Error exponents for Nakagami-m fading channels

Along with the channel capacity, the error exponent is one of the most important information-theoretic measures of reliability, as it sets ultimate bounds on the performance of communication systems employing codes of finite complexity. In this paper, we derive an exact analytical expression for the random coding error exponent, which provides significant insight regarding the ultimate limits to communications through Nakagami-m fading channels. An important fact about this error exponent is that it determines the behavior of error probability in terms of the transmission rate and the code length that reflects the coding complexity required to achieve a certain level of reliability. Moreover, from the derived analytical expression, we can easily compute the necessary codeword length without extensive Monte-Carlo simulation to achieve a predefined upper bound for error probability at a rate below the channel capacity. We also improve the random coding bound by expurgating bad codewords from the code ensemble, since random coding error exponent is determined by selecting codewords independently according to the input distribution where good and bad codewords contribute equally to the overall average error probability. Finally, we derive exact analytical expressions for the cutoff rate, critical rate, and expurgation rate and verify the analytical expressions via Monte-Carlo simulation.

[1]  Emre Telatar,et al.  Capacity of Multi-antenna Gaussian Channels , 1999, Eur. Trans. Telecommun..

[2]  S. Shamai,et al.  Error probabilities for the block-fading Gaussian channel , 1995 .

[3]  M. Nakagami The m-Distribution—A General Formula of Intensity Distribution of Rapid Fading , 1960 .

[4]  Muriel Médard,et al.  On Noncoherent MIMO Channels in the Wideband Regime: Capacity and Reliability , 2006, IEEE Transactions on Information Theory.

[5]  Achilleas Anastasopoulos,et al.  A New Upper Bound on the Average Error Exponent for Multiple-Access Channels , 2010, ArXiv.

[6]  Tharmalingam Ratnarajah,et al.  Random Coding Error Exponent for OSTBC Nakagami-m Fading MIMO Channel , 2011, 2011 IEEE 73rd Vehicular Technology Conference (VTC Spring).

[7]  R. Gallager Information Theory and Reliable Communication , 1968 .

[8]  Tharmalingam Ratnarajah,et al.  Error exponents for Nakagami-m fading keyhole MIMO channels , 2012, 2012 IEEE International Conference on Communications (ICC).

[9]  Hyundong Shin,et al.  Random Coding Exponent for MIMO Channels , 2008, VTC Spring 2008 - IEEE Vehicular Technology Conference.

[10]  Peter J. McLane,et al.  Random Coding Error Exponents for Two-Dimensional Flat Fading Channels with Complete Channel State Information , 1999, IEEE Trans. Inf. Theory.

[11]  R. Srikant,et al.  MIMO Channels in the Low-SNR Regime: Communication Rate, Error Exponent, and Signal Peakiness , 2007, IEEE Transactions on Information Theory.

[12]  Ali Abdi,et al.  Sum of gamma variates and performance of wireless communication systems over Nakagami-fading channels , 2001, IEEE Trans. Veh. Technol..

[13]  Joachim Speidel,et al.  Ergodic Capacity and Information Outage Probability of MIMO Nakagami-m Keyhole Channels with General Branch Parameters , 2007, 2007 IEEE Wireless Communications and Networking Conference.

[14]  Peiliang Qiu,et al.  Some extensions of Gallager's method to general sources and channels , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[15]  Caijun Zhong,et al.  On the capacity of non-uniform phase MIMO nakagami-m fading channels , 2010, IEEE Communications Letters.

[16]  Sean P. Meyn,et al.  Error exponents for channel coding with application to signal constellation design , 2006, IEEE Journal on Selected Areas in Communications.

[17]  John M. Cioffi,et al.  On the MIMO Channel Capacity for the Nakagami-$m$ Channel , 2007, IEEE Transactions on Information Theory.

[18]  M. Win,et al.  Gallager's exponent for MIMO channels: a reliability-rate tradeoff , 2006, IEEE Transactions on Communications.

[19]  Erik G. Larsson,et al.  Space-Time Block Coding for Wireless Communications , 2003 .

[20]  C. Shannon Probability of error for optimal codes in a Gaussian channel , 1959 .

[21]  Caijun Zhong,et al.  Capacity Bounds for MIMO Nakagami- $m$ Fading Channels , 2009, IEEE Transactions on Signal Processing.

[22]  Shlomo Shamai,et al.  Fading Channels: Information-Theoretic and Communication Aspects , 1998, IEEE Trans. Inf. Theory.

[23]  Lutz H.-J. Lampe,et al.  On the Random Coding Exponent for Differential Modulation and Detection , 2006, 2006 IEEE International Symposium on Information Theory.