Configuration of Base Station Antennas in Millimeter Wave MU-MIMO Systems

Transcription

1 Configuration of Base Station Antennas in Millimeter Wave MU-MIMO Systems Lu Liu, Yafei Tian and Yifan Xue School of Electronics and Information Engineering, Beihang University, Beijing, China {luliu, ytian, Abstract Millimeter wave (mmwave) communication is a significantly enabling technology in the fifth-generation (5G) cellular system to promote the capacity. In recent years, people have focused on studying the propagation characteristics of mmwave. However, the impact of mmwave channel on the system capacity, especially in multi-user multi-input and multi-output (MU- MIMO) scenarios, has not been fully investigated. In this paper, we first study a low-complexity numerical calculation method of the sum-capacity in MU-MIMO downlink scenarios, and then investigate the effect of base station (BS) antenna configurations on the system capacity. The antenna spacing, deployment position and array shape are respectively studied, and the cumulative distribution functions (CDFs) of user correlations are analyzed. We find insights to achieve higher channel capacity and to guide the deployment of BS antennas in practical cellular scenarios. Index Terms antenna array, array configuration, channel capacity, millimeter-wave, MU-MIMO. I. INTRODUCTION One of the primary technique in next generation cellular system is milimeter wave (mmwave) communications, which is the spectral frontier for wireless communication systems nowadays []. The mmwave band can provide much wider bandwidth comparing to today s cellular networks, and can promise much smaller base station (BS) antenna arrays due to its short wavelength [2]. The propagation characteristic of mmwave channel is different from the microwave channel. Due to the larger penetration loss, less scattering and diffraction, mmwave channel is more sparse both in time domain and space domain [3] [5]. Channel capacity not only depends on the propagation characteristics, but also depends on the antenna configurations. The sub-channel correlation greatly circumscribes the capacity of MIMO system [6]. One way to decrease the correlation is to equip antennas with different polarizations and radiation patterns. In [7], the author investigates MIMO systems where antenna arrays are composed of different number of dipoles in three axes (X,Y,Z) and then introduce diverse radiation patterns to reduce correlation. The result demonstrated that MlMO systems exploiting antenna pattern diversity allow for performance improving. In [8], genetic algorithm (GA) are utilized to design the antenna spacing of the linear array. Optimized results show that minute modification of the element positions can improve the array pattern, especially for the fewer elements case. In cellular system, the space constraint of BS antenna make it infeasible to deploy hundreds of antenna elements in one horizontal or vertical dimension. In order to cope with this limitation, people extend the line antenna array to twodimensional (2D) antenna arrays. The full-dimension MIMO (FD-MIMO) system was proposed in [9], where 2D active antennas are equipped on BSs. To accommodate the introduction of 2D antenna arrays, the channel model is extended to three dimensions (3D), where the azimuth and elevation angles are both taken into account. The simulation results show that the number or spacing of elements in two dimensions can impact the throughput gain. Referring to the factors of antennas, the geometry of antenna array has significant impact on the eigenvalues of single-user MIMO (SU-MIMO) channel [], and thus influences channel capacity. In addition, there are other studies on the formation, spacing and polarization of antenna arrays [] [3]. In this paper, we study the relationship between array factors and system performance in 3D mmwave multipleuser MIMO (MU-MIMO) channels. We first study a lowcomplexity numerical calculation method of the sum-capacity in MU-MIMO downlink scenarios. Then combining MU- MIMO with mmwave propagation, we analyze the impact of BS antenna factors in detail, including the spacing between antenna elements, the position of antennas, and the number of elements on the horizontal or vertical dimensions. The antenna spacing is changed with multiples of the wavelength from 8λ to 32λ to find the effect of different array spacing on the system performance. When BS antennas are deployed indoors, we considered two positions, such as the ceiling and the side wall, which are representative for the indoor layout. Changing the number of elements on the horizontal and vertical dimensions, we develop antenna arrays with kinds of configurations and then find their difference on the capacity as well as the user correlation coefficients. Since the relationship between each factor and the system capacity is complicated, it is intractable to derive a rigorous mathematical formula. Hence, Monte Carlo simulations are employed to observe cumulative distribution function (CDF) of channel correlation coefficients and the sum-capacity. After comprehensive study, we will obtain some general rules to guide the deployment and configuration of the BS antenna array in practical 5G systems. The rest of this paper is organized as follows. In Section II, we introduce channel models for two BS deployment positions. Then we formulate the calculation of channel capacity in Section III. In section IV, we investigate the impact of various

2 antenna factors on the system performance in detail. Finally, Section V concludes the paper. II. CHANNEL MODEL We consider a downlink MU-MIMO transmission scenario, where multiple users are randomly distributed in an indoor environment. Due to severe path losses, the mmwave environment is well characterized by a clustered channel model, i.e., the Saleh-Valenzuela model [4]. The channel matrix between the BS and one user is defined as N t N r N cl H = N cl N ray i= N ray l= α il a r (φ r il,θ r il)a H t (φ t il,θ t il). () In (), N t = is the number of transmitter antennas on the BS, and N r = N rh N rv is the number of receiver antennas on the user end. N xh and N xv represent the number of antenna elements on the horizontal and vertical dimensions respectively, where x {t, r} denotes the transmitter or receiver. When N xh = or N xv =, the array is linear, otherwise it is planar. N cl and N ray denote the number of clusters and the number of rays in each cluster. Generally, all of the clusters are consumed to be uniformly distributed, while the rays in one cluster follow Laplace distribution in their own angel spread. α il represents the gain of the l-th ray in the i-th cluster. We suppose that it is i.i.d. and follows the distribution CN(,σα,i 2 ) where σ2 α,i is the average power of the i-th cluster. a x (φ x il,θx il ) is the array response vector in which φx il is the azimuth angle and θil x is the elevation angle. The angles with superscript t denote AoDs and that with superscript r denote AoAs. The array response vector can be formulated as a x (φ x il,θil) x = [, e jvx il,...,e j(n xh )v x il ] T Nx (2) [, e jux il,...,e j(n xv )u x il ] T, where u x il and vx il are the phase difference between adjacent elements on the vertical and horizontal dimensions. The symbol denotes Kronecker product. While establishing the channel model, we found that the array response vector a x (φ x il,θx il ) is different in two kinds of antenna deployments, which will be shown in detail in the next two subsections. A. The Array Deployed on the Side Wall If the antenna array is on the side wall, we introduce the system schematic as shown in Fig. and take the link between the BS and one user as an example. In Fig., φ is the angle between the ray and the positive x-axis while θ is the angle between the ray and the x-y plane. Then the phase difference between adjacent elements on vertical and horizontal dimensions are u x il = 2πdx v λ sin θx il, (3) vil x = 2πdx h λ cos θx il cos φ x il, (4) where d v and d h are antenna spacing on the vertical and horizontal dimensions, and λ is the signal wavelength. X d V Z d2 H Y User Fig. : The system model with the array on the side wall. B. The Array Deployed on the Ceiling If the antenna array is on the ceiling, we establish the coordinate system as shown in Fig. 2. The definitions of angles keep the same, and the two kinds of phase difference are respectively X d Z u x il = 2πdx v λ cos θx il cos φ x il, (5) vil x = 2πdx h λ cos θx il sin φ x il. (6) V d2 H Y User Fig. 2: The system model with the array on the ceiling. III. CAPACITY ANALYSIS In the analysis of data rate, different precoding approaches could lead to different results. Considering that we mainly concentrate on the relationship between antenna factors and the system performance, to avoid the influence of specific precoding method, we directly analyze the channel sumcapacity. It is well known that dirty paper coding (DPC) can achieve the capacity region of MIMO broadcast channel (BC) [5], but the implementation of DPC has very high complexity. According to the duality between MIMO-BC and MIMO multiple-access channel (MAC), the capacity region of a MIMO-BC is the same as the capacity region of its dual

3 MIMO-MAC [6]. Considering a K-user Gaussian MIMO- BC, the capacity region is represented as C union (P, H H ) C MAC (P,...,P K ; H H ) K i= PiP = {(R,...,R K ): R i K i= Tr(Pi)P i S log I + H H, i P i H i S {,...,K} }. (7) i S In (7), H is the channel matrix between the BS and users where H =[H,...,H K ] T. H i is the channel matrix of the i-th user. We assume H is constant and is known perfectly at the transmitter and at all receivers. P i is individual power constraint for each transmitter in the dual MAC, and K i= P i P is the sum power constraint. The positive semidefinite matrix P i = V i Vi H represents every set of MAC covariance matrix, where V i denotes the precoding matrix for the i-th user equipment (UE i ). R i is the achievable rate of UE i. The calculation of capacity region involves a series of optimization problem of the sum-rate in any possible user set S. If we are interested in the sum-capacity instead of the whole capacity region, the problem will be much simplified. Consider a BS equipped with N t = antennas and UEs equipped with single antenna. Then, the matrix H i degrades to a vector h i. To find the sum-capacity of this MU-MIMO channel, (7) can be converted as the following optimization problem max P I + H H PH s.t. Tr(P) P, (8) where P = diag(p,...,p K ) and H =[h,...,h K ] T. Note that the sum-capacity optimization problem of MU-MIMO is different with the capacity optimization problem of SU- MIMO. In (8), if P is the covariance matrix of a single user precoding matrix, the optimal result will be obtained by singular value decomposition of H H and water filling power allocation. But for the MU-MIMO case, P is a diagonal matrix and only involves transmit power of each user. It has no rule on how to allocate the powers according to the channel matrix H, which is the combination of channel vectors of different users. The problem (8) is convex, which can be solved by standard convex optimization algorithm, like interior-point method [7]. However, the computational complexity of the interior-point method is still high for this specific problem. Furthermore, we cannot find the condition when the sum-capacity is achieved. Observing the optimization problem (8), we can recognize that the objective function is a quadratic form whose maximum value will increase along with the increasing of sum transmit power. Therefore, the optimal result should be obtained when the sum power reaches the constraint. In other words, the optimization problem (8) is the same as max P I + H H PH s.t. Tr(P) =P. (9) We propose a gradient based method to solve problem (9). Define L = I + H H PH, then the gradient over P is P = HH PH ( H(I + H H PH) H H) T. () The diagonal element of P is the derivative of L with respect to the diagonal elements of P. Theorem. In a K-user dual MIMO-MAC channel, where each user is equipped with single antenna, the maximum of its sum-capacity is achieved when the transmit powers of some users are zero and the gradients over the transmit powers of other users are equal. Proof. Assume that P has random initial values, P () = diag(p (),P () 2,...,P () K ), where Tr(P() ) = P. Then, we can get ( [ diag, P) ],..., P P 2 P K P=P () = P=P (). () Note that the partial derivative over one transmit power P i actually depends on itself and all other transmit powers. When P () is not the optimal one to get the maximum of sum-rate, there exists m and n leading to P m P n, (2) P=P () where m, n {, 2,...,K}. It means that the data rate grows with different speed over P m and P n.if P m > P n, L grows more rapidly over P m than over P n. In other words, adding a small amount of power ε to P m will be more effective to increase data rate than adding ε to P n. Considering that P m and P n are continuous, when ε is small enough, we can always get higher data rate by setting [ ] P () = diag P (),...,P m () + ε,...,p n () ε,...,p () K. (3) In this way, the sum-rate can keep increasing and the sumpower constraint is always satisfied. Since the transmit power cannot be less than, when one variable P i approaches, it will stay at and other variables keep to change. Finally, the gradients over all nonzero transmit powers will be equal, and the achieved sumrate is the sum-capacity of the given K-user MIMO-MAC channel. The proposed optimization algorithm follows the proof. At first, set initial values for P, for example P i = P K for all i. Then calculate the gradient matrix P using (). Find the transmit power P m with the maximal gradient component P m, and the transmit power P n with the minimal gradient

4 component P n. Update P using (3). If one transmit power P i approaches, find the second minimal gradient component and keep updating the transmit power matrix. The algorithm terminates when the gradients over all non-zero transmit powers are equal (within a given precision). In our statistics, the proposed algorithm can be about times faster than the interior-point method used in CVX toolbox, while simulation results of the two methods are almost the same. IV. FACTORS AFFECTING CAPACITY In this section, we strive to find factors of antenna arrays that affecting MU-MIMO system capacity in mmwave propagation environments. We simulate in a typical indoor office scenario described in 3GPP TR38.9 R4 [5], where the length is 2 m, the width is 5 m and the height is 3 m. There are 6 users uniformly distributed at the height of m, and the BS antennas are 3 m high. The channel parameters are given by N cl =8, N ray =and σα,i 2 =. The large scale pathloss is calculated according to [5] in the indoor-office scenario with 28 GHz center frequency. About the parameters of angles, in the cluster level, the angle spread of AoDs in line-of-sight (LOS) is 5 in the azimuth and is 7.5 in the elevation, in which central AoDs of each cluster in azimuth and elevation follow the uniform distribution. The central AoAs of each cluster in azimuth and elevation follow the uniform distribution varying from to 36. In the ray level, the AoDs and AoAs in azimuth and elevation of rays follow the Laplacian distribution. The angle spread of rays in each cluster is 5. Other simulation parameters are listed in Table I and II for various scenarios. A. The Antenna Spacing and Deployment Position If we enlarge or reduce the spacing between antenna elements such as d h and d v, the phase difference in (3)-(6) will change. Then the array response and the channel matrix will also change according to (2) and (). Therefore, the correlation of users and the channel capacity may be influenced. We want to find the relationship between the antenna spacing and the system performance. When the BS is at the side wall with 5 m width, the layout of indoor office scenario is shown in Fig. 3. The antenna array of the BS is square with =8, =8, totally 64 elements, while the antenna of the user is single. We change the BS antenna spacing by multiple of wavelength and study its influence on sum data rate. Set the simulation parameters as those in Table I. The sum-capacity is calculated with the proposed gradient algorithm, and is shown in Fig. 4. As a comparison, the sum data rate achieved by the zero-forcing (ZF) precoding method is also provided. The simulation results demonstrate that as the antenna spacing increases, the data rate continues to rise and eventually reaches a plateau. Therefore, in practical systems, when the antenna array is placed on the side wall, we can greatly improve the system capacity through increasing the antenna spacing appropriately. When the BS is at center of the ceiling, we set antenna arrays as those in the side wall scenario and apply the same method to study the effect of antenna spacing on the sum data rate. Fig. 5 displays the simulation results with the parameters in Table I. The trend of lines is similar to that in Fig. 4, but the achieved data rate is higher. When the antenna spacing is less than 4λ, the data rate increases more rapidly. In this scenario, when the antenna spacing is small, even though we enlarge it a little, the data rate can remarkably get higher. Therefore, widening antenna spacing will be more effective for the BS antennas on the ceiling than those on the side wall. In Fig. 4 and Fig. 5, we can also see that, there is a large gap between the sum-capacity and the achieved sum-rate of ZF precoding when the BS antenna array is on the side wall, while the gap greatly shrinks when the BS antenna array is deployed on the ceiling. Data Rate (bps/hz) TABLE I: Simulation Parameters BS array =8, =8 Receiver array N rh =,N rv = Antenna spacing 8 λ, 4 λ, λ, λ, 2λ, 4λ, 8λ, 6λ, 32λ 2 Nth Ntv BS 5m 3m User 2m Fig. 3: Layout of the indoor office scenario Antenna Spacing (λ) m Sum Capacity ZF Precoding Fig. 4: Data rate with different antenna spacing when the BS antenna array is on the side wall. B. The Configuration of the Antenna Array The BS antennas we considered are all 2D planar arrays which can be divided into horizontal and vertical dimensions.

5 Data Rate (bps/hz) Sum Capacity ZF Precoding Antenna Spacing (λ) Fig. 5: Data rate with different antenna spacing when the BS antenna array is on the ceiling. In this part, we change the number of antenna elements on these two dimensions to form different kinds of array configurations. For example, in the above subsections, the BS antenna arrays are square, and each array has 8 elements on both dimensions. However, what will happen on the system performance if the number of elements on vertical and horizontal dimensions are not equal? Firstly, take the BS array on the side wall as an example. When >, we call it fat array. Otherwise, if <, we call it thin array. Each array has 64 elements. Other parameters are listed in Table II. In Fig. 6, the sumcapacities are shown for seven kinds of array configurations. In general, when the array is on the side wall and the total number of elements is constant, the more elements are put on the horizontal dimension, the higher capacity that can be achieved. Additionally, the gap of the capacity curves of fat arrays is wider than that of thin arrays. Therefore, it will be more effective to increase the elements on horizontal dimension. Moreover, we plot the CDF of correlation among users in Fig. 7 when the antenna spacing is 4λ. It can be seen that when the array is fat, the correlation among users is smaller than that of the thin array, so the inter-user interference is weaker and the capacity is higher. Hence, if the antenna array is arranged on the side wall, we should increase elements on the horizontal dimension as much as possible. Then consider that the BS array is at the center of the ceiling. When >, we call it wide array. Otherwise, if <, we call it long array. We take the same method to get simulation results of capacity and correlation coefficients which are shown in Fig. 8 and Fig. 9. As can be seen from Fig. 8, the difference of capacity is most recognizable when the antenna spacing is 2λ. Thus, in Fig. 9 we assume the antenna spacing as 2λ to see the CDF of channel correlation coefficients. We can see that the wide arrays have advantages over the long arrays, so it is more effective to add elements on the horizontal dimension (parallel to the long side wall). But if the long array configuration has to be used, putting all elements on a vertical line (parallel to the short side wall) can achieve the highest capacity. Capacity (bps/hz) TABLE II: Simulation Parameters BS array Receiver array N rh =,N rv = Antenna spacing Side Ceiling 2 λ, λ, 2λ, 4λ, 8λ, 6λ λ, λ, λ, λ, 2λ, 4λ =64 = (Fat) =32 =2 (Fat) =6 =4 (Fat) =8 =4 =2 = =8 (Square) =6 (Thin) =32 (Thin) =64 (Thin) Antenna Spacing () Fig. 6: Capacity of BS antennas with various configurations on the side wall. CDF N =64 N = (Fat) th tv.4 N =32 N =2 (Fat) th tv.3 N =6 N =4 (Fat) th tv N =8 th N =8 (Square) tv.2 =4 =6 (Thin). N =2 th N =32 (Thin) tv N = th N =64 (Thin) tv Correlation Coefficient Fig. 7: CDF of users correlation for BS arrays on the side wall with various configurations and antenna spacing =4λ.

6 Capacity (bps/hz) =64 = (Wide) =32 =2 (Wide) =6 =4 (Wide) =8 =4 =2 = =8 (Square) =6 (Long) =32 (Long) =64 (Long) Antenna Spacing () Fig. 8: Capacity of BS antennas with various configurations on the ceiling. CDF =64 = (Wide) =32 =2 (Wide) =6 =4 (Wide) =8 =4 =2 = =8 (Square) =6 (Long) =32 (Long) =64 (Long) Correlation Coefficient Fig. 9: CDF of users correlation for BS arrays on the ceiling with various configurations and antenna spacing = 2 λ. V. CONCLUSION In this paper, we study how the array factors of BS antennas affect the system performance in millimeter-wave MU-MIMO systems. We first study the property of optimal power allocation when the sum-capacity is achieved, and then propose a low-complexity calculation method for the sumcapacity. Then we simulate the system performance under different kinds of deployments and configurations of the BS array in an indoor office scenario. In an appropriate range, with the antenna spacing increasing, the channel capacity will gradually increase. With a fixed height, deploying the BS antennas on the ceiling can achieve higher capacity. If the total number of elements in the array is constant, allocating more number of elements on the horizontal dimension is more effective. The channel correlation properties among users are also demonstrated. In a given propagation environment, the guidance of BS antenna configuration is trying every aspect to reduce the channel correlation among users. ACKNOWLEDGEMENTS This work was supported by the National Natural Science Foundation of China under Grant REFERENCES [] S. Rangan, T. S. Rappaport, and E. Erkip, Millimeter-wave cellular wireless networks: Potentials and challenges, Proceedings of the IEEE, vol. 2, no. 3, pp , March 24. [2] T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, Millimeter wave mobile communications for 5G cellular: It will work! IEEE Access, vol., pp , May 23. [3] T. S. Rappaport, G. R. MacCartney, M. K. Samimi, and S. Sun, Wideband millimeter-wave propagation measurements and channel models for future wireless communication system design, IEEE Transactions on Communications, vol. 63, no. 9, pp , Sept 25. [4] S. Hur, S. Baek, B. Kim, Y. Chang, A. F. Molisch, T. S. Rappaport, K. Haneda, and J. Park, Proposal on millimeter-wave channel modeling for 5G cellular system, IEEE Journal of Selected Topics in Signal Processing, vol., no. 3, pp , April 26. [5] 3GPP, Study on channel model for frequency spectrum above 6 GHz, in TR 38.9 V4.., June 26. [6] D.-S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, Fading correlation and its effect on the capacity of multielement antenna systems, IEEE Transactions on Communications, vol. 48, no. 3, pp , Mar 2. [7] L. Dong, H. Ling, and R. W. J. Heath, Multiple-input multiple-output wireless communication systems using antenna pattern diversity, in IEEE Global Telecommunications Conference, vol., Nov 22, pp [8] Y.-B. Tian and J. Qian, Improve the performance of a linear array by changing the spaces among array elements in terms of genetic algorithm, IEEE Transactions on Antennas and Propagation, vol. 53, no. 7, pp , July 25. [9] Y. H. Nam, B. L. Ng, K. Sayana, Y. Li, J. Zhang, Y. Kim, and J. Lee, Full-dimension MIMO (FD-MIMO) for next generation cellular technology, IEEE Communications Magazine, vol. 5, no. 6, pp , June 23. [] A. A. Abouda, H. M. El-Sallabi, and S. G. Haggman, Impact of antenna array geometry on MIMO channel eigenvalues, in IEEE 6th International Symposium on Personal, Indoor and Mobile Radio Communications, vol., Sept 25, pp [] C. B. Dietrich, K. Dietze, J. R. Nealy, and W. L. Stutzman, Spatial, polarization, and pattern diversity for wireless handheld terminals, IEEE Transactions on Antennas and Propagation, vol. 49, no. 9, pp , Sep 2. [2] K. Sulonen, P. Suvikunnas, L. Vuokko, J. Kivinen, and P. Vainikainen, Comparison of MIMO antenna configurations in picocell and microcell environments, IEEE Journal on Selected Areas in Communications, vol. 2, no. 5, pp , June 23. [3] L. Dong, H. Choo, R. W. Heath, and H. Ling, Simulation of MIMO channel capacity with antenna polarization diversity, IEEE Transactions on Wireless Communications, vol. 4, no. 4, pp , July 25. [4] T. S. Rappaport, R. W. Heath, R. Daniels, and J. N. Murdock, Millimeter Wave Wireless Communications. Prentice Hall: Pearson Education, 24. [5] H. Weingarten, Y. Steinberg, and S. S. Shamai, The capacity region of the Gaussian multiple-input multiple-output broadcast channel, IEEE Transactions on Information Theory, vol. 52, no. 9, pp , Sept 26. [6] S. Vishwanath, N. Jindal, and A. Goldsmith, Duality, achievable rates, and sum-rate capacity of Gaussian MIMO broadcast channels, IEEE Transactions on Information Theory, vol. 49, no., pp , Oct 23. [7] S. Boyd and L. Vandenberghe, Convex optimization. Cambridge: Cambridge University Press, 24.