Q-function

A plot of the Q-function.

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

If the underlying random variable is y, then the proper argument to the tail probability is derived as:

x={\frac {y-\mu }{\sigma }}

which expresses the number of standard deviations away from the mean.

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

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 normal Gaussian distribution.

The Q-function can be expressed in terms of the error function, or the complementary error function, as^[2]

{\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]

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.

The Q-function is not an elementary function. However, the bounds

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

become increasingly tight for large x, and are often useful.

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+{\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 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 ^[5]

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

A tight approximation of $Q(x)$ for $x\in [0,\infty )$ is given by Karagiannidis & Lioumpas (2007)^[6] 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\geq 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^{-1.4x}\right)e^{-{\frac {x^{2}}{2}}}}{1.135{\sqrt {2\pi }}x}},x\geq 0.

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 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, 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:^[7]

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.^[8]

References

↑ The Q-function, from cnx.org
1 2 Basic properties of the Q-function Archived March 25, 2009, at the Wayback Machine.
↑ Normal Distribution Function - from Wolfram MathWorld
↑ John W. Craig, A new, simple and exact result for calculating the probability of error for two-dimensional signal constellaions, Proc. 1991 IEEE Military Commun. Conf., vol. 2, pp. 571–575.
↑ Chiani, M., Dardari, D., Simon, M.K. (2003). New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels. IEEE Transactions on Wireless Communications, 4(2), 840–845, doi=10.1109/TWC.2003.814350.
↑ Karagiannidis, G. K., & Lioumpas, A. S. (2007). An improved approximation for the Gaussian Q-function. Communications Letters, IEEE, 11(8), 644-646.
↑ Savage, I. R. (1962). "Mills ratio for multivariate normal distributions". Journal Res. Nat. Bur. Standards Sect. B. 66: 93–96.
↑ Botev, Z. I. (2016). "The normal law under linear restrictions: simulation and estimation via minimax tilting". Journal of the Royal Statistical Society: Series B (Statistical Methodology). doi:10.1111/rssb.12162.

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