Rate Gap Analysis for Rate-adaptive Antenna Selection and Beamforming Schemes

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1 This full text pape was pee eviewed at the diection of IEEE Communications Society subject matte expets fo publication in the IEEE Globecom 00 poceedings. Rate Gap Analysis fo Rate-adaptive Antenna Selection and Beamfoming Schemes Kathikeyan Shanmugam and Sikishna Bhashyam Depatment of Electical Engineeing Indian Institute of Technology Madas Chennai , India Abstact We analyze the asymptotic pefomance of ate adaptation fo Tansmit Antenna Selection TAS and Maximum Eigenmode Beamfoming MEB schemes in Multiple- Input Multiple-Output MIMO systems unde impefect channel state infomation CSI and feedback delay. The ate is adapted accoding to a taget outage pobability. We deive lowe and uppe bounds to this ate. We also asymptotically chaacteize the multi-step pediction eo when MMSE pediction is used to combat feedback delay. Using the bounds and the pediction eo asymptotics, we show that the ate gap fom the ideal CSI scenaio asymptotically gows logaithmically with SNR. The slope is at most the taget outage pobability. We find that when the taget outage pobability is deceased faste than an identified gowth ate and pediction eo goes to zeo, then the ate gap emains bounded. I. INTRODUCTION Seveal adaptive tansmission schemes based on channel state infomation CSI have been poposed fo Multiple- Input Multiple-Output MIMO wieless systems. Two of those schemes ae Maximum Eigenmode Beamfoming MEB [] and Tansmit antenna selection TAS []. The MEB scheme involves beamfoming along the eigen vecto coesponding to the lagest singula value of the channel matix. The implementation of the MEB scheme equies feedback of at least the beamfoming vecto assuming all othe computations ae done at the eceive. The TAS scheme involves selecting the best tansmit antenna in tems of the maximum channel nom. The TAS scheme has educed complexity and equies feedback of only the index of the tansmit antenna to be chosen. Also, the TAS scheme has been shown to achieve full divesity asymptotically [3]. Rate o powe adaptation can be employed along with the above two schemes futhe to enhance pefomance [4]. We analyze the pefomance of ate-adaptive MEB and TAS systems in the pesence of impefections in CSI. When impefections in CSI at the eceive due to estimation eos and feedback delay ae intoduced, even ate adaptation cannot always esult in an outage fee tansmission due to the mismatch between estimates at the tansmitte and the eceive. In othe wods, the tansmitte gets delayed infomation about changes in the channel while eceive has infomation about both the cuent and the past channel conditions. Tansmitte adaptation has to take place, unde this uncetainty about CSI, at the tansmitte. One way to combat this poblem is to use pediction. Pobability of outage given a fixed ate at the tansmitte has been analysed fo the MEB scheme in [5] fo vaious impefect CSI assumptions. The aveage ate of a ate-adaptive TAS scheme based on a fixed taget outage pobability has been numeically calculated in [6]. In this pape, we deive analytical esults fo ate-adaptive MEB and TAS schemes. Fist, we fomally define the ate adaptation scheme applied to the MEB and TAS systems. We unify notation fo the MEB and TAS schemes and define the egodic ate gap to be the expected diffeence between the ate with pefect CSI and ate with impefect CSI and feedback delay. We deive lowe and uppe bounds to the adapted ate with impefect CSIT and show that the ate gap has a logsnr gowth. The slope of the ate gap is uppe bounded by the taget outage pobability. Fom this, we conclude that when the outage pobability is deceased with SNR at a logsnr ate faste than, and channel pediction dives the mismatch between CSI at the eceive CSIR and CSI at the tansmitte CSIT to zeo when SNR becomes high, the ate gap emains bounded. As pat of the above analysis, we also quantify the asymptotics of the mismatch pediction eo fom noisy past estimates between CSIR and CSIT unde multi-step MMSE pediction fo the Jakes fading model fo the channel. The asymptotics fo one-step pediction with past values in noise have been chaacteised in [7], [8] fo Dopple pocesses. We extend, by analytical calculations, the esult to multi-step pediction. We obseve that the exact asymptotic vaiation of pediction eo with SNR does not have any implications fo the ate gap asymptotics as long as the pediction eo goes to zeo. The oganisation of this pape is as follows. We pesent the system model fist in Section II, followed by ate adaptation fo both schemes in Section III. Then, we deive bounds on the adapted ate fo both the schemes in Section IV. We chaacteize the asymptotics of pediction eo with SNR in Section V. In Section VI, we analyze the egodic ate gap, chaacteise it asymptotically, and pesent some numeical esults. Conclusions ae dawn in Section VII /0/$ IEEE

2 This full text pape was pee eviewed at the diection of IEEE Communications Society subject matte expets fo publication in the IEEE Globecom 00 poceedings. II. SYSTEM MODEL A MIMO system with N t tansmit antennas and N eceive antennas is consideed. We assume a block ayleigh fading channel. The coelation between diffeent blocks follows the Jakes fading model. The channel matix fo a paticula block is denoted by H and has i.i.d enties distibuted as CN0,. The eceived vecto y N is given by: y = P Hx + z whee x is the tansmitted signal vecto and z CN0,σnI. The powe used pe taining symbol is P t and the powe used pe data symbol is P d. The estimated channel at eceive is CN0,. The CSIT is denoted H t and H ij t CN0,. We distinguish between two cases: Pefect CSIR and no feedback delay: Hee H = H t and = =. Impefect CSIR and feedback delay: The feedback delay is Δ blocks. H t is obtained using a Δ-step channel pediction fom past values of H.Letρbe the enty wise coelation denoted H. H ij between H and H t. = Pt P t+σ and σ n t = p H w, whee w is the L-tap Wiene filte used. p is the coss-coelation vecto between the cuent H ij and past values with delay Δ. The following elation holds H t and H ae jointly Gaussian [5]: [ ρ H = σ H t + ] ρ σ E t whee E ij CN0,. In the MEB scheme, the beamfoming vecto which coesponds to the lagest singula value of the channel matix CSIT H t is selected. Let this beamfoming vecto be u. Then, the tansmit vecto x = ux, whee x is the tansmitted data symbol. In the TAS scheme, only one antenna is selected fo tansmission. Theefoe, x has only one non-zeo enty coesponding to the selected antenna. III. RATE ADAPTATION Rate adaptation fo the TAS scheme was consideed in [6]. Similaly, we conside ate adaptation fo MEB scheme and unify the ate gap analysis fo both schemes. The tansmission ate is chosen based on a lowe bound on the mutual infomation and a taget outage pobability. Fo the MEB scheme, the mutual infomation achievable at the eceive can be lowe bounded by [9],[0]: Ix, y/h t, H log + Γu H H H H u 3 P whee Γ= d P d σ. Fo a given H e t, let the ate to be chosen +σ n by the tansmitte be RH t. The pobability of outage fo this ate can be uppe bounded as in [6]: +μ age/h t P A<β, 4 whee A = μ H t u + Eu, μ = σ ρ ρ, and β = e RH t Γ. In ode to ensue an uppe bound on the outage pobability, the ate RH t is decided by equating the uppe bound to a fixed outage pobability and is given by: σ R 0 H t =log +Γ + μ F nc,n P,δ out, 5 whee F nc,n,δ is the invese CDF of the non-cental distibution with N degees of feedom and centality H t u.theβ coesponding to the above R 0 is denoted β 0. The outage pobability bound fo the TAS scheme is also vey simila to the bound in equation 4. The only diffeence is in the expession fo A. H t u gets eplaced by H tsel coesponding to the maximum nom column of H t at the tansmitte and Eu gets eplaced by E sel. The statistics paamete δ = μ σ t of Eu and E sel ae identical. Both ae N ciculaly symmetic complex gaussian with vaiance pe dimension. Theefoe, conditioned on the CSIT H t, the distibution of andom vaiable A is identical fo both MEB and TAS schemes. Let Ĥ denote H tsel and H t u in thei espective cases. Let Ê denote E sel and Eu in thei espective schemes. The distibution of A conditioned on H t is a non-cental chi-squaed distibution with N degees of feedom and centality paamete δ = μ Ĥ. Since the mutual infomation lowe bound outage uppe bound is used fo calculating the ate, we have >Poutage/H t. 6 Let the pefect CSI H t = H = H ate be denoted by R ideal. Fo the MEB scheme, the pefect CSI ate is: R ideal = log + SNR H, 7 whee SNR = P d /σn, and H = Hu, u is the singula vecto coesponding to the maximum singula-value of H, the actual channel matix. Since Ĥ in the impefect CSI case and H in the pefect CSI case have the same PDF, we fist define ΔR fo a given H as follows: ΔR H =R ideal H age/ H R 0 H, 8 whee R ideal H is the pefect CSI ate when H = H, R 0 H is the impefect CSI ate when Ĥ = H, and age/ H accounts fo the possibility of outage with impefect CSIT. The egodic ate gap will be E H[ΔR H], whee H has the same PDF as H and Ĥ above. Similaly, fo the TAS scheme, the pefect CSI ate is again given by 7. Howeve, hee H is the maximum channel nom ove all tansmit antennas. Again, consideing the impefect CSI case in this scheme, the vaiable Ĥ is statistically same as H. Hence, ΔR H is given by the same expession 8. Although the ate gap expession is identical fo both schemes, the vaiables involved { H, Ĥ/} ae diffeent statistically. The TAS scheme is chaacteized by the statistics of the maximum channel nom. The MEB scheme is chaacteized by the statistics of the gaussian channel matix multiplied by the singula vecto coesponding to the lagest singula value /0/$ IEEE

3 This full text pape was pee eviewed at the diection of IEEE Communications Society subject matte expets fo publication in the IEEE Globecom 00 poceedings. Hence, when egodic ate gap is computed, the ate gap will be aveaged by diffeent pobability distibutions. The main esult of the pape is to show the following asymptotics fo both TAS and MEB schemes: E H[ΔR H] < logsnr+o. 9 IV. BOUNDS ON THE RATE To compute ΔR H, we need to chaacteise R 0 H o equivalently R 0 Ĥ. The invese CDF function in 5 is numeically computable but difficult to chaacteise analytically. Theefoe, to analytically chaacteize the asymptotics, we bound the ight-hand side of 4 in the following lemma. Lemma. The conditional ate R 0 Ĥ has the following bounds: R 0 < log +Γρ σ σ t Ĥ +Γ ρ F P out 0 and ρ R 0 > log +Γ F ρ σ Ĥ t whee F is the invese CDF of the cental distibution with N degees of feedom, and F is the invese CDF of the chi distibution with N degees of feedom. Poof: Conside the event E = { Ê +μ < β 0 σ μ Ĥ }. Since Ê is angulaly symmetic diection wise, P E + = {ReĤH Ê > 0} =P E = {ReĤH Ê < 0}. Also, we have and P E E <P +μ A<β 0, P E E=P E + E. Theefoe, using the above thee equations, we get +μ P E <P A<β 0 If the lowe bound on left hand side of the pevious equation is equated to the taget outage pobability, then a ate geate than R 0 will be obtained. Theefoe, we get the uppe bound as given in the lemma. The invese CDF F is due to the statistics of Ê. In ode to deived the ate lowe bound, conside the following tiangle inequality: Ê + μ Ĥ μ Ê Ĥ. Let P A< β0+μ σ be denoted by P A. The following uppe bound holds: P A < P μ Ê Ĥ +μ < β 0 μ = P Ê Ĥ +μ β 0 +μ μ P Ê β 0 + Ĥ. Equating the ight hand side of the above inequality to, we get the lowe bound on the ate. F is due to Ê. We denote the uppe bound by R upp Ĥ and lowe bound by R low Ĥ fom now on. V. ASYMPTOTIC MISMATCH BETWEEN CSIR AND CSIT In ode to chaateize the asymptotic behavio of R upp and R low, the asymptotics of ρ needs to be chaacteised in the impefect CSI case. In this section, we show the following assuming pediction using the entie past: ρ = OlnSNR Δ SNR π. 3 Each H ij t [n] is pedicted based on past CSIR {H ij [n Δ],H ij [n Δ ]..., H ij [n Δ L +]}. Note that H ij is the MMSE estimate of H ij. We assume that H ij is a Dopple pocess with spectum F e jω. H ij is nothing but a Dopple pocess H ij in noise. Fo the Dopple pocess in noise, the powe spectal density is given by: Pt F e jw Se jω = P t + σn + P t σn P t + σn ω < P t σn, P t + σn < ω <π 4 Specifically, fo the Jakes coelation model F e jω has the following fom: F e jω = 5 ω fo ω <. Since MMSE pediction is used, we have + σp =, whee σp is the pediction eo and ρ = σ. Theefoe, ρ = σ p σ. Since, σ = O at high SNR, ρ is dependent on the pediction eo. This chaacteizes the mismatch between CSIR and CSIT. The noise powe in the case whee H ij is nomalised with σ is +SNR. As noted befoe, fo Dopple pocesses, asymptotics of one step Δ = pediction eo in noise with powe SNR has been chaacteised in [7], [8]. This analysis is the best case scenaio when the entie past is used in pediction holds almost fo lage L. We quote the esult hee: ρ = σ p = OSNR π /0/$ IEEE

4 This full text pape was pee eviewed at the diection of IEEE Communications Society subject matte expets fo publication in the IEEE Globecom 00 poceedings. We now deive a simila esult fo multi-step pediction using esults fom []. The coelation model assumed in 4 obeys Paley-Wiene condition this can occu when F is absolutely log integable in its suppot which is: π π logse jω dω > It can be noted that the Jakes spectum in noise also satisfies this citeion. The following hold fo Se jω lnse jω = and, theefoe, we have Se jω = e c0 n= n= + c n e jωn n= = S + e jω S e jω. Let f n be defined such that: S + e jω = c n e jωn e c0 f n e jωn. n=0 + Expanding S + to get f n,wehave: k S + e jω = e c 0 c n e jωn n= k! k=0 n= c n e jωn = e c 0 [+c e jω + c + c e jω +..! n ] c i..c ik e jωn +... k! k= i +..i k =n Now, c n is evaluated as follows: c n = lnɛ + F e jωn e jωn dω+ π ω m π lnɛ e jωn dω + lnɛ e jωn dω π = O + Olnɛ = O + OlnSNR, 7 whee ɛ = Ptσ n P t+σ = OSNR is the noise vaiance. Let n E N denote the N- step pediction eo. Then, we have E N = N n=0 f n. 8 Also, e c0 = OSNR π. When f n is expessed in tems of c n s using equation 7, it is a summation of tems of the fom c i...c ik poduct of some c n s. The dominant tem in f n, fom tems like c i...c ik,isolnsnr n SNR π. This is due to the tem cn n!. Theefoe, we have ρ = σ p = E Δ = OlnSNR Δ SNR π 9 Hee Δ is the delay as mentioned befoe. When Δ=,we get back the existing esult in 6. VI. ERGODIC RATE GAP ANALYSIS In this section, we chaacteise asymptotics of ΔR H and show the esult stated in 9. The following lemma holds. Lemma. If R ideal H denotes the ate unde pefect CSI, and R low H and R upp H ae the uppe and lowe bounds on the ate fo impefect CSI R 0 H with delay Δ, then R ideal R low = R ideal R upp = log+ SNR SNR η whee P d = ηp t. Poof: Let x = SNR H, and y = Γρ H + Γ ρ F P out. We obseve that SNR R ideal R upp = log +x +y x y = log +. + y 0 Hence, it is enough to chaacteise the atio y x. Also, note that Γρ =Γ Γ ρ, SNR Γ SNR = +, and η SNR ρ =0. The last equality follows fom 9. Theefoe, the following holds: y SNR x = + η By eqns and 0, we have R ideal R upp = log + SNR η Similaly, the lowe bound esult can also be poved by H using y =Γ ρ F ρ. In the above esult, note that while the pediction eo goes to zeo, the ate at which the pediction eo tends to zeo does not matte. This is diffeent fom the divesity-multiplexing gain tadeoff in [5] whee the ate at which ρ tends to is impotant. The ate gap as defined in 8 can be bounded on both sides using 6 and R low,r upp as: R ideal H age/ H R upp H < ΔR H <R ideal H R low H age/ H R ideal + age/ H R ideal R upp < ΔR < R ideal + R ideal R low. The second tem of each bound above goes to a con /0/$ IEEE

5 This full text pape was pee eviewed at the diection of IEEE Communications Society subject matte expets fo publication in the IEEE Globecom 00 poceedings. 0 Egodic ate diff in nats/sec/hz =0.05 =0.08 =0.0*log0 0.8 /logsnr =0.0*log0 0.8 /logsnr =0.0*0 0.8 /SNR SNRin db Fig.. Egodic ate gap fo a system with =0.π,N = N t =, Δ=and L =0unde TAS scheme stant by Lemma. Theefoe, the gowth ate of ΔR is uppe bounded by logosnr and lowe bounded by age/ H logosnr, i.e., ΔRx < Kx logsnr fo SNR > Θ whee x is a ealisation of the vaiable H and Θ is independent of the vaiable x fom the definition of O. notation. Now, the egodic ate gap is uppe bounded as follows: ΔRxf H xdx < 0 logsnr+kxf H xdx fo SNR > Θ whee Θ is independent of the vaiable x. f H is the pdf accoding to the scheme chosen. On aveaging, the second tem esults in a constant. The fist tem integates out to logsnr. Hence the esult in 9 has been shown. Fom the above esults, we obseve that if outage taget is not a function of SNR, the egodic ate gap gows as logsnr. Howeve, if is decays faste than logsnr, then the ate gap is bounded. This means that at high SNR, the outage taget should be lowe. Note that Lemma still holds since all invese CDF functions also decease as deceases. We also show the ate gap behaviou with espect to the chosen outage pobability though simulations unde TAS scheme ef. Fig.. The egodic ate gap actually with substituted in 8 and invese CDFs of non-cental distibution calculated numeically is computed using monte calo simulations to aveage diffeent ealisations H and plotted fo vaious values of SNR. Δ =, N = N t = and =0.π ae the paametes used. We obseve that the ate gap gowth fo vey high SNR is linea in the constant case and the slope inceases when =0.05 is inceased to =0.08. Also, we note that fo the following gowth ates SNR, logsnr, logsnr, the ate gap is bounded asymptotically as pedicted by the theoy. VII. CONCLUSIONS We have compaed the achievable ates unde delay and impefect CSI of the ate adaptive TAS and MEB schemes with that of the pefect cases espectively. The analysis is common and the egodic ate gap in both cases ae shown to have a logsnr gowth. The slope depends on the taget outage pobability. The ate gap is shown to be bounded if the taget outage pobability is educed with SNR. It is instuctive to note that the analysis fo uplink MAC use selection is simila to the TAS scheme and the analysis fo downlink BC use selection is simila to that of the MEB scheme except that the beamfoming vecto is the unit channel vecto of the chosen use and the maximum nom would be the choosing citeia. The analysis pesented hee holds good when pediction eo goes to zeo and MMSE filte with lage numbe of taps almost ensues it. Futhemoe, the outage pobability has to be decided based on SNR and if suitably chosen can make ate gap bounded. ACKNOWLEDGEMENT This wok was suppoted in pat by the Depatment of Science and Technology, Govt. of India. REFERENCES [] S. Zhou and G. B. Giannakis, Optimal tansmitte eigen-beamfoming and space-time block coding based on channel mean feedback, IEEE Tans. on Sig. Poc., vol. 50, no. 0, pp , Octobe 00. [] A. F. Molisch, M. Z. Win, and J. H. Wintes, Capacity of MIMO systems with antenna selection, in Poc. Int. Conf. Communications, vol., June 00, pp [3] B. Vucetic, Z. Chen, and J. Yuan, Analysis of tansmit antenna selection/maximum-atio combining in ayleigh fading channels, IEEE Tansactions on Vehicula Technology, vol. 54, no. 4, pp. 3 3, Jul 005. [4] A. J. Goldsmith and P. P. Vaaiya, Capacity of fading channels with channel side infomation, IEEE Tansactions on Infomation Theoy, vol. 43, no. 6, pp , Novembe 997. [5] T. R. Ramya and S. Bhashyam, Eigen-beamfoming with delayed feedback and channel pediction, in Poceedings of IEEE Intenational Symposium on Infomation Theoy, 009., Seoul, Koea, Jun./Jul. 009, pp [6], Rate adaptation in MIMO antenna selection system with impefect CSIT, in Poceedings of IEEE COMSNETS 00 WISARD 00, Bangaloe, India, Jan 00. [7] G. Caie, N. Jindal, M. Kobayashi, and N. Ravindan, Quantized vs. analog feedback fo the MIMO boadcast channel: A compaison between zeo-focing based achievable ates, in IEEE Intenational Symposium on Infomation Theoy, 007 ISIT 007, 4-9 Jun 007, pp [8], Multiuse MIMO achievable ates with downlink taining and channel state feedback, IEEE Tansactions on Infomation Theoy, Dec 009, accepted fo publication. [Online]. Available: [9] B. Hassibi and B. Hochwald, How much taining is needed in multiple antenna wieless links? IEEE Tansactions on Infomation Theoy, vol. 49, no. 4, pp , ap 003. [0] T. Yoo and A. Goldsmith, Capacity and powe allocation fo fading MIMO channels with channel estimation eo, IEEE Tansactions on Infomation Theoy, vol. 5, no. 5, pp. 03 4, May 006. [] H. V. Poo, An Intoduction to Signal Detection and Estimation, nd ed. Spinge-Velag, /0/$ IEEE

Closed-form Formulas for Ergodic Capacity of MIMO Rayleigh Fading Channels

Closed-form Formulas for Ergodic Capacity of MIMO Rayleigh Fading Channels Closed-fom Fomulas fo Egodic Capacity of MIMO Rayleigh Fading Channels Hyundong Shin and Jae Hong Lee School of Electical Engineeing Seoul National Univesity Shillim-dong, Gwanak-gu, Seoul 151-742, Koea