Q-function#Inverse Q

Formally, the Q-function is defined as

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

Thus,

:$Q(x)\; =\; 1\; -\; Q(-x)\; =\; 1\; -\; \backslash Phi(x)\backslash ,\backslash !,$

where $\backslash Phi(x)$ 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

:$$

\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}

An alternative form of the Q-function known as Craig's formula, after its discoverer, is expressed as:{{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}}

:$Q(x)\; =\; \backslash frac\{1\}\{\backslash pi\}\; \backslash int\_0^\{\backslash frac\{\backslash pi\}\{2\}\}\; \backslash exp\; \backslash left(\; -\; \backslash frac\{x^2\}\{2\; \backslash sin^2\; \backslash theta\}\; \backslash right)\; d\backslash theta.$

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){{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}} 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$Q(x+y)\; =\; \backslash frac\{1\}\{\backslash pi\}\; \backslash int\_0^\{\backslash frac\{\backslash pi\}\{2\}\}\; \backslash exp\; \backslash left(\; -\; \backslash frac\{x^2\}\{2\; \backslash sin^2\; \backslash theta\}\; -\; \backslash frac\{y^2\}\{2\; \backslash cos^2\; \backslash theta\}\; \backslash right)\; d\backslash theta,\; \backslash quad\; x,y\; \backslash geqslant\; 0\; .$

• The Q-function is not an elementary function. However, it can be upper and lower bounded as,{{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.}}{{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.}}

::

:where $\backslash phi(x)$ 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:

::

:Similarly, using $\backslash phi\text{'}(u)\; =\; -\; u\; \backslash phi(u)$ and the quotient rule,

::$\backslash left(1+\backslash frac1\{x^2\}\backslash right)Q(x)\; =\backslash int\_x^\backslash infty\; \backslash left(1+\backslash frac1\{x^2\}\backslash right)\backslash phi(u)\backslash ,du\; >\backslash int\_x^\backslash infty\; \backslash left(1+\backslash frac1\{u^2\}\backslash right)\backslash phi(u)\backslash ,du\; =-\backslash biggl.\backslash frac\{\backslash phi(u)\}u\backslash biggr|\_x^\backslash infty$

=\frac{\phi(x)}x.

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

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

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

• Tighter bounds and approximations of $Q(x)$ can also be obtained by optimizing the following expression

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

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

• The Chernoff bound of the Q-function is

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

• Improved exponential bounds and a pure exponential approximation are {{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.}}

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

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

• The above were generalized by Tanash & Riihonen (2020),{{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}} who showed that $Q(x)$ can be accurately approximated or bounded by

::$\backslash tilde\{Q\}(x)\; =\; \backslash sum\_\{n=1\}^N\; a\_n\; e^\{-b\_n\; x^2\}.$

:In particular, they presented a systematic methodology to solve the numerical coefficients $\backslash \{(a\_n,b\_n)\backslash \}\_\{n=1\}^N$ that yield a minimax approximation or bound: $Q(x)\; \backslash approx\; \backslash tilde\{Q\}(x)$, $Q(x)\; \backslash leq\; \backslash tilde\{Q\}(x)$, or $Q(x)\; \backslash geq\; \backslash tilde\{Q\}(x)$ for $x\backslash geq0$. With the example coefficients tabulated in the paper for $N\; =\; 20$, the relative and absolute approximation errors are less than $2.831\; \backslash cdot\; 10^\{-6\}$ and $1.416\; \backslash cdot\; 10^\{-6\}$, respectively. The coefficients $\backslash \{(a\_n,b\_n)\backslash \}\_\{n=1\}^N$ for many variations of the exponential approximations and bounds up to $N\; =\; 25$ have been released to open access as a comprehensive dataset.{{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.}}

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

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

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

:: $\backslash \{A,\; B\backslash \}\; =\; \backslash underset\{\backslash \{A,B\backslash \}\}\{\backslash arg\; \backslash min\}\; \backslash frac\{1\}\{R\}\; \backslash int\_0^R\; |\; f(x;\; A,\; B)\; -\; \backslash operatorname\{erfc\}(x)\; |dx.$

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

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

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

: 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.{{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}}

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

:: $Q(x)\; \backslash approx\; e^\{-ax^2-bx-c\},\; \backslash qquad\; x\; \backslash ge\; 0.$

: The fitting coefficients $(a,b,c)$ can be optimized over any desired range of arguments in order to minimize the sum of square errors ($a\; =\; 0.3842$, $b\; =\; 0.7640$, $c\; =\; 0.6964$ for $x\; \backslash in\; [0,20]$) or minimize the maximum absolute error ($a\; =\; 0.4920$, $b\; =\; 0.2887$, $c\; =\; 1.1893$ for $x\; \backslash in\; [0,20]$). 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 $Q(x)$ 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 $x\; \backslash in\; [0,\; \backslash infty)$ was introduced by Abreu (2012){{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}} based on a simple algebraic expression with only two exponential terms:

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

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

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

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

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

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

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

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.

File:Q-factor vs BER.png

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.

{{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}}

The Q-function can be generalized to higher dimensions:{{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}}

:$Q(\backslash mathbf\{x\})=\; \backslash mathbb\{P\}(\backslash mathbf\{X\}\backslash geq\; \backslash mathbf\{x\}),$

where $\backslash mathbf\{X\}\backslash sim\; \backslash mathcal\{N\}(\backslash mathbf\{0\},\backslash ,\; \backslash Sigma)$ follows the multivariate normal distribution with covariance $\backslash Sigma$ and the threshold is of the form

$\backslash mathbf\{x\}=\backslash gamma\backslash Sigma\backslash mathbf\{l\}^*$ for some positive vector $\backslash mathbf\{l\}^*>\backslash mathbf\{0\}$ and positive constant $\backslash gamma>0$. 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 $\backslash gamma$ becomes larger and larger.{{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}}{{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 }}

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