Abstract
We consider distributed multiple-input–multiple-output (MIMO) antenna systems, along with their certain generalizations. We show that distributed MIMO configuration can be mapped to a semicorrelated (one side correlated) Wishart model. For a given set of large-scale fading parameters, associated with the path loss and shadow fading, we derive exact and closed-form results for the marginal density of eigenvalues of \(\mathbf{H}^\dag \mathbf{H}\) (or \(\mathbf{H} \mathbf{H} ^\dag\)), where \(\mathbf{H}\) is the channel matrix. We also obtain exact and closed-form expressions for the ergodic channel capacity with the aid of Meijer G-function. The ergodic capacity of semicorrelated Rayleigh fading channel follows as a special case. All analytical results are validated by comparison with Monte-Carlo simulations.
Similar content being viewed by others
Notes
Except, may be, the large-scale channel state information.
We use the notation \(\mathcal{L}_n^{(m)}(x)\) for the associated Laguerre polynomials, instead of the usual \(L_n^{(m)}(x)\), to avoid confusion with the number L of the antennas.
References
Foschini, G. J., & Gans, M. J. (1998). On limits of wireless communications in a fading environment when using multiple antennas. Wireless Personal Communications, 6(2), 311–335.
Telatar, I. E. (1999). Capacity of multi-antenna Gaussian channels. European Transactions on Telecommunication, 10(6), 585–595.
Heath, R., Peters, S., Wang, Y., & Zhang, J. (2013). A current perspective on distributed antenna systems for the downlink of cellular systems. IEEE Communications Magazine, 51(4), 161–167.
Saleh, A. A. M., Rustako, A. J., & Roman, R. (1987). Distributed antennas for indoor radio communications. IEEE Transactions on Communications, 35(12), 1245–1251.
Clark, M. V., et al. (2001). Distributed versus centralized antenna arrays in broadband wireless networks. In Proceedings of IEEE vehicular technology conference (Vol. 1, pp. 33–37).
Roh, W., & Paulraj, A. (2002). Outage performance of the distributed antenna systems in a composite fading channel. In Proceedings of IEEE vehicular technology conference (Vol. 3, pp. 1520–1524).
Roh, W., & Paulraj, A. (2002). MIMO channel capacity for the distributed antenna systems. In Proceedings of IEEE vehicular technology conference (Vol. 2, pp. 706–709).
Xiao, L., Dai, L., Zhuang, H., Zhou, S., & Yao, Y. (2003). Information-theoretic capacity analysis in MIMO distributed antenna systems. In Proceedings of IEEE vehicular technology conference (Vol. 1, pp. 779–782).
Zhuang, H., Dai, L., Xiao, L., & Yao, Y. (2003). Spectral efficiency of distributed antenna systems with random antenna layout. Electronics Letters, 39(6), 495–496.
Ni, Z., & Li, D. (2004). Effect of fading correlation on capacity of distributed MIMO. In Proceedings of IEEE personal, indoor and mobile radio communication conference (Vol. 3, pp. 1637–1641).
Choi, W., & Andrews, J. G. (2007). Downlink performance and capacity of distributed antenna systems in a multicell environment. IEEE Transactions on Wireless Communications, 6(1), 69–73.
Feng, Z., Jiang, Z., Pan, W., & Wang, D. (2008). Capacity analysis of generalized distributed antenna systems using approximation distributions. In Proceedings of 11th IEEE Singapore international conference on communication systems (pp. 828–830).
Feng, W., Zhang, X., Zhou, S., Wang, J., & Xia, M. (2009). Downlink power allocation for distributed antenna systems with random antenna layout. In Proceedings of IEEE vehicular technology conference (VTC 2009-Fall) (pp. 1–5).
Feng, W., Li, Y., Zhou, S., Wang, J., & Xia, M. (2009). Downlink capacity of distributed antenna systems in a multi-cell environment. In Proceedings of IEEE wireless communication networking conference (WCNC’09) (pp. 1–5).
Lee, S.-R., Moon, S.-H., Kim, J.-S., & Lee, I. (2012). Capacity analysis of distributed antenna systems in a composite fading channel. IEEE Transactions on Wireless Communications, 11(3), 1076–1086.
Lee, S.-R., Moon, S.-H., Kim, J.-S., & Lee, I. (2012). On the capacity of MIMO distributed antenna systems. In Proceedings on IEEE international conference on communications (ICC’12) (pp. 4824–4828).
Feng, W., Li, Y., Zhou, S., Wang, J., & Xia, M. (2009). On the optimal radius to deploy antennas in multi-user distributed antenna system with circular antenna layout. In Proceedings of WRI international conference on communications and mobile computing (CMC 2009) (pp. 56–59).
Wang, J.-B., Wang, J.-Y., & Chen, M. (2012). Downlink system capacity analysis in distributed antenna systems. Wireless Personal Communications, 67(3), 631–645.
Qian, Y., Chen, M., Wang, X., & Zhu, P. (2009). Antenna location design for distributed antenna systems with selective transmission. In Proceedings on international conference on wireless communications & signal processing (WCSP’09) (pp. 1–5).
Wen, Y.-P., Wang, J.-Y., & Chen, M. (2013). Ergodic capacity of distributed antenna systems over shadowed Nakagami-m fading channels. In Proceedings of international conference on wireless communications & signal processing (WCSP’13) (pp. 1–6).
Dai, L. (2014). An uplink capacity analysis of the distributed antenna system (DAS): From cellular DAS to DAS with virtual cells. IEEE Transactions on Wireless Communications, 13(5), 2717–2731.
Zhang, H., & Dai H. (2004). On the capacity of distributed MIMO systems. In Proceedings of conference on information sciences and systems (CISS’04) (pp. 1–5).
Dai, L., Zhou, S., & Yao, Y. (2005). Capacity analysis in CDMA distributed antenna systems. IEEE Transactions on Wireless Communications, 4(6), 2613–2620.
Chen, H. M., & Chen, M. (2009). Capacity of the distributed antenna systems over shadowed fading channels. In Proceedings of IEEE 69th vehicular technolology conference (pp. 1–4).
Feng, W., et al. (2011). On the deployment of antenna elements in generalized multi-user distributed antenna systems. Mobile Networks and Applications, 16(1), 35–45.
Heliot, F., Hoshyar, R., & Tafazolli, R. (2011). An accurate closed-form approximation of the distributed MIMO outage probability. IEEE Transactions on Wireless Communications, 10(1), 5–11.
Alfano, G., Tulino, A. M., Lozano, A., & Verdu, S. (2004). Capacity of MIMO channels with one-sided correlation. In Proceedings of IEEE international symposium on spread spectrum technology and applications (ISSSTA’04) (pp. 515–519).
Simon, S. H., Moustakas, A. L., & Marinelli, L. (2006). Capacity and character expansions: Moment-generating function and other exact results for MIMO correlated channels. IEEE Transactions on Information Theory, 52(12), 5336–5351.
Maaref, A., & Aïssa, S. (2007). Eigenvalue distributions of Wishart-type random matrices with application to the performance analysis of MIMO MRC systems. IEEE Transactions on Wireless Communications, 6(7), 2678–2689.
Maaref, A., & Aïssa, S. (2007). 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, 6(10), 3607–3619.
Chiani, M., & Win, M. Z. (2010). MIMO networks: The effects of interference. IEEE Transactions on Information Theory, 56(1), 336–349.
Recher, C., Kieburg, M., & Guhr, T. (2010). Eigenvalue densities of real and complex Wishart correlation matrices. Physical Review Letters, 105, 244101.
Recher, C., Kieburg, M., Guhr, T., & Zirnbauer, M. R. (2012). Supersymmetry approach to Wishart correlation matrices: Exact results. Journal of Statistical Physics, 148(6), 981–998.
Ivrlac, M. T., Utschick, W., & Nossek, J. A. (2003). Fading correlations in wireless MIMO communication systems. IEEE Journal on Selected Areas in Communications, 21, 819–828.
Smith, P. J., Roy, S., & Shafi, M. (2003). Capacity of MIMO systems with semicorrelated flat fading. IEEE Transactions on Information Theory, 49(10), 2781–2788.
Huang, X.-L., Wu, J., Hu, F., & Chen, H.-H. (2015). Optimal antenna deployment for multiuser MIMO systems based on random matrix theory. IEEE Transactions on Vehicular Technology. doi:10.1109/TVT.2015.2513005.
Wishart, J. (1928). The generalised product moment distribution in samples from a normal multivariate population. Biometrika, 20A, 32–52.
James, A. T. (1964). Distributions of matrix variates and latent roots derived from normal samples. The Annals of Mathematical Statistics, 35, 475–501.
Uhlig, H. (1994). On singular Wishart and singular multivariate beta distributions. The Annals of Statistics, 22, 395–405.
Muirhead, R. J. (2009). Aspects of multivariate statistical theory (Vol. 197, p. 218). Hoboken: Wiley.
Anderson, T. W. (2003). An introduction to multivariate statistical analysis (3rd ed., p. 251). Hoboken: Wiley.
Ratnarajah, T., & Vaillancourt, R. (2005). Complex singular matrices and applications. Computers and Mathematics with Applications, 50, 399–411.
Cook, R. D. (2011). On the mean and variance of the generalized inverse of a singular Wishart matrix. Electronic Journal of Statistics, 5, 146–158.
Nydick, S. W. (2012). The wishart and inverse wishart distributions. Electronic Journal of Statistics, 6, 1–19.
Wolfram Research Inc. (2013). Mathematica, Version 9.0, Champaign, IL.
Kumar, S., Pivaro, G. F., Fraidenraich, G., & Dias, C. F. (2015). On the exact and approximate eigenvalue distribution for sum of Wishart matrices. arXiv:1504.00222.
Prudnikov, A. A. P., Brychkov, Y. A., Brychkov, I. U. A., & Maričev, O. I. (1990). Integrals and series, Vol. 3: More special functions. London: Gordon and Breach Science Publishers.
Author information
Authors and Affiliations
Corresponding author
Appendices
Proofs for Eqs. (27) and (31)
As pointed out in Sect. 5, the MIMO DAS model and weighted sum of Wishart matrices, both can be mapped to a semicorrelated Wishart scenario. The eigenvalue statistics for weighted sum of Wishart matrices has already been worked out in [46]. As a consequence the proofs in the present case run parallel to those in [46]. However, for the sake of completeness, we reproduce the derivations here as well.
Exact result for the marginal density of eigenvalues for semicorrelated Wishart matrices is available from the notable works of several authors [27–33]. We employ here the determinantal expression obtained in [32, 33].
Consider \(n\times m\) dimensional complex matrices W taken from the distribution
where the \(n\times n\) dimensional covariance matrix is \(\hat{\mathbf{V }}={\text{diag}}(\hat{v}_1,\ldots ,\hat{v}_n)\). We assumed here that there are no multiplicities in the entries of \(\hat{\mathbf{V }}\), i.e., \(\hat{v}_1,\ldots ,\hat{v}_n\) are distinct. In such a scenario the marginal density of nonzero-eigenvalues of \({\mathbf{W}} {\mathbf{W}} ^\dag\) or \({\mathbf{W}} ^\dag {\mathbf{W}}\) is given by
Here \(\nu =\min (m,n)\), and \(\Delta _m(\{\hat{v}^{-1}\})=\prod _{j>k}(\hat{v}_j^{-1}-\hat{v}_k^{-1})\) is the Vandermonde determinant. Equation (27) follows from (36) by setting \(m=M\) and \(n=\mathsf{L}\) and using the well known result \(\Delta (\{r\})=\det [r_j^{k-1}]\).
We now proceed to derive the ergodic channel capacity using the relation (13). To this end we expand (27) using the first column and obtain
We note at this point that \(g_j(\lambda )=\lambda ^{M-j}/\varGamma (M-j+1)=0\) if \(j> M\) because of the diverging gamma function in the denominator. If \(\mathsf{L}\le M\) this situation is not encountered. However, if \(\mathsf{L}> M\) then \(g_j(\lambda )\) is nonzero only for \(1\le j\le M\). Thus in both cases we see that the nonzero terms in the summation in (37) involve \(j=1,\ldots ,\nu\), with \(\nu =\min (M,\mathsf{L})\). We incorporate this observation by changing the upper limit of the summation. We now bring in the \(g_\mu (\lambda )\) occurring before the determinants to the respective first rows, i.e., with \(f_k(\hat{v},\lambda )\), and obtain
This equation serves as yet another expression for the marginal density. We now use (13) and obtain the following expression for the ergodic capacity by interchanging the \(\lambda\)-integral and the summation:
The \(\lambda\)-integral can be introduced in the first row of the determinant, along with the term \(\log _2 \left( 1+\frac{\rho }{M}\lambda \right)\) to yield
where
This integral can be expressed in a closed form in terms of Meijer G-function or exponential-integral-functions as in (33) and (34), respectively. To obtain (33) we identify the following special cases of Meijer G-functions [47]:
We also employ the convolution integral for Meijer G-function:
The restrictions on the indices for this integration formula can be found in [47]. Next we perform row interchanges in the determinants to bring \(\mathcal{G}_{\mu ,k}\) in the respective \(\mu\)th row. Consequently, we have the expression for ergodic channel capacity as provided in (31).
Proofs for Eqs. (15), (20), (24), and (26)
To arrive at Eqs. (24) and (26) we need to set \(\hat{v_1}=\cdots =\hat{v}_{L_1}=v_1; \hat{v}_{L_1+1}=\cdots =\hat{v}_{L_2}=v_2\,;\ldots ; \hat{v}_{L_{(N-1)}+1}=\cdots =\hat{v}_{L_N}=v_N\) in (27) and (31). However, direct substitution of these values makes the determinant in the numerator, as well as the determinant in the denominator (contained in the normalization) to become zero. Therefore, we must invoke a limiting procedure to obtain the proper results, as described below.
Let us focus on the columns involving up to \(L_1\) in (27). The ratio of the determinants appears as
We take \(\hat{v}_k^{-1}=\hat{v}_1^{-1}+\epsilon _k\) with small \(\epsilon _k\) for \(k=2,3,\ldots\), and Taylor-expand up to the term \(\epsilon _k^{k-1}\):
Now, employing adequate column operations, we obtain
The factors containing \(\epsilon _k\) and factorial can be taken out of the columns, both from numerator and denominator, and cancelled out. This leaves us with
To evaluate the derivatives of \((\hat{v}_k^{-1})^M e^{\hat{v}_1^{-1}\lambda }\) we use Rodrigues’ formula for the associated Laguerre polynomials,
with adequate scaling of the variables. Similar steps are followed for rest of the columns and consequently we arrive at (24).
We employ a similar procedure in (31) to arrive at (26). To evaluate the derivative of Meijer G-function we use the result
which follows from the following more general expression [47]:
Finally, (15) and (20) follow trivially from, respectively, (24) and (26) by setting identical number of antennas at each port, i.e., \(L_1=\cdots =L_N=L\).
Rights and permissions
About this article
Cite this article
Kumar, S. On the Ergodic Capacity of Distributed MIMO Antenna Systems. Wireless Pers Commun 92, 381–397 (2017). https://doi.org/10.1007/s11277-016-3548-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11277-016-3548-6