Communications and Network, 2013, 5, 319326 http://dx.doi.org/10.4236/cn.2013.53B2059 Published Online September 2013 (http://www.scirp.org/journal/cn) Diversity–Multiplexing Tradeoff and Outage Performance for 2×2 DualPolarized Uncorrelated Rice MIMO Channels* Yanping Huang, Guangliang Ren State Key Lab. of Integrated Services Networks, Xidian University, Xi’an, China Email: huangyp207529@163.com Received July, 2013 ABSTRACT In this paper, diversitymultiplexing tradeoff (DMT) curve for 2×2 DualPolarized uncorrelated Rice MIMO chan nels is studied. Exact expressions for statistic information of mutual information exponent are derived. Impacts of channel pa rameters such as signal to noise ratio (SNR), kfactor and cross polarization discrimination (XPD) on mutual informa tion exponent are analyzed. Compared to conventional singlepolarized (SP) Rice MIMO systems, a qualitatively dif ferent behavior is observed for DP Rice systems. The work in this paper, allows to identify quantitatively for which channels (kfactor) and SNR levels the use of dual polarization becomes beneficial. Gamma or lognormal distribution are used to describe mutual information component, and a theoretical formulation for finiteSNR DMT curve in 2×2 DP uncorrelated Rice channels is presented for the first time, which is valid in low and medium SNRs when multiplexing gain is larger than 0.75. Keywords: DMT curve; DualPolarized; Uncorrelated Rice Channel; Mutual Information Exponent; kfactor; Outage Probablity Approximation 1. Introduction Due to the spacecost of the conventional Singlepolar ized (SP) multipleinput multipleoutput (MIMO) sys tems, dualpolarized (DP) MIMO has been receiving much attention as an attractive alternative for realizing MIMO architectures in compact devices [17]. Co mpared with SP MIMO, DP MIMO exhibits many different characteristics. For instance , in [1] it has been clearly illustrated that in Rice fading, after some kfactor (de fined as the ratio of the power in the fixed exponent to the power in the variable exponent), error probability of zeroforcing detection method for polarization multi plexing starts to decrease with increasing kfactor, while SP systems perform the opposite. Moreover, various lit eratures, such as in [14], an idea has been well devel oped that polarization diversity wo rks well only in corre lated Rayleigh fading or Rice fading channels with LOS components. It is necessary to note that measurements have been done to get real parameters of DP channels, which helps in getting more accurate polarized channel model [6]. To go further, channel correlation and capac ity are discussed in these literatures, proving that such dual polarization has decorrelation effect on correlated channels from a practical aspect. Nevertheless, this result does not extend to diversity systems, such as Almouti coded MIMO, where polarization confronts performance loss [2]. In conv entional MIMO systems, it is known that there exists a fundamental tradeoff between achievable diversity and multiplexing gains of any transmission over tr nn MIMO channel, i.e., diversitymultiplexing tradeoff (DMT), as has been clearly illustrated in [8] for sign al to noise ratio (SNR) approaching infinity. Moreover, it is also pointed out that DMT curve at finite SNR is quite different [913]. Under realistic propagation conditions, since SNR cannot reach infinity, it would be meaningful to study DMT behavior at finite SNRs that are practical in operating regimes. Up to now there are no literatures that investigate finiteSNR DMT for dualpolarized sys tems. In previous literatures [913], DMT curve is discussed based on the assumption that, elements of H follow Wishart distribution. However, for polarized MIMO, because of the asymmetric properties of the generalized channel matrix, random matrix theory results fo r Wishart matrices cannot be leveraged. Inspired by the idea pro posed in [14], which used gamma, lognormal or weibull *This work was supported in part by the State Natural Science Founda tion of China, Grant No.61072102 and National Major Specialized Project of Science and Technology, Grant No.2011ZX030010 0 7 01. C opyright © 2013 SciRes. CN
Y. P. HUANG, G. L. REN 320 distribution to approximate outage capacity for dual polarized MIMO in high SNR regime, by approximating mutual information exponent, we get theoretical DMT curve for DP in low and medium SNR regimes in 2×2 uncorrelated Rice channels. The rest of this paper is organized as follow. Section2 describes the channel model developed for a 2×2 DualPolarized uncorrelated Rice fading channels. Sec tion 3 discusses statistic characteristic of mutual infor mation exponen t, outage probability and DMT curve and their approximations. Section4 shows the simulation re sults. Finally, section5 is the conclusion. In this paper, and Ex Dx represents the ex pectation and variation of random variable x, respectively, * stands for the elementwise conjugation, H for conju gate transpose, det is the determinant of matrix A. 2. System Model and Definitions Consider a system with one dualpolarized transmit and one dualpolarized receive antenna. The channel is as sumed frequencyflat over the band of interest. The chan nel matrix is given by 11 21 12 22 hh Hhh (1) Assume that both transmitter and receiver employ the same polarization scheme, i.e. both of them employ horizontal/vertical or slanted polarization. Decomposing the channel matrix into the sum of a fixed exponent and a variable exponen t as 1 11 k H kk H (2) The elements of the matrix do not vary and satisfy 22 1122 1hh, 22 12 21 hh . The elements ij h of the matrix are complex random variables, which satisfy 22 22 11 221221 1,Eh EhEhEh (3) 111212 2212 211122 0; 0Ehh EhhEhh Ehh (4) where 01 f , 0 1 ,,;,1,2 11 ~1 ,,;, 11 ij f k Nijij kk hk Nij kk 1,2ij (5) In [8], conventional asymptotic definitions of multi plexing and diversity gains for a MIMO channel are given by: *log lim log R r (6) *log lim log out P d (7) where and represent the asymptotic multiplex ing and diversity gain respectively, * r* d is the average SNR per receive antenna, R is the system data rate and out is the outage probability. Assuming that no CSI is available at the transmitter, is defined by P out P P2 out PI RPW (8) where I is the mutual information between received and transmitted signals over the MIMO channels, and is the mutual information ex ponent satisfy W log W. The asymptotic DMT is given by the piecewise linear function connecting the points , where * ,id i * di is given by [8]: *, 0,..,min, rt r dinin iinn t n (9) r, t are numbers of receive and transmit antennas, respectively. Note that the asymptotic DMT describes situation where SNR approches infinity. However, for practical system design, it is desirable to characterize the diversitymultiplexing tradeoff at operational SNRs. The finiteSNR definitions for diversity and multiplexing gains can provide useful tool to characterize the DMT at real environment. The finiteSNR multiplexing gain r is defined as the ratio of to the capacity of an AWGN channel at SNR with array gain [11], n n RG min , rt Gn log 1 R rG (10) The finiteSNR outage probability , out Pr for a given and r is given 1 are related to the XPD for the fixed and variable exponent of the channel, respec tively. Good discrimination of orthogonal polarizations amounts to small values of and , and vice versa. Clearly, when f1, 1 the model becomes the conventional SP (single polarization) channel. For 2 × 2 DualPolarized uncorrelated Rice MIMO channels ij are complex Gaussian random variables whose parame ters are: h 1 P, 2r G out rPW (11) The finite SNR diversity gain ,dr is defined by the negative slope of the plot ,Pr out versus log : , , out out Pr dPr (12) Copyright © 2013 SciRes. CN
Y. P. HUANG, G. L. REN 321 3. Computation of DP finiteSNR DMT In this section, the DMT for 2×2 Dualpolarized Rice channels is examined. First, we derive an exact expres sion for the mean and variation of mutual information exponent, based on which some discussions on channel parameters are proposed to have a deeper insight into dualpolarization system. Second, using the expressions of statistic information derived in the first step, approxi mation equations of outage probability are presented. Finally, DMT for both asymptotic and finiteSNR in 2× 2 Dualpolarized Rice channels at are investigated. 3.1. Statistic Information of Mutual Information Exponent Consider that channel state information (CSI) is perfectly known at the receiver. The MIMO mutual information I conditioned on the channel realization is given by 1 logdet log1 r H H n rank HH i it IIHH N (13) where 11 H rank HH i it Wn , and i denotes the eigenvalues of H. For the case of or MIMO, mean and variances of W as a function of kfactor and 2 t n2r n, f are expressed be low: 2 1ii EW EE nn (14) 24 3 2, i ii WD WDD nn R n i (15) where , and min , rt nn n i E , i D , i E , , i R i are given in the Appendix 1, for the sake of space saving. The distribution of the mutual information exponent provides information about the available diversity in the system. describes the ergodic mutual informa tion exponent, which can be used to get upper bound of mutual information I. And presents some in formation about outage probability, i.e., the smaller the variance, the lower the probability of the outage error is when transmitting at a fixed rate [8]. EW DW From the analytical expression of and EW DW given in (14)(15), we find that both of them are influ enced by kfactor and SNR. With the existences of po larization indicators and , the influence are dif ferent. Let SP EW, W DP be mean of information exponent of SP and DP, we get E 2 2n 1 1 SP EWAk BkC k DP EW (16) Then kfactor fo r E DP EW W SP is divered: 24 2 BB AC k (17) where 2 22 2 1221 2 121 2 ff f An Bn Cn f (18) In suburban area, where XPD is measured to be 815dB range, let =0.4, 0.3 f . We find that in SP, mean of mutual information exponent decrease fast with the increase in kfactor, while for DP, declension is less. At 0dB , required kfactor to fill the gap between DP and SP is k= 0.5754 or 2.4376; 10dB , required k = 11.3118 or 0.4546; when , k = 4.0184 or 0.4266. 3.2. Approximating of Outage Probability Motivated by the work [14], in this section, we derive the approximation curve for outage probability at finite SNR for 2×2 dualpolarized uncorrelated Rice channels. The steps begin with the approximation of statistical information of mutual information exponent W. If we assume gamma distribution for W, i.e. /w p 1,0 Wp e fw w p w (19) 22 , ,, , ,, DWDW k kEWEW k EWEW k pk DWDW k (20) Then outage probablity at given multiplexing gain and SNR is 1 ,p , r out G Pr p (21) where is the incomplete gamma function. 1 0 ,xkt kxte dt If we assume lognormal distribution for W, i.e. Copyright © 2013 SciRes. CN
Y. P. HUANG, G. L. REN 322 2 2 ln 2 2 1, 2 wu W fwe w w 0 (22) 2 1 lnlnln 1 2 DW uE WEWEW (23) 2 2 lnln 1DW DW EW (24) Then ln 1 11 P, 22 2 out rGu rerf (25) where both and are given in section2. Note that format parameters of W are directly related to polarization parameters EW DW , f as well as kfactor, SNR. Corollary: When , for outage probability of DP, k P0,0 min, outr t rN N (26) In contrast, outage probability in SP is given by [12] 01 P1, 1 out r r (27) Proof: As , for conventional SP, the Rice fad ing channel approaches a rankone AWGN channel, such that the outage probability is 1 for , and 0 for ; However, for DP Rice fading channels, as , thanks to polarization orthogonality, channel matrix remains full rank. Thus, as k increases, channel approach two rankone AWGN channels. Therefore as , outage probability for both k 1r 0 1r k k DP P out 1r and . 1r 3.3. Asymptotic DMT for Rice Dual Polarized Channels Theorem: The asymptotic DMT curve for dualpolarized channels is independen t of , , which is identical to conventional asymptotic DMT in SP channels as de scribed in (9) [8 ]. Proof: The proof is given in appendix 2. 3.4. Diversity and Multiplexing Tradeoff at Finite SNR Simulated by the method in [11], we get finiteSNR DMT using (11). 1 0 12 P, ,, ,, r G out r wkdw Ar Ar (28) where for gamma approximation, /, ,1 , ,, ,, wk pk k e fw kwpk k (29) 1 1 1 20 ,1 1, ,,, r rr G , rrGGfG Arfw kdw k (30) For lognormal approximation, calculation step is similiar, which is omitted for the sake of space. 4. Simulation 4.1. Impact of kfactor and SNR on Mutual Information Exponent As it has been known that DP and SP systems perform rather diffident in Rice channels. In order to study how such a difference occurs, Figure 1 plots EW and DW as a function of the kfactor at 0dB and 10dB . Assume that for DP, 0.4 , 0.3 f , and without loss of generality ,take 1 ij h for ,1,ij 2 . The theory curves are identical to the ones by Monte Carlo simulations, which validate the derived expression of EW and DW in section 2. As expected, a quantitatively differen t behavior is observed for DP Rice system. Although either in SP or DP case, expectation and standard deviation of the mutual information expo nent drop dramatically with increasing kfactor, espe cially in low kfactor regime, where kfactor manifests the variation of W. It is clear that in DP, the drop is far less than that in SP both for and EW DW dB , since polarization can reduce the channel correlation brought by LOS component. Moreover, at 0 , no cross points for Wk SPDP at are found. But at EW E0 10dB , 11 k , a cross point appears ,matching the previous results from (17). Such a phenomenon can Figure 1. Comparison of mean and variation of W for SP and DP. Copyright © 2013 SciRes. CN
Y. P. HUANG, G. L. REN 323 be explained by the effect of eigenvalue of H. At medium SNR, minimum eigenvalue begins to affect channel information exponent. For conventional SP 2×2 Rice systems, when kfactor increases, channel matrix tends to be a rankdeficient matrix, leading the minimum eigenvalue to be smaller even approaching zero. In con trast, eigenvalues of DP systems nearly stays constant, without being hugely affected by varying kfactor. Hence, channel matrix does not become ill conditioned, i.e., not badly affected by LOS component. Thus, in seminars with strong LOS component, we suggested DP be used. To illustrate the different eigenvalues we can see Figure 2. 4.2. Impact of XPD on Mutual Information Exponent As a final parameter dependency study, we examine on mutual information exponent as a function of the XPD in LOS component. Using the analytical formation in sec tion2, Figure 3 plots plots and as a function of the EW DW at 0dB and 10dB , with fixed , 10 k0.4 . Figure 2. Eigenvalues of SP and DP for 2×2 uncorrelated Rice fading. Figure 3. XPD influence on mutual information exponent in 2×2 Rice uncorrelated channels with k=10. From Figure 3, it is clear that at low SNRs in creases with EW . However, at moderate SNR, starts to drop with improving EW . Conclusions can be made that XPD and SNR have impacts on the mutual information at the same time. It is then meaningful to find the optimal SNRs for different DP systems for opti mal code design. 4.3. Outage Probability in Finite SNR In this part, we study some plots of outage probability versus SNR in uncorrelated Rice fading with rt nn 2G . In Figure 4, given a fixed multiplexing gain 1r , outage probability versus SNR curves are plotted for SP and DP at 5,12k . It is seen that contrary to SP, outage probability of DP always drops as kfactor improves. At some SNR, negative gap of outage probability between SP and DP turns into positive, coinciding with previous analysis. In Figure 5, gamma or lognormal approximation are Figure 4. Outage probability SP VS DP for k =5 and k =12. Figure 5. Outage probability approximation for different multiplexing gains and kfactor DP. Copyright © 2013 SciRes. CN
Y. P. HUANG, G. L. REN 324 plotted as well as results of Monte Carlo simulation for outage probability in various multiplexing gain. The dash curves represent the approximation value, while the cir cle, square symbol represent the DP systems for respectively. When (in the plot), the gamma approximation matches the simulation well. At (), we can use the lognormal distribution instead, which works well in medium SNR 015dB. Note that the higher multiplexing gain, the more accuracy of the gamma approximation. By using this approximation method, it becomes simple to estimate the outage probability of DP Rice channels at or me dium SNRs without timecost simulation. 5, 10kk r 1 r1.5r 1r 10.75r 4.4. Diversity Gain at Finite SNR Figure 6 is a plot of diversity and multiplexing gain tradeoff in finite SNR for DP Rice channels, 10 k and . [0.75,2]r Obviously, the approximation curve agrees with the Monte Carlo simulation. For , as it has been indi cated in [11], the diversity gain in SP Rice fading chan nels approaches zero rapidly since the rankone LOS matrix limits the effective degrees of freedom in the channels. However, for DP Rice fading, a relatively high diversity gain can still be observed in . At 1r 1 r 10dB , , diversity gain can be as high as one. Explanations can be found from minimum eigenvalue of DP systems, that thanks to the decorrelation effect, minimum eigenvalue of DP do not approach zero despite of the exists of strong LOS component. 10 k For very high kfactor, the channel matrix only de pends on the Rice exponent. As , the channels tend to be AWGN, and the capacity increases only with SNR. For DP, asymptotic diversity gain becomes infi nite. k Figure 6. FiniteSNR DMT for DP Rice channels. 5. Conclusions In this paper, outage probablity and DMT for asymotic and finiteSNR are studied in 2×2 dualpolarized uncor related Rice fading channels. Exact expression mean and varition of mutual information in DP Rice channels are derived, based on which how channel paremeters as kfactors, SNR or XPD influence channel infromation exponent are discussed. Results show that in subran en vironments where 0.4 , 0.3 f , at 10dB , a 11k is required to fill the gap between erogotic mean of mutual information exponent of SP and DP. Outage probablity as well as asympotic and finiteSNR DMT are compared between of SP and DP. Using the gamma or lognormal distribution, their appromaxition curves for 2×2 dualpolarized uncorrelated Rice channels at 10k are given. The result in this paper, helps in finding the inner difference between DP and SP channels. And the appromaxiton approach for DMT in this paper, alough not so accurate in low multiplexing gain, can provide references in pratical code design in dualpolarized Rice systems, expecially in systems with large amouts of an tennas. REFERENCES [1] C. Degen and W. Keusgen, “Performance of Polarisation Multiplexing in Mobile Radio Systems,” Electronics Let ters, Vol. 38, No. 25, 2002, pp. 17301732. doi:10.1049/el:20021118 [2] R. U. Nabar, H. Bolskei, V. Erceg, D. Gesbert and A. J. Paulraj, “Performance of Multiantenna Signaling Tech nique in the Presence of Polarization Diversity,” IEEE Trans. Signal Process., Vol. 50, No. 10, 2002, pp. 25532562. doi:10.1109/TSP.2002.803322 [3] Y. Deng, A. Burr and G. White, “Performance of MIMO Systems with Combined Polarization Multiplexing and Transmit Diversity,” in proc.2005 Conf. Vehicular Tech nol. VTC 2005Spring. IEEE 61st, Vol. 2, pp. 869 873. [4] Mathini Sellathurai, Paul Guinand, and John Lodge “SpaceTime Coding in Mobile Satellite Communications Using DualPolarized Channels”, IEEE Trans, vol. 55, NO. 1, Jan 2006 [5] Claude Oestges, Bruno Clerckx, Maxime Guillaud, and M´erouane Debbah, “DualPolarized Wireless Commu nications:From Propagation Models to System Perform ance Evaluation”, IEEE Trans. Wireless, vol. 7, no. 10, oct, 2008 [6] M. Shafi, M. Zhang, A. Moustakas, P. Smith, A. Molisch, F. Tufvesson, and S. Simon, “Polarized MIMO channels in 3D: Models, measurements and mutual information,” IEEE Journal on Seleted Areas Commun. , vol. 24,no. 3, pp. 514–527, Mar. 2006. doi:10.1109/JSAC.2005.862398 [7] Y. B. Li, H. Q. Wang and X.G. Xia, “On QuasiOrthogonal SpaceTime Block Codes for DualPolarized MIMO Channels,” IEEE Transactions, Copyright © 2013 SciRes. CN
Y. P. HUANG, G. L. REN Copyright © 2013 SciRes. CN 325 Wireless Commun., Vol. 11, No. 1, 2012. [8] L. Zheng and D. Tse, “Diversity and Multiplexing: A Fun Damental Tradeoff in Multiple Antenna Channels,” IEEE Trans Info. Theory, Vol. 49, No. 5, 2003, pp. 10731096. [9] H. Yao and Wornell and W. Gregory, “Structured SpaceTime Block Codes With Optimal Diver sityMulti plexing Tradeo ff and Minimum Delay, ” in proc. Conf. Global Telecommunications (GLOBECOM), 2003, Vol. 4, pp. 19411945. [10] R. Narasimhan, A. Ekbal and J. Cioffi, “FiniteSNR Di versityMultiplexing Tradeoff of Spacetime Codes,” in Proc. 2005 , Int. Conf. Commun., 2005, pp. 458462. [11] R. Narasimhan, “FiniteSNRdiversitymultiplexing Trade off for Correlated Rayleigh and Rician MIMO Channels,” IEEE Trans. Inf. Theory, Vol. 52, No. 9, pp. 39653979, 2006. doi:10.1109/TIT.2006.880057 [12] W.Y. Shin, S.Y. Chung and Y. H. Lee, “Diver sityMultiplexing Tradeoff and Outage Performance for Rician MIMO Channels,” IEEE Transactions Informa tions Theory, Vol. 54, 2008, pp. 11861196. doi:10.1109/TIT.2007.915884 [13] C. Ammar El, Falou, W. Hamouda, C. Langlais, C. Abdel Nour and C. Douillard, “FiniteSNR Diver sityMultiplexing Tradeoff for Rayleigh MIMO,” IEEE, Communications Letters, Vol. 17, 2013, pp. 753756. doi:10.1109/LCOMM.2013.022213.130007 [14] F. Talebi and T. G. Pratt, “Approximating the Outage Capacity of Asymmetric 2×2 Dualpolarized MIMO at High SNR,” in proce Int. Conf. (ICNC), 2013, pp. 290 294. Appendix 1 , ii ii i R EEE i According to the distribution of channel elements (5), an (33) 2 H i HH which follows noncentral chisquare distribution. Mean we get and variance of i can be derived as : 2 2 2 2 2 2 2 12 11 1 2 21+ 11 1 1 21 11 1 ii f fff ff E k k k k kkk kk k kk kk k (34) 2 2 21 1 22 21 f i f i k Ek kk Dk 1 (31) As for i ，by 112 2 11 221221 rjsrjs hhhhr js ， i E and i D calculated as in [14] section 3 (11)(14). 22 2 2 11 f i kk Ek (32) Finally, using (15), exact expression of DW is given. According to (14),is derived. For, as EW DW Appendix 2 Using the method prosed in [9] we derive the proof for asymptotic DMT curve in dualpolarized uncorrelated
Y. P. HUANG, G. L. REN 326 Rice channels.The proof begins with Rayleigh fading cases. According to [9], let 2 logR . Firstly ,we de compose H as: 11 1211 12 22 22 ,00 rr rr HQRXR X rr (35) Since 2 11 :, 1rH, and is approximated as 2 11 r 2 11 4 r2 , similarly，22 12 2 r ， 2 22 2 r2 .Thus, for ，keep quadratic term 1r22 and neglect the other lower terms, we have 2 22 11 22 r out t PP rr n (36) As is a variable with a higher order than , and that small is mainly due to small , i.e., the main event that causes 2 11 r2 22 r 22 11 22 rr 2 22 r 2 22 11 22 r t rr n is 22 11 22 1r rr 2 2 occurring. Thus 222 11 22 1rr out PPr r 12r . When , diversity gain is derived as ,drr 2 1 (37) And when , neglecting the constant term, we 0r get 2 222 2 1112 22 2 22 11 22 t out r t rrr N PP rr N (38) Here, main events are 2 2122 12 1112 22 21 134 1 22 rr rr r rrr 1 1 (39) So that for 0r the diversity gain is derived as ,dr 3r 4，without any relationship with . Eventually, as LOS component do not affect the highSNR diversity gain [12], the asymptotic DMT analysis here hold on for Rice channels. Therefore, as ymptotic DMT curve for DP Rice channels at infinite SNR is the same as the conventional conclusion(9) in [8]. Copyright © 2013 SciRes. CN
