Abstract
Geometric mean decomposition (GMD) has emerged as an alternative method to design multiple-input multiple-output (MIMO) transceivers. The MIMO-GMD scheme decouples the MIMO channel into multiple independent links with identical gains. The GMD-based system with zero-forcing decision feedback equalizer (ZF-DFE) is known to minimize the bit error rate (BER) for high signal-to-noise ratios (SNRs). In addition, adaptive modulation has been widely used to enhance the average spectral efficiency (ASE) while maintaining a target BER and transmit power. In this paper, we present an analytic study of the adaptive modulation for GMD-ZF-DFE systems under Rayleigh flat fading correlated channels. In order to adjust the constellation size, the SNR at the equalizer output is sent back to the transmitter. The SNR at the DFE output is a function of the determinant of a Wishart complex matrix. The complementary cumulative distribution function (CCDF) is then an important key to our analysis. To evaluate the performance of the considered system, we use some bounds on the CCDF of the determinant and the trace of a Wishart matrix. Closed-form expressions of the BER, the ASE and the outage probability are derived and compared to Monte Carlo simulation results. Furthermore, we analyze the effect of the channel spatial correlation.
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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, 311–335.
Telatar, I. E. (1999). Capacity of multi-antenna Gaussian channels. European Transactions on Telecommunications, 10, 585–595.
Foschini, G. J., Golden, G. D., Valenzuela, R. A., & Wolniansky, P. W. (1999). Simplified processing for high spectral efficiency wireless communication employing multi-element arrays. IEEE Journal on Selected Areas in Communication, 17(11), 1841–1852.
Caire, G., & Shamai, S. (2003). On the achievable throughput of a multiantenna Gaussian broadcast channel. IEEE Transactions on, Information Theory, 49(7), 1691–1706.
Jiang, Y., Li, J., & Hager, W. W. (2005). Uniform channel decomposition for MIMO communications. IEEE Transactions on Signal Processing, 53(11), 4283–4294.
Jiang, Y., Li, J., & Hager, W. W. (2005). Joint transceiver design for MIMO communications using geometric mean decomposition. IEEE Transactions on Signal Processing, 53(10), 3791–3803.
Costa, M. (1983). Writing on dirty paper. IEEE Transactions on, Information Theory, 29(3), 439–441.
Xu, F., Davidson, T. N., Zhang, J. K., & Wong, K. M. (2006). Design of block transceivers with decision feedback detection. IEEE Transactions on Signal Processing, 54(3), 965–978.
Chen, C. Y., & Vaidyanathan, P. P. (2007). Precoded FIR and redundant V-BLAST systems for frequency-selective MIMO channels. IEEE Transactions on Signal Processing, 55(7), 3390–3404.
Liu, C. H., & Vaidyanathan, P. P. (2010). Zero-forcing DFE transceiver design over slowly time-varying MIMO channels using ST-GTD. IEEE Transactions on, Signal Processing, 58(11), 5779–5790.
Liu, C. H., & Vaidyanathan, P. P. (2012). Generalized geometric mean decomposition and DFE transceiver design—Part I: Design and complexity. IEEE Transactions on Signal Processing, 60(6), 3112–3123.
Goldsmith, A. J., & Chua, S. G. (1997). Variable-rate variable-power MQAM for fading channels. IEEE Transactions on Communications, 45(10), 1218–1230.
Webb, W., & Steele, R. (1995). Variable rate QAM for mobile radio. IEEE Transactions on, Communications, 43(7), 2223–2230.
Chung, S. T., & Goldsmith, A. J. (2001). Degrees of freedom in adaptive modulation: A unified view. IEEE Transactions on, Communications, 49(9), 1561–1571.
Zhou, Z., & Vucetic, B. (2011). Adaptive coded MIMO systems with near full multiplexing gain using outdated CSI. IEEE Transactions on Wireless Communications, 10(1), 294–302.
Weng, C. C., Chen, C. Y., & Vaidyanathan, P. P. (2010). MIMO transceivers with decision feedback and bit loading: Theory and optimization. IEEE Transactions on, Signal Processing, 58(3), 1334–1346.
Zhou, Z., Vucetic, B., Dohler, M., & Li, Y. (2005). MIMO systems with adaptive modulation. IEEE Transactions on, Vehicular Technology, 54(5), 1828–1842.
Huang, J., & Signell, S. (2009). On performance of adaptive modulation in MIMO systems using orthogonal space-time block codes. IEEE Transactions on Vehicular Technology, 58, 4238–4247.
Maaref, A., & Aissa, S. (2009). Optimized rate-adaptive PSAM for MIMO MRC systems with transmit and receive CSI imperfections. IEEE Transactions on Communications, 57(3), 821–830.
Torabi, M., Haccoun, D., & Ajib, W. (2010). Performance analysis of scheduling schemes for rate-adaptive MIMO OSFBC-OFDM systems. IEEE Transactions on Vehicular Technology, 59(5), 2363–2379.
Alouini, M. S., & Goldsmith, A. J. (2000). Adaptive modulation over Nakagami fading channels. Wireless Personal Communications, 13(1–2), 119–143.
Yee, M. S., & Hanzo, L. (2002). A wide-band radial basis function decision feedback equalizer-assisted burst-by-burst adaptive modem. IEEE Transactions on, Communications, 50(5), 693–697.
Ammari, M.L., Gagnon, F., Belzile, J. (2005). Adaptive M-QAM scheme and unbiased MMSE-DFE receiver. In Proceedings of the IEEE vehicular technology conference
Ammari, M.L., Gagnon, F. (2009). On combining adaptive modulation and unbiased MMSE-DFE receiver to increase the capacity of frequency selective channels. In Proceedings of the fifth advanced international conference telecommunications AICT ’09, pp. 203–208.
Nagaraj, S. (2009). Symbol-level adaptive modulation for coded OFDM on block fading channels. IEEE Transactions on, Communication, 57(10), 2872–2875.
Ye, S., Blum, R. S., & Cimini, L. J. (2006). Adaptive OFDM systems with imperfect channel state information. IEEE Transactions on Wireless Communications, 5(11), 3255–3265.
Lari, M., Mohammadi, A., & Abdipour, A. (2010). Cross layer design based on adaptive modulation and truncated ARQ in MIMO Rician channels. In Telecommunications (IST), 2010 5th international symposium on, pp. 318–323.
Ordonez, L., Palomar, D., & Fonollosa, J. (2009). Ordered eigenvalues of a general class of Hermitian random matrices with application to the performance analysis of MIMO systems. IEEE Transactions on, Signal Processing, 57(2), 672–689.
Zanella, A., Chiani, M., & Win, M. Z. (2009). On the marginal distribution of the eigenvalues of Wishart matrices. IEEE Transactions on, Communications, 57(4), 1050–1060.
Morales-Jimnez, D., Paris, J. F., Entrambasaguas, J. T., & Wong, K. K. (2011). On the diagonal distribution of a complex Wishart matrix and its application to the analysis of MIMO systems. IEEE Transactions on Communications, 59(12), 3475–3484.
Zhu, Y., yuen Kam, P., & Xin, Y. (2009). On the mutual information distribution of MIMO Rician fading channels. IEEE Transactions on, Communications, 57(5), 1453–1462.
Aniba, G., & Aissa, S. (2011). Cross-layer designed adaptive modulation algorithm with packet combining and truncated ARQ over MIMO Nakagami fading channels. IEEE Transactions on, Wireless Communications, 10(4), 1026–1031.
Goodman, N. R. (1963). The distribution of the determinant of a complex Wishart distributed matrix. The Annals of Mathematical Statistics, 34, 178–180.
Gradshteyn, I., & Ryzhik, I. (2007). Table of integrals, series, and products, 7th Ed. Academic Press, Waltham.
Cioffi, J. M., Dudevoir, G. P., Eyuboglu, M. V., & Forney, G. D. J. (1995). MMSE decision-feedback equalizers and coding. Part-I: Equalization results. IEEE Transactions on, Communications, 43(10), 2582–2594.
Yoo, T., & Goldsmith, A. (2006). Capacity and power allocation for fading MIMO channels with channel estimation error. IEEE Transactions on, Information Theory, 52(5), 2203–2214.
Paris, J. F., Goldsmith, A. J. (2006). Adaptive modulation for MIMO multiplexing under average BER constraints and imperfect CSI. In Communications, 2006. ICC ’06. IEEE international conference on, vol. 3, pp. 1318–1325.
Chen, Y., & Beaulieu, N. (2009). A simple polynomial approximation to the Gaussian Q-function and its application. IEEE, Communications Letters, 13(2), 124–126.
Choi, B., & Hanzo, L. (2003). Optimum mode-switching-assisted constant-power single- and multicarrier adaptive modulation. IEEE Transactions on, Vehicular Technology, 52(3), 536–560.
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Appendix: Average BER for Constellation \(M_i\)
Appendix: Average BER for Constellation \(M_i\)
The average BER for constellation \(M_i\) is
Using integration by parts, we have
where
and
To evaluate \(B\left( M_i, {v_i},v_{i+1} \right) \), we use the polynomial approximation of the Gaussian Q-function given in [38, eq. (4)]
The first derivative of \(Q(t)\) is
Hence, we have
To evaluate \( B\left( M_i, {v_i},v_{i+1} \right) \), we consider the CCDF bound of (18). Therefore, we have
where
Using the upper incomplete Gamma function, we can evaluate \( B\left( M_i, {v_i},v_{i+1} \right) \) as
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Ammari, M.L., Fortier, P. Performance Analysis of Adaptive Modulation for Precoded MIMO Systems with a GMD Zero-Forcing Transceiver. Wireless Pers Commun 77, 2075–2097 (2014). https://doi.org/10.1007/s11277-014-1625-2
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DOI: https://doi.org/10.1007/s11277-014-1625-2