Capacity Analysis for Correlated Multi-Hop MIMO Channels under Colored Noise
A capacity analysis for generally correlated wireless multi-hop multi-input multi-output (MIMO) channels is presented in this paper. The channel at each hop is spatially correlated, the source symbols are mutually correlated, and the additive Gaussian noises are colored. First, by invoking Karush-Kuhn-Tucker condition for the optimality of convex programming, we derive the optimal source symbol covariance for the maximum mutual information between the channel input and the channel output when having the full knowledge of channel at the transmitter. Secondly, we formulate the average mutual information maximization problem when having only the channel statistics at the transmitter. Since this problem is almost impossible to be solved analytically, the numerical interior-point-method is employed to obtain the optimal solution. Furthermore, to reduce the computational complexity, an asymptotic closed-form solution is derived by maximizing an upper bound of the objective function. Simulation results show that the average mutual information obtained by the asymptotic design is very closed to that obtained by the optimal design, while saving a huge computational complexity.
 I. E. Telatar, “Capacity of multi-antenna Gaussian channels,”European Trans. on Tele., vol. 10, pp. 585–595, 1999.
 I. S. Association, “IEEE 802.11: Wireless lan medium access control (MAC) and physical layer (PHY) specifications, 2007 edition,” http://standards.ieee.org/getieee802/download/802.11-2007.pdf, 2007.
 Linksys, “Linksys WRT54G,”http://www.speedguide.net/broadband-view.php?hw=36, 2006.
 Physical Layer Aspects of UTRA High Speed Downlink Packet Access (release 4). 2004.
 Physical Layer Standard for cdma2000 Spread Spectrum Systems. 1999.
 A. Doufexi, S. Armour, M. Butler, A. Nix, D. Bull, J. McGeehan, and P. Karlsson, “A comparison of the HIPERLAN/2 and IEEE 802.11a wireless LAN standards,” IEEE Commun. Mag., vol. 40, p. 172?80, May 2002.
 Local and Metropolitan Area Networks Part 16: Air Interface for Fixed Broadband Wireless Access Systems. 2002.
 J. P. Kermoal, L. Schumacher, P. E. Mogensen, and K. I. Pedersen, “Experimental investigation of correlation properties of MIMO radio channels for indoor picocell scenario,” in Proc. IEEE Veh. Technol. Conf. (VTC-Fall), pp. 14–21, Sep. 2000.
 H. Bolcskei, D. Gesbert, and A. J. Paulraj, “On the capacity of OFDM-based spatial multiplexing systems,” IEEE Trans. Commun., vol. 50, pp. 225–234, Feb. 2002.
 H. Zhang, Y. Li, A. Reid, and J. Terry, “Optimum training symbol design for MIMO OFDM in correlated fading channels,”IEEE Trans. Wireless Commun., vol. 5, pp. 2343–2347, Sep. 2006.
 H. Bolcskei, M. Borgmann, and A. J. Paulraj, “Impact of the propagation environment on the performance of spacefrequency coded MIMO-OFDM,” IEEE J. Selected Areas in Commun., vol. 21, pp. 427–439, Apr. 2003.
 M. Enescu, T. Roman, and V. Koivunen, “State-space approach to spatially correlated MIMO OFDM channel estimation,”Signal processing, ScienceDirect, vol. 87, Mar. 2007.
 D. Shiu, G. Foschini, M. Gans, and J. Kahn, “Fading correlation and its effect on the capacity of multielement antenna systems,” IEEE Trans. Commun., vol. 48, pp. 502–513, Mar. 2002.
 I. Akyildiz and X. Wang, “A Survey on Wireless Mesh Networks,”IEEE Radio Communications, pp. 23–30, Sept. 2005.
 K. Shi, E. Serpedin, and P. Cibat, “Decision- directed fine synchronization in ofdm systems,” IEEE Trans. Commun., vol. 53, pp. 408–412, Mar. 2005.
 T. Wong and B. Park, “Training sequence optimization in MIMO systems with colored interference,” IEEE Trans. Commun., vol. 12, pp. 1939–1947, Nov. 2004.
 Y. P. Lin and S. Phong, “Optimal ISI-Free DMT transceiver for distorted channels with colored noise,” IEEE Trans. Signal Process., vol. 49, pp. 2702–2712, Nov. 2001.
 C. D. Richmond, “Mean-squared error and threshold SNR prediction of maximum-likelihood signal parameter estimation with estimated colored noise covariances,” IEEE Trans. Inf. Theory, vol. 52, pp. 2146–2164, May 2006.
 N. N. Tran, H. D. Tuan, and H. H. Nguyen, “Training signal and precoder designs for OFDM under colored noise,” IEEE Trans. Veh. Technol., pp. 3911 – 3917, Nov. 2008.
 S. Jafar, S. Vishwanath, and A. Goldsmith, “Channel capacity and beamforming for multiple transmit and receive antennas with covariance feedback,” in Proc. Inter. Conf. Conmun. (ICC), 2001.
 A. Goldsmith, Wireless communications. Cambridge university press, 2005.
 S. L. Loyka, “Channel capacity of two-antenna BLAST architecture,”Electron. Letter, vol. 35, pp. 1421–1422, 1999.
 B. Wang, J. Zhang, and A. Hst-Madsen, “On the capacity of MIMO relay channels,” IEEE Trans. Inform. Theory., pp. 29 – 43, Jan. 2005.
 H. Bolcskei, R. Nabar, O. Oyman, and A. Paulraj, “Capacity scaling laws in MIMO relay networks,” IEEE Trans. Wireless Commun., p. 14331444, Jun. 2006.
 S. Borade, L. Zheng, and R. Gallager, “Amplify-and-forward in wireless relay networks: Rate, diversity, and network size,”IEEE Trans. Inform. Theory, p. 33023318, OCt. 2007.
 N. Fawaz, K. Zarifi, M. Debbah, and D. Gesbert, “Asymptotic capacity and optimal precoding strategy of multi-level precode & forward in correlated channels,” in IEEE ITW08, Information Theory Workshop, May 2008.
 J. Kermoal, L. Schumacher, K. I. Pedersen, P. E. Mogensen, and F. Frederiksen, “A stochastic MIMO radio channel model with experimental validation,” IEEE J. Selected Areas in Commun., vol. 20, pp. 1211–1226, Aug. 2002.
 K. Yu, M. Bengtsson, B. Ottersten, D. McNamara, and P. Karlsson, “Modeling of wideband MIMO radio channels based on NLoS indoor measurements,” IEEE Trans. Veh. Technol., vol. 53, pp. 655–665, May 2004.
 J. H. Kotecha and A. M. Sayeed, “Transmit signal design for optimal estimation of correlated MIMO channel,” IEEE Trans. Signal Process., vol. 52, pp. 546–557, 2004.
 H. Bolcskei, Principles of MIMO-OFDM wireless systems. CRC Handbook on Signal Processing for Communications, M. Ibnkahla, Ed., 2004, to appear, online at http://www.nari.ee.ethz.ch/commth/pubs/p/crc03.
 N. N. Tran, H. D. Tuan, and H. H. Nguyen, “Superimposed training designs for spatially correlated MIMO-OFDM systems,”IEEE Trans. Wireless Commun., pp. 876–880, Mar. 2010.
 T. Cover and J. Thomas, Elements of Information theory. Wiley, New York, 1991.
 R. A. Horn and C. R. Johnson, Matrix analysis. Cambridge University Press., 1985.
 A. Lewis, “Derivatives of spectral functions,” Mathematics of Operations Research, vol. 21, pp. 576–588, 1996.
 D. Luenberger, Linear and nonlinear programming. Springer, 2003.
 O. Romanko, I. P´ olik, and T. Terlaky, “The homepage for SeDuMi,” http://sedumi.mcmaster.ca/.
 K.-C. Toh, M. J. Todd, and R. H. Tutuncu, “SDPT3 version 4.0 (beta) - A MATLAB software for semidefinite-quadraticlinear programming,” http://www.math.nus.edu.sg/ mattohkc/sdpt3.html.
 C. Rao, Linear statistical inference and its applications. John Wiley and Sons, 1973.