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The Complementary Error Function
Frank R. Kschischang
Department of Electrical & Computer Engineering
University of Toronto
April 10, 2017
The complementary error function, erfc(x), is defined, for x ≥ 0, as
erfc(x) = 2
x
π
exp(−u^2 ) du.
The complementary error function represents the area under the two tails of a zero-mean Gaussian probability density function with variance σ^2 = 1/2, as illustrated in Fig. 1. The so-called “error function,” erf(x), is defined via
erf(x) = 1 − erfc(x).
From the fact that a probability density function has unit integral, we see that
erfc(0) = 1.
The complementary error function erfc(x) is plotted in Fig. 2, along with an upper and a lower bound (established in Appendix A). The bounds are asymptotically tight, i.e., the difference between the bound and the actual function converges to zero for large x.
−x x
u
√^1 π exp(−u
Figure 1: The complementary error function is defined as the area under the two Gaussian pdf tails shown.
− 8
10 −^7
10 −^6
10 −^5
10 −^4
10 −^3
10 −^2
10 −^1
exp(−x^2 ) x√π
√^ 2 exp(−x^2 ) π(x+√x^2 +2)
x
erfc(
x) and Bounds
Upper bound Lower bound erfc(x)
Figure 2: The function erfc(x) plotted together with an upper bound and a lower bound as indicated.
Table 1 gives a mapping from a desired value of erfc(
x) to the value of x that achieves this value. This table can often be used, in digital communications, to determine the signal-to- noise ratio needed to achieve a target error rate.
The complementary error function is part of the standard math library provided with the C programming language (simply #include <math.h>) and is also provided by standard math packages such as Matlab.
Some digital communications textbooks prefer to define error probabilities in terms of the so-called Q-function, defined, for x ≥ 0, via
Q(x) =
x
2 π
exp
u^2 2
du.
This is the area under a single tail of a zero-mean Gaussian of a zero-mean Gaussian prob- ability density function with unit variance. The Q-function and the complementary error function are obviously closely related; indeed
Q(x) =
erfc
√x 2
and erfc(x) = 2Q
x
A Bounds on the Complementary Error Function
Let X be a Gaussian random variable with probability density function
f (x) =
π
exp(−x^2 ),
and, for z ≥ 0, let
erfc(z) = 2P [X > z] =
π
z
exp(−x^2 ) dx.
Note that erfc(0) = 1. Throughout this appendix, we constrain z ≥ 0.
For n a non-negative integer, let Mn(z) = E[Xn^ | X > z] denote the conditional nth moment of X, given that X > z. Then
M 0 (z) = 1
M 1 (z) =
z √x π exp(−x
(^2) ) dx 1 2 erfc(z)^
π erfc(z)
z
2 x exp(−x^2 ) dx
exp(−z^2 ) √ π erfc(z)
and, for n ≥ 2, integrating by parts, taking
u = xn−^1 du = (n − 1)xn−^2 dx dv = 2x exp(−x^2 ) dx v = − exp(−x^2 ),
we get
Mn(z) =
z √xn π exp(−x
(^2) ) dx 1 2 erfc(z) =
π erfc(z)
z
2 x exp(−x^2 )xn−^1 dx
π erfc(z)
zn−^1 exp(−z^2 ) +
n − 1 2
z
2 xn−^2 exp(−x^2 ) dx
zn−^1 exp(−z^2 ) √ π erfc(z)
n − 1 2
Mn− 2 (z)
= zn−^1 M 1 (z) +
n − 1 2
Mn− 2 (z).
Thus, for example,
M 2 (z) = zM 1 (z) +
M 3 (z) = z^2 M 1 (z) + M 1 (z) = (z^2 + 1)M 1 (z),
M 4 (z) = z^3 M 1 (z) +
M 2 (z) = (z^3 +
z)M 1 (z) +
M 5 (z) = z^4 M 1 (z) + 2M 3 (z) = (z^4 + 2z^2 + 2)M 1 (z),
etc.
A simple upper bound on erfc(z) arises from the observation that M 1 (z) > z, from which it follows that
erfc(z) <
exp(−z^2 ) √ πz
A lower bound on erfc(z) arises from the observation that the conditional variance is positive, i.e., E((X − M 1 (z))^2 | X > z] = M 2 (z) − M 12 (z) > 0. We then have
zM 1 (z) +
− M 1 (z)^2 > 0
Since this parabola in M 1 (z) opens downwards, we have that M 1 (z) cannot exceed the largest parabolic zero crossing, i.e.,
M 1 (z) <
z +
z^2 + 2 2
from which it follows that
erfc(z) >
2 exp(−z^2 ) √ π(z +
z^2 + 2)
