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.

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. 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. The above proper integral form of Q-function, has been incorrectly credited to Craig. This form of Q-function was implied in earlier works by Wiesten , and explicitly stated by Pawula, Rice and Roberts . 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(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) 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 =u 2 /2, the upper bound is derived as follows: Q(x) =\int_x^\infty\phi(u)\,du 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 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). A pair of tight lower and upper bounds on the Gaussian Q-function for positive arguments x \in [0, \infty) was introduced by Abreu (2012) based on a simple algebraic expression with only two exponential terms: Q(x) \geq \frac{1}{12} e^{-x^2} + \frac{1}{\sqrt{2\pi} (x + 1)} e^{-x^2 / 2}, \qquad x \geq 0, Q(x) \leq \frac{1}{50} e^{-x^2} + \frac{1}{2 (x + 1)} e^{-x^2 / 2}, \qquad x \geq 0. These bounds are derived from a unified form Q_{\mathrm{B}}(x; a, b) = \frac{\exp(-x^2)}{a} + \frac{\exp(-x^2 / 2)}{b (x + 1)} , where the parameters a and b are chosen to satisfy specific conditions ensuring the lower ( a_{\mathrm{L}} = 12 , b_{\mathrm{L}} = \sqrt{2\pi} ) and upper ( a_{\mathrm{U}} = 50 , 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 Q^n(x) for positive integers n using the binomial theorem, maintaining their simplicity and effectiveness. ==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. 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.

Text derived from https://en.wikipedia.org/w/index.php?oldid=1363574830 under CC BY-SA 4.0; modifications have been made.