On the Modelling of Polarized MIMO Channel

Transcription

1 On the Modelling of Polarized MIMO Channel Lei Jiang, Lars Thiele and Volker Jungnickel Fraunhofer Institute for Telecommunications, einrich-ertz-institut Einsteinufer 37 D-587 Berlin, Germany Abstract This paper presents a simple model for polarized multiple-input multiple-output (MIMO) channels. At first, the model is desired for line-of-sight (LOS) scenario, and then generalized for non-los (NLOS) case. The analysis of the crosspolarization discrimination (XPD) for a MIMO systems shows that the XPD depends on both the antenna patterns and the polarization mismatch between the transmit and receive antenna pairs. In addition, this theoretical model can be easily integrated into the spatial channel model (SCM) to predict the dual-polarized MIMO channels for LOS and non-los (NLOS) scenarios. The comparisons between simulation results and measurements further proof this model s applicability. I. INTRODUCTION The multiple-input multiple-output (MIMO) system has been shown to dramatically increase the capacity of the wireless system and it draws increasing attentions in recent years [], []. An important condition for MIMO channel to achieve high capacity is that the environment provides sufficient multipath components. By exploiting the multipath, the MIMO link results in a high rank channel with improved capacity. owever, in a line-of-sight (LOS) scenario, the MIMO system exhibits a reduced performance since the LOS component overwhelms the multipath components in the received signal. In order to improve the performance of MIMO system in LOS scenario, techniques such as antenna polarization have been employed to obtain higher rank. Moreover, part of the existing infrastructure uses cross-polarized antenna elements, which could be used to support both polarization diversity and polarization multiplexing. Widely used channel models such as the spatial channel model (SCM) used by third generation partnership project (3GPP), have limited support for crosspolarized MIMO channel in LOS scenario. Therefore, it is important to explore how MIMO channel models for LOS scenario can properly support polarized antennas. Measurements have been made to investigate the performance and configurations of the polarized MIMO channels [3] [9]. [3] [5] show that the capacity of polarized MIMO system outperforms that of co-polarized MIMO system with constant signal to noise ration (SNR). [6] reported that there are two dominant singular values in LOS scenario for crosspolarized MIMO channel because of the polarization of the antennas. [5] presented a simple MIMO channel model that predicts the eigenvalue distribution and the capacity of dual and single polarized MIMO channel for specific antenna configurations. In [], polarization and scattering matrices are used to propose a stochastic geometry-based scattering model for the multipolarized MIMO channel. A 3-D polarized MIMO channel model is derived in []. [6] also developed an exact 3-D far-field model for the LOS scenario. They are precise for evaluation but may be complicated for implementation. For this, we need a generic approach which can be easily integrated into the existing models and provides a solid basis for performance evaluation. The idea here is to start from the standard -D channel, and to modify it according to the actual polarization mismatch between the transmit and receive antennas. In this paper, we show that the channel model for arbitrary polarized antennas can be derived from the model for copolarized antennas by element wise multiplication with a matrix containing the polarization mismatch loss between the transmit and receive antenna pairs as well as the effect of azimuthal direction of the terminal in the cell. The paper is organized as follows. In section II, the theoretical channel model considering the antenna polarization for MIMO system is formulated. Section III discusses the results and compares the theoretical channel model with the measured results. The integration of this model into SCM and comparisons with measurements are presented in section IV. Concluding remarks can be found in Section V. II. TEORY AND FORMULATION A. Channel Model for LOS Scenario For the simplicity of analysis, we assume that the distance between the transmitter and receiver D is large enough for the plane wave assumption to hold. Transmit and receive antennas are assumed to lie in the propagation plane. In the following, antennas are distinguished as co-polarized ( ), dual-polarized ( ) or cross-polarized (++). In Fig., there are N pairs of cross-polarized antennas at the receiver and M pairs at the transmitter. They are numbered as shown in the figure. The spacing between two neighboring pairs of antennas for the transmitter and receiver are denoted by d T and d R respectively. The transmitter array makes an angle ϕ T with the line connecting the base points of the transmitter and receiver arrays, while the receiver makes an angle ϕ R. For LOS component, the received field at the receiver from

2 Fig.. Cross-polarized MIMO channel. transmitter can be obtained according to Malus law as h = e jkd G R G T cos θ () d where θ is the polarization mismatch angle between the transmitter and the receiver. When transmit and receive antennas are co-polarized, θ =, and for orthogonally oriented antennas, θ = π/. G R and G T are antenna gains indicating field patterns of the receive and transmit elements, respectively. d is the distance between transmit and receive antenna elements. In the cross-polarized case as shown in Fig., the distance between the transmit antenna pair m and receive antenna pair n can be calculated as below d nm = [ ( ( N + ) ( M + ) D n d R cos ϕ R m ( (N + ) ( M + ) ) ] n d R sin ϕ R m d T sin ϕ T () The channel matrix for LOS component can be written as [ ] V V LOS = N M V N M V N M N M [ = N M AV N M V N M ] AV N M (3) N M AV N M N M A N M where denotes the element wise multiplication. N M is the LOS channel matrix for an N-receiver M-transmitter MIMO system with co-polarized antennas. A N M is the matrix taking into account the polarization mismatch. When transmit and receive antennas are all strictly aligned, A V N M V and A N M are all-one matrices (cos ), while AV N M and A V N M are all-zero matrices (cos π ). In realistic environments, transmitter and receiver are usually not in such ideal situation, we assume that the transmit elements are rotated at an an angle θ p compared with the receive antenna. Then the elements and A N M become cos θ p, and the elements of A V N M and AVN M become cos(π/ θ p). This angle is denoted as the polarization rotation angle. To get a clear view of equation (3), a cross-polarized MIMO channel is analyzed, where d nm = D. The channel of A V V N M matrix for LOS component is LOS = e jkd D e jkd D GV RG V T cos θ p e jkd ( D π G R G V T cos p) θ D e jkd ( π GV RG T cos p) θ G R G T cos θ p (4) Ignoring the small contribution of scattered components in LOS scenario, the XPD at the receiver can be calculated as h V V ( ) G V XP D V = h V = R cot θ p (5) G R h ( ) G XP D = h V = R cot θ p (6) where superscripts V and in the above symbols denote the vertical and horizontal polarization. In an ideal situation where transmit and receive elements are strictly orthogonal, i.e., θ p =, the XPD should thus be infinity. In realistic environment, the orientation of the mobile receiver changes from time to time, which results in a polarization mismatch. The XPD now depends on both the field pattern of receive antennas and the angle θ p. From equation (3) we found that the LOS channel matrix for a cross-polarized MIMO system can be written as the ) element wise multiplication of the LOS channel matrix for d T cos ϕ T + G V R co-polarized MIMO system with a matrix modelling the polarization relationship between the transmit and receive antennas. For implementation, it would be simpler to rewrite (3) as LOS = N M A N M (7) where N M is the LOS channel matrix for N-receiver and M-transmitter co-polarized MIMO system. The elements of A N M describe the polarization mismatch loss between the transmit and receive antenna pairs. To be more realistic, we take into account the azimuthal displacement of the transmit and receive antennas, i.e., the normal vectors of the transmit and the receive antenna do not lie along the LOS path. Assume that the displacement angle is ψ, a factor of cos ψ is added for all the horizontally polarized transmitting signals. To implement this, cos ψ is multiplied to A V N M and A N M in (3). According to the above formulation, the channel matrix for the cross-polarized MIMO system can now be modified as [ ] LOS = e jkd G V R G V T G V R G T D G R GV T G R G T [ ] (8) cos θp sin θ p cos ψ sin θ p cos θ p cos ψ The first term of (8) represents the co-polarized MIMO channel N M, which can be taken over from the existing channel models. While the second matrix is the additional contribution to the path loss due to the polarization mismatch

3 and azimuthal displacement. Both N and M equal to one in this case. Based on this idea, the model can be further extended to arbitrarily polarized M-transmitter N-receiver MIMO channel. The effects of polarization and azimuthal displacement can be represented by a N M matrix, and this matrix is then elementwisely multiplied to the co-polarized channel model. Consequently, we obtain the model for arbitrarily polarized MIMO channel. B. Channel Model for NLOS Scenario In a NLOS scenario, the signal transmits from the transmitter and undergoes some diffraction, reflection or scattering, then reaches the receiver as the dashed line shown in Fig.. It is well known that the signal at the cross-polarized receiver is statistically nonzero when a single vertical or horizontal polarization is transmitted [], [3]. This implies that the polarization is more or less rotated during the propagation because of the reflection, diffraction or scattering. Theoretically speaking, the polarization direction will rotate an arbitrary angle on [, π). ence there will be a polarization mismatch loss for each path. Same as before, a MIMO channel is analyzed as an example. The multipath channel can be written as NLOS = e jkli l i= i e jkli i= l i i= e jkli i= G V R,iG V T,iΓ i (φ i ) cos θ p,i G R,iG V T,iΓ i (φ i ) sin θ p,i e jkli G V l R,iG T,iΓ i (φ i ) sin θ p,i cos ψ i i l i G R,iG T,iΓ i (φ i ) cos θ p,i cos ψ i where Γ i (φ i ) represents the total amplitude and phase changes due to the reflection, diffraction and scattering during the whole propagation path. l i and θ p,i denote the path length and polarization rotation angle for each path respectively. N s is the number of multipath components. Absolute values are used when calculating the polarization mismatch loss since we assume that θ p,i is distributed on [, π). The elements of NLOS are the summations of all multipath fields for corresponding transmit and receive antenna pairs. In a rich scattering environment, N s is very large. According to the central limit theorem, the elements of NLOS can be approximated as Gaussian variables, which is in accordance with the well-known classical i.i.d. Gaussian model. ence, with rich multipath, not much impact of polarization on the channel statistics is expected. The general MIMO channel can be modelled as the weighted sum of the LOS and NLOS components [5], [], [4] as below = K K + LOS + (9) K + NLOS () where K is the Ricean K-factor. In a LOS scenario, the value of K factor can be very high, as K decreases, the contribution of scattered components to the channel becomes more significant while the effect of the LOS component becomes less important, until K reduces to zero, there will be no LOS component and it is a NLOS scenario. III. RESULTS AND DISCUSSION A simple measurement is made at 5. Gz in the laboratory to validate (8). ψ = during the measurement. Two crosspolarized path antennas are used and the real and imaginary part of the channel coefficients are measured as in Fig.. It is Fig.. Normalized Channel coefficients.. obvious that the channel coefficients change with polarization rotation angle θ p according to polarization mismatch matrix in (8). Careful inspection shows that cos θ p is the variation trend for the co-polarized components, while sin θ p are found for V- and -V components. The minor phase variation in measurements is attributed to unequal length of feeder cables of the antenna. Note that the measurement result is fully consistent with ((8)) if all the antenna gains are the same. The cumulative distributions of ordered singular values for a 6 MIMO channel are displayed in Fig. 3. The distance between the omnidirectional transmitter and receiver is chosen to be m, with a carrier frequency of 5. Gz. The element spacing for both transmit and receive antennas is λ. In this paper, we assume that the K factor does not change along the measuring track and equals to db. The average received SNR is selected as db. The polarization rotation angle is chosen as, and ψ is assumed to be 3. In the simulation results, there are two dominant singular values and eight smaller ones. Similar results have been observed in the measurement [6], they are due to the two polarization directions of the transmit and receive antennas. Note that the natural variations of the K-factor, causing fluctuations of singular

4 values and capacity, are not considered here but presented in measurements. The total number of singular values equals to the minimum number of transmit and receive antennas. In an ideal case, i.e., there is no azimuthal displacement and no scattered components, the two dominant singular values would be identical, and the other small singular values appear also in pairs with equal values. This symmetry is broken by the appearance of scattered components and the misalignment between the transmit and receive antennas..9.8 the number of antenna elements is relatively large, the size of the antenna will increase dramatically and become comparable with the distance D as the antenna element spacing increases, which results in a substantial path length difference among all the LOS paths, and hence leads to the significant phase and amplitude changes of the received signals. Consequently, the correlation of the LOS channel is reduced and the channel capacity is enhanced. If the number of elements is small or the distance between transmitter and receiver is very large, the effect of element spacing on the MIMO system capacity is not as obvious as mentioned hereinbefore. On the other hand, we can see that for the same number of antenna elements, the size of the cross-polarization will be half of that of co-polarized antenna..7 CDF Singular Values Fig. 3. Singular values for cross-polarized MIMO channel. Prob [ <abscissa] X 4X4 Mz bandwidth Narrowband 8X8 Prob [ <abscissa] d r =d t =λ d r =d t =3λ Fig. 4. Capacity comparison for LOS cross-polarized MIMO channel with different antenna element spacing. In the LOS scenario, when the number of transmit and receive antennas is large enough, increasing the antenna element spacing will result in the increase of system capacity. Fig. 4 shows the capacity for a 6 polarized MIMO system with different antenna element spacing. It is obvious that the capacity of the MIMO system with element spacing 3λ is larger than that of the system with element spacing λ. Intuitively, when Fig. 5. Capacity comparison for LOS cross-polarized MIMO channel with different antenna size. K= db. In Fig. 5, the comparisons between the wideband and narrowband MIMO channel capacity for different antenna array size are plotted. The numbers of transmit and receive antennas are assumed to be the same and increase from to 8. We find that both the relative and the absolute difference of the spread decreases as the number of antennas increases. Same result has been observed in [5]. The reason for this behavior is that the additional antennas already provide spatial diversity, so that the additional frequency diversity becomes less important. IV. INTEGRATION INTO TE SPATIAL CANNEL MODEL A. Dual-Polarized MIMO Channel for LOS Scenario In this section, we will briefly discuss how to integrate our theoretical model into the SCM introduced in [6]. In section II, we have mentioned that the arbitrary polarized MIMO channel model for LOS scenario can be derived from co-polarized model by multiplying a matrix which takes into account the polarization mismatch and the azimuthal displacement between the transmit and receive antennas. Based on this idea, the effects of the polarization can be most easily

5 integrated into SCM in case of the dual-polarized MIMO channel. Assume that the polarization mismatch angle θ p is and ψ equals to 3. Simply multiply these factors to the channel coefficient equation for LOS scenario model in SCM [6] (pp. 6) according to the desired transmit and receive antenna pairs, and the model can be immediately used to predict the dual-polarized MIMO channel. CDF Fig Singular Values Singular values for a 6 dual-polarized SCM for LOS scenario. In the SCM, co-polarized ULAs are used. In order to introduce dual-polarized antennas,we assume that half of the transmit and receive antennas are vertically polarized and half of them are horizontally polarized. In simulations, we have actually used the reference implementation of the extended SCM described in [7], which is available in [8]. Fig. 6 plots the cumulative distribution function (CDF) of the ordered singular values obtained from the SCM with dual-polarized antenna elements. The simulation is made for a 6 dualpolarized MIMO channel, where the element spacing is 4λ for base station and λ for mobile. Other SCM parameters are chosen following those in [6]. The urban micro scenario is selected with a carrier frequency of 5 Gz. LOS scenario is forced on, and the distance between the base station and the mobile receiver is m. ence K = db can be obtained from K = 3.3d(dB) [6], where d = m here. The receiver moves at a speed of 3 cm/s, and a m measuring track is implemented. The singular values in Fig. 6 are comparable with those from measurements [6] for the LOS scenario (I Pos. ). There are two dominant singular values and eight small ones. The slight differences between the simulation results and measurements are partly caused by the different types of antennas used. In the simulation, dual-polarized uniform linear arrays are used, while in the measurement, the cross-polarized transmit antennas are mounted on a star frame in a circle with 9 cm radius, and the receiver is a cube antenna. Moreover, as mentioned before, since the K factor is fixed during the simulation, the CDFs of simulated singular values are steeper than those obtained from measurements. Prob [ <abscissa] Narrowband Wideband Fig. 7. Comparison between the wideband and narrowband MIMO channel capacity for dual-polarized SCM for LOS scenario. The capacities of the wideband and narrowband MIMO channels are plotted in Fig. 7. It can be observed that there is a slight difference in the capacity as the bandwidth increases. As explained in section III, the frequency selectivity of the channel adds additional diversity. But since in LOS scenario, the multipath effect is not dominant, hence the difference in the capacity spread between wideband and narrowband is small. B. Dual-Polarized MIMO Channel for NLOS Scenario As discussed in section II, the theoretical model can also be used to predict the MIMO channel with polarized antennas for NLOS scenario by introducing a polarization mismatch factor. The distribution of θ p is an open issue, to our knowledge, no direct measurement of this distribution is available in the literature. Recent work has mainly concentrated on the XPD. The polarization direction rotations caused by the reflection, diffraction or scattering are different, so in different environments where the propagation mechanisms are dominated by reflection, diffraction or scattering, the distribution of θ p will be different. Furthermore, it is also known that the V- V transmission usually yields higher received power than - transmission []. Various measurement results [9], [9], [] and electromagnetic simulation results [] show that the polarization selectivity is in favor of vertical polarization. Therefore, the polarization direction rotation for vertically polarized signal will be less than that for horizontally polarized signal during the propagation. To model θ p, the above discussed factors should all be taken into account. To the best of authors knowledge, no previous work has been done to investigate the distribution of θ p. On the other hand, there is no indication that the distribution of θ p has a measurable impact on results. In this paper, for simplicity, we assume that θ p U[, π). Consequently, the polarization mismatch loss for V-V and - propagation path is cos θ p, and the

6 polarization mismatch loss for V- and -V propagation path is sin θ p. By multiplying these factors to each single path in SCM, with a random value of θ p, the model can now be extended for the dual-polarized MIMO channels in NLOS case. The azimuthal displacement angle is still assumed to be 3 as in previous section. CDF Fig. 8. Prob [ <abscissa] Singular Values Singular values for a 6 dual-polarized SCM for NLOS scenario Narrowband Wideband Fig. 9. Comparison between the wideband and narrowband MIMO channel capacity for dual-polarized SCM for NLOS scenario. The singular values and capacity for wideband and narrowband MIMO channels are plotted in Fig. 8 and Fig. 9. They are also comparable with the measurement in [6] (FT Pos. 5) for NLOS scenario. The differences between simulation results and measurements are for the same reason as mentioned in section IV-B, the types of transmit and receive antennas are different in simulation and measurement, and fixed K factor is used in the simulation along the measuring track. In Fig. 9, the distribution of the capacity becomes steeper as we increase bandwidth. Compared with Fig. 7, the spreading difference for NLOS scenario is much larger. Since there is no LOS component in NLOS scenario, the multipath scattering dominates the propagation mechanism, hence the frequencyselective fading is a distinctive feature. The capacity ranges from about 3 bps/z to 34 bps/z for NLOS scenario, while only from about 9.8 bps/z to bps/z for LOS scenario. V. CONCLUSION We propose a simple theoretical channel model for polarized MIMO communication system in this paper. The model for cross-polarized MIMO channel in LOS scenario is first derived from the co-polarized MIMO channel model by element wise multiplication with a matrix taking into account the polarization mismatch between the transmit and receive antenna pairs as well as the azimuthal displacement angle. Measurement result shows that the polarization mismatch matrix properly models the depolarization between the transmitter and receiver. The channel model for the NLOS scenario shows that not much impact of the polarization on the channel statistics is expected in a rich scattering environment. The singular value, capacity and XPD of the MIMO channel are further analyzed. In the LOS scenario, there will be two dominant singular values for the cross-polarized MIMO channel because of the two orthogonal polarization directions. The XPD of the crosspolarized MIMO channel depends on both the antenna patterns and the polarization rotation angle between the transmitter and the receiver. In addition, we have shown that this model can be easily integrated into SCM to predict the dual-polarized MIMO channels for both LOS and NLOS scenarios. Simulation shows that the modified SCM can provide realistic results compared to measurements. REFERENCES [] M. A. Jensen and J. W. Wallace, A review of antennas and propagation for MIMO wireless communicationss (invited paper), IEEE Trans. Antennas Propagat., vol. 5, pp. 8 84, Nov. 4. [] G. F. Foschini and M. J. Gans, On limits of wireless communication in a fading environment when using multiple antennas, Wireless Pers. Commun., vol. 6, no. 3, pp , 998. [3] P. R. King and S. Stavrou, Capacity improvement for a land mobile single satellite MIMO system, IEEE Antennas Wireless Propagat. Lett., vol. 5, pp. 98, 6. [4] K. Sulonen, P. Suvikunnas, L. Vuokko, J. Kivinen, and P. Vainikainen, Comparison of MIMO antenna configurations in picocell and microcell environments, IEEE J. Select. Areas Commun., vol., pp. 73 7, June 3. [5] V. Erceg,. Sampath, and S. Catreux-Erceg, Dual-polarization versus single-polarization MIMO channel measurement results and modeling, IEEE Trans. Wireless Commun., vol. 5, pp. 8 33, Jan. 6. [6] V. Jungnickel, S. Jaeckel, L. Thiele, U. Krueger, A. Brylka, and C. von elmolt, Capacity measurements in a multicell MIMO system, in Proc. IEEE Globecom, (San Francisco, CA), 7 Nov. Dec. 6. [7] V. Erceg, P. Soma, D. S. Baum, and S. Catreux, Multiple-input multipleoutput fixed wireless radio channel measurements and modeling using dual-polarized antennas at.5 Gz, IEEE Trans. Wireless Commun., vol. 3, pp , Nov. 4. [8] A. K. Jagannatham and V. Erceg, MIMO indoor WLAN channel measurements and parameter modeling at 5.5 Gz, in Proc. IEEE Veh. Tech. Conf., vol., pp. 6, Sept. 4. [9] P. Kyritsi, D. C. Cox, R. A. Valenzuela, and P. W. Wolniansky, Effect of antenna polarization on the capacity of a multiple element system in an indoor environment, IEEE J. Select. Areas Commun., vol., pp. 7 39, Aug..

7 [] C. Oestges, V. Erceg, and A. J. Paulraj, Propagation modeling of MIMO multipolarized fixed wireless channels, IEEE Trans. Veh. Technol., vol. 53, pp , May 4. [] M. Shafi, M. Zhang, A. L. Moustakas, P. J. Smith, A. F. Molisch, F. Tufvesson, and S.. Simon, Polarized MIMO channels in 3-D: Models, measurements and mutual information, IEEE J. Select. Areas Commun., vol. 4, pp , Mar. 6. [] W. C. Jakes, Microwave Mobile Communication. New York: IEEE Press, 994. [3] Y. Zhou, S. Rondineau, D. Popović, A. Sayeed, and Z. Popović, Virtual channel space-time processing with dual-polarized discrete lens antenna arrays, IEEE Trans. Antennas Propagat., vol. 53, pp , Aug. 5. [4] S. Wyne, A. Molisch, P. Almers, G. Eriksson, J. Karedal, and F. Tufvesson, Statistical evaluation of outdoor-to-indoor office MIMO measurements at 5. Gz, in Proc. IEEE Veh. Tech. Conf., vol., pp. 46 5, Spring 5. [5] A. F. Molisch, M. Steinbauer, M. Toeltsch, E. Bonek, and R. S. Thomä, Capacity of MIMO systems based on measured wireless channels, IEEE J. Select. Areas Commun., vol., pp , Apr.. [6] Spatial channel model for multiple input multiple output (MIMO) simulations. 3GPP TR V6.., Sept. 3. [7] D. S. Baum, J. Salo, G. D. Galdo, M. Milojevic, P. Kyösti, and J. ansen, An interim channel model for beyond-3g systems: Extending the 3GPP spatial channel model (SCM), in Proc. IEEE Veh. Tech. Conf., vol. 5, pp , 3 May Jun. 5. [8] D. S. Baum, J. Salo, M. Milojevic, P. Kyösti, and J. ansen, MATLAB implementation of the interim channel model for beyond-3g systems (SCME), May 5. [Online]. Available: [9] J. W. Wallace, M. A. Jensen, A. L. Swindlehurst, and B. D. Jeffs, Experimental characterization of the MIMO wireless channel: Data acquisition and analysis, IEEE Trans. Wireless Commun., vol., pp , Mar. 3. [] R. G. Vaughan, Polarization diversity in mobile communications, IEEE Trans. Veh. Technol., vol. 39, pp , Aug. 99. [] C. Oestges, A stochastic geometrical vector model of macro- and megacellular communication channels, IEEE Trans. Veh. Technol., vol. 5, pp , Nov..