ISSN 0032-9460, Problems of Information Transmission, 2014, Vol. 50, No. 2, pp. 144–170.
Pleiades Publishing, Inc., 2014.
Original Russian Text
J. Xue, M.Z.I. Sarkar, T. Ratnarajah, 2014, published in Problemy Peredachi Informatsii, 2014, Vol. 50, No. 2, pp. 31–59.
Error Exponents for Nakagami-m Fading Channels
J. Xue, M. Z. I. Sarkar, and T. Ratnarajah
Institute for Digital Communications (IDCOM), the University of Edinburgh, United Kingdom
J.Xue@ed.ac.uk email@example.com T.Ratnarajah@ieee.org
Received 12.02.2013; in ﬁnal form, 27.10.2013
Abstract—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 im-
portant 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 ensem-
ble, 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
The error exponent describes a decaying rate in the error probability as a function of the code-
word length, and hence serves to indicate how diﬃcult it may be to achieve a certain level of
reliability in communication at a rate below the channel capacity. In addition, the error expo-
nent is the logarithm of the error probability of fading channels over the length of the code .
Therefore, it is important to study how the probability of error drops as the length of the code
increases because of the limitation of code length in practice. Although it is more diﬃcult to ﬁnd
the exact error exponent than the channel capacity , its classical lower bound is available due to
Gallager . This lower bound is known as random coding error exponent (RCEE) or Gallager’s
exponent, and has been used to estimate the codeword length required to achieve a prescribed
upper bound for error probability [1, 3].
In , the RCEE was extensively studied for single-input single-output (SISO) and single-input
multiple-output (SIMO) Rayleigh ﬂat-fading channels with average or peak power constraint. With
perfect channel-state information, the expression of the RCEE for SIMO block-fading channels was
derived in  and showed that the error exponent decreases with reduction of eﬀective codeword
length. Moreover, it was shown that, although capacity is independent of channel coherence time,
the error exponent is aﬀected by the channel coherence time and was found to decrease with it.
Supported by the UK Engineering and Physical Sciences Research Council (EPSRC), grant no.
EP/IO37156/2, funded by the UK government.