Communications and Network, 2009, 2534 doi:10.4236/cn.2009.11004 Published Online August 2009 (http://www.scirp.org/journal/cn) Copyright © 2009 SciRes CN Approximate Analysis of Power Offset over Spatially Correlated MIMO Channels Guangwei YU, Xuzhen WANG School of Information and Communication Engineering, Beijing University of Posts and Telecommunications, Beijing, China Email: {yuguangwei1982, Gxitao}@gmail.com Abstract: Power offset is zeroorder term in the capacity versus signaltonoise ratio curve. In this paper, ap proximate analysis of power offset is presented to describe MIMO system with uniform linear antenna arrays of fixed length. It is assumed that the number of receive antenna is larger than that of transmit antenna. Spa tially Correlated MIMO Channel is approximated by tridiagonal toeplitz matrix. The determinant of tridi agonal toeplitz matrix, which is fitted by elementary curve, is one of the key factors related to power offset. Based on the curve fitting, the determinant of tridiagonal toeplitz matrix is mathematically tractable. Conse quently, the expression of local extreme points can be derived to optimize power offset. The simulation re sults show that approximation above is accurate in local extreme points of power offset. The proposed ex pression of local extreme points is helpful to approach optimal power offset. Keywords: MIMO, multiplexing gain, power offset, high snr region, toeplitz matrix 1. Introduction Multipleinput multipleoutput (MIMO) system is widely used in wireless communication to improve the perform ance. The spectral efficiency of MIMO channel is much higher than that over the conventional signal antenna channel. The research of MIMO includes two different perspectives: the first one concerns performance analysis in terms of error performance of practical systems, the second one concerns the study of channel capacity. The design of communication schemes was mainly considered in the former perspectives with the aid of theo retical analysis and simulation [1][2], and [3]. For the latter, important parameters such as diversity gain [4], multiplexing gain [5][6] (referred to degree of freedom in other literatures), and power offset [79], and [10] were emphatically analyzed in the high signaltonoise ratio (SNR) region. Furthermore, diversity and multiplexing tradeoff has been proposed in [11][12], and [13], that is to say diversity and multiplexing tradeoff can be obtained for a given multiple antenna channel. It is worth pointing out in [11] that the diversity and multiplexing tradeoff is essentially the tradeoff between the error probability and the data rate of a given system. The multiplexing gain is not sufficient to accurately characterize the property of MIMO capacity. A more ac curate representation of high SNR behavior in SNRca pacity curve is provided by an affine approximation to capacity, which includes both the multiplexing gain (i.e. slope) and power offset (i.e. zeroorder term) [7]. High SNR power offset has been analyzed in [8] over multiple antenna Ricean channels. It was shown in [8] that the im pact of the Ricean factor at high SNR region could be conveniently quantified through the corresponding power offset. In [9], high SNR power offset in multiple antenna communication was derived in detail. Achievable through put was compared between the optimal strategy of dirty paper coding and suboptimal linear precoding techniques (zeroforcing and block diagonalization) in [10] on appli cation of power offset. Hence, power offset is an impor tant parameter in multiple antenna communication. In many practical environments, signal correlation among the antenna array exists due to the scattering. Fad ing correlation and its effect on the capacity of multiele ment antenna system has been studied in [14]. Hence, analysis of spatially correlated MIMO channels has been another topic in the past few years, which necessitates the model of correlated channel. A general spacetime corre lation model for MIMO systems in mobile fading channel has been presented in [15]. The model in [15] was flexible and mathematically tractable. In [16], the correlated model of [15] has been used to investigate the capacity of spatially correlated MIMO Rayleighfading channels. Recently, application of different antenna arrays in MIMO system has also been exploited. With the consid eration of physical constraints imposed by maximum size of the antenna array, uniform linear array of fixed length has been used in [17] to analyze the asymptotic capacity of MIMO systems. The uniform linear array of fixed length is also used in this paper to analyze the power offset of MIMO system over spatially correlated channel. On application of curve fitting, the determinant of tridiagonal toeplitz matrix is mathematically tractable. Over spatially correlated chan
26 APPROXIMATE ANALYSIS OF POWER OFFSET OVER SPATIALLY CORRELATED MIMO CHANNELS nel, the main result of this paper is the derived expression of local extreme points of power offset, using the uni form linear array of fixed length. To the best of our knowledge, the main result has not been presented in other literatures. The rest parts are organized as follows. In Section 2, the basic definitions of multiplexing gain and power offset are presented. System model and correlation model are given in Section 3. Approximate analysis of power offset by fitting determinant curve of tridiagonal toeplitz matrix is put forward in Section 4. Section 5 shows the simulation results. Finally, a brief conclusion is given in Section 6. Notation: In the following context, matrices and vectors are denoted by boldface upper case symbols and boldface lower case symbols, respectively. The transpose and Hermitian transpose are denoted by and T , re spectively. The expected value is represented by E. is used for the Euclidian norm. 2. Basic Definition In the high SNR region, the capacity of singleuser MIMO system of coherent reception is given by [18][19] 2 ()min(,)log (1 TR CSNRn nSNRo) (1) where denote the number of transmit and receive antenna elements , respectively. is the signal to noise ratio. The MIMO capacity in high SNR region is a linear function of , i.e. the stationary slope in SNRcapacity curve. In other words, any increase in is immaterial, i.e. without any impact to the slope. The slope is the socalled maximum multiplexing gain (degree of freedom). , TR nn , ) TR nn SNR min( ,) TR nn max ( If is fixed to a given value (generally speaking, is less than six, expressed as T n T n T n in the following context), and T nn, the multiplexing gain equals to a fixed value without consideration of correlation of chan nel response and the number of receive antenna elements. So power offset is introduced to compare the impact of channel property with the same value of multiplexing gain, and (1) is replaced by [9] 2 ()(log() )(1 (1) 3 dB C SNRSSNRLo SNR SLo dB ) (2) where is multiplexing gain and is power offset in 3dB units. When ，the multiplexing gain and power offset can be computed by SL SNR 2 () lim log () SNR CSNR SSNR (3) 2 () limlog () SNR CSNR LSNR S (4) In the high SNR region, SNRcapacity curve is ap proximately determined by multiplexing gain and power offset. Most channels, having the same multiplexing gain, may have very different capacities because of various values of the power offset. Hence, power offset is an im portant parameter to describe the capacity behavior in high SNR region. 3. System Model and Correlation Mode The system model and channel correlation model are de fined in this section. For MIMO system, the general baseband model is given by yHxn (5) where is the channel response matrix, and is the input complex signal vector whose spatial covariance matrix normalized by the energy per dimension can be expressed as H 12 [ T T n xx xx] 2 () 1[] H T E E n xx Φ x (6) while and are the output vector and additive white Gaussian noise vector, respectively. Because of normalization and assumption of isotropic input, 12 [] R T n yy yy ()tr 12 [ R T n nn n n] T n Φ and , where is T n ΦIn I nn identity matrix. Kronecker model [16] is used to describe the correla tion of channel response. On the assumption of rich scat tering environments and having no line of sight, the cor related channel response matrix is denoted by TR ΣΣΣ (7) where is Kronecker product, and T Σ Σ are cor related matrix of transmit and receive antenna, respec tively. So the channel response matrix is given by H 1 2 RT HΣHΣ1 2 (8) where is channel response matrix whose elements are H independent and identical distribution random variables. Without loss of generality, we assume that the distance between transmit antenna elements is large enough to neglect the correlation at the transmit node. Consequently, without considering the antenna correlation at the transmit node, the correlation matrix is identity matrix, and the receive antenna correlation model T Σ Σ is given by [15][16]. Copyright © 2009 SciRes CN
APPROXIMATE ANALYSIS OF POWER OFFSET OVER SPATIALLY CORRELATED MIMO CHANNELS 27 Copyright © 2009 SciRes CN 222 0 0 44sin() (, )() ij ij R dj d ij I Σ (9) 02, 1 R R R L J n ap æö ÷ ç÷ =£ ç÷ ç÷ ç èø 10 L £ (12) where L is the fixed length of receive antenna array normalized by . In this paper, L is assumed to be in the range from 1 to 10. where [0, ) [,) is the socalled AOA(angle of arrival), is average value of AOA, ij ds normalized distance between antenna array elements, that is to say i ij ij dd he factual distance between i and j antenna array element, , ij d In the Figure 1 and 2, power offset approximation based on tridiagonal, fivediagonal, and sevendiagonal toeplitz matrix is obtained versus the number of receive antenna with various value of L. Furthermore, power offset over independent channel is also provided as lower bound for numerical comparison. It can be seen from Figure 1 and 2 that tridiagonal toeplitz matrix can be used to approximate the correlation model at the receive node. is t s wave length, i() n is norder modified Bessel function. It is obvious that is real symmetric toeplitz matrix. (, ) RijΣ On the assumption of isotropic scattering, the correla tion matrix at the receive node is simplified to 0 (, )(2) ij ij JdΣ (10) 4. Approximate Analysis of Power Offset where is n order Bessel function, and can be simplified to () n J(,) RijΣ tridiagonal toeplitz matrix through the simulation results illustrated by Figure 1 and Figure 2. Depending on the discussion in Section 3, tridiagonal toeplitz matrix can be used to approximate the correla tion model at the receive node. In this section, the deter minant curve of tridiagonal toeplitz matrix is fitted by an elementary function, which simplifies the approximate analysis of power offset. 1 R R nn Σ Expression of power offset for MIMO system over corre lated channel is derived in [9], and the expression is given by (11) 2 2 1 log(ln det()) ln 2 1log det() H TR T T T Ln E n n WΛW ΛP (13) where 2468 10 12 14 16 10 8 6 4 2 0 2 4 number of Nr power offset/dB power offset LR=5 Nt=2 det erminant tridiagonal app. fivediagonal app. sevendiagonal app. no correlation Figure 1. Approximate analysis of power offset based on 3, 5, and 7 diagonal toeplitz matrix with , 5 R L 2 T n
28 APPROXIMATE ANALYSIS OF POWER OFFSET OVER SPATIALLY CORRELATED MIMO CHANNELS 24 6 81012 14 16 1820 10 8 6 4 2 0 2 4 number of Nr power offset/dB power offset LR=6 Nt=2 determinant tridiagonal app. fivediagonal app. sevendiagonal app. no correlation Figure 2. Approximate analysis of power offset based on 3, 5, and 7 diagonal toeplitz matrix with , 6 R L 2 T n where and T Λ Λ are diagonal matrix whose ele ments are eigenvalues of and T Σ Σ Φ ) T P , respectively. is also diagonal matrix whose elements are eigenvalues of normalized spatial covariance matrix , so is iden tity matrix and is constant zero. P P 1/n2 g de T lo t(Λ Λcan be derived from (11). It is obvious that Λ is full rank matrix, and then (13) is simplified to 2 log(() ()) TRR Lnfngn (14) where () n and () n can be described by (lndet()) () ln 2 H R T E fn n WW (15) ln det() () ln 2 R T gn n Λ (16) Because is wishart matrix and nonsingular with probability 1, H WW () n is reduced to [9] 1 0 1 ()(log det()) ln 2 1() ln 2 T H Re T n R l T fn E n nl n WW (17) where () is digamma function and the definition of digamma function is 1 1 1 () n l nl (18) where is EulerMascheroni constant, and 1 1 limlog 0.5772 n e nl n l (19) When it comes to function () n, the derivation of () n can be expressed as 11 () lndet() lndet() ln 2ln 2 RR TT gn nn ΛΣ (20) It can be found from (20) that () n is only deter mined by the determinant of Σ, and the determinant of norder trid iagonal toeplitz matrix Σ can be com puted by [20]. 11 22 12 2 2 114 114 214 14 0 det 1 2 14 0 RR R R nn n R R n n Σ (21) With the aid of numerical computation and mathemati cal analysis, the curve of the determinant of tridiagonal toeplitz matrix can be fit by some tractable elementary Copyright © 2009 SciRes CN
APPROXIMATE ANALYSIS OF POWER OFFSET OVER SPATIALLY CORRELATED MIMO CHANNELS 29 function. When in (21), det 2 140 Σcan be ap proximated by the fitted curve as follow: 22 0 ()1515 (2) 1 R R R L hnJ n (22) The approximation between the fitted curves and de terminant curves of tridiagonal toeplitz matrix are illus trated in Figure 3 and Figure 4. From Figure 3 and 4, we can find that the fitting curve can exactly approach the extreme points of the determinant of tridiagonal toeplitz matrix. However, the expression is much simpler. In other words, the further derivation using the fitted curve is tractable using wellknown mathematical software such as WOLFRAM MATHEMATICA. In the following context, power offset is analyzed in detail. The approximate analysis of power offset is based on the piecewise function of n. The condition in (22) is necessary due to , so the number of receiver antenna 2 15 0 det 0 R Σ n can not be large than . Moreover, it shows from the conclusion in [21] that the effect of correlation can be neglected when the normalized length of receive antenna array 41 R L L 1 RR nL is larger than one wave length, that is to say . Hence, the considered range of the number of receive antenna is . 1 RR Ln 4 R L 1 With the discussion above, when , the normalized length of receive antenna array 1 RR nL L is large than one wave length and the effect of correlation can be neglected, that is to say and the power offset is expressed as ()0 R gn 1 2 0 1 log( ) ln 2 T n T l T Ln n n R l 1 (23) When it comes to the range of , the normalized length of receive antenna array 14 RRR LnL L ()0 R gn is smaller than one wave length and the effect of correlation can not be neglected. In other words, . However, for a given value of T n, the firstorder derivative (() n () is assumed to be continuous) of n is obtained by ' 1 0 1 1 0 () 1() ln 2 1() ln 2 T T n R R l RT n R l T df nnl dn n nl n 0 (24) where 1() is trigamma function. Hence, it shown from (24) that () n approximately equal to the con stant (approximation accuracy is related to the value of C T n). The power offset is expressed as 2 log( ) TR Lngn C (25) 2 4 6 81012 14 16 18 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 number of Nr determinant curve fitted of determinant of tridiagonal toeplitz matrix LR=5 determinant curve fitted Figure 3. Approximation of determinant of tridiagonal toeplitz matrix by curve fitting and 5 R L C opyright © 2009 SciRes CN
30 APPROXIMATE ANALYSIS OF POWER OFFSET OVER SPATIALLY CORRELATED MIMO CHANNELS 051015 20 25 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1 number of Nr determinant curve fitted of determinant of tridiagonal toeplitz matrix LR=6 determinan t curve fitted Figure 4. Approximation of determinant of tridiagonal toeplitz matrix by curve fitting and 6 R L Without loss of generality, the constant can be neglected in the approximate computation. Hence, the power offset in (25) is only determined by C () n 2 . From (22), the curve of the determinant of tridiagonal toeplitz matrix can be fitted by ()1 R hn 5 , where 0 2 ( 1) R L Jn . The first order derivative (() hn is as sumed to be continuous) of () hn can be computed by 01 2 22 4()( ()1 1 (1) RR R RR RR LL LJ J dh nnn dn n ) R (26) where , that is to say 14 RRR LnL And then, the secondorder derivative of () hn can be computed by 01 2 23 22 2 1 4 22 00 2 4 22 8()() ()1 1 (1) 2 8() 1 (1) 22 2 4()()( 11 (1) RR R RRR RR R R R R RR R RR R R LL LJ J dhnn n dn n L LJn n LL L LJ JJ nn n n ) 1 R 3 1 22 21 R L (27) Using (27), the secondorder derivative in points ,1,2, i Pi is obtained as follow: R n . In the range above, function 0 2 () 1 R Jn L in (26) has two zero points, i.e. 1 22.4048 1 R L 2 122 22.4048 1 ( )4.51 :0 R R L RR n dhn Pdn L 2 (28) R Pn and 2 25.5201 1 R R L Pn , and function 1 2 () 1 R L Jn only has one zero point, i.e. 3 23.8317 1 R 2 222 25.5201 1 () 53.75 :0 R R L RR n dhn Pdn L 2 (29) R L Pn . Consequently, () hn has three extreme points in the range of 2 21 R R L n 2 322 23.8317 1 () 17.48 :0 R R L RR n dhn Pdn L 2 (30) 2 3 , de scribed by . ,1 i Pi ,2, With the discussion above, it is obvious that point and are local maximum points of 1 P 2 P() hn and point Copyright © 2009 SciRes CN
APPROXIMATE ANALYSIS OF POWER OFFSET OVER SPATIALLY CORRELATED MIMO CHANNELS 31 3 P is local minimum point. Furthermore, it is obvious from (20) that the extreme points of () hn is same as ones of () n. Consequently, in the range of 1 R L t , the extreme points of power offset in (25) can be approximately computed by 41 R L arg mi arg ma R R n n P R n R n n x ,for local minit ,for local maxiin L L mal poin mum po (31) where is defined as follows: 22 1,1 ,1,2,3= RR ii LL i PP pp ìü êúéù ïï ïï êúêú + + íý êúêú ïï ïï ëûêú îþ P= (32) where êú ëû and éù êú denote the maximum integer smaller than and minimum integer larger than x, re spectively. is the extreme points com puted above. (31) can be used to approximately compute the number of receive antenna, in order to achieve the optimal power offset. ,1,2,3 iPi 5. Simulation Results In the Figure 1 and 2, with the assumption of 2 T n , power offset is approximately analyzed based on tridi agonal, fivediagonal, and sevendiagonal toeplitz versus the number of receive antenna, with in Figure 1 and in Figure 2. Furthermore, power offset in independent channel is also provided as lower bound for numerical comparison. The approximation trend can be 5 R L 6 R L found from Figure 1 and 2. The tridiagonal toeplitz ma trix can be assumably used to characterize the property of power offset over spatially correlated channel. Hence, tridiagonal toeplitz matrix can be used to approximate the correlation model at the receive node. The approximation of fitting the curve of tridiagonal toeplitz matrix determinant is illustrated in Figure 3 and Figure 4 to compare the difference between fitted curves and determinant curves, with in Figure 3 and 5 R L 6 R L in Figure 4. From Figure 3 and 4, we can find that the fitting curve can exactly approach the extreme points of the determinant of tridiagonal toeplitz matrix. However, the expression is mathematically tractable. With the same assumption of 2 T n, approximation of power offset is shown in Figure 5 and 6, with 5 R L in Figure 5 and 6 R L in Figure 6. From Figure 5 and 6, we can find the position of two local minimal points and one local maximum point. With the condition of 5 R L and T n2 , the local extreme points computed by (31) is 9 R n for local maximum point and 7, R n15 6 for local minimal points. With the condition of R L and 2 T n , the local extreme points com puted by (31) is 11 R n for local maximum point and 8, R n16 for local minimal points. The computation results based on (31) coincide with the illustration results in Figure 5 and 6. Hence, all the approximation in this paper is reasonable, including approximation of correla tion model and approximation of determinant of tridi agonal toeplitz matrix. 24 6 81012 1416 18 20 10 8 6 4 2 0 2 4 number of Nr power offset/dB er offsetLR=5 Nt=2 pow determ inant curve fitted no correlation Figure 5. Power offset comparation between accurate and approximate determinant of toeplitz matrix with , 5 R L 2 T n C opyright © 2009 SciRes CN
32 APPROXIMATE ANALYSIS OF POWER OFFSET OVER SPATIALLY CORRELATED MIMO CHANNELS 05 10 15 20 25 12 10 8 6 4 2 0 2 4 number of Nr power offset/dB power offsetLR=6 Nt=2 det erminant curve fitted no correlation Figure 6. Power offset comparation between accurate and approximate determinant of toeplitz matrix with , 6 R L 2 T n Similar results are presented in Figure 7 and 8, with the assumption of and in Figure 7 while in Figure 8. With the condition of 4 T n5 R L 8 R L5 R L and 4 T 9 n R , the local extreme points computed by (31) is for local maximum point and for local minimal points. With the condition of n7, R R L 15 8 n and , the local extreme points computed by (31) is for local maximum point and for local minimal points. The computation results based on (31) coincide with the illustration results in Figure 7 and 8. T n n 4 14 R10,22 R n Consequently, it is presented in Figure 5 and 7 that the effect caused by the number T n can be neglected. In other words, for small value of T n and in the range of number of receive antenna , the assumption of 14 RRR LnL 1 () R nC is tenable with neglectable loss of precision. The extreme points of power offset is only determined by () n. 468 1012 141618 12 10 8 6 4 2 0 2 4 6 8 Nr power offset/dB power offset LR=5 Nt=4 determ i nant curve fitted no correlation Figure 7. Power offset comparation between accurate and approximate determinant of toeplitz matrix with , 5 R L 4 T n Copyright © 2009 SciRes CN
APPROXIMATE ANALYSIS OF POWER OFFSET OVER SPATIALLY CORRELATED MIMO CHANNELS 33 510 1520 25 30 12 10 8 6 4 2 0 2 4 6 8 Nr power offset/dB power offset LR=8 Nt=4 determin ant curve fitted no correlation Figure 8. Power offset comparation between accurate and approximate determinant of toeplitz matrix with , 8 R L 4 T n 6. Conclusions Approximate analysis of power offset over spatially cor related channel is proposed to optimize the number of receive antenna with antenna array of fixed length. An tenna correlation matrix is approximated by tridiagonal toeplitz matrix. The determinant of tridiagonal toeplitz matrix is reduced to simple style by curve fitting with neglectable loss of precision. Based on the approxima tion above, the expression of local extreme points of power offset is derived. Simulation results shows that the derived expression is accurate in the local extreme points of power offset, that is to say, all the approximation in this paper is reasonable. REFERENCES [1] TAROKH V, JAFARKHANI H, CALDERBANK A R. Spacetime block code from orthogonal designs. IEEE Transactions on Information Theory, Jul. 1999, 45(5): 14561467. [2] BOUBAKER N, LETAIEF K B, MURCH R D. Performance of BLAST over frequencyselective wireless communication channles. IEEE Transactions on Communication, Feb. 2002, 50(2): 196199. [3] LOZANO A, PAPADIAS C. Layered spacetime receivers for frequencyselective wireless channels. IEEE Transactions on Communication, Jan. 2002, 50(1): 6573. [4] BAKIR L, FROLIK J. Diversity gains in tworay fading channels. IEEE Transactions on Wireless Communications, Feb. 2009, 8(2): 968977. [5] POON A, BRODERSEN R, TSE D. N. C. Degrees of freedom in multipleantenna channels: A signal space approach. IEEE Transactions on Information Theory, Feb. 2006, 51(2): 523536. [6] JAFAR S A, SHAMAI S. Degrees of freedom region of the MIMO X channel. IEEE Transactions on Information Theory, Jan. 2008, 54(1): 151170. [7] JINDAL N. High SNR analysis of MIMO broadcast channels. ISIT’05, 2005, 23102314. [8] TULINO A M, LOZANO A, VERDU S. HighSNR power offset in multiantenna Ricean channels. ICC’05, 2005, 683687. [9] LOZANO A, TULINO A M, VERDU S. HighSNR power offset in multiantenna communication. IEEE Transactions on Information Theory, Dec. 2005, 51(12): 41344151. [10] LEE J, JINDAL N. High SNR analysis for MIMO broadcast channels: Dirty paper coding versus linear precoding. IEEE Transactions on Information Theory, Dec. 2007, 53(12): 47874792. [11] ZHENG L Z, TSE D N C. Diversity and multiplexing: A fundamental tradeoff in multipleantenna channels. IEEE Transactions on Information Theory, May 2003, 49(5): 10731096. [12] TSE D N C, VISWANATH P, ZHENG L Z. Diversity multiplexing tradeoff in multipleaccess channels. IEEE Transactions on Information Theory, Sep. 2004, 50(9): 18591874. [13] WAGNER J, LIANG Y C, ZHANG R. On the balance of multiuser diversity and spatial multiplexing gain in random beamforming. IEEE Transactions on Wireless Communication, Jul. 2008, 7(7): 25122525. [14] SHIU D, FOSCHINI G J, GANS M J, ET AL. Fading correlation and its effect on the capacity of multielement antenna system. IEEE Transactions on Communication, Mar. 2000, 48(3): 502513. C opyright © 2009 SciRes CN
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