Performance bounds of space-time block coding in Rician and log-normal fading channels

Transcription

1 Performance bounds of sace-time block coding in Rician and log-normal fading channels GD Paadooulos, GK Karagiannidis and FN Pavlidou Abstract: Analytical exressions concerning the caacity and bit error rate (BER) of multile-inut multile-outut systems with sace-time block coding (STBC) are derived Two fading environments are examined, log-normal and Rician channels A tight closed-form uer bound is resented for the BER of systems oerating in log-normal fading environments in addition to an uer bound for the caacity of this tye of systems The latter bound alies to systems that oerate under Rician fading as well The analytical results were validated against amle numerical simulations for three STBC schemes and three hase-shift-keying s The roosed bounds roved to be a tractable way to evaluate the system erformance when no closed-form exression for the robability density function or the moment generating function is known 1 Introduction The ever-increasing need for low-cost broadband communications has become the driving force behind the extensive research in the field of wireless communications Wireless local area networks (WLANs) used for wideband transmission have attracted the interest of the scientific community with the bulk of studies focusing on the investigation of coding as well as techniques The need to examine the channel caacity and the bit error rate (BER) with mathematical analysis can become burdensome deending on the environment that is investigated In this aer, two channel models are taken into account; the Rician and the log-normal channel models Exerimental results at several frequencies suort the fact that the Rician model describes indoor (see Babich and Lombardi [1] and references therein) as well as outdoor environments, whereas, the log-normal distribution family is articularly true for indoor radio roagation environments, where terminals with low mobility have to rely on macroscoic diversity to overcome the shadowing from the indoor obstacles and moving human bodies In slowly varying lognormal fading channels, the small- and large-scale effects tend to get mixed, and the log-normal statistics tend to dominate and to accurately describe the distribution of the channel ath gain [2] In most scattering environments, antenna diversity is a ractical and convenient method aiming at ameliorating the detrimental effects of multiath fading A simle transmit diversity scheme for two transmitting antennas was first introduced by Alamouti [3] and generalised to an arbitrary number of antennas as sace time block coding (STBC) by Tarokh et al [4] Considerable research efforts have been devoted in recent years for the average caacity and the BER erformance, when alying multile-inut # The Institution of Engineering and Technology 2007 doi:101049/iet-com: Paer first received 3rd June 2005 and in revised form 30th June 2006 The authors are with the Telecommunications Division, Deartment of Electrical and Comuter Engineering, Aristotle University of Thessaloniki, Paneistimiouolis, Thessaloniki 54124, Greece ageo@authgr 86 multile-outut (MIMO) diversity and STBC to fading channels [5 12] The Shannon average caacity rovides imortant information for the maximum transmission rate of wireless communications systems On writing this aer, the caacity limits of wireless communications fading channels are of the utmost interest, because they reresent an otimistic bound for ractical communications systems The Shannon caacity of a MIMO channel is not always easy to estimate, usually because of the difficulty that arises when it is necessary to find the distribution of the sum of the random variables (RVs) that reresent the aths of the MIMO channel To overcome this difficulty, some caacity bounds are roosed in the literature; Cui et al [13] have suggested an uer and a lower bound for the caacity of MIMO correlated Rician fading channels, while Loyka and Kouki [14] have roosed an uer bound on the mean MIMO channel caacity The MIMO caacity in fading environments has been shown to grow significantly if the number of antennas is increased both at the transmitter and at the receiver [8, 15] As already known, the achieved information data rate of an STBC system is well below the theoretical caacity limit of the MIMO channel [9] Further, the number of transmitting and receiving antennas, N T and N R, resectively, are combined to roduce a diversity gain of order N T N R Regarding the BER erformance of a STBC system in a Rice distribution environment, the BER can be easily evaluated Nevertheless, when log-normal distribution is considered, there is difficulty in analytically evaluating the exact average BER arising from the fact that not closed-form exression for the moment generating function (MGF) is known; it can only be aroximated as yet Research has therefore concentrated on aroximations for this unsolvable roblem with several aroximate methods so far suggested [16, 17] In the work of Slimane [18], some erformance bounds have been suggested for the distribution function and recently, Berggren and Slimane [19] have roosed a lower bound for the outage robability In [20], closed-form exressions as well as bound aroximations are investigated for the outage robability, the average allocated ower, the achievable sectral efficiency and the BER over Nakagami-m fading channels in a single-inut single-outut (SISO) environment IET Commun, 2007, 1, (1), 86 91

2 In this aer, an invertible and closed-form uer bound for the average caacity in Rician or log-normal fading environments is derived using Jensen s inequality This bound is then tested in two different STBC schemes, to corroborate the roosed mathematical analysis An uer bound for the BER of a log-normal fading channel is roosed, based on the arithmetic-geometric mean inequality and on the caability of a MIMO channel that can be equal to a SISO one This bound is tested in three different STBC schemes and in three different hase-shift-keying (PSK) s Furthermore, we examine by analysis and simulation the exact BER erformance of a Rician fading channel model, testing it in the same STBC schemes and PSK s as in the log-normal fading channel BER analysis 2 System and channel model We consider a STBC system with N T transmitting and N R receiving antennas, hereafter referred to as STBC N T N R, oerating in a quasistatic flat fading channel, so that the ath gains are constant over a frame eriod and vary indeendently from one frame to another The system transmits a block of K symbols of energy E s, roduced from a comlex or real signalling, during a eriod K of T timeslots The symbols fx k g k¼1 are encoded by a STBC, creating a T N T column orthogonal transmission matrix with linear combinations of x 1, x 2,, x K and their conjugates The code rate R of STBC is determined by R ¼ K/T It should be mentioned that STBC achieves the maximum ossible transmission rate for any number of transmitting antennas when using any arbitrary real constellation (eg M-PAM) On the other hand, for an arbitrary comlex constellation (eg M-PSK and M-QAM), full-rate exists only for two transmitting antennas, while, for other cases, R is below unity (3/4 for three and for four transmitting antennas and 1/2 for all the other transmitting antenna schemes, ie N T 5) [4] The channel gain from the ith (i ¼ 1, 2,, N T ) transmitting to the jth ( j ¼1, 2,, N R ) receiving antenna is denoted as h i,j and is modelled as a comlex RV with its hase uniformly distributed within [0, 2), whereas its enveloe c i,j ¼ jh i,j j is distributed according to one of the well-known Rice or log-normal distribution families The corresonding robability density functions (PDFs) are given by [21, eq (216) and (253)] ð1 þ KÞe K ð1 þ KÞg g ðgþ ¼ ex g g s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ð1 þ KÞKg I 0 2 ; g 0 ð1þ g for the Rice distribution, where K is the Rician factor and I 0 () is the zero-order modified Bessel function of the first kind, and j g ðgþ ¼ffiffiffiffiffiffi ex ð10 log 10 g mþ 2 2 sg 2s 2 ð2þ for the log-normal distribution, where j ¼ 10/ln 10 ¼ 43429, and m (db) and s (db) are the mean and the standard deviation of 10 log 10 g, resectively Moreover, the instantaneous inut signal-to-noise ratio (SNR) for the STBC system is defined as g i;j ¼ E s N 0 jh i;j j 2 ð3þ and the channel gain matrix H as 2 3 h 1;1 h 2;1 h NT ;1 h 1;2 h 2;2 h NT ;2 H W h 1;NR h 2;NR h NT ;N R The entries of H are assumed to be uncorrelated, but not necessarily identically distributed, with arbitrary values for the fading severity arameters Furthermore, it should be noted that erfect knowledge of the channel matrix H is assumed at the receiver 3 Uer bound for the average caacity The evaluation of the average caacity, in a generalised fading environment, needs statistical averaging over the PDF of the instantaneous SNR However, in many cases, this PDF is either unknown (eg at the outut of equalgain-combining (EGC) receivers), or is in such a comlicated form (eg Rice fading) that it does not lend itself to this averaging To overcome this difficulty, we resent an invertible tight closed-form uer bound in terms of the average SNR For a signal s s transmission bandwidth BW and energy E s ¼ 1kjsj 2 l (with 1kl denoting averaging) over the additive white Gaussian noise (AWGN) channel with a single-sided ower sectral density (PSD) N 0, the Shannon caacity is given by C g ¼ BW log 2 ð1 þ gþ where g indicates the constant received SNR er symbol When the same signal s is transmitted over a fading channel, it exeriences multilicative fading of the enveloe of h Hence, g is a RV and is written as g ¼ E s N 0 jhj 2 The Shannon caacity can also be considered as a RV, because C g is straightforwardly connected to g in (5) The average channel caacity can be obtained by averaging C g over the PDF of g at the outut of the receiver [22], that is C g ¼ BW ð 1 0 log 2 ð1 þ gþf g ðgþ dg To overcome the already mentioned difficulties (nescience or comlicated form of f g ()) of evaluating the average caacity in closed form, a tight bound for C g can be derived using Jensen s inequality, because log 2 (1 þ x) isa concave function for x [ [0, 1) 1k log 2 ð1 þ gþl log 2 ð1 þ 1kglÞ Hence, by alying (8) in (5), C g can be uer bounded as given by the following simle exression C g BW log 2 ð1 þ gþ where ḡ is the average SNR er symbol at the outut of the receiver, ḡ ¼ VE s /N 0, with V ¼ 1kjhj 2 l being the average fading ower From (9), we observe that this uer bound is invertible, meaning that the average SNR can simly be exressed in terms of average caacity as g 2 C g =BW 1 ð4þ ð5þ ð6þ ð7þ ð8þ ð9þ ð10þ IET Commun, Vol 1, No 1, February

3 Also, comaring (9) with (5), we can see that the caacity of a fading channel is always less than the caacity of an AWGN channel with the same average ower [23] Hence, (9) rovides an uer bound for the average channel caacity, requiring only knowledge of a closedform exression for the received outut SNR From this it can be concluded that, if we consider a regular adative transmission system, in which the users adat their rates according to the instantaneous SNR, then the average caacity of these users will be a function of the average SNR that they are exeriencing In this case, the bound roosed in (9) gives an accurate evaluation of this average caacity as a function of the average SNR The convenient general STBC caacity exression is given in matrix form by [7, 8] C stbc ¼ BW R log 2 det I NR þ E s HH y ð11þ where I NR is an N R N R identity matrix and H denotes the transose conjugate of the channel matrix H Following the analysis of [8, 9]! C stbc ¼ BW R log 2 1 þ E X NT X N R s jh N T RN i;j j 2 ð12þ 0 and based on the inequality (9), the bound for C stbc measured at the outut of STBC system can be exressed as a function of the channel comonents where, based on [24] j¼1 C stbc BW R log 2 ð1 þ g stbc Þ g stbc ¼ E s X NT X N R j¼1 jh i;j j 2 ð13þ ð14þ which corresonds to the equivalent Gaussian SISO model for the average outut SNR [9] 4 Error-rate analysis 41 Rician fading As far as the Rician channel is concerned, the BER erformance is quite an easy task to comute, using the well-known formula for the MGF of the Rice fading model [21, equation (217)] M g ðsþ ¼ ð1 þ KÞ ð1 þ KÞ sg e Ksg=ð1 þ K sgþ ð15þ Thus, to evaluate the BER we need to find the distribution of a sum of Rician RVs Regarding the instantaneous SNR and the MGF, we have and g ¼ XN T N R g i M g ðsþ ¼ YN T N R M gi ðsþ ð16þ ð17þ Knowing this MGF, the BER for M-PSK is calculated using the integral [21, equation (578)] P e ¼ 1 ð ðm 1Þ=M M g a2 2 sin 2 du ð18þ u 88 0 where a 2 is a constant that deends on the secific / detection combination and is equal to a 2 ¼ 2sin 2 (/M) 42 Log-normal fading When the fading channel is log-normal, as no closed-form exression is known, the MGF can only be aroximated by [21, equation (254)] M g ðsþ 1 ffiffiffi XN n¼1 h H xn ex 10ð ffiffi 2 s xn þm Þ=10 s i ð19þ where x n are the zeros of the N -order Hermite olynomial, H xn are the weight factors of the N -order Hermite olynomial, and m (in db) and s (in db) are the mean and the standard deviation of 10 log 10 g, resectively, as in (2) Based on the arithmetic-geometric mean inequality, which says that the geometric mean is less than or equal to the arithmetic mean a 1 þ a 2 þa N Nða 1 a 2 a N Þ 1=N ð20þ and on the assumtion that a log-normal distribution is generally roduced by a normal one, ie if x is a normal variable, then e x is a log-normal one, from (14) and (16) we obtain for the instantaneous SNR E s g stbc ¼ N XT N R l¼1 E s jh l j 2 ¼ E Y s ðn N T RN T N R Þ 0 l¼1 ¼ E sn R N 0 R ex 2 N T N R N T N R ðe x l Þ 2 " # N XT N R l¼1 x l N XT N R l¼1 1=ðNT N R Þ ðe x l Þ 2 ð21þ where x l are normal variables with mean m and variance y The argument of the exonential factor is also equivalent to a normal variable, with mean m b ¼ 2m and standard deviation 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s b ¼ y ðn N T N T N R Þ R ð22þ ð23þ Finally, from (18), (19), (21), (22) and (23), we obtain the uer bound for the BER of a log-normal fading channel when STBC is alied ^P e ¼ 1 ð ( ðm 1Þ=M 1 XN ffiffiffi 0 n¼1 ffiffi H xn ex 10 ð 2sb x n þm Þ=10 b E sn R a2 N 0 R 2sin 2 du u ¼ 1 ð XN ðm 1Þ=M h ffiffi H 3=2 xn ex 10 ð 2sb x n þm b Þ=10 n¼1 0 E sn R N 0 R a 2 2sin 2 du ð24þ u The integral in the last equation is easy to evaluate when alying binary-hase-shift-keying (BPSK) IET Commun, Vol 1, No 1, February 2007

4 Fig 1 STBC caacity for Rician fading channel Fig 3 BER analysis for Rician fading channel with BPSK (M ¼ 2), so (24) reduces to ^P e ¼ 1 2 ffiffiffi XN H xn n¼1 1 erf s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!! ffiffi 10 ð 2sb x n þm b Þ=10 E sn R N 0 R ð25þ However, in the case that quadrature hase-shift keying (QPSK) (M ¼ 4) and 8-PSK (M ¼ 8) s are alied, the integral in (24) can only be evaluated with numerical analysis methods In our work, the integral is numerically evaluated with the hel of Male and Matlab and the results are resented in Section 5 5 Numerical evaluation and simulation results The results of the uer bound for the normalised to BW average STBC caacity (C STBC /BW ) and the BER erformance are resented here The results were obtained by running multile simulations, to minimise the statistical errors and to assure the validity of the results The simulation curves were created by evaluating a system, transmitting 2 22 ( ) data bits to achieve illustration of low BER The STBC systems examined here are the STBC 2 1, STBC 31 and STBC 3 2, and are tested in the two aforementioned environments; the Rician and the lognormal fading channels The STBC 21 is examined because it is the simlest STBC alication The STBC 31 and 32 are taken into consideration, to examine some other STBC schemes aart from Alamouti s roosal, with lower than unitary rate (R ¼ 3/4) It should be mentioned here that full rate full diversity STBC exists for any number of transmit antennas when real orthogonal designs, like BPSK, is considered The STBC scheme of rate R ¼ 3/4 considered for three transmit antennas is introduced in [4], thus we omit illustrating the transmission matrix Furthermore, it should be mentioned that the techniques taken into account are the BPSK, QPSK and 8-PSK 51 Caacity The arameters for the evaluated Rician channel are the Rician factor K, which is chosen to be 3 db, and the variance, which is set to 01 On the other hand, for the normal sequence used to generate the log-normal channel, the mean and the standard deviation have been chosen to be 0 and 01, resectively Based on the mathematical analysis in Section 3, we obtain the results for the analytical bound and the simulation of two STBC schemes (STBC 21 and STBC 3 2) in Fig 2 STBC caacity for log-normal fading channel Fig 4 BER analysis for Rician fading channel with QPSK IET Commun, Vol 1, No 1, February

5 Fig 5 BER analysis for Rician fading channel with 8-PSK Fig 7 BER analysis for log-normal fading channel with QPSK each of Figs 1 and 2 regarding Rician and log-normal fading channel, resectively The symbols without line corresond to the caacity of the simulated system, while the solid line corresonds to the uer bound From the resented results, it can be seen that the roosed bound is very tight, regardless of the environment and the STBC scheme that is examined Even though in both Figures the simulations symbols are very close to the uer bound curves, they are always below the analytical curves in all the SNR range 52 Bit error rate The BER erformance for the Rician channel is exhibited in Figs 3 5, alying BPSK, QPSK and 8-PSK s, resectively The solid line shows the analytical erformance of the systems under study, stemming from the mathematical analysis given in Section 4, and the symbols without line show the simulations results The simulations symbols have some fluctuations with resect to the analytical curves, because of the statistical errors generated from the noninfinite number of the simulated bits In Figs 6 8, the credibility of the uer bound for the BER in a log-normal fading environment is examined, when alying BPSK, QPSK and 8-PSK s, Fig 6 BER analysis for log-normal fading channel with BPSK 90 Fig 8 BER analysis for log-normal fading channel with 8-PSK resectively Again, with the solid line, we show the analytical results of the roosed uer bound and with the symbols without line we show the simulation results in each STBC scheme The arameters chosen for the BER analysis in both channel environments are the same with the ones taken into consideration in the caacity analysis However, for the uer bound curves of the BER of the log-normal environment, equations (21) (23) indicate the arameters for the log-normal distribution, which are for the STBC 2 1, the mean is equal to 0 and the standard deviation is equal to 01414; for the STBC 3 1, the mean is equal to 0 and the standard deviation is equal to 01155, and, for the STBC 32, the mean is equal to 0 and the standard deviation is equal to Finally, the abscissas and the weight factors for the N -order Hermite olynomial are taken from [25, Table 2510], with the order of the olynomial chosen to be N ¼ 20, so as to achieve better accuracy From all the resented Figures, comaring the erformance evaluation results to accurate comuter simulation ones, it is evident that the roosed bounds for both the average STBC caacity and the BER are very tight Hence, the roosed formulations rovide accurate exressions for evaluating these erformance characteristics IET Commun, Vol 1, No 1, February 2007

6 6 Conclusions In this work, we examined the caacity and BER of STBC wireless communications systems In articular, a closedform uer bound for the caacity of wireless systems oerating in either log-normal or Rician fading environments is derived As far as the BER erformance is concerned, albeit that it has been the subject of extensive research over the years, to the best of the authors knowledge, there exists no study that grales with this meaningful issue in lognormal fading environments to date Toward this end, a tight uer bound for the BER has been resented The validity of the roosed analysis was reinforced by numerical simulations for different STBC schemes and different PSK s The analysis resented in this aer aves the way for an accurate evaluation of the erformance of wireless systems in several fading environments 7 References 1 Babich, F, and Lombardi, G: Statistical analysis and characterization of the indoor roagation channel, IEEE Trans Commun, 2000, 48, (3), Alouini, M-S, and Simon, MK: Dual diversity over correlated log-normal fading channels, IEEE Trans Commun, 2002, 50, (12), Alamouti, SM: A simle transmit diversity technique for wireless communications, IEEE J Select Areas Commun, 1998, 16, (8), Tarokh, V, Jafarkhani, H, and Calderbank, AR: Sace-time block codes from orthogonal designs, IEEE Trans Inf Theory, 1999, 45, (5), Zhang, QT, and Liu, DP: A simle caacity formula for correlated diversity rician fading channels, IEEE Commun Lett, 2002, 6, (11), Goldsmith, A, Jafar, SA, Jindal, N, and Vishwanath, S: Caacity limits of MIMO channels, IEEE J Select Areas Commun, 2003, 21, (5), Sandhu, S, and Paulraj, A: Sace-time block codes: a caacity ersective, IEEE Commun Lett, 2000, 4, (12), Foschini, GJ, and Gans, MJ: On limits of wireless communications in a fading environment when using multile antennas, Wire Personal Commun, 1998, 6, Carrasco, H, Fonollosa, JR, and Penin, JAD: Channel caacity of sace-time block coding Proc 1st Mobile Commun Summit 2001, Setember 2001, Barcelona, Sain 10 Chuah, C-N, Tse, DNC, Kahn, JM, and Valenzuela, RA: Caacity scaling in MIMO wireless systems under correlated fading, IEEE Trans Inf Theory, 2002, 48, (3), Zeng, XN, and Ghrayeb, A: Performance bounds for sace-time block codes with receive antenna selection, IEEE Trans Inf Theory, 2004, 50, (9), Femenias, G: BER erformance of linear STBC from orthogonal designs over MIMO correlated Nakagami-m fading channels, IEEE Trans Veh Technol, 2004, 53, (2), Cui, XW, Zhang, QT, and Feng, ZM: Generic rocedure for tightly bounding the caacity of MIMO correlated Rician fading channels, IEEE Trans Commun, 2005, 53, (5), Loyka, S, and Kouki, A: New comound bound on MIMO channel, IEEE Commun Lett, 2002, 6, (3), Telatar, E: Caacity of multi-antenna Gaussian channels, Eur Trans Telecommun, 1999, 10, (6), Beaulieu, NC, Abu-Dayya, AA, and McLane, PJ: Estimating the distribution of a sum of indeendent lognormal random variables, IEEE Trans Commun, 1995, 43, (12), Tellambura, C, and Annamalai, A: An unified numerical aroach for comuting the outage robability for mobile radio systems, IEEE Commun Lett, 1999, 3, (4), Slimane, SB: Bounds on the distribution of a sum of indeendent lognormal random variables, IEEE Trans Commun, 2001, 49, (6), Berggren, F, and Slimane, SB: A simle bound on the outage robability with lognormally distributed interferers, IEEE Commun Lett, 2004, 8, (5), Alouini, M-S, Tang, X, and Goldsmith, AJ: An adative scheme for simultaneous voice and data transmission over fading channels, IEEE J Select Areas Commun, 1999, 17, (5), Simon, MK, and Alouini, M-S: Digital communication over fading channels (Wiley-Interscience, New Jersey, 2005) 22 Lee, WCY: Estimate of channel caacity in Rayleigh fading environment, IEEE Trans Veh Technol, 1990, 39, (3), Goldsmith, AJ, and Varaiya, PP: Caacity of fading channels with channel side information, IEEE Trans Inf Theory, 1997, 43, (6), Bauch, G, and Hagenauer, J: Analytical evaluation of sace-time transmit diversity with FEC coding Proc IEEE GLOBECOM 2001 Conference, No 1, San Antonio Texas, USA, November 2001, Abramowitz, M, and Stegun, IA: Handbook of mathematical functions with formulas, grahs and mathematical tables (Dover Publications, New York, 1970, 9th edn) 26 Hanzo, L, Liew, TH, and Yea, BL: Turbo coding, turbo equalisation and sace-time coding for transmission over fading channels (IEEE Press, John Wiley & Sons, 2002) IET Commun, Vol 1, No 1, February