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


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

Thus,


 * $$Q(x) = 1 - Q(-x) = 1 - \Phi(x)\,\!,$$

where $$\Phi(x)$$ 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:


 * $$Q(x) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \exp \left( - \frac{x^2}{2 \sin^2 \theta} \right) d\theta.$$

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:
 * Q function complex plot plotted with Mathematica 13.1 ComplexPlot3D.svg$$Q(x+y) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \exp \left( - \frac{x^2}{2 \sin^2 \theta} - \frac{y^2}{2 \cos^2 \theta} \right) d\theta, \quad x,y \geqslant 0 .$$

Bounds and approximations

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


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


 * where $$\phi(x)$$ 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:


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


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


 * $$\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. $$


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


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


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


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


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


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


 * The Chernoff bound of the Q-function is


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


 * Improved exponential bounds and a pure exponential approximation are


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


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


 * 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: $$Q(x) \approx \tilde{Q}(x)$$, $$Q(x) \leq \tilde{Q}(x)$$, or $$Q(x) \geq \tilde{Q}(x)$$ for $$x\geq0$$. 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 $$x \in [0,\infty)$$ is given by Karagiannidis & Lioumpas (2007) who showed for the appropriate choice of parameters $$\{A, B\}$$ that


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


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


 * $$\{A, B\} = \underset{\{A,B\}}{\arg \min} \frac{1}{R} \int_0^R | f(x; A, B) - \operatorname{erfc}(x) |dx.$$


 * 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 $$\forall x \ge 0.$$


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


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


 * 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 $$x \in [0,\infty)$$ is given by López-Benítez & Casadevall (2011) based on a second-order exponential function:


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


 * 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:


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

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


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

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.

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.

Generalization to high dimensions
The Q-function can be generalized to higher dimensions:


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

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