644 IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 8, AUGUST 2007
An Improved Approximation for the Gaussian Q-Function
George K. Karagiannidis, Senior Member, IEEE, and Athanasios S. Lioumpas Student Member, IEEE
Abstract— We present a novel, simple and tight approximation
for the Gaussian Q-function and its integer powers. Compared to
other known closed-form approximations, an accuracy improve-
ment is achieved over the whole range of positive arguments.
The results can be efficiently applied in the evaluation of the
symbol error probability (SEP) of digital modulations in the
presence of additive white Gaussian noise (AWGN) and the
average SEP (ASEP) over fading channels. As an example we
evaluate in closed-form the ASEP of differentially encoded QPSK
in Nakagami-m fading.
Index Terms— Gaussian Q-function, Differentially encoded
QPSK, Nakagami-m fading.
I. INTRODUCTION
T
HE Gaussian Q-function, the directly related error func-
tion erf(·) and its complement function erfc(·)areof
great importance in communication theory problems where
the noise is often characterized by the Gaussian distribution
[1]. Although, efficient numerical methods and infinite series
have been proposed for the calculation of the Q-function [2],
[3], no exact and simple expression is known, appropriate
for mathematical manipulations. Towards this concept, several
approximations have been derived in [4], [5]. In the former,
two approximations are presented; the first is highly accurate
but its mathematical form is complicated in order to be useful
for mathematical manipulations, and the second is simple but
with insufficient accuracy. In [5], a useful and simple approx-
imation that involves the sum of two exponential functions
was presented and the results were applied to the general
problem of evaluating the average symbol error probability
(ASEP) in fading channels. However, its accuracy vanishes
for small arguments.
In this letter, we present a novel, simple and tighter approx-
imation for the Gaussian Q-function and its integer powers,
which can be applied in the evaluation of the ASEP of
digital modulations in additive white Gaussian (AWGN) as
well as fading channels. Contrary to other known closed-form
approximations, a sufficient accuracy is guaranteed for all
positive arguments. This fact is important while -as mentioned
in [5]- bounds or approximations for the Q-function are not
generally suitable for application to average error-probability
Manuscript received March 29, 2007. The associate editor coordinating the
review of this paper and approving it for publication was Dr. Charalambos
Charalambous. This work was performed within the framework of the Satellite
Communications Network of Excellence (SatNEx) project (IST-507052) and
its Phase-II, SatNEx-II (IST- 27393), funded by the European Commission
(EC) under its FP6 program.
The authors are with the Electrical and Computer Engineering Department,
Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece (e-mail:
{geokarag, alioumpa}@auth.gr).
Digital Object Identifier 10.1109/LCOMM.2007.070470.
evaluation, because of the need to average over the signal-
to-noise ratio (SNR) between zero and infinity. By evaluating
in closed-form the ASEP of the differentially encoded QPSK
(DE-QPSK) in Nakagami-m fading, we show that the approx-
imation presented in this letter, can be efficiently applied for
this type of problems, without loss in accuracy.
II. A
N IMPROVED APPROXIMATION FOR THE
GAUSSIAN Q-FUNCTION
The Gaussian Q-function is defined as
Q(x)=
1
2
erfc
x
√
2
(1)
where
erfc(x)=
2
√
π
∞
x
e
−t
2
dt (2)
is the complementary error function [6, (8.2.50)].
The erfc(x) can be represented by a continued fraction as
[7, (7.1.14)]
erfc(x)=
e
−x
2
√
π
1
x+
1/2
x+
1
x+
3/2
x+
2
x+
...
, Re(x) > 0
(3)
A simple upper bound can be obtained by truncation at the
second term as
erfc(x) <h(x)=
e
−x
2
√
πx
. (4)
Compared to other well-known upper bounds, such as the
Chernoff-Rubin [5], (4) is much worse for small arguments,
but much tighter for large arguments. However, it is observed
that by multiplying (4) with a monotonically increasing func-
tion of a specific form, a tightness in the small arguments
region could be also achieved without losing the accuracy
in the large arguments region. For this purpose, we utilize
a function of the form
g(x)=
1 − e
−Ax
B
(5)
where A, B are constant real numbers. In Fig.1, we illustrate
this concept by plotting erfc(x), h(x) and g(x). Thus, the
problem is to find the parameters A and B of the function
f(x, A, B)=
1 − e
−Ax
e
−x
2
B
√
πx
≈ erfc(x) (6)
in order to minimize the integral of the absolute error in the
range of values of interest, i.e.
{A, B} = arg min
{A,B}
1
R
R
0
|f(x, A, B) −erfc(x)|dx (7)
1089-7798/07$25.00
c
2007 IEEE
KARAGIANNIDIS and LIOUMPAS: AN IMPROVED APPROXIMATION FOR THE GAUSSIAN Q-FUNCTION 645
Fig. 1. An approximation of the erfc(x) can be obtained by multiplying g(x)
and h(x).
Fig. 2. Comparison between the erfc(x), (8), (9) and (6).
where [0,R] is the arguments region of interest and |·|
denotes absolute value. The optimal values of A and B depend
on the range in which the integral of the absolute error is
minimized. In other words, the accuracy of the approximation
can be improved for a specific range of values, depending on
the application. In this letter, the optimum values for A, B
were derived so that the erfc(x) is accurately approximated
over the range of positive arguments, which is of practical
interest in most communications systems applications. Thus,
for R =20, it can be found numerically that the integral of
the absolute error is minimized for A =1.98 and B =1.135.
At this point, we should note that the proposed approximation
is more accurate compared to [5, (14)], [4, (9)] for values
close to the origin with lim
x−→0
f(x, A, B)=0.984 as well.
Nonetheless, depending on the application, the accuracy at
the origin could be further improved, by adding the constraint
lim
x−→0
f(x, A, B)=1at the minimization problem given in
(7); this, however, comes the cost of loss in accuracy for the
rest of the region of x.
In Fig. 2 and Table I we compare t he erfc(x) against the
Fig. 3. The relative error of the approximations given by (6) and [5, Eq.
14].
TABLE I
C
OMPARISON BETWEEN ERFC(X), (8), (9) AND (6)
x Erfc(x) Chiani et al. B
¨
orjesson et al. (6)
0.1 0.887537 0.658386 0.394983 0.884027
0.3 0.671373 0.595782 0.365303 0.678248
0.5 0.479500 0.488066 0.314897 0.486562
0.7 0.322199 0.362259 0.255808 0.326246
0.9 0.203092 0.243941 0.197780 0.204350
1 0.157299 0.193112 0.171099 0.157618
2 0.0046777 0.0054665 0.0241455 0.0044654
4 1.54173e-8 1.90275e-8 0.00003246 1.39798e-8
5 1.53746e-12 2.31633e-12 2.9157e-7 1.38062e-12
10 2.08849e-45 6.20013e-45 7.65641e-24 1.84919e-45
approximation given in [5, (14)] as
erfc(x) ≈
1
6
e
−x
2
+
1
2
e
−
4
3
x
2
, (8)
that given in [4, (9)] as
erfc(x) ≈
e
−
x
2
2
√
2π
√
1+x
2
, (9)
and the proposed one given in
(6). As it is observed, for a
wide range of arguments and especially in the low arguments
region, the approximation in (6) is more accurate than (8) and
(9). In Fig. 3, we compare the relative error (RE) i.e.
RE =
|F (x) − erfc(x)|
erfc(x)
(10)
of (8) and (6), where F (x) is the approximated function. The
RE of (9) is too high and cannot be plotted in the same figure.
III. I
NTEGER POW E R S O F T HE Q-FUNCTION
The expression in (6) can be also applied to derive efficient
approximative values for the integer powers of the Gaussian
Q-function, i.e.
Q
n
(x) ≈
1
2
n
f
n
(
x
√
2
,A,B) (11)
646 IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 8, AUGUST 2007
which can be written after the binomial expansion as
Q
n
(x) ≈
1
2
n
n
k=0
n
k
−1)
n−k
e
−A(n−k)
x
√
2
e
−An
x
2
√
2
(B
√
πx)
n
(12)
where
n
k
denotes the binomial coefficient.
A. Evaluation of the ASEP of Differentially Encoded QPSK
in Nakagami-m fading
The integer powers of the Gaussian Q-function are involved
in several communications problems, including the evaluation
of the error probability in AWGN, and the ASEP of coherent
detection of 4-ary orthogonal signaling and DE-QPSK over
fading channels. Simon, in his pioneering work [8], derived
the Craig-type forms up to fourth power of the Gaussian Q-
function and applied them to evaluate the ASEP of DE-QPSK
in fading channels. However, for Nakagami-m fading, the final
formula involves definite integrals and is valid for integer
values of the m parameter [9, (8.124)].
Next, we apply the approximate expression in (12) to eval-
uate in closed-form the ASEP of the DE-QPSK in Nakagami-
m fading. The SNR per symbol, γ, is distributed according to
gamma distribution with a probability density function (pdf)
given by [9, (2.21)]
p
γ
(γ)=
m
m
γ
m−1
γ
m
Γ(m)
e
−
mγ
γ
(13)
where m is the fading parameter, which ranges from 0.5 to
∞,
γ is the average SNR per symbol and Γ(·) denotes the
gamma function [7]. The SEP of DE-QPSK on the AWGN
channel is given by [8]
P
s
(E)=4Q (
√
γ)−8Q
2
(
√
γ)+8Q
3
(
√
γ)−4Q
4
(
√
γ) (14)
The ASEP is obtained by averaging (14) over p
γ
(γ), i.e.
P
s
(E)=
∞
0
P
s
(E)p
γ
(γ)dγ (15)
By substituting (12) in (15), integrals of the form
I
1
=
∞
0
x
c
e
−ax
dx (16)
and
I
2
=
∞
0
e
−ax
e
−b
√
x
x
c
dx (17)
arise, where a>0,b>0,c>−1, which implies that m>2.
These integrals can be evaluated using [6, 3.351.3] and [6,
3.462.1] as
I
1
= a
−c−1
Γ(c +1) (18)
and
I
2
= a
−
3
2
−c
√
aΓ(c +1)
1
F
1
c +1;
1
2
;
b
2
4a
−
bΓ
c +
3
2
1
F
1
c +
3
2
;
3
2
;
b
2
4a
where
1
F
1
(·; ·; ·) denotes the confluent hypergeometric func-
tion [6, 9.21], finally resulting in a closed-form solution for
P
s
(E) which is omitted here due to space limitations.
TABLE II
T
HE COMPARISON OF THE EXACT AND APPROXIMATED ASEP OF THE
DEFERENTIALLY ENCODED
QPSK
m =2.5
γ Exact Chiani et al. B
¨
orjesson et al. (6)
0 0.481372 0.506094 0.400987 0.484639
5 0.213309 0.243585 0.235053 0.213539
10 0.0432554 0.0512244 0.0727512 0.0428847
15 0.00440965 0.00525679 0.0104954 0.00434423
20 0.000309771 0.000369446 0.000877668 0.000304364
25 0.000018772 0.000022386 0.000056820 0.000018427
m =3.5
0 0.476344 0.504489 0.399072 0.479782
5 0.194979 0.226913 0.226411 0.194906
10 0.0288924 0.0347995 0.0597309 0.0284718
15 0.00150192 0.0018022 0.00544383 0.00146504
20 0.000040809 0.000048692 0.00020369 0.000039628
25 8.39755e-7 9.99544e-7 4.76981e-6 8.14168e-7
B. Numerical Results and Discussion
In Table II, we compare the exact ASEP of the DE-
QPSK, calculated by numerical integration using (15) with
the approximated ones using (12) and the corresponding ones
in (8), (9) and (6). It is observed that (12) results in a tighter
approximation of the ASEP over the whole range of the SNR
values.
We should note here that as mentioned in [8], although for
the AWGN channel the first two terms of (14) are typically
sufficient for a good approximation of the error performance
in high SNR applications, in fading channel all four terms
are important because of the need to average over the SNR
between zero and infinity. This fact strengthens the argument
that (6) can be efficiently applied for the evaluation of error
probabilities in fading channels, since a declination from the
exact value would lead in approximation errors, especially in
the low SNR region.
R
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![Fig. 3. The relative error of the approximations given by (6) and [5, Eq. 14].](/figures/fig-3-the-relative-error-of-the-approximations-given-by-6-3ddyqg2j.webp)