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Generic Procedure for Tightly Bounding the Capacity of MIMO Correlated Rician Fading Channels X. W. Cui, Q. T. Zhang, Senior Member, IEEE, and Z. M. Feng
Abstract—No systematic procedure for tightly bounding the average capacity of multiple-input–multiple-output (MIMO) correlated Rician fading channels is available in the literature. In addition to the involvement of a highly nonlinear log-determinant operator in the conditional capacity expression, the difficulty arises from the complicated noncentral Wishart distribution of channel sample matrix. In this paper, we tackle the problem with arbitrary antenna correlation existing either at the transmitter or at the receiver, but allowing for the numbers of the transmit and receive antennas to be arbitrary. By introducing an exact determinant expansion and by finding an explicit expression for the general moment of the determinant of the channel sample matrix, we obtain a general upper bound for the average channel capacity. To obtain a general lower bound, we construct and prove a multivariate convex function with each of its variables being the log-determinant function of a complex noncentral Wishartdistributed matrix. We further show that the general bounds so obtained can be simplified to explicit expressions for Rician fading channels with arbitrary semicorrelation and a mean matrix of rank one. The new results are simple, easy to be used, and superior in tightness as evidenced by intensive numerical examples. Index Terms—Channel capacity, correlated Rician fading multiple-input–multiple-output (MIMO) channels, lower bound, upper bound.
I. INTRODUCTION
S
TIMULATED by their extraordinary channel gains, multiple-input–multiple-output (MIMO) wireless communication systems have rapidly become a focus of many researchers. According to information theory, channel capacity is defined as the maximum mutual information over all possible input distributions. Under the assumption that all channels that link any pair of transmit and receive antennas are subject to flat Rayleigh fading with independent identical distribution (i.i.d.), one can achieve the channel capacity by simply transmitting Gaussian signal vectors of i.i.d. entries [1], [2]. In a correlated MIMO Rayleigh fading environment, the attainment of the channel capacity requires a matrix transformation on transmitted vectors to match the channel statistical characteristics
Paper approved by P. Y. Kam, the Editor for Modulation and Detection for Wireless Systems of the IEEE Communications Society. Manuscript received October 3, 2003; revised June 11, 2004 and November 4, 2004. This work was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China, under Project CityU 1248/02E. This paper was presented in part at the ICC’2004, Paris, France, June 2004. X. W. Cui was with City University of Hong Kong, Kowloon, Hong Kong. He is now with the Department of Electronic Engineering, Tsinghua University, Beijing 100084, China (e-mail:
[email protected]). Q. T. Zhang is with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail:
[email protected]). Z. M. Feng is with the Department of Electronic Engineering, Tsinghua University, Beijing 100084, China (e-mail:
[email protected]). Digital Object Identifier 10.1109/TCOMM.2005.847128
[3]. In many practical situations, however, channel statistical characteristics can change from time to time or from one location to another due to the mobility of a vehicle, rendering itself difficult or even impossible to be available at the transmitter. As such, most of the design strategies for MIMO wireless systems, including V-BLAST [4], are based on such an i.i.d. Rayleigh fading assumption. A major concern is, therefore, the robustness of the system so designed in exploiting the channel capacity when the system operates in a mismatched fading environment. The channel capacity expression obtained by assuming i.i.d. Rayleigh fading physically represents the mutual information achievable at the receiver, independent of operational environments. We will keep, for convenience, the term channel capacity for this mutual information, in the same way as other authors [3], [10]. In this paper, we study the capacity variation of a MIMO wireless system, which is designed for i.i.d. Rayleigh fading channels but operates in a general Rician fading environment with arbitrary spatial correlation among the transmit or receive antennas. Little analytical work has been done in the literature for the channel capacity of MIMO correlated Rician fading channels. The case of i.i.d. Rician channels is investigated in [3] by averaging the conditional channel capacity over the joint distribution of the sample eigenvalues of a noncentral Wishart distributed matrix; the result takes the form of a very complicated multifold integral. A semi-analytical method for the analysis of i.i.d. MIMO Rician channels is proposed in [11] by relying on the Gaussian approximation where the theoretic mean and variance are approximately estimated from computer simulation. The authors of [10] directly work on the direct path and diffuse components of the channel sample matrix and drop the cross terms of the two components in the conditional capacity expression, ending up with an upper bound for the average capacity of correlated Rician channels. The result obtained may not be really a capacity upper bound, since neglecting the cross terms is justified only when the number of transmit antennas approaches infinity. Some other papers [12], [13] address the same issue, but mainly by resorting to computer simulation. Recently, the mean and variance of the capacity for independent but not identically distributed MIMO Rician channels are obtained in the form of a single integral [15]. Indeed, it is difficult to determine the exact average capacity of correlated Rician MIMO channels or even its tight bounds for two reasons. First, we need to express the determinant of form in terms of channel sample matrix as accurately as possible. A commonly used technique relies on the Mincowski’s inequality [14] which, however, can lead to a very poor approximation. This occurs when the sample covariance
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CUI et al.: GENERIC PROCEDURE FOR TIGHTLY BOUNDING THE CAPACITY OF MIMO CORRELATED RICIAN FADING CHANNELS
matrix is not of full rank or close to degeneration, due to either an insufficient number of the transmit antennas in use or high correlation among antennas. A simpler approach is to difunction to obtain rectly exploit the concavity of the an upper bound for the average capacity [16], [17]. Second, jointly correlated Rician channels lead to a sample channel matrix of the complex noncentral Wishart distribution. The latter is a multidimensional extension of noncentral chi-square distribution and has only very limited results available even in mathematical literature. Let us outline the idea used in this paper. We introduce an exact determinant expansion to represent the conditional capacity as a linear sum of determinants of noncentral Wishart distributed submatrices, thereby enabling us to construct multivariate convex and concave functions. We then determine, by using some mathematical skills, the expected values of the determinant and log-determinant function of such submatrices whereby very tight upper and lower bounds are obtained. The rest of the paper is organized as follows. Section II formulates the problem, followed by the derivation of the determinant expansion and its properties in Section III. The detailed derivation of the upper and lower bound are presented in Sections IV and V, respectively. Numerical results are presented for illustrating the tightness of the new bounds in Section VI, followed by concluding remarks in Section VII. II. FORMULATION receive and Consider a wireless MIMO system with transmit antennas having the total transmit signal-to-noise ratio denote the channel matrix. The (SNR) of . Let received -by-1 vector is related to the -by-1 transmitted symbol vector by
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For simplicity, we only consider the correlation at one end of the transceiver. The first expression is convenient if correlation among receive antennas is considered, and the second is preferred if correlation exists among transmit antennas. The use of semicorrelation can be justified for several reasons. First, it allows for a relatively easy treatment of the capacity problem in the framework of multivariate statistical theory. Second, recent measurements were conducted in downtown Helsinki, Finland [7] demonstrating the validity of the semi-correlation model for certain urban environments. In fact, such a model has been used previously in, for example, [8]. Since the treatment of the two cases in (3) is similar, we consider correlation at receiver end, are mutually independent and assume that and each follows the complex Gaussian distribution with mean and covariance matrix . Symbolically, we can write (4) In this paper, we allow to have distinct mean vectors but the same covariance structure. The mean channel matrix is then given by Note that matrix so defined is quite general in the sense that it is not necessarily of full rank. We further let (5) denote a subset of such that . If we collect the entries from with the same indices and order as specified by the subset , the resulting -by-1 vector . We denote its mean vector and covariance is denoted as matrix by and , respectively. Note that is a and is a submatrix of . Clearly, we subvector of have (6)
(1) We further define the where is normalized to have unit energy, and denotes the additive white Gaussian noise (AWGN) vector at the receive antennas with unit variance. Clearly, the th row of represents the gain vector resulting from linking the th receive antenna to transmitters and will simply be denoted by for the subsequent use. Similarly, the th column of , denoted here by , signifies the gain vector of paths that connect the th transreceivers. The conditional channel capacity can be mitter to expressed as [2] (2) where represents the average SNR at each transmit antenna, and the sample matrix can be defined by two equivalent expressions, as shown by
(3)
mean matrix,
, as (7)
To determine the average capacity, we need to average (2) over the joint probability density function (pdf) of . The is usually difficult to handle. noncentral distribution of The difficulty is further gravitated by the complicated nonoperator involved in the conditional capacity. linear Throughout the paper, we will use and interchangeto ably to denote the determinant operator and use denote the rank of a matrix. III. DETERMINANT EXPANSION AND PROPERTIES Let us begin with the expansion of in terms of the fine structure of . This can be done by invoking the rule by Lovitt [24], [25], as stated below. denote the th entry of an maLemma 1: Let can be trix . If is a scalar, then determinant expressed as
where signifies the identity matrix with dimensions compatand superscript denotes Hermitian transposition. ible to Authorized licensed use limited to: CityU. Downloaded on January 7, 2009 at 01:21 from IEEE Xplore. Restrictions apply.
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with
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 5, MAY 2005
denoting (9)
where
in the next section. To begin with, let us explicitly represent the pdf of as [5]
is defined by is used to denote the trace of a matrix and where defined as .. .
.. .
.. .
(17) is matrix (18)
(10) . and , defining in a Applying Lemma 1 to , and using the index subset notation way similar to defined the previous section, we can rewrite
is the complex multivariate gamma function given by (19) where
(11) Since and some of its higher order submatrices can be degendenote the maximum value of for which is erate, let of full rank. It is easy to find that (12)
is
the
gamma function, and is the hypergeometric function with matrix argument . Such a hypergeometric function is usually defined in terms of zonal polynomials [5]; can be see (A1) in Appendix A for details. The pdf of represented in a manner similar to (17). Then we use the definition of the th moment and the pdf expression of to write
Thus, we can re-express (11) as (13) All submatrices in this expression are of full rank and can be represented as (14) by using the subvector notations defined in the previous secfollows the noncentral tion. It follows from (6) and [5] that Wishart distribution, as shown by where the subscript signifies the dimension of the sample masignifies the sample size. trix and IV. UPPER BOUND
(20) The integral in (20) is taken over a set of Hermitian matrices. This integral involves a matrix argument and its calculation is quite involved. We therefore only outline the idea and steps for its simplification and leave the details in Appendix A. The basic idea is to represent the hypergeometric function as an expansion of zonal polynomials with a matrix argument, thereby enabling us to determine the integration term by term. The integral for each term can be determined by taking an appropriate orthogonal transform and employing the reproductive property of zonal polynomials. The final result is given by
The use of (13) and the Jensen’s inequality allows us to bound the average channel capacity, as shown by
(21)
(15)
For a detailed proof, the reader is referred to Appendix A. Having obtained (21), we can rewrite the upper bound (16) yielding (22), shown at the bottom of the next page. The general upper bound given in (22) can be further simpliis of rank one. This condified if the mean channel matrix tion is certainly satisfied when all channel vectors have the same mean value. It is also satisfied in some other practical scenarios as discussed in [9], [10], and [13] where the LOS component of the channel matrix approaches the scaled all-ones matrix as long as the transmitter and receiver are well separated relative to both transmit and receive antenna spacing. We first simplify the ratio of two complex Gamma functions, ending up with
where the upper bound
is given by (16)
As expected, the upper bound depends on the SNR and the fine structures of the antenna covariance matrix. The key issue to simplifying (16) is to determine the expected value on the righthand side; namely, to determine the expected value of the determinant of a complex noncentral Wishart-distributed matrix. For the problem at hand, we need to determine the expectation where and is of the form is what we need in (16). We a real number. The case of consider here the general case since the result will also be used
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CUI et al.: GENERIC PROCEDURE FOR TIGHTLY BOUNDING THE CAPACITY OF MIMO CORRELATED RICIAN FADING CHANNELS
We next combine (18) and (7) to write (24) Given mean matrix of rank one, the rank of its submatrix is one or zero [21] and the same is true of . When has rank one, its trace can be represented by its unique nonzero , so that we obtain eigenvalue, denoted here by
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multivariate function and, thus, can be considered as a natural generalization of its counterpart for a univariate function, which uses a single second-order derivative to determine the convexity. The result can be stated as follows. of multiple Lemma 2: Define a vector , the real function variables. Then, for arbitrary (30)
(25) The rank-one condition also allows us to simplify the hypergeometric function in (22) to give (26) Expressions (25) and (26) are derived for the case of rank one. It is easy to examine that they are still valid for the case of rank zero, for which . See Appendix B for a detailed derivation. By inserting (23), (25), and (26) into (22), it follows that
. is convex on A detailed proof has been given in [22] and is briefly outlined in Appendix C for completeness. The Jensen’s inequality is a special application of the properties of a convex or concave function. The latter leads to the former as long as we use the probability as weighting coefficients. In our case, we can write (31) The crucial issue is to determine the expectation of the function of a complex noncentral Wishart-distributed matrix, which involves a nonlinear log function. A useful skill to avoid handling the log function is to use the equality that (32)
(27) This is the upper bound for MIMO Rician correlated channels of rank one. Under the same condition, it with mean matrix also follows that
which, in turn, is based on the fact that and that both differentiation and expectation are linear operators. Denote . By combining (32), (21), and (24), it yields
(28) . It will be used subsewhere quently for the lower bound derivation. whereby the general lower bound can be represented as
V. LOWER BOUND To derive a lower bound for
(33)
, let us rewrite (13) as
(34) (29)
If we consider each exponent as a variable, then is a multiis a univariate convex variate function. Recall that function which, along with the Jensen’s inequality, is commonly used to derive lower bounds for channel capacity in the literature. The crucial issue now is to prove that the function shown on the right of (29) is a multivariate convex function. A criterion for deciding the convexity of a multivariate function is that its Hessian matrix is nonnegative definite [28]. The Hessian matrix is constructed from the second-order partial derivatives of the
The general lower bound given in (34) can be simplified if the mean channel matrix has rank one. To this end, let us consider where is of rank one. Denote . From (28) and (32), it follows that
(35) Three factors on the right are function of . We determine their derivatives respectively. The first factor involves complex
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Gamma function. Directly working on itself is somewhat difficult. Therefore, we work on its logarithm instead, producing
(27) and (41) with the exact capacity derived in [15] for independent Rician channels. The exact result, as shown in [15, eqs. (13–14)], takes the form of integral involving confluent hypergeometric functions in its integrand. Our result, on the contrary, allows for branch correlation and yet only contains simple elementary functions. VI. NUMERICAL RESULTS
(36) where
is Digamma function [20, p. 258] defined as , and, for integer , we can write [27] (37)
and is the Euler’s constant. The treatment of the second factor is trivial, resulting in . The treatment of the third factor is also straightforward, ending up with
(38) We insert (36) and (38) into (35) and simplify, yielding (39) with
defined by (40)
with the lower The use of (39) allows us to obtain bound given by (41), shown at the bottom of the page, which, along with (27), provides simple, yet tight, upper and lower bounds for the average channel capacity of correlated Rician has rank one. At this point, channels under the condition that it is interesting to compare our upper and lower bounds given in
As an illustration, let us consider a MIMO wireless system operating on correlated Rician channels such that are mutually independent, and the column vectors of each follows the -dimensional Gaussian distribution . Here, the Rician is defined as the power ratio of the line-of-sight factor component to a multipath component. Vector with denoting the all-one column vector, and is the correlation matrix of each column vector of accounting for the spatial correlation among receive antennas. The mean vector and covariance matrix has been so scaled that each entry of has a total power of unity. We consider two correlation models in our numerical evaluation. The first model is specified by th entry of the exponential-type correlation function: the correlation matrix, where denotes the absolute value of correlation coefficient between two adjacent antennas, and stands for the antenna separation normalized by the wavelength. The variation of average channel capacity with channel correlation , . The upper is shown in Fig. 1, for which and lower bounds are labeled with dotted and dash–dotted lines, respectively. The results for true capacity obtained by a Monte Carlo approach are also included for comparison. In this section, the true capacity for each SNR point is obtained through 20 000 independent computer runs. It is observed that the channel capacity decreases as the channel correlation increases. It is also observed that both bounds in general, and the lower bound in particular, demonstrate superior tightness in bounding the true average channel capacity. The tightness of the upper bound improves with increased . We are also affects the average interested in the way the Rician factor , capacity; the results are depicted in Fig. 2 where , and three sets of curves are shown for 1, 5, and 10, respectively. Again, the new bounds demonstrate excellent tightness in bounding the true capacity. Interestingly enough, an increase in the Rician factor results in a reduction in the average channel capacity. This observation, however, is not difficult to understand. Recall that multipath fading contains tremendous potential channel capacity and is thus the major
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CUI et al.: GENERIC PROCEDURE FOR TIGHTLY BOUNDING THE CAPACITY OF MIMO CORRELATED RICIAN FADING CHANNELS
Fig. 1. Influence of the channel correlation on the average capacity: exponential model.
Fig. 2.
Variation of average capacity with Rician factor: exponential model.
motivation to the use of a wireless MIMO system. For a given transmission power, the larger the Rician factor, the smaller the multipath component and, thus, the lower the channel capacity. to maintain In the previous numerical results, we chose the full rank of . This restriction is unnecessary. Now we set and , and the results are graphed in Fig. 3 for which and . For comparison, the result for is also included. Clearly, our new bounds are applicable equally well regardless of whether is of full rank or degenerate. To demonstrate the wide applicability of our new bounds, let us consider the second correlation model specified by the Jakes’ model. Specifically, we assume and . Parameter depends on the distance between the transmitter and receiver and on the incident angle of the , which is the same wavefront; it was assumed to be value as used in [26]. The influence of and on the average
Fig. 3. New bounds work equally well for a degenerate two 5-by-1 channel vectors: exponential model.
Fig. 4.
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S constructed from
Variation of average capacity with Rician factor: Jakes’ model.
capacity is shown in Figs. 4 and 5, respectively. Once again, superior tightness of the new bounds is observed. VII. CONCLUSION In this paper, we proposed a generic procedure for tightly bounding the average capacity of wireless MIMO systems on general correlated Rician channels. We expanded the conditional capacity as a linear sum of determinants of noncentral complex Wishart-distributed matrices of varying orders by resorting to a lemma of Lovitt. We explicitly determined the moment of such a determinant whereby an upper bound on the average channel capacity is obtained. By constructing a multivariate convex function and elegantly determining the of a noncentral Wishart-distributed matrix, we obtained a lower capacity bound on correlated Rician channels in simple closed form. Both upper and lower bounds are very tight, as supported by various numerical examples.
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We invoke the reproductive property of the zonal polynomial that
(46) to simplify the expression yielding
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APPENDIX B PROOF OF (26) Fig. 5. New bounds work equally well for a degenerate two 5-by-1 channel vectors: Jakes’ model.
S constructed from
We represent the hypergeometric function as (48)
APPENDIX A PROOF OF (21) The complex hypergeometric function of matrix argument is defined as [5]
(42) is a
is the zonal polynomial with an argument of a where matrix . depends on matrix argument only through its eigenvalues. can be represented as a hoAccording to James [5], of mogeneous polynomial of order in eigenvalues such that [6, p. 228] (49)
Hermitian matrix, where is a partition of into not more than parts, such that integers and , and denotes the summation over all of the partitions of the nonnegative integer . is the zonal polynomial, and is the complex multivariate hypergeometric coefficient defined as
where is a constant determined by the partition. Recall that, equals one, it has only one nonzero eigenwhen the rank of value, say . We can thus assert that, for any given , only one zonal polynomial that corresponds to the partition is nonzero whereas all terms in (49) for other partitions always contain factors of zero. It is clear that the only nonzero zonal polynomial can be written as
(43)
(50)
depends only on the eigenvalues of , the defSince inition of hypergeometric function can be extended to a nonHermitian matrix. We represent the hypergeometric function in (20) in terms of zonal polynomials, yielding
It remains to determine the unknown coefficient . To this end, we invoke the property [6, eq. (3)], which states that (51) where the summation is taken over all of the partitions of . It follows from combining the two expressions above that (52) Furthermore, for partition
, we have (53)
(44)
which, along with (52), allows us to simplify (48) to obtain where we have changed the argument expression of . Then the zonal polynomials. Let and whereby
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CUI et al.: GENERIC PROCEDURE FOR TIGHTLY BOUNDING THE CAPACITY OF MIMO CORRELATED RICIAN FADING CHANNELS
.. .
.. .
is of rank zero, it is easy to obtain . This result is included in (54) and can be . obtained from the latter by letting
When
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(57)
.. .
. We next add where to the remaining rows, respectively, and simplify, yielding
APPENDIX C PROOF OF LEMMA 2 Define second-order derivatives
.. .
. It is straightforward to obtain the
.. .
.. .
(61)
(55) is the
The Hessian matrix of
matrix defined by (56)
To prove the nonnegative definitiveness of , it suffices to verify that the determinant of an arbitrary main submatrix of is non, an arbitrary ordernegative. By denoting main submatrix can be written as (57), shown at the top of the page. To determine its determinant, we take the common factor of from the th row for and simplify, resulting in
.. .
.. .
.. . (58)
where we denote (59) The above expression can be further simplified. We add columns to column 1 so that the first column becomes the vector of equal elements, as shown by
.. .
.. .
.. .
(60)
Since
,
, and , it follows that
, thus com-
pleting the proof. REFERENCES [1] G. J. Foschini and M. J. Gans, “On limits of wireless communication in a fading environment when using multiple antennas,” Wireless Pers. Commun., vol. 6, no. 3, pp. 311–335, Mar. 1998. [2] E. Telatar, “Capacity of multi-antenna Gaussian channels,” AT&T Bell Labs., Tech. Rep. BL0 112 170-950 615-07TM, 1995. [3] S. K. Jayaweera and H. V. Poor, “On the capacity of multi-antenna systems in the presence of Rician fading,” in Proc. 56th IEEE Vehicular Technology Conf., vol. 4, Sep. 22–28, 2002, pp. 1963–1967. [4] G. J. Foschini, G. D. Golden, R. A. Valenzuela, and P. W. Wolniansky, “Simplified processing for high spectral efficiency wireless communication employing multi-element arrays,” IEEE J. Sel. Areas Commun., vol. 17, no. 11, pp. 1841–1852, Nov. 1999. [5] A. T. James, “Distributions of matrix variates and latent roots derived from normal samples,” Ann. Math. Statist., vol. 35, pp. 475–501, 1964. [6] R. J. Muirhead, Aspects of Multivariate Statistical Aspects. New York: Wiley, 1982. [7] J. Laurila, K. Kalliola, M. Toeltsch, K. Hugel, P. Vainikainen, and E. Bonek, “Wideband 3-d characterization of mobile radio channels in urban environments,” IEEE Trans. Antennas Propag., vol. 50, no. 2, pp. 233–243, Feb. 2002. [8] P. J. Smith, S. Roy, and M. Shafi, “Capacity of MIMO systems with semicorrelated flat fading,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2781–2788, Oct. 2003. [9] D. Gesbert, H. Bolcskei, D. Gore, and A. Paulraj, “Outdoor MIMO wireless channels: Models and performance prediction,” IEEE Trans. Commun., vol. 50, no. 12, pp. 1926–1934, Dec. 2002. [10] J. Ayadi, A. A. Hutter, and J. Farserotu, “On the multiple input multiple output capacity of Ricean channels,” in Proc. 5th Int. Symp. Wireless Personal Multimedia Communications, vol. 2, Oct. 2002, pp. 402–406. [11] P. J. Smith and M. Shafi, “On a Gaussian approximation of the capacity of wireless MIMO systems,” in Proc. Int. Conf. Communications, New York, Apr. 2002, pp. 406–410. [12] V. Jungnickel, V. Pohl, H. Nguyen, U. Kruger, T. Haustein, and C. von Helmolt, “High capacity antennas for MIMO radio systems,” in Proc. 5th Int. Symp. Wireless Personal Multimedia Communications, vol. 2, Oct. 2002, pp. 407–411.
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[13] M. A. Khalighi, J.-M. Brossier, G. Jourdain, and K. Raoof, “On the capacity of Ricean MIMO channels,” in Proc. 12th IEEE Int. Symp. Personal, Indoor, and Mobile Radio Communications, vol. 1, Sep. 2001, pp. A150–A154. [14] O. Oyman, R. U. Nabar, H. Bolcskei, and A. J. Paulraj, “Tight lower bound on the ergodic capacity of Rayleigh fading MIMO channels,” in Proc. IEEE Telecommunications Conf., vol. 2, 2002, pp. 1172–1176. [15] M. Kang and M.-S. Alouini, “On the capacity of MIMO Rician channels,” in Proc. 40th Annu. Allerton Conf. Communications, Control, and Computing, Allerton, Monticello, IL, Oct. 2002, pp. 936–945. [16] S. L. Loyka, “Channel capacity of MIMO architecture using the exponential correlation,” IEEE Commun. Lett., vol. 5, no. 9, pp. 369–371, Sep. 2001. , “New compound upper bound on MIMO channel capacity,” IEEE [17] Commun. Lett., vol. 6, no. 3, pp. 96–98, Mar. 2002. [18] D. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fading correlation and its effect on the capacity of multielement antenna systems,” IEEE Trans. Commun., vol. 48, no. 3, pp. 502–513, Mar. 2000. [19] C.-N. Chuah, D. N. C. Tse, J. M. Kahn, and R. A. Valenzuela, “Capacity scaling in MIMO wireless systems under correlated fading,” IEEE Trans. Inf. Theory, vol. 48, no. 3, pp. 637–650, Mar. 2002. [20] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions. New York: Dover, 1972. [21] A. R. Rao and P. Bhimasankaram, Linear Algebra, 2nd ed. New Delhi, India: Hindustan Book Agcy., 2000. [22] Q. T. Zhang, X. W. Cui, and X. M. Li, “Very tight cpacity bounds for MIMO correlated Rayleigh fading channels,” IEEE Trans. Wireless Commun., vol. 4, no. 2, pp. 681–688, Mar. 2005, to be published. [23] C. R. Rao, Linear Statistical Inference and its Applications, 2nd ed. New York: Wiley, 1973. [24] W. V. Lovitt, Linear Integral Equations. New York: Dover, 1950, sec. 15, p. 24. [25] D. Middleton, An Introduction to Statistical Communication Theory. New York: IEEE Press, 1996, p. 727. [26] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [27] P. M. Lee, Bayesian Statistics: An Introduction. London, U.K.: Arnold, 1998. [28] R. Webster, Convexity. London, U.K.: Oxford Univ. Press, 1994, pp. 217, 230–231.
X. W. Cui received the B.S. and Ph.D. degrees in electrical engineering from Tsinghua University, Beijing, China, in 1999 and 2005, respectively. His research interests include information theory and statistical signal processing for wireless communication.
Q. T. Zhang (S’84–M’85–SM’95) received the B.Eng. degree from Tsinghua University, Beijing, China, in wireless communications, the M.Eng. degree from South China University of Technology, Guangzhou, China, in wireless communications, and the Ph.D. degree in electrical engineering from McMaster University, Hamilton, ON, Canada. After graduation from McMaster in 1986, he held a research position and Adjunct Assistant Professorship at the same institution. In January 1992, he joined the Spar Aerospace Ltd., Satellite and Communication Systems Division, Montreal, QC, Canada, as a Senior Member of Technical Staff. At Spar Aerospace, he participated in the development and manufacturing of the Radar Satellite (Radarsat). He was subsequently involved in the development of the advanced satellite communication systems for the next generation. He joined Ryerson Polytechnic University, Toronto, ON, Canada, in 1993 and became a Full Professor in 1999. In 1999, he took one-year sabbatical leave at the National University of Singapore and is now with the City University of Hong Kong, Kowloon. His research interests are transmission and reception over fading channels with a current focus on wireless MIMO and UWB systems. Prof. Zhang is presently an Associate Editor for the IEEE COMMUNICATIONS LETTERS.
Z. M. Feng received the B.Eng. and M.Eng. degrees from Tsinghua University, Beijing, China, in 1970 and 1981, respectively, both in radar signal processing. He has been with the Department of Electronic Engineering, Tsinghua University, since 1970, where he became a Professor in 2000. Before 1990, his research direction was radar signal processing. Now his research interests include broadband access networks, satellite positioning and navigation, and wireless communications.
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