Abstract
Eigenvalue densities of complex noncentral Wishart matrices are investigated to study an open problem in information theory. Specifically, the largest, smallest, and joint eigenvalue densities of complex noncentral Wishart matrices are derived. These densities are expressed in terms of complex zonal polynomials and invariant polynomials. A connection between the complex Wishart matrix theory and information theory is given. This facilitates evaluation of the most important information-theoretic measure, the so-called ergodic channel capacity. In particular, the capacity of multiple-input multiple-output (MIMO) Rician distributed channels is investigated. We consider both spatially correlated and uncorrelated MIMO Rician channels and derive exact and easily computable tight upper bound formulas for ergodic capacities. Numerical results are also given, which show how the channel correlation degrades the capacity of the communication system.
Similar content being viewed by others
REFERENCES
Telatar, I.E., Capacity of Multi-Antenna Gaussian Channels, Eur. Trans. Telecom., 1999, vol. 10, pp. 585–595.
Muirhead, R.J., Aspects of Multivariate Statistical Theory, New York: Wiley, 1982.
James, A.T., Distributions of Matrix Variate and Latent Roots Derived from Normal Samples, Ann. Math. Statist., 1964, vol. 35, pp. 475–501.
Khatri, C.G., Non-central Distributions of ith Largest Characteristic Roots of Three Matrices Concerning Complex Multivariate Normal Populations, Ann. Inst. Statist. Math., 1969, vol. 21, pp. 23–32.
Ratnarajah, T., Vaillancourt, R., and Alvo M., Eigenvalues and Condition Numbers of Complex Random Matrices, SIAM J. Matrix Anal. Appl., 2005, vol. 26, no.2, pp. 441–456.
Davis, A.W., Invariant Polynomials with Two Matrix Arguments Extending the Zonal Polynomials: Applications to Multivariate Distribution Theory, Ann. Inst. Statist. Math., 1979, vol. 31, pp. 465–485.
Davis, A.W., Invariant Polynomials with Two Matrix Arguments Extending the Zonal Polynomials, Multivariate Analysis-V: Proc. 5th Int. Sympos. on Multivariate Analysis, Univ. of Pittsburgh, 1978, Krishnaiah, P.R., Ed., Amsterdam: North-Holland, 1980, pp. 287–299.
Ash, R.B., Information Theory, New York: Dover, 1965.
Ratnarajah, T., Vaillancourt, R., and Alvo, M., Complex Random Matrices and Rayleigh Channel Capacity, Commun. Inf. Syst., 2003, vol. 3, no.2, pp. 119–138. Available from http://www.ims.cuhk.edu.hk/~cis/.
Foschini, J.G. and Gans, M.J., On Limits of Wireless Communications in a Fading Environment when Using Multiple Antennas, Wireless Personal Commun., 1998, vol. 6, pp. 311–335.
Foschini, J.G., Layered Space-Time Architecture for Wireless Communication in a Fading Environment when Using Multi-Element Antennas, Bell Labs Tech. J., 1996, pp. 41–59.
Sellathurai, M. and Foschini, G., A Stratified Diagonal Layered Space-Time Architecture: Information Theoretic and Signal Processing Aspects, IEEE Trans. Signal Process., 2003, vol. 51, pp. 2943–2954.
Chuah, C.N., Tse, D., Kahn, J.M., and Valenzuela, R.A., Capacity Scaling in MIMO Wireless Systems under Correlated Fading, IEEE Trans. Inform. Theory, 2002, vol. 48, no.3, pp. 637–650.
Shiu, D.S., Foschini, G.F., Gans, M.G., and Kahn, J. M., Fading Correlation and Its Effect on the Capacity of Multielement Antenna Systems, IEEE Trans. Commun., 2000, vol. 48, pp. 502–513.
Shin, H. and Lee, J.H., Capacity of Multiple-Antenna Fading Channels: Spatial Fading Correlation, Double Scattering, and Keyhole, IEEE Trans. Inform. Theory, 2003, vol. 49, no.10, pp. 2636–2647.
Ratnarajah, T. and Vaillancourt, R., Quadratic Forms on Complex Random Matrices and Multiple-Antenna Channel Capacity, in Proc. 12th Ann. Workshop on Adaptive Sensor Array Processing, MIT Lincoln Laboratory, 2004.
Ratnarajah, T. and Vaillancourt, R. Quadratic Forms on Complex Random Matrices and Channel Capacity, in Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, Montreal, 2004, vol. 4, pp. 385–388.
Kang, M. and Alouini, M.-S., Performance Analysis of MIMO MRC Systems over Rician Fading Channels, in Proc. IEEE Veh. Tech. Conf., Vancouver, 2002, pp. 869–873.
Kang, M. and Alouini, M.-S., Largest Eigenvalue of Complex Wishart Matrices and Performance Analysis of MIMO MRC Systems, IEEE J. Sel. Areas Commun., 2003, vol. 21, pp. 406–417.
Kang, M. and Alouini, M.-S., Impact of Correlation on the Capacity of MIMO Channels, in Proc. IEEE Int. Conf. Commun., Anchorage, 2003, pp. 2623–2627.
Kang M., Alouini, M.-S., On the Capacity of MIMO Rician Channel, in Proc. 40th Ann. Allerton Conf. on Communication, Control, and Computing, Monticello, 2002.
Kang, M., Yang, L., and Alouini, M.-S., Capacity of MIMO Rician Channels with Multiple Correlated Rayleigh Co-Channel Interferers, in Proc. IEEE Global Telecom., San Francisco, 2003, pp. 1119–1123.
Jayaweera, S. and Poor, H., On the Capacity of Multi-Antenna Systems in the Presence of Rician Fading, in Proc. IEEE Veh. Tech. Conf., Vancouver, 2002, pp. 1963–1967.
Zhang, Q.T. and Liu, D.P., A Simple Capacity Formula for Correlated Diversity Rician Fading Channels IEEE Commun. Lett., 2002, vol. 6, pp. 481–483.
Ratnarajah, T., Vaillancourt, R., and Alvo, M., Jacobians and Hypergeometric Functions in Complex Multivariate Analysis, Canad. Appl. Math. Quart., to appear.
Khatri, C.G., On Certain Distribution Problems Based on Positive Definite Quadratic Functions in Normal Vectors, Ann. Math. Statist., 1966, vol. 37, no.2, pp. 468–479.
Constantine, A.G., Some Noncentral Distribution Problems in Multivariate Analysis, Ann. Math. Statist., 1963, vol. 34, pp. 1270–1285.
Chikuse, Y. and Davis, A.W., Some Properties of Invariant Polynomials with Matrix Arguments and Their Applications in Econometrics, Ann. Inst. Statist. Math., 1986, vol. 38, pp. 109–122.
Miller, K.S., Complex Stochastic Processes: An Introduction to Theory and Application, NewYork: Addison-Wesley, 1974.
Author information
Authors and Affiliations
Additional information
Translated from Problemy Peredachi Informatsii, No. 1, 2005, pp. 3–27.
Original Russian Text Copyright © 2005 by Ratnarajah, Vaillancourt, Alvo.
This work was partially supported by the Natural Sciences and Engineering Research Council of Canada.
Rights and permissions
About this article
Cite this article
Ratnarajah, T., Vaillancourt, R. & Alvo, M. Complex random matrices and Rician channel capacity. Probl Inf Transm 41, 1–22 (2005). https://doi.org/10.1007/s11122-005-0006-6
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11122-005-0006-6