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1 Definitions
The Q-function is tail integral of a unit-Gaussian pdf, and is defined as
Q(z) =∆
∫ (^) ∞
z
2 π
e
− 2 x 2 dx.
The Q-function has the following properties:
zlim→∞ Q(z) = 0 lim z→−∞ Q(z) = 1 Q(0) = 1/ 2 Q(−z) = 1 − Q(z).
There are several other common notations used to denote this integral function or a close relative. The Q-function is sometime referred to as the “Gaussian Integral Function” and denoted GIF(z). Other functions which are closely related are the erf(·) (error function) and erfc(·) (complementary error function):
erf(z) ∆ =
∫ (^) z
0
π
e−x
2 dx z ≥ 0
erfc(z) ∆ =
∫ (^) ∞
z
π
e−x
2 dx = 1 − erf(z) z ≥ 0
The Q-function is related to these functions by
Q(z) =
[ 1 − erf
( z √ 2
)]
erfc
( z √ 2
) z ≥ 0.
It is clear that if X(u) is a mean zero, unit variance Gaussian random variable, that
Q(z) = 1 − FX(u)(z).
A useful relation is that if Y (u) is Gaussian with mean m and variance σ^2 , then
Pr {Y (u) > a} = Q
( (^) a − m
σ
) .
2 Numerical Computation
The Q-function must be evaluated numerically; there is no closed form solution for the inte- gral. All numerical methods are the result of a trade-off between computational complexity
By George K. Karagiannidis Electrical & Computer Engineering Dept Aristotle University of Thessaloniki
and accuracy. The range for z over which the approximation is valid is also a concern. The numerical approximation which I find most useful is given by^1
Q(z) ≈ (a 1 t + a 2 t^2 + a 3 t^3 + a 4 t^4 + a 5 t^5 )e
−z^2 (^2) z ≥ 0 ,
where
t =
1 + Bz
B = 0. 231641888
a 1 = 0. 127414796 a 2 = − 0. 142248368 a 3 = 0. 7107068705 a 4 = − 0. 7265760135 a 5 = 0. 5307027145.
The associated approximation error is guaranteed to be less than 1. 5 × 10 −^7. I have found that this approximation is acceptable for all practical values of z.
Another useful concept is a simple over-bound. This allows a “worst-case” scenario to be quickly evaluated. The most common overbound is
Q(z) ≤
2 πz
e
− 2 z 2 z > 0.
This bound becomes quite “tight” for large z.
The Q-function and the over-bound are plotted in Figures ??-??. The plot of Figure ?? is on a log-scale to emphasize the behavior for large z.
The Q-function is tabulated in Table 1 for z = 0 to 10. The values of z for which Q(z) = 10−k^ for k = 1, 2... 10 are also given.^2
(^1) This is adapted from the erf(·) approximation of equation 7.1.26 in M. Abramowitz and A. Stegun,
Handbook of Mathematical Functions, Dover. Less complex approximations can also be found therein. (^2) The values in Table 1 were calculated using the approximation for z < 4. The values for z ≥ 4 as well
as those in the inverse Q table were taken from Albert Leon-Garcia, Probability and Random Processes for Electrical Engineering, Addison Wesley, 1989.
- 0.0 5.000e-01 3.0 1.350e- z Q(z) z Q(z)
- 0.1 4.602e-01 3.1 9.677e-
- 0.2 4.207e-01 3.2 6.872e-
- 0.3 3.821e-01 3.3 4.835e-
- 0.4 3.446e-01 3.4 3.370e-
- 0.5 3.085e-01 3.5 2.327e-
- 0.6 2.743e-01 3.6 1.591e-
- 0.7 2.420e-01 3.7 1.078e-
- 0.8 2.119e-01 3.8 7.237e-
- 0.9 1.841e-01 3.9 4.812e-
- 1.0 1.587e-01 4.0 3.17e-
- 1.1 1.357e-01 4.5 3.40e-
- 1.2 1.151e-01 5.0 2.87e-
- 1.3 9.680e-02 5.5 1.90e-
- 1.4 8.076e-02 6.0 9.87e-
- 1.5 6.681e-02 6.5 4.02e-
- 1.6 5.480e-02 7.0 1.28e-
- 1.7 4.457e-02 7.5 3.19e-
- 1.8 3.593e-02 8.0 6.22e-
- 1.9 2.872e-02 8.5 9.48e-
- 2.0 2.275e-02 9.0 1.13e-
- 2.1 1.786e-02 9.5 1.05e-
- 2.2 1.390e-02 10.0 7.62e-
- 2.3 1.072e-
- 2.4 8.198e-
- 2.5 6.210e-
- 2.6 4.661e-
- 2.7 3.467e-
- 2.8 2.555e-
- 2.9 1.866e-
- 1e-01 1. Q(z) z
- 1e-02 2.
- 1e-03 3.
- 1e-04 3.
- 1e-05 4.
- 1e-06 4.
- 1e-07 5.
- 1e-08 5.
- 1e-09 5.
- 1e-10 6.
