Channel Capacity Performance of Transmit Antenna Selective MIMO System in Weibull Fading

Transcription

1 Channel Capacity Perfrmance f ransmit Antenna Selective MIMO System in Weibull Fading Mzabalaz Lupupa, Mqhele E. Dldl University f Cape wn Department f Electrical Engineering Private Bag, Rndebsch, 7701 el: , Fax: mzielupps@yah.cm; mqhele.dldl@uct.ac.za Abstract Multiple-input multiple-utput (MIMO) systems play a great rle in imprving cmmunicatin in wireless systems. But an increase in the number f antennas used in a cmmunicatin system results in increased system cmplexity. address this drawback assciated with MIMO systems we prpse the use f a transmit antenna selective MIMO system. In this the best perfrming transmit antenna is selected frm all the available antennas fr cmmunicatin. he signalt-nise rati (SNR) perfrmance f the prpsed system is studied as the number f antennas used in the system is increased. In studying the SNR perfrmance, maximal-rati-cmbining (MRC) is used as the ptimal cmbining technique at the receiver end. Average channel perfrmance is als investigated as we vary the number f antennas used. he Weibull fading channel is cnsidered in studying the perfrmance f the transmit antenna selective MIMO system. We further illustrate the effect f the Weibull fading parameter n the average channel capacity perfrmance. Index erms Average channel capacity, maximalrati-cmbing, transmit antenna selectin, Weibull fading. I. INRODUCION he increasing demand fr high-rate data and/r better quality services has created a great need fr imprved bandwidth efficiency in wireless cmmunicatin systems. Several diversity techniques have been prpsed t try t imprve the perfrmance f wireless cmmunicatin system. hese techniques help in cmbating the effects assciated with fading and multipath prpagatin. hey include amngst thers time diversity, frequency diversity and antenna diversity. Out f all these, antenna diversity has been bserved t be bandwidth efficient and this has led t the develpment f multiple-input multiple utput (MIMO) systems. MIMO systems are able t prvide imprved data rates r better quality signals thrugh the use f multiple transmit and multiple receive antennas [1]. his imprvement in perfrmance is assciated with encding the mdulated signals using space-time cding techniques and then transmitting the encded signals using different antennas []. imprve n data rates the signals are encded using Bell Labratries Layered Space-ime (BLAS). Whilst t imprve n the quality, the signals are encded using space-time trellis (S) cding r space-time blck (SB) cding. SB cding is cnsidered in this study due t the fact that it has less decder cmplexity cmpared t S cding. his is because the simple maximum likelihd technique can be used t decde signals which have been encded using SB cding. he perfrmance f SB cdes was first studied by Alamuti wh shwed that diversity gain culd be achieved by using tw transmit antennas. Diversity gain implies that it is highly unlikely fr all the signals t simultaneusly fall in the deep fade regin. His study was later extended t any number f transmit antennas by ark et al. [] One drawback with MIMO systems is the increase in cmplexity assciated with an increase in the number f antennas used in the system [], []. help slve this prblem, antenna selectin was prpsed [], [7]. he idea with antenna selectin is t select the best perfrming antenna(s) and using this instead f all the available antennas fr cmmunicatin. In this way the number f radi frequency chains t be decded at the receiver is reduced. Antenna selectin is applicable at the transmitter and/r receiver end. In this crrespndence transmit antenna selectin is cnsidered. With transmit antenna selectin the single best perfrming transmit antenna is selected. he selectin is such that the antenna that maximizes the signal s SNR is chsen. he signals are then cmbined at the receiver end using several cmbining techniques, namely, maximal-raticmbining (MRC), equal-gain-cmbining (EGC) and selectin cmbing (SC). Of all these cmbining techniques, maximal-rati-cmbining is the best perfrming and it s the ne we will cnsider when studying the perfrmance f the transmit antenna selective MIMO system [8]. In this paper we cnsider the Weibull fading channel in investigating the SNR perfrmance with an increase in the number f antennas in the system and when MRC has been used. We als study the average channel capacity

2 perfrmance f the transmit antenna selective MIMO system. he rest f the paper is rganized as fllws. he system mdel is presented in sectin II. In sectin III the Weibull fading channel is presented. Sectins IV and V prvide the SNR and average channel capacity analysis fr the transmit antenna selective MIMO system respectively. In sectin VI, simulatin results are presented. Finally, the paper is cncluded in sectin VII. II. SYSEM MODEL We cnsider an ( n,1; n ), Fig. 1 wireless link in a flat R Weibull fading envirnment where n is the number transmit antennas, nr the number f receive antennas and 1 is the selected single transmit antenna. he fading cefficients h,1 i n,1 j n are mdeled as R independent samples f cmplex Gaussian randm variables with a zer mean and the variance f 0. per dimensin. It is assumed that the channel state infrmatin is perfectly knwn at the receiver end and partially knwn at the transmitter end thrugh a feedback channel. At any time, nly ne f the ttal n antennas is chsen and activated fr further transmissin. cntinue until the BER falls t a set threshld, after which all the transmit antennas will be allwed t start the transmissin prcess and then the selectin takes place again. Having selected the best transmit antenna, the signals at the receiver end are then cmbined using MRC. III. WEIBULL FADING In a MIMO system with flat fading wireless channel, the received signal can be mdelled as fllws: y Hx n () where x is an ( n 1) transmitted signal vectr, with n being the number f transmitters. y is an ( nr 1) received signal vectr, with n being the number f receivers. H is R the ( nr n ) channel matrix and n is an ( nr 1) additive white Gaussian nise vectr. he channel matrix H is given as h h h h h h H hn R 1 h n 1 n nrn () where h describes the channel gain between the ith receiving antenna and the j th transmitting antenna. Fr the Weibull fading mdel, the cmplex envelpe h can be written as a functin f the Gaussian in-phase X and quadrature Y elements f the multipath cmpnents [11] / h ( X jy ) () where j 1 and is the fading parameter. Let Z be the magnitude f h, i.e. Z h, the Weibull fading mdel, Z can be expressed as a pwer transfrmatin f a Rayleigh distributed randm variable (RV), R X jy as [1] Fig. 1. ransmit antenna selective MIMO system mdel. Z / R () he selectin is such that nly the single transmit antenna with the highest signal, SNR is selected frm all the available transmit antennas. he single selected transmit antenna, dented asx, is determined by [9], [10] x arg max x h nr i (1) 1 jn i1 After the selectin, all the transmit pwer is then cncentrated n this particular antenna. ransmissin will Frm (), the PDF f Z can be given as [1] 1 r fz ( r) r exp with E( Z ). is the fading parameter expressing the fading severity ( >0) and is the average fading pwer. As increases, the effect f fading decreases, ()

3 while fr the special case f, the Weibull PDF f Z reduces t the Rayleigh PDF. Whilst fr 1 the Weibull PDF f Z reduces t the well knwn negative expnential PDF. he crrespnding CDF f Z can be expressed as r Fz ( r) 1 exp IV. SIGNAL-O-NOISE RAIO In Weibull fading the instantaneus signal-t-nise rati at the input f the receiver is given by [1] s N (7) E Z (8) and the average SNR is then given as [1] E ( Z E E ) 1 N N s / s Based n an interesting prperty f the Weibull distributin, that the n pwer f a Weibull distributed th randm variable with parameters (, ) is anther Weibull distributed randm variable with parameters ( / n, ) [1]. Frm the abve mentined prperty it can then be cncluded that is als a Weibull randm variable with parameters where a 1/ 1 / (9) / ( /, ( a ) ). he PDF f can then be derived frm () by replacing / a with ( ) as [1] with / and / ( /) 1 a p ( ) / exp a (10) When MRC is applied at the receiver end the instantaneus SNR measured at the utput f the tw receivers is given as (11) MRC 1 j j where 1 j dentes the instantaneus SNR between the first receive antenna and the selected jth transmit antenna and j dentes the instantaneus SNR between the secnd receive antenna and the selected jth transmit antenna. he average SNR is written as (1) MRC 1 j j where 1 j dentes the average SNR between the first receive antenna and the selected jth transmit antenna and j dentes the average SNR between the secnd receive antenna and the selected jth transmit antenna. his average SNR will then be used in studying the average channel capacity perfrmance f the transmit antenna selective MIMO system under the Weibull fading channel. V. AVERAGE CHANNEL CAPACIY he standard frmula fr the Shannn capacity is [1] C BW lg (1 ) (1) he instantaneus capacity f a MIMO system with n CSI at the transmitter is given by [1] [18] C BW lg det In R HH (1) n where E / N is the average SNR per receiving antenna, with s s E as the average pwer at the utput f each f the receiving antennas and N the crrespnding nise pwer. I is the ( n R n R ) identity matrix and dentes n R the cnjugate and transpsitin and BW is the bandwidth. We then cnsider the average channel capacity, which in Shannn s sense is given as [1], [19] lg (1 ) ( ) (1) 0 C BW p d where ( ) is the PDF f the signal-t-nise rati and is p given by (10), and BW the bandwidth. Substituting (10) int (1), fr Weibull fading the average channel capacity can be written as BW C / a ln() 0 ( / ) 1 ( / ) / exp a d ln(1 ) (1) he abve integral can be evaluated in clsed-frm by expressing the lgarithmic and expnential integrands as Meer s G-functins as [1], [0] ln(1 ) and 1, 1,1 G, 1,0 1,0 a G a / / exp / 0,1 / 0.

4 he average channel capacity can then be written in clsedfrm as ( /) a BW kl C ln() G 1 kl /,,,1 k a I l I l k (,0),,,, I k I l I l k l, l l, kl k (17) 1 n 1 where I ( n, ),,..., with as an n n n arbitrary real value and n as a psitive integer. k Furthermre, where k and l are psitive integers, l depending n the value f, e.g. fr 1. we chse k and l and fr we chse k and l. end. hus MIMO systems play a great rle in mitigating the effects f fading in wireless cmmunicatin systems. Signal-t-nise rati (db) MRC EGC SC Number f antennas Fig.. SNR perfrmance fr MRC, EGC and SC with increasing number f antennas. VI. SIMULAION RESULS In this sectin we present the SNR and average channel capacity perfrmance f the transmit antenna selective MIMO system. he effect n system perfrmance when changing the number f transmit antennas and als when changing the fading parameter, is illustrated. study these perfrmance parameters, simulatins were dne using MALAB and Mathematica. he signals were assumed t be mdulated using binary-phase-shift keying (BPSK) and encded using SB cding. he antennas were assumed t be sufficiently spaced s as t avid any interference between neighburing antennas. he channel state infrmatin (CSI) was assumed t be available at the receiver end and partially available at the transmitter end thrugh the feedback channel. he ttal number f transmit antennas used ranged between tw and fur whilst the receive antennas were kept at tw. he transmit antenna selective MIMO system is presented as ( n,1; n ), where nly ne f the ttal n transmit antennas is chsen and activated fr further transmissin and nr is the number f receive antennas. Fig. is a plt f number f antennas used in a wireless cmmunicatin MIMO system against the signal-t-nise rati fr three cmbining techniques, namely, maximalrati-cmbining (MRC), equal-gain-cmbining (EGC) and selective cmbining (SC). he MRC technique is bserved t utperfrm the ther tw cmbining techniques. Fr this reasn it was selected as the ptimum cmbining technique fr studying the perfrmance f the transmit antenna MIMO system. Literature tells us that the SNR in a MIMO system can be imprved by increasing the number f antennas used in the system. Fig. illustrates the SNR imprvement with increase in the number f antennas used. his imprvement in SNR leads t enhanced perfrmance as the signals can be decded with reduced prbability f errr at the receiver R Using (17), Fig. and Fig. were pltted. hese are plts f the average SNR against the nrmalized average channel capacity fr different transmit antenna selective MIMO systems with different values f fading parameter. he SISO system is included just fr cmparisn purpses. As the ttal number f transmit antennas is increased in the transmit antenna selective MIMO system the channel capacity perfrmance als imprves, i.e. (,1;) (,1;) (,1;). his shws that MIMO systems can be used t imprve the channel capacity withut bandwidth expansin thus leading t imprvement in cmmunicatin. Nrmalised Average Channel Capacity (bits/sec/hz) (,1;) (,1;) (,1;) (1;1) Average Signal-Nise Rati (db) Fig.. Average SNR as a functin f the nrmalized average channel capacity fr 1..

5 Nrmalised Average Channel Capacity (bits/sec/hz) (,1;) (,1;) (,1;) (1;1) Average Signal-Nise Rati (db) Fig. and Fig. are plts f average SNR against the nrmalized average channel. In each graph the same transmit antenna selective MIMO system is cnsidered but fr different values f fading parameter. As the value f increases the average channel capacity is imprved. An increase in results in a decrease in the effects f fading, thus imprvement in the perfrmance f a wireless cmmunicatin system. Nrmalised Average Channel Capacity (bits/sec/hz) Fig.. Average SNR as a functin f the nrmalized average channel capacity fr Average Signal-Nise Rati (db) Fig.. Average SNR as a functin f the nrmalized average channel capacity fr the (,1;) transmit antenna selective MIMO system, with 1., and. Nrmalised Average Channel Capacity (bits/sec/hz) Average Signal-Nise Rati (db) Fig.. Average SNR as a functin f the nrmalized average channel capacity fr the (,1;) transmit antenna selective MIMO system, with 1., and. VII. CONCLUSION he perfrmance f the transmit antenna selective MIMO was investigated. he SNR imprvement with MRC and increasing the number f antennas in the system was illustrated. It was shwn that using multiple antennas in wireless system imprves the SNR. Imprvement in SNR implies signals can be received at the receiver end with minimum errrs thus allwing fr easier decding. he average channel capacity perfrmance was als demnstrated fr varius system cnfiguratins. he prpsed transmit antenna selective MIMO system was shwn t utperfrm the SISO system. Channel capacity perfrmance was als demnstrated t imprve by increasing the fading parameter. his is because an increase in the fading parameter results t a decrease in the effects f fading n the wireless cmmunicatin system. REFERENCES [1] S. M. Alamuti, A simple transmit diversity technique fr wireless cmmunicatins, IEEE Jurnal n Select Areas in Cmmunicatins, vl. 1, n. 8, Oct []. D. Mavares and R. P. rres, Space time cde selectin fr transmit antenna diversity systems, IEEE rans. Veh. echnl., vl. 7, n. 1, pp. 0-9, Jan [] V. arkh, H. Jafarkhani, and A. Calderbank, Spacetime blck cdes frm rthgnal designs, IEEE rans. Infrm. hery, vl., pp. 1-17, July [] A. Ghrayeb and. M. Duman, Perfrmance analysis f MIMO systems with antenna selectin ver quasi static fading channels, IEEE rans. Veh. echnl., vl., n., pp , Mar. 00. [] R. Gaspa and J. R. Fnllsa, Cmparisn f different transmit diversity space-time cde algrithms, Prc. IS Mbile Cmmunicatins Summit 000, pp. 0-08, Oct [] X. N. Zeng and A. G. Ghrayeb, Perfrmance bunds fr space-time blck cdes with receive antenna

6 selectin, IEEE rans. Infrm. hery, vl. 0, n. 9, pp , Sep. 00. [7].H. Liew and L. Hanz, Space-time cdes and cncatenated channel cdes fr wireless cmmunicatins, Prc. IEEE, vl. 90, n., pp , Feb. 00. [8] A. F. Mlisch, M. Z. Win, and J. H. Winters, Capacity f MIMO systems with antenna selectin, in Prc. IEEE Int. Cnf. Cmmunicatins, vl., pp. 70-7, June 001. [9] Z. Chen, J. Yuan, and B. Vucetic, Analysis f transmit antenna selectin/maximal-rati cmbining in Rayleigh fading channels, IEEE rans. Veh. echnl., vl., n., pp , Jul. 00. [10] Z. Chen, J. Yuan, B. Vucetic, and Z. Zhu, Perfrmance f Alamuti scheme with transmit antenna selectin, Electrn. Lett., vl. 9, n., Nv. 00. [11] N.C. Sagias, and G.K. Karagiannidis, Gaussian class multivariate Weibull distributins: thery and applicatins in fading channels, IEEE rans. Inf. hery, vl. 1, n. 10, pp , Oct. 00. [1] M. H. Ismail and M. M. Matalgah, Perfrmance f dual maximal rati cmbining diversity in nnidentical crrelated Weibull fading channels using Padé apprximatin, IEEE rans. Cmmun., vl., n., pp. 97-0, March 00. [1] N. C. Sagias, D. A. Zgas, G. K. Karagiannidis, and G. S. mbras, Channel capacity and secnd-rder statistics in Weibull fading, IEEE Cmmun. Lett., vl. 8, n., pp , Jun. 00. [1] N. C. Sagias, and G. S. mbras, On the cascaded Weibull fading channel mdel, Jurnal f the Franklin Institute,, pp. 1-11, 007. [1] B. Sklar, Digital Cmmunicatins Fundamentals and Applicatins, Prentice-Hall PR, 00. [1] H. Zhang, and. A Gulliver, Capacity and errr prbability analysis fr rthgnal space-time blck cdes ver fading channels, IEEE rans. Cmmun., vl., n., Mar. 00. [17] L. Jiandng, L. Zhu, P. Jiyng, and C. Liang, he impact f training sequence n the capacity f MIMO system, China Cmm., pp. -7, Apr. 00. [18] M. Chiani, M.Z. Win, A. Zanella, On the capacity f spatially crrelated MIMO Rayleigh-fading channels IEEE rans. n Inf. her., vl. 9, pp. 71, Oct. 00. [19] X. Hailin, N. Zaiping, and Y. Shiwen, Channel capacity and digital mdulatin schemes in crrelated Weibull fading channels with nn-identical statistics, Jurnal f Systems Engineering and Electrnics, vl. 8, n., pp. 0-09, 007. [0] N. C. Sagias, Capacity f dual-branch selectin diversity receivers in crrelative Weibull fading, Eur. rans. elecmms., 17, pp.7-, 00. Mzabalaz Lupupa (S 08) was brn in Manzini, Swaziland in 198. He received his B.Eng. degree in electrnics at the University f Swaziland in 00. He is currently ding his Master f Science Engineering degree at the University f Cape wn in Suth Africa.