Q-function

A plot of the Q-function.

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

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

$P(Y>y)=P(X>x)=Q(x)$

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

Other definitions of the Q-function, all of which are simple transformations of the normal cumulative distribution function, are also used occasionally.

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

$Q(x)=\frac{1}{\sqrt{2\pi }}\underset{x}{\overset{\mathrm{\infty }}{\int }}\exp (-\frac{u^2}{2})du.$

Thus,

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

where $\mathrm{\Phi }(x)$ 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

$\begin{array}{cc}Q(x)& =\frac{1}{2}(\frac{2}{\sqrt{\pi }}\underset{x/\sqrt{2}}{\overset{\mathrm{\infty }}{\int }}\exp (-t^2)dt)\\ & =\frac{1}{2}-\frac{1}{2}\mathrm{erf}\left(\frac{x}{\sqrt{2}}\right)\text{ -or-}\\ & =\frac{1}{2}\mathrm{erfc}\left(\frac{x}{\sqrt{2}}\right).\end{array}$

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

$Q(x)=\frac{1}{\pi }\underset{0}{\overset{\frac{\pi }{2}}{\int }}\exp (-\frac{x^2}{2{\sin }^2\theta })d\theta .$

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

the Q-function plotted in the complex plane

$Q(x+y)=\frac{1}{\pi }\underset{0}{\overset{\frac{\pi }{2}}{\int }}\exp (-\frac{x^2}{2{\sin }^2\theta }-\frac{y^2}{2{\cos }^2\theta })d\theta ,x,y⩾0.$

Bounds and approximations

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

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

where $\varphi (x)$ 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:

$Q(x)=\underset{x}{\overset{\mathrm{\infty }}{\int }}\varphi (u)du<\underset{x}{\overset{\mathrm{\infty }}{\int }}\frac{u}{x}\varphi (u)du=\underset{\frac{x^2}{2}}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-v}}{x\sqrt{2\pi }}dv=-\frac{e^{-v}}{x\sqrt{2\pi }}{|}_{\frac{x^2}{2}}^{\mathrm{\infty }}=\frac{\varphi (x)}{x}.$

Similarly, using ${\varphi }^{\prime }(u)=-u\varphi (u)$ and the quotient rule,

$(1+\frac{1}{x^2})Q(x)=\underset{x}{\overset{\mathrm{\infty }}{\int }}(1+\frac{1}{x^2})\varphi (u)du>\underset{x}{\overset{\mathrm{\infty }}{\int }}(1+\frac{1}{u^2})\varphi (u)du=-\frac{\varphi (u)}{u}{|}_x^{\mathrm{\infty }}=\frac{\varphi (x)}{x}.$

Solving for Q(x) provides the lower bound.

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

$Q(x)\approx \frac{\varphi (x)}{\sqrt{1+x^2}},x\ge 0.$

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

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

For $x\ge 0$, 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 $a=1/\pi$ and $b=2\pi$ with maximum absolute relative error of 1.17%.

• The Chernoff bound of the Q-function is

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

• Improved exponential bounds and a pure exponential approximation are

$Q(x)\le \frac{1}{4}{e}^{-x^2}+\frac{1}{4}{e}^{-\frac{x^2}{2}}\le \frac{1}{2}{e}^{-\frac{x^2}{2}},x>0$

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

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

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

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

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

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

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

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

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

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

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

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.

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

$Q(x)\approx e^{-a{x}^2-bx-c},x\ge 0.$

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 $x\in [0,20]$) or minimize the maximum absolute error ($a=0.4920$, $b=0.2887$, $c=1.1893$ for $x\in [0,20]$). 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 $Q(x)$ is trivial and does not alter the algebraic form of the approximation).

Inverse Q

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

$Q^{-1}(y)=\sqrt{2}{\mathrm{e}\mathrm{r}\mathrm{f}}^{-1}(1-2y)=\sqrt{2}{\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}}^{-1}(2y)$

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

$\mathrm{Q}\text{-}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}=20{\log }_{10}(Q^{-1}(y))\mathrm{d}\mathrm{B}$

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).

Values

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.

Q(0.0)	0.500000000	1/2.0000
Q(0.1)	0.460172163	1/2.1731
Q(0.2)	0.420740291	1/2.3768
Q(0.3)	0.382088578	1/2.6172
Q(0.4)	0.344578258	1/2.9021
Q(0.5)	0.308537539	1/3.2411
Q(0.6)	0.274253118	1/3.6463
Q(0.7)	0.241963652	1/4.1329
Q(0.8)	0.211855399	1/4.7202
Q(0.9)	0.184060125	1/5.4330

Q(1.0)	0.158655254	1/6.3030
Q(1.1)	0.135666061	1/7.3710
Q(1.2)	0.115069670	1/8.6904
Q(1.3)	0.096800485	1/10.3305
Q(1.4)	0.080756659	1/12.3829
Q(1.5)	0.066807201	1/14.9684
Q(1.6)	0.054799292	1/18.2484
Q(1.7)	0.044565463	1/22.4389
Q(1.8)	0.035930319	1/27.8316
Q(1.9)	0.028716560	1/34.8231

Q(2.0)	0.022750132	1/43.9558
Q(2.1)	0.017864421	1/55.9772
Q(2.2)	0.013903448	1/71.9246
Q(2.3)	0.010724110	1/93.2478
Q(2.4)	0.008197536	1/121.9879
Q(2.5)	0.006209665	1/161.0393
Q(2.6)	0.004661188	1/214.5376
Q(2.7)	0.003466974	1/288.4360
Q(2.8)	0.002555130	1/391.3695
Q(2.9)	0.001865813	1/535.9593

Q(3.0)	0.001349898	1/740.7967
Q(3.1)	0.000967603	1/1033.4815
Q(3.2)	0.000687138	1/1455.3119
Q(3.3)	0.000483424	1/2068.5769
Q(3.4)	0.000336929	1/2967.9820
Q(3.5)	0.000232629	1/4298.6887
Q(3.6)	0.000159109	1/6285.0158
Q(3.7)	0.000107800	1/9276.4608
Q(3.8)	0.000072348	1/13822.0738
Q(3.9)	0.000048096	1/20791.6011
Q(4.0)	0.000031671	1/31574.3855

Generalization to high dimensions

The Q-function can be generalized to higher dimensions:

$Q(𝐱)=\mathbb{P}(𝐗\ge 𝐱),$

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