Improving the Performance of HALE UAV Communication Link Through MIMO Cooperative Relay Strategy

Abstract

High-altitude long endurance unmanned aerial vehicles (HALE UAVs) are military and strategic UAVs that fly above the ground in the stratosphere. The ability of stealth technology and long flight time, has led to use of these UAVs for various military missions such as interception and spying, control guidance, remote sensing, navigation, surveillance and others. In this paper, improving the performance of HALE UAVs communication through multiple input multiple output (MIMO) cooperative relay with amplitude-and-forward strategy is investigated. In clear sky (without rain), two-hop system with line of sight (LoS) channels from source to relay and relay to destination is considered. In LoS-MIMO channels, due to correlation between the sub-channels, neither high-rank MIMO channel nor maximum capacity are achieved. However, based on antennas optimum placement that provide orthogonally between the received signals, maximum capacity will be obtained. The proposed scheme in this paper dramatically increases capacity relative to LoS-SISO channel and dual-hop MIMO Rayleigh up to 6 b/s/Hz and 2 b/s/Hz respectively. Also, simulation results verify exactness analytical expressions. However, rain as one of the most important ambient conditions causes the signal to be scattered in different directions. Therefore, LoS channel is changed to fading (Rayleigh) channel by increasing rainfall. In this case, telecommunication range is proposed as a meaningful metric, and outage probability (P_outage) based on telecommunication range for N-hop channel is extracted. In rainy conditions, simulation results show, the telecommunication range dramatically increases by increasing the number of relay UAVs for specified outage probability so that in the long-range, the outage probability decreases up to 50% with the increasing number of relays.

Appendices

Appendix 1

We derive a simplified criterion for the design of an orthogonal MIMO (2 × 2 × 2) dual-hop architecture in pure LoS as shown in Fig. 1. Starting from Eqs. (17), (22)

$$\left| {{d}_{11}}-{{d}_{12}}-{{d}_{21}}+{{d}_{22}} \right|=\left(2m+1 \right)\frac{\lambda}{2}\quad m=0,1,2,\ldots$$

(42)

The Euclidean distances between all elements for the first hop configuration shown in Fig. 1 are given by

$$\begin{aligned}{{d}_{11}}&=D=\frac{\left| {{H}_{R1}}-{{H}_{T}} \right|}{sin{{\beta}_{1}}}\\{{d}_{12}}&=\sqrt{{{\left(D+{{S}_{R}}cos{{\theta}_{R}}+{{S}_{R}}cos{{\varphi}_{R}} \right)}^{2}}+{{\left({{S}_{R}}sin{{\theta}_{R}} \right)}^{2}}+{{\left({{S}_{R}}cos{{\theta}_{R}}sin{{\varphi}_{R}} \right)}^{2}}}\\{{d}_{21}}&=\sqrt{{{\left(D-{{S}_{S}}cos{{\theta}_{S}} \right)}^{2}}+{{\left({{S}_{S}}sin{{\theta}_{S}} \right)}^{2}}}\\{{d}_{22}}&=\sqrt{{{\left(D+{{S}_{R}}cos{{\theta}_{R}}+{{S}_{R}}cos{{\varphi}_{R}}-{{S}_{S}}cos{{\theta}_{S}} \right)}^{2}}+{{\left({{S}_{R}}sin{{\theta}_{R}}-{{S}_{S}}sin{{\theta}_{S}} \right)}^{2}}+{{\left({{S}_{R}}cos{{\theta}_{R}}sin{{\varphi}_{R}} \right)}^{2}}}\end{aligned}$$

(43)

Equations (43) can be simplified using the following first-order Taylor series approximation

$$\sqrt{{{\left(A+B \right)}^{2}}+{{C}^{2}}}\approx \left(A+B \right)+\frac{{{C}^{2}}}{2\left(A+B \right)}$$

(44)

This approximation is valid for ${{\left(A+B \right)}^{2}}\gg {{C}^{2}}$, which is a reasonable assumption for the targeted application, providing that $d \gg S$. Then, (43) become, respectively

$$\begin{aligned}{{d}_{12}}&=D+{{S}_{R}}cos{{\theta}_{R}}+{{S}_{R}}cos{{\varphi}_{R}}+\frac{{{\left({{S}_{R}}sin{{\theta}_{R}} \right)}^{2}}+{{\left({{S}_{R}}cos{{\theta}_{R}}sin{{\varphi}_{R}} \right)}^{2}}}{2D}\\{{d}_{21}}&=D-{{S}_{S}}cos{{\theta}_{S}}+\frac{{{\left({{S}_{S}}sin{{\theta}_{S}} \right)}^{2}}}{2D}\\{{d}_{22}}&=D+{{S}_{R}}cos{{\theta}_{R}}+{{S}_{R}}cos{{\varphi}_{R}}-{{S}_{S}}cos{{\theta}_{S}}+\frac{{{\left({{S}_{R}}sin{{\theta}_{R}}-{{S}_{S}}sin{{\theta}_{S}} \right)}^{2}}+{{\left({{S}_{R}}cos{{\theta}_{R}}sin{{\varphi}_{R}} \right)}^{2}}}{2D}\end{aligned}$$

(45)

Finally, using Eq. (42) becomes

$${{S}_{S}}{{S}_{R}}=\left(m+\frac{1}{2} \right)\frac{ \lambda\left|{{H}_{R1}}-{{H}_{T}} \right|}{sin{{\beta}_{1}}sin{{\theta}_{S}}sin{{\theta}_{R}}} \quad m=0,1,2,\ldots$$

(46)

In the same way

$$\left| d_{11}^{\prime}+d_{22}^{\prime}-d_{12}^{\prime}-d_{21}^{\prime} \right|=\left(2{m}^{\prime}+1 \right)\frac{\lambda}{2}$$

(47)

The Euclidean distances between all elements for the second hop configuration shown in Fig. 1 are given by

$$\begin{aligned} d_{11}^{\prime}&={{D}^{\prime}}=\frac{\lambda\left| {{H}_{R2}}-{{H}_{R1}} \right|}{sin{{\beta}_{2}}}\\ d_{12}^{\prime}&=\sqrt{{{\left({{D}^{\prime}}+{{S}_{D}}cos{{\theta}_{D}}+{{S}_{D}}cos{{\varphi}_{D}} \right)}^{2}}+{{\left({{S}_{D}}sin{{\theta}_{D}} \right)}^{2}}+{{\left({{S}_{D}}cos{{\theta}_{D}}sin{{\varphi}_{D}} \right)}^{2}}}\\ d_{21}^{\prime}&=\sqrt{{{\left({{D}^{\prime}}-{{S}_{R}}cos{{\theta}_{R}}-{{S}_{R}}cos{{\varphi}_{R}} \right)}^{2}}+{{\left({{S}_{R}}sin{{\theta}_{R}} \right)}^{2}}+{{\left({{S}_{R}}cos{{\theta}_{R}}sin{{\varphi}_{R}} \right)}^{2}}} \\ d_{22}^{\prime}&=\sqrt{{{\left({{D}^{\prime}}+{{S}_{D}}cos{{\theta}_{D}}+{{S}_{D}}cos{{\varphi}_{D}}-{{S}_{R}}cos{{\theta}_{R}}-{{S}_{R}}cos{{\varphi}_{R}} \right)}^{2}}+{{\left({{S}_{D}}sin{{\theta}_{D}}-{{S}_{R}}sin{{\theta}_{R}} \right)}^{2}}}+\ldots \\&\quad+{{\left({{S}_{D}}cos{{\theta}_{D}}sin{{\varphi}_{D}}-{{S}_{R}}cos{{\theta}_{R}}sin{{\varphi}_{R}} \right)}^{2}} \end{aligned}$$

(48)

$$\begin{aligned} & d_{12}^{\prime } = D^{\prime } + S_{D} cos\theta_{D} + S_{D} cos\varphi_{D} + \frac{{\left( {S_{D} sin\theta_{D} } \right)^{2} + \left( {S_{D} cos\theta_{D} sin\varphi_{D} } \right)^{2} }}{{2D^{\prime } }} \\ & d_{21}^{'} = D^{\prime } - S_{R} cos\theta_{R} - S_{R} cos\varphi_{R} + \frac{{\left( {S_{R} sin\theta_{R} } \right)^{2} + \left( {S_{R} cos\theta_{R} sin\varphi_{R} } \right)^{2} }}{{2D^{\prime } }} \\ & d_{22}^{\prime } = D^{\prime } + S_{D} cos\theta_{D} + S_{D} cos\varphi_{D} - S_{R} cos\theta_{R} - S_{R} cos\varphi_{R} \\ & \qquad + \frac{{\left( {S_{D} sin\theta_{D} - S_{R} sin\theta_{R} } \right)^{2} + \left( {S_{D} cos\theta_{D} sin\varphi_{D} - S_{R} cos\theta_{R} sin\varphi_{R} } \right)^{2} }}{{2D^{\prime } }} \\ \end{aligned}$$

Use Eq. (47) Finally,

$${{S}_{R}}{{S}_{D}}=\left({m}^{\prime}+\frac{1}{2} \right)\frac{\lambda\left|{{H}_{R2}}-{{H}_{R1}} \right|}{sin{{\beta}_{2}}\left(sin{{\theta}_{R}}sin{{\theta}_{D}}+cos{{\theta}_{R}}cos{{\theta}_{D}}sin{{\varphi}_{R}}sin{{\varphi}_{D}} \right)} \quad {m}^{\prime}=0,1,2,\ldots$$

(49)

Appendix 2

Lemma 1

To compute the cumulative distribution of Y = g(X) in terms of the cumulative distribution of X, note that

$${{F}_{Y}}\left(y \right)=P\left(Y\le y \right)=P\left\{g\left(X \right)\le y \right\}=P\left\{X\le {{g}^{-1}}\left(y \right) \right\}={{F}_{X}}\left({{g}^{-1}}\left(y \right) \right)$$

(50)

For g decreasing within the range of X:

$${{F}_{Y}}\left(y \right)=P\left(Y\le y \right)=P\left\{g\left(X \right)\le y \right\}=P\left\{X\ge {{g}^{-1}}\left(y \right) \right\}=1-{{F}_{X}}\left({{g}^{-1}}\left(y \right) \right)$$

(51)

Appendix 3

For N = 1 and N = 2, the ${\text{G}}_{{1,\,\,\,{\text{N}} + 1}}^{{{\text{N}},\,\,1}} \left( . \right)$ function in Eq. (37) reduces to a Rayleigh and double-Rayleigh distribution, respectively. When N = 1, the following equation results are obtained:

[31, (Eq. §07.34.03.0273.01)]

$$G\begin{array}{*{20}c} {1,1} \\ {1,2} \\ \end{array} \left( {z\left| {\begin{array}{*{20}c} {\frac{1}{2}} \\ {\frac{1}{2}, - \frac{1}{2}} \\ \end{array} } \right.} \right) = e^{ - z} z^{{\frac{1}{2}}} \varGamma \left( 0 \right)L_{ - 1}^{1} \left( z \right)\left| {z = \left( {2\left( {\frac{4\pi d}{\lambda }} \right)^{2} } \right)^{ - 1} \left( {\frac{A}{{\gamma_{eq} }}} \right)^{2} } \right.$$

(52)

In Eq. (52) $\text{L}_{\vartheta}^{\uplambda}\left(\text{z} \right)$ is the Laguerrel function, that is, [31, (Eq. § 07.03.26.0002.01)]

$$L_{\vartheta}^{\lambda}\left(z \right)=\frac{{{\left(\lambda +1 \right)}_{\vartheta}}}{\varGamma\left(\vartheta +1 \right)} 1{{F}_{1}}\left(-\vartheta;\lambda;z \right) ; -\lambda \in {{N}^{+}}$$

(53)

For $\uptheta =-1$ and $\uplambda = 1$, Eq. (53) reduces to $\frac{{\Gamma \left( 1 \right)}}{{\Gamma \left( 0 \right)}}1{\text{F}}_{1} \left( {1;1;{\text{z}}} \right)$. Moreover, for $\text{a}=1,\text{b}=1$, the hypergeometric function $1{{\text{F}}_{1}}\left(\text{a};\text{b};\text{z} \right)$ is: $-\frac{1}{\text{z}}\left(1-{{\text{e}}^{\text{z}}} \right)$ [31, (Eq. § 07.20.03.0026.01)]. Hence For N = 1:

$${{F}_{{{L}_{f}}}}\left(d \right)=2{{\left({{2}^{1}}{{\left(\frac{4\pi d}{\lambda} \right)}^{2}} \right)}^{-\frac{1}{2}}} \left(\frac{A}{{{\gamma}_{req}}} \right)\left(1-{{e}^{-z}} \right)\left| z=2{{\left(\frac{4\pi d}{\lambda} \right)}^{2}})^{-1}{{\left(\frac{A}{{{\gamma}_{eq}}} \right)}^{2}} , \right. \quad {{P}_{outage}}=1-{{F}_{{{L}_{f}}}}\left(d \right)$$

(54)

For N = 2,

$$f_{{L_{f} }} \left( d \right) = 2\left( {2^{2} \left( {\frac{4\pi d}{\lambda }} \right)^{2} } \right)^{{ - \frac{1}{2}}} G_{0,\;\;2}^{2,\;\;0} \left( {\left( {2^{2} \left( {\frac{4\pi d}{\lambda }} \right)^{2} } \right)^{ - 1} \left( {\frac{A}{{\gamma_{eq} }}} \right)^{2} \left| {\begin{array}{*{20}c} - \\ {\frac{1}{2},\frac{1}{2}} \\ \end{array} } \right.} \right)$$

(55)

From [31, (Eq. § 07.34.03.0605.01)]:

$$\begin{aligned} G\begin{array}{*{20}c} {2,0} \\ {0,2} \\ \end{array} \left( {z\left| {\begin{array}{*{20}c} - \\ {\frac{1}{2},\frac{1}{2}} \\ \end{array} } \right.} \right) & = 2z^{{\frac{1}{2}\left( {b + c} \right)}} K_{b - c} \left( {2\sqrt z } \right)\left| {b,c = \frac{1}{2}} \right. \\ & = 2\sqrt z K_{0} \left( {2\sqrt z } \right)\left| {z = \left( {2^{2} \left( {\frac{4\pi d}{\uplambda}} \right)^{2} } \right)^{ - 1} \left( {\frac{A}{{\gamma_{eq} }}} \right)^{2} } \right. \\ \end{aligned}$$

(56)

In Eq. (56), ${{\text{K}}_{\text{v}}}\left(. \right)$ is the modified Bessel function of the second kind.

Replacing G-function in Eq. (55) with Eq. (56) and integrating the result, we obtain the following equation.

$${{F}_{{{L}_{f}}}}\left(d \right)=\left. -2{{\left({{2}^{2}}{{\left(\frac{4\pi d}{\lambda} \right)}^{2}} \right)}^{-\frac{1}{2}}} \left(\frac{A}{{{\gamma}_{req}}} \right)2\sqrt{z}{{K}_{1}}\left(2\sqrt{z} \right) \right|z={{\left({{2}^{2}}{{\left(\frac{4\pi d}{\lambda} \right)}^{2}} \right)}^{-1}}{{\left(\frac{A}{{{\gamma}_{eq}}} \right)}^{2}} ,\quad {{P}_{outage}}=1-{{F}_{{{L}_{f}}}}\left(d \right)$$

(57)

Applying N = 3, 4, and 5 to Eq. (37), will be achieved Eqs. (58), (59), and (60), respectively:

For N = 3:

$$\begin{aligned} F_{{L_{f} }} \left( d \right) & = \frac{z}{2}\mathop \sum \limits_{k = 0}^{\infty } \frac{{\left( { - 1} \right)^{k} }}{{\left( k \right)!^{3} \left( {k + 1} \right)}} \times \left. {\left[ {ln^{2} \left( z \right) - 2ln\left( z \right)C_{3} \left( k \right) + C_{3}^{\left( 1 \right)} \left( k \right) + \left[ {C_{3} \left( k \right)} \right]^{2} } \right]z^{k} } \right|z \\ & = \left( {2^{3} \left( {\frac{4\pi d}{\lambda }} \right)^{2} } \right)^{ - 1} \left( {\frac{A}{{\gamma_{eq} }}} \right)^{2} \\ \end{aligned}$$

(58)

For N = 4:

$$\begin{aligned} {\text{F}}_{{{\text{L}}_{\text{f}} }} \left( {\text{d}} \right) & = \frac{{\text{z}}}{6}\mathop \sum \limits_{{{\text{k}} = 0}}^{\infty } \frac{1}{{\left( {\text{k}} \right)!^{4} \left( {{\text{k}} + 1} \right)}} \\ &\quad \times \left[ { - { \ln }^{3} \left( {\text{z}} \right) + 3\ln^{2} \left( {\text{z}} \right){\text{C}}_{4} \left( {\text{k}} \right) - 3\ln \left( {\text{z}} \right)\left[ {{\text{C}}_{4}^{\left( 1 \right)} \left( {\text{k}} \right) + \left[ {{\text{C}}_{4} \left( {\text{k}} \right)} \right]^{2} } \right] + {\text{C}}_{4}^{\left( 2 \right)} \left( {\text{k}} \right) + 3{\text{C}}_{4}^{\left( 1 \right)} \left( {\text{k}} \right){\text{C}}_{4} \left( {\text{k}} \right) + \left[ {{\text{C}}_{4} \left( {\text{k}} \right)} \right]^{3} } \right]\left. {{\text{z}}^{\text{k}} } \right|{\text{z}} \\ & = \left( {2^{4} \left( {\frac{{4{{\uppi d}}}}{\lambda }} \right)^{2} } \right)^{ - 1} \left( {\frac{{\text{A}}}{{{{\upgamma }}_{\text{eq}} }}} \right)^{2} \\ \end{aligned}$$

(59)

For N = 5:

$$\begin{aligned} {\text{F}}_{{{\text{L}}_{\text{f}} }} \left( {\text{d}} \right) & = \frac{{\text{z}}}{24}\mathop \sum \limits_{{{\text{k}} = 0}}^{\infty } \frac{{\left( { - 1} \right)^{\text{k}} }}{{\left( {\text{k}} \right)!^{5} \left( {{\text{k}} + 1} \right)}} \\ & \quad \times \left[ {{ \ln }^{4} \left( {\text{z}} \right) - 4\ln^{3} \left( {\text{z}} \right){\text{C}}_{5} \left( {\text{k}} \right) + 6\ln^{2} \left( {\text{z}} \right)\left[ {{\text{C}}_{5}^{\left( 1 \right)} \left( {\text{k}} \right) + \left[ {{\text{C}}_{5} \left( {\text{k}} \right)} \right]^{2} } \right] - 4\ln \left( {\text{z}} \right)\left[ {{\text{C}}_{5}^{\left( 2 \right)} \left( {\text{k}} \right) + 3{\text{C}}_{5}^{\left( 1 \right)} \left( {\text{k}} \right){\text{C}}_{5} \left( {\text{k}} \right) + \left[ {{\text{C}}_{5} \left( {\text{k}} \right)} \right]^{3} } \right] + {\text{C}}_{5}^{\left( 3 \right)} \left( {\text{k}} \right) + {\text{C}}_{5}^{\left( 2 \right)} \left( {\text{k}} \right){\text{C}}_{5} \left( {\text{k}} \right) + 3\left[ {{\text{C}}_{5}^{\left( 1 \right)} \left( {\text{k}} \right)} \right]^{2} + 6\left[ {{\text{C}}_{5} \left( {\text{k}} \right)} \right]^{2} {\text{C}}_{5}^{\left( 1 \right)} \left( {\text{k}} \right) + \left[ {{\text{C}}_{5} \left( {\text{k}} \right)} \right]^{4} } \right]{\text{z}}^{\text{k}} \left| {\text{z}} \right. \\ = \left( {2^{5} \left( {\frac{{4{{\uppi d}}}}{\lambda }} \right)^{2} } \right)^{ - 1} \left( {\frac{{\text{A}}}{{{{\upgamma }}_{\text{eq}} }}} \right)^{2} \\ \end{aligned}$$

(60)

where ${\text{C}}_{\text{N}}^{{\left( {\text{l}} \right)}} \left( {\text{k}} \right) = {\text{B}}_{\text{N}}^{{\left( {\text{l}} \right)}} \left( {\text{k}} \right) + \frac{{{\text{l}}!}}{{\left( {{\text{k}} + 1} \right)^{{{\text{l}} + 1}} }}$. In addition, ${\text{B}}_{\text{N}}^{{\left( {\text{l}} \right)}} \left( {\text{k}} \right)$ is:

$${\text{B}}_{\text{N}}^{{\left( {\text{l}} \right)}} \left( {\text{k}} \right) = {\text{N}}\left[ {\uppsi^{{\left( {\text{l}} \right)}} \left( 1 \right) + \left( { - 1} \right)^{{{\text{l}} + 1}} \left[ {\uppsi^{{\left( {\text{l}} \right)}} \left( 1 \right) -\uppsi^{{\left( {\text{l}} \right)}} \left( {{\text{k}} + 1} \right)} \right]} \right]\quad {\text{for}}\quad {\text{N}} \ge 2$$

(61)

In Eq. (38), $\uppsi^{{\left( {\text{p}} \right)}} \left( {\text{x}} \right) = \left( {\frac{{{\text{d}}^{\text{p}} }}{{{\text{x}}^{\text{p}} }}} \right)\uppsi\left( {\text{x}} \right) = \left( {\frac{{{\text{d}}^{{{\text{p}} + 1}} }}{{{\text{x}}^{{{\text{p}} + 1}} }}} \right)\ln \left( {\text{x}} \right)$ is the pth polygamma function.

Appendix 4

In fact, Eq. (7) is a mathematical equation defined to simplify the capacity formula in Eq. (4). In this respect, the amount of $\psi$ defined by comparing Eqs. (4) and (6)

$$\psi =H_{RD}^{\dagger}{R}_{n}^{-{1}}{{H}_{RD}}$$

(62)

By utilizing the singular value decomposition (SVD) of ${\text{H}}_{\text{RD}}$

$${{H}_{RD}}={{U}_{RD}}\Lambda_{RD}^{1/2}V_{RD}^{\dagger}$$

(63)

where $\Uplambda_{\text{RD}}^{1/2}=\text{diag}\left\{\sqrt{\lambda_{1}},\ldots,\sqrt{\lambda_{ \hbox{min} \left({{\text{N}}_{\text{D}}},{{\text{N}}_{\text{R}}} \right)}} \right\}$ is a ${{\text{N}}_{\text{D}}}\times {{\text{N}}_{\text{R}}}$ diagonal matrix of singular values $({{\sigma}_{\text{i}}})$ of ${{\text{H}}_{\text{RD}}}$. ${{\text{U}}_{\text{RD}}}\in {{\text{C}}^{{{\text{N}}_{\text{D}}}\times {{\text{N}}_{\text{D}}}}}$ and ${{\text{V}}_{\text{RD}}}\in {{\text{C}}^{{{\text{N}}_{\text{R}}}\times {{\text{N}}_{\text{R}}}}}$ are unitary matrices containing the respective eigenvectors. As defined in manuscript for the ${{\text{R}}_{\text{n}}}$

$$R_{n} = I_{N_{D} } + aH_{RD} H_{RD}^{\dag } = I_{{N_{D} }} + a U_{RD} \varLambda_{RD} V_{RD}^{\dag } = \left[ \begin{array}{*{20}c} {1 + a \lambda_{1} } & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & {1 + a \lambda_{{N_{R} }} } \\ \end{array} \right].$$

(64)

When $\hbox{min} \left({{\text{N}}_{\text{D}}}, {{\text{N}}_{\text{R}}} \right)={{\text{N}}_{\text{R}}}\,\,{\text{or}}\,\, {{\text{N}}_{\text{R}}}\le {{\text{N}}_{\text{D}}}$ and

$$R_{n} = \left[ {\begin{array}{*{20}c} {1 + a\lambda_{1} } & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & {1 + a\lambda_{{N_{D} }} } \\ \end{array} } \right] .$$

(65)

When $\hbox{min} \left({{\text{N}}_{\text{D}}},{{\text{N}}_{\text{R}}} \right)={{\text{N}}_{\text{D}}}\,\,{\text{or}} \,\,{{\text{N}}_{\text{R}}} > {{\text{N}}_{\text{D}}}$, finally ${\text{R}}_{\text{n}}^{ - 1}$ according to the theorem

diag(a₁,…, a_n)⁻¹ = diag(a⁻¹₁,…, a⁻¹_n) is equal to

$$R_{n}^{ - 1} = \left[ {\begin{array}{*{20}c} {\frac{1}{{ 1 +{\text{a}}\lambda_{ 1} }}} & {\quad \cdots } & {\quad 0} \\ \vdots & \quad \ddots & {\quad \vdots } \\ 0 & {\quad \cdots } & {\quad \frac{1}{{ 1 +{\text{a}}\lambda_{{{\text{N}}_{\text{R}} }} }}} \\ \end{array} } \right] = \begin{array}{*{20}c} {{\text{diag}}\left\{ {\frac{ 1}{{ 1 +{\text{a}}\lambda_{ 1} }} , \ldots ,\frac{ 1}{{ 1 +{\text{a}}\lambda_{{{\text{N}}_{\text{R}} }} }}} \right\}} \\ {{\text{N}}_{\text{R}} \le {\text{N}}_{\text{D}} } \\ \end{array}$$

(66)

or

$$R_{n}^{ - 1} = \left[ {\begin{array}{*{20}c} {\frac{1}{{ 1 + {\text{a}}\lambda_{ 1} }}} & {\quad \cdots } & {\quad 0} \\ \vdots & {\quad \ddots } & {\quad \vdots } \\ 0 & {\quad \cdots } & {\quad \frac{1}{{ 1 +{\text{a}}\lambda_{{{\text{N}}_{\text{D}} }} }}} \\ \end{array} } \right] = \begin{array}{*{20}c} {{\text{diag}}\left\{ {\frac{ 1}{{ 1 +{\text{a}}\lambda_{ 1} }} , \ldots ,\frac{ 1}{{ 1 +{\text{a}}\lambda_{{{\text{N}}_{\text{D}} }} }}} \right\}} \\ {{\text{N}}_{\text{R}} {\text{ > N}}_{\text{D}} } \\ \end{array}$$

(67)

Finally

$$\Psi = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{\text{diag}}\left\{ {\frac{{\lambda_{ 1} }}{{ 1 +{\text{a}}\lambda_{ 1} }} ,\ldots ,\frac{{\lambda_{{{\text{N}}_{\text{R}} }} }}{{ 1 +{\text{a}}\lambda_{{{\text{N}}_{\text{R}} }} }}} \right\}} \\ {{\text{N}}_{\text{R}} \le {\text{N}}_{\text{D}} } \\ \end{array} } \\ {\begin{array}{*{20}c} {{\text{diag}}\left\{ {\frac{{\lambda_{ 1} }}{{ 1 +{\text{a}}\lambda_{ 1} }} ,\ldots ,\frac{{\lambda_{{{\text{N}}_{\text{D}} }} }}{{ 1 +{\text{a}}\lambda_{{{\text{N}}_{\text{D}} }} }},\overbrace {0, \ldots ,0}^{{{\text{N}}_{\text{R}} - {\text{N}}_{\text{D}} }}} \right\}} \\ {{\text{N}}_{\text{R}} {\text{ > N}}_{\text{D}} } \\ \end{array} } \\ \end{array} } \right.$$

(68)

In condition ${\text{N}}_{\text{R}} {\text{ > N}}_{\text{D}}$, ${{\text{N}}_{\text{R}}}-{{\text{N}}_{\text{D}}}$ zero added to the diagonal matrix to be the same rank with ${{I}_{{{N}_{R}}}}$. For MIMO (2 × 2) system and ${{\text{N}}_{\text{R}}}\le {{\text{N}}_{\text{D}}}$, the ψ value is simplified as follows

$$\Psi = \left[ {\begin{array}{*{20}c} {\frac{{\lambda_{1} }}{{1 + a\lambda_{1} }}} & {\quad 0} \\ 0 & {\quad \frac{{\lambda_{2} }}{{1 + a\lambda_{2} }}} \\ \end{array} } \right]$$

(69)