# On Marginal Distributions of the Ordered Eigenvalues of Certain Random Matrices

- Haochuan Zhang
^{1}Email author, - Shi Jin
^{2}, - Xin Zhang
^{1}and - Dacheng Yang
^{1}

**2010**:957243

https://doi.org/10.1155/2010/957243

© Haochuan Zhang et al. 2010

**Received: **27 November 2009

**Accepted: **2 July 2010

**Published: **18 July 2010

## Abstract

This paper presents a general expression for the marginal distributions of the ordered eigenvalues of certain important random matrices. The expression, given in terms of matrix determinants, is compacter in representation and more efficient in computational complexity than existing results in the literature. As an illustrative application of the new result, we then analyze the performance of the multiple-input multiple-output singular value decomposition system. Analytical expressions for the average symbol error rate and the outage probability are derived, assuming the general double-scattering fading condition.

## Keywords

## 1. Introduction

Random matrix theory, since its inception, has been known as a powerful tool for solving practical problems arising in physics, statistics, and engineering [1–3]. Recently, an important aspect of random matrix theory, that is, the distribution of the eigenvalues of random matrices, has been successfully applied to the analysis and design of wireless communication systems [4]. These applications, mostly concerning the multiple-input multiple-output (MIMO) systems, can be summarized as follows. In single-user MIMO systems, the eigenvalue distributions of Wishart matrices (a Wishart matrix [1] is formed by multiplying a Gaussian random matrix (of the size ) with its Hermitian transposition (given that ). If , the product matrix was termed the pseudo-Wishart matrix [5]) were widely applied to the analysis of MIMO channel capacity [6–11] and specific MIMO techniques, such as MIMO maximum ratio combining (MIMO MRC) (MIMO MRC is a technique that transmits signals along the strongest eigen-direction of the channel. It was also known as maximum-ratio transmission [12], transmit-receive diversity [13], and MIMO beamforming [14]) [15–17] and, MIMO singular value decomposition (MIMO SVD) (MIMO SVD, also known as MIMO multichannel beamforming [18], and spatial multiplexing MIMO [19, 20], is a generalization of MIMO MRC. It transmits multiple data streams along several strongest eigen-directions of the channel). [18–21], given that the MIMO channel was Rayleigh/Rician faded. For channels that are not Rayleigh/Rician faded (e.g., the double-scattering [22] fading channel to be discussed in Section 4), the eigenvalue distributions of Wishart matrices also played an essential role in the performance analysis of MIMO systems [23–30]. Even for relay channels, statistical distributions of the eigenvalues were shown very useful in the derivation of the channel capacity [31, 32]. In multiuser MIMO systems, the eigenvalue distribution of a random matrix (characterized by the channel matrix of the desired user and that of the interferers) was applied to the performance analysis of MIMO optimum combing (MIMO OC) [33–41]. Furthermore, in cognitive radio networks, the eigenvalue distributions of random matrices were recently applied to devise effective algorithms for spectrum sensing [42–44].

Given its importance in various applications, the eigenvalue distribution of random matrices is arguably one of the hottest topic in communication engineering. During the past two years, general methods for obtaining these eigenvalue distributions were developed, applying for a general class of random matrices. To be specific, Ordóñez et al*.* [20] presented a general expression for the marginal distributions of the ordered eigenvalues, while Zanella et al*.* [45, 46] proposed alternative expressions for the same distributions. The results, however, need separate expressions to cover the Wishart and pseudo-Wishart matrices. This problem was later avoided in the new expression of Chiani and Zanella [21], which was given in terms of the "determinant" of the rank-
tensor. After that, a simpler expression for the eigenvalue distribution was presented by Sun et al*.* [41], where only conventional (
-dimensional) determinants were involved.

In this paper, we aim at finding a new expression for the eigenvalue distribution, which is even simpler than Sun's result. To that end, we first show that many important random matrices, especially those in the summary above, share a common structure on the joint distributions of their (nonzero) eigenvalues. Based on the common structure, we then derive the marginal distributions of the ordered eigenvalues, using a classical result from the theory of order statistics, along with the multilinear property of the determinant. It turns out that the new expression we obtained is compacter in representation and more efficient in computational complexity, when comparing with existing results. The new result can unify the eigenvalue distributions of Wishart and pseudo-Wishart matrices with only a single expression. Moreover, it is given in conventional ( -dimensional) determinants, and importantly, it replaces many functions in Sun's result with constant numbers, greatly improving the computational efficiency. As an illustrative application of the new expression, we analyze the performance of MIMO SVD systems, assuming the (uncorrelated) double-scattering [22] fading channels. It is worth noting that, different from the Rayleigh/Rician fading channels, where the performance of MIMO SVD were well-studied in [18], the behaviors of MIMO SVD in double-scattering channels is still not clear (expect for some primary results in [47] by the authors). In this context, we derive first the joint eigenvalue distribution of the MIMO channel matrix, using the law of total probability. Then, based on the joint distribution, we apply the general result to get the marginal distribution for each ordered eigenvalue. After that, we analyze the performance of the MIMO SVD. Analytical expressions for the average SER and the outage probability of the system are derived and validated (with numerical simulations). As the simulation results illustrate, the analytical expressions agree perfectly with the Monte Carlo results.

The rest of this paper is organized as follows. Section 2 presents the common structure of the joint eigenvalue distributions. Based on the common structure, Section 3 derives the general expression for the marginal eigenvalue distributions. Then, in Section 4, we analyze the performance of the MIMO SVD in double-scattering channels, by applying the general result. Finally, we summarize the paper in Section 5. Next, we list the notations used throughout this paper: all vectors and matrices are represented with bold symbols; denotes the transposition of a matrix; denotes the Hermitian transposition of a matrix; denotes an matrix with only zero elements; denotes the identity matrix; denotes that is an complex matrix; is the th element of a matrix ; denotes the determinant of a matrix; is the determinant of a matrix whose th element is ; is the expectation of a random variable with respect to ; denotes that is an complex Gaussian matrix with a mean value , a row correlation , and a column correlation .

## 2. Joint Distributions of Ordered Eigenvalues

In this section, we show that the random matrices discussed in Section 1 share a common structure on the joint probability density functions (PDFs) of their eigenvalues. (Although the common structure can be found in various random matrices (Rayleigh, Rician, and double-scattering, etc.), it is not true that all random matrices have this structure on the joint PDF of their eigenvalues. A good example in this point is the Nakagami-Hoyt channel, whose joint eigenvalue PDF of the channel matrix is different from (1), see [48, Equation ( )], for more details. It is also worth noting that, for non-Gaussian random matrices, obtaining exact expressions on their joint eigenvalue distributions is generally difficult. Very few results can be found in the literature. In this paper, we focus on exact eigenvalue distributions, and thus, we consider mainly Gaussian and Gaussian-related random matrices.) Indeed, this common structure (formulated as the proposition below) was previously reported in [20, 45, 49] among others.

Proposition 1.

with being an arbitrary constant, 's and 's being a generic function.

Next, we verify the proposition above with random matrices discussed in Section 1. (Let and denote two mutually independent complex Gaussian matrices).

Clearly, the joint PDF above is in the form of (1). For semicorrelated Rayleigh and uncorrelated Rician fading channels, one can also verify that the joint PDFs are in the same form as (1), see [18, 20, 45, 46].

with being the modified Bessel function of the second kind [50, Equation ( )].

Proof.

See Appendix A.

Again, the joint distribution fits well in the from of Proposition 1. More results pertaining to double-scattering channels can be found in [47, Lemma ].

with being the generalized hypergeometric function [50, Equation ( )].

Obviously, the joint PDF here also belongs to the class defined by Proposition 1. For more examples, see [40, 51].

with being the upper incomplete Gamma function [50, Equation ( )].

Again, the joint PDF has the same form as (1). More results can be found in [39].

In summary, the random matrices discussed in Section 1 share a common structure on the joint distributions of their eigenvalues. Based on this common structure, we derive in the following section a general result for the marginal distribution of each ordered eigenvalue.

## 3. Marginal Distributions of Ordered Eigenvalues

### 3.1. General Expression for the Marginal Distribution

Theorem 1.

where , is a permutation of that satisfies and . The second summation is over all permutations, that is, in total.

Proof.

See Appendix B.

where is an matrix with each element being a function of , and is identical to , except that all elements in the th column are replaced by their derivatives with respect to .

In the literature, exact expressions on the marginal distributions of the ordered eigenvalues were reported in [20, 45, 46]. (The expression obtained in [45, 46] was given in the form of a sum of terms. That form allows closed-form evaluation of moments and characteristic functions of the eigenvalues.) These results, however, needed separate expressions to represent the eigenvalue distributions of Wishart (i.e., ) and pseudo-Wishart (i.e., ) matrices. In contrast, Theorem 1 unifies the two cases ( and ) with only a single expression. It is also worth noting that, although another unified expression could be found in [21], the result there was given in terms of the determinant of rank- tensor . (Letting be a rank- tensor, that is, for , the "determinant" of , denoted by , is given by [7] where and are permutations of the integers , the summation is over all possible permutations, and is the sign of the permutation.) which was computationally complex, especially comparing to our new result in a conventional ( -dimensional) determinant form. Perhaps the most related work in the literature is [41]. To see the difference between [41] and Theorem 1 above, we rewrite [41, Lemma ] in the following proposition. After comparing the two results, one can clearly see that our expression is much more efficient in computational complexity, since the functions in (15) are replaced by constant numbers in (13).

Proposition 2.

Proof.

where . Substituting (1) into (17) and invoking the generalized Cauchy-Binet formula [41, Lemma ] the multi-nested integration can be carried out analytically. As such, we get the desired result.

It is also worth noting that the work of this paper can be viewed as an interesting proof for the equivalence between (13) and (15), because both Theorem 1 and Proposition 2 represent the same eigenvalue distribution.

### 3.2. Specific Eigenvalue Distributions

As a simple application of the general result, we particularize into the eigenvalue distribution of the double-scattering channel matrix.

Corollary 1.

Proof.

The integral above can be written in a closed form by invoking [50, Equations ( ) and ( )]. The results are given in (20). Substituting (5) into (13) and using (21) completes the proof.

The marginal CDF of the largest eigenvalue (i.e., ) of the double-scattering channel matrix was reported earlier in [14]. The expression above extends this result to marginal distributions of all ordered eigenvalues. We also note that marginal CDFs of the ordered eigenvalues were also investigated in the authors' previous work [47]. However, the result there were derived based on Proposition 2 above.

## 4. Performance Analysis of MIMO SVD Systems

where , , , , and are the numbers of transmit antennas, receive antennas, and the scatterers, respectively. The matrix represents the fading channel between the transmitter and the scatterers, while represents the channel between the scatterers and the receiver. The introduction of the double-scattering model is due to the fact that [53] MIMO channels exhibits a rank deficient behavior when there is not enough scattering around the transmitter and receiver (a typical example is the keyhole/pinhole channel [54], where the MIMO channel matrix has rank one regardless of the number of transmit and receive antennas, since only one scatterer exists in the environment). In this model, the MIMO channel matrix is characterized by the product (concatenation) of two Gaussian matrices, representing the channel from the transmitter to the scatterers, and the channel from the scatterers to the receiver, respectively. Varying the number of the scatterers, the double-scattering model describes a broad family of practical channels, ranging from conventional Rayleigh channel (infinite scatterers) to degenerate keyhole channel (only one scatterer). In the rest of this section, we use notations , , , and as they were defined in (6).

### 4.1. System Model

Clearly, the performance of MIMO SVD depends directly on the eigenvalues s.

It is worth noting that, although the capacity-achievable power allocation for MIMO SVD is water-filling [6], exact analysis of such allocation strategy is very difficult (in water-filling, each allocated power is a function of all eigenvalues , leading to an intractable SER expression of each subchannel [18], Ft. 1). For this reason, earlier researches on MIMO SVD generally considered fixed (but not necessarily uniform) power allocation [18, 20]. (Indeed, given a sufficiently high SNR, the water-filling power strategy tends to a uniform power allocation, that is, a special case of the fixed allocation [20].) Following this direction, we consider here fixed power allocation, but it worth noting hat the results obtained can serve as a starting point for the analysis of channel-dependent power allocations [19], as well as the analysis of diversity-multiplexing tradeoff [55].

### 4.2. Performance Analysis

### 4.3. Numerical Examples

In this subsection, numerical simulations are used to verify the theoretical results above. For notational convenience, we denote the double-scattering channel with transmit antennas, receive antennas, and scatterers by a three-tuple . We also assume that all subchannels are active (i.e., ), upon which equal power allocation is employed (i.e., for all ).

## 5. Conclusion

The eigenvalue distribution of random matrices has long been known as a powerful tool for analyzing and designing communication systems. In this paper, we derived a new expression for the marginal distributions of the ordered eigenvalues of certain important random matrices. The new expression was compacter in representation and more efficient in computational complexity, when comparing to existing results in the literature. As an illustrative application, we then used the general result to analyze the performance of MIMO SVD systems, under the assumption of double-scattering fading channels. Joint and marginal eigenvalue distributions of the channel matrix were presented, which further yielded analytical expressions on the average SER and outage probability of the system. Finally, the theoretical results were verified with numerical simulations.

## Declarations

### Acknowledgments

The work of H. Zhang, X. Zhang, and D. Yang was supported by National Science and Technology Major Project of China under Grant no. 2008ZX03003-001. The work of S. Jin was supported by National Natural Science Foundation of China under Grant no. 60902009 and 60925004, and National Science and Technology Major Project of China under Grant no. 2009ZX03003-005.

## Authors’ Affiliations

## References

- Wishart J: The generalised product moment distribution in samples from a normal multivariate population.
*Biometrika*1928, 20(1-2):32-52.View ArticleMATHGoogle Scholar - Wigner E: Characteristic vectors of bordered matrices with infinite dimensions.
*Annals of Mathematics*1955, 62: 546-564.MathSciNetView ArticleMATHGoogle Scholar - Marčenko VA, Pastur LA: Distributions of eigenvalues for some sets of random matrices.
*Mathematics of the USSR-Sbornik*1967, 1: 457-483. 10.1070/SM1967v001n04ABEH001994View ArticleGoogle Scholar - Tulino AM, Verdú S:
*Random Matrix Theory and Wireless Communications*. now publishers, Boston, Mass, USA; 2004.MATHGoogle Scholar - Mallik RK: The pseudo-Wishart distribution and its application to MIMO systems.
*IEEE Transactions on Information Theory*2003, 49(10):2761-2769. 10.1109/TIT.2003.817465MathSciNetView ArticleMATHGoogle Scholar - Telatar IE: Capacity of multi-antenna Gaussian channels.
*European Transactions on Telecommunications*1999, 10(6):585-595. 10.1002/ett.4460100604View ArticleGoogle Scholar - Chiani M, Win MZ, Zanella A: On the capacity of spatially correlated MIMO Rayleigh-fading channels.
*IEEE Transactions on Information Theory*2003, 49(10):2363-2371. 10.1109/TIT.2003.817437MathSciNetView ArticleMATHGoogle Scholar - Smith PJ, Roy S, Shafi M: Capacity of MIMO systems with semicorrelated flat fading.
*IEEE Transactions on Information Theory*2003, 49(10):2781-2788. 10.1109/TIT.2003.817472MathSciNetView ArticleMATHGoogle Scholar - Simon SH, Moustakas AL, Marinelli L: Capacity and character expansions: moment-generating function and other exact results for MIMO correlated channels.
*IEEE Transactions on Information Theory*2006, 52(12):5336-5351.MathSciNetView ArticleMATHGoogle Scholar - Shin H, Win MZ, Lee JH, Chiani M: On the capacity of doubly correlated MIMO channels.
*IEEE Transactions on Wireless Communications*2006, 5(8):2253-2264.View ArticleGoogle Scholar - Alfano G, Lozano A, Tulino AM, Verdú S: Mutual information and eigenvalue distribution of MIMO Ricean channels.
*Proceedings of the International Symposium on Information Theory and Its Applications (ISITA '04), October 2004, Parma, Italy*Google Scholar - Lo TKY: Maximum ratio transmission.
*IEEE Transactions on Communications*1999, 47(10):1458-1461. 10.1109/26.795811View ArticleGoogle Scholar - Dighe PA, Mallik RK, Jamuar SS: Analysis of transmit-receive diversity in Rayleigh fading.
*IEEE Transactions on Communications*2003, 51(4):694-703. 10.1109/TCOMM.2003.810871View ArticleGoogle Scholar - Jin S, McKay MR, Wong K-K, Gao X: Transmit beamforming in Rayleigh product MIMO channels: capacity and performance analysis.
*IEEE Transactions on Signal Processing*2008, 56(10):5204-5221.MathSciNetView ArticleGoogle Scholar - Kang M, Alouini M-S: Largest eigenvalue of complex wishart matrices and performance analysis of MIMO MRC systems.
*IEEE Journal on Selected Areas in Communications*2003, 21(3):418-426. 10.1109/JSAC.2003.809720View ArticleGoogle Scholar - McKay MR, Grant AJ, Collings IB: Performance analysis of MIMO-MRC in double-correlated Rayleigh environments.
*IEEE Transactions on Communications*2007, 55(3):497-507.View ArticleGoogle Scholar - Jin S, McKay MR, Gao X, Collings IB: Asymptotic SER and outage probability of MIMO MRC in correlated fading.
*IEEE Signal Processing Letters*2007, 14(1):9-12.View ArticleGoogle Scholar - Jin S, Mckay MR, Gao X, Collings IB: MIMO multichannel beamforming: SER and outage using new eigenvalue distributions of complex noncentral Wishart matrices.
*IEEE Transactions on Communications*2008, 56(3):424-434.View ArticleGoogle Scholar - Ordóñez LG, Palomar DP, Pagès-Zamora A, Fonollosa JR: High-SNR analytical performance of spatial multiplexing MIMO systems with CSI.
*IEEE Transactions on Signal Processing*2007, 55(11):5447-5463.MathSciNetView ArticleGoogle Scholar - Ordóñez LG, Palomar DP, Fonollosa JR: Ordered eigenvalues of a general class of Hermitian random matrices with application to the performance analysis of MIMO systems.
*IEEE Transactions on Signal Processing*2009, 57(2):672-689.MathSciNetView ArticleGoogle Scholar - Chiani M, Zanella A: Joint distribution of an arbitrary subset of the ordered eigenvalues of wishart matrices.
*Proceedings of the IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC '08), September 2008, Cannes, France*1-6.Google Scholar - Gesbert D, Bölcskei H, Gore DA, Paulraj AJ: Outdoor MIMO wireless channels: models and performance prediction.
*IEEE Transactions on Communications*2002, 50(12):1926-1934. 10.1109/TCOMM.2002.806555View ArticleGoogle Scholar - Shin H, Lee JH: Capacity of multiple-antenna fading channels: spatial fading correlation, double scattering, and keyhole.
*IEEE Transactions on Information Theory*2003, 49(10):2636-2647. 10.1109/TIT.2003.817439MathSciNetView ArticleMATHGoogle Scholar - Maaref A, Aïssa S: Impact of spatial fading correlation and keyhole on the capacity of MIMO systems with transmitter and receiver CSI.
*IEEE Transactions on Wireless Communications*2008, 7(8):3218-3229.View ArticleGoogle Scholar - Loyka S, Kouki A: On MIMO channel capacity, correlations, and keyholes: analysis of degenerate channels.
*IEEE Transactions on Communications*2002, 50(12):1886-1888. 10.1109/TCOMM.2002.806543View ArticleGoogle Scholar - Müller A, Speidel J: Characterization of mutual information of spatially correlated MIMO channels with keyhole.
*Proceedings of the IEEE International Conference on Communications (ICC '07), June 2007, Glasgow, Scotland*750-755.Google Scholar - Shin H, Win MZ: MIMO diversity in the presence of double scattering.
*IEEE Transactions on Information Theory*2008, 54(7):2976-2996.MathSciNetView ArticleMATHGoogle Scholar - Levin G, Loyka S: Multi-keyhole MIMO channels: asymptotic analysis of outage capacity.
*Proceedings of the IEEE International Symposium on Information Theory (ISIT '06), July 2006, Seattle, Wash, USA*1305-1309.Google Scholar - Yang S, Belfiore J-C: On the diversity of Rayleigh product channels.
*Proceedings of the IEEE International Symposium on Information Theory (ISIT '07), June 2007, Nice, France*1276-1280.Google Scholar - Yang S, Belfiore J-C: Diversity-multiplexing tradeoff of double-scattering MIMO channels.
*IEEE Transactions on Information Theory*. http://arxiv.org/abs/cs/0603124. - Jin S, McKay MR, Zhong C, Wong K-K: Ergodic capacity analysis of amplify-and-forward MIMO dual-hop systems.
*IEEE Transactions on Information Theory*2010, 56(5):2204-2224.MathSciNetView ArticleGoogle Scholar - Firag A, Smith PJ, McKay MR: Capacity analysis for MIMO two-hop amplify-and-forward relaying systems with the source to destination link.
*Proceedings of the IEEE International Conference on Communications (ICC '09), June 2009, Dresden, Germany*1-6.Google Scholar - Winters JH: Optimum combining in digital mobile radio with cochannel interference.
*IEEE Journal on Selected Areas in Communications*1984, 2(4):528-539. 10.1109/JSAC.1984.1146095View ArticleGoogle Scholar - Chiani M, Win MZ, Zanella A, Mallik RK, Winters JH: Bounds and approximations for optimum combining of signals in the presence of multiple cochannel interferers and thermal noise.
*IEEE Transactions on Communications*2003, 51(2):296-307. 10.1109/TCOMM.2003.809265View ArticleGoogle Scholar - Chiani M, Win MZ, Zanella A: Error probability for optimum combining of
*M*-ary PSK signals in the presence of interference and noise.*IEEE Transactions on Communications*2003, 51(11):1949-1957. 10.1109/TCOMM.2003.819197View ArticleGoogle Scholar - Chiani M, Win MZ, Zanella A: On optimum combining of
*M*-PSK signals with unequal-power interferers and noise.*IEEE Transactions on Communications*2005, 53(1):44-47. 10.1109/TCOMM.2004.840640View ArticleGoogle Scholar - Mckay MR, Zanella A, Collings IB, Chiani M: Error probability and SINR analysis of optimum combining in Rician fading.
*IEEE Transactions on Communications*2009, 57(3):676-687.View ArticleGoogle Scholar - Kang M, Alouini M-S: Quadratic forms in complex Gaussian matrices and performance analysis of MIMO systems with cochannel interference.
*IEEE Transactions on Wireless Communications*2004, 3(2):418-431. 10.1109/TWC.2003.821188View ArticleGoogle Scholar - Kang M, Yang L, Alouini M-S: Outage probability of MIMO optimum combining in presence of unbalanced co-channel interferers and noise.
*IEEE Transactions on Wireless Communications*2006, 5(7):1661-1668.View ArticleGoogle Scholar - Jin S, McKay MR, Wong K-K, Gao X: MIMO multichannel beamforming in interference-limited ricean fading channels.
*Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM '08), December 2008, New Orleans, La, USA*1012-1016.Google Scholar - Sun L, McKay MR, Jin S: Analytical performance of MIMO multichannel beamforming in the presence of unequal power Cochannel Interference and Noise.
*IEEE Transactions on Signal Processing*2009, 57(7):2721-2735.MathSciNetView ArticleGoogle Scholar - Zeng Y, Liang Y-C: Eigenvalue-based spectrum sensing algorithms for cognitive radio.
*IEEE Transactions on Communications*2009, 57(6):1784-1793.View ArticleGoogle Scholar - Cardoso LS, Debbah M, Bianchi P, Najim J: Cooperative spectrum sensing using random matrix theory.
*Proceedings of the 3rd International Symposium on Wireless Pervasive Computing (ISWPC '08), May 2008, Santorini, Greece*334-338.Google Scholar - Penna F, Garello R, Figlioli D, Spirito MA: Exact non-asymptotic threshold for eigenvalue-based spectrum sensing.
*Proceedings of the 4th International Conference on Cognitive Radio Oriented Wireless Networks and Communications (CROWNCOM '09), June 2009, Hannonver, Vietnam*1-5.Google Scholar - Zanella A, Chiani M, Win MZ: On the marginal distribution of the eigenvalues of wishart matrices.
*IEEE Transactions on Communications*2009, 57(4):1050-1060.View ArticleGoogle Scholar - Zanella A, Chiani M: The PDF of the 1th largest eigenvalue of central wishart matrices and its application to the performance analysis of MIMO systems.
*Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM '08), December 2008, New Orleans, La, USA*1062-1067.Google Scholar - Zhang H, McKay MR, Yang D: MIMO multi-channel beamforming in double-scattering channels.
*Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '09), April 2009, Taipei, Taiwan*2093-2096.Google Scholar - Kumar S, Pandey A: Random matrix model for Nakagami-Hoyt fading.
*IEEE Transactions on Information Theory*2010, 56(5):2360-2372.MathSciNetView ArticleGoogle Scholar - Alfano G, Tulino AM, Lozano A, Verdú S: Eigenvalue statistics of finite-dimensional random matrices for MIMO wireless communications.
*Proceedings of the IEEE International Conference on Communications (ICC '06), June 2006, Istanbul, Turkey*9: 4125-4129.Google Scholar - Gradshteyn IS, Ryzhik IM:
*Table of Integrals, Series, and Products*. 5th edition. Academic Press, New York, NY, USA; 1996.MATHGoogle Scholar - Zhong C, Jin S, Wong K-K: Performance analysis of a Rayleigh-product MIMO channel with receiver correlation and cochannel interference.
*Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '08), April 2008, Las Vegas, Nev, USA*2877-2880.Google Scholar - Meyer CD:
*Matrix Analysis and Applied Linear Algebra*. SIAM, Philadelphia, Pa, USA; 2000.View ArticleGoogle Scholar - Almers P, Tufvesson F, Molisch AF: Keyhole effect in MIMO wireless channels: measurements and theory.
*IEEE Transactions on Wireless Communications*2006, 5(12):3596-3604.View ArticleGoogle Scholar - Chizhik D, Foschini GJ, Gans MJ, Valenzuela RA: Keyholes, correlations, and capacities of multielement transmit and receive antennas.
*IEEE Transactions on Wireless Communications*2002, 1(2):361-368. 10.1109/7693.994830View ArticleGoogle Scholar - Ordóñez LG, Pagès-Zamora A, Fonollosa JR: Diversity and multiplexing tradeoff of spatial multiplexing MIMO systems with CSI.
*IEEE Transactions on Information Theory*2008, 54(7):2959-2975.View ArticleMathSciNetMATHGoogle Scholar - Proakis JG:
*Digital Communications*. 4th edition. McGraw–Hill, New York, NY, USA; 2001.MATHGoogle Scholar - Ordóñez LG, Palomar DP, Pagès-Zamora A, Fonollosa JR: Minimum BER linear MIMO transceivers with adaptive number of substreams.
*IEEE Transactions on Signal Processing*2009, 57(6):2336-2353.MathSciNetView ArticleGoogle Scholar - Maaref A, Aïssa S: Eigenvalue distributions of wishart-type random matrices with application to the performance analysis of MIMO MRC systems.
*IEEE Transactions on Wireless Communications*2007, 6(7):2678-2689.View ArticleGoogle Scholar - Maaref A, Aïssa S: Joint and marginal eigenvalue distributions of (Non)central complex wishart matrices and PDF-based approach for characterizing the capacity statistics of MIMO ricean and rayleigh fading channels.
*IEEE Transactions on Wireless Communications*2007, 6(10):3607-3619.View ArticleGoogle Scholar - David HA, Nagaraja HN:
*Order Statistics*. 3rd edition. John Wiley & Sons, Hoboken, NJ, USA; 2003.View ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.