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Definition and basic properties

Formally, the Q-function is defined as

Q ( x ) = 1 2 π ∫ x ∞ exp ⁡ ( − u 2 2 ) d u . {\displaystyle 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 − Φ ( x ) , {\displaystyle Q(x)=1-Q(-x)=1-\Phi (x)\,\!,}

where Φ ( x ) {\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, as⁴

Q ( x ) = 1 2 ( 2 π ∫ x / 2 ∞ exp ⁡ ( − t 2 ) d t ) = 1 2 − 1 2 erf ⁡ ( x 2 )      -or- = 1 2 erfc ⁡ ( x 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:⁵

Q ( x ) = 1 π ∫ 0 π 2 exp ⁡ ( − x 2 2 sin 2 ⁡ θ ) d θ . {\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)⁶ for the Q-function of the sum of two non-negative variables, as follows:

Q ( x + y ) = 1 π ∫ 0 π 2 exp ⁡ ( − x 2 2 sin 2 ⁡ θ − y 2 2 cos 2 ⁡ θ ) d θ , x , y ⩾ 0. {\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,⁷⁸

( x 1 + x 2 ) ϕ ( x ) < Q ( x ) < ϕ ( x ) x , x > 0 , {\displaystyle \left({\frac {x}{1+x^{2}}}\right)\phi (x)<Q(x)<{\frac {\phi (x)}{x}},\qquad x>0,} where ϕ ( x ) {\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: Q ( x ) = ∫ x ∞ ϕ ( u ) d u < ∫ x ∞ u x ϕ ( u ) d u = ∫ x 2 2 ∞ e − v x 2 π d v = − e − v x 2 π | x 2 2 ∞ = ϕ ( x ) x . {\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 ϕ ′ ( u ) = − u ϕ ( u ) {\displaystyle \phi '(u)=-u\phi (u)} and the quotient rule, ( 1 + 1 x 2 ) Q ( x ) = ∫ x ∞ ( 1 + 1 x 2 ) ϕ ( u ) d u > ∫ x ∞ ( 1 + 1 u 2 ) ϕ ( u ) d u = − ϕ ( u ) u | x ∞ = ϕ ( x ) x . {\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 Q ( x ) {\displaystyle Q(x)} : Q ( x ) ≈ ϕ ( x ) 1 + x 2 , x ≥ 0. {\displaystyle Q(x)\approx {\frac {\phi (x)}{\sqrt {1+x^{2}}}},\qquad x\geq 0.}

• Tighter bounds and approximations of Q ( x ) {\displaystyle Q(x)} can also be obtained by optimizing the following expression ⁹

Q ~ ( x ) = ϕ ( x ) ( 1 − a ) x + a x 2 + b . {\displaystyle {\tilde {Q}}(x)={\frac {\phi (x)}{(1-a)x+a{\sqrt {x^{2}+b}}}}.} For x ≥ 0 {\displaystyle x\geq 0} , the best upper bound is given by a = 0.344 {\displaystyle a=0.344} and b = 5.334 {\displaystyle b=5.334} with maximum absolute relative error of 0.44%. Likewise, the best approximation is given by a = 0.339 {\displaystyle a=0.339} and b = 5.510 {\displaystyle b=5.510} with maximum absolute relative error of 0.27%. Finally, the best lower bound is given by a = 1 / π {\displaystyle a=1/\pi } and b = 2 π {\displaystyle b=2\pi } with maximum absolute relative error of 1.17%.

• The Chernoff bound of the Q-function is

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

• Improved exponential bounds and a pure exponential approximation are ¹⁰

Q ( x ) ≤ 1 4 e − x 2 + 1 4 e − x 2 2 ≤ 1 2 e − x 2 2 , x > 0 {\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} Q ( x ) ≈ 1 12 e − x 2 2 + 1 4 e − 2 3 x 2 , 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),¹¹ who showed that Q ( x ) {\displaystyle Q(x)} can be accurately approximated or bounded by

Q ~ ( x ) = ∑ n = 1 N a n e − b n x 2 . {\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 { ( a n , b n ) } n = 1 N {\displaystyle \{(a_{n},b_{n})\}_{n=1}^{N}} that yield a minimax approximation or bound: Q ( x ) ≈ Q ~ ( x ) {\displaystyle Q(x)\approx {\tilde {Q}}(x)} , Q ( x ) ≤ Q ~ ( x ) {\displaystyle Q(x)\leq {\tilde {Q}}(x)} , or Q ( x ) ≥ Q ~ ( x ) {\displaystyle Q(x)\geq {\tilde {Q}}(x)} for x ≥ 0 {\displaystyle x\geq 0} . With the example coefficients tabulated in the paper for N = 20 {\displaystyle N=20} , the relative and absolute approximation errors are less than 2.831 ⋅ 10 − 6 {\displaystyle 2.831\cdot 10^{-6}} and 1.416 ⋅ 10 − 6 {\displaystyle 1.416\cdot 10^{-6}} , respectively. The coefficients { ( a n , b n ) } n = 1 N {\displaystyle \{(a_{n},b_{n})\}_{n=1}^{N}} for many variations of the exponential approximations and bounds up to N = 25 {\displaystyle N=25} have been released to open access as a comprehensive dataset.¹²

• Another approximation of Q ( x ) {\displaystyle Q(x)} for x ∈ [ 0 , ∞ ) {\displaystyle x\in [0,\infty )} is given by Karagiannidis & Lioumpas (2007)¹³ who showed for the appropriate choice of parameters { A , B } {\displaystyle \{A,B\}} that

f ( x ; A , B ) = ( 1 − e − A x ) e − x 2 B π x ≈ erfc ⁡ ( x ) . {\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 f ( x ; A , B ) {\displaystyle f(x;A,B)} and erfc ⁡ ( x ) {\displaystyle \operatorname {erfc} (x)} over the range [ 0 , R ] {\displaystyle [0,R]} is minimized by evaluating { A , B } = arg ⁡ min { A , B } 1 R ∫ 0 R | f ( x ; A , B ) − erfc ⁡ ( x ) | d x . {\displaystyle \{A,B\}={\underset {\{A,B\}}{\arg \min }}{\frac {1}{R}}\int _{0}^{R}|f(x;A,B)-\operatorname {erfc} (x)|dx.} Using R = 20 {\displaystyle R=20} and numerically integrating, they found the minimum error occurred when { A , B } = { 1.98 , 1.135 } , {\displaystyle \{A,B\}=\{1.98,1.135\},} which gave a good approximation for ∀ x ≥ 0. {\displaystyle \forall x\geq 0.} Substituting these values and using the relationship between Q ( x ) {\displaystyle Q(x)} and erfc ⁡ ( x ) {\displaystyle \operatorname {erfc} (x)} from above gives Q ( x ) ≈ ( 1 − e − 1.98 x 2 ) e − x 2 2 1.135 2 π x , x ≥ 0. {\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.¹⁴

• A tighter and more tractable approximation of Q ( x ) {\displaystyle Q(x)} for positive arguments Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle x \in [0,\infty)} is given by López-Benítez & Casadevall (2011)¹⁵ based on a second-order exponential function:

Q ( x ) ≈ e − a x 2 − b x − c , x ≥ 0. {\displaystyle Q(x)\approx e^{-ax^{2}-bx-c},\qquad x\geq 0.} The fitting coefficients ( a , b , c ) {\displaystyle (a,b,c)} can be optimized over any desired range of arguments in order to minimize the sum of square errors ( a = 0.3842 {\displaystyle a=0.3842} , b = 0.7640 {\displaystyle b=0.7640} , c = 0.6964 {\displaystyle c=0.6964} for x ∈ [ 0 , 20 ] {\displaystyle x\in [0,20]} ) or minimize the maximum absolute error ( a = 0.4920 {\displaystyle a=0.4920} , b = 0.2887 {\displaystyle b=0.2887} , c = 1.1893 {\displaystyle c=1.1893} for x ∈ [ 0 , 20 ] {\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 Q ( x ) {\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 x ∈ [ 0 , ∞ ) {\displaystyle x\in [0,\infty )} was introduced by Abreu (2012)¹⁶ based on a simple algebraic expression with only two exponential terms:

Q ( x ) ≥ 1 12 e − x 2 + 1 2 π ( x + 1 ) e − x 2 / 2 , x ≥ 0 , {\displaystyle 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 ) ≤ 1 50 e − x 2 + 1 2 ( x + 1 ) e − x 2 / 2 , x ≥ 0. {\displaystyle 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 B ( x ; a , b ) = exp ⁡ ( − x 2 ) a + exp ⁡ ( − x 2 / 2 ) b ( x + 1 ) {\displaystyle Q_{\mathrm {B} }(x;a,b)={\frac {\exp(-x^{2})}{a}}+{\frac {\exp(-x^{2}/2)}{b(x+1)}}} , where the parameters a {\displaystyle a} and b {\displaystyle b} are chosen to satisfy specific conditions ensuring the lower ( a L = 12 {\displaystyle a_{\mathrm {L} }=12} , b L = 2 π {\displaystyle b_{\mathrm {L} }={\sqrt {2\pi }}} ) and upper ( a U = 50 {\displaystyle a_{\mathrm {U} }=50} , b U = 2 {\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 Q n ( x ) {\displaystyle Q^{n}(x)} for positive integers n {\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:

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

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

Q - f a c t o r = 20 log 10 ( Q − 1 ( y ) )   d B {\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.

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:¹⁷

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

where X ∼ N ( 0 , Σ ) {\displaystyle \mathbf {X} \sim {\mathcal {N}}(\mathbf {0} ,\,\Sigma )} follows the multivariate normal distribution with covariance Σ {\displaystyle \Sigma } and the threshold is of the form x = γ Σ l ∗ {\displaystyle \mathbf {x} =\gamma \Sigma \mathbf {l} ^{*}} for some positive vector l ∗ > 0 {\displaystyle \mathbf {l} ^{*}>\mathbf {0} } and positive constant γ > 0 {\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 γ {\displaystyle \gamma } becomes larger and larger.¹⁸¹⁹

References

1. "The Q-function". cnx.org. Archived from the original on 2012-02-29. https://web.archive.org/web/20120229030808/http://cnx.org/content/m11537/latest/ ↩

2. "Basic properties of the Q-function" (PDF). 2009-03-05. Archived from the original (PDF) on 2009-03-25. https://web.archive.org/web/20090325160012/http://www.eng.tau.ac.il/~jo/academic/Q.pdf ↩

3. Normal Distribution Function – from Wolfram MathWorld http://mathworld.wolfram.com/NormalDistributionFunction.html ↩

4. "Basic properties of the Q-function" (PDF). 2009-03-05. Archived from the original (PDF) on 2009-03-25. https://web.archive.org/web/20090325160012/http://www.eng.tau.ac.il/~jo/academic/Q.pdf ↩

5. Craig, J.W. (1991). "A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations" (PDF). MILCOM 91 - Conference record. pp. 571–575. doi:10.1109/MILCOM.1991.258319. ISBN 0-87942-691-8. S2CID 16034807. 0-87942-691-8 ↩

6. Behnad, Aydin (2020). "A Novel Extension to Craig's Q-Function Formula and Its Application in Dual-Branch EGC Performance Analysis". IEEE Transactions on Communications. 68 (7): 4117–4125. doi:10.1109/TCOMM.2020.2986209. S2CID 216500014. /wiki/Doi_(identifier) ↩

7. Gordon, R.D. (1941). "Values of Mills' ratio of area to bounding ordinate and of the normal probability integral for large values of the argument". Ann. Math. Stat. 12 (3): 364–366. doi:10.1214/aoms/1177731721. /wiki/Doi_(identifier) ↩

8. Borjesson, P.; Sundberg, C.-E. (1979). "Simple Approximations of the Error Function Q(x) for Communications Applications". IEEE Transactions on Communications. 27 (3): 639–643. doi:10.1109/TCOM.1979.1094433. /wiki/Doi_(identifier) ↩

9. Borjesson, P.; Sundberg, C.-E. (1979). "Simple Approximations of the Error Function Q(x) for Communications Applications". IEEE Transactions on Communications. 27 (3): 639–643. doi:10.1109/TCOM.1979.1094433. /wiki/Doi_(identifier) ↩

10. Chiani, M.; Dardari, D.; Simon, M.K. (2003). "New exponential bounds and approximations for the computation of error probability in fading channels" (PDF). IEEE Transactions on Wireless Communications. 24 (5): 840–845. doi:10.1109/TWC.2003.814350. http://campus.unibo.it/85943/1/mcddmsTranWIR2003.pdf ↩

11. Tanash, I.M.; Riihonen, T. (2020). "Global minimax approximations and bounds for the Gaussian Q-function by sums of exponentials". IEEE Transactions on Communications. 68 (10): 6514–6524. arXiv:2007.06939. doi:10.1109/TCOMM.2020.3006902. S2CID 220514754. /wiki/ArXiv_(identifier) ↩

12. Tanash, I.M.; Riihonen, T. (2020). "Coefficients for Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials [Data set]". Zenodo. doi:10.5281/zenodo.4112978. https://zenodo.org/record/4112978 ↩

13. Karagiannidis, George; Lioumpas, Athanasios (2007). "An Improved Approximation for the Gaussian Q-Function" (PDF). IEEE Communications Letters. 11 (8): 644–646. doi:10.1109/LCOMM.2007.070470. S2CID 4043576. http://users.auth.gr/users/9/3/028239/public_html/pdf/Q_Approxim.pdf ↩

14. Tanash, I.M.; Riihonen, T. (2021). "Improved coefficients for the Karagiannidis–Lioumpas approximations and bounds to the Gaussian Q-function". IEEE Communications Letters. 25 (5): 1468–1471. arXiv:2101.07631. doi:10.1109/LCOMM.2021.3052257. S2CID 231639206. /wiki/ArXiv_(identifier) ↩

15. Lopez-Benitez, Miguel; Casadevall, Fernando (2011). "Versatile, Accurate, and Analytically Tractable Approximation for the Gaussian Q-Function" (PDF). IEEE Transactions on Communications. 59 (4): 917–922. doi:10.1109/TCOMM.2011.012711.100105. S2CID 1145101. http://www.lopezbenitez.es/journals/IEEE_TCOM_2011.pdf ↩

16. Abreu, Giuseppe (2012). "Very Simple Tight Bounds on the Q-Function". IEEE Transactions on Communications. 60 (9): 2415–2420. doi:10.1109/TCOMM.2012.080612.110075. /wiki/Doi_(identifier) ↩

17. Savage, I. R. (1962). "Mills ratio for multivariate normal distributions". Journal of Research of the National Bureau of Standards Section B. 66 (3): 93–96. doi:10.6028/jres.066B.011. Zbl 0105.12601. https://doi.org/10.6028%2Fjres.066B.011 ↩

18. Botev, Z. I. (2016). "The normal law under linear restrictions: simulation and estimation via minimax tilting". Journal of the Royal Statistical Society, Series B. 79: 125–148. arXiv:1603.04166. Bibcode:2016arXiv160304166B. doi:10.1111/rssb.12162. S2CID 88515228. /wiki/ArXiv_(identifier) ↩

19. Botev, Z. I.; Mackinlay, D.; Chen, Y.-L. (2017). "Logarithmically efficient estimation of the tail of the multivariate normal distribution". 2017 Winter Simulation Conference (WSC). IEEE. pp. 1903–191. doi:10.1109/WSC.2017.8247926. ISBN 978-1-5386-3428-8. S2CID 4626481. 978-1-5386-3428-8 ↩