Saddlepoint Approximation of
Random–Coding Bounds
Alfonso Martinez
Centrum Wiskunde & Informatica
1090 GB Amsterdam, The Netherlands
alfonso.martinez@ieee.org
Albert Guille´n i Fa`bregas
Department of Engineering
University of Cambridge
Cambridge, CB2 1PZ, UK
guillen@ieee.org
Abstract— This paper considers random-coding bounds to the
decoding error probability with maximum-metric mismatched
decoders. Their error exponents are determined and the saddle-
point approximation to the corresponding probability is derived.
This approximation is accurate and allows for simple numerical
evaluation, as verified for several channels of interest.
I. INTRODUCTION
Two of the fundamental problems in the field of channel
coding are characterizing the error probability attained by a
given code construction and finding the largest achievable rate
with vanishing error probability. While research has chiefly
focused on the second problem, that of finding the channel
capacity, recently, spurred by the construction of near-capacity
achieving codes, renewed attention has been paid to the error
probability in the finite-length regime. In particular, Polyan-
skiy et al. [1] have derived a number of new results, such as the
random-coding union (RCU) bound, the dependence-testing
bound (DT), and the κβ bound among others. A key quantity
in their development is the information density, defined as
i(x, y) = log
PY |X(y|x)
PY (y)
(1)
where PY |X(y|x) is the vector channel transition probability
and x, y are the channel input and output sequences, re-
spectively. Moreover, these bounds have been coupled with
Strassen’s Gaussian approximation [2] to the error probability
around capacity, thereby providing an estimate of the effective
capacity for finite block length and non-zero error probability.
Glossing over the details, the key observation is that, for
memoryless channels, the information density is expressed as a
sum of random variables, which suggests the application of the
central limit theorem and leads to a Gaussian approximation.
In this paper, we rederive the RCU and DT bounds within
the framework of mismatched decoding. Whereas maximum
information-density decoders apply a maximum-likelihood de-
coding rule, a decoder is said mismatched [3], [4] if it selects
the message vˆ with largest decoding metric q(x(v), y), i. e.
vˆ = arg max
v
q(x(v), y), (2)
This work has been supported by the International Joint Project 2008/R2
of the Royal Society, the FP7 Network of Excellence in Wireless Communi-
cations (NEWCOM++), and the Isaac Newton Trust.
where q(x(v), y) need not be the channel likelihood metric.
As we shall see, the information density is naturally replaced
by a generalized information density is(x, y), given by
is(x, y) = log
q(x, y)s
E[q(X ′, y)s]
, (3)
where s ≥ 0. The cumulant generating function of this
generalized information density is closely related to Gallager’s
E
0
(ρ, s) function [5]. Indeed, for an i.i.d. codebook and a
memoryless channel with metric q(x, y) =
∏n
i=1 q(xi, yi),
letting τ be an arbitrary complex number, we have that
κ(τ) = log E[eτis(X,Y )] (4)
=
n
∑
i=1
log E
[
(
q(Xi, Yi)s
E[q(X ′i, Yi)s|Yi]
)τ
]
(5)
= −nEˆ
0
(−τ, s), (6)
where Eˆ
0
(ρ, s) is the Gallager function for mismatched decod-
ing [3]. Setting q(x, y) = PY |X(y|x) gives the usual E0(ρ, s).
More precisely, we express the RCU and DT bounds as the
tail probability of a random variable, a form which allows us
to determine the error exponent attained by these bounds in
terms of Gallager’s E
0
(ρ, s) function. Moreover, this form as
a tail probability allows us to use the saddlepoint (or Laplace)
approximation. While this approximation is essentially as easy
to compute as the error exponent or the Gaussian approxima-
tion, it turns out to be more accurate, and thus provides an
efficient method to estimate the effective capacity for finite
block length and non-zero error probability.
Notation: Random variables are denoted by capital letters
and their realization by small letters. Sequences are identified
by a boldface font. The probability of an event is denoted
by Pr{·} and the expectation operator is denoted by E[·].
Logarithms are in natural units and information rates in nats,
except in the examples, where bits are used.
II. UPPER BOUNDS TO THE ERROR PROBABILITY
We adopt the conventional setup in channel coding.
First, and for a given information message v, with v ∈
{1, 2, . . . ,M}, the encoder outputs a codeword of length n
x(v) ∈ Xn, where X is the symbol channel input alphabet.
One could consider more general vector alphabets and the