Multiuser Scheduling for MIMO-OFDM Systems with Continuous-Rate ...
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2008 proceedings.
Multiuser Scheduling for MIMO-OFDM Systems with Continuous-Rate Adaptive Modulation Mohammad Torabi Dept. of Electrical Engineering ´ Ecole Polytechnique de Montr´eal Montr´eal, QC, Canada
Wessam Ajib Dept. of Computer Science Universit´e du Queb´ec a´ Montr´eal Montr´eal, QC, Canada
Abstract—In this paper, we present a multiuser scheduling technique for MIMO-OFDM system over multipath frequencyselective fading channels to exploit the multiuser, space and frequency diversities. A continuous-rate adaptive modulation is employed to increase the spectral efficiency of the system. A proportional fair scheduler is also considered to maintain the fairness among the users while exploiting the multiuser diversity. We also use a scheme to reduce the required feedback channel information. Using mathematical analysis and numerical simulations, the significant advantages of the proposed scheme have been shown. It is also shown that when the number of active users is moderately high and is above 30, even 10% feedback load is sufficient to get the benefits of the proposed scheme.
I. I NTRODUCTION Orthogonal frequency division multiplexing (OFDM) has been considered as a promising technique for the application in the future broadband wireless communications. It is considered in many existing systems and standards such as IEEE802.11 wireless local area networks, IEEE802.16 (WiMax), and a candidate in ultra-wideband (UWB) standard. On the other hand, multiple-input multiple-output (MIMO) technology has been recognized as a key approach for improving the system performance and the channel capacity of wireless communication systems. In particular, MIMO-OFDM system has been considered as an attractive solution for broadband wireless communications [1]. Since OFDM converts the frequencyselective fading channel into several parallel flat-fading subchannels, it allows to implement the MIMO-related algorithms on each subcarrier. OFDM can facilitate the adaptive modulation which can increase both the system performance and the throughput [2]. A significant advantage of MIMO-OFDM systems is that they allow a rate and power allocation (adaptive modulation) and dynamic resource allocation to the system. In the multiuser (MU) transmission scenario, the MUMIMO-OFDM system is capable of exploiting the frequency, space and users diversities. An important issue in these systems is the user scheduling that is the allocation of the OFDM subcarriers among the users. In multiuser MIMOOFDM, users can share the OFDM symbols by employing a subcarrier of an OFDM symbol. Both fixed and dynamic resource allocation can be considered. A fixed resource allocation such as time division multiple access (TDMA) or frequency division multiple access (FDMA), will assign a time-slot or a frequency-slot (OFDM subcarrier) to each user,
David Haccoun Dept. of Electrical Engineering ´ Ecole Polytechnique de Montr´eal Montr´eal, QC, Canada
regardless of the channel condition. Considering the channel state information, a dynamic resource allocation scheme that allocates time or frequency slots to the users adaptively may provide a higher spectral efficiency and a better system performance. To do this, two types of optimal techniques for dynamic resource allocation known as rate adaptive (RA) and margin adaptive (MA) schemes have been considered [3], [4]. Although, the issue of rate-adaptive modulation with multiuser scheduling has been studied in the past, especially for single-input single-output (SISO) systems, most of the work was focus on flat-fading channels [5], [6]. However, for frequency-selective fading channels, adaptive modulation in MU-MIMO-OFDM has attracted research interests and continues to be an open research area. In this case, transmission power and rate (modulation modes) along with user allocation can be adapted for every subcarrier. Most of the related work which has been done for variable-rate and variablepower allocation in these systems introduces a high system complexity, specially with using the well-known water-filling technique [7]–[11]. For the conventional OFDM systems, it is shown that the optimum power adaptation provides a small spectral efficiency gain in the order of 1 dB compared to the constant-power variable-rate system [12]. Therefore, it is recommended to use a constant-power spectrum in order to save computational complexity in the adaptive modulation. In this paper, we present a continuous-rate adaptive modulation for increasing the spectral efficiency of the MUMIMO-OFDM systems with the constraint on maintaining a pre-defined target bit error rate (BER). Our objective is to apply the rate-adaptive modulation strategy to the MU-MIMOOFDM system and to study the resulting improvement in the system performance. We also study the effects of number of users, and number of receive antennas on the system performance. Although the scheme is general, but we consider the popular orthogonal space-frequency block coded OFDM (OSFBC-OFDM) where the orthogonal structure leads to a low-complexity receiver. The contribution of this paper can be divided in three points. First, we derive the expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the signal-to-noise-ratio (SNR) for each OFDM subchannel for user scheduling for a MIMO channel. Second, we express the PDF and the CDF for the SNRbased user scheduling scheme which enable us to establish a mathematical analysis and formulation for the average spectral
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2008 proceedings.
In every time slot, all users can have access to the N subcarriers of OFDM. However, each frequency slot (subcarrier) can be dedicated to only one user (selected user). The multiuser scheduler can select the best user based on the channel quality of the users in each frequency-slot according to full-feedback or limited-feedback channel information. Similar to the expression in [14] for the received signal in STBC, the received signal for OSFBC-OFDM after OSFBC decoding can be written as s[k, n] = c H[k, n]2F s[k, n] + η[k, n],
Fig. 1.
Down-link multiuser diversity with MIMO-OFDM scheduling.
efficiency and the average capacity of the system under study. We consider two scenarios: full-feedback and limited-feedback of the channel information to the transmitter. Finally, we study the continuous-rate adaptive algorithm using spectral efficiency and average capacity analysis. We also consider proportional fair scheduling to maintain the fairness among the users while exploiting the multiuser diversity. Numerical results are also provided showing the significant advantages of the proposed scheme. The rest of this paper is organized as follows. Section II presents the MU-MIMO-OFDM system model and the used assumptions. The PDF and CDF expressions for the user scheduling scheme in the multiuser OSFBC-OFDM system is presented in Section III. In Section IV, the proposed continuous-rate adaptive modulation scheme is presented. Then a proportional fair scheduling (PFS) technique is presented for the system under study. Both average spectral efficiency and average capacity of the proposed MU-OSFBCOFDM system are analytically evaluated. Numerical results are presented in Section V. Finally, Section VI concludes this paper. II. S YSTEM M ODEL
(1)
where s[k, n] is the output signal of OSFBC decoder, s[k, n] squared Frobenius is the transmitted symbol, .2F isthe nR nT 2 norm of a matrix, H[k, n]2F = j=1 i=1 |Hj,i [k, n]| , and η[k, n] is a complex Gaussian noise with distribution 2 0 N (0, cN 2 H[k, n]F ) per dimension and c is a constant that depends on the OSTBC. For example, for the orthogonal space-time block codes given in [13], c = 1 for the rate 1 code G2 and for the rate 3/4 codes H3 , and H4 , and c = 2 for the rate 1/2 codes G3 and G4 . The total energy of the symbol transmitted through the nT antennas can be normalized to nT and, therefore, similar to the expression in [14], we can express the instantaneous SNR per symbol at the receiver of the kth user as γ H[k, n]2F , (2) γ[k, n] = nT Rc where γ is the average receive SNR per antenna, and Rc is the code rate of OSTBC. III. PDF AND CDF E XPRESSIONS In this section, we express the PDF and the CDF for the SNR-based user scheduling scheme for the multiuser OSFBCOFDM system, which enable us to establish a mathematical analysis and formulation for the average spectral efficiency and the average capacity of the system under study. For the feedback channel we consider two scenarios: full-feedback and limited-feedback of the channel information to the transmitter. For both scenarios, we also express the PDF and the CDF for the SNR of the best and scheduled user. A. PDF and CDF for the OSFBC-OFDM
The system model is illustrated in Fig. 1. We consider a multiuser MIMO-OFDM system with an orthogonal spacefrequency block coded-OFDM (OSFBC-OFDM), employing nT transmit antennas at the base station, and K users, each with nR receive antennas. We assume OFDM with N subcarriers so that the MIMO channel between the kth user and the base station on nth subcarrier can be expressed by H[k, n] matrix of size nR ×nT , with elements Hj,i [k, n] corresponding to the discrete Fourier transform (DFT) of the hj,i (k, t); the latter corresponds to the complex channel gain of the channel response between the ith transmit and jth receive antennas. Depending on the employed orthogonal space-time code, the number of transmit antennas nT will be different. For example in the orthogonal space-time codes G2 , G3 , H3 G4 , and H4 , nT will be 2, 3, 3, 4, and 4 respectively [13].
In the MIMO-OFDM systems, the subchannel fading, i.e., |Hj,i [k, n]| can be considered as a Rayleigh flat-fading, there2 fore |Hj,i [k, n]| for each user in each subchannel is a Chisquared distributed random variable. Since H[k, n]2F is 2 the sum of nT nR i.i.d |Hj,i [k, n]| random variables, then H[k, n]2F is Chi-squared distributed random variable with 2nT nR degrees of freedom. We omit the index [k, n] in γ for simplicity. Thus, using a change in variables, we can show that the probability density function (PDF) and cumulative distribution function (CDF) of the received SNR for each subchannel of each user in OSFBC-OFDM; given in (2), can be expressed as
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fγ (γ) =
(nT Rc )nT nR γ nT nR −1 e−nT Rc γ/γ (nT nR − 1)! γ nT nR
(3)
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−nT Rc γ/γ
Fγ (γ) = 1 − e
nT nR −1
(nT Rc γ/γ)l . l!
l=0
(4)
D. Normalized average feedback load The normalized average feedback load F is defined as the ratio of the average load per time-slot over the total number of active users K. It can be expressed in terms of the CDF as [5] F = 1 − Fγ (γth ), (9)
B. Full feedback load Scenario Assuming that full channel information is sent back to the base station, the scheduler selects the best user for the nth subcarrier such that m = arg maxk {γ}. This means that the user with the best channel SNR will be scheduled for the transmission in that frequency-slot. Therefore, assuming that the users’ SNR are i.i.d, using the theory of order statistics [15], the PDF and CDF of the best user (with SNR γm ) selected from K available users can be obtained from
where 0 ≤ F ≤ 1 and F = 1 corresponds to a full feedback load. Therefore, for a given feedback load, the corresponding SNR threshold, γth depends on the MU-MIMO-OFDM system settings such as the number of transmit and receive antennas. This threshold can be obtained from
fγm (γ) = K fγ (γ) [Fγ (γ)]K−1 ,
(5)
where Fγ−1 (.) is the inverse function of the CDF given in (4) and it can be calculated numerically.
Fγm (γ) = [Fγ (γ)]K ,
(6)
IV. A DAPTIVE M ODULATION The general idea of adaptive modulation is to choose a set of suitable modulation parameters based on the feedback information regarding the full or partial channel state information (CSI). The goal of adaptive modulation is to ensure that the most efficient mode is always used over varying channel conditions based on a mode selection criterion to achieve high spectral efficiency. In contrast, non-adaptive systems with fixed-mode modulation are designed for the worst-case channel conditions, resulting in insufficient utilization of the full channel capacity. Both variable-rate and variable-power adaptive modulation are possible. We assume the use of a constant-power spectrum in order to save computational complexity and since the optimum power adaptation provides a small spectral efficiency gain in the order of 1 dB, compared to the constant-power variable-rate system [12]. Throughout this paper, we consider an adaptive modulation with constant-power and adaptive-rate transmission. We consider a continuous-rate adaptive modulation. The term continuousrate means that the number of bits per symbol is not restricted to integer values. In the following, we present the adaptive modulation and provide a mathematical analysis for the proposed system. Although, the analysis are general and applicable for both fullfeedback and limited-feedback scenarios, but the results will be somehow different by considering the associated PDF and CDF for each scenario.
and
where fγ (γ) and Fγ (γ) are defined in (3) and (4), respectively. C. Reduced feedback load Scenario In the full-feedback communications, all the users should send their channel SNR to the base station. Although, the base station only requires a feedback from the user with the best channel quality, but the users are not aware of the channel condition of the other users. Therefore, for the purpose of scheduling and resource allocation, the base station needs a feedback from all the active users. In order to reduce the amount of feedback load, in [5] a scheme has been presented for the multiuser scheduling for the SISO channels. Here we use a similar algorithm and extend it to the MU-MIMO-OFDM system for the MIMO channels. In this scheme, in each frequency-slot only a set of active users whose channel SNR is greater than a pre-defined threshold (γ > γth ) should feedback their channel information to the base station, other users remain silent. If none of the users has a SNR above the threshold (γ ≤ γth ), the scheduler in the base station selects a random user. Assuming that the users’ SNR are i.i.d, we can express the CDF of the selected user’s SNR as Fγm (γ) = Fγ (γ) [Fγ (γth )]K−1 , Fγm (γ) =
γ ≤ γth
K K [Fγ (γth )]K−k k
k=1
k
. [Fγ (γ) − Fγ (γth )] ,
γ > γth
(7)
Therefore, the PDF fγm (γ) can be obtained by taking derivative of the CDF Fγm (γ) in (9) with respect to γ. Then, it can be written as fγm (γ) = fγ (γ) [Fγ (γth )]K−1 , fγm (γ) =
γ ≤ γth
K K k fγ (γ) [Fγ (γth )]K−k k
k=1
. [Fγ (γ) − Fγ (γth )]
k−1
,
γ > γth
(8)
where fγ (.) and Fγ (.) are defined in (3) and (4), respectively.
γth = Fγ−1 (1 − F ),
(10)
A. Continuous-Rate Adaptive Modulation The goal is to find the number of bits/symbol (β[k, n]) to be assigned on each subcarrier, as a function of the target BER. We use an exponential approximation for the BER in an ”invertible” form as a function of the β[k, n] and the SNR. It is shown that the expression for the instantaneous BER of the nth subchannel and kth user of OSFBC-OFDM (square M -QAM with Gray bit mapping on each subcarrier, where M = 2β[k,n] ) over a frequency-selective fading channel can be approximately expressed as [16] 1.6 γ H[k, n]2F . (11) BER[k, n] 0.2 exp − nT Rc 2 β[k,n] − 1
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By inverting (11), the suitable modulation scheme and the corresponding number of bits in OSFBC-OFDM to obtain the target BER, (BERt ) can be calculated from 2 γ H[k, n] 1.6 F β[k, n] = log2 1 + . (12) 0.2 nT Rc ln BER t Using (12), a continuous bit allocation can be performed on each subcarrier and for each user. In the single user scenario, the average modulation throughput (bits/sec/Hz) for the nth OFDM subcarrier will be given by R(n) = E {β[k, n]}, where E {.} denotes the expectation operator. Taking the OSFBC code rate into account, the effective average spectral efficiency (ASE) will be equal to 2 1.6 γ H[k, n]F
, (13) ASE(n) = Rc E log2 1 + n R ln 0.2 T
c
BERt
in terms of bits/sec/Hz. The average spectral efficiency over all the N subcarriers is given by N −1 1 ASE = ASE(n). (14) N n=0 In the multiuser scenario, and assuming that only the best user (mth user) is allowed for the transmission at the nth frequencyslot, the instantaneous spectral efficiency can be obtained from 1.6 γ m
, (15) R[m, n] = Rc log2 1 + 0.2 ln BER t where γm = γ[m, n] = γH[m, n]2F /nT Rc is the SNR of the selected (best) user in the nth subcarrier. Therefore, the average spectral efficiency in this case will be given by ASE(n) = E {R[m, n]} ∞ Rc log2 1 + = 0
ln
1.6 γm
fγm (γm ) dγm , 0.2 BERt
(16)
where the expression of the fγm (γ) is given in (5) for fullfeedback and in (8) for limited-feedback scenario. Then, we can obtain the overall average spectral efficiency using (14) which requires a numerical integration of (16). B. Average channel capacity analysis The channel capacity of the kth user in OSFBC-OFDM for the nth subcarrier can be written as γ 2 (17) H[k, n]F . C[k, n] = Rc log2 1 + nT Rc Therefore, the capacity achieved by the best user can be expressed as C[m, n] = Rc log2 (1 + γm ).
(18)
Therefore, the average capacity will be given by ∞ C(n) = E {C[m, n]} = Rc log2 (1 + γm ) fγm (γm ) dγm . 0
(19) Using the expression of the fγm (γm ), (PDF of the best user) we can simply obtain the corresponding average capacity using numerical integration. By averaging over N subchannels, the overall average channel capacity (AC) can be obtained from AC =
N −1 1 C(n). N n=0
(20)
C. Proportional Fair Scheduling (PFS) From the practical point of view, user fairness and feedback delay are two main issues that should be considered with the scheduling techniques. In the ideal case, when the statistics of users are the same, the scheduling technique maximizes the total throughput and also the throughput of individual users. In reality, the statistics of the users are not the same. The scheduler presented in the previous Section always selects the user with the highest SNR and therefore the highest throughput at each frequency-slot. By employing power control in the multiuser system, all users can have the same average SNR, 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. This gives an unfair resource allocation among the users. To overcome this, a proportional fair scheduling (PFS) technique has been proposed to provide a good compromise between the fairness and throughput [17]. PFS technique tries to schedule (t) a user whose ratio of instantaneous throughput R [k, n] to (t) its own average throughput T [k, n] over the past window (t)
of length tc , ( R(t) [k,n] ) is the largest. In time-slot t, the PFS, T [k,n] selects the user m with the largest value of that ratio among all users in the system, i.e. (t) R [k, n] . (21) m = arg max k T (t) [k, n] (t)
The value of T [k, n] is then updated as follows:
(t) (t) 1 T [k, n] + t1c R [k, n], 1 − (t+1) tc (t) T [k, n] = 1 − 1 T [k, n], tc
k=m
k = m (22) By adjusting tc , the desired tradeoff between fairness and throughput can be achieved. In general, higher value of tc , provides larger total throughput and more unfairness among the users. (t) (t) 1.6 γ H [k,n]2F for We consider R [k, n]=Rc log2 1 + n R ln 0.2 ( BERt ) T c (t) the analysis efficiency and R [k, n] = of average(t)spectral Rc log2 1 + nTγRc H [k, n]2F for channel capacity analysis with PFS.
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V. S IMULATION AND N UMERICAL R ESULTS 14 Formula Simulation
Average Spectral Efficiency (b/s/Hz)
12
10
8
6
K = 1, 8, 32, 512 Users
4
2
0 0
5
10
15
20 SNR (dB)
25
30
35
40
Fig. 2. Average spectral efficiency for continuous-rate adaptive OSFBCOFDM (2Tx,1Rx) multiuser scheduling (BERt = 10−4 ).
4.5
MIMO (2,8)
4
Average Spectral Efficiency (b/s/Hz)
In this section, we provide the results obtained from the Monte Carlo simulation and from the mathematical expressions for the multiuser MIMO-OFDM system with user scheduling over a frequency-selective fading channel. We assume an OFDM with N =64 subcarriers, where the subchannel gains are independent identically distributed for each user. Fig. 2 shows the average spectral efficiency of OSFBCOFDM system in terms of average SNR, using code G2 with two transmit and one receive antennas in a multiuser case. Multiuser scheduler simply assign the resource to a user with highest instantaneous SNR that satisfies a target BER of 10−4 . The user having the highest SNR will be scheduled for the transmission, then based on its SNR, a suitable bit allocation will be made. The first set of curves is provided by calculating the ASE form (13) with selection of the best user with the highest SNR from K active users and then averaging over several realizations of MIMO channel. The second set of curves is obtained from the expression in (16). As can be seen simulation results are matched with the results obtained from the formula. It can be observed that increasing the number of available users improves the average spectral efficiency. Fig. 3 shows the average spectral efficiency in terms of number of users for the average SNR of 10 dB for the same system explained for Fig. 2. We assume different number of receive antennas nR and a target BER of 10−5 . As can be observed by increasing either the number of receive antennas or the number of users, average spectral efficiency will be increased. In order to provide more fairness in the user scheduling, we have also used the PFS technique with the latency scale of tc = 50. As can be observed, providing fairness with PFS decreases slightly the average spectral efficiency. Fig. 4 shows the average channel capacity of the system under study with two transmit antennas and a multiuser scheduling for the average SNR of 10 dB. The results have been obtained from the expression in (8) for the various MIMO cases. It is shown that the average capacity can be increased by increasing either the number of receive antennas or the number of active users. This also shows that the diversities in space and user can be utilized at the same time. The simulation results are also provided for the case when PFS technique (with tc = 50) has been employed. Again, it can be seen that at the price of a slight loss in the transmission capacity, scheduling fairness among the users can be guaranteed. Finally, Fig. 5 and Fig. 6 show the average spectral efficiency and average channel capacity for the same system in terms of the number of users in the reduced-feedback load scenario for the average SNR of 10 dB. Also different system configurations with different feedback loads (F = 1, 0.5, 0.2 and 0.1) have been considered. As can be observed reducing the feedback load by 50% (F = 0.5), the system performance remains almost the same as that of in full-feedback load (F = 1). For a smaller feedback load such as F = 0.1 a performance loss is negligible for a moderate-to-high number of users. This suggests that when the number of users K is higher than 30, a feedback greater than 10% is not necessary.
3.5 MIMO (2,4)
3 MIMO (2,2)
2.5 MISO (2,1)
2
1.5
1 0
Formula Simulation PFS
5
10
15 20 25 Number of Users
30
35
40
Fig. 3. Average spectral efficiency for continuous-rate adaptive OSFBCOFDM (2Tx,nR Rx) multiuser scheduling (SNR = 10 dB and BERt = 10−5 ) with and without PFS.
VI. C ONCLUSION In this paper, we have presented and analyzed a multiuser scheduling technique for MIMO-OFDM system over multipath frequency-selective MIMO fading channels. A continuous-rate adaptive modulation has been employed to increase the spectral efficiency of the system. Two channel feedback scenarios have been considered: full-feedback and limited feedback. For both scenarios, a performance evaluation using mathematical analysis and numerical simulation has been performed to show the significant advantages of the proposed scheme. It was shown that adaptive modulation and user scheduling can increase the average spectral efficiency. It was observed that
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6
7.5 MIMO (2,8)
7
MIMO (2,2)
6.5 MIMO (2,4)
6
Capacity (bits/sec/Hz)
Average Spectral Efficiency (bits/s/Hz)
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MIMO (2,2)
5.5 5
MISO (2,1)
4.5
5
MISO (2,1)
4.5
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4
3.5
F=1 F = 0.5 F = 0.2 F = 0.1
3.5
Formula PFS, Simulation 3 0
3 0
100
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60 40 Number of Users
20
Fig. 4. Average capacity for OSFBC-OFDM (2Tx,nR Rx) with multiuser scheduling (SNR = 10 dB) with and without PFS. 3.5
Average Spectral Efficiency (bits/s/Hz)
MIMO (2,2)
3
MISO (2,1)
2.5
2
1.5
F=1 F = 0.5 F = 0.2 F = 0.1 1 0
10
20
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60 50 40 Number of Users
70
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Fig. 5. Average spectral efficiency for continuous-rate adaptive OSFBCOFDM (2Tx,nR Rx) with multiuser scheduling for limited feedback scenario (SNR = 10 dB and BERt = 10−5 ).
using the proposed user scheduling with reduced-feedback load scenario, reduced up to 90%, the channel capacity and average spectral efficiency of the system under study remain almost the same as that of in full-feedback load scenario when the number of users is greater that 30. This suggests that the feedback load greater than 10% is not necessary when the number of users is high. R EFERENCES [1] H. Yang, “A road to future broadband wireless access: MIMO-OFDMBased air interface,” IEEE Commun. Mag., vol. 43, no. 1, pp. 53–60, Jan. 2005. [2] T. Keller and L. Hanzo, “Adaptive multicarrier modulation: a convenient framework for time-frequency processing in wireless communications,” Proc. IEEE, vol. 88, no. 5, pp. 611–640, May 2000.
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Fig. 6. Average capacity for OSFBC-OFDM (2Tx,nR Rx) with multiuser scheduling for limited feedback scenario (SNR = 10 dB). [3] C. Wong, R. Cheng, K. Lataief, and R. Murch, “Multiuser OFDM with adaptive subcarrier, bit, and power allocation,” IEEE J. Select. Areas Commun., vol. 17, no. 10, pp. 1747–1758, Oct. 1999. [4] W. Rhee and J. Cioffi, “Increase in capacity of multiuser OFDM system using dynamicsubchannel allocation,” in Proc. IEEE Vehicular Technology Conference, VTC00-Spring, 2000, pp. 1085–1089. [5] D. Gesbert and M. Alouini, “How much feedback is multi-user diversity really worth?” in Proc. IEEE International Conference on Communications, ICC04, 2004, pp. 234–238. [6] B. Holter, M.-S. Alouini, and G. Oien, “Multiuser switched diversity transmission,” in Proc. IEEE Vehicular Technology Conference, VTC2004-Fall, 2004, pp. 2038–2043. [7] Y. Zhang and K. Letaief, “Multiuser adaptive subcarrier-and-bit allocation with adaptive cell selection for OFDM systems,” IEEE Trans. Wireless Commun., vol. 3, no. 5, pp. 1566–1575, Sept. 2004. [8] P. Chan and R. Cheng, “Capacity Maximization for Zero-Forcing MIMO-OFDMA Downlink Systems with Multiuser Diversity,” IEEE Trans. Wireless Commun., vol. 6, no. 5, pp. 1880–1889, May 2007. [9] S. Thoen, L. Van der Perre, M. Engels, and H. De Man, “Adaptive loading for OFDM/SDMA-based wireless networks,” IEEE Trans. Commun., vol. 50, no. 11, pp. 1798–1810, Nov. 2002. [10] Y. Zhang and K. Letaief, “An efficient resource-allocation scheme for spatial multiuser access in MIMO/OFDM systems,” IEEE Trans. Commun., vol. 53, no. 1, pp. 107–116, Jan. 2005. [11] J. Jang and K. Lee, “Transmit power adaptation for multiuser OFDM systems,” IEEE J. Select. Areas Commun., vol. 21, no. 2, pp. 171–178, Feb. 2003. [12] A. Czylwik, “Adaptive ofdm for wideband radio channels,” in Proc. IEEE Global Telecommunications Conference, Globecom’96, 1996, pp. 713–718. [13] V. Tarokh, H. Jafarkhani, and A. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45, no. 5, pp. 1456–1467, July 1999. [14] H. Shin and J. Lee, “Performance analysis of space-time block codes over keyhole Nakagami-m fading channels,” IEEE Trans. Veh. Technol., vol. 53, no. 2, pp. 351–362, March 2004. [15] A. Papoulis, Probability, random variables, and stochastic process. McGraw-Hil, 1991. [16] M. Torabi, S. Aissa, and M. Soleymani, “On the BER Performance of Space-Frequency Block Coded OFDM Systems in Fading MIMO Channels,” IEEE Trans. Wireless Commun., vol. 6, no. 4, pp. 1366– 1373, April 2007. [17] P. Viswanath, D. Tse, and R. Laroia, “Opportunistic beamforming using dumb antennas,” IEEE Trans. Inform. Theory, vol. 48, no. 6, pp. 1277– 1294, June 2002.
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Multiuser Scheduling for MIMO-OFDM Systems with Continuous-Rate Adaptive Modulation Mohammad Torabi Dept. of Electrical Engineering ´ Ecole Polytechnique de Montr´eal Montr´eal, QC, Canada
Wessam Ajib Dept. of Computer Science Universit´e du Queb´ec a´ Montr´eal Montr´eal, QC, Canada
Abstract—In this paper, we present a multiuser scheduling technique for MIMO-OFDM system over multipath frequencyselective fading channels to exploit the multiuser, space and frequency diversities. A continuous-rate adaptive modulation is employed to increase the spectral efficiency of the system. A proportional fair scheduler is also considered to maintain the fairness among the users while exploiting the multiuser diversity. We also use a scheme to reduce the required feedback channel information. Using mathematical analysis and numerical simulations, the significant advantages of the proposed scheme have been shown. It is also shown that when the number of active users is moderately high and is above 30, even 10% feedback load is sufficient to get the benefits of the proposed scheme.
I. I NTRODUCTION Orthogonal frequency division multiplexing (OFDM) has been considered as a promising technique for the application in the future broadband wireless communications. It is considered in many existing systems and standards such as IEEE802.11 wireless local area networks, IEEE802.16 (WiMax), and a candidate in ultra-wideband (UWB) standard. On the other hand, multiple-input multiple-output (MIMO) technology has been recognized as a key approach for improving the system performance and the channel capacity of wireless communication systems. In particular, MIMO-OFDM system has been considered as an attractive solution for broadband wireless communications [1]. Since OFDM converts the frequencyselective fading channel into several parallel flat-fading subchannels, it allows to implement the MIMO-related algorithms on each subcarrier. OFDM can facilitate the adaptive modulation which can increase both the system performance and the throughput [2]. A significant advantage of MIMO-OFDM systems is that they allow a rate and power allocation (adaptive modulation) and dynamic resource allocation to the system. In the multiuser (MU) transmission scenario, the MUMIMO-OFDM system is capable of exploiting the frequency, space and users diversities. An important issue in these systems is the user scheduling that is the allocation of the OFDM subcarriers among the users. In multiuser MIMOOFDM, users can share the OFDM symbols by employing a subcarrier of an OFDM symbol. Both fixed and dynamic resource allocation can be considered. A fixed resource allocation such as time division multiple access (TDMA) or frequency division multiple access (FDMA), will assign a time-slot or a frequency-slot (OFDM subcarrier) to each user,
David Haccoun Dept. of Electrical Engineering ´ Ecole Polytechnique de Montr´eal Montr´eal, QC, Canada
regardless of the channel condition. Considering the channel state information, a dynamic resource allocation scheme that allocates time or frequency slots to the users adaptively may provide a higher spectral efficiency and a better system performance. To do this, two types of optimal techniques for dynamic resource allocation known as rate adaptive (RA) and margin adaptive (MA) schemes have been considered [3], [4]. Although, the issue of rate-adaptive modulation with multiuser scheduling has been studied in the past, especially for single-input single-output (SISO) systems, most of the work was focus on flat-fading channels [5], [6]. However, for frequency-selective fading channels, adaptive modulation in MU-MIMO-OFDM has attracted research interests and continues to be an open research area. In this case, transmission power and rate (modulation modes) along with user allocation can be adapted for every subcarrier. Most of the related work which has been done for variable-rate and variablepower allocation in these systems introduces a high system complexity, specially with using the well-known water-filling technique [7]–[11]. For the conventional OFDM systems, it is shown that the optimum power adaptation provides a small spectral efficiency gain in the order of 1 dB compared to the constant-power variable-rate system [12]. Therefore, it is recommended to use a constant-power spectrum in order to save computational complexity in the adaptive modulation. In this paper, we present a continuous-rate adaptive modulation for increasing the spectral efficiency of the MUMIMO-OFDM systems with the constraint on maintaining a pre-defined target bit error rate (BER). Our objective is to apply the rate-adaptive modulation strategy to the MU-MIMOOFDM system and to study the resulting improvement in the system performance. We also study the effects of number of users, and number of receive antennas on the system performance. Although the scheme is general, but we consider the popular orthogonal space-frequency block coded OFDM (OSFBC-OFDM) where the orthogonal structure leads to a low-complexity receiver. The contribution of this paper can be divided in three points. First, we derive the expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the signal-to-noise-ratio (SNR) for each OFDM subchannel for user scheduling for a MIMO channel. Second, we express the PDF and the CDF for the SNRbased user scheduling scheme which enable us to establish a mathematical analysis and formulation for the average spectral
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In every time slot, all users can have access to the N subcarriers of OFDM. However, each frequency slot (subcarrier) can be dedicated to only one user (selected user). The multiuser scheduler can select the best user based on the channel quality of the users in each frequency-slot according to full-feedback or limited-feedback channel information. Similar to the expression in [14] for the received signal in STBC, the received signal for OSFBC-OFDM after OSFBC decoding can be written as s[k, n] = c H[k, n]2F s[k, n] + η[k, n],
Fig. 1.
Down-link multiuser diversity with MIMO-OFDM scheduling.
efficiency and the average capacity of the system under study. We consider two scenarios: full-feedback and limited-feedback of the channel information to the transmitter. Finally, we study the continuous-rate adaptive algorithm using spectral efficiency and average capacity analysis. We also consider proportional fair scheduling to maintain the fairness among the users while exploiting the multiuser diversity. Numerical results are also provided showing the significant advantages of the proposed scheme. The rest of this paper is organized as follows. Section II presents the MU-MIMO-OFDM system model and the used assumptions. The PDF and CDF expressions for the user scheduling scheme in the multiuser OSFBC-OFDM system is presented in Section III. In Section IV, the proposed continuous-rate adaptive modulation scheme is presented. Then a proportional fair scheduling (PFS) technique is presented for the system under study. Both average spectral efficiency and average capacity of the proposed MU-OSFBCOFDM system are analytically evaluated. Numerical results are presented in Section V. Finally, Section VI concludes this paper. II. S YSTEM M ODEL
(1)
where s[k, n] is the output signal of OSFBC decoder, s[k, n] squared Frobenius is the transmitted symbol, .2F isthe nR nT 2 norm of a matrix, H[k, n]2F = j=1 i=1 |Hj,i [k, n]| , and η[k, n] is a complex Gaussian noise with distribution 2 0 N (0, cN 2 H[k, n]F ) per dimension and c is a constant that depends on the OSTBC. For example, for the orthogonal space-time block codes given in [13], c = 1 for the rate 1 code G2 and for the rate 3/4 codes H3 , and H4 , and c = 2 for the rate 1/2 codes G3 and G4 . The total energy of the symbol transmitted through the nT antennas can be normalized to nT and, therefore, similar to the expression in [14], we can express the instantaneous SNR per symbol at the receiver of the kth user as γ H[k, n]2F , (2) γ[k, n] = nT Rc where γ is the average receive SNR per antenna, and Rc is the code rate of OSTBC. III. PDF AND CDF E XPRESSIONS In this section, we express the PDF and the CDF for the SNR-based user scheduling scheme for the multiuser OSFBCOFDM system, which enable us to establish a mathematical analysis and formulation for the average spectral efficiency and the average capacity of the system under study. For the feedback channel we consider two scenarios: full-feedback and limited-feedback of the channel information to the transmitter. For both scenarios, we also express the PDF and the CDF for the SNR of the best and scheduled user. A. PDF and CDF for the OSFBC-OFDM
The system model is illustrated in Fig. 1. We consider a multiuser MIMO-OFDM system with an orthogonal spacefrequency block coded-OFDM (OSFBC-OFDM), employing nT transmit antennas at the base station, and K users, each with nR receive antennas. We assume OFDM with N subcarriers so that the MIMO channel between the kth user and the base station on nth subcarrier can be expressed by H[k, n] matrix of size nR ×nT , with elements Hj,i [k, n] corresponding to the discrete Fourier transform (DFT) of the hj,i (k, t); the latter corresponds to the complex channel gain of the channel response between the ith transmit and jth receive antennas. Depending on the employed orthogonal space-time code, the number of transmit antennas nT will be different. For example in the orthogonal space-time codes G2 , G3 , H3 G4 , and H4 , nT will be 2, 3, 3, 4, and 4 respectively [13].
In the MIMO-OFDM systems, the subchannel fading, i.e., |Hj,i [k, n]| can be considered as a Rayleigh flat-fading, there2 fore |Hj,i [k, n]| for each user in each subchannel is a Chisquared distributed random variable. Since H[k, n]2F is 2 the sum of nT nR i.i.d |Hj,i [k, n]| random variables, then H[k, n]2F is Chi-squared distributed random variable with 2nT nR degrees of freedom. We omit the index [k, n] in γ for simplicity. Thus, using a change in variables, we can show that the probability density function (PDF) and cumulative distribution function (CDF) of the received SNR for each subchannel of each user in OSFBC-OFDM; given in (2), can be expressed as
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fγ (γ) =
(nT Rc )nT nR γ nT nR −1 e−nT Rc γ/γ (nT nR − 1)! γ nT nR
(3)
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2008 proceedings.
−nT Rc γ/γ
Fγ (γ) = 1 − e
nT nR −1
(nT Rc γ/γ)l . l!
l=0
(4)
D. Normalized average feedback load The normalized average feedback load F is defined as the ratio of the average load per time-slot over the total number of active users K. It can be expressed in terms of the CDF as [5] F = 1 − Fγ (γth ), (9)
B. Full feedback load Scenario Assuming that full channel information is sent back to the base station, the scheduler selects the best user for the nth subcarrier such that m = arg maxk {γ}. This means that the user with the best channel SNR will be scheduled for the transmission in that frequency-slot. Therefore, assuming that the users’ SNR are i.i.d, using the theory of order statistics [15], the PDF and CDF of the best user (with SNR γm ) selected from K available users can be obtained from
where 0 ≤ F ≤ 1 and F = 1 corresponds to a full feedback load. Therefore, for a given feedback load, the corresponding SNR threshold, γth depends on the MU-MIMO-OFDM system settings such as the number of transmit and receive antennas. This threshold can be obtained from
fγm (γ) = K fγ (γ) [Fγ (γ)]K−1 ,
(5)
where Fγ−1 (.) is the inverse function of the CDF given in (4) and it can be calculated numerically.
Fγm (γ) = [Fγ (γ)]K ,
(6)
IV. A DAPTIVE M ODULATION The general idea of adaptive modulation is to choose a set of suitable modulation parameters based on the feedback information regarding the full or partial channel state information (CSI). The goal of adaptive modulation is to ensure that the most efficient mode is always used over varying channel conditions based on a mode selection criterion to achieve high spectral efficiency. In contrast, non-adaptive systems with fixed-mode modulation are designed for the worst-case channel conditions, resulting in insufficient utilization of the full channel capacity. Both variable-rate and variable-power adaptive modulation are possible. We assume the use of a constant-power spectrum in order to save computational complexity and since the optimum power adaptation provides a small spectral efficiency gain in the order of 1 dB, compared to the constant-power variable-rate system [12]. Throughout this paper, we consider an adaptive modulation with constant-power and adaptive-rate transmission. We consider a continuous-rate adaptive modulation. The term continuousrate means that the number of bits per symbol is not restricted to integer values. In the following, we present the adaptive modulation and provide a mathematical analysis for the proposed system. Although, the analysis are general and applicable for both fullfeedback and limited-feedback scenarios, but the results will be somehow different by considering the associated PDF and CDF for each scenario.
and
where fγ (γ) and Fγ (γ) are defined in (3) and (4), respectively. C. Reduced feedback load Scenario In the full-feedback communications, all the users should send their channel SNR to the base station. Although, the base station only requires a feedback from the user with the best channel quality, but the users are not aware of the channel condition of the other users. Therefore, for the purpose of scheduling and resource allocation, the base station needs a feedback from all the active users. In order to reduce the amount of feedback load, in [5] a scheme has been presented for the multiuser scheduling for the SISO channels. Here we use a similar algorithm and extend it to the MU-MIMO-OFDM system for the MIMO channels. In this scheme, in each frequency-slot only a set of active users whose channel SNR is greater than a pre-defined threshold (γ > γth ) should feedback their channel information to the base station, other users remain silent. If none of the users has a SNR above the threshold (γ ≤ γth ), the scheduler in the base station selects a random user. Assuming that the users’ SNR are i.i.d, we can express the CDF of the selected user’s SNR as Fγm (γ) = Fγ (γ) [Fγ (γth )]K−1 , Fγm (γ) =
γ ≤ γth
K K [Fγ (γth )]K−k k
k=1
k
. [Fγ (γ) − Fγ (γth )] ,
γ > γth
(7)
Therefore, the PDF fγm (γ) can be obtained by taking derivative of the CDF Fγm (γ) in (9) with respect to γ. Then, it can be written as fγm (γ) = fγ (γ) [Fγ (γth )]K−1 , fγm (γ) =
γ ≤ γth
K K k fγ (γ) [Fγ (γth )]K−k k
k=1
. [Fγ (γ) − Fγ (γth )]
k−1
,
γ > γth
(8)
where fγ (.) and Fγ (.) are defined in (3) and (4), respectively.
γth = Fγ−1 (1 − F ),
(10)
A. Continuous-Rate Adaptive Modulation The goal is to find the number of bits/symbol (β[k, n]) to be assigned on each subcarrier, as a function of the target BER. We use an exponential approximation for the BER in an ”invertible” form as a function of the β[k, n] and the SNR. It is shown that the expression for the instantaneous BER of the nth subchannel and kth user of OSFBC-OFDM (square M -QAM with Gray bit mapping on each subcarrier, where M = 2β[k,n] ) over a frequency-selective fading channel can be approximately expressed as [16] 1.6 γ H[k, n]2F . (11) BER[k, n] 0.2 exp − nT Rc 2 β[k,n] − 1
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By inverting (11), the suitable modulation scheme and the corresponding number of bits in OSFBC-OFDM to obtain the target BER, (BERt ) can be calculated from 2 γ H[k, n] 1.6 F β[k, n] = log2 1 + . (12) 0.2 nT Rc ln BER t Using (12), a continuous bit allocation can be performed on each subcarrier and for each user. In the single user scenario, the average modulation throughput (bits/sec/Hz) for the nth OFDM subcarrier will be given by R(n) = E {β[k, n]}, where E {.} denotes the expectation operator. Taking the OSFBC code rate into account, the effective average spectral efficiency (ASE) will be equal to 2 1.6 γ H[k, n]F
, (13) ASE(n) = Rc E log2 1 + n R ln 0.2 T
c
BERt
in terms of bits/sec/Hz. The average spectral efficiency over all the N subcarriers is given by N −1 1 ASE = ASE(n). (14) N n=0 In the multiuser scenario, and assuming that only the best user (mth user) is allowed for the transmission at the nth frequencyslot, the instantaneous spectral efficiency can be obtained from 1.6 γ m
, (15) R[m, n] = Rc log2 1 + 0.2 ln BER t where γm = γ[m, n] = γH[m, n]2F /nT Rc is the SNR of the selected (best) user in the nth subcarrier. Therefore, the average spectral efficiency in this case will be given by ASE(n) = E {R[m, n]} ∞ Rc log2 1 + = 0
ln
1.6 γm
fγm (γm ) dγm , 0.2 BERt
(16)
where the expression of the fγm (γ) is given in (5) for fullfeedback and in (8) for limited-feedback scenario. Then, we can obtain the overall average spectral efficiency using (14) which requires a numerical integration of (16). B. Average channel capacity analysis The channel capacity of the kth user in OSFBC-OFDM for the nth subcarrier can be written as γ 2 (17) H[k, n]F . C[k, n] = Rc log2 1 + nT Rc Therefore, the capacity achieved by the best user can be expressed as C[m, n] = Rc log2 (1 + γm ).
(18)
Therefore, the average capacity will be given by ∞ C(n) = E {C[m, n]} = Rc log2 (1 + γm ) fγm (γm ) dγm . 0
(19) Using the expression of the fγm (γm ), (PDF of the best user) we can simply obtain the corresponding average capacity using numerical integration. By averaging over N subchannels, the overall average channel capacity (AC) can be obtained from AC =
N −1 1 C(n). N n=0
(20)
C. Proportional Fair Scheduling (PFS) From the practical point of view, user fairness and feedback delay are two main issues that should be considered with the scheduling techniques. In the ideal case, when the statistics of users are the same, the scheduling technique maximizes the total throughput and also the throughput of individual users. In reality, the statistics of the users are not the same. The scheduler presented in the previous Section always selects the user with the highest SNR and therefore the highest throughput at each frequency-slot. By employing power control in the multiuser system, all users can have the same average SNR, 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. This gives an unfair resource allocation among the users. To overcome this, a proportional fair scheduling (PFS) technique has been proposed to provide a good compromise between the fairness and throughput [17]. PFS technique tries to schedule (t) a user whose ratio of instantaneous throughput R [k, n] to (t) its own average throughput T [k, n] over the past window (t)
of length tc , ( R(t) [k,n] ) is the largest. In time-slot t, the PFS, T [k,n] selects the user m with the largest value of that ratio among all users in the system, i.e. (t) R [k, n] . (21) m = arg max k T (t) [k, n] (t)
The value of T [k, n] is then updated as follows:
(t) (t) 1 T [k, n] + t1c R [k, n], 1 − (t+1) tc (t) T [k, n] = 1 − 1 T [k, n], tc
k=m
k = m (22) By adjusting tc , the desired tradeoff between fairness and throughput can be achieved. In general, higher value of tc , provides larger total throughput and more unfairness among the users. (t) (t) 1.6 γ H [k,n]2F for We consider R [k, n]=Rc log2 1 + n R ln 0.2 ( BERt ) T c (t) the analysis efficiency and R [k, n] = of average(t)spectral Rc log2 1 + nTγRc H [k, n]2F for channel capacity analysis with PFS.
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V. S IMULATION AND N UMERICAL R ESULTS 14 Formula Simulation
Average Spectral Efficiency (b/s/Hz)
12
10
8
6
K = 1, 8, 32, 512 Users
4
2
0 0
5
10
15
20 SNR (dB)
25
30
35
40
Fig. 2. Average spectral efficiency for continuous-rate adaptive OSFBCOFDM (2Tx,1Rx) multiuser scheduling (BERt = 10−4 ).
4.5
MIMO (2,8)
4
Average Spectral Efficiency (b/s/Hz)
In this section, we provide the results obtained from the Monte Carlo simulation and from the mathematical expressions for the multiuser MIMO-OFDM system with user scheduling over a frequency-selective fading channel. We assume an OFDM with N =64 subcarriers, where the subchannel gains are independent identically distributed for each user. Fig. 2 shows the average spectral efficiency of OSFBCOFDM system in terms of average SNR, using code G2 with two transmit and one receive antennas in a multiuser case. Multiuser scheduler simply assign the resource to a user with highest instantaneous SNR that satisfies a target BER of 10−4 . The user having the highest SNR will be scheduled for the transmission, then based on its SNR, a suitable bit allocation will be made. The first set of curves is provided by calculating the ASE form (13) with selection of the best user with the highest SNR from K active users and then averaging over several realizations of MIMO channel. The second set of curves is obtained from the expression in (16). As can be seen simulation results are matched with the results obtained from the formula. It can be observed that increasing the number of available users improves the average spectral efficiency. Fig. 3 shows the average spectral efficiency in terms of number of users for the average SNR of 10 dB for the same system explained for Fig. 2. We assume different number of receive antennas nR and a target BER of 10−5 . As can be observed by increasing either the number of receive antennas or the number of users, average spectral efficiency will be increased. In order to provide more fairness in the user scheduling, we have also used the PFS technique with the latency scale of tc = 50. As can be observed, providing fairness with PFS decreases slightly the average spectral efficiency. Fig. 4 shows the average channel capacity of the system under study with two transmit antennas and a multiuser scheduling for the average SNR of 10 dB. The results have been obtained from the expression in (8) for the various MIMO cases. It is shown that the average capacity can be increased by increasing either the number of receive antennas or the number of active users. This also shows that the diversities in space and user can be utilized at the same time. The simulation results are also provided for the case when PFS technique (with tc = 50) has been employed. Again, it can be seen that at the price of a slight loss in the transmission capacity, scheduling fairness among the users can be guaranteed. Finally, Fig. 5 and Fig. 6 show the average spectral efficiency and average channel capacity for the same system in terms of the number of users in the reduced-feedback load scenario for the average SNR of 10 dB. Also different system configurations with different feedback loads (F = 1, 0.5, 0.2 and 0.1) have been considered. As can be observed reducing the feedback load by 50% (F = 0.5), the system performance remains almost the same as that of in full-feedback load (F = 1). For a smaller feedback load such as F = 0.1 a performance loss is negligible for a moderate-to-high number of users. This suggests that when the number of users K is higher than 30, a feedback greater than 10% is not necessary.
3.5 MIMO (2,4)
3 MIMO (2,2)
2.5 MISO (2,1)
2
1.5
1 0
Formula Simulation PFS
5
10
15 20 25 Number of Users
30
35
40
Fig. 3. Average spectral efficiency for continuous-rate adaptive OSFBCOFDM (2Tx,nR Rx) multiuser scheduling (SNR = 10 dB and BERt = 10−5 ) with and without PFS.
VI. C ONCLUSION In this paper, we have presented and analyzed a multiuser scheduling technique for MIMO-OFDM system over multipath frequency-selective MIMO fading channels. A continuous-rate adaptive modulation has been employed to increase the spectral efficiency of the system. Two channel feedback scenarios have been considered: full-feedback and limited feedback. For both scenarios, a performance evaluation using mathematical analysis and numerical simulation has been performed to show the significant advantages of the proposed scheme. It was shown that adaptive modulation and user scheduling can increase the average spectral efficiency. It was observed that
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6
7.5 MIMO (2,8)
7
MIMO (2,2)
6.5 MIMO (2,4)
6
Capacity (bits/sec/Hz)
Average Spectral Efficiency (bits/s/Hz)
5.5
MIMO (2,2)
5.5 5
MISO (2,1)
4.5
5
MISO (2,1)
4.5
4
4
3.5
F=1 F = 0.5 F = 0.2 F = 0.1
3.5
Formula PFS, Simulation 3 0
3 0
100
80
60 40 Number of Users
20
Fig. 4. Average capacity for OSFBC-OFDM (2Tx,nR Rx) with multiuser scheduling (SNR = 10 dB) with and without PFS. 3.5
Average Spectral Efficiency (bits/s/Hz)
MIMO (2,2)
3
MISO (2,1)
2.5
2
1.5
F=1 F = 0.5 F = 0.2 F = 0.1 1 0
10
20
30
60 50 40 Number of Users
70
80
90
100
Fig. 5. Average spectral efficiency for continuous-rate adaptive OSFBCOFDM (2Tx,nR Rx) with multiuser scheduling for limited feedback scenario (SNR = 10 dB and BERt = 10−5 ).
using the proposed user scheduling with reduced-feedback load scenario, reduced up to 90%, the channel capacity and average spectral efficiency of the system under study remain almost the same as that of in full-feedback load scenario when the number of users is greater that 30. This suggests that the feedback load greater than 10% is not necessary when the number of users is high. R EFERENCES [1] H. Yang, “A road to future broadband wireless access: MIMO-OFDMBased air interface,” IEEE Commun. Mag., vol. 43, no. 1, pp. 53–60, Jan. 2005. [2] T. Keller and L. Hanzo, “Adaptive multicarrier modulation: a convenient framework for time-frequency processing in wireless communications,” Proc. IEEE, vol. 88, no. 5, pp. 611–640, May 2000.
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