Q-function

Bounds and approximations

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

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

where ${\displaystyle \varphi (x)}$ 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:

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

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

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

Solving for Q(x) provides the lower bound.

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

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

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

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

For ${\displaystyle x\ge 0}$, 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

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

• Improved exponential bounds and a pure exponential approximation are

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

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

• The above were generalized by Tanash & Riihonen (2020), who showed that ${\displaystyle Q(x)}$ 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: ${\displaystyle Q(x)\approx \tilde{Q}(x)}$, ${\displaystyle Q(x)\le \tilde{Q}(x)}$, or ${\displaystyle Q(x)\ge \tilde{Q}(x)}$ for ${\displaystyle x\ge 0}$. 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.

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

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

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

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

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 ${\displaystyle \mathrm{\forall }x\ge 0.}$

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

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

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 ${\displaystyle Q(x)}$ for positive arguments ${\displaystyle x\in [0,\mathrm{\infty })}$ is given by López-Benítez & Casadevall (2011) based on a second-order exponential function:

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

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 ${\displaystyle x\in [0,\mathrm{\infty })}$ was introduced by Abreu (2012) based on a simple algebraic expression with only two exponential terms:

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

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

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

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

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

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

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

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.

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:

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

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