Globeco - Wireless Counications Syposiu Receive Antenna Subset Selection For Tie-Varying Channels Using Slepian Subspace Projections Hassan A Abou Saleh and Steven D Blostein Departent of Electrical and Cop Eng, Queen s University, ingston, Canada hassanabousaleh, stevenblostein @queensuca Abstract Receive antenna selection AS preserves the diversity benefits of ultiple antennas with considerable reduction of hardware coplexity and costs We propose a receive AS ethod for tie-varying channels which utilizes low-coplexity Slepian subspace projection techniques The proposed ethod uses Doppler bandwidth nowledge and taes into account practical liitations such as training, pacetization and antenna switching Results show that the proposed AS ethod outperfors ideal conventional systes with perfect channel state inforation CSI but no AS at the receiver and AS using the conventional Fourier estiation/prediction ethod A closed-for expression for the sybol error probability SEP of M-ary phase-shift eying MPS with receive AS is derived I INTRODUCTION Receive antenna selection AS reduces hardware coplexity by using liited nuber of radio-frequency RF chains at the receiver of a wireless syste [] Algoriths and perforance analysis for AS systes are reported in nuerous previous studies [] [5] It is only recently that a liited nuber of recent studies have investigated practical issues such as pilot-based training and ipleentation of AS [6] In the above references, perfect channel nowledge is assued However, the wireless channel is tievarying which results in outdated CSI at the receiver The ipact of iperfect channel nowledge on the perforance of AS systes is studied in [7] [] The perforance of AS systes with CSI feedbac delay are studied in [] and [] Weighted AS rules for tie-varying channels which use the teporal correlation nowledge are proposed in [3] and [4] In [3] and [4], only channel gain estiates obtained fro the AS training phase are used in the selection and decoding processes This results in a signal-to-noise ratio SNR loss Motivated by the above observations, [5], [6] recently proposed a practical training-based receive AS algorith for tie-varying channels It uses CSI nowledge of the data transission phase in selection and decoding processes by utilizing low-training overhead Slepian prediction [7] and estiation [8] The optial Wiener predictor requires detailed correlation nowledge whereas the Slepian estiator/predictor only requires nowledge of the Doppler bandwidth [7] However, only the sipler proble of selecting a single antenna at the receiver is considered in [5] The paper s contributions are suarized as follows: This research has been supported by the Natural Sciences and Engineering Research Council of Canada NSERC Discovery Grant 473 A single receive AS ethod for tie varying channels based on Slepian subspace projections [5], [6] is extended to accoodate the selection of ultiple receive antennas A closed-fro expression for the sybol-error probability SEP of M-ary phase-shift eying MPS with receive AS is provided, and verified with siulations Extensive siulation results are presented to copare the perforance of the proposed ethod to that of ideal conventional SIMO systes with perfect CSI but no AS at the receiver as well as AS based on conventional Fourier basis prediction/estiation II SYSTEM MODEL Consider a syste with one transit and receive antennas equipped only with RF chains Depending on the AS switching tie, either per-pacet or sybol-by-sybol AS can be used For exaple, icroelectroechanical syste MEMS switches enable only per-pacet switching with negligible attenuations Solid-state switches can enable switching of antennas between sybols, but with non-negligible attenuations [9] A Antenna Selection Training Phase In total L pilot sybols are transitted in L rounds of transission, as depicted in Fig In each round the transitter sends out pilots, where each pilot is received by at ost receive AEs Two consecutive pilots are spaced T p αts, where T s is the sybol duration and α Therefore, two consecutive pilots transitted for each of the antenna subsets are separated in tie by Tr Tp The AS training pilots are received by AE at ties Tsp, where Tsp [ ] α + l, l L The observations over the AS training pilots are necessary to perfor Slepian channel prediction [7] for each AE over the data transission phase I dt Based on the predicted channel gains ĥsp I dt, the receiver then selects its receive AEs subset S AE, AE,, AE and connects the to the RF chains in a duration of T p T s Therefore, the AS training phase spans the discrete tie 488
A E A E A E 3 A E 4 T s T p T r L L L L A S training phase T p-t s L ' L ' Selection & Sw itching D ata trans ission phase ti e Fig Antenna selection cycle for the proposed per-pacet AS ethod AEs and 3 are selected, 4, L, L, and T p T s interval I tr,,, M, where M α L In sybol-by-sybol AS, the ost suitable receive AEs subset S AE, AE,, AE is selected for each sybol at tie B Data Transission Phase Per-Pacet AS: The data transission phase spans I dt M, M +,, M + N, where the transitter sends out a length-n pacet containing L pilots tie-ultiplexed with N L data sybols as [6] P dp l N L + N L l L The received signal at AE, for, is h d + n, I dt \T dp y h p + n, T dp 3 where d and p denote the transitted data and pilot sybols, respectively The L pilot sybols are received by antenna subset S at ties T dp, where T dp M + l N L + N L 4 Thus, in total, L tp L + L pilot sybols are received by each selected AE at ties T tp T sp T dp, 5 with T sp and T dp given in and 4, respectively Fro these L tp pilots, refined channel gain estiates ĥse I dt are obtained to decode data Sybol-By-Sybol AS: After selection, the transitter sends out a length-n pacet which consists of N L data sybols and L pilot sybols The L pilot sybols are needed to obtain refined channel gain estiates ĥse I dt This is because in sybol-by-sybol AS, for each sybol an antenna subset S is selected at tie Since different antenna subsets ight be selected during the data transission phase, L pilots should be sent to each of the antenna subsets The sybol locations in the pacet that carry the L pilots for AE, for, are given by Pdp + l N L + N L 6 Thus L tp L + L pilots are received by each AE, for, at ties T tp T sp T dp, 7 where Tdp M + + l N L + N L 8 The data sybols received by S are given by y h d + n 9 where I dt \ Tdp T dp III SLEPIAN BASIS EXPANSION MODEL A Slepian Estiator The true channel vector h [ h [],, h [M ] ] T of size M is estiated as [7] h ĥse U ˆγ D i ˆγ i u i [ where the Slepian function u i ui [],, u i [M ] ] T consists of tie-liited discrete prolate spheroidal DPS sequences corresponding to eigenvalue λ i The DPS sequences u i Z M i are defined as [8] M l sin ν ax l l u i [l] λ i u i, Z where i I bl,,, M and ν ax is the noralized Doppler bandwidth In, U [ u,, u D ] is an M D atrix The coefficient vector ˆγ [ˆγ, ˆγ,, ˆγ D ] T is estiated using the J interleaved pilots p [l] l J, received at ties l J, via [8] ˆγ G l J y [l] p [l] f [l] where y [l] is the observation over the transitted pilot sybol p [l], f [l] [ u [l],, u D [l] ] T, and G is a D D atrix given by G l J In, D is given by [7] D argin d,,j f [l] f [l] 3 ν ax J where N is the noise variance J λ i + d J N id 4 489
B Slepian Predictor The Slepian predictor approxiates h as [7] ĥ SP f T ˆγ D i ˆγ i u i, Z \ I bl 5 where u i Z \ I bl M i can be calculated fro IV RECEIVE ANTENNA SELECTION ALGORITHM We propose the following training-based out of receive AS algorith for tie-varying channels: Every antenna subset of the total subsets is trained using L pilot sybols The spacing between consecutive pilot sybols transitted for each of the antenna subsets is Tr α Ts The receiver then perfors channel prediction over I dt via 5 D ĥ SP f T ˆγ ˆγ,i u i 6 i ] T where ˆγ [ˆγ,, ˆγ,,, ˆγ,D is obtained via with Tsp replacing J a Per-Pacet AS: Selects its receive antenna subset S which axiizes the post-processing SNR over I dt, ie, the first order statistics of M+N ĥsp M b Sybol-By-Sybol AS: Selects its instantaneous antenna subset S which consists of the first order statistics of ĥ SP 3 a Per-Pacet AS: The transitter sends a length-n data pacet in which L pilots are tie ultiplexed with N L data sybols as b Sybol-By-Sybol AS: The transitter sends out a length-n data pacet, which consists of N L data sybols and L pilots interleaved as 6 4 a Per-Pacet AS: Refined channel gain estiates ĥse I dt for S are obtained via where ĥse ĥ SE D U ˆγ ˆγ,i u i 7 i [ĥse [M],, ĥse [M + N ]] T, U [ u,, u D ] is the N D subatrix of the coplete M + N D Slepian sequences atrix U, and u i [ u i [M],, u i [M + N ] ] T b Sybol-By-Sybol AS: To decode data the receiver obtains ĥ SE I dt for S via ĥ SE D i ˆγ,i u i 8 V SYMBOL ERROR PROBABILITY ANALYSIS The soft estiate is obtained by axiu-ratio-cobining MRC of the received soft sybols fro S as ĥse r S y 9 ĥsp ĥse Conditioned on, and d, using standard results on oents of conditional Gaussian RVs [] it can be shown that r S in 9 is a coplex Gaussian RV whose conditional ean µ rs depends on ĥse both ĥsp and ĥse which aes the analysis analytically intractable We next derive an SEP expression which provides insights for both sybol-by-sybol and per-pacet AS as described in Sec IV The first and second conditional oents of 9 are conditioned only on ĥse Theore With MRC decision variable in 9 conditioned ĥse only on the refined channel gain estiates, the MPS SEP for a sybol received at tie for a syste with one transit and receive antennas eploying the out of sybol-by-sybol receive AS in Sec IV is [ SEP η ] Γ P Γ Pr P P M M r sin θ sin θ + σ X SE sin θ + σ X SE sin θ + σ X SE + Ϝ dθ where η Es N is SNR, P is a perutation of set,,, and P the set of all! perutations σ X SE +σ e SE, bse ρ ĥ SE, ĥsp sin +η M denote the correlation coefficient of ĥse where, and ρĥse, ĥsp and ĥsp [ Also Ϝ Γ P r ] for, and Ϝ r [ ] Γ P r for > where Γ r ρ ĥ SE, ĥsp y Proof: 9 can be expressed as ĥse d ese d + n 483
where ĥse h + e SE with ese the estiation error Conditioned on ĥse and d, y in is a coplex Gaussian RV whose conditional ean µ y and variance σ y are given by µ y E y ĥse, d σ y ĥse var y d ζse ĥse, d + η 3 The axiu-lielihood ML soft estiate is obtained by conducting MRC on the received soft sybols fro S as ĥse r S y + η 4 where the scaling factor explicitly taes account of the channel estiation uncertainty, ie, the variances in 3 ĥse Conditioned on and d, r S in 4 is a coplex Gaussian RV whose conditional ean µ r S and variance σr are given by S µ r S σ r S E r S var ĥse ĥse, d ζ SE d 5 + η r S ĥse ĥse, d + η 6 The SEP of an MPS sybol received at tie conditioned on ĥsp SEP ĥsp ĥse, by SEP Ξ, is [5], [], [] SEP Ξ where µ r S σ r S M M M M ĥse and, denoted for brevity µ r S sin M exp σr sin dθ θ S Y SE S exp sin dθ 7 θ ĥse and the last equality follows fro YS SE ĥse +η X SE where X SE ĥse The SEP averaged over SEP ĥsp, denoted for brevity by SEP ϖ, is SEP ϖ M M M κ sin dθ 8 θ where M κ is the oent generating function MGF of Y SE M YS SE ĥ SP S conditioned on Now conditioned on ĥsp ĥsp Gaussian RV with conditional ean µ X SE are given by σ X SE µ X SE σ X SE E X SE bse var ĥsp ζ SP X SE ĥsp ρ ĥ SE, ie,, X SE is a coplex and variance ρĥse, ĥsp ĥsp, ĥsp 9 Therefore YS SE follows a non-central Chi-squared distribution with conditional MGF given by [] M κ x σ x X SE exp Substituting 3 in 8, yields SEP ϖ M M where ξ Υ ξ exp ζ SP ĥsp Γ E Υ ξ µ X SE σ X SE sin θ + σ X SE ĥsp sin θ + σ X SE ρ ĥ SE, ĥsp x 3 x sin θ dθ 3 Denoting by the exponential RV with ean ρ ĥ SE, ĥsp Averaging over ĥ SP and using the virtual branch cobining VBC technique [3], yield the desired result 483
Pacet error rate PER 3, 6 proposed AS algorith perfect CSI & no AS, 6 perfect CSI & AS Slepian prediction & no AS, 6 Slepian prediction & AS, 6 no prediction & AS, 6 DFT ethod & AS 4 5 5 SNR db Sybol error probability SEP 3, 4 sybol by sybol AS Theore, 4 sybol by sybol AS si, 4 per pacet AS si 4 4 6 8 4 6 8 SNR db Fig PER perforance of the proposed AS ethod for a, 6 syste QPS, data pacet length N 4, training pilots L, post-selection pilots L, and T p 3 T s Fig 4 SEP for the first 8PS data sybol for a, 4 syste Data pacet length N 4, AS training pilots L, L, and T p 3T s Sybol error probability SEP, 4 sybol by sybol AS Theore, 4 sybol by sybol AS si, 4 per pacet AS si 3 4 6 8 4 6 8 SNR db Fig 3 SEP for the -th 8PS data sybol for a, 4 syste Data pacet length N 4, AS training pilots L, L, and T p 3T s VI SIMULATIONS A syste with one transit and receive antennas out of which is selected, denoted by,, is siulated The carrier frequency f c GHz and the user oves with radial velocity v ax /h The pacet consists of N 4 MPS sybols each of duration T s 57 µs [6] These paraeters give a Doppler bandwidth ν ax 38 3 The realizations of the tie-varying channel are generated as h P p P exp j ν ax cos α p + ψ p 3 where the nuber of propagation paths P 3, and the path angles α p and ψ p are independent and uniforly distributed over [ The pacet error rate PER of the proposed AS approach for a, 6 syste is illustrated in Fig For coparison, we also show the PER perforance of i a receive antenna syste with perfect CSI using MRC and no AS, ii a, 6 syste eploying AS without channel prediction, and iii a, 6 syste eploying discrete Fourier transfor DFT basis expansion ethod [8] Inspection of Fig reveals that the, 6 syste eploying the proposed per-pacet AS algorith achieves an SNR perforance gain in excess of db over the receive antenna syste with perfect CSI and no AS at a PER 3 The perforance of the sae proposed, 6 syste is about 7 db worse than, 6 syste eploying AS with perfect CSI at PER of 3 The coplexity of the proposed AS ethod is higher than that of a syste eploying AS with DFT expansion odel The generation of length-m Slepian sequences requires the use of singular value decoposition SVD to calculate the eigenvectors of the M M atrix C with l, entry defined as C [l, ] sinνaxl l This requires O M 3 coplex ultiplications [4] However, the Fourier basis functions, which do not require the nowledge of the Doppler spread and SVD, can be stored in eory using pre-calculated looup tables The SEP of the -th and first 8PS sybols as a function of average SNR for a, 4 syste eploying the proposed receive AS algorith are depicted in Figs 3 and 4, respectively A gap can be observed in Fig 4 between the perpacet and sybol-by-sybol AS curves at oderate to high SNRs This is because the channel prediction for the first sybol is better than that for the -th sybol Also, there are upward transitions in the curves which are the result of an increase of the subspace diension D in 4 to avoid error-floors The SEP of the -th and first 4PS sybols as a function of average SNR for a, 6 syste eploying the proposed receive AS algorith are also depicted in Figs 5 and 6, respectively Fro Figs 3 6 and fro other siulations not included, we also observe that Theore reasonably approxiates the SEP perforance of systes eploying the sybol-by-sybol AS algorith in Sec IV 483
Sybol error probability SEP 3, 6 sybol by sybol AS Theore, 6 sybol by sybol AS si, 6 per pacet AS si 4 4 6 8 4 SNR db Fig 5 SEP for the -th 4PS data sybol for a, 6 syste Data pacet length N 4, AS training pilots L, L, and T p 3T s Sybol error probability SEP 3 4, 6 sybol by sybol AS Theore, 6 sybol by sybol AS si, 6 per pacet AS si 5 4 6 8 4 SNR db Fig 6 SEP for the first 4PS data sybol for a, 6 syste Data pacet length N 4, AS training pilots L, L, and T p 3T s VII CONCLUSIONS We have proposed a receive antenna subset selection approach which uses Slepian basis expansion for prediction and estiation It taes into account practical constraints iposed by next-generation wireless standards such as training and pacet reception for antenna selection AS We have derived a closed-for expression for the MPS SEP with the receive AS ethod It is shown that the proposed AS schee outperfors ideal conventional systes with perfect channel nowledge but no AS at the receiver and conventional coplex basis based estiation REFERENCES [4] R W Heath and D J Love, Multiode antenna selection for spatial ultiplexing systes with linear receivers, IEEE Trans Signal Process, vol 53, pp 34 356, Aug 5 [5] Z Xu, S Sfar, and R S Blu, Analysis of MIMO systes with receive antenna selection in spatially correlated Rayleigh fading channels, IEEE Trans Veh Technol, vol 58, pp 5 6, Jan 9 [6] H Zhang, A F Molisch, and J Zhang, Applying antenna selection in WLANs for achieving broadband ultiedia counications, IEEE Trans Broadcast, vol 5, pp 475 48, Dec 6 [7] Zhang and Z Niu, Adaptive receive antenna selection for orthogonal space-tie bloc codes with channel estiation errors with antenna selection, in Proc IEEE Globeco, 5 [8] W Xie, S Liu, D Yoon, and J-W Chong, Ipacts of Gaussian error and Doppler spread on the perforance of MIMO systes with antenna selection, in Proc WiCOM, 6 [9] R Annavajjala and L B Milstein, Perforance analysis of optiu and suboptiu selection diversity schees on Rayleigh fading channels with iperfect channel estiates, IEEE Trans Veh Technol, vol 56, pp 9 3, May 7 [] W M Gifford, M Z Win, and M Chiani, Antenna subset diversity with non-ideal channel estiation, IEEE Trans Wireless Coun, vol 7, pp 57 539, May 8 [] S Han and C Yang, Perforance analysis of MRT and transit antenna selection with feedbac delay and channel estiation error, in Proc IEEE WCNC, 7, pp 35 39 [] T R Raya and S Bhashya, Using delayed feedbac for antenna selection in MIMO systes, IEEE Trans Wireless Coun, pp 659 667, Dec 9 [3] V riste, N B Mehta, and A F Molisch, Optial receive antenna selection in tie-varying fading channels with practical training constraints, IEEE Trans Coun, vol 58, pp 3 34, Jul [4], Training and voids in receive antenna subset selection in tievarying channels, IEEE Trans Wireless Coun, vol, pp 99 3, Jun [5] H A Saleh, A F Molisch, T Zeen, S D Blostein, and N B Mehta, Antenna selection for tie-varying channels based on Slepian subspace projections, in Proc IEEE ICC, [6], Receive antenna selection for tie-varying channels using discrete prolate spheroidal sequences, IEEE Trans Wireless Coun, vol, pp 66 67, Jul [7] T Zeen, C F Meclenbräuer, F altenberger, and B H Fleury, Miniu-energy band-liited predictor with dynaic subspace selection for tie-variant flat-fading channels, IEEE Trans Signal Process, vol 55, pp 4534 4548, Sep 7 [8] T Zeen and C F Meclenbräuer, Tie-variant channel estiation using discrete prolate spheroidal sequences, IEEE Trans Signal Process, vol 53, pp 3597 367, Sep 5 [9] G L Stüber, J R Barry, S W McLaughlin, Y G Li, M A Ingra, and T G Pratt, Broadband MIMO-OFDM wireless counications, Proc IEEE, vol 9, pp 7 94, Feb 4 [] V riste, N B Mehta, and A F Molisch, A novel, balanced, and energy-efficient training ethod for receive antenna selection, IEEE Trans Wireless Coun, vol 9, pp 74 753, Sep [] M-S Alouini and A Goldsith, A unified approach for calculating error rates of linearly odulated signals over generalized fading channels, IEEE Trans Coun, vol 47, pp 34 334, Sep 999 [] M Di Renzo and H Haas, Space shift eying SS- MIMO over correlated Rician fading channels: Perforance analysis and a new ethod for transit-diversity, IEEE Trans Coun, vol 59, pp 6 9, Jan [3] M Z Win and J H Winters, Analysis of hybrid selection/axialratio cobining of diversity branches with unequal SNR in rayleigh fading, in Proc IEEE VTC, 999 [4] G H Golub and C F Van Loan, Matrix Coputations Baltiore, USA: The Johns Hopins University Press, 996 [] S Sanayei and A Nosratinia, Antenna selection in MIMO systes, IEEE Coun Mag, vol 4, pp 68 73, Oct 4 [] M Z Win and J H Winters, Virtual branch analysis of sybol error probability for hybrid selection/axial-ratio cobining in Rayleigh fading, IEEE Trans Coun, vol 49, pp 96 934, Nov [3] A Ghrayeb and T M Duan, Perforance analysis of MIMO systes with antenna selection over quasi-static fading channels, IEEE Trans Veh Technol, vol 5, pp 8 88, Mar 3 4833