- Email: [email protected]
Ergodic channel capacity of the spatial correlated rayleigh MIMO channel
THE JOURNAL OF CHINA UNIVERSITIES OF POSTS AND TELECOMMUNICATIONS Volume 14. Issue 4. December 2007
ZHANG Hui-ping, WU Ping, LIU Ai-jun
Ergodic channel capacity of the spatial correlated rayleigh MlMO channel CLC number TN911
Document A
Abstract The theoretical capacity of the spatial correlated Rayleigh multiple input multiple output (MIMO) channel is an important issue in MIMO technology. In this article, an ergodic channel capacity formula of the spatial correlated rayleigh MlMO channel is provided, which is deduced when two antennas exist at either the transmitter or the receiver. The multi-dimensional least-squares fit algorithm is employed to narrow the difference between the theoretical formula capacity and the practical capacity. Simulation results show that the theoretical capacity approaches the practical one closely.
Article ID
1005-8885 (2007) 04-0032-04
Keywords MIMO, rayleigh fading channel, Shannon capacity
The main contribution of this article is that it provides a close-form formula of the ergodic channel capacity for the spatial correlated rayleigh MIMO channel, which has two antennas at either the transmitter or the receiver. The proposed formula can be used for MIMO theoretical analysis and engineering application conveniently. Simulations show that the capacity calculated by the formula and the capacity obtained by monte carlo simulation coincide perfectly. This article is organized as follows: Section 2 introduces the Kronecker spatial correlated MIMO channel model. Sections 3 and 4 provide the capacity formula. Section 5 provides the simulation results. Finally, Section 6 concludes the article.
1 lntroductlon
2
The MIMO system is considered as a key technology owing to the considerable capacity or spectrum efficiency [I-31. Channel capacity is an important issue in the MIMO system. A great deal of study has been camed out in this filed, e.g. the theoretical capacity of the uncorrelated rayleigh MIMO channel has been studied sufficiently [4-71. However, in the MIMO system, when the antennas are not separated sufficiently and/or there is lack of scattering, spatial correlation will exist. The research of the correlated MIMO channel capacity has traditionally been considered as a difficult problem [8-101. Semi-correlated channel capacity has been studied in Ref. [9]. The distribution of the correlated channel capacity has been discussed in Ref. [ll], and some capacity bounds of the correlated channel have been provided in Ref. [12]. However, a close-form formula of the correlated MIMO capacity has not been proposed.
Received date: 2007-04- I7 ZHANG Hui-ping, LIU Ai-jun School of Languages, Beijing University of Posts and Telecommunications, Beijing 100876, China WUPing(7) Key Laboratory of Universal Wireless Communications (Beijing University of Posts and Telecommunications) Ministry of Education, Wireless Technology Innovation Institute, Beijing 100876, China E-mail: [email protected]
Spatlal comlated channel model We consider that the MIMO system has N, transmitting
antennas and N, receiving antennas, and the channel matrix is H. The Kronecker spatial correlated MIMO channel can be modeled as [ 10- 131: y = Hs + n = Rf/2H,R:/2s+ n (1) where [-]I/*represents the matrix square root; H w is a N , x N, complex matrix, whose elements are independently identically distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance; and x , y , and n represent the transmitting, the receiving, and the noise vector, respectively. We assume that the spatial correlation at the transmitter is independent of the correlation at the receiver. R, is M , x M , transmitting correlation matrix and R, is M , x M ,receiving correlation matrix. R, , R, are written as:
1 r
R, =
( N, -k
,.(N, -1)’
r 1
... ...
,.(A’,
... ...... -8)’
,.(N, -1)’
...... ......
. . . . . . ,.(N,-I)‘ .........
...
...
I
r
...
r
1
...
... ,.(N, -1)’
33
ZHANG Hui-Dine. et al.: Ereodic channel caDacitv of the sDatial correlated ravleigh MIMO channel
No. 4
multi-dimensional least-squares fit algorithm is employed to approach fmodify. The approach function is noted as jmadify=ar + bt + cp + d . Then, we obtain the practical f d f y ( q 9 t j 9 P k )= z ~ , using ~ , ~
1
...
...
t ( N t -1)'
...
...
1
t(Nt-j12
...
t
1
where r is the receiving correlation coefficient between the nearest receiving antennas, and t is the transmitting correlation coefficient between the nearest transmitting antennas.
monte car10 method based on Eq.(2) and Eq. (4). We can then obtain the coefficients a, b, c, d as: ( a , b, c , dIT= S-'y = S - ' ( Y , ,Y , , y 3 , y J T S, y are given as follows:
(5)
M
8 Ergodlc channel capachy
Nkkqtj S=
LNtti
j=l
1-1
L$ttjpk
j=l
j=l k = l
LNctj j=l
When the channel information is not available at the transmitter, and equal-power allocation is adopted, the MIMO channel capacity in Sect. 2 can be written as [6]: C = log,(det(ZN,+ pHH ')) = log,(det(ZN,+ pH 'H )) (2 ) where [ .IHdenotes the matrix conjugate transposition; ZNr is the N , x N , identity matrix; det(0) is the matrix determinant; and p is the average signal noise ratio (SNR). We consider that the MIMO system has 2 antennas at either the transmitting or the receiving side. Without lose of generality, we assume H, R, are 2 x N , , 2 x 2 matrices,
and A,,
4
obtain: E(det(1,
+ pR:'2H,R,H:R:'2))
are the eigenvalues of pHHH; then, we can = 1 + E(trace(R,H,R,H,H))+
det(R,)E(det(H,R,HLH)) = 1 + 2 N , p d 2 + (1 - r 2 ) N,' - N,
(
-2
C
N, -I
ir2(N,-1)'
,=I
[
1 1 p264
(3)
where H , = f i H w , E ( - ) denotes the expectation, and
trace
(0)
Y=
represents the matrix trace. log,(x) is a concave function
of x, therefore, log,(E(x))~E(log,(n)) and log,(E(x) approaches E(log, ( x ) ) closely. Consequently, according to Eq. ( 2 ) and Eq.(3), we obtain: E ( C )= E(log,(det(Z, +pHHH)))= log,(E(det(Z, pHH'))) N, -I
2 c it2"I)' ,=I
fmdfY
[
+
[
=log, 1 + 2 N , p 8 + ( 1 - r 2 ) N , ' - N , -
1 1
P2d4 - fmad,fy(r'f'P",)
(4)
where fmdlfy is the modify function, which is used to narrow the difference between E(Iog,(x)) and l o g , ( E ( x ) ) , and the method to obtain
fmodlfY
where L, M ,N are the sampling numbers of variables r , t , p, respectively. Based on Eqs. (5)-(7), we can obtain the coefficients a, b, c, d of jmmodify(~, t,, p k ) under different conditions as shown in Table 1, Table 1 Coefficients of 2 a - 0.134 [O,lO)dB b -0.134 c 0.04 d 0.373 a -0,269 b - 0.269 "*201dBc 0.003 d 0.998
can be approached by linear function precisely. The
4 -0.135 0.247 0.024 0.215 -0.07 0.472 O.OOO7 0.331
6
- 0.095 0.345 0.014 0.127 -0.028 0.494 O.OOO3 0.168
8 -0.067 0.343 0.009 0.085 -0.015 0.439 O.OOO1 0.106
Then, the ergodic channel capacity of the 2xN or Nx2 correlated MIMO Rayleigh fading channel can be given as:
1 + 2 N , p d 2+ ( 1 - r 2 ) 2xir2(N,-i)' i=l
&ify
N,
P
is provided in Sect. 4.
4 Method of approachlng fmodirr
jmmodlfy (I; ,t j , p k)
T h e Journal of CHUPT
34
where coefficients a, b, c, d can be obtained as in Table 1
2007
As seen in Fig. 1 and Fig. 2, the ergodic channel capacity decreases
I Slmulatlon m u t t s
when
the
spatial
correlation
increases.
The
theoretical capacities calculated using Eq. (8) are close to the
Simulations were carried out to compare the theoretic capacity calculated using Eq. (8) and the practical capacity calculated using Eq. (2). Based on Eq. (2), we calculated the practical channel capacity for each realization of channel matrix H . T h e ergodic capacity can be given as:
(9) where N is the number of the generated channel matrix, and C, is the capacity of the ith channel matrix The simulations compare the capacities of Eq. (8) and Eq. (9). Figure 1 shows the simulation results when the transmitting spatial correlation is 0, and Fig. 2 depicts the results when the transmitting spatial correlation is 0.7.
monte carlo capacities obtained using Eq. (9) tightly under different antenna numbers and spatial correlations.
6
Concludons In this article, w e provide a close-form formula of the
ergodic capacity for spatial correlated Rayleigh fading MIMO channel, which has two antennas at either the transmitter or the receiver. T h e simulation results demonstrate that the capacity calculated using the proposed formula is close to the capacity obtained using the monte carlo method. In addition, the provided formula is analytical, which can be used for further theoretical analysis and direct capacity calculation.
10 I
Acknowledgements This work is supported by the National Natural Science Foundation of China (60702051), the Hi-Tech Research and Development Program of Chma (2006AAOlZ260), and Huawei Company.
References 1 . Zhang Ping. Some research issues for beyond 3G mobile systems.
Journal of Beijing University of Posts and Telecommunications, 41 0
1
1
I
1
I
I
j
0.2 0.3 0 4 0 5 0 6 0.7 0.8 0 9 Receiving correlation coetlicient + J h > ? . 5 dH. theor) -e-Jb>2.Y dl3. theon 4 - 2 b b 2 . 7 df3. theor) -e-Jh)2.5 dB. Monte -e-Jh>Z.Y dU. Monte -A-2h)?.7 dB. Monte *4b!2.7 dB. theor) 4 - 2 h 5 2 . 7 dR. thcon -+?b>Z.S dH. theor) -*-4b>2.7 dR. Montc -8-2bb2.7 dH. Monte -b-2bb2.5 dR. Monte 0.1
2002, 25(3): 1-6 (in Chinese) 2. Jin Yi-dan, Wu Wei-ling. Reduced complexity turbo equalization for turbo coded MIMO/OFDM systems. The Journal of China Universities of Posts and Telecommunications, 2006, 13(1): 93-98
Fig. 1 Ergodic capacity, 2x2 and 4x2 MIMO. SNR 1-IOdB, J2 = I , transmitting spatial correlation is O 10 I
3. Xiao Zheng-rong, Yu Zhi, Wu Wei-ling. New development of MIMO wireless communication system. Journal of Chongqing University of Posts and Telecommunications: Natural Science Edition, 2004, 16(4): 25-29 (in Chinese) 4. Telatar I E, Capacity of multi-antenna Gaussian channels. Tech Report, AT & T Bell Labs, 1995 5. Chao W, Sjunjun W U, Linrang Z, et al. Capacity evaluation of MIMO systems by monte-carlo methods. Proceedings of IEEE International Conference on Neural Networks and Signal Processing: Vol. 2, Dec 14-17, 2003, Xanjing, Chma. Piscataway,
41
n
' 02
NJ. USA: IEEE, 2003: 1464-1466 "
1
1
I
0.3 0.4 0.5 0.6 0.7 0.8 0.9 Recei\ ing correlation coefficient +4b>2.5 dB. theon. + J h ) 2 . 9 dB. t h e o n &2h!2.7 dB. theon -8-4b)2.5 dB. Monte - e - l h > 2 . 9 dB. Monte -A-2b>2.7 dB. Monte -4bb2.7 dB. theon e 2 h k 2 . 7 dB. theon -+2b)?.5 dB. theon -*-4b>?.7 dB. Monte - s - Z b > 2 . 7 dB. Monte - , - ? b y 2 3 dB. Monte 0.1
Fig. 2 Ergodic capacity, 2x2 and 4x2 MIMO, SNR 1-lOdB, J2 = 1 , transmitting spatial correlation is 0.7
6. Khalighi M A, Brossier J M, Jourdain G, et al. Water filling capacity of rayleigh MIMO channels. Proceedings of 12th International Conference on Personal, Indoor and Mobile Radio Communications: Vol. 1, Sept 3 W c t 3, 2001 San Diego, CA, USA. Piscataway, NJ, USA: IEEE, 2001: 155-158 7. Perera R R, Pollock T S, Abhayapala T D. Upper bound on
No. 4
ZHANG Hui-Dine. et al.: Ereodic channel caDacitv of the sDatial correlated ravleieh MIMO channel
non-coherent MIMO channel capacity in rayleigh fading. Proceedings of 2005 Asia-Pacific Conference on Communications, Oct 3-5, 2005, Perth, Australia. Piscataway, NJ. USA: IEEE, 2005: 72-76 8. Kang M, Alouini M S. Water-filling capacity and beamfoming performance of MIMO systems with covariance feedback.
35
MIMO-correlated rayleigh fading channels. IEEE Transactions on Wireless Communications, 2005,4(2): 681-688 13. Zhao Lan, Dubey V K. Transmit diversity and combining scheme for spatial multiplexing over correlated channels. Proceedings of
59th Vehicular Technology Conference: Vol. I , May 17-19,2003, Milan, Italy. Piscataway, NJ, USA: IEEE, 2003: 380-383
Proceedings of IEEE Workshop on Signal Processing Advances in Wireless Communications, Jun 15-18, 2003, Rome, Italy. Piscataway, NJ, USA: IEEE, 2003: 556-560 9. Kang M, Alouini M S. Capacity of correlated MIMO rayleigh channels. TEEE Transactions on Wireless Communications, 2006, 5(1): 143-155
Biographies: ZHANG Hui-ping, master Candidate in Beijing University of Posts and Telecommunications, School of Languages, interested in the research on english teaching, MIMO, orthogonal frequency division multiplex (OFDM) technology.
10. Oestges C, Clerckx B, Vanhoenacker D, et al. Impact of diagonal correlations on MIMO capacity: applica-tion to geometrical scattering models. Proceedings of 58th E E E Vehicular Technology
Lnl Ai-jun, associate professor in Beijing University
Conference: Vol I , Oct 6-9, 2003, Orlando, FL, USA. Piscataway,
of Posts and Telecommunications, School of Languages, interested in the research on English
NJ, USA: IEEE, 2003: 394-398 11. Chiani M, Win M Z, Zanella A. On the capacity of spatially
correlated MIMO rayleigh fading channels. IEEE Transactions on Information Theory, 2003,49( 10): 2363-2371 12. Zhang Q T, Cui X W, Li X M. Very tight capacity bounds for
teaching, MIMO, orthogonal frequency division multiplex (OFDM) technology.
ZHANG Hui-ping, WU Ping, LIU Ai-jun
Ergodic channel capacity of the spatial correlated rayleigh MlMO channel CLC number TN911
Document A
Abstract The theoretical capacity of the spatial correlated Rayleigh multiple input multiple output (MIMO) channel is an important issue in MIMO technology. In this article, an ergodic channel capacity formula of the spatial correlated rayleigh MlMO channel is provided, which is deduced when two antennas exist at either the transmitter or the receiver. The multi-dimensional least-squares fit algorithm is employed to narrow the difference between the theoretical formula capacity and the practical capacity. Simulation results show that the theoretical capacity approaches the practical one closely.
Article ID
1005-8885 (2007) 04-0032-04
Keywords MIMO, rayleigh fading channel, Shannon capacity
The main contribution of this article is that it provides a close-form formula of the ergodic channel capacity for the spatial correlated rayleigh MIMO channel, which has two antennas at either the transmitter or the receiver. The proposed formula can be used for MIMO theoretical analysis and engineering application conveniently. Simulations show that the capacity calculated by the formula and the capacity obtained by monte carlo simulation coincide perfectly. This article is organized as follows: Section 2 introduces the Kronecker spatial correlated MIMO channel model. Sections 3 and 4 provide the capacity formula. Section 5 provides the simulation results. Finally, Section 6 concludes the article.
1 lntroductlon
2
The MIMO system is considered as a key technology owing to the considerable capacity or spectrum efficiency [I-31. Channel capacity is an important issue in the MIMO system. A great deal of study has been camed out in this filed, e.g. the theoretical capacity of the uncorrelated rayleigh MIMO channel has been studied sufficiently [4-71. However, in the MIMO system, when the antennas are not separated sufficiently and/or there is lack of scattering, spatial correlation will exist. The research of the correlated MIMO channel capacity has traditionally been considered as a difficult problem [8-101. Semi-correlated channel capacity has been studied in Ref. [9]. The distribution of the correlated channel capacity has been discussed in Ref. [ll], and some capacity bounds of the correlated channel have been provided in Ref. [12]. However, a close-form formula of the correlated MIMO capacity has not been proposed.
Received date: 2007-04- I7 ZHANG Hui-ping, LIU Ai-jun School of Languages, Beijing University of Posts and Telecommunications, Beijing 100876, China WUPing(7) Key Laboratory of Universal Wireless Communications (Beijing University of Posts and Telecommunications) Ministry of Education, Wireless Technology Innovation Institute, Beijing 100876, China E-mail: [email protected]
Spatlal comlated channel model We consider that the MIMO system has N, transmitting
antennas and N, receiving antennas, and the channel matrix is H. The Kronecker spatial correlated MIMO channel can be modeled as [ 10- 131: y = Hs + n = Rf/2H,R:/2s+ n (1) where [-]I/*represents the matrix square root; H w is a N , x N, complex matrix, whose elements are independently identically distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance; and x , y , and n represent the transmitting, the receiving, and the noise vector, respectively. We assume that the spatial correlation at the transmitter is independent of the correlation at the receiver. R, is M , x M , transmitting correlation matrix and R, is M , x M ,receiving correlation matrix. R, , R, are written as:
1 r
R, =
( N, -k
,.(N, -1)’
r 1
... ...
,.(A’,
... ...... -8)’
,.(N, -1)’
...... ......
. . . . . . ,.(N,-I)‘ .........
...
...
I
r
...
r
1
...
... ,.(N, -1)’
33
ZHANG Hui-Dine. et al.: Ereodic channel caDacitv of the sDatial correlated ravleigh MIMO channel
No. 4
multi-dimensional least-squares fit algorithm is employed to approach fmodify. The approach function is noted as jmadify=ar + bt + cp + d . Then, we obtain the practical f d f y ( q 9 t j 9 P k )= z ~ , using ~ , ~
1
...
...
t ( N t -1)'
...
...
1
t(Nt-j12
...
t
1
where r is the receiving correlation coefficient between the nearest receiving antennas, and t is the transmitting correlation coefficient between the nearest transmitting antennas.
monte car10 method based on Eq.(2) and Eq. (4). We can then obtain the coefficients a, b, c, d as: ( a , b, c , dIT= S-'y = S - ' ( Y , ,Y , , y 3 , y J T S, y are given as follows:
(5)
M
8 Ergodlc channel capachy
Nkkqtj S=
LNtti
j=l
1-1
L$ttjpk
j=l
j=l k = l
LNctj j=l
When the channel information is not available at the transmitter, and equal-power allocation is adopted, the MIMO channel capacity in Sect. 2 can be written as [6]: C = log,(det(ZN,+ pHH ')) = log,(det(ZN,+ pH 'H )) (2 ) where [ .IHdenotes the matrix conjugate transposition; ZNr is the N , x N , identity matrix; det(0) is the matrix determinant; and p is the average signal noise ratio (SNR). We consider that the MIMO system has 2 antennas at either the transmitting or the receiving side. Without lose of generality, we assume H, R, are 2 x N , , 2 x 2 matrices,
and A,,
4
obtain: E(det(1,
+ pR:'2H,R,H:R:'2))
are the eigenvalues of pHHH; then, we can = 1 + E(trace(R,H,R,H,H))+
det(R,)E(det(H,R,HLH)) = 1 + 2 N , p d 2 + (1 - r 2 ) N,' - N,
(
-2
C
N, -I
ir2(N,-1)'
,=I
[
1 1 p264
(3)
where H , = f i H w , E ( - ) denotes the expectation, and
trace
(0)
Y=
represents the matrix trace. log,(x) is a concave function
of x, therefore, log,(E(x))~E(log,(n)) and log,(E(x) approaches E(log, ( x ) ) closely. Consequently, according to Eq. ( 2 ) and Eq.(3), we obtain: E ( C )= E(log,(det(Z, +pHHH)))= log,(E(det(Z, pHH'))) N, -I
2 c it2"I)' ,=I
fmdfY
[
+
[
=log, 1 + 2 N , p 8 + ( 1 - r 2 ) N , ' - N , -
1 1
P2d4 - fmad,fy(r'f'P",)
(4)
where fmdlfy is the modify function, which is used to narrow the difference between E(Iog,(x)) and l o g , ( E ( x ) ) , and the method to obtain
fmodlfY
where L, M ,N are the sampling numbers of variables r , t , p, respectively. Based on Eqs. (5)-(7), we can obtain the coefficients a, b, c, d of jmmodify(~, t,, p k ) under different conditions as shown in Table 1, Table 1 Coefficients of 2 a - 0.134 [O,lO)dB b -0.134 c 0.04 d 0.373 a -0,269 b - 0.269 "*201dBc 0.003 d 0.998
can be approached by linear function precisely. The
4 -0.135 0.247 0.024 0.215 -0.07 0.472 O.OOO7 0.331
6
- 0.095 0.345 0.014 0.127 -0.028 0.494 O.OOO3 0.168
8 -0.067 0.343 0.009 0.085 -0.015 0.439 O.OOO1 0.106
Then, the ergodic channel capacity of the 2xN or Nx2 correlated MIMO Rayleigh fading channel can be given as:
1 + 2 N , p d 2+ ( 1 - r 2 ) 2xir2(N,-i)' i=l
&ify
N,
P
is provided in Sect. 4.
4 Method of approachlng fmodirr
jmmodlfy (I; ,t j , p k)
T h e Journal of CHUPT
34
where coefficients a, b, c, d can be obtained as in Table 1
2007
As seen in Fig. 1 and Fig. 2, the ergodic channel capacity decreases
I Slmulatlon m u t t s
when
the
spatial
correlation
increases.
The
theoretical capacities calculated using Eq. (8) are close to the
Simulations were carried out to compare the theoretic capacity calculated using Eq. (8) and the practical capacity calculated using Eq. (2). Based on Eq. (2), we calculated the practical channel capacity for each realization of channel matrix H . T h e ergodic capacity can be given as:
(9) where N is the number of the generated channel matrix, and C, is the capacity of the ith channel matrix The simulations compare the capacities of Eq. (8) and Eq. (9). Figure 1 shows the simulation results when the transmitting spatial correlation is 0, and Fig. 2 depicts the results when the transmitting spatial correlation is 0.7.
monte carlo capacities obtained using Eq. (9) tightly under different antenna numbers and spatial correlations.
6
Concludons In this article, w e provide a close-form formula of the
ergodic capacity for spatial correlated Rayleigh fading MIMO channel, which has two antennas at either the transmitter or the receiver. T h e simulation results demonstrate that the capacity calculated using the proposed formula is close to the capacity obtained using the monte carlo method. In addition, the provided formula is analytical, which can be used for further theoretical analysis and direct capacity calculation.
10 I
Acknowledgements This work is supported by the National Natural Science Foundation of China (60702051), the Hi-Tech Research and Development Program of Chma (2006AAOlZ260), and Huawei Company.
References 1 . Zhang Ping. Some research issues for beyond 3G mobile systems.
Journal of Beijing University of Posts and Telecommunications, 41 0
1
1
I
1
I
I
j
0.2 0.3 0 4 0 5 0 6 0.7 0.8 0 9 Receiving correlation coetlicient + J h > ? . 5 dH. theor) -e-Jb>2.Y dl3. theon 4 - 2 b b 2 . 7 df3. theor) -e-Jh)2.5 dB. Monte -e-Jh>Z.Y dU. Monte -A-2h)?.7 dB. Monte *4b!2.7 dB. theor) 4 - 2 h 5 2 . 7 dR. thcon -+?b>Z.S dH. theor) -*-4b>2.7 dR. Montc -8-2bb2.7 dH. Monte -b-2bb2.5 dR. Monte 0.1
2002, 25(3): 1-6 (in Chinese) 2. Jin Yi-dan, Wu Wei-ling. Reduced complexity turbo equalization for turbo coded MIMO/OFDM systems. The Journal of China Universities of Posts and Telecommunications, 2006, 13(1): 93-98
Fig. 1 Ergodic capacity, 2x2 and 4x2 MIMO. SNR 1-IOdB, J2 = I , transmitting spatial correlation is O 10 I
3. Xiao Zheng-rong, Yu Zhi, Wu Wei-ling. New development of MIMO wireless communication system. Journal of Chongqing University of Posts and Telecommunications: Natural Science Edition, 2004, 16(4): 25-29 (in Chinese) 4. Telatar I E, Capacity of multi-antenna Gaussian channels. Tech Report, AT & T Bell Labs, 1995 5. Chao W, Sjunjun W U, Linrang Z, et al. Capacity evaluation of MIMO systems by monte-carlo methods. Proceedings of IEEE International Conference on Neural Networks and Signal Processing: Vol. 2, Dec 14-17, 2003, Xanjing, Chma. Piscataway,
41
n
' 02
NJ. USA: IEEE, 2003: 1464-1466 "
1
1
I
0.3 0.4 0.5 0.6 0.7 0.8 0.9 Recei\ ing correlation coefficient +4b>2.5 dB. theon. + J h ) 2 . 9 dB. t h e o n &2h!2.7 dB. theon -8-4b)2.5 dB. Monte - e - l h > 2 . 9 dB. Monte -A-2b>2.7 dB. Monte -4bb2.7 dB. theon e 2 h k 2 . 7 dB. theon -+2b)?.5 dB. theon -*-4b>?.7 dB. Monte - s - Z b > 2 . 7 dB. Monte - , - ? b y 2 3 dB. Monte 0.1
Fig. 2 Ergodic capacity, 2x2 and 4x2 MIMO, SNR 1-lOdB, J2 = 1 , transmitting spatial correlation is 0.7
6. Khalighi M A, Brossier J M, Jourdain G, et al. Water filling capacity of rayleigh MIMO channels. Proceedings of 12th International Conference on Personal, Indoor and Mobile Radio Communications: Vol. 1, Sept 3 W c t 3, 2001 San Diego, CA, USA. Piscataway, NJ, USA: IEEE, 2001: 155-158 7. Perera R R, Pollock T S, Abhayapala T D. Upper bound on
No. 4
ZHANG Hui-Dine. et al.: Ereodic channel caDacitv of the sDatial correlated ravleieh MIMO channel
non-coherent MIMO channel capacity in rayleigh fading. Proceedings of 2005 Asia-Pacific Conference on Communications, Oct 3-5, 2005, Perth, Australia. Piscataway, NJ. USA: IEEE, 2005: 72-76 8. Kang M, Alouini M S. Water-filling capacity and beamfoming performance of MIMO systems with covariance feedback.
35
MIMO-correlated rayleigh fading channels. IEEE Transactions on Wireless Communications, 2005,4(2): 681-688 13. Zhao Lan, Dubey V K. Transmit diversity and combining scheme for spatial multiplexing over correlated channels. Proceedings of
59th Vehicular Technology Conference: Vol. I , May 17-19,2003, Milan, Italy. Piscataway, NJ, USA: IEEE, 2003: 380-383
Proceedings of IEEE Workshop on Signal Processing Advances in Wireless Communications, Jun 15-18, 2003, Rome, Italy. Piscataway, NJ, USA: IEEE, 2003: 556-560 9. Kang M, Alouini M S. Capacity of correlated MIMO rayleigh channels. TEEE Transactions on Wireless Communications, 2006, 5(1): 143-155
Biographies: ZHANG Hui-ping, master Candidate in Beijing University of Posts and Telecommunications, School of Languages, interested in the research on english teaching, MIMO, orthogonal frequency division multiplex (OFDM) technology.
10. Oestges C, Clerckx B, Vanhoenacker D, et al. Impact of diagonal correlations on MIMO capacity: applica-tion to geometrical scattering models. Proceedings of 58th E E E Vehicular Technology
Lnl Ai-jun, associate professor in Beijing University
Conference: Vol I , Oct 6-9, 2003, Orlando, FL, USA. Piscataway,
of Posts and Telecommunications, School of Languages, interested in the research on English
NJ, USA: IEEE, 2003: 394-398 11. Chiani M, Win M Z, Zanella A. On the capacity of spatially
correlated MIMO rayleigh fading channels. IEEE Transactions on Information Theory, 2003,49( 10): 2363-2371 12. Zhang Q T, Cui X W, Li X M. Very tight capacity bounds for
teaching, MIMO, orthogonal frequency division multiplex (OFDM) technology.