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, *γ*_{th}, below which we declare occurrences of outage.

### Outage probability bounds for MRC

We express the OP for MRC with respect to a signal threshold, *γ*_{th}, 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 2*m**N*-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 *f*_{b}(**b**)≈*f*_{b}(**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 2*m**N* 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 *f*_{b}(**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 *f*_{b}(**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 *O*_{LM}≤*f*_{b}(**b**)≤*O*_{UM}, 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, *γ*_{th}, 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 *f*_{b}(**b**)≈*f*_{b}(**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 I*R*_{EGC}(*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 *f*_{b}(**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 *O*_{LE}≤*f*_{b}(**b**)≤*O*_{UE} 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, *γ*_{th}, 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 *f*_{b}(**b**)≈*f*_{b}(**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 2*m*-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 2*m*-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, I*R*_{SC}(*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 *f*_{b}(**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 *O*_{LS}≤*f*_{b}(**b**)≤*O*_{US} 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.