Performance of spatial multiplexing_polarization diversity

Performance of spatial multiplexing_polarization diversity...

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Unformatted text preview: PERFORMANCE OF SPATIAL MULTIPLEXING IN THE PRESENCE OF POLARIZATION DIVERSITY Helmet Bo'lcskeill, Rohit U. Noborll, V. Erceg2), D. Gesbert2), and Arogyaswami J. Pauli-of) 1) Information Systems Laboratory, Stanford University Packard 223, 350 Serra Mall, Stanford, CA 94305-9510 Phone: (650)-724-3640, Fax: (650)-723-8473, email: [email protected] 2) Iospan (formerly Gigabit) Wireless Inc., 3099 North First Street, San Jose, CA 95134 Abstract—In practice large antenna spacings are needed to achieve high capacity gains in multiple-input multiple- output (MIMO) wireless systems. The use of dual-polarized antennas is a promising cost effective alternative where two spatially separated antennas can be replaced by a single antenna element employing orthogonal polarizations. This paper investigates the performance of spatial multi- plexing in MIMO wireless systems with dual-polarized an- tennas. We compute estimates of the symbol error rate as a function of cross-polarization discrimination (XPD) and spatial fading correlations. Using these estimates, we show that dual-polarized antennas can significantly improve the performance of spatial multiplexing systems. It is demon— strated that improvements in terms of symbol error rate of up to an order of magnitude are possible. We furthermore find that in general for a given SNR there is an optimum XPD for which the symbol error rate is minimum. Finally, we present simulation results and we show that our esti- mates closely match the numerical results. 1. INTRODUCTION AND OUTLINE The use of multiple antennas at both ends of a wireless link has recently been shown to have the potential of drasti- cally increasing capacity through a technique called spatial multiplexing [1H5]. This capacity gain depends strongly on transmit and receive antenna spacing. In practice anten- na spacings of several wavelengths are required in order to achieve significant multiplexing gain. Unfortunately, large antenna spacing increases both size and cost of base stations and renders the use of multiple antennas in handsets very difficult. The use of dual—polarized antennas is a promis— ing cost effective alternative where two spatially separated antennas can be replaced by a single antenna element em— ploying orthogonal; polarizations. Contributions. In this paper, we investigate the per— formance of uncoded spatial multiplexing in systems em- ploying dual-polarized antennas, Although our techniques are generally applicable, for the sake of simplicity, we con- sider a link with one dual-polarized transmit and one dual- polarized receive antenna. Our contributions are as follows. The work of H. Bolcskei was supported by F‘WF‘ grant .11868- TEC. R. Nabar’s work was supp0rted by the Dr. T. J. Rodgers Stanford Graduate Fellowship. H. Bdlcskei is on leave from the Institut fiir Nachrichtentechnik und IIochfrequenztechnik, Tech- nische Universitat VVien, Vienna, Austria. 0-7803—7041-4/01/$10.00 ©2001 IEEE 2437 We introduce a channel model for a dual-polarized single-input single-output link taking into account spatial fading correlations and cross-polarization dis- crimination (XPD). We propose a method for computing estimates of the uncoded symbol error rate of spatial multiplexing in the presence of polarization diversity. We identify the propagation conditions where the use of polarization diversity is beneficial from an error- probability point of View, and we show that improve- ments in terms of symbol error rate of up to an order of magnitude are possible. We demonstrate that our symbol error estimates closely match the simulation results. Our method can therefore be used to predict performance trends analytically and helps avoiding time-consuming com- puter simulations. Organization of the paper. The rest of this paper is organized as follows. Section 2 introduces the channel mod- el for a dual-polarized single-input single—output link and states our assumptions. In Section 3, we derive estimates for the uncoded symbol error rate of spatial multiplexing as a function of spatial fading correlations, XPD, and SNR. Section 4 provides simulation results and demonstrates that our estimates closely match the simulation results. Finally, Section 5 contains our conclusions. 2. THE CHANNEL MODEL We consider a system with one dual-polarized transmit and one dual-polarized receive antenna. The channel is as- sumed to be flat over the frequency-band of interest. The input-output relation is therefore given by1 r=\/E,Hx+n, (1) where x [no $1]T is the 2 X 1 transmit signal vector whose elements are taken from a finite (complex) constella- tion chosen such that the average energy of the constellation elements is 1, r = [7'0 r1]T is the 2 x 1 receive signal vector, n is complex-valued gaussian noise with £{nnH } = (7312, hon her] hm h1,1 1The superscripts T and H stand for transpose and conjugate transpose, respectively. H=l is the channel transfer matrix which is also the polarization matrix, and \/E_,, is a normalization factor. The polariza- tion matrix describes the degree of suppression of individual co- and cross—polarized components, cross correlation, and cross coupling of energy from one polarization state to the other polarization state. In practice, the following polar— izations are generally considered: horizontal, vertical, and 145° slanted polarization. In this paper, we assume that the transmitter and the receiver employ the same polar- izations, i.e., both of them employ horizontal and vertical polarization for example. The signals 20 and :51 are trans- mitted on the two different polarizations, and To and 1'1 are the signals received on the corresponding polarizations. We emphasize that although we are dealing with one physical transmit and one physical receive antenna, the underlying channel is a 2—input 2—output channel, since each polar- ization mode is treated as an independent physical channel. We assume that the channel is purely Rayleigh fading, i.e., the matrix H consists of (in general correlated) complex gaussian random variables with zero mean. The more gen- eral case taking into account a line-of-sight component is treated in The correlation between the elements of the matrix H and the variances of the elements depend on the propagation conditions and the choice of polarizations, re- spectively. Throughout the paper, we assume2 8{Iho,o[2} = £{lh1,1|2} = 1 S{lho.1I2} = £{Ih1,oI2} = a, where 0 < or s 1 depends on the XPD. Good XPD yields small a and vice versa. The case a = 1 can also be in- terpreted as having two physical antennas on each side of the link employing the same polarization. We furthermore define the following correlation coefficients3 Eihophaul = 5{h1,ohi,i} J5 x/E _ fi_fi. For the sake of simplicity, throughout the paper, we as- sume that S{ho,ohi‘,1} = £{h1,oh3‘1} = 0. Measured values of XPD and correlation coefficients have been reported for example in [7H9]. 3. DERIVATION OF SYMBOL ERROR RATE In this section, we first discuss the impact of polariza— tion diversity on spatial multiplexing and then compute an estimate of the symbol error rate. 3.1. Impact of Polarization Diversity Spatial multiplexing [1]—[5] has the potential to dramat— ically increase the capacity of wireless radio links with no additional power or bandwidth consumption. The basic idea is that if scattering in the multi—antenna channel is rich enough independent parallel spatial data pipes are created within the same bandwidth, which ideally yields a linear (in the number of antennas) capacity increase. Traditionally, 25 stands for the expectation operator. 3The superscript ’ stands for complex conjugate. 243 8 the ability to perform spatial multiplexing has been related to a rich enough scattering environment. In the present case virtual multiple antennas are created by employing differ- ent polarizations and the MIMO channel matrix is replaced by the polarization matrix. It is therefore not clear a pri- ori how spatial multiplexing will perform in the presence of polarization diversity and how the exact values of XPD and fading correlations will influence the performance. We can, however, quantitatively establish the benefit of polar— ization diversity. It is well known that the multiplexing gain is maximized if the condition number of the channel matrix is 1. Now, it is intuitively clear that for small a the individ- ual realizations of H tend to have lower condition number. In fact, in the limiting case a = 0 (i.e. perfect XPD) every realization of H yields orthogonal columns and hence high multiplexing gain can be expected. 3.2. Error Probability Throughout the paper, we assume that the channel is unknown in the transmitter, perfectly known in the receiver, and that maximum—likelihood (ML) decoding is performed. The receiver computes the ML estimate according to :2 = arg minllr — VEaHxllz, X where the minimization is performed over the set of all pos— sible codevectors. Let c and e be two different codevectors of size 2 x 1 and assume that c was transmitted. For a given channel realization H, the probability that the receiver decides er- roneously in favor of the vector e is given by [10] E3 203, P(c —) ell-I) = Q( d’(c,e|H)) , (2) where d2(c.eIH) = “H(c - e)||2- Upon defining y = H(c — e) we get d2(c,e|H) = Ilyll2 and hence using the Chernoif bound Q(:c) S (5“2/2 it follows from (2) that E 2 P(c —) elH) g e 4735"” . (3) Since H was assumed to be gaussian it follows that the 2 X 1 vector y is gaussian as well. The average over all channel realizations of the right—hand side in (3) is fully characterized by the eigenvalues of the 2 x 2 covariance matrix of y [11] defined as Cy = E{yyH}. In particular, denoting the eigenvalues of C, as z\.-(C,,), we get 1 1 1 + AdCflfié— 1 + A2(Cy)fi§’ where P(c —) e) = £H{P(c ——) e|H)} is the pairwise error probability averaged over all channel realizations. Straight- forward manipulations reveal that4 C,, = log —- eol2 + a|c1 — ell2 + 2R{(co — eo)(c1— el)‘t\/r_x} r*\/E(|co — eol2 +|C1 — 61'?) Tx/aflco — Col? + [61 r ell?) alco — col2 +IC1 — 6112 + 2R{(co — eo)(C1 — myth/5} ' The eigenvalues of C3, are given by P(c ——> e) g (4) 472M} stands for the real part of a. a+dzt (a—d)2+4bc Mo = 2 with a = [co—eo|2+a|cl —e1|2 + 2R{(co — eo)(c1 — myth/E} b = n/Eflco — 80]2 + Ici — 51'2) c = rK/Eflco — eol2 + I61 — 61'2) d = a|co — eo|2 + [c1 — el|2 + 2’R{(co — eo)(01 — efi‘tfi}. If no polarization diversity is used (Le. 01 = 1) and the channel matrix is i.i.d., we have )‘1 = A2 = (|co — (20]2 + Ici — ella). In this case the error rate behavior is governed by error events where only one out of the two symbols is in error, say (co -— 60) 7E 0 with |co — eol2 = din-1, where dmin denotes the minimum distance of the scalar constella- tion used. Clearly, the error rate will decay for increasing (1mm. In the general case, where o < 1 and the individual entries in H are correlated the error events governing the performance of spatial multiplexing depend on the channel geometry induced by the correlation coefficients and the value of a. In order to avoid having to find those error events for a particular channel geometry, we average over all possible error events including a weighting which takes into account that different vector error events cause a dif— ferent number of scalar symbol error events. In particular, we want to study the influence of XPD on the error rate of spatial multiplexing systems. It is therefore crucial to reveal how the error probability behaves as a function of XPD for a given SNR and given t and r. We assume that 4-PSK is employed. This implies that there are 240 error events. The individual scalar error events (0, — er) can take values from the set {0, idmimijdmmghdminfl + j), :bdm,,,(1 — j)}. Now, with the relative frequency of an error event 5; where c: - ei = [(co,.- — €O.i) (cm — 81.0]T given by "‘i ‘ 240 with 4, :c = O w(;1;)= 2, .13 = idmin; ijdmin ) { 1, idmin(1+j),idmin(1 we estimate the average symbol error rate as 13 = Z n5,P(E,-)3(Ei)- (5) Here, P(e.;) is (4) evaluated for [(co,,- — cod) (01,,- — e1,,')]T and 3(6‘) 2 l 0. In the next section, P is shown to reveal all the relevant trends and a close match between the exact symbol error rate and P is found. We note that (4) can be used to study the impact of t, r and a on P analytically. For example, it follows immediate- ly from (4) and (5) that for t = r = 0 the quantity P will be minimum for a = 1, i.e., for the case where no polarization diversity is employed. Indeed, we will find in the next sec- tion that polarization diversity improves the multiplexing gain only in the presence of high correlations. 2. 1 (Cod —eo,.') 95 0, (Cm -81.i) 9'5 0 (60,; — 80,.) = 0 or (cm- —ei,.-) 3 2439 4. SIMULATION RESULTS In this section, we provide simulation results demon- strating the performance of spatial multiplexing in the p- resence of polarization diversity and spatial fading corre- lation. In particular, we show that P reveals the impact of polarization diversity on the performance of spatial mul- tiplexing quite accurately. We simulated a system with 1 dual—polarized transmit and l dual-polarized receive anten- na using 4-PSK and employing an ML receiver. The signal— to—noise-ratio (SNR) was defined as SNR = 10 log Simulation Example The first simulation example serves to demonstrate that P provides an accurate estimate of the symbol error rate for high SNR. For t = 0.5, 'r = 0.3 and a = 0.4, Fig. 1 shows the symbol error rate obtained using Monte Carlo simulations along with the estimated symbol error rate P. It can be seen that especially in the high S_NR regime the two curves match well. Note, however, that P is not a strict upper bound on the symbol error rate. 10" ,. ., ID- Symbolotrurma 1n‘ _ “1...; 10" i D 10 SNde Fig. 1. Symbol error rate as a function of SNR. Simulation Example 2. The second simulation ex- ample serves to demonstrate the benefit of polarization di- versity. For an SNR_of 15dB, Fig. 2 shows the symbol er- ror rate along with P as a function of a for various values of t and for 1‘ = 0. It can be seen that for low trans- mit correlation the use of polarization diversity leads to a performance degradation or equivalently reduced multiplex- ing gain. (Recall that a = 1 can be interpreted as having two physical antennas on each side of the link all of which employ the same polarization.) This result conforms with the investigations in the last paragraph of Sec. 3.2. When the transmit correlation starts to increase and the condi- tion number of the channel matrix realizations increases or equivalently the angle between the realizations of the two columns decreases, polarization diversity yields improved spatial separation and hence increases the multiplexing gain. We found that starting at t = 0.85 polarization di- versity, i.e., a < 1 starts improving the multiplexing gain. We can also see that in the case of fully correlated transmit antennas, i.e., t = 1, there is an optimum value of a for which the symbol error rate is minimum. Observe that this minimum is accurately predicted by P. Fig. 2 furthermore shows that for high transmit correlation the use of polar- ization diversity can improve the symbol error rate by up to an order of magnitude. - +Slmulaiadl=01 . .. +srnutaiaai=a7 .. l , @— Simatadl=| -e- EsmedlSUJ + Estmuladqu e Estimad|=1 0.9 0.1 Fig. 2. Symbol error rate as a function of a for various values of the transmit correlation coeflicient t. Simulation Example 3. In the last simulation ex- ample, we study the performance of spatial multiplexing with and without polarization diversity in the presence of receive correlation only. Fig. 3 shows the symbol error rate along with P for t = 0 and various values of r. We observe that for r = 1 the use of polarization diversity increases the multiplexing gain or equivalently reduces the symbol error rate. This effect, however, is much less pronounced than in the case of transmit correlation only. The reason for this is that in the presence of transmit correlation there are code vectors which “tend to excite the null space of the channel matrix” and hence yield very high probability of error. This does not happen in the case of receive correlation only. A more detailed description of this observation is provided in [12]. For r = 1, the optimum value of (1 seems to be the same as in the case of transmit correlation only with t = 1. 10_‘ . .. .l. . —e— slimlaladr:o.1 '+ simulatedr=OJ -e— Simulllodr=1 '5 Esmladr=0.1 -v * Estimatedr=01 e. Bum-led”! y 0.8 [1.7 0.6 Fig. 3. Symbol error rate as a function of a for various values of the receive correlation coefficient r. 5. CONCLUSION We studied the use of polarization diversity for spatial multiplexing and found that in the presence of high spa- tial fading correlation dual-polarized antennas can yield a 2440 significantly improved multiplexing gain. In particular, we computed an estimate of the uncoded symbol error rate which was found to be very accurate in the high SNR regime and which allows to study the impact of polarization di— versity on the performance of spatial multiplexing without having to resort to time-consuming computer simulations. We demonstrated that especially in the presence of transmit correlation the use of polarization diversity can yield signif- icant improvements in terms of symbol error rate. Further— more, we found that for high spatial fading correlation and for a given SNR in general there is an optimum value of oz for which the symbol error rate is minimum. Our symbol error estimate accurately predicts this point. Finally, we provided simulation results. 6. REFERENCES [1] A. J. Paulraj and T. Kailath, “Increasing capac- ity in wireless broadcast systems using distribut- ed transmission/directional reception,” U. S. Patent, no. 5,345,599, 1994. G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech. J., p- p. 41—59, Autumn 1996. I. E. Telatar, “Capacity of multi—antenna gaussian Channels,” Tech. Rep. #BL0112170-950615—07TM, AT & T Bell Laboratories, 1995. G. G. Raleigh and J. M. 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