An overview of generic tools for information-theoretic secrecy performance analysis over wiretap fading channels

Kong, Long; Ai, Yun; Lei, Lei; Kaddoum, Georges; Chatzinotas, Symeon; Ottersten, Björn

doi:10.1186/s13638-021-02065-4

EURASIP Journal on Wireless Communications and Networking

Table 3 Fox’s H-equivalents of typical and generalized statistical models for instantaneous received SNR \(\gamma _i, i \in \{B,E \}\), and \({\bar{\gamma }}_i\) is the average SNR

From: An overview of generic tools for information-theoretic secrecy performance analysis over wiretap fading channels

Model	\({\boldsymbol{{\mathcal {K}}}}\)	\(\boldsymbol{\lambda}\)	m n	\({\boldsymbol{\mathfrak {a}}}\)	\({\boldsymbol{\mathfrak {b}}}\)
Model	\({\boldsymbol{{\mathcal {K}}}}\)	\(\boldsymbol{\lambda}\)	p q	\({\boldsymbol{\mathscr {A}}}\)	\({\boldsymbol{\mathscr {B}}}\)
Rayleigh	\(\frac{1}{{\bar{\gamma }}_i}\)	\(\frac{1}{{\bar{\gamma }}_i}\)	1 0	–	0
Rayleigh	\(\frac{1}{{\bar{\gamma }}_i}\)	\(\frac{1}{{\bar{\gamma }}_i}\)	0 1	–	1
Nakagami	\(\frac{m}{\Gamma (m){\bar{\gamma }}_i}\)	\(\frac{m}{{\bar{\gamma }}_i}\)	1 0	–	\(m -1\)
Nakagami	\(\frac{m}{\Gamma (m){\bar{\gamma }}_i}\)	\(\frac{m}{{\bar{\gamma }}_i}\)	0 1	–	1
Weibull	\(\frac{\Gamma (1+\frac{2}{\alpha })}{{\bar{\gamma }}_i}\)	\(\frac{\Gamma (1+\frac{2}{\alpha })}{{\bar{\gamma }}_i}\)	1 0	–	\(1 - \frac{2}{\alpha }\)
Weibull	\(\frac{\Gamma (1+\frac{2}{\alpha })}{{\bar{\gamma }}_i}\)	\(\frac{\Gamma (1+\frac{2}{\alpha })}{{\bar{\gamma }}_i}\)	0 1	–	\(\frac{2}{\alpha }\)
\(\alpha\)-\(\mu\)	\(\frac{\Gamma (\mu + \frac{2}{\alpha })}{\Gamma (\mu )^2{\bar{\gamma }}_i}\)	\(\frac{\Gamma (\mu +\frac{2}{\alpha })}{\Gamma (\mu ){\bar{\gamma }}_i}\)	1 0	–	\(\mu - \frac{2}{\alpha }\)
\(\alpha\)-\(\mu\)			0 1	–	\(\frac{2}{\alpha }\)
Maxswell	\(\frac{3}{\sqrt{\pi }{\bar{\gamma }}_i}\)	\(\frac{3}{2{\bar{\gamma }}_i}\)	1 0	–	\(\frac{1}{2}\)
Maxswell	\(\frac{3}{\sqrt{\pi }{\bar{\gamma }}_i}\)	\(\frac{3}{2{\bar{\gamma }}_i}\)	0 1	–	1
\(N*\)(\(\alpha\)-\(\mu )\)	\(\prod \limits _{i=1}^N\frac{\Gamma (\mu _i + \frac{2}{\alpha _i})}{\Gamma (\mu _i)^2{\bar{\gamma }}_i}\)	\(\prod \limits _{i=1}^N\frac{\Gamma (\mu _i + \frac{2}{\alpha _i})}{\Gamma (\mu _i){\bar{\gamma }}_i}\)	N 0	–	\((\mu _1 - \frac{2}{\alpha _1},\cdots ,\mu _N - \frac{2}{\alpha _N})\)
\(N*\)(\(\alpha\)-\(\mu )\)			0 N	–	\((\frac{2}{\alpha _1},\cdots , \frac{2}{\alpha _N})\)
Fisher-Snedecor \({\mathcal {F}}\)	\(\frac{m}{m_s{\bar{\gamma }}_i \Gamma (m)\Gamma (m_s)}\)	\(\frac{m}{m_s{\bar{\gamma }}_i }\)	1 1	\(-m_{s}\)	1
Fisher-Snedecor \({\mathcal {F}}\)	\(\frac{m}{m_s{\bar{\gamma }}_i \Gamma (m)\Gamma (m_s)}\)	\(\frac{m}{m_s{\bar{\gamma }}_i }\)	1 1	\(m-1\)	1
Generalized-\({\mathcal {K}}\)	\(\frac{m_l m_{sl}}{\Gamma (m_l)\Gamma (m_{sl}){\bar{\gamma }}_i}\)	\(\frac{m_1m_2}{{\bar{\gamma }}_i}\)	2 0	–	\((m_l-1 ,m_{s1} - 1)\)
Generalized-\({\mathcal {K}}\)		\(\frac{m_1m_2}{{\bar{\gamma }}_i}\)	0 2	–	(1, 1)
EGK	\(\frac{\Gamma (m + \frac{1}{\xi }) \Gamma (m_s + \frac{1}{\xi _s})}{{\bar{\gamma }}_i\Gamma (m)^2\Gamma (m_s)^2}\)	\(\frac{\Gamma (m + \frac{1}{\xi }) \Gamma (m_s + \frac{1}{\xi _s})}{{\bar{\gamma }}_i\Gamma (m)\Gamma (m_s)}\)	2 0	–	\((m - \frac{1}{\xi }, m_{s} - \frac{1}{\xi _{s}})\)
EGK			0 2	–	\((\frac{1}{\xi }, \frac{1}{\xi _{s}})\)

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