13, we use he following esul fom [12, Appendix B]: I n µ = ln1 + n 1 e µ d, µ>0, n = 1, 2, 0 n = n 1!e µ Γ n + j,µ µ j 14 j=1 whee Γa, is he complemenay incomplee gamma funcion defined by [20, Eq ] Γa, = e u u a 1 du. 15 Accoding o 14, he inegal I is evaluaed as I = β α + i! β α+i+1 e / β α+i+1 j=1 Γ β + α i 1 + j, / / j. 16 The exponenial inegal E n of 10 is he special case of he complemenay incomplee gamma funcion, i.e., E n = n 1 Γ1 n,. 17 Applying he ideniy 17 o 16, we have β α+i I = β α + i!e / E j Inseing 18 ino 13, he egodic capaciy C, in bis/s/h can be expessed in closed fom as 9. Example 1: Conside = = n. Fom 9 wih α = β = n, he capaciy of a MIMO channel wih n anennas a boh he ansmie and he eceive is given by n 1 C n,n = e n/ log 2 e 1 i 2k 2l k l 2l i 2 2k i 2l l i n E j+1 }. 19 Example 2: Conside = n and = 1 MISO channel. Fom 9 wih α = 1andβ = n, he capaciy of a MISO channel wih n ansmi anennas is given by 673 n 1 n C n,1 = e n/ log 2 e E j Example 3: Conside = 1and = n SIMO channel. Fom 9 wih α = 1andβ = n, he capaciy of a SIMO channel wih n eceive anennas is given by n 1 1 C 1,n = e 1/ log 2 e E j Noe ha inceasing he numbe of eceive anennas fom n 1on in he SIMO channel yields an addiional capaciy advanage equal o e 1/ log 2 ee n 1/ bis/s/h and 21 is in ageemen wih he fomely known esul of he exac capaciy fomula fo Rayleigh fading channels wih ecepion divesiy if applying he ideniy 17 o [12, Eq. 40]. 4. Capaciy Raes A naual quesion o ask is How does he capaciy gow wih he numbe of anennas. We define he capaciy ae as his ae of gowh as follows. Definiion 1: Fo a MIMO channel wih ansmi and eceive anennas, he capaciy ae, denoed by R MIMO,is defined as is egodic capaciy nomalied by min, }. The asympoic capaciy ae fo he MIMO channel is defined as he asympoic behavio of R MIMO, i.e., R MIMO τ lim, / τ C, min, } 22 which implies he asympoic gowh ae of he MIMO egodic capaciy when and end o infiniy in such a way / ends o a limi τ. Fo MISO and SIMO channels, he capaciy aes, denoed by R MISO and R SIMO, ae defined as hei egodic capaciy nomalied by he numbe of eceive o ansmi anennas, especively. The asympoic capaciy aes fo he MISO and SIMO channels ae defined as he asympoic behavio of R MISO and R SIMO, i.e., and R MISO R SIMO C,1 lim lim 23 C 1,. 24 Using he limi heoem on he disibuion of he eigenvalues of lage dimensional andom maices [19], he asympoic behavio of he MIMO capaciy has been epoed in [1] and [4]. The following heoem deemines he asympoic capaciy ae fo MIMO channels and is used o esablish appoximae fomulas fo 9 in he nex secion. Theoem 2: If and end o infiniy in such a way ha / τ and le τ = τ sgnτ 1, aτ = τ 1 2, bτ = τ + 1 2,and ν = τ τ < 1 τ 1,

4 674 IEICE TRANS. COMMUN., VOL.E87 B, NO.3 MARCH 2004 hen he asympoic capaciy ae RMIMO τ isgivenby R MIMO τ = 1 2π bτ aτ log ν bτ } = τ log ν Jν, τ } + log ν τ Jν, τ 1 1 aτ d 25 log 2 e Jν, τ 26 ν whee Ju,w 1 u 2 w u w Poof: The well-known esul fom he RMT [19] says ha as β/α τ o / τ, he asympoic eigenvalue densiy of 1 αw conveges in pobabiliy o bτ p = aτ, 2π aτ bτ which is someimes called he defomed quae cicle law o Macenko-Pasu law. Fom 6 and p, he asympoic capaciy ae defined in 22 can be wien as RMIMO τ = 1 2π = 1 2π bτ aτ bτ aτ log α p d log ν p d. Noe ha he closed-fom expession 26 fo he inegal of 25 was pesened in he conex of he capaciy analysis fo code-division muliple-access CDMA sysems wih andom speading sequences [13], [14]. In [13], 26 was indiecly obained by using he fac ha he pefec successive cancele wih minimum mean-squae-eo MMSE pefileing achieves asympoically he same capaciy as he maximum-likelihood decode. On he ode hand, Rapajic and Popescu [14] deived he closed-fom expession diecly fom he inegal and poved he esuls in [13]. When =,i.e.,τ = 1, 26 educes o RMIMO C, 1 = lim = 2log log 2 e Theoem 3: Fo MISO and SIMO channels, he asympoic capaciy ae is equal o eo, ha is, RMISO = R SIMO = 0. Poof: As ges lage, 1 HH fo fixed conveges almos suely o I by he law of lage numbes. Theefoe, he egodic capaciy fo lage and fixed becomes C, E [ log 2 de I ] = log Similaly, as ges lage, 1 H H fo fixed conveges almos suely o I by he law of lage numbes, and he egodic capaciy fo fixed and lage becomes [ C, E log 2 de I + = log ]. 30 Fom 23, 24, 29 and 30, i is saighfowad ha RMISO = R SIMO = 0. Figue 1 shows he capaciy aes fo MIMO, MISO, and SIMO channels vesus he numbe of anennas n a =10 db fo he following five cases: a = n and = 2n; b = 2n and = n; c = = n; d = 1and = n SIMO; e = n and = 1 MISO. If he numbe of anennas is inceased a boh he ansmie and he eceive, i.e., he cases a, b, and c, he MIMO capaciy ae R MIMO conveges vey fas o is asympoic value accoding o Theoem 2 as n inceases. The asympoic capaciy aes fo a, b, and c ae , , and bis/s/h, especively. On he ode hand, fo SIMO and MISO channels wih n eceive o ansmi anennas, i.e., he cases d and e, especively, he capaciy ae R SIMO and R MISO decease wih n and appoach eo as n. This implies Fig. 1 Capaciy aes fo MIMO, MISO, and SIMO channels vesus he numbe of anennas n a =10 db. Thee exiss a mino ypo in [4, Eq. 9] he em 2 in28 was missed ou.

5 SHIN and LEE: ON THE CAPACITY OF MIMO WIRELESS CHANNELS ha if he numbe of anennas is inceased only a he ansmie o he eceive, he asympoic capaciy ae is equal o eo as shown in Theoem 3. Also, we see ha due o he oal ansmied powe consain, R MISO conveges o eo fase han R SIMO as he numbe of anennas inceases. 5. Appoximae Fomulas fo he Capaciy Fom he fac ha he capaciy ae fo MIMO channels wih a ceain fixed aio τ conveges vey quickly o is asympoic value RMIMO τ as he numbe of anennas inceases see Fig. 1, i is obvious ha he capaciy of MIMO channels is vey well appoximaed by a linea funcion of min, } wih a ae of RMIMO τ fo a given aveage SNR. Fo example, if = = n, he capaciy appoximaely inceases RMIMO 1 bis/s/h fo each incease in n and hus C n,n C 1,1 +n 1 RMIMO 1. Theefoe, he following appoximaions ae saighfowad. The egodic capaciy of a MIMO channel wih n anennas a boh he ansmie and he eceive in 19 is simply appoximaed by [ + 1 log J } + log J,, } log 2 e J, ]. 33 To assess he accuacy of he appoximae expessions fo he egodic capaciy of MIMO channels, we compae he appoximaions wih he exac calculaions in Figs Figue 2 shows he exac egodic capaciy of a MIMO channel wih = = n and is appoximaion fom 31 vesus aveage SNR fo vaious n. Figue 3 shows he exac egodic capaciy of a MIMO channel wih = n and = κ n and is appoximaion fom 32 vesus aveage SNR fo n = 2, 6, 10, and κ = 1.5 and 3. Figue 4 shows he exac 675 C n,n C 1,1 + n 1 RMIMO 1 1 = e 1/ log 2 ee 1 +n 1 2log log 2 e }. 31 Fo <, an appoximae fomula fo he egodic capaciy is given by C, C 1, / + 1 RMIMO / 1 1 = e 1/ log 2 e E j+1 [ + 1 log J, } + log log 2 e J, J ], } 32 whee epesens he smalles inege geae han o equal o. Similaly, when >, he egodic capaciy is appoximaed by C, C /,1 + 1 RMIMO / 1 = e / / log 2 e / E j+1 Fig. 2 Egodic capaciy and is appoximaion fo a MIMO channel wih = = n vesus aveage SNR. Fig. 3 Egodic capaciy and is appoximaion fo a MIMO channel wih = n and = κ n vesus aveage SNR.

6 676 IEICE TRANS. COMMUN., VOL.E87 B, NO.3 MARCH 2004 Fig. 4 Egodic capaciy and is appoximaion fo a MIMO channel wih = κ n and = n vesus aveage SNR. egodic capaciy fo = κ n and = n and is appoximaion fom 33 vesus aveage SNR fo he same values of n and κ as hose in Fig. 3. Fom hese plos we can see ha he appoximaions closely mach he exac egodic capaciy 9 fo he enie ange of aveage SNRs and ge moe accuae if he numbe of ansmi anennas is an inegal muliple of ha of eceive anennas and vice vesa. These closed-fom appoximaions can heefoe be safely used o pedic he capaciy of MIMO channels. 6. Conclusions In his pape we obained a closed-fom expession and an accuae appoximaion fo he egodic capaciy of i.i.d. Rayleigh fla-fading MIMO channels unde oal ansmi powe consain and equal powe allocaion. By using hese expessions, we can easily pedic he capaciy pefomance of MIMO channels wihou any numeical inegaions o saisical simulaions wih numeical maix calculaions. We also defined he capaciy ae as he gowh ae of he egodic capaciy wih espec o he numbe of anennas. The capaciy ae conveges asympoically o a ceain noneo consan as he numbe of anennas a boh he ansmie and eceive ends o infiniy, while i appoaches eo if he numbe of anennas ends o infiniy only a ansmie o he eceive. Acknowledgemen [2] G.J. Foschini and M.J. Gans, On limis of wieless communicaions in a fading envionmen when using muliple anennas, Wieless Pesonal Commun., vol.6, no.3, pp , Mach [3] G.G. Raleigh and J.M. Cioffi, Spaio-empoal coding fo wieless communicaions, IEEE Tans. Commun., vol.46, no.3, pp , Mach [4] C.-N. Chuah, D. Tse, J.M. Kahn, and R.A. Valenuela, Capaciy scaling in MIMO wieless sysems unde coelaed fading, IEEE Tans. Inf. Theoy, vol.48, no.3, pp , Mach [5] G.J. Foschini, Layeed space-ime achiecue fo wieless communicaion in a fading envionmen when using muli-elemen anennas, Bell Labs. Tech. J., vol.1, no.2, pp.41 59, [6] V. Taokh, N. Seshadi, and A.R. Caldebank, Space-ime codes fo high daa ae wieless communicaion: Pefomance cieion and code consucion, IEEE Tans. Inf. Theoy, vol.44, no.2, pp , Mach [7] J.-C. Guey, M.P. Fi, M.R. Bell, and W.-Y. Kuo, Signal design fo ansmie divesiy wieless communicaion sysems, IEEE Tans. Commun., vol.47, no.4, pp , Apil [8] S.M. Alamoui, A simple ansmi divesiy echnique fo wieless communicaions, IEEE J. Sel. Aeas Commun., vol.16, no.8, pp , Oc [9] V. Taokh, H. Jafakhani, and A.R. Caldebank, Space-ime block codes fom ohogonal designs, IEEE Tans. Inf. Theoy, vol.45, no.5, pp , July [10] W.C.Y. Lee, Esimae of channel capaciy in Rayleigh fading envionmen, IEEE Tans. Veh. Technol., vol.39, no.3, pp , Aug [11] C.G. Gunhe, Commen on esimae of channel capaciy in Rayleigh fading envionmen, IEEE Tans. Veh. Technol., vol.45, no.2, pp , May [12] M.-S. Alouini and A.J. Goldsmih, Capaciy of Rayleigh fading channels unde diffeen adapive ansmission and divesiycombining echniques, IEEE Veh. Technol., vol.48, no.4, pp , July [13] S. Vedu and S. Shamai, Specal efficiency of CDMA wih andom speading, IEEE Tans. Inf. Theoy, vol.45, no.2, pp , Mach [14] P.B. Rapajic and D. Popescu, Infomaion capaciy of a andom signaue muliple-inpu muliple-oupu channel, IEEE Tans. Commun., vol.48, no.8, pp , Aug [15] T.M. Cove and J.A. Thomas, Elemens of Infomaion Theoy, Wiley, New Yok, [16] R.J. Muihead, Aspecs of Mulivaiae Saisical Theoy, John Wiley & Sons, New Yok, [17] V.L. Giko, Theoy of Random Deeminans, Kluwe, Nowell, MA, [18] A. Edelman, Eigenvalues and Condiion Numbes of Random Maices, Ph.D. Thesis, Dep. Mahemaics, M.I.T., Cambidge, MA, May [19] J.W. Silvesein, Song convegence of he empiical disibuion of eigenvalues of lage dimensional andom maices, J. Mulivaiae Anal., vol.55, pp , [20] I.S. Gadsheyn and I.M. Ryhik, Table of Inegals, Seies, and Poducs, 6h ed., Academic, San Diego, CA, This wok was suppoed in pa by he Naional Reseach Laboaoy NRL Pogam of Koea. Refeences [1] I.E. Telaa, Capaciy of muli-anenna Gaussian channels, Euopean Tans. Telecommun. ETT, vol.10, no.6, pp , Nov./Dec. 1999, oiginally published as Tech. Memo., Bell Laboaoies, Lucen Technologies, Oc

7 SHIN and LEE: ON THE CAPACITY OF MIMO WIRELESS CHANNELS 677 Hyundong Shin eceived he B.S. degee in eleconics and adio engineeing fom Kyunghee Univesiy, Koea, in 1999, and he M.S. degee in elecical engineeing fom Seoul Naional Univesiy, Seoul, Koea, in He is cuenly woking owad he Ph.D. degee in elecical engineeing a Seoul Naional Univesiy. His cuen eseach ineess include wieless communicaion and coding heoy, ubo codes, space-ime codes, and muliple-inpu muliple-oupu MIMO sysems. He was he ecipien of he 2001 SNU/EECS Disinguished Disseaion Awad. Jae Hong Lee eceived he B.S. and M.S. degees in eleconics engineeing fom Seoul Naional Univesiy SNU, Seoul, Koea, in 1976 and 1978, especively. He eceived he Ph.D. degee in elecical engineeing fom he Univesiy of Michigan, Ann Abo, Michigan, in Fom 1978 o 1981 he was wih he Depamen of Eleconics Engineeing of he Republic of Koea Naval Academy, Jinhae, Koea, as an Insuco. In 1987 he joined he faculy of SNU whee he is cuenly a Pofesso of he School of Elecical Engineeing. He was a Membe of Technical Saff a he AT&T Bell Laboaoies, Whippany, New Jesey, fom 1991 o Fom 1992 o 1994 he seved as he Chaiman of he Depamen of Eleconics Engineeing of SNU. Fom 2001 o 2002 he seved as he Associae Dean fo Suden Affais of he College of Engineeing of SNU. Cuenly he is he Dieco of he Insiue of New Media and Communicaions of SNU. His cuen eseach ineess include communicaion and coding heoy, CDMA, OFDM, and hei applicaion o wieless and saellie communicaions. He is a membe of IEEE, IEEK, KICS, KSBE, and Tau Bea Pi.