Q-function

In statistics, the Q-function is the tail distribution function of the standard normal distribution.^[1][2] In other words, ${\displaystyle Q(x)}$ is the probability that a normal (Gaussian) random variable will obtain a value larger than ${\displaystyle x}$ standard deviations. Equivalently, ${\displaystyle Q(x)}$ is the probability that a standard normal random variable takes a value larger than ${\displaystyle x}$.

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

${\displaystyle P(Y>y)=P(X>x)=Q(x)}$

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

Other definitions of the Q-function, all of which are simple transformations of the normal cumulative distribution function, are also used occasionally.^[3]

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

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

Thus,

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

where ${\displaystyle \Phi (x)}$ 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^[2]

${\displaystyle {\begin{aligned}Q(x)&={\frac {1}{2}}\left({\frac {2}{\sqrt {\pi }}}\int _{x/{\sqrt {2}}}^{\infty }\exp \left(-t^{2}\right)\,dt\right)\\&={\frac {1}{2}}-{\frac {1}{2}}\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)~~{\text{ -or-}}\\&={\frac {1}{2}}\operatorname {erfc} \left({\frac {x}{\sqrt {2}}}\right).\end{aligned}}}$

An alternative form of the Q-function known as Craig's formula, after its discoverer, is expressed as:^[4]

${\displaystyle Q(x)={\frac {1}{\pi }}\int _{0}^{\frac {\pi }{2}}\exp \left(-{\frac {x^{2}}{2\sin ^{2}\theta }}\right)d\theta .}$

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)^[5] for the Q-function of the sum of two non-negative variables, as follows:

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

Bounds and approximations

• The Q-function is not an elementary function. However, it can be upper and lower bounded as,^[6][7]

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

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

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

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

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

Solving for Q(x) provides the lower bound.

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

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

• Tighter bounds and approximations of ${\displaystyle Q(x)}$ can also be obtained by optimizing the following expression ^[7]

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

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

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

• Improved exponential bounds and a pure exponential approximation are ^[8]

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

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

• The above were generalized by Tanash & Riihonen (2020),^[9] 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: ${\displaystyle Q(x)\approx {\tilde {Q}}(x)}$, ${\displaystyle Q(x)\leq {\tilde {Q}}(x)}$, or ${\displaystyle Q(x)\geq {\tilde {Q}}(x)}$ for ${\displaystyle x\geq 0}$. 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.^[10]

• Another approximation of ${\displaystyle Q(x)}$ for ${\displaystyle x\in [0,\infty )}$ is given by Karagiannidis & Lioumpas (2007)^[11] who showed for the appropriate choice of parameters ${\displaystyle \{A,B\}}$ that

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

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

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

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 ${\displaystyle \forall x\geq 0.}$

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

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

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.^[12]

• A tighter and more tractable approximation of ${\displaystyle Q(x)}$ for positive arguments ${\displaystyle x\in [0,\infty )}$ is given by López-Benítez & Casadevall (2011)^[13] based on a second-order exponential function:

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

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).

Inverse Q

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

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

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

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

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:^[14]

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

where ${\displaystyle \mathbf {X} \sim {\mathcal {N}}(\mathbf {0} ,\,\Sigma )}$ follows the multivariate normal distribution with covariance ${\displaystyle \Sigma }$ and the threshold is of the form ${\displaystyle \mathbf {x} =\gamma \Sigma \mathbf {l} ^{*}}$ for some positive vector ${\displaystyle \mathbf {l} ^{*}>\mathbf {0} }$ 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.^[15][16]

References