Int. J. Communications, Network and System Sciences, 2010, 3, 266272 doi:10.4236/ijcns.2010.33034 blished Online March 2010 (http://www.SciRP.org/journal/ijcns/). Copyright © 2010 SciRes. IJCNS Pu Performance of Block Diagonalization Broadcasting Scheme for Multiuser MIMO System Operating in Presence of Spatial Correlation and Mutual Coupling Feng Wang, Marek E. Bialkowski, Xia Liu School of Information Technology & Electrical Enginnering, University of Queensland, Brisbane, Australia Email: {fwang, meb, xialiu}@itee.uq.edu.au Received December 31, 2009; revised January 30, 2010; accepted February 27, 2010 Abstract In this paper, the capacity of a multiuser Multiple Input Multiple Output (MIMO) system employing the block diagonalization broadcasting scheme in presence of spatial correlation and mutual coupling is investi gated. It is shown by computer simulations that, in general, the presence of spatial correlation decreases the capacity of a multiuser MIMO system. However, for some particular antenna element spacing mutual cou pling decreases the spatial correlation rendering an increased capacity. The optimized diagonalization broadcasting technique with a twostage power allocation scheme is proposed and verified. The presented simulations results confirm the advantage of the proposed broadcasting scheme. Keywords: Multiuser MIMO, Block Diagonalizaiton, Mutual Coupling, Dirty Paper Coding 1. Introduction It has been shown via theoretical derivations as well as by experiment that using multiple element antennas with a suitable signal transmission scheme in a rich scattering propagation environment can enhance peertopeer com munication without the use of extra frequency bandwidth [1,2]. This potential of multiple input multiple output (MIMO) communication systems can be used to advan tage using two alternative approaches. In one approach, the signal transmission quality via diversity can be im proved. Alternatively for a chosen quality factor such as bit error rate (BER), the data rate can be increased by a stream multiplexing transmission. Most recent studies on MIMO focus on multiuser sys tems. For a multiuser MIMO system, allocation of the channel resources among independent users either in the form of multiple accesses (uplink) or broadcasting (dow nlink) is considered. The information theory hints that the broadcasting case is by far the most challenging. In this case, an interuser interference occurs due to the spatially multiplexed transmitted signals at a base station (BS). For the Gaussian MIMO broadcasting channels, it has been proved that dirty paper coding (DPC) [3] can achieve the available capacity [4]. However, to deploy DPC in a real system is challenging due to the high com plexity and computational burden on successive encod ing and decoding. An alternative strategy is the block diagonalization (BD) [5,6]. Compared with DPC sche me, BD is a suboptimal technique with much reduced complexity. Using this technique, signals are transmitted only to desired users. In turn, null steering is applied to other users by decomposing the multiuser channel into a group of parallel single user MIMO channels. To achieve such decomposition, BS needs to select a suitable beam forming matrix for each user. The matrix is vertical to the space spanned by other users’ channels matrices. If the channel matrices of all the scheduled users are per fectly known at the transmitter, the interuser interfer ence can be eliminated by BD, rendering a simple re ceiver structure. Because of its simplicity and good performance, BD is under constant research. In [7], the imperfect channel state information (CSI) assumption was used while in vestigating BD. The effect of outdated CSI at transmitter on multiuser MIMO system with BD was reported. In [8], a BD algorithm that accounts for the presence of other cell interferences was proposed under the assumption that the transmitter has full CSI and the information about the interference plus noise covariance matrix for incell users. Most of the research on BD for multiuser MIMO systems was done by neglecting interactions within the transmitting and receiving array antennas and between the array antennas and scatterers. When a
F. WANG ET AL. 267 MIMO transceiver has to be of compact size interactions within the transmitting or receiving array antennas have to be taken into account. The small interelement spacing in the antenna array in such transceivers renders mutual coupling. The effect of mutual coupling on a point topoint MIMO system has been investigated in many works, such as [9,10,11]. In this paper, a BD algorithm that accounts for the effect of spatial correlation and mutual coupling in array antennas is presented and its performance is evaluated with respect to the overall sys tem capacity. The paper is organized as follows. Section 2 describes a multiuser MIMO system model including the channel model with spatial correlation. Section 3 describes inter actions between the array elements and scatterers in the propagation environment in which mutual coupling ef fects cannot be neglected. Section 4 gives details of the BD algorithm that accounts for the effect of mutual cou pling. Section 5 quantifies the effect of mutual coupling by presenting numerical results. Section 6 summarizes the findings of the undertaken research. 2. System Model 2.1. Signal Model A narrowband multiuser system is assumed. It is postu lated that it is created around a base station (BS) with L downlink mobile users. The base station includes N transmitting antennas. At time t, K mobile stations (MS) from L available users are scheduled to be serviced by BS. The kth mobile station (MS) employs Mk antennas. The transmitted signal intended for the kth mobile sta tion is denoted by a Qk × 1 dimensional vector xk which is weighted by an N × Qk preprocessing matrix Wk be fore transmission. Qk is the number of parallel data symbols transmitted simultaneously to the kth MS. The MIMO channel between the BS the kth MS is described by the complex matrix Hk, whose (i,j)th entries represent the complex gain between the jth transmit antenna at BS and ith antenna at kth MS. It is assumed that different MS experience independent fading. The received signal at kth MS can be presented by 1 1, K kk kkkk k K kkkkk jjjk jjk yH WExn HWExHWEx n (1) where trace(EkE† k) = pk is the power transmitted to the kth MS. nk is the additive Gaussian white noise (AWGN) vector, whose elements are independent identical distri bution (i.i.d.) zeromean circularly symmetric complex Gaussian random variables with variance σn 2. 2.2. Channel Model The channel matrix Hk describing the channel properties between BS and the kth MS is influence by the transmit ting and receiving antenna array configurations and a signal propagation environment. It is assumed that the BS and MSs are equipped with wire dipoles arranged in liner arrays. The length of each dipole element is as sumed to be half wavelength. Also, the links between BS and different MSs do not share the same scattering envi ronment. This assumption confirms the earlier assumed independent signal fading for different MSs. For each link, the Kronecker channel model [5,12] is assumed. In this model, the correlations at transmitter and receiver sides are independent and the channel matrix Hk is rep resented as 1/2 1/2k kMS H HR GR BS (2) where GH is a matrix with i.i.d. Gaussian entries with zero mean and unit variance and k S R and RBS are spa tial correlation matrices at the kth MS and BS, respec tively. In a rich scattering environment, the correlation for any pair of dipole element with spacing dm,n can be obtained using Clark’s model and are given by a Bessel function ,0, ( mn mn Jkd ) (3) Using (3), the correlation matrix for the kth MS can be generated as 1,1 1, ,1 , k kk M k MS MM k M R (4) In turn, the correlation matrix for BS can be obtained from 1,1 1, ,1 , N BS NN N R (5) 3. Mutual Coupling For the array formed by linear parallel wire dipoles, the mutual coupling matrix can be expressed using electro magnetic and circular theory described in [9] 1 (ΖΖ)( ) AT TM ZCI (6) where ZA = 73 + j42.5[Ω] is the element impedance in isolation and ZT is impedance of the receiver at each ele ment. It is chosen to be the complex conjugate of ZA to obtain the impedance match. Z is the mutual impedance C opyright © 2010 SciRes. IJCNS
F. WANG ET AL. 268 matrix with all the diagonal elements equal to ZA + ZT, its nondiagonal elements Znm are decided by the physical parameters including dipole length, the horizontal dis tance between the two dipoles. For a sidebyside array configuration and dipole length l equals to 0.5λ, Znm is given by [9,10] 012 012 300.5722ln(2 )(2 ) 30(2) , 30 2()()() 302() () () i i mn iii iii lCl mn jS l ZCuCuCumn jSuSuSu (7) where β=2π⁄λ is the wave number and Ci(u) and Si(u) are the cosine and sine integral, respectively, given as 0 cos( ) () u i u i x Cu dx x sinx Su dx x (8) and the constants are given by [10] 0 22 1 22 2 h h h ud udl udl l l (9) where dh is the horizontal distance between the two di pole antennas. 4. Block Diagonalizaiton with Mutual Coupling We assume at time t, K mobile stations (MS) from L available users are scheduled to be serviced by BS. To ensure the sufficient freedom for BS to perform BD over the K scheduled MSs, it is assumed that 1 K k k N (10) With spatial correlation and mutual coupling taken into account, the received signal at kth MS described by (1) can be rewritten as 1/ 21/2 1 1/ 21/ 2 1/ 21/2 1, K kk kMSMSBSBSkkkk k kk MSMSBSBSkk k K kk SMSBSBSjjj k jjk H H H yCRGRCWExn CR GRCWEx CR GRCWExn (11) where, CBS and k S C are the mutual coupling matrices for the dipole element array at BS and the kth MS, re spectively. Wk is the beamforming matrix at BS for the kth MS. To eliminate the interference from the signals transmitted to other MSs, the key idea in the block di agonalization is to zeroforce the interference by impos ing the following condition 1/ 21/2(,1, kk MSMSBS BSjjk kjK H CR GRCW0) (12) when the mutual coupling and correlation is taken into account, 1/ 21/ 2 12 12 [, ,,] [, ,,] kk kMSMSBSB T K K H HCR GRC HHH H WWW W S (13) where (•)T donates the matrix transpose operation. By including the condition given by (12), the effective chan nel matrix for the multiuser MIMO system with K MSs can be represented by a ∑Mk × ∑Mk matrix, given as 11121 21 222 1 K K KK K DHW HW HWHW HW HWHW HW HW (14) By using (12), Equation (14) can be rewritten as 11 22 KK HW0 00 0HW0 0 D00 0 000HW (15) At this point it is important to comment whether the condition (12) can be met in practice. From the theory of antennas it is known that an Nelement array antenna is capable of forming N1 nulls. This means that in the strict sense, the BS having an Nelement array is able only to null up to N1 MSs. For a larger number of MSs, the condition (12) has to be compromised. In such a case, the BS can direct low sidelobes instead of nulls towards undesired users. In further considerations, it is assumed that the number of MSs served by BS is such that the condition (12) is met. 4.1. Calculation of Beamforming Matrices In order to transmit a signal only to the desired MS while steering nulls to the remaining MSs, the beamforming matrix for the desired MS should be orthogonal to the space spanned by the channel matrices of the undesired MSs. We define the channel matrix as 111 [,,, T kkk HHHH H ] K (16) Copyright © 2010 SciRes. IJCNS
F. WANG ET AL. 269 which is obtained by removing the channel matrix for the kth MS from . Performing the eigenvalue decomposi tion (EVD) over the N × N nonnegative Hermitian Ma trix, one obtains H † † † [] k kk kk k Σ0V HHVV 00V (17) where (•) † denotes the conjugate transpose operation. It can bee seen that Vk is a matrix with the dimension of N × Mk. Its columns correspond to those zero eigen values. By letting Wk = Vk, a perfect null steering to all the undesired K1 MSs can be achieved. By repeating the steps represented by Equation (15) and (16), all the K beamforming matrices can be obtained. In this way, as shown in Equation (14), the multiuser MIMO downlink system is decomposed into K independent singleuser MIMO systems. 4.2. Overall Capacity of Multiuser MIMO Broadcasting with Block Diagonalization For the case of a multiuser MIMO downlink system which is decomposed into K independent singleuser MIMO systems by block diagonalization, the overall capacity can be obtained as a sum of individual links capacities, as expressed by ††† 22 1 1 log det K k umk kkk kk kn C xx IEWHRHWE (18) With mutual coupling and spatial correlations taken into account, (18) can be rewritten as ††† 22 1 1 log det K k umk kkk kk kn C xx IEWHRHWE (19) where is the kth MS’s input covariance matrix. The capacity for kth MS is k xx R ††† 22 1 log detk kkkk n C xx IEWHRHWE kkk (20) We assume that the signal intended for the kth MS is a Gaussian signal. As a result, (20) can be simplified to †† 22 †† †† 22 log det log det k kkkkk n k kBSkMSMSkBSk n p C p IWHHW IWCHCCHCW For high SNRs, (21) can be further simplified and given by †† †† 22 †† 22 †† †† † 22 2 † log det log det det det log detlog det det k kkBSk MSMSk BSk n k kkkk n MS MSBS BS k kkkkMSMS n BS BS p C p p WC HCCHCW WHHW CC CC WHHWC C CC (22) The last part of Equation (22) shows three terms con tributing to the capacity. The first term represents the broadcasting capacity for the kth MS without the effect of mutual coupling at BS and MS. The second and third terms represent the mutual coupling at kth MS and BS. The effect of these terms on capacity depends on the coupling matrices at the MS and BS ends. If the product of the determinants of the mutual coupling matrices is larger than one, the effect of mutual coupling on capacity is positive. Otherwise, it is negative. 4.3. Power Allocation Scheme The most straightforward power allocation scheme from BS to different MSs is accomplished by transmitting equal power to each MS. That is † trace() T kkk P p EE (23) where PT is the total transmitted power at BS and pk is the power allocated to the kth MS. This power allocation scheme is simple to realize in practice. However, it does not always provide the best performance with respect to capacity. To maximize the capacity, a twostage power allocation scheme is pre sented. At the first stage, the power allocation is accom plished according to the objective function at the users’ level, as expressed by 12 12 ,,, 1 Max,, , K sum K pp p K kT k Cpp p Subject topP (24) (21) The result of (24) is the optimized power allocation for different users under service. This is the capacitygreedy power allocation scheme and is nonlinear. The solution can be obtained by applying a Lagrange method. At the second stage, the transmit power for each user can be optimized at an antenna level by using a wa terfilling scheme. At this stage, the power is allocated to C opyright © 2010 SciRes. IJCNS
F. WANG ET AL. 270 different transmit antennas according to the objective function, which is described by 2 1 ,1,2,, in kk i k r i kk i pi Subject topp r (25) where (z)+=max(0,z) and μk is chosen to obey the power constraint for the kth MS and r is the rank of the effec tive channel matrix between BS and the kth MS 1/2 1/2 rank kk SMS BSBSk rH CR GRCW (26) By applying the waterfilling scheme, the capacity for kth MS is 2 2 1 1 log 1 r i kkk in C n (27) 5. Numerical Results Using the presented theory, computer simulations are performed for a multiuser MIMO system with 8 trans mit antennas at BS and 3 MSs each equipped with 2 re ceive antennas. It is assumed that the three MSs are scheduled and served by BS at the same time. As a result, this system is referred to as a 3 × (2 × 8) system. Figure 1 presents the possible impact of spatial corre lation and mutual coupling on the broadcasting through put. In simulations, the dipole spacing at BS and MS is assumed to be fixed at 1.0λ and 0.5λ, respectively. As observed from the results presented in Figure 1, Dirty Paper Coding, where effects of spatial correlation and mutual coupling are neglected, offers the largest 5 05 10 15 20 0 5 10 15 20 25 30 35 40 45 SNR[dB] Sum Throughput[bps/Hz] Dirty Paper Coding BD with Rayleigh Channel BD with Spatial Correlation BD with Mu tu a l Coup li n g Figure 1. Broadcasting throughput for a 3 × (2 × 8) system. scheme in which spatial correlation and mutual coupling are neglected shows a reduced throughput. The through put for BD with mutual coupling or spatial correlation included in calculations further reduces the system throu ghput. The differences are most pronounced at larger levels of SNR. Figure 2 shows the effect of spatial correlation and mutual coupling on the broadcasting throughput for a 3× (2×8) system. The SNR is set to 10 dB and the unit for dipole spacing is the wavelength, represented by λ. The solid lines represent CDF of broadcasting throughput with spatial correlation only and the dotted lines are for the CDF of broadcasting throughput with spatial correla tion and mutual coupling combined. It can bee seen form Figure 2 that the presence of spatial correlation and mu tual coupling results in a degraded broadcasting throug hput in comparison with an idealized Rayleigh channel. . In general, spatial correlation is regarded as a negative factor in a MIMO communication system. However, mutual coupling can be seen as a positive factor at some dipole spacing. As observed in Figure 2, for the dipole spacing of 0.2λ and 0.3λ, the existence of mutual cou pling results in a higher capacity. It is interesting to note that the curve of the capacity with and without mutual coupling merge at the point of dipole spacing equal to 0.4λ. When the spacing is increased to 0.6λ, the plot rep resenting the capacity with mutual coupling is on the left side of the curve for the capacity with correlation only. This is the case for which the presence of mutual cou pling leads to a lower capacity. Figures 3 and 4 show comparisons between capacity with spatial correlation only, and with spatial correlation plus mutual coupling, as a function of antenna element spacing. In the presented simulation results, the SNR is set to 67 8 91011 12 13 1415 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1 Sum Throughput[bps/Hz] CDF Dipole Spacing =0.2 No M C Dipole Spacing =0.2 Wit h M C Dipole Spacing =0.3 No M C Dipole Spacing =0.3 Wit h M C Dipole Spacing =0.4 No M C Dipole Spacing =0.4 Wit h M C Dipole Spacing =0.6 No M C Dipole Spacing =0.6 Wit h M C Rayl ei gh Channel Dipole Spaci ng =0.2 Dipole Spacing =0.3 Dipole S pacing =0.6 Dipole S pac i ng = 0. 4 Figure 2. Broadcasting throughput CDF for a 3 × (2 × 8) system. Copyright © 2010 SciRes. IJCNS
F. WANG ET AL. 271 0.10.2 0.30.4 0.50.60.7 0.8 0.91 6 6. 5 7 7. 5 8 8. 5 9 9. 5 Dipole spacing Sum Throughput[bps/Hz] BD with Spatial Correlation BD with Mutual Coupling Figure 3. Broadcasting throughput vs. MS array interele ment spacing for a 3 × (2 × 8) system. 00.2 0.4 0.6 0.8 1 5 0 5 10 15 20 0 5 10 15 20 25 MS Int erelem ent spaci ng SNR(dB) Sum T hroughput(bps/Hz ) 5 10 15 20 BD with Spatial Correlation BD with Mutual Coupling Figure 4. Broadcasting throughput vs. MS array interele ment spacing and SNR for a 3 × (2 × 8) system. 10 dB and the unit for dipole spacing is the wavelength, as represented by λ. The dipole spacing ranges from 0.0λ to 1.0λ. We can see that the curves for BD with spatial correlation only and BD with spatial correlation plus mu tual coupling cross at 0.4λ and 0.95λ. For the dipole spacing range from 0.4λ to 0.95λ, mutual coupling in creases the spatial correlation level and results in a de creased capacity. In turn, when the dipole spacing ranges from 0.1λ to 0.4λ, mutual coupling decreases the spatial correlation level and renders an increased capacity. The results presented in Figures 5 and 6 verify the twostage power allocation scheme described in Section 6. One can see from results presented in Figures 5 and 6 that with or without mutual coupling, the optimized power allocation scheme leads to a higher capacity than the nonoptimized one over the SNR range from 5 dB to 20 dB and the antenna spacing from 0.1λ to 1λ. The 0 0. 2 0. 4 0.6 0.8 1 5 0510 15 20 0 5 10 15 20 25 30 MS Interel ement spac ing SNR(d B) Sum T hroughput(bps/Hz) 5 10 15 20 25 Opt im i zed Broadc asting Throughput Nonoptim i zed Broadc asting Throughtput Figure 5. Comparison of optimized and nonoptimized broadcasting throughput vs. MS array interelement spacing and SNR for a 3 × (2 × 8) system in the presence of spatial correlation only. 0 0. 2 0. 4 0. 6 0. 8 1 5 0 5 10 15 20 0 10 20 30 S NR(dB) MS Int erelem ent spac i ng S um T hroughput(bps/ Hz) 5 10 15 20 25 Opt i m i zed Broadc asti ng Throughut Nonopti m i zed Broadc asti ng Throughut Figure 6. Comparison of optimized and nonoptimized broadcasting throughput vs. MS array interelement spac ing and SNR for a 3 × (2 × 8) system in the presence of spa tial correlation and mutual coupling. optimized scheme improves capacity in the presence of spatial correlation and mutual coupling. This achieve ment is more apparent at higher values of SNR and lar ger interelement antenna spacing. 6. Conclusions In this paper, investigations into the capacity of a multi user MIMO system with block diagonalization broad casting scheme in the presence of spatial correlation and mutual coupling have been presented. The effect of spa tial correlation and mutual coupling on the broadcasting throughput for block diagonalization broadcasting has been analyzed. It has been shown by the performed computer simulations that the presence of spatial correla C opyright © 2010 SciRes. IJCNS
F. WANG ET AL. Copyright © 2010 SciRes. IJCNS 272 tion leads to a decreased capacity. However, mutual cou pling may have negative or positive influence of capacity. For some particular dipole spacing range, mutual cou pling decreases the spatial correlation level, rendering an increased capacity. The optimized diagonalization broad casting technique with a twostage power allocation scheme has been proposed and verified. The presented simulations results have demonstrated a positive impact of this optimized BD scheme. 7. Acknowledgment One of the authors (F. Wang) acknowledges the support of the University of Queensland in the form of Interna tional Postgraduate Research Scholarship (IPRS). 8. References [1] G. J. Foschini and M. J. Gans, “On limits of wireless co mmunications in a fading environment when using mul tiple antennas,” Wireless Personal Communications, Vol. 6, pp. 311–335, 1998. [2] E. Telatar, “Capacity of multiantenna Gaussian chann els,” European Transactions on Telecommunications, Vol. 10, No. 6, pp. 585–596, November 1999. [3] M. Costa, “Writing on dirty paper,” IEEE Transactions on Information Theory, Vol. 49, No. 3, pp. 439–441, May 1983. [4] W. Weingarten, Y. Steinberg, and S. Shamai, “The capa city region of the Gaussian multipleinput multipleoutput broadcast channel,” IEEE Transactions on Information Theory, Vol. 52, No. 9, pp. 3936–3964, September 2006. [5] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Transactions on Infor mation Theory, Vol. 42, No. 3, pp. 461–471, February 2004. [6] L. U. Choi and R. D. Murch, “A transmit preprocssing technique for multiuser MIMO systems using a decom position approch,” IEEE Transactions on Wireless Com munications, Vol. 3, No. 1, pp. 20–24, January 2004. [7] K. Zhang and Z. Niu, “Multiuser MIMO downlink trans mission over timevaring channels,” Proceedings Interna tional Conference on Communications, pp. 5514–5518, June 2007. [8] S. Shim, J. S. Kwak, R. W. Heath, and J. Andrews, “Block diagonalization for multiuser MIMO with othercell interference,” IEEE Transactions on Wireless Commun ications, Vol. 7, No. 7, July 2008. [9] S. Durrani and M. E. Bialkowski, “Effect of mutual coupling on the interference rejection capabilities of linear and circular arrays in CDMA systems,” IEEE Transactions on Antennas and Propagation, Vol. 52, No. 4, pp. 1130–1134, April 2004. [10] M. E. Bialkowski, P. Uthansakul, K. Bialkowski, and S. Durrani, “Investigating the performance of MIMO sys tems from an electromagenetic perspective,” Microwave and Optical Technology Letters, Vol. 48, No. 7, pp. 1233 –1238, July 2006. [11] F. Wang, M. E. Bialkowski, and X. Liu, “Investigating the effect of mutual coupling on SVD based beam forming over MIMO channels,” International Journal on Signal Processing, Vol. 3, No. 4, pp. 73–82, July 2009. [12] C. N. Chuah, D. N. C. Tse, and J. M. Kahn, “Capacity scaling in MIMO wireless systems under correlated fad ing,” IEEE Transactions on Information Theory, Vol. 48, pp. 637–650, March 2002.
