Q-function

In statistics, the Q-function is the tail distribution function 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. Equivalently, is the probability that a standard normal random variable takes a value larger than .

A plot of the Q-function.

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 propertiesEdit

Formally, the Q-function is defined as

 

Thus,

 

where   is the cumulative distribution function of the standard 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.

Craig's formula was later extended by Behnad (2020)[5] for the Q-function of the sum of two non-negative variables, as follows:

 

Bounds and approximationsEdit

  • The Q-function is not an elementary function. However, the bounds, where   is the density function of the standard normal distribution,[6]
 
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 geometric mean of the upper and lower bound gives a suitable approximation for  :
 
  • Tighter bounds and approximations of   can also be obtained by optimizing the following expression [6]
 
For  , the best upper bound is given by   and   with maximum absolute relative error of 0.44%. Likewise, the best approximation is given by   and   with maximum absolute relative error of 0.27%. Finally, the best lower bound is given by   and   with maximum absolute relative error of 1.17%.
 
  • Improved exponential bounds and a pure exponential approximation are [7]
 
 
  • The above were generalized by Tanash & Riihonen (2020),[8] who showed that   can be accurately approximated or bounded by
 
In particular, they presented a systematic methodology to solve the numerical coefficients   that yield a minimax approximation or bound:  ,  , or   for  . With the example coefficients tabulated in the paper for  , the relative and absolute approximation errors are less than   and  , respectively. The coefficients   for many variations of the exponential approximations and bounds up to   have been released to open access as a comprehensive dataset.[9]
  • Another approximation of   for   is given by Karagiannidis & Lioumpas (2007)[10] 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
 
  • A tighter and more tractable approximation of   for positive arguments   is given by López-Benítez & Casadevall (2011)[11] based on a second-order exponential function:
 
The fitting coefficients   can be optimized over any desired range of arguments in order to minimize the sum of square errors ( ,  ,   for  ) or minimize the maximum absolute error ( ,  ,   for  ). This approximation offers some benefits such as a good trade-off between accuracy and analytical tractability (for example, the extension to any arbitrary power of   is trivial and does not alter the algebraic form of the approximation).

Inverse QEdit

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).

ValuesEdit

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 dimensionsEdit

The Q-function can be generalized to higher dimensions:[12]

 

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.[13][14]

ReferencesEdit

  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. ^ Craig, J.W. (1991). "A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations" (PDF). MILCOM 91 - Conference record. pp. 571–575. doi:10.1109/MILCOM.1991.258319. ISBN 0-87942-691-8. S2CID 16034807.
  5. ^ Behnad, Aydin (2020). "A Novel Extension to Craig's Q-Function Formula and Its Application in Dual-Branch EGC Performance Analysis". IEEE Transactions on Communications. 68 (7): 4117–4125. doi:10.1109/TCOMM.2020.2986209. S2CID 216500014.
  6. ^ a b Borjesson, P.; Sundberg, C.-E. (1979). "Simple Approximations of the Error Function Q(x) for Communications Applications". IEEE Transactions on Communications. 27 (3): 639–643. doi:10.1109/TCOM.1979.1094433.
  7. ^ Chiani, M.; Dardari, D.; Simon, M.K. (2003). "New exponential bounds and approximations for the computation of error probability in fading channels" (PDF). IEEE Transactions on Wireless Communications. 24 (5): 840–845. doi:10.1109/TWC.2003.814350.
  8. ^ Tanash, I.M.; Riihonen, T. (2020). "Global minimax approximations and bounds for the Gaussian Q-function by sums of exponentials". IEEE Transactions on Communications. 68 (10): 6514–6524. arXiv:2007.06939. doi:10.1109/TCOMM.2020.3006902. S2CID 220514754.
  9. ^ Tanash, I.M.; Riihonen, T. (2020). "Coefficients for Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials [Data set]". Zenodo. doi:10.5281/zenodo.4112978.
  10. ^ Karagiannidis, George; Lioumpas, Athanasios (2007). "An Improved Approximation for the Gaussian Q-Function" (PDF). IEEE Communications Letters. 11 (8): 644–646. doi:10.1109/LCOMM.2007.070470. S2CID 4043576.
  11. ^ Lopez-Benitez, Miguel; Casadevall, Fernando (2011). "Versatile, Accurate, and Analytically Tractable Approximation for the Gaussian Q-Function" (PDF). IEEE Transactions on Communications. 59 (4): 917–922. doi:10.1109/TCOMM.2011.012711.100105. S2CID 1145101.
  12. ^ Savage, I. R. (1962). "Mills ratio for multivariate normal distributions". Journal of Research of the National Bureau of Standards Section B. 66 (3): 93–96. doi:10.6028/jres.066B.011. Zbl 0105.12601.
  13. ^ Botev, Z. I. (2016). "The normal law under linear restrictions: simulation and estimation via minimax tilting". Journal of the Royal Statistical Society, Series B. 79: 125–148. arXiv:1603.04166. Bibcode:2016arXiv160304166B. doi:10.1111/rssb.12162. S2CID 88515228.
  14. ^ Botev, Z. I.; Mackinlay, D.; Chen, Y.-L. (2017). "Logarithmically efficient estimation of the tail of the multivariate normal distribution". 2017 Winter Simulation Conference (WSC). IEEE. pp. 1903–191. doi:10.1109/WSC.2017.8247926. ISBN 978-1-5386-3428-8. S2CID 4626481.