2013 International Conference on Communication Systems and Network Technologies
Gaussian Q-function and its Approximations Vinay Kumar Pamula, Member, IEEE, Department of ECE, University College of Engineering, JNTUK, Kakinada, India 533003.
[email protected]
Sirisha Malluri, Member, IEEE, Department of ECE, DVR & Dr.HS MIC College of Technology, Kanchikacherla, India 521180.
[email protected]
paper presents some useful approximations to the Gaussian Q-function with an application to computation of bit error probability (BEP) of M-ary phase shift keying (MPSK) modulation scheme. Computational results are provided to compare various approximations to Gaussian Q-function.
The Q-function and complementary error function are related by [1]
Keywords- Absolute error; bit error probability; Nakagami fading; phase shift keying; Q-function.
A pair of upper and lower bounds on the Gaussian Qfunction are presented in [3]. The following are the tight bounds on the Q(x), valid only for all x 0. The Upper bound and lower bound are, respectively, given by [3]
Abstract—This
I.
Q( x) =
INTRODUCTION
The Gaussian Q-function, popular in communication theory literature, is frequently found in the analysis of multiantenna communication systems over fading channels [1]. These functions are tabulated, and often available as built-in functions in mathematical software tools. However, in many cases it is useful to have closed-form bounds or approximations instead of the exact expression. In fact, these approximations are particularly useful in evaluating the bit error probability (BEP) in many communication theory problems. The Gaussian Q-function’s definition in the form of an improper integral makes it hard to conduct exact analyses for communication systems. Thus it would be highly desirable to obtain a closed-form using elementary functions. However no such solution is possible. The only option has been to approximate. A number of approximations have been proposed by mathematicians, but the search continuous. The remaining part of the paper is organized as: section II provides some approximations to Q-function. In Section III, an example on the application of the above approximations is discussed and the average BEP of MPSK is computed. In section IV, the absolute error for the approximations is discussed evaluated. And concluding remarks are offered in section V. II.
Q( x) ≥ e
Q( x) ≤ e
Q( x) =
2π
x
978-0-7695-4958-3/13 $26.00 © 2013 IEEE DOI 10.1109/CSNT.2013.25
³e
−ξ 2
dξ .
50
).
(2)
2
+e
− x 2 /2
.
1
(3)
2( x + 1)
− x2
1
.
12
+e
− x 2 /2
1
.
2π ( x + 1)
.
(4)
Another approximation to the Q-function is given in terms of erfc in [4, (14)] as erfc ( x ) ≈
e
− x2
6
+
1
+
1
e
4 − x2 3
(5)
2
i.e., Q( x) ≈
e
− x 2 /2
12
e
2 − x2 3
.
(6)
4
The approximation (5) works well for some problems, but large errors on small arguments limit its application, which was denoted as the CDS approximation in [5]. The approximation (7) is called GKAL approximation of Karagiannidis & Lioumpas [6] gives smaller approximation error and is given by
The Gaussian Q-function, which relates the complementary error function erfc(x), is important in performance analysis of wireless communication systems over fading channels. The Q-function and erfc are related by [2, (4.1.1)] ∞
2
1
.
x
erfc (
and
Q-FUNCTION APPROXIMATIONS
1
− x2
1
(1)
Q( x) ≈
§ · − x 2 /2 ¨1− 1 e− ax ¸ e ¨ ¸ 2 © ¹
b 2π x 74
(7)
Pe = AQ ( Bλ )
where a = 1.98 and b = 1.135. The accuracy with which (7) represents the actual Q-function is quite remarkable. However the presence of x in the denominator of (7) makes it difficult to evaluate the probability of error [7]. Another approximation to Q-function [8, (8)] is
Q( x) ≈
e
where A and B are constants. The value of A is typically a function of number of symbols and B is proportional to minimum Euclidean distance between the source symbols. The values of A and B for MPSK are 2 and
− x 2 /2 2
1.64 + 0.76 x + 4
.
(8)
2
III.
Bλ − § − Bλ · m m −1 − m λ · 2 e e ¨ ¸§§ m · λ Pe = ³ e λ ¸d λ + ¨¨ ¨ ¸ ¸ ¨ ¸ 0 ¨ 12 2π ( Bλ + 1) ¸ © © λ ¹ Γ ( m ) ¹ © ¹ m m −1 − m λ · ∞ § e − Bλ · § § m · λ e λ ¸d λ = ³¨ ¨ ¸ ¨ ¸ ¸ 0 © 50 ¹ ¨ © λ ¹ Γ ( m ) © ¹ ∞
2
/2 .
(9)
A. Importance of Nakagami fading channel There are several probability distributions that have been used to model the statistical characteristics of the fading channel. When there are a large number of scatterers in the channel that contribute to the signal at the receiver, as is the case in ionospheric or tropospheric signal propagation, application of the central limit theorem leads to a Gaussian process model for the channel impulse response. If the process is zero-mean, then the envelope of the channel impulse response at any instant has a Rayleigh probability distribution and an alternative statistical model to represent channel response in the Nakagami-m distribution [10]. In contrast to the Rayleigh distribution, which has a single parameter that can be used to match the fadingchannel statistics, the Nakagami-m is a two parameter distribution, with the parameter distribution, with the parameters m and . As a consequence this distribution provides more flexibility and accuracy in matching the observed single statistics. The Nakagami-m distribution can be used to model fading channel conditions that are either more or less severe than the Rayleigh distribution as a special case (m = 1) and this distribution provides the best fit for data signals received in urban radio channels [10]. As an application to fading channel, Nakagami fading will be considered in this paper. The probability density function of instantaneous received SNR on Nakagami fading channel is given by [6, (13)]
Bλ 2
e
.
(12)
The first term of (12) is reduced into closed-form as [12] A§
m
· ¨ ¸ . The second term of (12) will not reduce 50 © m + B λ ¹ m
into closed-form, in terms of elementary functions. Numerical integral technique will be used to solve it. The Average BEP with lower bound (4) is given by Bλ − § − Bλ · m m −1 − m λ · 2 e e ¨ ¸§§ m · λ Pe = ³ e λ ¸d λ + ¨¨ ¨ ¸ ¸ ¸ 0¨ m 12 Γ 2π ( Bλ + 1) ¸ © © λ ¹ ( ) ¨ ¹ © ¹ m m −1 − m λ · ∞ § e − Bλ · § § m · λ e λ ¸d λ = ³¨ ¨ ¸ ¨ ¸ ¸ 0 © 12 ¹ ¨ © λ ¹ Γ ( m ) © ¹ ∞
+³
λ
m −1 − m §m· λ e λ ¸ m Γ λ ( ) © ¹
−
§ § m ·m λ m −1 − m λλ · e +³ ¨¨ ¸ ¸¸ d λ 0 2( Bλ + 1) ¨ © λ ¹ Γ ( m ) © ¹ ∞
∞
f (λ ) = ¨
The average BEP with Q-
function approximation (3) is given by [10, (3)]
APPLICATION TO FADING CHANNELS
m
§π · ¸ respectively. ©M ¹
2 sin ¨
And one better known approximation is the Chernoff bound [9]
Q ( x ) ≈ 1 e− x 2
(11)
0
(10)
−
Bλ 2
§ § m ·m λ m−1 − m λλ · e ¨¨ ¸ ¸¸ d λ 2π ( Bλ + 1) ¨© © λ ¹ Γ ( m ) ¹ Ae
. (13) The closed-form expression to the first term of (12) is
where m is fading factor, λ is average output received SNR.
A§
m
· ¨ ¸ . And there is no closed-form solution for 12 © m + B λ ¹
B. Average bit error probability The average BEP for MPSK will be derived in this section. The conditional probability of error for MPSK is given by [11]
m
the second term in terms of elementary functions, so the second term integration is solved by numerical integration
75
m
technique. The average BEP using approximation (6) is given by
§ − B2λ ¨e Pe = A ³ 0 ¨ 12 ¨ © § − Bλ ∞ e 2 = A³ ¨ 0¨ ¨ 12 © ∞
+
e
−
2 Bλ 3
· m m −1 − m λ ¸§ m · λ e λ dλ ¨ ¸ 4 ¸¸ © λ ¹ Γ ( m ) ¹
· m m −1 − m λ · ¸§§ m · λ e λ ¸d λ ¨ ¸ ¸¸ ¨ ¨© λ ¸¹ Γ ( m ) © ¹ ¹
§ − 2 B3λ e +A³ ¨ ¨ 0 ¨ 4 © ∞
Pe =
. (14)
The closed-form representation of (14) is m
§ · ¸ A¨ m ¸ + ¨ 4¨ 2Bλ ¸ ¸ ¨m+ ¹ 3 ©
IV.
m
· ¸ ¸ . ¸ ¸ ¹
Pe =
m
∞
¨ ¸ ³λ
Γ(m) © λ ¹
m −1
(1 − e
− a Bλ
2
b 2π Bλ
0
)e (
)
− B 2+ m λ λ
CALCULATION OF ABSOLUTE ERROR
Although all the approximations to Q-function are may be accurate, but cannot be applied practically, because in some applications accuracy is not a desired parameter of interest. And in some applications accuracy becomes primary parameter of interest but at a cost of complexity in the approximate function. One of the measures of accuracy of these approximations is absolute error. The absolute error can be calculated using [13]
(15)
The average probability of error using Q-function approximation (7) is given by A §m·
'
Qabs = Q ( x ) − Q ( x )
dλ .
(16) The integration in (16) will not be reduced into closed-form for all values of m. This can also be solved by numerical integration technique. The average probability of error using (8) is given by
Pe =
A §m·
m
∞
¨ ¸ ³λ Γ(m) © λ ¹ 0
m −1
e
§ B m· −¨ + ¸λ ©2 λ¹
1.64 + 0.76 Bλ + 4
d λ . (17)
The integration in (17) will not be reduced into closed-form for all values of m and hence numerical integration has to be used. The average BEP, using Q-function approximation (9), is given as
§ Ae − Bλ Pe = ³ ¨ 0© 2 ∞
m λ · § § m · λ m−1 − m λ · e dλ . ¸¨¨ ¸ ¸¸ ¹ ©¨ © λ ¹ Γ ( m ) ¹
(19)
The average BEP of MPSK signals over the Nakagami channel using all the Q-function approximations (3), (4), (6), (7), (8), (9) is plotted in Fig. 1. The average BEP of MPSK signals over the Nakagami-m channel using all the Q-function approximations, (3), (4), (6), (7), (8), (9) is plotted in Fig. 1. Variation of the average BEP for all approximations for a range of values of average signal-tonoise ratio (SNR) is shown. The average BEP decreases with increase in average SNR. Average BEP with approximation (3) has small value at origin when compared to remaining approximations and it is large with approximation (8). It is also clear that average BEP computed with (3), (4), (6) and (8) is almost similar to that of exact average BEP computed using numerical integration (NI) method. Average BEP using approximations (6) and (8) is exactly same.
· m m −1 − m λ · ¸§§ m · λ e λ ¸d λ ¸ ¸ ¨¨ ¨© λ ¸¹ Γ ( m ) ¸© ¹ ¹
§ A¨ m Pe = ¨ 12 ¨ Bλ ¨m+ © 2
A§ · m . 2 ¨© m+ B λ /2 ¸¹
(18) Figure 1. Average BEP Vs Average SNR (dB) with m = 1, 10.
The closed-form expression to (18) is given by
76
(20)
REFERENCES [1] [2]
[3] [4]
[5] [6]
[7]
[8]
[9] Figure 2. Absolute Error Vs x..
V.
CONCLUSION
[10]
Some useful approximations to Gaussian Q-function are presented in this paper. As an application of Q-function approximations, the average BEP of MPSK modulation scheme over Nakagami-m fading channels is computed. Absolute error is also computed to study the accuracy of these approximations.
[11]
[12] [13]
77
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