Q-function

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

If ${\displaystyle Y}$ is a Gaussian random variable with mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2$ , then ${\displaystyle X={\frac {Y-\mu }{\sigma$ is standard normal and

${\displaystyle P(Y>y)=P(X>x)=Q(x)}$

where ${\displaystyle x={\frac {y-\mu }{\sigma$ .

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

${\displaystyle Q(x)={\frac {1}{\sqrt {2\pi \int _{x}^{\infty }\exp \left(-{\frac {u^{2{2\right)\,du.}$

Thus,

${\displaystyle Q(x)=1-Q(-x)=1-\Phi (x)\,\!,}$

where ${\displaystyle \Phi (x)}$ 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]

${\displaystyle {\begin{aligned}Q(x)&={\frac {1}{2\left({\frac {2}{\sqrt {\pi \int _{x/{\sqrt {2^{\infty }\exp \left(-t^{2}\right)\,dt\right)\\&={\frac {1}{2-{\frac &#1$

An alternative form of the Q-function known as Craig's formula, after its discoverer, is expressed as:^[4]

${\displaystyle Q(x)={\frac {1}{\pi \int _{0}^{\frac {\pi }{2\exp \left(-{\frac {x^{2{2\sin ^{2}\theta \right)d\theta .}$

The above proper integral form of Q-function, has been incorrectly credited to Craig. This form of Q-function was implied in earlier works by Wiesten^[5] , and explicitly stated by Pawula, Rice and Roberts^[6].

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)^[7] for the Q-function of the sum of two non-negative variables, as follows:

the Q-function plotted in the complex plane ${\displaystyle Q(x+y)={\frac {1}{\pi \int _{0}^{\frac {\pi }{2\exp \left(-{\frac {x^{2{2\sin ^{2}\theta -{\frac {y^{2{2\cos ^{2}\theta \right)d\theta ,\quad x,y\geqslant 0.}$

Bounds and approximations#edit

The Q-function is not an elementary function. However, it can be upper and lower bounded as,^[8]^[9]

${\displaystyle \left({\frac {x}{1+x^{2\right)\phi (x)<Q(x)<{\frac {\phi (x)}{x,\qquad x>0,}$

where ${\displaystyle \phi (x)}$ is the density function of the standard normal distribution, and the bounds become increasingly tight for large x.

Using the substitution v =u²/2, the upper bound is derived as follows:

${\displaystyle Q(x)=\int _{x}^{\infty }\phi (u)\,du<\int _{x}^{\infty }{\frac {u}{x\phi (u)\,du=\int _{\frac {x^{2{2^{\infty }{\frac {e^{-v{x{\sqrt {2\pi \,dv=-$

Similarly, using ${\displaystyle \phi '(u)=-u\phi (u)}$ and the quotient rule,

${\displaystyle \left(1+{\frac {1}{x^{2\right)Q(x)=\int _{x}^{\infty }\left(1+{\frac {1}{x^{2\right)\phi (u)\,du>\int _{x}^{\infty }\left(1+{\frac {1}{u^{2\right)\phi (u)\,du&#6$

Solving for Q(x) provides the lower bound.

The geometric mean of the upper and lower bound gives a suitable approximation for ${\displaystyle Q(x)}$ :

${\displaystyle Q(x)\approx {\frac {\phi (x)}{\sqrt {1+x^{2,\qquad x\geq 0.}$

Tighter bounds and approximations of ${\displaystyle Q(x)}$ can also be obtained by optimizing the following expression ^[9]

${\displaystyle {\tilde {Q(x)={\frac {\phi (x)}{(1-a)x+a{\sqrt {x^{2}+b.}$

For ${\displaystyle x\geq 0}$ , the best upper bound is given by ${\displaystyle a=0.344}$ and ${\displaystyle b=5.334}$ with maximum absolute relative error of 0.44%. Likewise, the best approximation is given by ${\displaystyle a=0.339}$ and ${\displaystyle b=5.510}$ with maximum absolute relative error of 0.27%. Finally, the best lower bound is given by ${\displaystyle a=1/\pi }$ and ${\displaystyle b=2\pi }$ with maximum absolute relative error of 1.17%.

The Chernoff bound of the Q-function is

${\displaystyle Q(x)\leq e^{-{\frac {x^{2{2,\qquad x>0}$

Improved exponential bounds and a pure exponential approximation are ^[10]

${\displaystyle Q(x)\leq {\tfrac {1}{4e^{-x^{2+{\tfrac {1}{4e^{-{\frac {x^{2{2\leq {\tfrac {1}{2e^{-{\frac {x^{2{2,\qquad x>0}$

${\displaystyle Q(x)\approx {\frac {1}{12e^{-{\frac {x^{2{2+{\frac {1}{4e^{-{\frac {2}{3x^{2,\qquad x>0}$

The above were generalized by Tanash & Riihonen (2020),^[11] who showed that ${\displaystyle Q(x)}$ can be accurately approximated or bounded by

${\displaystyle {\tilde {Q(x)=\sum _{n=1}^{N}a_{n}e^{-b_{n}x^{2.}$

In particular, they presented a systematic methodology to solve the numerical coefficients ${\displaystyle \{(a_{n},b_{n})\}_{n=1}^{N$ that yield a minimax approximation or bound: ${\displaystyle Q(x)\approx {\tilde {Q(x)}$ , ${\displaystyle Q(x)\leq {\tilde {Q(x)}$ , or ${\displaystyle Q(x)\geq {\tilde {Q(x)}$ for ${\displaystyle x\geq 0}$ . With the example coefficients tabulated in the paper for ${\displaystyle N=20}$ , the relative and absolute approximation errors are less than ${\displaystyle 2.831\cdot 10^{-6$ and ${\displaystyle 1.416\cdot 10^{-6$ , respectively. The coefficients ${\displaystyle \{(a_{n},b_{n})\}_{n=1}^{N$ for many variations of the exponential approximations and bounds up to ${\displaystyle N=25}$ have been released to open access as a comprehensive dataset.^[12]

Another approximation of ${\displaystyle Q(x)}$ for ${\displaystyle x\in [0,\infty )}$ is given by Karagiannidis & Lioumpas (2007)^[13] who showed for the appropriate choice of parameters ${\displaystyle \{A,B\$ that

${\displaystyle f(x;A,B)={\frac {\left(1-e^{-Ax}\right)e^{-x^{2{B{\sqrt {\pi x\approx \operatorname {erfc} \left(x\right).}$

The absolute error between ${\displaystyle f(x;A,B)}$ and ${\displaystyle \operatorname {erfc} (x)}$ over the range ${\displaystyle [0,R]}$ is minimized by evaluating

${\displaystyle \{A,B\}={\underset {\{A,B\{\arg \min {\frac {1}{R\int _{0}^{R}|f(x;A,B)-\operatorname {erfc} (x)|dx.}$

Using ${\displaystyle R=20}$ and numerically integrating, they found the minimum error occurred when ${\displaystyle \{A,B\}=\{1.98,1.135\},}$ which gave a good approximation for ${\displaystyle \forall x\geq 0.}$

Substituting these values and using the relationship between ${\displaystyle Q(x)}$ and ${\displaystyle \operatorname {erfc} (x)}$ from above gives

${\displaystyle Q(x)\approx {\frac {\left(1-e^{\frac {-1.98x}{\sqrt {2\right)e^{-{\frac {x^{2{2{1.135{\sqrt {2\pi x,x\geq 0.}$

Alternative coefficients are also available for the above 'Karagiannidis–Lioumpas approximation' for tailoring accuracy for a specific application or transforming it into a tight bound.^[14]

A tighter and more tractable approximation of ${\displaystyle Q(x)}$ for positive arguments ${\displaystyle x\in [0,\infty )}$ is given by López-Benítez & Casadevall (2011)^[15] based on a second-order exponential function:

${\displaystyle Q(x)\approx e^{-ax^{2}-bx-c},\qquad x\geq 0.}$

The fitting coefficients ${\displaystyle (a,b,c)}$ can be optimized over any desired range of arguments in order to minimize the sum of square errors ( ${\displaystyle a=0.3842}$ , ${\displaystyle b=0.7640}$ , ${\displaystyle c=0.6964}$ for ${\displaystyle x\in [0,20]}$ ) or minimize the maximum absolute error ( ${\displaystyle a=0.4920}$ , ${\displaystyle b=0.2887}$ , ${\displaystyle c=1.1893}$ for ${\displaystyle x\in [0,20]}$ ). 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 ${\displaystyle Q(x)}$ is trivial and does not alter the algebraic form of the approximation).

A pair of tight lower and upper bounds on the Gaussian Q-function for positive arguments ${\displaystyle x\in [0,\infty )}$ was introduced by Abreu (2012)^[16] based on a simple algebraic expression with only two exponential terms:

${\displaystyle Q(x)\geq {\frac {1}{12e^{-x^{2+{\frac {1}\sqrt {2\pi (x+1)e^{-x^{2}/2},\qquad x\geq 0,}$

${\displaystyle Q(x)\leq {\frac {1}{50e^{-x^{2+{\frac {1}{2(x+1)e^{-x^{2}/2},\qquad x\geq 0.}$

These bounds are derived from a unified form ${\displaystyle Q_{\mathrm {B} }(x;a,b)={\frac {\exp(-x^{2})}{a+{\frac {\exp(-x^{2}/2)}{b(x+1)$ , where the parameters ${\displaystyle a}$ and ${\displaystyle b}$ are chosen to satisfy specific conditions ensuring the lower ( ${\displaystyle a_{\mathrm {L} }=12}$ , ${\displaystyle b_{\mathrm {L} }={\sqrt {2\pi$ ) and upper ( ${\displaystyle a_{\mathrm {U} }=50}$ , ${\displaystyle b_{\mathrm {U} }=2}$ ) bounding properties. The resulting expressions are notable for their simplicity and tightness, offering a favorable trade-off between accuracy and mathematical tractability. These bounds are particularly useful in theoretical analysis, such as in communication theory over fading channels. Additionally, they can be extended to bound ${\displaystyle Q^{n}(x)}$ for positive integers ${\displaystyle n}$ using the binomial theorem, maintaining their simplicity and effectiveness.

Inverse Q#edit

The inverse Q-function can be related to the inverse error functions:

${\displaystyle Q^{-1}(y)={\sqrt {2\ \mathrm {erf} ^{-1}(1-2y)={\sqrt {2\ \mathrm {erfc} ^{-1}(2y)}$

The function ${\displaystyle Q^{-1}(y)}$ finds application in digital communications. It is usually expressed in dB and generally called Q-factor:

${\displaystyle \mathrm {Q{\text{-factor} =20\log _{10}\!\left(Q^{-1}(y)\right)\!~\mathrm {dB} }$

where y is the bit-error rate (BER) of the digitally modulated signal under analysis. For instance, for quadrature phase-shift keying (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). Image: source and license

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:^[17]

${\displaystyle Q(\mathbf {x} )=\mathbb {P} (\mathbf {X} \geq \mathbf {x} ),}$

where ${\displaystyle \mathbf {X} \sim {\mathcal {N(\mathbf {0} ,\,\Sigma )}$ follows the multivariate normal distribution with covariance ${\displaystyle \Sigma }$ and the threshold is of the form ${\displaystyle \mathbf {x} =\gamma \Sigma \mathbf {l} ^{*$ for some positive vector ${\displaystyle \mathbf {l} ^{*}>\mathbf {0} }$ and positive constant ${\displaystyle \gamma >0}$ . 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 ${\displaystyle \gamma }$ becomes larger and larger.^[18]^[19]

References#edit

^ "The Q-function". cnx.org. Archived from the original on 2012-02-29.
^ a b "Basic properties of the Q-function" (PDF). 2009-03-05. Archived from the original (PDF) on 2009-03-25.
^ Normal Distribution Function – from Wolfram MathWorld
^ 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. Archived from the original (PDF) on 2012-04-03. Retrieved 2011-11-16.
^ Weinstein, F. "Simplified relationships for the probability distribution of the phase of a sine wave in narrow-band normal noise (Corresp.)". IEEE Transactions on Information Theory. 20 (5): 658–661. doi:10.1109/TIT.1974.1055272. ISSN 1557-9654.
^ Pawula, R.; Rice, S.; Roberts, J. "Distribution of the Phase Angle Between Two Vectors Perturbed by Gaussian Noise". IEEE Transactions on Communications. 30 (8): 1828–1841. doi:10.1109/TCOM.1982.1095662. ISSN 1558-0857.
^ 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.
^ Gordon, R.D. (1941). "Values of Mills' ratio of area to bounding ordinate and of the normal probability integral for large values of the argument". Ann. Math. Stat. 12 (3): 364–366. doi:10.1214/aoms/1177731721.
^ 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.
^ 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. Archived from the original (PDF) on 2014-10-20. Retrieved 2014-10-20.
^ 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.
^ 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.
^ 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.
^ Tanash, I.M.; Riihonen, T. (2021). "Improved coefficients for the Karagiannidis–Lioumpas approximations and bounds to the Gaussian Q-function". IEEE Communications Letters. 25 (5): 1468–1471. arXiv:2101.07631. doi:10.1109/LCOMM.2021.3052257. S2CID 231639206.
^ 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.
^ Abreu, Giuseppe (2012). "Very Simple Tight Bounds on the Q-Function". IEEE Transactions on Communications. 60 (9): 2415–2420. doi:10.1109/TCOMM.2012.080612.110075.
^ 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.
^ 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.
^ 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.

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