The OP and error rate of diversity combining schemes are two important performance metrics that can be used to quantify the effects of correlated fading. We analyze the performance of this new fading model here with arbitrary channel correlation by computing the bounds according to [17]. Here, we set an SNR threshold, γth, below which we declare occurrences of outage.
Outage probability bounds for MRC
We express the OP for MRC with respect to a signal threshold, γth, as
$$ P_{o}^{\text{MRC}}\left({{\gamma_{th}}} \right) = {\text{Pr}}\left\{ {{\gamma_{{\text{MRC}}}} \le {\gamma_{{\text{th}}}}} \right\}. $$
(21)
Substituting (12) into (11a) and then substituting the result into (21) leads to
$$\begin{array}{*{20}l}\notag P_{o}^{\text{MRC}}\left({{\gamma_{th}}} \right) &= {\text{Pr}}\left\{ {\sum\limits_{n = 1}^{N} {\sum\limits_{i = 1}^{2m} {b_{n,i}^{2}} \le {\gamma_{\text{th}}}}} \right\} \\ &= \int\limits_{{\gamma_{{\text{MRC}}}} \le {\gamma_{{\text{th}}}}} {{f_{\mathbf{b}}}\left(\mathbf{b} \right)} d\mathbf{b}. \end{array} $$
(22)
It should be noted that the above integral is a 2mN-fold integral, but for notational simplicity, it is expressed with just one integral sign.
Asymptotic outage probability approximation for MRC
The asymptotic OP approximation is obtained by substituting fb(b)≈fb(0) into (22), where
$$\begin{array}{*{20}l}\notag {f_{\mathbf{b}}}({\mathbf{0}})= \frac{1}{{\sqrt {{{\left({2\pi} \right)}^{2~\text{mN}}}\left| {{\mathbf{R}_{\mathbf{b}}}} \right|} }} \exp \left({ - \frac{1}{2}{{\boldsymbol{\mu }}}^{T}}\mathbf{R}_{\mathbf{b}}^{- 1}{\boldsymbol{\mu }}_{\mathbf{b}} \right) \end{array} $$
and simplifying to give
$$\begin{array}{*{20}l} P_{o,\infty }^{\text{MRC}}\left({{\gamma_{\text{th}}}} \right) = {f_{\mathbf{b}}}\left({\mathbf{0}} \right)\int\limits_{{\gamma_{\text{MRC}}} \le {\gamma_{\text{th}}}} {d\mathbf{b}} \end{array} $$
(23)
where \({\gamma _{\text {MRC}}} = \sum \limits _{n = 1}^{N} {\sum \limits _{i = 1}^{2m} {b_{n,i}^{2}}} \). The integral in (23) is the volume of a 2mN dimensional sphere with a radius of \(r = \sqrt {{\gamma _{\text {th}}}}\). Ultimately, Eq. (23) can be expressed as
$$\begin{array}{*{20}l} &P_{o,\infty }^{\text{MRC}}\left({{\gamma_{\text{th}}}} \right) = {f_{\mathbf{b}}}\left({\mathbf{0}} \right)\frac{{{{\left({2\sqrt {{\gamma_{th}}}} \right)}^{2~\text{mN} + 1}}{\pi^{\text{mN}}}\Gamma \left({\text{mN} + 1} \right)}}{{\left({2~\text{mN} + 1} \right)!}}. \end{array} $$
(24)
Lower bound outage probability for MRC
The lower bound to the OP is obtained by replacing fb(0) with the lower bound of the pdf obtained in (20) into (23). Substituting the result into (23) gives
$$\begin{array}{*{20}l} P_{o,\text{LM}}^{\text{MRC}}\left({{\gamma_{\text{th}}}} \right) = {O_{LM}}\int\limits_{{\gamma_{\text{MRC}}} \le {\gamma_{\text{th}}}} {d\mathbf{b}} \end{array} $$
(25)
where
$$ {O_{\text{LM}}} = {f_{\mathbf{b}}} \times \left({\mathbf{0}} \right)\exp \left({ - \frac{1}{2}\left({{\lambda_{\text{max}}}{\gamma_{\text{th}}} + 2\left\| {{\boldsymbol{\mu}_{\mathbf{b}}}^{T}\mathbf{R}_{\mathbf{b}}^{- 1}} \right\|\sqrt {{\gamma_{\text{th}}}}} \right)} \right). $$
Eq. 25 can be further simplified to
$$\begin{array}{*{20}l} &P_{o,\text{LB}}^{\text{MRC}}\left({{\gamma_{\text{th}}}} \right) = {O_{\text{LM}}} \times \left[ {\frac{{{{\left({2\sqrt {{\gamma_{\text{th}}}}} \right)}^{\left({2\text{mN} + 1} \right)}}{\pi^{\text{mN}}}\Gamma \left({\text{mN} + 1} \right)}}{{\left({2\text{mN} + 1} \right)!}}} \right]. \end{array} $$
(26)
Upper bound outage probability for MRC
The upper bound to the OP is obtained by replacing fb(0) with the upper bound of the pdf obtained in (20) and by substituting the result into (23). This gives
$$\begin{array}{*{20}l} P_{o,UB}^{\text{MRC}}\left({{\gamma_{\text{th}}}} \right) = {O_{\text{UM}}}\int\limits_{{\gamma_{\text{MRC}}} \le {\gamma_{\text{th}}}} {d\mathbf{b}} \end{array} $$
(27)
where \( {{O}}_{\text {UM}} = {f_{\mathbf {b}}}\left ({\mathbf {0}} \right)\exp \left ({\left \| {{{\boldsymbol {\mu }}_{\mathbf {b}}}^{T}\mathbf {R}_{\mathbf {b}}^{- 1}} \right \|\sqrt {{\gamma _{\text {th}}}}} \right){{ }}\). Eq. 27 is further simplified to
$$\begin{array}{*{20}l} P_{o,UB}^{\text{MRC}}\left({{\gamma_{\text{th}}}} \right) = {O_{\text{UM}}} \times \left[ {\frac{{{{\left({2\sqrt {{\gamma_{\text{th}}}}} \right)}^{\left({2~\text{mN} + 1} \right)}}{\pi^{\text{mN}}}\Gamma \left({\text{mN} + 1} \right)}}{{\left({2~\text{mN} + 1} \right)!}}} \right]. \end{array} $$
(28)
We compare the results obtained in (24), (26), and (28), observing that OLM≤fb(b)≤OUM, which is the same as (20). It is seen that the difference between the bounds lies in the exponential component of the equations. An increase to the average SNR per branch leads to the lower and upper bounds of the OP converging to the asymptotic approximation.
Outage probability bounds for EGC
We express the OP for EGC with respect to a signal threshold, γth, as
$$\begin{array}{*{20}l} P_{o}^{\text{EGC}}\left({{\gamma_{\text{th}}}} \right) &= \rm{{Pr}}\left\{ {{\gamma_{EGC}} \le {\gamma_{th}}} \right\}. \end{array} $$
(29)
Substituting (12) into (11b) and then substituting the result into (29) gives
$$\begin{array}{*{20}l}\notag P_{o}^{\text{EGC}}\left({{\gamma_{\text{th}}}} \right) &= {\rm{Pr}}\left\{ {\frac{1}{N}{{\left({\sum\limits_{n = 1}^{N} {\sqrt {\sum\limits_{i = 1}^{2m} {b_{n,i}^{2}}} }} \right)}^{\rm{2}}} \le {\gamma_{\text{th}}}} \right\} \\ &= \int\limits_{{\gamma_{\text{EGC}}} \le {\gamma_{\text{th}}}} {{f_{\mathbf{b}}}\left(\mathbf{b} \right)} d\mathbf{b}. \end{array} $$
(30)
Asymptotic outage probability approximation for EGC
The asymptotic OP approximation is obtained by substituting fb(b)≈fb(0) in (30) and simplifying to give
$$\begin{array}{*{20}l} P_{o,\infty }^{\text{EGC}}\left({{\gamma_{\text{th}}}} \right) = {f_{\mathbf{b}}}\left({\mathbf{0}} \right)\int\limits_{{\gamma_{\text{EGC}}} \le {\gamma_{\text{th}}}} {d\mathbf{b}} \end{array} $$
(31)
where \({\gamma _{\text {EGC}}} = \frac {1}{N}{\left ({\sum \limits _{n = 1}^{N} {\sqrt {\sum \limits _{i = 1}^{2m} {b_{n,i}^{2}}} }} \right)^{2}}\). The integral in (31) is obtained as [17]
$$\begin{array}{*{20}l} \int\limits_{{\gamma_{\text{EGC}}} \le {\gamma_{\text{th}}}} {d\mathbf{b}} = {\left({\frac{{2m{\pi^{m}}}}{{\Gamma \left({m + 1} \right)}}} \right)^{N}}\frac{{{\Gamma^{N}}\left({2m} \right)}}{{\left({2mN} \right)}}{\left({N{\gamma_{\text{th}}}} \right)^{\text{mN}}}. \end{array} $$
(32)
This result in (32) is substituted into (31) to give the asymptotic OP approximation as
$$\begin{array}{*{20}l} &P_{o,\infty }^{\text{EGC}}\left({{\gamma_{\text{th}}}} \right) = {f_{\mathbf{b}}}\left({\mathbf{0}} \right){\left({\frac{{2m{\pi^{m}}}}{{\Gamma \left({m + 1} \right)}}} \right)^{N}}\frac{{{\Gamma^{N}}\left({2m} \right)}}{{\left({2~\text{mN}} \right)}}{\left({N{\gamma_{\text{th}}}} \right)^{\text{mN}}}. \end{array} $$
(33)
Lower bound outage probability for EGC
We compare the integral region for EGC to that of MRC. These integral regions can be expressed as
$$\begin{array}{*{20}l}\notag I{R_{\text{EGC}}}\left(r \right) \buildrel \Delta \over = \left\{ {\frac{1}{N}\sum\limits_{n = 1}^{N} {\sum\limits_{i = 1}^{2m} {b_{n,i}^{2}}} \le {r^{2}}} \right\} \end{array} $$
and
$$\begin{array}{*{20}l} \notag \mathrm{I{R_{MRC}}}\left(r \right) \buildrel \Delta \over = \left\{ {\sum\limits_{n = 1}^{N} {\sum\limits_{i = 1}^{2m} {b_{n,i}^{2}}} \le {r^{2}}} \right\}. \end{array} $$
Note that \(\sum \limits _{n = 1}^{N} {\sum \limits _{i = 1}^{2m} {b_{n,i}^{2}}} \le N{r^{2}}\) can be derived from \(\frac {1}{N}\sum \limits _{n = 1}^{N} {\sum \limits _{i = 1}^{2m} {b_{n,i}^{2}}} \le {r^{2}}\). Thus, the integral region IREGC(r) lies within \(\mathrm {I{R_{MRC}}}\left ({\sqrt N r} \right)\). We then obtain a bound on the pdf for EGC by substituting r with \(\sqrt {N}r\) in (20) as
$$\begin{array}{*{20}l}\notag &{f_{\mathbf{b}}}\left({\mathbf{0}} \right)\exp \left({ - \frac{1}{2}\left({{\lambda_{\text{max}}}Nr^{2} + 2\left\| {{{\boldsymbol{\mu }}_{\mathbf{b}}}^{T}\mathbf{R}_{\mathbf{b}}^{- 1}} \right\| \sqrt{N}r} \right)} \right) \\ &\le {f_{\mathbf{b}}}\left(\mathbf{b} \right) \le {f_{\mathbf{b}}}\left({\mathbf{0}} \right)\exp \left({\left\| {{{\boldsymbol{\mu }}_{\mathbf{b}}}^{T}\mathbf{R}_{\mathbf{b}}^{- 1}} \right\| \sqrt{N}r} \right) \end{array} $$
(34)
where \(r=\sqrt {\gamma _{\text {th}}}\). Substituting the lower bound of the result obtained in (34) into (31), the lower bound OP for EGC is obtained as
$$\begin{array}{*{20}l} P_{o,\text{LB}}^{\text{EGC}}\left({{\gamma_{\mathrm{\text{th}}}}} \right) = {O_{\text{LE}}} \times \int\limits_{{\gamma_{\text{EGC}}} \le {\gamma_{\text{th}}}} {d\mathbf{b}} \end{array} $$
(35)
where
$${O_{\text{LE}}} = {f_{\mathbf{b}}}\left({\mathbf{0}} \right) \times \exp \left({ - \frac{1}{2}\left({{\lambda_{\text{max}}}N{\gamma_{\mathrm{\text{th}}}} + 2\left\| {{{\boldsymbol{\mu }}_{\mathbf{b}}}^{T}\mathbf{R}_{\mathbf{b}}^{- 1}} \right\|\sqrt {N{\gamma_{th}}}} \right)} \right). $$
This result can be expressed as
$$\begin{array}{*{20}l} P_{o,\text{LB}}^{\text{EGC}}\left({{\gamma_{\text{th}}}} \right) = {O_{\text{LE}}} \times {\left({\frac{{2m{\pi^{m}}}}{{\Gamma \left({m + 1} \right)}}} \right)^{N}}\frac{{{\Gamma^{N}}\left({2m} \right)}}{{\left({2~\text{mN}} \right)}}{\left({N{\gamma_{\text{th}}}} \right)^{\text{mN}}}. \end{array} $$
(36)
Upper bound outage probability for EGC
The upper bound to the OP of EGC is obtained by replacing fb(0) in (31) by the upper bound of the pdf in (34), which gives
$$\begin{array}{*{20}l} P_{o,\text{UB}}^{\text{EGC}}\left({{\gamma_{\text{th}}}} \right) = {O_{\text{UE}}} \times \int\limits_{{\gamma_{\text{EGC}}} \le {\gamma_{\text{th}}}} {d\mathbf{b}} \end{array} $$
(37)
where \({O_{\text {UE}}} = {f_{\mathbf {b}}}\left ({\mathbf {0}} \right)\exp \left ({\left \| {{{\boldsymbol {\mu }}_{\mathbf {b}}}^{T}\mathbf {R}_{\mathbf {b}}^{- 1}} \right \|\sqrt {N{\gamma _{\text {th}}}}} \right)\). This result can be expressed as
$$\begin{array}{*{20}l} &P_{o,\text{UB}}^{\text{EGC}}\left({{\gamma_{\text{th}}}} \right) = {O_{\text{UE}}} \times {\left({\frac{{2m{\pi^{m}}}}{{\Gamma \left({m + 1} \right)}}} \right)^{N}}\frac{{{\Gamma^{N}}\left({2m} \right)}}{{\left({2~\text{mN}} \right)}}{\left({N{\gamma_{\text{th}}}} \right)^{\text{mN}}}. \end{array} $$
(38)
We compare the results obtained in (33), (36), and (38), observing that OLE≤fb(b)≤OUE is the same as (34). It is seen again that the difference between the bounds lies in the exponential components of the equations. An increase to the average SNR per branch leads to the lower and upper bounds of the OP converging to the asymptotic approximation, as was found previously.
Outage probability bounds for SC
We express the OP for SC with respect to a signal threshold, γth, as
$$\begin{array}{*{20}l} P_{o}^{\text{SC}}\left({{\gamma_{\text{th}}}} \right) = {\rm{Pr}}\left\{ {{\gamma_{\text{SC}}} \le {\gamma_{{{\text{th}}}}}} \right\}. \end{array} $$
(39)
Substituting (12) into (11c) gives
$$\begin{array}{*{20}l}\notag P_{o}^{\text{SC}}\left({{\gamma_{\text{th}}}} \right) &= {\rm{Pr}}\left\{ {\mathop {\max }\limits_{n} \left\{ {\sum\limits_{{{i = 1}}}^{2m} {b_{{{n}},{{i}}}^{{2}}}} \right\} \le {\gamma_{{{\text{th}}}}}} \right\} \\ &= \int\limits_{{\gamma_{\text{SC}}} \le {\gamma_{\mathrm{{th}}}}} {{f_{\mathbf{b}}}\left(\mathbf{b} \right)} d\mathbf{b}. \end{array} $$
(40)
Asymptotic outage probability approximation for SC
The asymptotic OP approximation is obtained by substituting fb(b)≈fb(0) into (40) and simplifying to give
$$\begin{array}{*{20}l} P_{o,\infty }^{\text{SC}}\left({{\gamma_{\text{th}}}} \right) = {f_{\mathbf{b}}}\left({\mathbf{0}} \right)\prod\limits_{n = 1}^{N} {\int\limits_{\sum\limits_{i = 1}^{2m} {b_{n,i}^{2}} \le {\gamma_{\text{th}}}} {d{\mathbf{b}_{n}}} } \end{array} $$
(41)
where \(d{\mathbf {b}_{n}} = \Pi _{n = 1}^{N}d{b_{n,i}}\). The integral in (41) is the 2m-dimensional volume of a ball with a radius of \(r = \sqrt {{\gamma _{\text {th}}}}\). Ultimately, Eq. (41) can be expressed as
$$\begin{array}{*{20}l} P_{o,\infty }^{\text{SC}}\left({{\gamma_{\text{th}}}} \right) = {f_{\mathbf{b}}}\left({\mathbf{0}} \right) \times{\left[ {\frac{{{{\left({2\sqrt {{\gamma_{\text{th}}}}} \right)}^{\left({2m + 1} \right)}}{\pi^{m}}\Gamma \left({m + 1} \right)}}{{\left({2m + 1} \right)!}}} \right]^{N}}. \end{array} $$
(42)
Lower bound outage probability for SC
We compare the integral region for SC to that of MRC. These integral regions can be expressed as
$$\begin{array}{*{20}l} \notag \mathrm{I{R_{SC}}}\left(r \right) \buildrel \Delta \over = \left\{ {\mathop {\max }\limits_{n} \left\{ {\sum\limits_{i = 1}^{2m} {b_{n,i}^{2}}} \right\} \le {r^{2}}} \right\} \end{array} $$
and
$$\begin{array}{*{20}l} \notag \mathrm{I{R_{MRC}}}\left(r \right) \buildrel \Delta \over = \left\{ {\sum\limits_{n = 1}^{N} {\sum\limits_{i = 1}^{2m} {b_{n,i}^{2}}} \le {r^{2}}} \right\} \end{array} $$
where r is the radius. We can therefore bound the 2m-dimensional ball in (41) by the hypersphere in (23). In addition, we note that \(\sum \limits _{n = 1}^{N} {\sum \limits _{i = 1}^{2m} {b_{n,i}^{2}}} \le Nr\) can be derived from \(\mathop {\max }\limits _{n} \left \{ {\sum \limits _{i = 1}^{2m} {b_{n,i}^{2}}} \right \} \le {r}\); therefore, IRSC(r) lies within \(\mathrm {I{R_{MRC}}}\left ({\sqrt N r} \right)\). Equation 34 is therefore an applicable bound to SC. Substituting the lower bound of the result obtained in (34) into (41) gives
$$\begin{array}{*{20}l} P_{o,\text{LB}}^{\text{SC}}\left({{\gamma_{\text{th}}}} \right) &= {{O}}_{\text{LS}} \int\limits_{{\gamma_{{{\mathrm{\mathrm{\text{SC}}}}}}} \le {\gamma_{\text{th}}}} {d\mathbf{b}} \end{array} $$
(43)
where
$$\begin{array}{*{20}l} {O_{\text{LS}}} = {f_{\mathbf{b}}}\left({\mathbf{0}} \right) \times\exp \left({ - \frac{1}{2}\left({{\lambda_{\text{max}}}N{\gamma_{\text{th}}} + 2\left\| {{{\boldsymbol{\mu }}_{\mathbf{b}}}^{T}\mathbf{R}_{\mathbf{b}}^{- 1}} \right\|\sqrt {N{\gamma_{\text{th}}}}} \right)} \right). \end{array} $$
Equation 43 can be further simplified to
$$\begin{array}{*{20}l} P_{o,\text{LB}}^{\text{SC}}\left({{\gamma_{\text{th}}}} \right) = {{O}}_{\text{LS}} \times {\left[ {\frac{{{{\left({2\sqrt {{\gamma_{\text{th}}}}} \right)}^{\left({2m + 1} \right)}}{\pi^{m}}\Gamma \left({m + 1} \right)}}{{\left({2m + 1} \right)!}}} \right]^{N}}. \end{array} $$
(44)
Upper bound outage probability for SC
The upper bound to the OP is obtained by replacing fb(0) in (40) with the upper bound of the pdf obtained in (34). This gives
$$\begin{array}{*{20}l} &P_{o,\text{UB}}^{\text{SC}}\left({{\gamma_{\text{th}}}} \right) = {{O}}_{\text{US}} \int\limits_{{\gamma_{{{\text{SC}}}}} \le {\gamma_{\text{th}}}} {d\mathbf{b}} \end{array} $$
(45)
where \({O_{\text {US}}} = {f_{\mathbf {b}}}\left ({\mathbf {0}} \right)\exp \left ({\left \| {{{\boldsymbol {\mu }}_{\mathbf {b}}}^{T}\mathbf {R}_{\mathbf {b}}^{- 1}} \right \|\sqrt {N{\gamma _{\text {th}}}}} \right)\). Equation 45 is further simplified to
$$\begin{array}{*{20}l} P_{o,\text{UB}}^{\text{SC}}\left({{\gamma_{\text{th}}}} \right) = {{O}}_{\text{US}} \times {\left[ {\frac{{{{\left({2\sqrt {{\gamma_{\text{th}}}}} \right)}^{\left({2m + 1} \right)}}{\pi^{m}}\Gamma \left({m + 1} \right)}}{{\left({2m + 1} \right)!}}} \right]^{N}}. \end{array} $$
(46)
We compare the results obtained in (42), (44), and (46), observing that OLS≤fb(b)≤OUS is the same as (34). It is once again seen that the difference between the bounds lies in the exponential components of the equations. An increase to the average SNR per branch leads to the lower and upper bounds of the OP converging to the asymptotic approximation, as seen previously.