ISSN 0032-9460, Problems of Information Transmission, 2015, Vol. 51, No. 1, pp. 1–19.
Pleiades Publishing, Inc., 2015.
Original Russian Text
J. Xue, T. Ratnarajah, C. Zhong, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51, No. 1, pp. 3–22.
Error Exponents for Multi-Keyhole MIMO Channels
Institute for Digital Communications, School of Engineering,
The University of Edinburgh, Edinburgh, United Kingdom
e-mail: J.Xue@ed.ac.uk, T.Ratnarajah@ieee.org
Institute of Information and Communication Engineering, Zhejiang University, Zhejiang, China
National Mobile Communications Research Laboratory, Southeast University, Nanjing, China
Received January 27, 2014; in ﬁnal form, November 29, 2014
Abstract—Along with the channel capacity, the error exponent is one of the most important
information-theoretic measures of reliability, because it sets ultimate bounds on the perfor-
mance of communication systems employing codes of ﬁnite complexity. In this paper, we derive
closed-form expressions for the Gallager random coding and expurgated error exponents for
multi-keyhole multiple-input multiple-output (MIMO) channels, which provide insights into a
fundamental tradeoﬀ between the communication reliability and information rate. We inves-
tigate the eﬀect of keyholes on the error exponents and cutoﬀ rate. Moreover, without an
extensive Monte-Carlo simulation we can easily compute the codeword length necessary to
achieve a predeﬁned error probability at a given rate, which quantiﬁes the eﬀects of the number
of antennas, channel coherence time, and the number of keyholes. In addition, we derive exact
closed-form expressions for the ergodic capacity and cutoﬀ rate based on the easily computable
Meijer G-function. Finally, we extend our study to Rayleigh-product MIMO channels and
keyhole MIMO channels.
One of the critical information-theoretic measures of wireless communication systems is the
channel capacity, which, however, only provides the knowledge of the maximum error-free trans-
mission rate achieved with an inﬁnitely long code. However, in practice, we are more interested in
scenarios where ﬁnite length codes are used; hence, a natural question arises of how to understand
the fundamental relationship between the reliability and information rate. In fact, such a rela-
tionship can be characterized by the error exponent, which is a function of the code length L and
information rate R. In general, it is very diﬃcult to obtain the exact error exponent of a particular
channel. Nevertheless, a tight lower bound of the error exponent, also refereed to as the random
coding error exponent (RCEE), was proposed by Gallager .
Being a function of both the transmission rate and the code length, the RCEE provides an
alternative measure to study the fundamental tradeoﬀ between the communication reliability and
information rate of communication systems. Therefore, it has gained enormous attentions from
Supported in part by the UK Engineering and Physical Sciences Research Council (EPSRC), grant
no. EP/I037156/2, funded by the UK government.
Supported in part by the National High-Tech. R&D Program of China under grant 2014AA01A705, the Na-
tional Natural Science Foundation of China (61201229), the Zhejiang Science and Technology Department
Public Project (2014C31051), the Fundamental Research Funds for Central Universities (2014QNA5019),
and the open research fund of National Mobile Communications Research Laboratory, Southeast Univer-
sity (no. 2013D06).