Q-function

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A plot of the Q-function.

In statistics, the Q-function is the tail probability of the standard normal distribution .[1][2] In other words, is the probability that a normal (Gaussian) random variable will obtain a value larger than standard deviations above the mean. Equivalently, is the probability that a standard normal random variable takes a value larger than .

If is a Gaussian random variable with mean and variance , then is standard normal and

where .

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[edit]

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 fixed and finite.

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.
  • Improved exponential bounds and a pure exponential approximation are [5]
  • A tight approximation of for is given by Karagiannidis & Lioumpas (2007)[6] 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 related to the inverse error functions:

The function finds application in digital communications. It is usually expressed in dB and generally called Q-factor:

where y is the bit-error rate (BER) of the digitally modulated signal under analysis. For instance, for QPSK in additive white Gaussian noise, the Q-factor defined above coincides with the value in dB of the signal to noise ratio that yields a bit error rate equal to y.

Q-factor vs. bit error rate (BER).

Values[edit]

The Q-function is well tabulated and can be computed directly in most of the mathematical software packages such as R and those available in Python, MATLAB and Mathematica. Some values of the Q-function are given below for reference.

Generalization to high dimensions[edit]

The Q-function can be generalized to higher dimensions:[7]

where follows the multivariate normal distribution with covariance and the threshold is of the form for some positive vector and positive constant . As in the one dimensional case, there is no simple analytical formula for the Q-function. Nevertheless, the Q-function can be approximated arbitrarily well as becomes larger and larger.[8]

References[edit]

  1. ^ The Q-function, from cnx.org
  2. ^ a b Basic properties of the Q-function Archived March 25, 2009, at the Wayback Machine.
  3. ^ Normal Distribution Function - from Wolfram MathWorld
  4. ^ John W. Craig, A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations, Proceedings of the 1991 IEEE Military Communication Conference, vol. 2, pp. 571–575.
  5. ^ Chiani, M., Dardari, D., Simon, M.K. New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels (2003). IEEE Transactions on Wireless Communications, 4(2), pp. 840–845. doi=10.1109/TWC.2003.814350.
  6. ^ Karagiannidis, G. K., & Lioumpas, A. S. An improved approximation for the Gaussian Q-function. 2007. Communications Letters, IEEE, 11(8), pp. 644-646.
  7. ^ Savage, I. R. (1962). "Mills ratio for multivariate normal distributions". Journal Res. Nat. Bur. Standards Sect. B. 66: 93–96. 
  8. ^ Botev, Z. I. (2016). "The normal law under linear restrictions: simulation and estimation via minimax tilting". Journal of the Royal Statistical Society, Series B (Statistical Methodology). doi:10.1111/rssb.12162.