imported>Getuar-WCASP: Additional tight, yet simple, upper and lower bound for Q function

2025-03-13T18:40:04Z

Additional tight, yet simple, upper and lower bound for Q function

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{{Short description|Statistics function}}
{{For|the phase-space function representing a quantum state|Husimi Q representation}}
[[Image:Q-function.png|thumb|right|400px|A plot of the Q-function.]]
In [[statistics]], the '''Q-function''' is the [[Cumulative distribution function#Complementary cumulative distribution function (tail distribution)|tail distribution function]] of the [[Normal distribution#Standard normal distribution|standard normal distribution]].<ref>{{cite web|url=http://cnx.org/content/m11537/latest/|title=The Q-function|website=[[cnx.org]]|archive-url=https://web.archive.org/web/20120229030808/http://cnx.org/content/m11537/latest/|archive-date=2012-02-29}}</ref><ref name="jo">{{cite web|url=http://www.eng.tau.ac.il/~jo/academic/Q.pdf|title=Basic properties of the Q-function|archive-url=https://web.archive.org/web/20090325160012/http://www.eng.tau.ac.il/~jo/academic/Q.pdf|archive-date=2009-03-25 |date=2009-03-05 }}</ref> In other words, <math>Q(x)</math> is the probability that a normal (Gaussian) [[random variable]] will obtain a value larger than <math>x</math> standard deviations. Equivalently, <math>Q(x)</math> is the probability that a standard normal random variable takes a value larger than <math>x</math>.

If <math>Y</math> is a Gaussian random variable with mean <math>\mu</math> and variance <math>\sigma^2</math>, then <math>X = \frac{Y-\mu}{\sigma}</math> is [[Normal distribution#Standard normal distribution|standard normal]] and

:<math>P(Y > y) = P(X > x) = Q(x)</math>

where <math>x = \frac{y-\mu}{\sigma}</math>.

Other definitions of the ''Q''-function, all of which are simple transformations of the normal [[cumulative distribution function]], are also used occasionally.<ref>[http://mathworld.wolfram.com/NormalDistributionFunction.html Normal Distribution Function – from Wolfram MathWorld]</ref>

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

:<math>Q(x) = \frac{1}{\sqrt{2\pi}} \int_x^\infty \exp\left(-\frac{u^2}{2}\right) \, du.</math>

Thus,

:<math>Q(x) = 1 - Q(-x) = 1 - \Phi(x)\,\!,</math>

where <math>\Phi(x)</math> is the [[Standard normal distribution#Cumulative distribution function|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<ref name="jo"/>

:<math>
\begin{align}
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}{2} \operatorname{erf} \left( \frac{x}{\sqrt{2}} \right) ~~\text{ -or-}\\
&= \frac{1}{2}\operatorname{erfc} \left(\frac{x}{\sqrt{2}} \right).
\end{align}
</math>

An alternative form of the ''Q''-function known as Craig's formula, after its discoverer, is expressed as:<ref>{{cite book |doi=10.1109/MILCOM.1991.258319 |chapter-url=http://wsl.stanford.edu/~ee359/craig.pdf|chapter=A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations|title=MILCOM 91 - Conference record|pages=571–575|year=1991|last1=Craig|first1=J.W.|isbn=0-87942-691-8|s2cid=16034807}}</ref>

:<math>Q(x) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \exp \left( - \frac{x^2}{2 \sin^2 \theta} \right) d\theta.</math>

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)<ref>{{cite journal |doi=10.1109/TCOMM.2020.2986209 |title=A Novel Extension to Craig's Q-Function Formula and Its Application in Dual-Branch EGC Performance Analysis|journal=IEEE Transactions on Communications |volume=68|issue=7|pages=4117–4125|year=2020|last1=Behnad|first1=Aydin|s2cid=216500014}}</ref> for the ''Q''-function of the sum of two non-negative variables, as follows:

:[[File:Q function complex plot plotted with Mathematica 13.1 ComplexPlot3D.svg|alt=the Q-function plotted in the complex plane|thumb|the Q-function plotted in the complex plane]]<math>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 .</math>

==Bounds and approximations==
*The ''Q''-function is not an [[elementary function]]. However, it can be upper and lower bounded as,<ref name = "Gordon">{{Cite journal |doi = 10.1214/aoms/1177731721|title = Values of Mills' ratio of area to bounding ordinate and of the normal probability integral for large values of the argument| journal = Ann. Math. Stat.|volume = 12|issue = 3|pages = 364–366|year = 1941|last = Gordon|first = R.D.}}</ref><ref name = "Borjesson">{{Cite journal |doi = 10.1109/TCOM.1979.1094433|title = Simple Approximations of the Error Function Q(x) for Communications Applications|journal = IEEE Transactions on Communications|volume = 27|issue = 3|pages = 639–643|year = 1979|last1 = Borjesson|first1 = P.|last2 = Sundberg|first2 = C.-E.}}</ref>

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

:where <math>\phi(x)</math> is the density function of the standard normal distribution, and the bounds become increasingly tight for large ''x''.

:Using the [[integration by substitution|substitution]] ''v'' =''u''<sup>2</sup>/2, the upper bound is derived as follows:

::<math>Q(x) =\int_x^\infty\phi(u)\,du <\int_x^\infty\frac ux\phi(u)\,du =\int_{\frac{x^2}{2}}^\infty\frac{e^{-v}}{x\sqrt{2\pi}}\,dv=-\biggl.\frac{e^{-v}}{x\sqrt{2\pi}}\biggr|_{\frac{x^2}{2}}^\infty=\frac{\phi(x)}{x}.</math>

:Similarly, using <math>\phi'(u) = - u \phi(u)</math> and the [[quotient rule]],

::<math>\left(1+\frac1{x^2}\right)Q(x) =\int_x^\infty \left(1+\frac1{x^2}\right)\phi(u)\,du >\int_x^\infty \left(1+\frac1{u^2}\right)\phi(u)\,du =-\biggl.\frac{\phi(u)}u\biggr|_x^\infty
=\frac{\phi(x)}x. </math>

:Solving for ''Q''(''x'') provides the lower bound.

:The [[geometric mean]] of the upper and lower bound gives a suitable approximation for <math>Q(x)</math>:

::<math>Q(x) \approx \frac{\phi(x)}{\sqrt{1 + x^2}}, \qquad x \geq 0. </math>

* Tighter bounds and approximations of <math>Q(x)</math> can also be obtained by optimizing the following expression <ref name = "Borjesson"/>

:: <math> \tilde{Q}(x) = \frac{\phi(x)}{(1-a)x + a\sqrt{x^2 + b}}. </math>

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

*The [[Chernoff bound]] of the ''Q''-function is

::<math>Q(x)\leq e^{-\frac{x^2}{2}}, \qquad x>0</math>

*Improved exponential bounds and a pure exponential approximation are <ref>{{cite journal |url=http://campus.unibo.it/85943/1/mcddmsTranWIR2003.pdf |doi=10.1109/TWC.2003.814350|title=New exponential bounds and approximations for the computation of error probability in fading channels|journal=IEEE Transactions on Wireless Communications|volume=24|issue=5|pages=840–845|year=2003|last1=Chiani|first1=M.|last2=Dardari|first2=D.|last3=Simon|first3=M.K.}}</ref>

::<math>Q(x)\leq \tfrac{1}{4}e^{-x^2}+\tfrac{1}{4}e^{-\frac{x^2}{2}} \leq \tfrac{1}{2}e^{-\frac{x^2}{2}}, \qquad x>0</math>

:: <math>Q(x)\approx \frac{1}{12}e^{-\frac{x^2}{2}}+\frac{1}{4}e^{-\frac{2}{3} x^2}, \qquad x>0 </math>

*The above were generalized by Tanash & Riihonen (2020),<ref>{{cite journal |doi=10.1109/TCOMM.2020.3006902|title=Global minimax approximations and bounds for the Gaussian Q-function by sums of exponentials|journal=IEEE Transactions on Communications|year=2020|last1=Tanash|first1=I.M.|last2=Riihonen|first2=T.|volume=68|issue=10|pages=6514–6524|arxiv=2007.06939|s2cid=220514754}}</ref> who showed that <math>Q(x)</math> can be accurately approximated or bounded by

::<math>\tilde{Q}(x) = \sum_{n=1}^N a_n e^{-b_n x^2}.</math>

:In particular, they presented a systematic methodology to solve the numerical coefficients <math>\{(a_n,b_n)\}_{n=1}^N</math> that yield a [[minimax approximation algorithm|minimax]] approximation or bound: <math>Q(x) \approx \tilde{Q}(x)</math>, <math>Q(x) \leq \tilde{Q}(x)</math>, or <math>Q(x) \geq \tilde{Q}(x)</math> for <math>x\geq0</math>. With the example coefficients tabulated in the paper for <math>N = 20</math>, the relative and absolute approximation errors are less than <math>2.831 \cdot 10^{-6}</math> and <math>1.416 \cdot 10^{-6}</math>, respectively. The coefficients <math>\{(a_n,b_n)\}_{n=1}^N</math> for many variations of the exponential approximations and bounds up to <math>N = 25</math> have been released to open access as a comprehensive dataset.<ref>{{cite journal |doi=10.5281/zenodo.4112978|title=Coefficients for Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials [Data set]|url=https://zenodo.org/record/4112978|website=Zenodo|year=2020|last1=Tanash|first1=I.M.|last2=Riihonen|first2=T.}}</ref>

*Another approximation of <math>Q(x)</math> for <math>x \in [0,\infty)</math> is given by [[George Karagiannidis|Karagiannidis]] & Lioumpas (2007)<ref>{{cite journal |doi=10.1109/LCOMM.2007.070470 |url=http://users.auth.gr/users/9/3/028239/public_html/pdf/Q_Approxim.pdf|title=An Improved Approximation for the Gaussian Q-Function|journal=IEEE Communications Letters|volume=11|issue=8|pages=644–646|year=2007|last1=Karagiannidis|first1=George|last2=Lioumpas|first2=Athanasios|s2cid=4043576}}</ref> who showed for the appropriate choice of parameters <math>\{A, B\}</math> that

:: <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>

: 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

:: <math>\{A, B\} = \underset{\{A,B\}}{\arg \min} \frac{1}{R} \int_0^R | f(x; A, B) - \operatorname{erfc}(x) |dx.</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>

: Substituting these values and using the relationship between <math>Q(x)</math> and <math>\operatorname{erfc}(x)</math> from above gives

:: <math> 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 \ge 0. </math>

: 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.<ref>{{cite journal |doi=10.1109/LCOMM.2021.3052257|title=Improved coefficients for the Karagiannidis–Lioumpas approximations and bounds to the Gaussian Q-function|journal=IEEE Communications Letters|year=2021|last1=Tanash|first1=I.M.|last2=Riihonen|first2=T.|volume=25|issue=5|pages=1468–1471|arxiv=2101.07631|s2cid=231639206}}</ref>

*A tighter and more tractable approximation of <math>Q(x)</math> for positive arguments <math>x \in [0,\infty)</math> is given by López-Benítez & Casadevall (2011)<ref>{{cite journal |doi=10.1109/TCOMM.2011.012711.100105 |url=http://www.lopezbenitez.es/journals/IEEE_TCOM_2011.pdf|title=Versatile, Accurate, and Analytically Tractable Approximation for the Gaussian Q-Function|journal=IEEE Transactions on Communications|volume=59|issue=4|pages=917–922|year=2011|last1=Lopez-Benitez|first1=Miguel|last2=Casadevall|first2=Fernando|s2cid=1145101}}</ref> based on a second-order exponential function:

:: <math> Q(x) \approx e^{-ax^2-bx-c}, \qquad x \ge 0. </math>

: The fitting coefficients <math> (a,b,c) </math> can be optimized over any desired range of arguments in order to minimize the sum of square errors (<math>a = 0.3842</math>, <math>b = 0.7640</math>, <math>c = 0.6964</math> for <math>x \in [0,20]</math>) or minimize the maximum absolute error (<math>a = 0.4920</math>, <math>b = 0.2887</math>, <math>c = 1.1893</math> for <math>x \in [0,20]</math>). 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 <math>Q(x)</math> 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 <math>x \in [0, \infty)</math> was introduced by Abreu (2012)<ref>{{cite journal |doi=10.1109/TCOMM.2012.080612.110075 |title=Very Simple Tight Bounds on the Q-Function |journal=IEEE Transactions on Communications |volume=60 |issue=9 |pages=2415–2420 |year=2012 |last=Abreu |first=Giuseppe}}</ref> based on a simple algebraic expression with only two exponential terms:

:: <math> Q(x) \geq \frac{1}{12} e^{-x^2} + \frac{1}{\sqrt{2\pi} (x + 1)} e^{-x^2 / 2}, \qquad x \geq 0, </math>

:: <math> Q(x) \leq \frac{1}{50} e^{-x^2} + \frac{1}{2 (x + 1)} e^{-x^2 / 2}, \qquad x \geq 0. </math>

These bounds are derived from a unified form <math>Q_{\mathrm{B}}(x; a, b) = \frac{\exp(-x^2)}{a} + \frac{\exp(-x^2 / 2)}{b (x + 1)}</math>, where the parameters <math>a</math> and <math>b</math> are chosen to satisfy specific conditions ensuring the lower (<math>a_{\mathrm{L}} = 12</math>, <math>b_{\mathrm{L}} = \sqrt{2\pi}</math>) and upper (<math>a_{\mathrm{U}} = 50</math>, <math>b_{\mathrm{U}} = 2</math>) 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 <math>Q^n(x)</math> for positive integers <math>n</math> using the binomial theorem, maintaining their simplicity and effectiveness.

==Inverse ''Q''==

The inverse ''Q''-function can be related to the [[error function#Inverse functions|inverse error functions]]:

:<math>Q^{-1}(y) = \sqrt{2}\ \mathrm{erf}^{-1}(1-2y) = \sqrt{2}\ \mathrm{erfc}^{-1}(2y)</math>

The function <math>Q^{-1}(y)</math> finds application in digital communications. It is usually expressed in [[Decibel#Field quantities and root-power quantities|dB]] and generally called '''Q-factor''':

:<math>\mathrm{Q\text{-}factor} = 20 \log_{10}\!\left(Q^{-1}(y)\right)\!~\mathrm{dB}</math>

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#Decibels|signal to noise ratio]] that yields a bit error rate equal to ''y''.
[[File:Q-factor vs BER.png|thumb|none|400px|Q-factor vs. bit error rate (BER).]]

== Values ==
The ''Q''-function is well tabulated and can be computed directly in most of the mathematical software packages such as [[R (programming language)|R]] and those available in [[Python (programming language)|Python]], [[MATLAB]] and [[Wolfram Mathematica|Mathematica]]. Some values of the ''Q''-function are given below for reference.


{{col-begin}}
{{col-4}}
{|class="wikitable"
! scope="row" | ''Q''(0.0)
| 0.500000000 || 1/2.0000
|-
! scope="row" | ''Q''(0.1)
| 0.460172163 || 1/2.1731
|-
! scope="row" | ''Q''(0.2)
| 0.420740291 || 1/2.3768
|-
! scope="row" | ''Q''(0.3)
| 0.382088578 || 1/2.6172
|-
! scope="row" | ''Q''(0.4)
| 0.344578258 || 1/2.9021
|-
! scope="row" | ''Q''(0.5)
| 0.308537539 || 1/3.2411
|-
! scope="row" | ''Q''(0.6)
| 0.274253118 || 1/3.6463
|-
! scope="row" | ''Q''(0.7)
| 0.241963652 || 1/4.1329
|-
! scope="row" | ''Q''(0.8)
| 0.211855399 || 1/4.7202
|-
! scope="row" | ''Q''(0.9)
| 0.184060125 || 1/5.4330
|}
{{col-4}}
{|class="wikitable"
! scope="row" | ''Q''(1.0)
| 0.158655254 || 1/6.3030
|-
! scope="row" | ''Q''(1.1)
| 0.135666061 || 1/7.3710
|-
! scope="row" | ''Q''(1.2)
| 0.115069670 || 1/8.6904
|-
! scope="row" | ''Q''(1.3)
| 0.096800485 || 1/10.3305
|-
! scope="row" | ''Q''(1.4)
| 0.080756659 || 1/12.3829
|-
! scope="row" | ''Q''(1.5)
| 0.066807201 || 1/14.9684
|-
! scope="row" | ''Q''(1.6)
| 0.054799292 || 1/18.2484
|-
! scope="row" | ''Q''(1.7)
| 0.044565463 || 1/22.4389
|-
! scope="row" | ''Q''(1.8)
| 0.035930319 || 1/27.8316
|-
! scope="row" | ''Q''(1.9)
| 0.028716560 || 1/34.8231
|}
{{col-4}}
{|class="wikitable"
! scope="row" | ''Q''(2.0)
| 0.022750132 || 1/43.9558
|-
! scope="row" | ''Q''(2.1)
| 0.017864421 || 1/55.9772
|-
! scope="row" | ''Q''(2.2)
| 0.013903448 || 1/71.9246
|-
! scope="row" | ''Q''(2.3)
| 0.010724110 || 1/93.2478
|-
! scope="row" | ''Q''(2.4)
| 0.008197536 || 1/121.9879
|-
! scope="row" | ''Q''(2.5)
| 0.006209665 || 1/161.0393
|-
! scope="row" | ''Q''(2.6)
| 0.004661188 || 1/214.5376
|-
! scope="row" | ''Q''(2.7)
| 0.003466974 || 1/288.4360
|-
! scope="row" | ''Q''(2.8)
| 0.002555130 || 1/391.3695
|-
! scope="row" | ''Q''(2.9)
| 0.001865813 || 1/535.9593
|}
{{col-4}}
{|class="wikitable"
! scope="row" | ''Q''(3.0)
| 0.001349898 || 1/740.7967
|-
! scope="row" | ''Q''(3.1)
| 0.000967603 || 1/1033.4815
|-
! scope="row" | ''Q''(3.2)
| 0.000687138 || 1/1455.3119
|-
! scope="row" | ''Q''(3.3)
| 0.000483424 || 1/2068.5769
|-
! scope="row" | ''Q''(3.4)
| 0.000336929 || 1/2967.9820
|-
! scope="row" | ''Q''(3.5)
| 0.000232629 || 1/4298.6887
|-
! scope="row" | ''Q''(3.6)
| 0.000159109 || 1/6285.0158
|-
! scope="row" | ''Q''(3.7)
| 0.000107800 || 1/9276.4608
|-
! scope="row" | ''Q''(3.8)
| 0.000072348 || 1/13822.0738
|-
! scope="row" | ''Q''(3.9)
| 0.000048096 || 1/20791.6011
|-
! scope="row" | ''Q''(4.0)
| 0.000031671 || 1/31574.3855
|}
{{col-end}}

== Generalization to high dimensions ==
The ''Q''-function can be generalized to higher dimensions:<ref>{{cite journal|last1=Savage|first1=I. R.|title=Mills ratio for multivariate normal distributions|journal=Journal of Research of the National Bureau of Standards Section B|date=1962|volume=66|issue=3|pages=93–96|doi=10.6028/jres.066B.011|zbl=0105.12601|doi-access=free}}</ref>

:<math>Q(\mathbf{x})= \mathbb{P}(\mathbf{X}\geq \mathbf{x}),</math>
where <math>\mathbf{X}\sim \mathcal{N}(\mathbf{0},\, \Sigma) </math> follows the multivariate normal distribution with covariance <math>\Sigma </math> and the threshold is of the form
<math>\mathbf{x}=\gamma\Sigma\mathbf{l}^*</math> for some positive vector <math> \mathbf{l}^*>\mathbf{0}</math> and positive constant <math>\gamma>0</math>. As in the one dimensional case, there is no simple analytical formula for the ''Q''-function. Nevertheless, the ''Q''-function can be [http://www.mathworks.com/matlabcentral/fileexchange/53796 approximated arbitrarily well] as <math>\gamma</math> becomes larger and larger.<ref>{{cite journal|last1=Botev|first1=Z. I.|title=The normal law under linear restrictions: simulation and estimation via minimax tilting|journal=Journal of the Royal Statistical Society, Series B|volume=79|pages=125–148|date=2016|doi=10.1111/rssb.12162|arxiv=1603.04166|bibcode=2016arXiv160304166B|s2cid=88515228}}</ref><ref name="bmc17">{{cite book |chapter=Logarithmically efficient estimation of the tail of the multivariate normal distribution |last1=Botev |first1=Z. I. |last2=Mackinlay |first2=D. |last3=Chen |first3=Y.-L. |date=2017 |publisher=IEEE |isbn=978-1-5386-3428-8 |title= 2017 Winter Simulation Conference (WSC)|pages=1903–191 |doi= 10.1109/WSC.2017.8247926 |s2cid=4626481 }}
</ref>

== References ==
<references />

[[Category:Normal distribution]]
[[Category:Special functions]]
[[Category:Functions related to probability distributions]]
[[Category:Articles containing proofs]]

Q-function - Revision history

imported>Getuar-WCASP: Additional tight, yet simple, upper and lower bound for Q function