Performance Analysis of Multiuser MIMO Systems with Scheduling and Antenna Selection

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1 Performance Analysis of Multiuser MIMO Systems with Scheduling and Antenna Selection Mohammad Torabi Wessam Ajib David Haccoun Dept. of Electrical Engineering Dept. of Computer Science Dept. of Electrical Engineering École Polytechnique de Montréal Université duquebéc a Montréal École Polytechnique de Montréal Montréal, QC, Canada Montréal, QC, Canada Montréal, QC, Canada Abstract In this paper, we present a performance analysis of the user scheduling for the multiuser MIMO systems exploiting the multiuser and antenna diversities while maintaining the fairness among the users. We present different scheduling schemes including absolute throughput-based scheduling, normalized throughput-based scheduling, absolute SNR-based scheduling, and normalized SNR-based scheduling schemes. We use an antenna selection scheme to overcome the drawback of channel hardening in multiuser MIMO systems. This also improves the system performance and reduces the system complexity. Using mathematical analysis and numerical simulations, we compare the presented schemes and show their significant advantages. I. INTRODUCTION In the multiuser wireless communications, scheduling techniques have been proposed to exploit the multiuser diversity. While user scheduling aims to maximize the system throughput, the fairness in resource allocation among the users is a key parameter and should be taken into consideration. As a full-fair scheduling scheme, round-robin scheduler is a simple and efficient scheme, where all users have the same priority for accessing the channel, but it does not exploit the multiuser diversity. On the other hand, the best-user scheduler the so called opportunistic or greedy scheduler, selects the user with the highest throughput. The greedy scheduler achieves a higher throughput than round-robin scheduler does, but at the price of unfairness in resource allocation among the users. In the greedy scheduling scheme, some of the users who are closer to the base station, will have more chance to access the channel, causing an unfairness among the users. The proportional fair scheduler (PFS has been proposed for exploiting multiuser diversity while maintaining the fairness among the users. In PFS, there is a tradeoff between the total throughput and the fairness. In stead of selecting the best user with highest absolute-throughput or absolute-snr, PFS chooses the user with the highest normalized-throughput or normalized-snr, (normalized to its own average [1], [2]. In Multiple-input multiple-output (MIMO systems, a multiuser scheduler can be used to exploit the antenna and multiuser diversities at the same time and to increase the system performance [1] [3]. An overview of user selection and scheduling algorithms in MIMO systems is presented in []. It has been observed that in the MIMO systems due to the channel hardening the full advantages of both multiuser and antenna diversities can not be exploited [], []. In order to overcome this, antenna selection can be employed to reduce the channel hardening []. It is shown that a significant gain from both multiuser and antenna diversities can be obtained at the same time. In this paper, we present a performance analysis for multiuser MIMO systems employing orthogonal space-time block coding (OSTBC, with proportional fair scheduling. In order to reduce the effects of MIMO channel hardening, we use transmit antenna selection. This can improve the system performance while reducing the hardware complexity. We derive the expressions for the probability density function (PDF and cumulative distribution function (CDF of the signal-to-noiseratio (SNR in a MIMO channel with antenna selection for several multiuser scheduling schemes which enable us to establish a mathematical analysis and formulation for the average capacity of the system under study. We consider two scenarios: Heterogenous (independent non-identically distributed SNRsi.ni.d. and Homogenous (independent identically distributed SNRs- i.i.d. cases, and derive the expressions for the PDF and CDF of the channel SNR for the users in each case. Then using numerical simulations and calculation of the mathematical formulas, we compare the presented schemes and show their significant advantages. The rest of this paper is organized as follows. Section II presents the system model and the used assumptions. The PDF and CDF expressions for the user scheduling in the multiuser OSTBC system is presented in Section III. In Section IV, several multiuser scheduling schemes with antenna selection is presented. An average capacity analysis for the presented schemes is given in Section V. Numerical results are presented in Section VI. Finally, Section VII concludes this paper. II. SYSTEM MODEL We consider a multiuser MIMO system, employing m T transmit antennas at the base station, and K users, each with n R receive antennas. We assume that m T transmit antennas are subdivided in L groups each containing n T transmit antennas, such that m T = Ln T. We assume that the MIMO channel between the uth user and each group of antennas at the base station is a Rayleigh fading MIMO channel and can be expressed by H u,a matrix of size n R n T, with elements h (j,i u,a corresponding to the complex channel gain of the channel response between the i-th transmit and j-th receive antennas for the u-th user and a-th group of transmit antennas. Depending on the employed orthogonal space-time code, the number of transmit antennas n T will be different. For example in the orthogonal space-time codes G 2, G 3, H 3 G, and H, n T will be 2, 3, 3,, and respectively [7]. The multiuser scheduler can select the best user based on the channel quality of the users in each time-slot according to their absolute or normalized SNR (or Throughput values /08/$ IEEE 1910

2 The expression for the received signal for each user in OSTBC after the OSTBC decoding can be written as [8] s u,a = c H u,a 2 F s u,a + η u,a, (1 where s u,a is the output signal of OSTBC decoder, s u,a is the transmitted symbol,. 2 F is the squared Frobenius norm of a matrix, H u,a 2 F = n R nt j=1 i=1 h (j,i u,a 2, and η u,a is a complex Gaussian noise with distribution N (0, cn0 2 H u,a 2 F per dimension and c is a constant that depends on the OSTBC. For example, for the orthogonal space-time block codes given in [7], c =1for the rate 1 (R c =1 code G 2 and for the rate 3/ codes H 3, and H, and c =2for the rate 1/2 codes G 3 and G. The total energy of the symbol transmitted through the n T antennas can be normalized to n T and, therefore, similar to the expression in [8], we can express the instantaneous SNR per symbol at the receiver of the u-th user as u,a = u,a H u,a 2 F, (2 where u,a is the average receive SNR per antenna, and R c is the code rate of OSTBC. III. PDF AND CDF EXPRESSIONS In this section, we express the PDF and the CDF for the SNR-based user scheduling schemes for the multiuser OSTBC system, which enable us to establish a mathematical analysis and formulation for the average channel capacity of the system under study. A. PDF and CDF for the OSTBC We assumed that in the MIMO system the channel fading, i.e., h (j,i u,a is a Rayleigh flat-fading. Therefore, h (j,i u,a 2 for each user will be a Chi-squared distributed random variable. Since H u,a 2 F is the sum of 2n Tn R independent and identically distributed (i.i.d. h (j,i u,a 2 random variables, then H u,a 2 F is distributed with 2n T n R degrees of freedom. Thus, using a change in variables, we can show that the PDF and CDF of the received SNR for each user in OSTBC; given in (2, can be expressed as f u,a ( = ( ntnr ntnr 1 e ntrc/ u,a (3 (n T n R 1! u,a ntnr F u,a ( =1 e ntrc/ u,a n Tn R 1 l=0 ( / u,a l. ( l! IV. MULTIUSER SCHEDULING WITH ANTENNA SELECTION Considering that there are K users, and L group of transmit antennas, first the best transmit antenna group for each user will be selected. Assuming the channel state information (CSI is known at the receiver, the antenna selection process can be made at the receiver side, and then only the index of the selected group of transmit antennas will be fedback to the transmitter at the base station with the SNR information of the corresponding users of the selected antenna group (instead of feedbacking the CSI of L channel matrices. Then the base station selects the best user among all the active users. The process of user selection can be performed by different scheduling techniques. A. Antenna Selection Assume that each user employs n R antennas at the receiver and there are m T available antennas at the transmitter. The m T available transmit antennas is subdivided in L groups each including n T antennas, such that m T = Ln T. Then, we select the best group (b-th group with the highest channel SNR. The antenna selection scheme is to choose a group of transmit antennas such that b = arg max { u,a(t}, ( a A where A = {1, 2,...,L}, u,a (t is the instantaneous SNR, in the time slot t for the a-th group of transmit antennas. This antenna selection also maximizes the throughput. In this scheme, the receiver needs only to feedback the index of the selected group of transmit antennas to the transmitter, instead of full CSI of all groups. This scheme is of interest, since it significantly reduces the amount of the required feedback to the transmitter, that is only log 2 L bits of information. Using the order statistics [9], and assuming that the SNR values corresponding to the transmit antennas are independent non-identically distributed (i.ni.d., we can express the CDF and PDF of the SNR for the best group of transmit antennas as L F ( = F u,a (, ( f ( = a=1 L f u,a ( a=1 L j=1,j a F u,j (, (7 where f u,a ( and F u,a ( are defined in (3 and (. For the i.i.d. SNRs, they can be written as F ( =[F u,a (] L, (8 and f ( =Lf u,a ( [F u,a (] L 1 (9 The F ( and f ( are the obtained CDF and the PDF of SNR in a multiuser MIMO system when antenna has been performed. B. User Selection After selecting the best group of transmit antennas (b-th group, the goal is to select the best user (m-th user among all the active users. In the following, we consider different multiuser scheduling schemes and explain them separately. We first consider the case where the channel SNR values for the users are independent and identically distributed (i.i.d.. Then, we consider the case where the short-term average SNR values of users are independent and non-identically distributed (i.ni.d.. For this case, we present the scheduling schemes including maximum absolute SNR-base scheduling (MASS, proportional fair scheduling (PFS with maximizing the normalized-throughput, and PFS with maximizing the normalized-snr. We then explain each scheme separately. Homogeneous Case In the homogeneous case, the statistics of users are the same, and it is assumed that the users SNR are independent identically distributed (i.i.d.. In this case, the scheduler selects the best user (m-th user of b-th group of transmit antennas, with either the maximum absolute-snr, i.e. 1911

3 { (t}, (10 or the maximum absolute-throughput, i.e. {R (t}, (11 which they are equivalent for OSTBC systems. U = {1, 2,...,K}, (t and R (t are the instantaneous SNR and instantaneous throughput (capacity, respectively for the user u of the b-th transmit antenna group in the time slot t. This means that the user with the best channel SNR will be scheduled for the transmission in that time-slot. Therefore, using the theory of order statistics [9], the CDF and PDF of the best user (with SNR m,b selected from K available users can be obtained from F m,b ( =[F (] K, (12 and f m,b ( =Kf ( [F (] K 1, (13 Considering the fact that the i,b s, i {1, 2,...,K} are independent random variables, it can be written as F m,b ( = Pr ( = F (, (1 and therefore, the PDF of m,b can be obtained by taking the derivative of the corresponding CDF in (1 with respect to as follows f m,b ( = f ( F j,b (, (17 where f ( and F ( are defined in (-(9. This scheduler provides the highest throughput at each timeslot. By employing power control in the multiuser system, all users can have the same average SNR, throughput, and therefore the maximum throughput and fairness among the users can be maintained. However without power control, users suffering from bad channel conditions may starve and will not be given a chance to transmit their information. This results an unfairness among the users for resource allocation. 2 Normalized throughput-based proportional fair scheduling (NT-PFS: From the practical point of view, user fairness is an important issue that should be considered with the scheduling techniques. To guarantee the fairness among the users for resource allocation, a proportional fair scheduling (PFS technique has been proposed to provide a good compromise between the fairness and throughput [1]. PFS technique tries to schedule a user whose ratio of instantaneous throughput R (t to its own average throughput T (t over the past window of length t c, ( R (t T (t is the largest. In the time-slot t, the PFS, selects the user m with the largest value where f ( and F ( are defined in (-(9. of that ratio among all users in the system, i.e. Heterogenous Case { } R (t In the heterogenous case, the statistics of users are not. (18 the same. This is due to the fact that the faded signals T (t of users that arrive from different paths can cause different The value of T (t is then updated as follows: average SNR values for the different users. In this case, users are independent and non-identically distributed (i.ni.d.. (1 Therefore, in the heterogenous case, the scheduling technique 1tc T (t+ 1 t T (t +1= c R (t, u = m can maximize the total throughput and also the throughput of (1 1tc T (t, u m individual users. In the following, we present some possible (19 scheduling schemes for the heterogenous case where the shortterm average SNR values of users are i.ni.d. throughput can be achieved. In general, higher value of t c, By adjusting t c, the desired tradeoff between fairness and 1 Maximum absolute SNR-based scheduling (MASS: A provides larger total throughput and more unfairness among maximum absolute SNR-based scheduler (MASS, always the users and also (18 will be approximately equal to (11. selects a user such that { For the average throughput analysis in (19, we use the (t}, (1 OSTBC channel capacity expression as follows. ( where U = {1, 2,...,K}, (t is the i.ni.d. instantaneous R (t =R c log 2 1+ H (t 2 F. (20 SNR, respectively for the user u in the time slot t. Therefore, using the theory of order statistics [9], the CDF of the best 3 Normalized SNR-based Proportional Fair Scheduling user (with SNR m,b selected from K available users can be (NS-PFS: Considering that users are independent and nonidentically distributed (i.ni.d., an alternative criterion for the obtained from F m,b ( =Pr ( m,b user scheduling with proportional fair scheduling, is defined =Pr ( m,b ; m,b all other (u m as the ratio of the instantaneous SNR of each user to its own =Pr ( average SNR, [2], i.e., m,b ; 1,b ; 2,b ;...; K,b. (1 { (t }, (21 (t where (t and (t are the instantaneous and the shortterm average SNR values, respectively for the user u in the time slot t. This scheduler has been called as a normalized SNR-based scheduler [10]. When the average SNR of all users are the same, (users are i.i.d. this scheduler will be the same as the maximum absolute SNR-based scheduler stated earlier. In this scheduling scheme, the base station selects the users with the largest normalized SNR value. Therefore, similar to the method in [10], the CDF of the best user (with SNR m,b 1912

4 selected from K available users can be obtained from F m,b ( =Pr ( m,b = = Pr ( m,b ; m,b = ( Pr which can be written as F m,b ( = 0 f (x ; max F j,b (xdx (22 (23 where f ( and F j,b ( are defined in (-(9. Therefore, the PDF f m,b ( can be obtained by taking derivative of the CDF F m,b ( in (23 with respect to. Then, it can be written as ( K ( 1 f m,b ( = f F j,b. (2 The F m,b ( and f m,b ( are the CDF and the PDF of SNR of the best user and the best group of transmit antennas in a multiuser MIMO system when both antenna and user selection have been performed. Note that although the SNRs of users, { (t are } non-identically distributed, their normalized SNRs, are identically distributed and thus are i.i.d. (t (t Therefore, all the users will have equal chance for accessing the channel. This clearly states that the normalized SNR-based scheduling is a completely fair scheduling scheme. V. AVERAGE CHANNEL CAPACITY ANALYSIS The channel capacity of the u-th user in OSTBC can be written as ( C u,a = R c log 2 1+ u,a H u,a 2 F. (2 Therefore, for the SNR-based scheduling schemes, the capacity achieved by the best user in the best group of transmit antennas can be expressed as C m,b = R c log 2 (1 + m,b, (2 where m,b is the SNR of the the best user in the best group of transmit antennas. Therefore, the average capacity will be given by C = E {C m,b } = R c log 2 (1 + m,b f m,b ( m,b d m,b. 0 (27 Using the expression of the f m,b ( m,b of each scheduling scheme, given in (13, (17, and (2, we can obtain the corresponding average capacity using numerical integration. VI. SIMULATION AND NUMERICAL RESULTS In this section, we provide the results obtained from the Monte Carlo simulation and from the mathematical expressions for the multiuser MIMO OSTBC system using code G 2 (R c =1, employing two transmit and two receive antennas (n T =2, n R =2 with user scheduling and antenna selection over a MIMO Rayleigh fading channel. The setup in the figures is as follows. At every time slot, the short-term average SNR values, u,a (t, foru {1,...,K} and a {1,...,L}, are considered to be i.i.d. or i.ni.d. random variables. In the i.i.d. case, we assume u,a (t = SNR, (longterm average SNR. The number of users is increased by two each time. In the i.ni.d. case, we generate u,a (t for two users from a uniform distribution between 0 and 1. We then normalize them so that their sum to be equal to 2 SNR. Therefore, for the K users, the total SNR is KSNR. The unsmoothness of the curves is due to the fact that the short-term average SNR values are random variables. However, smoother curves can be obtained by averaging several values of the average capacity. We consider the following cases. We first consider an i.i.d. case for users and antennas. In this case both absolute-snr and normalized-snr based scheduling are the same. The second case is i.ni.d. users with i.i.d. antennas. In this case, absolute-snr and normalized-snr based scheduling schemes operate differently. Finally, we consider a case with i.ni.d. users and i.ni.d. antennas. In all cases, a system with or without transmit antenna selection (TAS is also considered. Fig. 1 shows the average channel capacity of system under study in terms of bits per second per hertz, versus number of users, for SNR= 10dB, for different user scheduling schemes and with transmit antenna selection (TAS with L = 2 or without transmit antenna selection (no-tas, i.e. L =1. It can be observed that, in all cases, increasing the number of users increases the system capacity. As can be seen, non-identically distributed u,a (t values cause a loss in channel capacity for normalized-snr based scheduling scheme. This is clear since normalized-snr based scheduling scheme provides the complete fairness among the users but at the expense of a slight loss in the system capacity. On the other hand, it can be observed that the absolute SNR-based scheduling improves the capacity of the system at the cost of more un-fairness among the users. Furthermore, we can see that antenna selection can improve the system capacity in each case. In the case of i.i.d. antennas, this improvement is almost the same for all user scheduling schemes. We note that in the homogeneous case (i.i.d. users and antennas, SNR-based and throughput-based scheduling schemes have the same average capacity. However, in the case of i.ni.d. users, i.i.d. antennas, the results of normalized-snr and normalized-throughput are different. The results in the later case, depend on the value of t c in (19, which has been chosen as t c = 100. Fig.s 2-3 show the average channel capacity versus average long-term SNR, for K = 10. Again, we observe that the absolute SNR-based scheduling provides a superior system capacity compared to that of i.i.d. SNR scheduling, while the normalized SNR-based scheduling provides an inferior capacity. It is also shown that antenna selection improves the capacity for each scheme. In Fig. 3, we observe that for the case of i.ni.d. antennas the improvement due to the antenna selection is higher than that of i.i.d. antennas case for normalized SNR-based user scheduling and is very low for 1913

5 . SNR = 10 db Absolute SNR, i.ni.d. users, i.ni.d. tx antennas, TAS, L=2 Absolute SNR, i.ni.d. users, i.ni.d. tx antennas, no TAS, L=1 Normalized SNR, i.ni.d. users, i.ni.d. tx antennas, TAS, L=2 Normalized SNR, i.ni.d. users, i.ni.d. tx antennas, no TAS, L=1. Absolute SNR, i.ni.d. users, i.i.d. tx antennas, TAS, L=2 Absolute SNR, i.ni.d. users, i.i.d. tx antennas, no TAS, L=1. Normalized SNR, i.ni.d. users, i.i.d. tx antennas, TAS, L=2 Normalized SNR, i.ni.d. users, i.i.d. tx antennas, no TAS, L=1 Normalized Throughput, i.ni.d. users, i.i.d. tx antennas, no TAS, L= Number of Users 7 3 K = 10 users SNR (db Fig. 1. Average Capacity versus K for SNR = 10 db. Fig. 3. Average Capacity versus SNR for K = Absolute SNR, i.ni.d. users, i.i.d. tx antennas, TAS, L=2 Absolute SNR, i.ni.d. users, i.i.d. tx antennas, no TAS, L=1 Normalized SNR, i.ni.d. users, i.i.d. tx antennas, TAS, L=2 Normalized SNR, i.ni.d. users, i.i.d. tx antennas, no TAS, L=1..2 SNR = 10 db K = 20 users K = 10 users SNR (db Fig. 2. Average Capacity versus SNR for K =10. absolute SNR-based scheduling. This is intuitively clear, since the i.ni.d. antenna selection can be considered as an absolute SNR-based antenna selection, and therefore, it improves the capacity of normalized-based user scheduling more compared to the absolute SNR-based user scheduling. Finally, Fig., shows the average channel capacity versus the number of available group of transmit antennas at the base station (L when K =20and SNR=10 db. It can be seen that increasing the number of available antennas for the antenna selection, improves the capacity of system. VII. CONCLUSION In this paper, we have presented a performance analysis of the user scheduling for the multiuser MIMO systems exploiting the multiuser and antenna diversities while maintaining the fairness among the users. Absolute throughputbased scheduling, normalized throughput-based scheduling, absolute SNR-based scheduling and normalized SNR-based scheduling schemes have been presented. An antenna selection scheme has been used to overcome the drawback of channel hardening in multiuser MIMO systems, and also to improve the system performance. Using mathematical analysis and numerical simulations, the significant advantages of the presented schemes have been shown. It was shown that the normalized SNR-based scheduling guarantees the fairness among the users Fig...2 Absolute SNR, i.ni.d. users, i.i.d. tx antennas i.i.d. users, i.i.d. tx antennas Normalized SNR, i.ni.d. users, i.i.d. tx antennas Number of Group of Transmit Antennas, L Average Capacity versus L for K =20, SNR = 10 db. while the slight performance loss due to the fair scheduling can be compensated with an antenna selection at the transmitter. REFERENCES [1] P. Viswanath, D. Tse, and R. Laroia, Opportunistic beamforming using dumb antennas, IEEE Trans. Inf. Theory, vol. 8, no., pp , June [2] N. Sharma and L. Ozarow, A study of opportunism for multiple-antenna systems, IEEE Trans. Inf. Theory, vol. 1, no., pp , May 200. [3] J. Jiang, R. Buehrer, and W. Tranter, Antenna diversity in multiuser data networks, IEEE Trans. Commun., vol. 2, no. 3, pp , Mar [] W. Ajib and D. Haccoun, An overview of scheduling algorithms in MIMO-based fourth-generation wireless systems, IEEE Network Mag., vol. 19, no., pp. 3 8, September/October 200. [] B. Hochwald, T. Marzetta, and V. Tarokh, Multi-antenna channel hardening and its implication for rate feedback and scheduling, IEEE Trans. Inf. Theory, vol. 0, no. 9, pp , Sept [] D. Bai, P. Mitran, S. Ghassemzadeh, R. Miller, and V. Tarokh, Channel hardening and the scheduling gain of antenna selection diversity schems, in Proc. IEEE Int. Symp. on Inf. Theory, ISIT2007, 2007, pp [7] V. Tarokh, H. Jafarkhani, and A. Calderbank, Space-time block codes from orthogonal designs, IEEE Trans. Inf. Theory, vol., no., pp. 1 17, July [8] H. Shin and J. Lee, Performance analysis of space-time block codes over keyhole Nakagami-m fading channels, IEEE Trans. Veh. Technol., vol. 3, no. 2, pp , March 200. [9] A. Papoulis, Probability, random variables, and stochastic process. McGraw-Hil, [10] L. Yang and M.-S. Alouini, Performance analysis of multiuser selection diversity, IEEE Trans. Veh. Technol., vol., no. 3, pp , May