Effects of Antenna Correlation on Spatial Diversity and Multiuser ...
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2008 proceedings.
Effects of Antenna Correlation on Spatial Diversity and Multiuser Diversity Haelyong Kim, Wan Choi, and Hyuncheol Park School of Engineering Information and Communications University (ICU) 119 Munjiro, Yuseong-gu, Daejeon, 305-732, KOREA E-mail: {seamirr, wchoi, hpark}@icu.ac.kr Abstract— This paper investigates the effects of antenna correlations on spatial diversity and multiuser diversity. Using an upper bound on achievable capacity from order statistic theory, we quantify the interactions between spatial correlation, spatial diversity and multiuser diversity. Our theoretical analysis and simulation results demonstrate a positive influence of spatial correlations on achievable capacity of spatial diversity techniques combined with multiuser diversity, while it has been known that spatial diversity limits a multiuser diversity gain without spatial correlation.
I. I NTRODUCTION Along with spatial diversity techniques using multiple antennas, multiuser diversity has been considered as a key technology to achieve high spectral efficiency in wireless communications. Multiple antennas can be used for enhancing channel fluctuations and hence amplifying multiuser diversity gains [1], [2]. However, several studies have reported that there is a fundamental conflict between spatial and multiuser diversity – the channel hardening effect by spatial diversity limits the multiuser diversity gain so single-input single-output (SISO) even outperforms some spatial diversity techniques if multiuser opportunistic scheduling is employed [3], [4], [5], [6]. Spatial correlations impose negative impacts on multipleinput multiple-output (MIMO) communications [7], [8]. Specifically, channel fluctuations by fading are not effectively removed by spatial diversity techniques in correlated channels. Interestingly, these channel fluctuations, however, enhance multiuser diversity gains. From this viewpoint, it is necessary to understand the interaction among channel correlation, spatial diversity, and multiuser diversity. A referential study on this interaction can be found in [9] where it was shown that capacity of particular beamforming schemes combined with multiuser opportunistic scheduling rather increases with spatial correlation. Even though [9] provides a key clue to explain the interaction between spatial correlation, spatial diversity, and multiuser diversity, it fails to draw a generalized conclusion on the interaction. Also the analytical results of [9] are not effective for a small number of users since they rely on asymptotic approaches. This paper analyzes and quantifies the relations between spatial correlation, spatial diversity, and multiuser diversity through order statistic theory. We use an upper bound on
the achievable capacity to intuitively explain the interaction. Contrary to the asymptotic analysis, the upper bound from order statistic theory is rather tighter for smaller number of users so it quantifies more accurately the interaction. Our analysis indicates that the lost diversity gain by spatial correlation is fully compensated by the multiuser diversity gain. As a result, capacities of diversity techniques increase with correlation if multiuser diversity is combined, whereas they have been known to decrease with correlation in a point-to-point communication. Furthermore, SISO does not outperform diversity techniques in correlated channels while it does without correlation. II. S YSTEM MODEL We focus on downlink communications with K users where the base station is equipped with M transmit antennas and each user has a single receive antenna. The received signal of user k is expressed as yk = hH k x + nk , k = 1, 2, · · · , K,
(1)
2
where x is a transmit vector with E[x ] = P and · is the norm of a vector, hk = [hk,1 , hk,2 , · · · , hk,M ]T is a channel gain vector of user k, which is modeled by a circularly symmetric complex Gaussian random vector whose elements follow CN (0, 1), and nk is an additive white Gaussian noise following CN (0, 1). At the receiver, P corresponds to the average SNR without channel effects. The correlation matrix for transmit antennas is given as
1 ρ .. .
RT = ρM −2 ρM −1
ρ 1 .. .
M −3
ρ ρM −2
· · · ρM −2 · · · ρM −3 .. .. . . ··· 1 ··· ρ
ρM −1 ρM −2 .. , . ρ 1
(2)
where 0 ≤ ρ < 1. The correlation matrix RT is decomposed into RT = UΛUH where (·)H is a conjugate transpose operation, Λ = diag[λ1 , λ2 , · · · , λM ], and λi is an eigenvalue of RT . U is an unitary matrix consisting of corresponding eigenvectors. Then the correlated channel vector of user k is given by 1/2 (3) hk = RT wk ,
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1/2
where RT = UΛ1/2 UH , and wk = [wk,1 , wk,2 , · · · , T wk,M ] is a circularly symmetric complex Gaussian random vector consisting of independent and identically distributed (i.i.d.) elements. The vectors {wk } are independent for index k, and hence channel gain vectors {hk } are assumed to be independent on each other. We also assume that each user can perfectly estimate its channel gain vector and feeds the required information back to the transmitter. The transmitter adopts a max-scheduler where the user with the highest achievable capacity is served at each time instant.
where tr(·) is a trace operation, and tr(R2T ) increases with correlation. In case of two antennas, the variance becomes 2 Var[γk ] = P2 (1 + ρ2 ) since Λ = diag[1 − ρ, 1 + ρ] and hk 2 = (1 − ρ)|vk,1 |2 + (1 + ρ)|vk,2 |2 . It should be noted that diversity effects reduce the variance of the received SNR but correlations contribute to an increase of the variance.
III. E FFECTS OF S PATIAL C ORRELATION ON S PATIAL D IVERSITY
CM RT = log2 1 + Phk 2 .
C. Maximum ratio combining at transmitter (MRT) MRT is a diversity technique when perfect CSIT is available. Its achievable capacity is given by
It has been known that spatial correlations impose negative effects on spatial diversity [7], [8]. Spatial diversity techniques effectively reduce channel fluctuations while spatial correlations limit diversity effects. The impacts of correlation on diversity techniques are well captured by the first and the second moments of the received SNR without multiuser opportunistic scheduling.
where γk = Phk 2 corresponds to the received SNR, and the norm of a channel vector is the same as that of TD. Correspondingly, the mean and variance of MRT are obtained by E[γk ] = P
A. Single-input single-output (SISO) Achievable capacity of SISO is given by CSISO = log2 1 + P|hk,1 |2 .
Var[γk ] = P =P
(5)
(7) 2
and vk = [vk,1 , vk,2 , · · · , vk,M ]T = UH wk . The statistical distribution of vk is the same as wk since the multiplication of an unitary matrix does not change the distribution of a Gaussian random vector [10]. Correspondingly, the mean and variance of TD are given by M P P tr (RT ) = P and λm = M m=1 M
M P2 2 P2 λm = 2 tr R2T Var[γk ] = 2 M m=1 M
M −1 P2 2m = 2 M+ , 2(M − m)ρ M m=1
.
(12)
Phase steering or coherent beamforming is a diversity technique making the received signals from transmit antennas cophased [12]. PS achieves multiple antenna diversity gains with much less feedback information than MRT since the required feedback information is just the channel phases of each user. Achievable capacity of PS is given by
= λ1 |vk,1 | + λ2 |vk,2 | + · · · + λM |vk,M | ,
E[γk ] =
2(M − m)ρ
D. Phase steering (PS)
corresponds to the received SNR and 2
2m
The statistical properties of MRT are the same as TD except that MRT attains an array gain – the factor M in the mean reflects the array gain. As in TD, the variance of MRT also increases with correlation.
TD is a diversity technique when channel state information at the transmitter (CSIT) is not available. Achievable capacity is given by
P 2 hk . (6) CT D = log2 1 + M
2
M+
M −1 m=1
B. Transmit diversity (TD)
hk 2 = wkH UΛUH wk = vkH Λvk
2
(11)
λ2m = P 2 · tr R2T
The received SNR of SISO corresponds to γk = P|hk,1 | , which is an exponential random variable with
P 2 M hk
λm = P · tr (RT ) = M P and
m=1 M 2 m=1
2
where γk =
M
(4)
E[γk ] = P and Var[γk ] = P 2 .
(10)
CP S
(8)
M
2 P = log2 1 + |hk,m | M m=1
M P ≈ 2 log2 |hk,m | . M m=1
(13)
M Hence, γk = P/M m=1 |hk,m | plays the equivalent role of the received SNR and its mean value is (9)
E[γk ] =
66
M P E [|hk,m |] = M m=1
√
πM P , 2
(14)
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2008 proceedings.
where |hk,m | is a Rayleigh distributed random variable with √ the mean value of π/2. The variance is obtained by M P Var |hk,m | (15) Var[γk ] = M m=1 =
scheduling maxk γk . However, it is still difficult to obtain an exact closed form of the mean value after scheduling. Since {γk } are independent, we use an upper bound from order statistics as in [6], [11]: K −1 Var[γk ] (19) E max γk ≤ E[γk ] + √ k=1,2,··· ,K 2K − 1
M M M P P Var [|hk,m |] + Cov [|hk,m |, |hk,n |] , M m=1 M m=1 n=1,
This bound intuitively quantifies the effects of correlation on spatial and multiuser diversity – the mean value after scheduling, E[maxk γk ], increases with not only the number of users but also the variance of received SNR of each user, Var[γk ], given in terms of spatial correlation. Since Var[γk ] increases with correlation, E[maxk γk ] also increase with correlation. That is, correlation rather impose positive effects on spatial diversity if opportunistic multiuser scheduling is adopted, while correlation has been known to impose negative impacts on spatial diversity in a point-to-point communication. Table II summarizes the mean values of diversity techniques after scheduling when K = 20 and P = 10dB. It is verified that a larger variance by higher correlation positively affects on E[maxk γk ]. By comparing Fig. 1 and Fig. 2, we can also find that SISO outperforms TD in uncorrelated channels as reported in [3], [4], [5], [6], but they achieve almost the same capacity in highly correlated channels.
n=m
where Var[|hk,m |] = 1 − π/4, and Cov[R1 , R2 ] = E[R1 R2 ] − E[R1 ]E[R2 ] for random variables R1 = |hk,m | and R2 = |hk,n |. Since R1 and R2 are Rayleigh distributed random variables, the mean value of R = R1 R2 is expressed as [13]
3 3 3 2 2 2 2|m−n| E[R1 R2 ] = Γ (1 − ρ ) Φ , ; 1; ρ , (16) 2 2 2 l ∞ l (b)l d is a Gaussian hyperwhere Φ(a, b; c; d) = 1 + l=1 (a)(c) l l! geometric function, and (t)l = t × (t + 1) × · · · × (t + l − 1) for a real value t. Consequently, (15) is rewritten as
π P M π P− (17) Var[γk ] = 1 − 4 2M 2
M M 2 3 3 P 2 3 Γ 1 − ρ2 , ; 1; ρ2|m−n| . Φ + M 2 2 2 m=1 n=1
V. S IMULATION RESULTS
n=m
This section presents theoretical and simulation results. We assume that the average SNR P is 10dB, the number of user K is 20 for multiuser diversity, and the number of transmit antennas M is 2. The performance of SISO is not affected by spatial correlation since there is a single antenna but is provided as a reference. Fig. 3 shows the negative impacts of antenna correlations on spatial diversity in point-to-point communications. All the diversity techniques degrade as correlation increases as previously known. On the other hand, Fig. 4 shows a positive effects of correlation on spectral efficiency when multiuser opportunistic scheduling is employed. In this figure, spectral efficiencies of diversity techniques increase with correlation since correlation limits diversity effects and hence enhances multiuser diversity gains. The enhanced multiuser diversity gains from spatial correlation perfectly compensate for performance degradation by correlation, and eventually yield an increase of spectral efficiencies. Fig. 5 and Fig. 6 show spectral efficiencies of diversity techniques according to SNR when ρ = 0 and ρ = 0.9, respectively. In Fig. 5, SISO outperforms TD in uncorrelated channels when multiuser opportunistic scheduling is employed as reported in [3], [4], [5], [6]. On the other hand, SISO and TD achieve almost the same spectral efficiency in highly correlated channels as shown in Fig. 6 since spatial correlation amplifies a multiuser diversity gain. Another interesting observation from these figures is that PS achieves similar spectral efficiency to MRT in highly correlated channels although the required feedback information of PS is much less than that of MRT.
where it should be noted that the third term of (17) is an increasing function with respect to a correlation value ρ, which is proven in Appendix. Therefore, the variance of PS increases with correlation and correspondingly its spatial diversity gain decreases. The mean and variance of each diversity technique without multiuser scheduling when M = 2 and P = 10dB are summarized in Table I. Even though the mean and variance of SISO are not affected by correlation, they are provided as references. Table I verifies the fact that variances of SNR increase with correlation. IV. E FFECTS OF S PATIAL C ORRELATION ON M ULTIUSER D IVERSITY If we adopt opportunistic multiuser scheduling to exploit a multiuser diversity gain, the achievable capacity of each diversity technique is given by Cmax = E max log2 (1 + γk ) k
≤ log2 1 + E max γk . (18) k
where the upper bound comes from Jensen’s inequality. The bound becomes very tight in the high SNR region where SNR after scheduling typically is. Note that we apply a maxscheduler which selects the user with the highest achievable capacity to clarify the effects of correlation on spatial and multiuser diversity. Owing to Jensen’s inequality, the mean value after scheduling well characterizes the achievable capacity without the knowledge of exact distribution of SNR after
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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.
VI. C ONCLUSION
[7] C. N. Chuah, D. N. C. Tse, J. M. Kahn, and R. A. Valenzuela, “Capacity scaling in MIMO wireless systems under correlated fading,” IEEE Transactions on Information Theory, vol. 48, no. 3, pp. 637-650, Mar. 2002. [8] D. S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fading correlation and its effect on the capacity of multielement antenna systems,” IEEE Transactions on Communications, vol. 48, no. 3, Mar. 2000. [9] D. Park and S. Park, “Effect of transmit antenna correlation on multiuser diversity,” IEEE International Symposium on Information Theory (ISIT), Sep. 2005. [10] E. Teletar, “Capacity of multi-antenna Gaussian channels,” European Transactions on Telecommunications, vol. 10, no. 6, pp. 585-595, Nov./Dec. 1999. [11] W. Choi, N. Himayat, S. Talwar, and M. Ho, “The effects of cochannel interference on spatial diversity techniques,” IEEE Wireless Communications and Networking Conference (WCNC), Mar. 2007. [12] W. Choi and J. G. Andrews, “Downlink performance and capacity of distributed antenna systems in a multicell environment,” IEEE Transactions on Wireless Communications, vol. 6, no. 1, pp. 69-73, Jan. 2007. [13] M. K. Simon, Probability distributions involving Gaussian random variables: A handbook for engineers and scientists, 1st ed., Kluwer Academic Publishers, 2002.
This paper has studied interaction between spatial correaltion, spatial diversity, and multiuser diversity. Using an upper bound from order statistic theory, we have intuitively quantified the effects of correlations on spatial and multiuser diversity. The analytical and simulation results have shown that spatial correlation imposes positive impacts on spatial diversity techniques if opportunistic multiuser scheduling is employed although spatial correlations have been considered drawbacks in point-to-point communications. A PPENDIX We define f (ρ) and g(ρ) in the third term of (17), respectively, as
2 3 1 − ρ2 (20) f (ρ) = Γ 2
M M 3 3 2|m−n| , ; 1; ρ , (21) Φ g(ρ) = 2 2 m=1
1
n=1,n=m
0.9
where f (ρ) is a decreasing function (but f (ρ) ≥ 0 for 0 ≤ ρ < 1) and function with respect to ρ. g(ρ) is an increasing Since Φ 32 , 32 ; 1; ρ2|m−n| is bounded by
3 3 3 2|m−n| 2|m−n| Φ , ; 1; ρ , 1; 1; ρ ≥Φ (22) 2 2 2 ∞ 3 2|m−n|l 2 lρ (23) =1+ l! l=1 ∞ 3 l 2|m−n|l ρ 2 (24) ≥1+ l! l=1
3 2|m−n| = exp ρ , (25) 2
0.8 0.7 0.6 CDF
SISO MRT
0.5 0.4
PS
TD
0.3 0.2 K=1
0.1 0 −20
−15
Fig. 1.
g(ρ) is exponentially increasing whereas f (ρ) is a decreasing polynomial. Therefore, f (ρ)g(ρ) is an increasing function with respect to ρ since the multiplication of a decreasing polynomial f (ρ) and an exponentially increasing function g(ρ) is increasing with ρ.
−10
−5
K=20
0 5 10 Received SNR [dB]
15
20
25
30
CDFs for uncorrelated channel (ρ = 0, P = 10dB)
1 0.9 0.8
R EFERENCES
0.7
[1] P. Viswanath, D. N. C. Tse, and R. Laroia, “Opportunistic beamforming using dumb antennas,” IEEE Transactions on Information Theory, vol. 48, no. 6, pp. 1277-1294, June 2002. [2] M. Sharif and B. Hassibi, “On the capacity of MIMO broadcast channel with partial side information,” IEEE Transactions on Information Theory, vol. 51, no. 2, pp. 506-522, Feb. 2005. [3] R. Gozali, R. M. Buehrer, and B. D. Woerner, “The impact of multiuser diversity on space-time block coding,” IEEE Communications Letters, vol. 7, no. 5, pp. 213-215, May 2003. [4] E. G. Larsson, “On the combination of spatial diversity and multiuser diversity,” IEEE Communications Letters, vol. 8, no. 8, pp. 517-519, Aug. 2004. [5] B. Hochwald, T. Marzetta, and V. Tarokh, “Multiple-antenna channel hardening and its implications for rate feedback and scheduling,” IEEE Transactions on Information Theory, vol. 50, no. 9, Sep. 2004. [6] W. Choi, et al., “Interactions between multiuser diversity and spatial diversity techniques in an interference-limited environment,” IEEE Wireless Communications and Networking Conference (WCNC), Mar. 2007.
CDF
0.6 K=20 0.5
K=1
0.4 PS, MRT
SISO 0.3 TD
0.2 0.1 0 −20
−15
Fig. 2.
68
−10
−5
0
5 10 15 Received SNR [dB]
20
25
30
CDFs for correlated channel (ρ = 0, P = 10dB)
35
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2008 proceedings.
TABLE I M EAN AND VARIANCE BEFORE SCHEDULING (P = 10dB) ρ=0 E[γ] in dB Var[γ] in dB 10.00 20.01 10.01 16.98 12.53 22.23 13.02 23.00
SISO TD PS MRT
ρ = 0.9 E[γ] in dB Var[γ] in dB 10.00 20.01 10.00 19.60 12.92 25.59 13.01 25.62
TABLE II M EAN AND VARIANCE AFTER SCHEDULING (P = 10dB, K=20) ρ=0 E[maxk γk ] in dB Var[maxk γk ] in dB Simulation Upper bound 22.04 15.56 16.07 17.41 14.35 14.99 22.91 16.95 18.51 23.43 17.36 18.00
SISO TD PS MRT
ρ = 0.9 E[maxk γk ] in dB Var[maxk γk ] in dB Simulation Upper bound 22.04 15.56 16.07 21.58 15.40 15.98 27.60 18.38 19.94 27.60 18.41 18.92
5
6 SISO TD PS MRT
4.5
SISO TD PS MRT
5
Throughput[bps/Hz]
Throughput[bps/Hz]
K=20 4
3.5
3
4
3
2 K=1
2.5
2
1
0
Fig. 3.
0.1
0.2
0.3
0.4
0.5 0.6 Correlation[ρ]
0.7
0.8
0.9
0
1
Throughput according to the correlation before scheduling
2
3
4
5 SNR[dB]
6
7
8
9
10
Throughput according to the average SNR (ρ = 0)
7
SISO TD PS MRT
SISO TD PS MRT
6
5
6 Throughput[bps/Hz]
Throughput[bps/Hz]
1
Fig. 5.
7
6.5
0
5.5
5
K=20 4
3
2 K=1
4.5
4
1
0
0
Fig. 4.
0.2
0.4 0.6 Correlation[ρ]
0.8
1
0
Fig. 6.
Throughput according to the correlation after scheduling
69
1
2
3
4
5 SNR[dB]
6
7
8
9
10
Throughput according to the average SNR (ρ = 0.9)
Effects of Antenna Correlation on Spatial Diversity and Multiuser Diversity Haelyong Kim, Wan Choi, and Hyuncheol Park School of Engineering Information and Communications University (ICU) 119 Munjiro, Yuseong-gu, Daejeon, 305-732, KOREA E-mail: {seamirr, wchoi, hpark}@icu.ac.kr Abstract— This paper investigates the effects of antenna correlations on spatial diversity and multiuser diversity. Using an upper bound on achievable capacity from order statistic theory, we quantify the interactions between spatial correlation, spatial diversity and multiuser diversity. Our theoretical analysis and simulation results demonstrate a positive influence of spatial correlations on achievable capacity of spatial diversity techniques combined with multiuser diversity, while it has been known that spatial diversity limits a multiuser diversity gain without spatial correlation.
I. I NTRODUCTION Along with spatial diversity techniques using multiple antennas, multiuser diversity has been considered as a key technology to achieve high spectral efficiency in wireless communications. Multiple antennas can be used for enhancing channel fluctuations and hence amplifying multiuser diversity gains [1], [2]. However, several studies have reported that there is a fundamental conflict between spatial and multiuser diversity – the channel hardening effect by spatial diversity limits the multiuser diversity gain so single-input single-output (SISO) even outperforms some spatial diversity techniques if multiuser opportunistic scheduling is employed [3], [4], [5], [6]. Spatial correlations impose negative impacts on multipleinput multiple-output (MIMO) communications [7], [8]. Specifically, channel fluctuations by fading are not effectively removed by spatial diversity techniques in correlated channels. Interestingly, these channel fluctuations, however, enhance multiuser diversity gains. From this viewpoint, it is necessary to understand the interaction among channel correlation, spatial diversity, and multiuser diversity. A referential study on this interaction can be found in [9] where it was shown that capacity of particular beamforming schemes combined with multiuser opportunistic scheduling rather increases with spatial correlation. Even though [9] provides a key clue to explain the interaction between spatial correlation, spatial diversity, and multiuser diversity, it fails to draw a generalized conclusion on the interaction. Also the analytical results of [9] are not effective for a small number of users since they rely on asymptotic approaches. This paper analyzes and quantifies the relations between spatial correlation, spatial diversity, and multiuser diversity through order statistic theory. We use an upper bound on
the achievable capacity to intuitively explain the interaction. Contrary to the asymptotic analysis, the upper bound from order statistic theory is rather tighter for smaller number of users so it quantifies more accurately the interaction. Our analysis indicates that the lost diversity gain by spatial correlation is fully compensated by the multiuser diversity gain. As a result, capacities of diversity techniques increase with correlation if multiuser diversity is combined, whereas they have been known to decrease with correlation in a point-to-point communication. Furthermore, SISO does not outperform diversity techniques in correlated channels while it does without correlation. II. S YSTEM MODEL We focus on downlink communications with K users where the base station is equipped with M transmit antennas and each user has a single receive antenna. The received signal of user k is expressed as yk = hH k x + nk , k = 1, 2, · · · , K,
(1)
2
where x is a transmit vector with E[x ] = P and · is the norm of a vector, hk = [hk,1 , hk,2 , · · · , hk,M ]T is a channel gain vector of user k, which is modeled by a circularly symmetric complex Gaussian random vector whose elements follow CN (0, 1), and nk is an additive white Gaussian noise following CN (0, 1). At the receiver, P corresponds to the average SNR without channel effects. The correlation matrix for transmit antennas is given as
1 ρ .. .
RT = ρM −2 ρM −1
ρ 1 .. .
M −3
ρ ρM −2
· · · ρM −2 · · · ρM −3 .. .. . . ··· 1 ··· ρ
ρM −1 ρM −2 .. , . ρ 1
(2)
where 0 ≤ ρ < 1. The correlation matrix RT is decomposed into RT = UΛUH where (·)H is a conjugate transpose operation, Λ = diag[λ1 , λ2 , · · · , λM ], and λi is an eigenvalue of RT . U is an unitary matrix consisting of corresponding eigenvectors. Then the correlated channel vector of user k is given by 1/2 (3) hk = RT wk ,
65 1525-3511/08/$25.00 ©2008 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2008 proceedings.
1/2
where RT = UΛ1/2 UH , and wk = [wk,1 , wk,2 , · · · , T wk,M ] is a circularly symmetric complex Gaussian random vector consisting of independent and identically distributed (i.i.d.) elements. The vectors {wk } are independent for index k, and hence channel gain vectors {hk } are assumed to be independent on each other. We also assume that each user can perfectly estimate its channel gain vector and feeds the required information back to the transmitter. The transmitter adopts a max-scheduler where the user with the highest achievable capacity is served at each time instant.
where tr(·) is a trace operation, and tr(R2T ) increases with correlation. In case of two antennas, the variance becomes 2 Var[γk ] = P2 (1 + ρ2 ) since Λ = diag[1 − ρ, 1 + ρ] and hk 2 = (1 − ρ)|vk,1 |2 + (1 + ρ)|vk,2 |2 . It should be noted that diversity effects reduce the variance of the received SNR but correlations contribute to an increase of the variance.
III. E FFECTS OF S PATIAL C ORRELATION ON S PATIAL D IVERSITY
CM RT = log2 1 + Phk 2 .
C. Maximum ratio combining at transmitter (MRT) MRT is a diversity technique when perfect CSIT is available. Its achievable capacity is given by
It has been known that spatial correlations impose negative effects on spatial diversity [7], [8]. Spatial diversity techniques effectively reduce channel fluctuations while spatial correlations limit diversity effects. The impacts of correlation on diversity techniques are well captured by the first and the second moments of the received SNR without multiuser opportunistic scheduling.
where γk = Phk 2 corresponds to the received SNR, and the norm of a channel vector is the same as that of TD. Correspondingly, the mean and variance of MRT are obtained by E[γk ] = P
A. Single-input single-output (SISO) Achievable capacity of SISO is given by CSISO = log2 1 + P|hk,1 |2 .
Var[γk ] = P =P
(5)
(7) 2
and vk = [vk,1 , vk,2 , · · · , vk,M ]T = UH wk . The statistical distribution of vk is the same as wk since the multiplication of an unitary matrix does not change the distribution of a Gaussian random vector [10]. Correspondingly, the mean and variance of TD are given by M P P tr (RT ) = P and λm = M m=1 M
M P2 2 P2 λm = 2 tr R2T Var[γk ] = 2 M m=1 M
M −1 P2 2m = 2 M+ , 2(M − m)ρ M m=1
.
(12)
Phase steering or coherent beamforming is a diversity technique making the received signals from transmit antennas cophased [12]. PS achieves multiple antenna diversity gains with much less feedback information than MRT since the required feedback information is just the channel phases of each user. Achievable capacity of PS is given by
= λ1 |vk,1 | + λ2 |vk,2 | + · · · + λM |vk,M | ,
E[γk ] =
2(M − m)ρ
D. Phase steering (PS)
corresponds to the received SNR and 2
2m
The statistical properties of MRT are the same as TD except that MRT attains an array gain – the factor M in the mean reflects the array gain. As in TD, the variance of MRT also increases with correlation.
TD is a diversity technique when channel state information at the transmitter (CSIT) is not available. Achievable capacity is given by
P 2 hk . (6) CT D = log2 1 + M
2
M+
M −1 m=1
B. Transmit diversity (TD)
hk 2 = wkH UΛUH wk = vkH Λvk
2
(11)
λ2m = P 2 · tr R2T
The received SNR of SISO corresponds to γk = P|hk,1 | , which is an exponential random variable with
P 2 M hk
λm = P · tr (RT ) = M P and
m=1 M 2 m=1
2
where γk =
M
(4)
E[γk ] = P and Var[γk ] = P 2 .
(10)
CP S
(8)
M
2 P = log2 1 + |hk,m | M m=1
M P ≈ 2 log2 |hk,m | . M m=1
(13)
M Hence, γk = P/M m=1 |hk,m | plays the equivalent role of the received SNR and its mean value is (9)
E[γk ] =
66
M P E [|hk,m |] = M m=1
√
πM P , 2
(14)
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2008 proceedings.
where |hk,m | is a Rayleigh distributed random variable with √ the mean value of π/2. The variance is obtained by M P Var |hk,m | (15) Var[γk ] = M m=1 =
scheduling maxk γk . However, it is still difficult to obtain an exact closed form of the mean value after scheduling. Since {γk } are independent, we use an upper bound from order statistics as in [6], [11]: K −1 Var[γk ] (19) E max γk ≤ E[γk ] + √ k=1,2,··· ,K 2K − 1
M M M P P Var [|hk,m |] + Cov [|hk,m |, |hk,n |] , M m=1 M m=1 n=1,
This bound intuitively quantifies the effects of correlation on spatial and multiuser diversity – the mean value after scheduling, E[maxk γk ], increases with not only the number of users but also the variance of received SNR of each user, Var[γk ], given in terms of spatial correlation. Since Var[γk ] increases with correlation, E[maxk γk ] also increase with correlation. That is, correlation rather impose positive effects on spatial diversity if opportunistic multiuser scheduling is adopted, while correlation has been known to impose negative impacts on spatial diversity in a point-to-point communication. Table II summarizes the mean values of diversity techniques after scheduling when K = 20 and P = 10dB. It is verified that a larger variance by higher correlation positively affects on E[maxk γk ]. By comparing Fig. 1 and Fig. 2, we can also find that SISO outperforms TD in uncorrelated channels as reported in [3], [4], [5], [6], but they achieve almost the same capacity in highly correlated channels.
n=m
where Var[|hk,m |] = 1 − π/4, and Cov[R1 , R2 ] = E[R1 R2 ] − E[R1 ]E[R2 ] for random variables R1 = |hk,m | and R2 = |hk,n |. Since R1 and R2 are Rayleigh distributed random variables, the mean value of R = R1 R2 is expressed as [13]
3 3 3 2 2 2 2|m−n| E[R1 R2 ] = Γ (1 − ρ ) Φ , ; 1; ρ , (16) 2 2 2 l ∞ l (b)l d is a Gaussian hyperwhere Φ(a, b; c; d) = 1 + l=1 (a)(c) l l! geometric function, and (t)l = t × (t + 1) × · · · × (t + l − 1) for a real value t. Consequently, (15) is rewritten as
π P M π P− (17) Var[γk ] = 1 − 4 2M 2
M M 2 3 3 P 2 3 Γ 1 − ρ2 , ; 1; ρ2|m−n| . Φ + M 2 2 2 m=1 n=1
V. S IMULATION RESULTS
n=m
This section presents theoretical and simulation results. We assume that the average SNR P is 10dB, the number of user K is 20 for multiuser diversity, and the number of transmit antennas M is 2. The performance of SISO is not affected by spatial correlation since there is a single antenna but is provided as a reference. Fig. 3 shows the negative impacts of antenna correlations on spatial diversity in point-to-point communications. All the diversity techniques degrade as correlation increases as previously known. On the other hand, Fig. 4 shows a positive effects of correlation on spectral efficiency when multiuser opportunistic scheduling is employed. In this figure, spectral efficiencies of diversity techniques increase with correlation since correlation limits diversity effects and hence enhances multiuser diversity gains. The enhanced multiuser diversity gains from spatial correlation perfectly compensate for performance degradation by correlation, and eventually yield an increase of spectral efficiencies. Fig. 5 and Fig. 6 show spectral efficiencies of diversity techniques according to SNR when ρ = 0 and ρ = 0.9, respectively. In Fig. 5, SISO outperforms TD in uncorrelated channels when multiuser opportunistic scheduling is employed as reported in [3], [4], [5], [6]. On the other hand, SISO and TD achieve almost the same spectral efficiency in highly correlated channels as shown in Fig. 6 since spatial correlation amplifies a multiuser diversity gain. Another interesting observation from these figures is that PS achieves similar spectral efficiency to MRT in highly correlated channels although the required feedback information of PS is much less than that of MRT.
where it should be noted that the third term of (17) is an increasing function with respect to a correlation value ρ, which is proven in Appendix. Therefore, the variance of PS increases with correlation and correspondingly its spatial diversity gain decreases. The mean and variance of each diversity technique without multiuser scheduling when M = 2 and P = 10dB are summarized in Table I. Even though the mean and variance of SISO are not affected by correlation, they are provided as references. Table I verifies the fact that variances of SNR increase with correlation. IV. E FFECTS OF S PATIAL C ORRELATION ON M ULTIUSER D IVERSITY If we adopt opportunistic multiuser scheduling to exploit a multiuser diversity gain, the achievable capacity of each diversity technique is given by Cmax = E max log2 (1 + γk ) k
≤ log2 1 + E max γk . (18) k
where the upper bound comes from Jensen’s inequality. The bound becomes very tight in the high SNR region where SNR after scheduling typically is. Note that we apply a maxscheduler which selects the user with the highest achievable capacity to clarify the effects of correlation on spatial and multiuser diversity. Owing to Jensen’s inequality, the mean value after scheduling well characterizes the achievable capacity without the knowledge of exact distribution of SNR after
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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.
VI. C ONCLUSION
[7] C. N. Chuah, D. N. C. Tse, J. M. Kahn, and R. A. Valenzuela, “Capacity scaling in MIMO wireless systems under correlated fading,” IEEE Transactions on Information Theory, vol. 48, no. 3, pp. 637-650, Mar. 2002. [8] D. S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fading correlation and its effect on the capacity of multielement antenna systems,” IEEE Transactions on Communications, vol. 48, no. 3, Mar. 2000. [9] D. Park and S. Park, “Effect of transmit antenna correlation on multiuser diversity,” IEEE International Symposium on Information Theory (ISIT), Sep. 2005. [10] E. Teletar, “Capacity of multi-antenna Gaussian channels,” European Transactions on Telecommunications, vol. 10, no. 6, pp. 585-595, Nov./Dec. 1999. [11] W. Choi, N. Himayat, S. Talwar, and M. Ho, “The effects of cochannel interference on spatial diversity techniques,” IEEE Wireless Communications and Networking Conference (WCNC), Mar. 2007. [12] W. Choi and J. G. Andrews, “Downlink performance and capacity of distributed antenna systems in a multicell environment,” IEEE Transactions on Wireless Communications, vol. 6, no. 1, pp. 69-73, Jan. 2007. [13] M. K. Simon, Probability distributions involving Gaussian random variables: A handbook for engineers and scientists, 1st ed., Kluwer Academic Publishers, 2002.
This paper has studied interaction between spatial correaltion, spatial diversity, and multiuser diversity. Using an upper bound from order statistic theory, we have intuitively quantified the effects of correlations on spatial and multiuser diversity. The analytical and simulation results have shown that spatial correlation imposes positive impacts on spatial diversity techniques if opportunistic multiuser scheduling is employed although spatial correlations have been considered drawbacks in point-to-point communications. A PPENDIX We define f (ρ) and g(ρ) in the third term of (17), respectively, as
2 3 1 − ρ2 (20) f (ρ) = Γ 2
M M 3 3 2|m−n| , ; 1; ρ , (21) Φ g(ρ) = 2 2 m=1
1
n=1,n=m
0.9
where f (ρ) is a decreasing function (but f (ρ) ≥ 0 for 0 ≤ ρ < 1) and function with respect to ρ. g(ρ) is an increasing Since Φ 32 , 32 ; 1; ρ2|m−n| is bounded by
3 3 3 2|m−n| 2|m−n| Φ , ; 1; ρ , 1; 1; ρ ≥Φ (22) 2 2 2 ∞ 3 2|m−n|l 2 lρ (23) =1+ l! l=1 ∞ 3 l 2|m−n|l ρ 2 (24) ≥1+ l! l=1
3 2|m−n| = exp ρ , (25) 2
0.8 0.7 0.6 CDF
SISO MRT
0.5 0.4
PS
TD
0.3 0.2 K=1
0.1 0 −20
−15
Fig. 1.
g(ρ) is exponentially increasing whereas f (ρ) is a decreasing polynomial. Therefore, f (ρ)g(ρ) is an increasing function with respect to ρ since the multiplication of a decreasing polynomial f (ρ) and an exponentially increasing function g(ρ) is increasing with ρ.
−10
−5
K=20
0 5 10 Received SNR [dB]
15
20
25
30
CDFs for uncorrelated channel (ρ = 0, P = 10dB)
1 0.9 0.8
R EFERENCES
0.7
[1] P. Viswanath, D. N. C. Tse, and R. Laroia, “Opportunistic beamforming using dumb antennas,” IEEE Transactions on Information Theory, vol. 48, no. 6, pp. 1277-1294, June 2002. [2] M. Sharif and B. Hassibi, “On the capacity of MIMO broadcast channel with partial side information,” IEEE Transactions on Information Theory, vol. 51, no. 2, pp. 506-522, Feb. 2005. [3] R. Gozali, R. M. Buehrer, and B. D. Woerner, “The impact of multiuser diversity on space-time block coding,” IEEE Communications Letters, vol. 7, no. 5, pp. 213-215, May 2003. [4] E. G. Larsson, “On the combination of spatial diversity and multiuser diversity,” IEEE Communications Letters, vol. 8, no. 8, pp. 517-519, Aug. 2004. [5] B. Hochwald, T. Marzetta, and V. Tarokh, “Multiple-antenna channel hardening and its implications for rate feedback and scheduling,” IEEE Transactions on Information Theory, vol. 50, no. 9, Sep. 2004. [6] W. Choi, et al., “Interactions between multiuser diversity and spatial diversity techniques in an interference-limited environment,” IEEE Wireless Communications and Networking Conference (WCNC), Mar. 2007.
CDF
0.6 K=20 0.5
K=1
0.4 PS, MRT
SISO 0.3 TD
0.2 0.1 0 −20
−15
Fig. 2.
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−10
−5
0
5 10 15 Received SNR [dB]
20
25
30
CDFs for correlated channel (ρ = 0, P = 10dB)
35
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2008 proceedings.
TABLE I M EAN AND VARIANCE BEFORE SCHEDULING (P = 10dB) ρ=0 E[γ] in dB Var[γ] in dB 10.00 20.01 10.01 16.98 12.53 22.23 13.02 23.00
SISO TD PS MRT
ρ = 0.9 E[γ] in dB Var[γ] in dB 10.00 20.01 10.00 19.60 12.92 25.59 13.01 25.62
TABLE II M EAN AND VARIANCE AFTER SCHEDULING (P = 10dB, K=20) ρ=0 E[maxk γk ] in dB Var[maxk γk ] in dB Simulation Upper bound 22.04 15.56 16.07 17.41 14.35 14.99 22.91 16.95 18.51 23.43 17.36 18.00
SISO TD PS MRT
ρ = 0.9 E[maxk γk ] in dB Var[maxk γk ] in dB Simulation Upper bound 22.04 15.56 16.07 21.58 15.40 15.98 27.60 18.38 19.94 27.60 18.41 18.92
5
6 SISO TD PS MRT
4.5
SISO TD PS MRT
5
Throughput[bps/Hz]
Throughput[bps/Hz]
K=20 4
3.5
3
4
3
2 K=1
2.5
2
1
0
Fig. 3.
0.1
0.2
0.3
0.4
0.5 0.6 Correlation[ρ]
0.7
0.8
0.9
0
1
Throughput according to the correlation before scheduling
2
3
4
5 SNR[dB]
6
7
8
9
10
Throughput according to the average SNR (ρ = 0)
7
SISO TD PS MRT
SISO TD PS MRT
6
5
6 Throughput[bps/Hz]
Throughput[bps/Hz]
1
Fig. 5.
7
6.5
0
5.5
5
K=20 4
3
2 K=1
4.5
4
1
0
0
Fig. 4.
0.2
0.4 0.6 Correlation[ρ]
0.8
1
0
Fig. 6.
Throughput according to the correlation after scheduling
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1
2
3
4
5 SNR[dB]
6
7
8
9
10
Throughput according to the average SNR (ρ = 0.9)
