MIMO Ricean Channel Capacity

Transcription

1 MIMO Ricean Channel Capaciy Guillaume Lebrun, Michael Faulkner Ausralian Telecommunicaion CRC Cenre for Telecom Micro-elecronics Vicoria Universiy, Melbourne, Ausralia Mansoor Shafi Telecom New Zeal Ld 9- Tory Sree Wellingon, New Zeal Peer J Smih Dep of lecrical Compuer ng Universiy of Canerbury Privae Bag 800 Chrischurch, New Zeal Absrac This paper presens asympoic bounds limis for he ergodic channel capaciy of MIMO sysems under Ricean channel condiions I is shown ha he ergodic capaciy per dimension decreases as he K facor increases in value approaches a value equal o ha of he underlying scaering channel when he number of anennas are large The accuracy of he bounds is verified by simulaions In addiion, a variey of resuls for he MIMO Ricean channel are brough ogeher o give an overview of he curren knowledge in his area I INTRODUCTION Since he work of Foschini [1] Telaar [], here has been inense research aciviy in he area of MIMO sysems Mos of his research effor has been focussed on he flafading Rayleigh channel, which corresponds o a wireless propagaion environmen where he number of scaerers is large I is now well undersood ha he capaciy of he MIMO Rayleigh channel increases linearly wih he number of anennas for a fixed raio of ransmi o receive anenna numbers Therefore, i is convenien o define a normalized capaciy, he capaciy of he channel divided by he minimum of he number of anennas a he receiver he number of anennas a he ransmier I has been repored recenly ha he sardized Rayleigh channel capaciy ends o a Gaussian rom variable as he number of ransmi receive anennas ends o infiniy [] Furher, he capaciy disribuion is close o Gaussian even for small anenna numbers A good summary of asympoic resuls for he independen Rayleigh case is given in [] In his paper we consider he ergodic capaciy for he more general case of a Ricean channel In he Ricean case, he fla-fading channel is composed of a Line Of Sigh (LOS componen a Rayleigh componen The choice of he Ricean K-facor varies he Ricean channel from a Rayleigh channel (K = db o a pure LOS channel (K =+ db I has been shown [] ha he MIMO capaciy decreases wih increasing power of he LOS componen (he K-facor Relaed work is found in [] [7] The ouline of he paper follows: he sysem model is described in Secion II The capaciy of LOS Rayleigh channels is discussed in Secion III Secion IV presens he main conribuion of his paper: we show ha he ergodic normalized capaciy of he Ricean MIMO channel approaches he corresponding normalized capaciy of he underlying scaering channel when he anenna numbers are large Simulaion resuls are presened o confirm he accuracy of his resul II SYSTM MODL Consider a single-user MIMO sysem Transmission is over a fla-fading Ricean channel wih anennas a he ransmier r anennas a he receiver If x is a vecor of he inpu symbols (x C, H he channel marix (H C r, n a vecor of addiive whie Gaussian noise (AWGN on he receiving anennas (n C r, he vecor of received symbols can be expressed as y = Hx + n (1 In Ricean fading he elemens of H are non-zero mean complex Gaussians Hence we can express H in marix noaion as [] H = ah sp + bh sc ( where he specular scaered componens of H are denoed by superscrips, a>0, b>0 a + b =1 The enries of H sc =(h i,j are independen idenically disribued (iid complex Gaussian rom variables wih zero mean uni magniude variance A common model for H sp is H sp = v r v [] where v r v have modulus one elemens are he specular array responses a he receiver he ransmier respecively Here, we assume H sp is an r marix of uni enries since his definiion gives exacly he same disribuion for he eigenvalues of HH as he sard model The superscrip denoes he ranspose conjugae The Ricean K-facor is defined as 10log 10 (a /b db Assuming equal power uncorrelaed sources he capaciy is C = log ( I r + ρ HH, ( where I r is he ideniy marix of dimension r, ρ is he SNR on each receiving anenna denoes he deerminan of a marix The ergodic capaciy is commonly defined as he expeced value of C in ( The normalized capaciy is defined as he ergodic capaciy divided by m where m Alhough his paper considers he baseline case of equal power ransmission, more complex sysems migh use channel informaion o vary hese powers according o some performance crierion 99

2 III CAPACITY OF RAYLIGH AND LOS CHANNLS In his Secion we summarize resuls on he exreme cases of a pure LOS channel a pure Rayleigh channel Some new resuls are also given o complee he summary A Pure LOS Channel : K =+ In general, a MIMO LOS channel has a capaciy of C(K =+,,r,ρ = log (1 + ρr ( Since he channel is no rom, he capaciy is fixed he ergodic capaciy he capaciy are equal I should be noed ha he capaciy does no depend on he number of ransmi anennas, only increases logarihmically wih he number of receive anennas In he special case = r =1, he channel reduces o a Single Inpu Single Oupu (SISO Addiive Whie Gaussian Noise (AWGN channel B Pure Rayleigh Channel : K =, r =1or =1 For he Rayleigh channel, H i,j is a χ variae (chisquared variae wih wo degrees of freedom bu normalized so ha ( H i,j =1, where ( denoes he expecaion denoes he absolue value For one ransmi anenna, he channel capaciy is [1] C(K =,=1,r,ρ = log (1 + ρχ r, ( using one receive anenna he channel capaciy is [1] Noice ha C(K =,,r =1,ρ = log (1 + (ρ/χ ( (1 + (ρ/ χ = (1+ρ (1 + ρχ r = (1+ρr, log ( is a convex funcion, ha is z >0 (log (z log ((z Therefore (C(K =,,r =1,ρ (C(K =+,,r =1,ρ (8 (C(K =,=1,r,ρ (C(K =+,=1,r,ρ, (9 Hence, for a Single Inpu Muliple Oupu (SIMO or Muliple Inpu Single Oupu (MISO channel, he ergodic capaciy is higher in a LOS case han in a Rayleigh case (see Fig 1 C K =, r, /r = α When H is Rayleigh (K = he number of anennas is large, he normalized capaciy can be approximaed by a Gaussian rom variable [] Suppose r, wih /r = α, hen he mean is given by [] (C/m = (7 (log (w + ρ+ (1 α log (1 w w α ln max(1,β (10 where m, β 1/α, w ± (w ± w /α/ (11 The variance of C is also given in [] as, wih w 1+ 1 α + 1 ρ (1 σ C = log e log 1 q p β (1 q ρ (β 1 1/ρ + (β 1 ρ +β/ρ p ρ (1 β 1/ρ + (1 β ρ +/ρ (1 From hese resuls, i is obvious ha he mean capaciy grows linearly wih he number of anennas IV ASYMPTOTIC CAPACITY OF A RICAN CHANNL While he capaciies of LOS Rayleigh channels are well undersood, he capaciy of he Ricean channel is no sraighforward since capaciy is no a linear operaor Some exac resuls are now emerging for finie numbers of anennas [8] bu in his Secion we will look a he limiing case where r, /r = α Since Rayleigh capaciy grows linearly wih m LOS capaciy only grows logarihmically, i is inuiively obvious ha he normalized Ricean ergodic capaciy will approach ha of he underlying Rayleigh channel when he number of anennas (, r grows large Also he Ricean ergodic capaciy should be greaer han ha of he underlying Rayleigh channel Neiher of hese resuls appear o be available so we prove hem in his Secion To begin, noe ha log I r + ρ HH = log I r + b ρ Hsc (H sc + ρ F, (1 wih a, b defined by ( F is he r r hermiian marix, F = ab(h sc (H sp + H sp (H sc +a H sp (H sp (1 From Appendix I, F is a marix of maximum rank wo, wih one negaive one posiive eigenvalue The posiive eigenvalue, denoed λ 1 (F, behaves as below λ 1 (F /r a (17 Hence he posiive eigenvalue of F grows quadraically wih he number of anennas Despie his, we expec F o have a negligible effec in (1 for large numbers of anennas, since F only has eigenvalues whereas he scaering erm has m This is shown below A Lower bound We demonsrae in Appendix II ha r,, ρ, K, ( ( C(K,, r, ρ C(K =,,r,b ρ (18 90

3 7 K= Upper Bound Simulaion Lower Bound (C/ (C/ K= Fig 1 rgodic Capaciy per anenna wih /r = 1, Ricean fading SNR=0dB B Upper bound We demonsrae in Appendix III ha as, r, ( ( C(K,, r, ρ C(K =,,r,b ρ +, (19 where 0 as, r Hence, for Ricean channels ha are no pure LOS (K +, he normalized ergodic capaciy ends o he normalized ergodic capaciy of he scaering componen Hence, ( C(K,, r, ρ C Simulaion resuls ( C(K =,,r,b ρ (0 Fig 1 plos he average normalized capaciy wih α =1 for an increasing number of anennas differen K-facors As indicaed in our analysis, for =1, he capaciy of Ricean channels is higher han he capaciy of Rayleigh channels This rend is invered for, r > 1 ForK = 1000, he capaciy converges rapidly o a limi as, as indicaed in [] For oher values of K, he capaciy decreases wih he number of anennas over his range As soon as >1, he capaciy of he Ricean channel is a decreasing funcion of K Fig shows he behaviour of he normalized capaciy of a Ricean channel as he number of anennas grows large For all values of K, he normalized capaciy of he Ricean channel ends o he normalized capaciy of is scaering componen (he lower bound on he capaciy This lower bound is igher when K is smaller, for K = 1000 i is impossible o discern he simulaion from he lower bound The upper bound converges slowly o he lower bound is igh for large values of K m An explanaion for he slowness of convergence can be found in ( where i is shown ha ends o zero like log(m + ρa r/, which iself converges very slowly Alhough he upper bound is Fig rgodic Capaciy per anenna wih /r = 1, Ricean fading SNR=0dB (C/min(,r 8 7 Upper Bound Simulaion, min(,r=100 Lower Bound α Fig rgodic Capaciy per anenna as, r wih Ricean fading SNR=0dB only sricly valid for a large number of anennas (see he assumpions in (, in he simulaions i is sill correc for values of as low as 0, for K 1 Fig shows he behaviour of he normalized capaciy, for varying α, in he asympoic case of a large number of anennas ( = 100 As in Fig, he lower bound is igh for small K, whereas he upper bound is igh for large K Noe ha he ighness of he bounds appears uniform across all α values, indicaing ha he ighness depends on he raio /r, no on heir acual values The resuls demonsrae ha he upper lower bounds provide a fas reliable way o bound he ergodic capaciy of a Ricean channel for large m, wihou exensive simulaions Furhermore, depending on he K-facor, i is sraighforward o deduce which of he bounds is he ighes K= 91

4 V CONCLUSIONS The capaciy of he Rayleigh LOS channels have been sudied exensively are well known, boh for a small number of anennas in he asympoic case of a large number of anennas For a Ricean channel, he capaciy is more difficul o derive For a large number of anennas, he normalized capaciy of a Ricean channel ends o he normalized capaciy of is Rayleigh componen Precisely, he capaciy of he Ricean channel is lower bounded by he capaciy of is Rayleigh componen upper bounded by a quaniy ha ends o he capaciy of is Rayleigh componen when he number of anennas grows large The lower bound is valid for any number of anennas, depending on he choice of a consan M, he upper bound can be valid for any number of anennas, or only for a large number of anennas (in which case he upper bound is igher when he number of anennas grows large The lower bound is igher when he K-facor is smaller, whereas he upper bound is igher wih increasing K The wo bounds allow us o esimae he capaciy of a Ricean channel wihou exensive simulaions The limiing resuls for he Ricean channel are useful o observe he speed of convergence owards asympoic behavior when moderae anenna numbers are employed APPNDIX I IGNVALUS OF F We assume hroughou his Appendix ha a 0 A Singular Value Decomposiions (SVDs The marices H sp (H sp H sc (H sp can be wrien, H sp (H sp = (1 r,r (1 H sc (H sp =( h i,k i=1r,j=1r ( Boh are rank one marices have he following SVDs, H sp (H sp =( v 1 r( v 1 ( H sc (H sp =( u 1 σ( v 1 ( The singular values are r σ he singular vecors are v 1 u 1 These are defined below, v 1 = 1 r (1 r,1 ( The singular vecor u 1 is given by u 1 = x 1 / x 1, where x 1 =( h 1,k, h,k,, h r,k T ( r σ = r i=1 h i,k, (7 B igenvalues of F Using he SVDs above, we can wrie F in (1 as F = a r( v 1 ( v 1 +abσ(( v 1 ( u 1 +( u 1 ( v 1 Hence rank(f since F is he sum of wo rank 1 marices, a r( v 1 ( v 1 +abσ( v 1 ( u 1 abσ( u 1 ( v 1 By consrucion, i follows ha any eigenvecor, k,off, associaed wih he non-zero eigenvalue κ saisfies he following, { β 1,β such ha k = β 1 v 1 + β u 1 F k = κ k, (8 Subsiuing for F k equaing coefficiens in (8 gives: { β1 abσ( u 1 v 1 +β abσ + β 1 a r + β a r( v 1 u 1 =κβ 1 β 1 abσ + β abσ( v 1 u 1 =κβ (9 Defining o = v 1 u 1, solving (9 for κ gives, κ = a r+abσ(o+o ± (a r+abσ(o+o +(abσ (1 oo which defines he possibly non-zero eigenvalues of F C Asympoic eigenvalues of F (0 quaion (7 indicaes ha σ 0 (σ =r Furhermore, o 1, soforr,, (a r + abσ(o + o (abσ (1 oo a r abσ(o + o wih probabiliy 1 Hence, one soluion of (0 is posiive he oher negaive Since, all oher eigenvalues are zero we have he orederd eigenvalues denoed by λ r (F < 0=λ r 1 (F = = λ (F <λ 1 (F Taking he posive square roo in (0 gives λ 1 (F a r + abσ(o + o in he limi λ 1 (F /r a (1 APPNDIX II DRIVATION OF TH LOWR BOUND Since HH is a non-cenral complex Wishar marix we can use Barle s decomposiion [9] o give HH = b L L ( where L is upper riangular wih diagonal elemens denoed L 1,L,L r which are independen of all oher elemens We assume ha r bu he proof can easily be adaped o r> also The disribuion of L 1 is non-cenral chi-squared, L 1 χ (δ wih δ =(a /b race(h sp (H sp Forj > 1 he disribuions are cenral chi-squared, L j χ j+ Hence, we have D = I r + ρ HH = [I r [ b ρ L ] Using he Cauchy-Bine heorem gives D = γ A γ A γ = γ I r b ρ L ] ( A γ, ( 9

5 b where A γ is an r r submarix of [I ρ r L ] γ is a subse of r columns from (1,,,r Now we spli he summaion ino wo pars, over γ 1 where he deerminans A γ1 do no involve L 1 over γ where he deerminans A γ do involve L 1 Hence D = γ 1 A γ1 + γ A γ ( The only choice of columns which gives deerminans involving L 1 are hose where column r +1 is seleced column 1 is omied Hence he A γ marices are of he form 0 0 A γ = D γ1 0 0 b ρ L D γ ( Hence A γ = b ρ L 1 D γ where D γ =[D γ1 D γ ] D = γ 1 A γ1 + b ρ L 1 γ D γ = X+L 1Y xacly he same analysis holds for he Rayleigh case, excep L 1 χ To summarize, D Ricean = X + χ (δy D Rayleigh = X + χ Y (7 where X, Y are posiive rom variables wih X, Y independen of he χ variables Hence (C(H = (log (X + (log (1 + χ (δy/x (C(bH sc = (log (X + (log (1 + χ Y/X (8 Now χ (δ is sochasically greaer han χ Hence (f(χ (δ (f(χ for any increasing funcion f (C(H (C(bH sc as required Defining APPNDIX III DRIVATION OF TH UPPR BOUND A = I r + b ρ Hsc (H sc, (9 we have he normalized capaciy as C = 1 log ( A + F = 1 r log ( λ i (A + F, (0 i=1 where F = ρ F λ i(a + F are he eigenvalues of he hermiian posiive definie marix A + F, ordered so ha 0 λ r (A + F λ 1 (A + F Combining Weyl s heorem [10] resuls from Appendix I leads o λ r (A + F λ r 1 (A+λ ( F =λ r 1 (A λ r 1 (A + F λ r (A+λ ( F =λ r (A λ (A + F λ 1 (A+λ ( F =λ 1 (A λ 1 (A + F λ 1 (A+λ 1 ( F (1 Therefore, C 1 log (λ r 1 (A λ 1 (A(λ 1 (A+λ 1 ( F = 1 log ( r i=1 λ i(a + 1 log ( λ1(a+λ1( F λ r(a ( Now wrie = 1 log (, since λ j (A 1 for any j, λ 1 (A+λ 1 ( F, ( λ r (A 1 log (λ 1 (A+ ρ λ 1(F ( I is known ha he eigenvalues of A are bounded as r, [11] Therefore, M such ha λ 1 (A M when, 1 log (M + ρr(λ 1 (F /(r 0, ( since λ 1 (F /(r a This concludes he demonsraion From [11] we know ha λ 1 (A 1+b ρ(1 + / max(, r, ( as, r wih /r = α This provides he smalles value for M ha can be used gives he bound ha we use in he simulaions RFRNCS [1] GJ Foschini, MJ Gans, "On limis of wireless communicaion in a fading environmen when using muliple anennas," Wireless Personal Communicaions, vol, no, pp 11-, March 1998 [] I Telaar, "Capaciy of muli-anenna gaussian channels," uropean Transacions on Telecommunicaions, vol 10, no, pp 8-9, 1999 [] PJ Smih M Shafi, "On a Gaussian approximaion o he capaciy of wireless MIMO sysems," Proc I Inernaional Conference on Communicaions (ICC00, pp 0-10, New York, April 8-May, 00 [] Biglieri Giorgio Tarico, "Large-Sysem Analysis of Muliple- Anenna Sysem capaciies," Journal of Commiunicaions Neworks, vol, No, pp 9-10, June 00 [] PF Driessen G J Foschini, "On he capaciy formula for mulipleinpu-muliple-oupu wireless channels: a geomeric approach," I Trans Commun, vol 7, No, pp 17-17, Feb 1999 [] F R Farrokhi, G J Foschini, A Lozano RA Valenzuela, "Linkopimal space-ime processing wih muliple ransmi receive anennas," I Communicaions Leers, vol, no, pp 8-87, March 001 [7] SK Jayaweera HV Poor, "On he capaciy of muli-anenna sysems in he presence of Ricean fading," Proc I Vehicular Technology Conference (VTC00, pp , Vancouver, Sepember -8, 00 [8] PJ Smih LM Garh, xac capaciy disribuion for Dual MIMO sysems in Ricean fading, in press I Communicaions Leers [9] RJ Muirhead, Aspecs of Mulivariae Saisical Theory, NewYork, John Wiley & Sons Inc, 198 [10] RA Horn CR Johnson, Marix Theory, Cambridge, Cambridge Universiy Press, 198 [11] A delman, igenvalues Condiion Numbers of Rom Marices, PhD hesis, MIT,