AN APPLICATION OF A GENERALISED JAKES MODEL FOR MIMO CHANNELS

Transcription

1 AN APPLICATION OF A GENERALISED JAKES MODEL FOR MIMO CHANNELS Davi B. Smith (1) (1) Faculty of Engineering (Telecommunications), University of Technology Syney PO Box 13 Broaway NS 007 Australia avism@eng.uts.eu.au ABSTRACT Jakes moel for frequency flat faing processes in mobile raio systems is extene to allow for better moelling of a space-time Rayleigh faing multiple-input multiple-output (MIMO) channel. A conventional four transmit an two receive (4,) MIMO raio channel is analyse, an a ring of scatterers moel is use to fin the faing channel istortions. A base station antenna spacing for appropriate partially correlate faing is assume. Differential spacetime moulation is use over the channel. A general tren is presente showing optimality of a range of frame lengths over the MIMO channel for rapi ecoing in this ifferential space-time application. The importance of restricting the frame lengths within certain limits is also emonstrate. INTRODUCTION Diversity is one effective means for proviing performance improvements over faing channels, principally by mitigating the faing that occurs. The combination of temporal an spatial iversity can significantly improve the communication quality in a rich scattering environment. This combination has been emonstrate through proposals for space-time coe moulations. The majority of these proposals have relie on accurate channel state information for ecoing. hen assuming space-time moulation it is often assume that spatial channels are uncorrelate when consiering time-varying faing; a conition which is har to satisfy. Base on this a space-time moel for a two transmit an one receive (,1) multiple-input single-output (MISO) channel is further generalise to a typical (4,) MIMO raio channel in a partially correlate faing environment [1]. The moel use is a space-time generalisation of Jakes moel []. For this application space-time moulation is use base on recent proposals for ifferential space-time moulation schemes which require no channel-state information. Some of these use linear ispersive iagonal coes that can be easily generate [3]. It has recently been shown how to rapily ecoe these ifferential moulation schemes over flat faing channels [4]. This paper emonstrates optimal transmission frame lengths for newly evelope rapi approximate ifferential ecoing over a Rayleigh fast-faing MIMO channel. The optimal transmission frame lengths are emonstrate relating to non-optimise block error rates for flat faing channels, using the generalise Jakes moel evelope. CHANNEL MODEL A typical scenario for Rayleigh faing MIMO channels is as follows; two, N R =, co-locate mobile station (MS) antennas an four, M = 4, base station (BS) antennas moelle as Fig. 1, where a scatterer ring is place aroun the MS to moel the multipath reflectors. The scatterers are uniformly istribute on the ring, an each scatterer has an inepenent, uniformly istribute initial phase over [-S,S). The moel is meant to represent an average channel for the purpose of macroscopic system esign, an not to escribe iniviual channel realisations. The antennas in the moel are assume to be omniirectional. The moel is also not esigne to apply to a fixe wireless system where the Doppler sprea is mainly ue to the motion of the scatterers. It is possible, however, that this moel coul be extene to the case of the fixe system. For the purposes of this analysis the co-locate mobile antennas are spatially ecorrelate. The general expression for the flat faing channel istortions from each of the BS transmit antennas, m = 1 4, to either of the MS receive antennas, n r =1,, without normalisation allowing for the number of scatterers (i.e. multipath reflectors) is given by:-

2 v MS1, a [ D n,1 P n,1 T U s 11 s 1 s 31 s 41 E c BS1 BS BS3 BS4 sp sp sp Fig.1. The MIMO transmission moel between mobile an the base station cs ^j> f t [ D ` N m, n ( t) exp D cos n n n n ' r, r, I r n, m, n 1 S (1) where, in reference to Fig. 1, N is the number of reflectors, f D is the Doppler sprea cause by vehicle movement, Dn, is the angle of the nth reflector on the scatterer ring relate to the n r th MS antenna; Dn,1 = Sn/N, Dn, = S(n+0.5)/N, [ is the angle of mobile motion, In, is the initial phase of the nth scatterer relate to either MS antenna receive at the secon BS antenna, an 'In,m, is the phase ifference cause by the path length ifference from the nth reflector to the mth BS antenna with respect to the secon BS antenna. The 'In,m, are eterministic an can be evaluate by 'In,m, = S(s, s m, )/O these are efine in Appenix 1. Both phases In,1 an In, will be moelle as uniformly istribute on [-S,S), but are ecorrelate ue to offset. Due to approximations in the (4,) MIMO moel use, an the generality of the macroscopic system moel, in orer to moel each of the faing channel istortions the variance of the expression given in (1) for each of the faing channel istortions is not exactly N. Obtaining the variances for approximate spatial ecorrelation, at the appropriate BS antenna spacing, sp, for each of the channel istortions enables better channel moelling. This gives an approximate *5(0,1) moel of the channel by normalisation by these variances when the faing coefficients between the antennas are spatially inepenent. This variance is enote var(cs_) m,, ; giving the proper normalise faing channel istortions as c V m, n t cs r m, var( cs _ ) m, t () where the variance of the channel, V = 1 in this paper. Equation () allows for the moelling of coherent etection, although this is not require in the following sections, in which non-coherent etection is analyse. The space-time cross-correlation between ajacent BS antennas can be foun in a manner similar to [1] for the case of the MIMO channel. hen = 0, the approximate optimal spatial correlation can be erive for inepenent faing; similarly an appropriate sp for partial spatial correlation can be foun. If one assumes = 1000O, a = 5O, E = S/4, [ = S/4, the optimal BS antenna spacing is foun to be O. For a spatial correlation of 0.7 the BS antenna spacing sp is 10O, for a spatial correlation of 0.5, sp 14O. In reference to Fig. 1, E is the mobile position angle with respect to the en-fire of the BS antennas an is the istance from the mobile to the centre of the four BS antennas.

3 DIFFERENTIAL SPACE-TIME MODULATION In the introuction a moulation scheme which requires no channel-state information was iscusse briefly. The moulation scheme, calle unitary space-time moulation, is ieally suite for Rayleigh fast faing environments. It oes not require the receiver to know or learn the propagation coefficients. The complex-value signals are orthonormal with respect to time among the transmitter antennas. Importantly, when viewe as vector functions of time, the signals carry the message information entirely in their irections [3,5]. The signals transmitte over the M antennas in the MIMO channel will be groupe in time blocks of size M, as in [3,4]; will be use to inex the time blocks. The transmitte signals are organise in an MuM matrix S where the column inices represent the ifferent antennas an the row inices represent the time samples t = M,,M+M 1. The matrices are power normalise so that the total transmitte power oes not epen on M. The moel for the channel can be written compactly as X U S H for 0,1, (3) where X is the MuN R matrix of receive signals, x ;m,n. is an MuN R matrix of aitive inepenent *5(0,1) receiver noise. The MuN R matrix H contains the faing coefficients which are given by the flat faing channel istortions () in the previous section, note that across blocks h ;m, an h ;m, are not inepenent an the faing coefficients are also not time-invariant within a block, U is the effective SNR at the receiver. One block takes up M uses of the channel, so that a rate R requires L = RM ifferent signals. Each signal is an MuM unitary matrix V l from a constellation = of L such matrices. The bits to be transmitte are packe into an integer ata sequence z 1 z {0,1,, L 1}. The relevant transmission equation, [3], is S V S, 1,, with S I (4) z 1 0 M The specifie constellations in [3,4], with full iversity an high iversity prouct, are use, which are given by V l V l 1, where V 1 iag j u L j u L j um L > e, e 0 l L S 1 S S (5) where u m are integers between 0 an L-1 an u 1 = 1. The constellation is thus entirely efine by u u M. Because = forms a group every transmitte matrix S belongs to =. This implies that at any given time only one antenna transmits a phase-shift keying (PSK) symbol. Differential space-time moulation is typically ecoe using maximum-likelihoo (ML) ecoing, which in this case is given by ˆ (6) z ml arg min X Vl X 1 l 0,..., L1 where F represents the Frobenius norm, A F = tr(a A) an represents the ajoint operator. TRANSMISSION FRAME LENGTHS FOR RAPID DIFFERENTIAL DECODING Using the nature of the transmitte signals one can obtain a goo approximation of ML ecoing by a form of pseuo-differential PSK (DPSK) ecoing. This is base on the approximation of the cosine representation of the ML ecoer use for fast lattice ecoing [4]. The cosine representation is ML l N R A, cos u l M 1m 1 m m F zˆ arg max S L (7) M m,

4 where A m, = x ;m,. x ;m, 1/ _x represents the geometric mean of the moulus of the signals, an Mm,=arg(x ;m,/ x ;m,)l/(s), where arg(x) has the range [-S,S), represents their phase-ifference in units of S/L. However this is not a fast lattice ecoing proceure, [4,6], but is an even quicker proceure because the process of basis reuction is not require. A better ecoer output, ẑ ec, is obtaine (with little computational overhea) for large symbol constellations, by checking a much smaller subset of symbols by ML ecoing aroun the original ecoer answer [6]. The ifferential space-time moulation schemes escribe in the previous section have greater utility at higher signal-to-noise ratios (SNR). The optimality of transmission frame lengths can be foun using rapi ifferential ecoing, which closely approximate to those for a moification of fast lattice ecoing to MIMO channels [6]. This optimality can be emonstrate by the general tren of the block-error rate versus transmission frame lengths at a fixe SNR, in Fig. this is 17.5 B. The block-error rate is the number of times that ẑ ec is not equal to z. The effect of varying the SNR on the optimality of three particular transmission frame lengths is shown in Fig. 3; blockerror rate is again use as the performance measure. In both Fig. an Fig. 3 the faing parameter, f D T = (where T is the symbol perio). The channel moel use is the generalise Jakes (4,) MIMO channel moel as escribe in the previous section, with the same moel parameters use as specifie for spatial correlation, U 0.5. The scatterer ring contains N = 34 reflectors. A rate R = ifferential unitary space-time coe is use for an L = 56 symbol constellation. The pseuo-dpsk scheme is use for rapi ecoing. In Fig. 3, three ifferent frame lengths are use, 00, 1 an 0, corresponing to 50, 53 an 55 blocks being transmitte in one frame, for a range of SNR from 1.5 B to 30 B Block Error Rate Frame Length Fig.. Performance Tren of Frame Lengths for Rapi Differential Decoing over a MIMO channel, SNR = 17.5 B, using Block Error Rate as Performance Measure Frame Length = 00 Frame Length = 1 Frame Length = 0 Block Error Rate SNR (B) Fig. 3. Performance of three particular Frame Lengths for Rapi Differential Decoing over a MIMO channel, for high SNR.

5 It is clear that there is a significant rapi increase in block-error rate for frame lengths from ~160 to ~40 in Fig., i.e. for number of blocks transmitte from ~40 to ~60. The general tren of transmission frame length optimality fluctuates, i.e. the curve of block-error rate shown in Fig. is not always increasing. This is espite the general tren of egraation in block-error rate as might be expecte. The results in Fig. show clearly the effect of a wie range of transmission frame lengths in a typical MIMO channel using rapi ifferential ecoing. They show the importance of restricting the frame lengths, an thus the numbers of blocks transmitte within certain limits to optimise the performance of the system. Fig. 3 provies a comparison with the results of Fig., emonstrating what happens at three critical transmission frame lengths in this ifferential space-time ecoing application. A high range of SNR is use for purposes of vali comparison; it is clear that the performance iminishes rapily as the block length (number of blocks in one frame) increases from 50 to 55 for M = 4 transmit antennas. The improvement in performance as the SNR increases also iminishes significantly as the frame length increases. It shoul be note that the block-error rate is orers of magnitue larger than what the corresponing bit-error rate woul be, an is thus a strict performance measure. CONCLUSIONS The application of a space-time generalisation of Jakes moel for a typical Rayleigh faing MIMO channel to fast ecoing of ifferential unitary space-time moulation has emonstrate some significant trens. These trens give some guielines an insight in relation to esirable transmission frame lengths for this application. This shoul be an important consieration in the esign of a number of applications using space-time coing. It is clearly emonstrate that transmission frame lengths for this typical (4,) MIMO channel shoul be restricte to less than 00 for goo performance at high SNR. For better performance, frame length shoul be restricte to less than 160. For further optimisation of performance below a frame length of 160, some consieration coul also be given to fluctuations in performance at ifferent frame lengths. The results from the generalise Jakes moel applie to a (4,) MIMO channel has possibly uncovere some limitations in the uses of fast ifferential ecoing of unitary space-time moulation. Further, the statistical complexity of this new space-time generalise Jakes moel is still yet to be fully unerstoo, an warrants further analysis. It is postulate that it is more rigorous than previous applications of Jakes moel to MIMO or MISO channels, hence better suite to obtain goo performance measures. However the level of rigour in this MIMO channel moel requires further investigation. ACKNOLEDGEMENTS This work was carrie out with the support of a octoral scholarship from the University of Technology, Syney (UTS). The author thanks Dr Tim Aubrey of the Faculty of Engineering at UTS, for his useful critique of this work an his continue supervision of octoral research. REFERENCES [1] Chen, T.A., Fitz, M.P., Kuo,.Y., Zoltowski, M.D., Grimm, J.H. A Space-Time Moel for Frequency Nonselective Rayleigh Faing Channels with Applications to Space-Time Moems, IEEE Journal on Selecte Areas in Communications, vol. 18, pp , July 000. [] Jakes,.C., Microwave Mobile Communications, n e., IEEE Press, New York, [3] Hochwal, B.M., Swelens,., Differential Unitary Space-Time Moulation, IEEE Transactions on Communications, vol. 48, pp , Dec 000. [4] Clarkson, K.L., Swelens,., Zheng, A., Fast Multiple-Antenna Differential Decoing, IEEE Trans. Comm., vol. 49, pp , Feb 001. [5] Hochwal, B.M., Marzetta, T.L., Unitary Space-Time Moulation for Multiple-Antenna Communications in Rayleigh Flat Faing, IEEE Trans. Inform. Theory, vol. 46, pp , Mar 000. [6] Smith, D.B., Fast Differential Decoing for a MIMO Raio Channel, Submitte to Electronics Letters Sep. 001.

6 APPENDIX 1 In this appenix the expressions for path-length ifferences from the nth reflector in the scatterer ring aroun the colocate MS antennas to the four BS antennas is given, with reference to Fig. 1. From these the relevant phaseifferences, 'In,m,, (1), can be erive. By using far-fiel assumptions, which are typical for wireless communications, whereby 1,, 3, 4 S a, T U, cost 1, ( + 3 ) an ( 3 ) sp cose, an making further appropriate simplifications, we obtain the following approximations for the path length ifferences sp a sp s #, s1, sp cos sin E sin E sin E D n, ¹ E (8) sp s, s3, # sp cos E sin E sin E D n, a (9) sp a sp s, s4, # sp cos E sin E sin E sine D n, 'I n,m,nr can then be evaluate by S(s, s m, )/O (10)