Q-function: Difference between revisions
m Correctly formatted the formula used in reference 6 for the argument minimum over {A, B} for the approximation of Q(x). |
Clarified the expression for the measurement of error in the approximation for Q(x). |
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: <math>f(x; A, B) = \frac{\left(1 - e^{-Ax}\right)e^{-x^2}}{B\sqrt{\pi} x} \approx \operatorname{erfc} \left(x\right).</math> |
: <math>f(x; A, B) = \frac{\left(1 - e^{-Ax}\right)e^{-x^2}}{B\sqrt{\pi} x} \approx \operatorname{erfc} \left(x\right).</math> |
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: The absolute error |
: The absolute error between <math>f(x; A, B)</math> and <math>\operatorname{erfc}(x)</math> over the range <math>[0, R]</math> is minimized by evaluating |
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: <math>\{A, B\} = \underset{\{A,B\}}{arg\ min} \frac{1}{R} \int_0^R | f(x; A, B) - \operatorname{erfc}(x) |dx.</math> |
: <math>\{A, B\} = \underset{\{A,B\}}{arg\ min} \frac{1}{R} \int_0^R | f(x; A, B) - \operatorname{erfc}(x) |dx.</math> |
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: Using <math>R = 20</math> and numerically integrating, they found the minimum error occurred when <math>\{A, B\} = \{1.98, 1.135\},</math> which gave a good approximation for <math>\forall x \ge 0.</math> |
: Using <math>R = 20</math> and numerically integrating, they found the minimum error occurred when <math>\{A, B\} = \{1.98, 1.135\},</math> which gave a good approximation for <math>\forall x \ge 0.</math> |
Revision as of 23:28, 26 September 2015
In statistics, the Q-function is the tail probability of the standard normal distribution .[1][2] In other words, Q(x) is the probability that a normal (Gaussian) random variable will obtain a value larger than x standard deviations above the mean.
If the underlying random variable is y, then the proper argument to the tail probability is derived as:
which expresses the number of standard deviations away from the mean.
Other definitions of the Q-function, all of which are simple transformations of the normal cumulative distribution function, are also used occasionally.[3]
Because of its relation to the cumulative distribution function of the normal distribution, the Q-function can also be expressed in terms of the error function, which is an important function in applied mathematics and physics.
Definition and basic properties
Formally, the Q-function is defined as
Thus,
where is the cumulative distribution function of the normal Gaussian distribution.
The Q-function can be expressed in terms of the error function, or the complementary error function, as[2]
An alternative form of the Q-function known as Craig's formula, after its discoverer, is expressed as:[4]
This expression is valid only for positive values of x, but it can be used in conjunction with Q(x) = 1 − Q(−x) to obtain Q(x) for negative values. This form is advantageous in that the range of integration is finite.
- The Q-function is not an elementary function. However, the bounds
- become increasingly tight for large x, and are often useful.
- Using the substitution v =u2/2, the upper bound is derived as follows:
- Similarly, using and the quotient rule,
- Solving for Q(x) provides the lower bound.
- The Chernoff bound of the Q-function is
- Improved exponential bounds and a pure exponential approximation are [5]
- A tight approximation of for is given by Karagiannidis & Lioumpas (2007)[6][failed verification] who showed for the appropriate choice of parameters that
- The absolute error between and over the range is minimized by evaluating
- Using and numerically integrating, they found the minimum error occurred when which gave a good approximation for
- Substituting these values and using the relationship between and from above gives
Inverse Q
The inverse Q-function can be trivially related to the inverse error function:
Values
The Q-function is well tabulated and can be computed directly in most of the mathematical software packages such as Matlab and Mathematica. Some values of the Q-function are given below for reference.
Q(0.0) = 0.500000000 = 1/2.0000 |
Q(1.0) = 0.158655254 = 1/6.3030 |
Q(2.0) = 0.022750132 = 1/43.9558 |
Q(3.0) = 0.001349898 = 1/740.7967 |
This article needs additional citations for verification. (September 2011) |
References
- ^ The Q-function, from cnx.org
- ^ a b Basic properties of the Q-function
- ^ Normal Distribution Function - from Wolfram MathWorld
- ^ John W. Craig, A new, simple and exact result for calculating the probability of error for two-dimensional signal constellaions, Proc. 1991 IEEE Military Commun. Conf., vol. 2, pp. 571–575.
- ^ Chiani, M., Dardari, D., Simon, M.K. (2003). New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels. IEEE Transactions on Wireless Communications, 4(2), 840–845, doi=10.1109/TWC.2003.814350.
- ^ Karagiannidis, G. K., & Lioumpas, A. S. (2007). An improved approximation for the Gaussian Q-function. Communications Letters, IEEE, 11(8), 644-646.