On the diversity gain region of the dynamic decode-and-forward relay-assisted Z-channel

Abstract

In an interference-limited system, the interference forwarding by a relay enhances the interference level and thereby enables the cancellation of the interference. In this work, interference forwarding by a half-duplex dynamic decode-and-forward (HD DDF) relay in a two-user Z-channel is considered. In the two-user Z-channel, one user is interference-limited while the other user is interference-free. The diversity gain region (DGR), which characterizes the tradeoff between the achievable diversity orders between the two users, is an appropriate performance metric for the Z-channel. Closed-form expression for the achievable DGR with the interference forwarding by the HD DDF relay is presented. The multiplexing gain regions (MGRs) where the HD DDF protocol achieves better DGR over the direct transmission scheme, full-duplex decode-and-forward (FD DF) and FD partial DF relay assisted Z- channel are identified. The HD DDF protocol is shown to achieve better DGR than the FD DF and FD PDF relay for a large range of MGR. The achievable DGRs for the HD DDF, FD DF, and FD PDF relay-assisted Z-channel and direct transmission scheme are presented for various interference levels and multiplexing gain pairs.

Appendix: Proof of Theorem 1

Let $v_{ij}$, for $\left( i,j\right) \in \left\{ 1,2,3\right\}$, be the exponential order of the channel gain $h_{ij}$ between the terminals i and j. It is defined as [25]

$$\begin{aligned} v_{ij}\, {\mathop{=} ^{{\varDelta }}}\, - \lim _{\gamma \rightarrow \infty } \frac{\log \left( |h_{ij}|^2\right) }{\log \left( \gamma \right) }. \end{aligned}$$

(32)

The probability density function (pdf) of $v_{ij}$ is [14]

$$\begin{aligned} f_{v_{ij}}(v) = {\left\{ \begin{array}{ll} \gamma ^{- v} \quad \text {for} \quad v \ge 0 \\ 0 \quad \text {for} \quad v < 0. \end{array}\right. } \end{aligned}$$

First, few notations are introduced. Later, using these notations, the achievable DGR with the HD DDF protocol is presented. Let ${\mathbf{r}} = [r_1 \quad r_2]$ and $r_s = r_1+r_2$. Let ${\mathcal{G}}_i({\mathbf{r}},\beta _{21},f)$, for $i \in \left\{ 1,2,3\right\}$, and ${\mathcal{G}}_4({\mathbf{r}},f)$ be defined as

$$\begin{aligned} {\mathcal{G}}_1({\mathbf{r}},\beta _{21},f)&= \left[ \frac{1-r_s-f\left( 1-\beta _{21}\right) }{1-f}\right] ^++\left[ 1-\frac{r_2}{f}\right] ^+, \end{aligned}$$

(33)

$$\begin{aligned} {\mathcal{G}}_2({\mathbf{r}},\beta _{21},f)&= \left[ \frac{1-r_s}{f}\right] ^++\left[ \frac{1-r_s}{f}+\beta _{21}-1\right] ^+ +\left[ 1-\frac{r_2}{f}\right] ^+, \end{aligned}$$

(34)

$$\begin{aligned} {\mathcal{G}}_3({\mathbf{r}},\beta _{21},f)&= \left[ 1-\beta _{21}+\frac{[1-r_s]^+-f(1-\beta _{21})}{f}\right] ^++\left[ \frac{[1-r_s]^+-f(1-\beta _{21})}{f}\right] ^+ \nonumber \\&\quad+ \left[ 1-\frac{r_2}{f}\right] ^+, \end{aligned}$$

(35)

$$\begin{aligned} {\mathcal{G}}_4(r_2,f)&= 1+\left[ \frac{1-r_2-f}{1-f}\right] ^+ + \left[ 1-\frac{r_2}{f}\right] ^+. \end{aligned}$$

(36)

1.1 Calculation of $d^{DDF_{hd}}_{FUE}$:

The FUE is in outage if any one of the inequalities in (6) and (7) is not satisfied. Let ${\mathcal{O}}^{DDF_{hd}}_{FUE,1}$ and ${\mathcal{O}}^{DDF_{hd}}_{FUE,2}$ be the outage events related to the inequalities (6) and (7), respectively. Let $Pr\left( {\mathcal{O}}^{DDF_{hd}}_{FUE,i}\right)$, for $i=\left\{ 1,2\right\}$, be the probability of the outage corresponding to the event ${\mathcal{O}}^{DDF_{hd}}_{FUE,i}$. Using the union bound, $Pr\left( {\mathcal{O}}^{DDF_{hd}}_{FUE}\right)$ can be evaluated as

$$\begin{aligned} Pr\left( {\mathcal{O}}^{DF_{hd}}_{FUE}\right) &\le Pr\left( {\mathcal{O}}^{DDF_{hd}}_{FUE,1}\right) +Pr\left( {\mathcal{O}}^{DDF_{hd}}_{FUE,2}\right) , \end{aligned}$$

(37)

$$\begin{aligned} \text {where }{\mathcal{O}}^{DDF_{hd}}_{FUE,1}&= \left\{ {\mathbf{h}}:\log \left( 1+\eta ^2_{11}|h_{11}|^2P_1\right) \le R_1\right\} , \end{aligned}$$

(38)

$$\begin{aligned} {\mathcal{O}}^{DDF_{hd}}_{FUE,2}&= \left\{ {\mathbf{h}}:f \log \left( 1+\eta ^2_{11}|h_{11}|^2P_1+\eta ^2_{21}|h_{21}|^2P_2\right) \right. \nonumber \\&\left. \quad +(1-f) \log \left( 1+\eta ^2_{11}|h_{11}|^2P_1+\eta ^2_{21}|h_{21}|^2P_2 + \eta ^2_{31}|h_{31}|^2P_3 \right) \right. \nonumber \\&\left. \le R_1+R_2 \right\} . \end{aligned}$$

(39)

Here, ${\mathbf{h}} = [h_{11}\;h_{21}\;h_{31}]$. Let

$$\begin{aligned} Pr\left( {\mathcal{O}}^{DDF_{hd}}_{FUE,i}\right) \stackrel{.}{=} \gamma ^{-d^{DDF_{hd}}_{FUE,i}}, \text { for } i=\left\{ 1,2\right\} . \end{aligned}$$

(40)

Using (17), (37) and (40), the achievable DMT at the FUE with HD DDF protocol can be obtained as shown in (12). The calculation of $d^{DDF_{hd}}_{FUE,i}$, for $i=\left\{ 1,2\right\}$ is presented below.

1.1.1 Evaluation of $d^{DDF_{hd}}_{FUE,1}$

The high SNR approximation of ${\mathcal{O}}^{DDF_{hd}}_{FUE,1}$ can be written as

$$\begin{aligned} {\mathcal{O}}^{DDF_{hd}}_{FUE,1} = \left\{ v_{11}:[1-v_{11}]^+ \le r_1\right\} . \end{aligned}$$

(41)

As shown in [26], the $Pr\left( {\mathcal{O}}^{DDF_{hd}}_{FUE,1}\right)$ at high SNR can be obtained as

$$\begin{aligned} Pr\left( {\mathcal{O}}^{DDF_{hd}}_{FUE,1}\right) \stackrel{.}{=} \gamma ^{-[1-r_1]^+}. \end{aligned}$$

(42)

From (40) and (42), the $d^{DDF_{hd}}_{FUE,1}$ can be obtained as shown in (14).

1.1.2 Evaluation of $d^{DDF_{hd}}_{FUE,2}$

The $Pr\left( {\mathcal{O}}^{DDF_{hd}}_{FUE,2}\right)$ can be evaluated as

$$\begin{aligned} Pr\left( {\mathcal{O}}^{DDF_{hd}}_{FUE,2}\right)&= Pr\left( {\mathcal{O}}^{DDF_{hd}}_{FUE,2}|f=1\right) Pr(f=1) \nonumber \\&\quad+ Pr\left( {\mathcal{O}}^{DDF_{hd}}_{FUE,2}|f<1\right) Pr(f<1). \end{aligned}$$

(43)

Here, $f < 1$ means, the relay can decode and retransmit the signal, and the FUE can make use of the retransmitted signal. $f = 1$ means, the relay fails to decode the data from the MNB. In the HD DDF protocol, if the relay fails to decode the data, the FUE is declared to be in outage. Hence,

$$\begin{aligned} Pr\left( {\mathcal{O}}^{DDF_{hd}}_{FUE,2}|f=1\right) = 1. \end{aligned}$$

(44)

The exponential orders of $Pr(f =1 )$, $Pr\left( {\mathcal{O}}^{DDF_{hd}}_{FUE,2}|f < 1\right)$ and $Pr(f <1)$ are defined as

$$\begin{aligned}Pr(f =1 )& \stackrel{.}{=} \gamma ^{-d^{DDF_{hd}}_{Relay}}, \end{aligned}$$

(45)

$$\begin{aligned} Pr\left( {\mathcal{O}}^{DDF_{hd}}_{FUE,2}|f < 1\right) &\stackrel{.}{=} \gamma ^{-\tilde{d}^{DDF_{hd}}_{FUE,2}} , \end{aligned}$$

(46)

$$\begin{aligned} Pr(f <1 ) &\stackrel{.}{=} \gamma ^{-\tilde{d}^{DDF_{hd}}_{Relay}}. \end{aligned}$$

(47)

Using (40), (43), (44), (45), (46) and (47), $d^{DDF_{hd}}_{FUE,2}$ can be obtained as shown in (15). Next, $d^{DDF_{hd}}_{Relay}$, $\tilde{d}^{DDF_{hd}}_{Relay}$ and $\tilde{d}^{DDF_{hd}}_{FUE,2}$ are evaluated.

From (5), the $Pr(f =1 )$ can be evluated as

$$\begin{aligned} Pr(f=1) &= Pr\left( \log \left( 1+\eta ^2_{23}|h_{23}|^2P_2\right) < R_2\right) \stackrel{.}{=} \gamma ^{-[1-r_2]^+}. \end{aligned}$$

(48)

From (45) and (48), the $d^{DDF_{hd}}_{Relay}$ can be evaluated as in (16). $Pr(f<1)$ can be evaluated as

$$\begin{aligned} Pr(f<1) = 1-Pr(f=1). \end{aligned}$$

(49)

Using (48) and (49), it can be shown that $\tilde{d}^{DDF_{hd}}_{Relay} = 0$.

Next, The evaluation of $\tilde{d}^{DDF_{hd}}_{FUE,2}$ is presented.

For $f < 1$, the high SNR approximation of (39) can be written as

$$\begin{aligned} {\mathcal{O}}^{DDF_{hd}}_{FUE,2}&= \left\{ {\mathbf{v}}:f \max \left\{ [1-v_{11}]^+,[\beta _{21}-v_{21}]^+\right\} \right. \\&\quad\left. +\,(1-f) \max \left\{ [1-v_{11}]^+,[\beta _{21}-v_{21}]^+,[1-v_{31}]^+ \right\} \le r_s \right\} \end{aligned}$$

Using the Laplace integration technique [25], $\tilde{d}^{DDF_{hd}}_{FUE,2}$ can be obtained as

$$\begin{aligned} \tilde{d}^{DDF_{hd}}_{FUE,2}&= \min \left\{ v_{11}+v_{21}+ v_{31}+ v_{23}\right\} \nonumber \\&\text {subject to: } v_{ij} > 0 \; \; \forall \; \;\left\{ i,j\right\} \in \left\{ 1,2,3\right\} ,\nonumber \\&f \max \left\{ [1-v_{11}]^+,[\beta _{21}-v_{21}]^+\right\} \nonumber \\&\qquad + (1-f) \max \left\{ [1-v_{11}]^+,[\beta _{21}-v_{21}]^+,[1-v_{31}]^+\right\} \le r_s, \nonumber \\&f = \frac{ r_2}{\left[ 1-v_{23}\right] ^+}. \end{aligned}$$

(50)

In the above problem, the equality constraint is obtained from the high-SNR parametrization of (5).

As the constraints related to $v_{23}$ and $[v_{11}\; v_{21}\; v_{31}]$ are not coupled, the above optimization problem can be equivalently written as [27]

$$\begin{aligned} \tilde{d}^{DDF_{hd}}_{FUE,2}&= \min _{f} \left\{ \min _{\left( v_{11},v_{21},v_{31}\right) } \left\{ v_{11}+v_{21}+ v_{31}\right\} + \min _{v_{23}}\left\{ v_{23}\right\} \right\} \nonumber \\&\text {subject to: } v_{ij} > 0 \; \; \forall \; \;\left\{ i,j\right\} \in \left\{ 1,2,3\right\} , \nonumber \\&f \max \left\{ [1-v_{11}]^+,[\beta _{21}-v_{21}]^+\right\} \nonumber \\&\qquad + (1-f) \max \left\{ [1-v_{11}]^+,[\beta _{21}-v_{21}]^+,[1-v_{31}]^+\right\} \le r_s, \nonumber \\&f = \frac{ r_2}{\left[ 1-v_{23}\right] ^+}. \end{aligned}$$

(51)

The above problem can be equivalently written as

$$\begin{aligned} \tilde{d}^{DDF_{hd}}_{FUE,2}&= \min _{f} \left\{ d_{inner}(f)+ \left[ 1-\frac{r_2}{f}\right] ^+\right\} \nonumber \\&\text {subject to: }r_2< f < 1, \nonumber \\&\text {where }d_{inner}(f) = \min _{\left( v_{11},v_{21},v_{31}\right) } \,\left\{ v_{11}+v_{21}+ v_{31}\right\} \nonumber \\&\text { subject to } v_{ij} > 0 \; \; \forall \; \;\left\{ i,j\right\} \in \left\{ 1,2,3\right\} ,\nonumber \\&f \max \left\{ [1-v_{11}]^+,[\beta _{21}-v_{21}]^+\right\} \nonumber \\&\qquad \qquad \qquad + (1-f) \max \left\{ [1-v_{11}]^+,[\beta _{21}-v_{21}]^+,[1-v_{31}]^+\right\} \le r_s. \end{aligned}$$

(52)

The inner optimization problem, $d_{inner}(f)$, can be solved by dividing the constraint region into several subregions as follows.

• Case - I: $[1-v_{11}]^+ > [\beta _{21}-v_{21}]^+$ and $[1-v_{11}]^+ > [1-v_{31}]^+$.

• Case - II: $[\beta _{21}-v_{21}]^+ > [1-v_{11}]^+$ and $[\beta _{21}-v_{21}]^+ > [1-v_{31}]^+$.

• Case - III: $[1-v_{31}]^+ > [1-v_{11}]^+$, $[1-v_{31}]^+ > [\beta _{21}-v_{21}]^+$, and $[1-v_{11}]^+ > [\beta _{21}-v_{21}]^+$.

• Case - IV: $[1-v_{31}]^+ > [1-v_{11}]^+$, $[1-v_{31}]^+ > [\beta _{21}-v_{21}]^+$, and $[1-v_{11}]^+ < [\beta _{21}-v_{21}]^+$.

First, $d_{inner}(f)$ needs to be evaluated for each case. Later, the outer optimization, i.e., the minimization over f, needs to be done for each case. Let $d^{FUE}_{I}$, $d^{FUE}_{II}$, $d^{FUE}_{III}$, and $d^{FUE}_{IV}$ be the optimal values corresponding to Case I, Case II, Case III, and Case IV, respectively. As the calculation of these parameters involve the standard linear programming techniques, the evaluation is not shown here. The $\tilde{d}^{DDF_{hd}}_{FUE,2}$ can be obtained by taking the minimum of all these values as

$$\begin{aligned} \tilde{d}^{DDF_{hd}}_{FUE,2}&= \min \left\{ d^{FUE}_{I}, d^{FUE}_{II}, d^{FUE}_{III}, d^{FUE}_{IV}\right\} . \end{aligned}$$

(53)

where

$$\begin{aligned} d^{FUE}_{I}&= 2\left[ 1-r_s\right] ^+ + \left[ \beta _{21}-1+[1-r_s]^+\right] ^+ \text { if } r_2 \le 1, \end{aligned}$$

(54)

$$\begin{aligned} d^{FUE}_{II}&= \left[ 1-r_s+[r_s-\beta _{21}]^+\right] ^+ + [\beta _{21}-r_s]^++\left[ 1-\beta _{21}+[\beta _{21}-r_s]^+\right] ^+ \nonumber \\&\text { if } r_2 \le 1, \end{aligned}$$

(55)

$$\begin{aligned} d^{FUE}_{III}&= \min \left\{ d^{FUE}_{III,1}, d^{FUE}_{III,2}, d^{FUE}_{III,3}, d^{FUE}_{III,4}\right\} , \end{aligned}$$

(56)

$$\begin{aligned} d^{FUE}_{III,1}= {\left\{ \begin{array}{ll}& \left[ 1-r_s\right] ^+ + \left[ 1-r_2\right] ^+ \text { if } r_1+2r_2< 1, \\ &\quad \beta _{21}\le r_s\le 1,\max \left\{ \frac{1}{2},(1-r_s),\left( \frac{1-r_s}{1-\beta _{21}}\right) ,r_2\right\}< 1 \\ &\left[ \frac{1-r_s}{\max \left\{ \frac{1}{2},(1-r_s),\left( \frac{1-r_s}{1-\beta _{21}}\right) ,r_2\right\} }\right] ^+ + \left[ 1-\frac{r_2}{\max \left\{ \frac{1}{2},(1-r_s),\left( \frac{1-r_s}{1-\beta _{21}}\right) ,r_2\right\} }\right] ^+ \\ & \quad \text { if }\quad r_1 + 2r_2 \ge 1, \beta _{21}\le r_s\le 1,\max \left\{ \frac{1}{2},(1-r_s),\left( \frac{1-r_s}{1-\beta _{21}}\right) ,r_2\right\} < 1, \\ \end{array}\right. } \end{aligned}$$

(57)

$$\begin{aligned} d^{FUE}_{III,2}&= \left[ 1-\beta _{21}\right] ^++\min \left\{ {\mathcal{G}}_1\left( {\mathbf{r}},\beta _{21}, \max \left\{ \frac{1}{2},(1-r_s),r_2\right\} \right) , \right. \nonumber \\&\left. {\mathcal{G}}_1\left( {\mathbf{r}},\beta _{21}, \min \left\{ \frac{2}{3},\frac{1-r_s}{1-\beta _{21}}\right\} \right) \right\} \nonumber \\&\text { if }\,\,\beta _{21}\le r_s\le 1, \max \left\{ \frac{1}{2},(1-r_s),r_2\right\} \le \min \left\{ \frac{2}{3},\frac{1-r_s}{1-\beta _{21}}\right\} . \end{aligned}$$

(58)

$$\begin{aligned} d^{FUE}_{III,3}&= \min \left\{ {\mathcal{G}}_2\left( {\mathbf{r}},\beta _{21},\min \left\{ 1,\frac{1-r_s}{1-\beta _{21}}\right\} \right) ,\right. \nonumber \\&\quad \left. {\mathcal{G}}_2\left( {\mathbf{r}},\beta _{21},\max \left\{ \frac{2}{3},(1-r_s),r_2\right\} \right) \right\} \nonumber \\&\quad \text { if } r_s\le 1, \max \left\{ \frac{2}{3},(1-r_s),r_2\right\} \le \min \left\{ 1,\frac{1-r_s}{1-\beta _{21}}\right\} . \end{aligned}$$

(59)

$$\begin{aligned} d^{FUE}_{III,4}&= [1-\beta _{21}]^++\min \left\{ {\mathcal{G}}_1\left( {\mathbf{r}},\beta _{21}, \max \left\{ \frac{1}{2},r_2\right\} \right) , \right. \nonumber \\&\quad \left. {\mathcal{G}}_1\left( {\mathbf{r}},\beta _{21}, \min \left\{ \frac{2}{3},[1-r_s]^+\right\} \right) \right\} \nonumber \\&\quad \text { if }\,\,\beta _{21}\le r_s\le 1, \max \left\{ \frac{1}{2},r_2\right\} \le \min \left\{ \frac{2}{3},[1-r_s]^+\right\} , \end{aligned}$$

(60)

$$\begin{aligned} d^{FUE}_{IV}&= \min \left\{ d^{FUE}_{IV,1},d^{FUE}_{IV,2}, d^{FUE}_{IV,3}\right\} \end{aligned}$$

(61)

$$\begin{aligned} d^{FUE}_{IV,1}&= [1-\beta _{21}]^++\min \left\{ {\mathcal{G}}_1\left( {\mathbf{r}},\beta _{21},r_2\right) , \right. \nonumber \\&\quad \left. {\mathcal{G}}_1\left( {\mathbf{r}},\beta _{21},\min \left\{ \frac{1}{2},\frac{1-r_s}{1-\beta _{21}}\right\} \right) \right\} \nonumber \\&\quad \text { if }\,\,\beta _{21}\le r_s\le 1, r_2 \le \min \left\{ \frac{1}{2},\frac{1-r_s}{1-\beta _{21}}\right\} , \end{aligned}$$

(62)

$$\begin{aligned}&d^{FUE}_{IV,2} = [1-\beta _{21}]^+ +\min \left\{ {\mathcal{G}}_1\left( {\mathbf{r}},\beta _{21},\max \left\{ r_2,\frac{1}{2}\right\} \right) , \right. \nonumber \\&quad \left. {\mathcal{G}}_1\left( {\mathbf{r}},\beta _{21},\min \left\{ \frac{2}{3},\frac{(1-r_s)}{(1-\beta _{21})}\right\} \right) \right\} \nonumber \\&\quad \text { if }\,\,\beta _{21}\le r_s\le 1, \max \left\{ r_2,\frac{1}{2}\right\} \le \min \left\{ \frac{2}{3},\frac{(1-r_s)}{(1-\beta _{21})}\right\} , \end{aligned}$$

(63)

$$\begin{aligned} d^{FUE}_{IV,3}&= \min \left\{ {\mathcal{G}}_3\left( {\mathbf{r}},\beta _{21},\max \left\{ \frac{2}{3},[1-r_s]^+,r_2\right\} \right) ,\right. \nonumber \\&\quad \left. {\mathcal{G}}_3\left( {\mathbf{r}},\beta _{21},\frac{(1-r_s)}{(1-\beta _{21})}\right) \right\} \nonumber \\&\quad \text { if }\,\,\beta _{21}\le r_s\le 1, \max \left\{ \frac{2}{3},[1-r_s]^+,r_2\right\} \le \frac{(1-r_s)}{(1-\beta _{21})}, \end{aligned}$$

(64)

1.2 Achievable DMT at the MUE with the HD DDF protocol

From (8), the ${\mathcal{O}}^{DDF_{hd}}_{MUE}$ can be defined as

$$\begin{aligned} {\mathcal{O}}^{DDF_{hd}}_{MUE} &= \left\{ {\mathbf{h}}:f \log \left( 1+\eta ^2_{22}|h_{22}|^2P_2\right) + (1-f) \log \left( 1+\eta ^2_{22}|h_{22}|^2P_2 + \eta ^2_{32}|h_{32}|^2P_3\right) \le R_2 \right\} \end{aligned}$$

The $Pr\left( {\mathcal{O}}^{DDF_{hd}}_{MUE}\right)$ can be written as

$$\begin{aligned} Pr\left( {\mathcal{O}}^{DDF_{hd}}_{MUE}\right) &= Pr\left( {\mathcal{O}}^{DDF_{hd}}_{MUE}|f=1\right) Pr(f=1) + Pr\left( {\mathcal{O}}^{DDF_{hd}}_{MUE}|f<1\right) Pr(f<1) \end{aligned}$$

(65)

If the relay cannot decode the data within the frame duration, outage is declared at the MUE . Hence,

$$\begin{aligned} Pr\left( {\mathcal{O}}^{DDF_{hd}}_{MUE}|f=1\right) = 1 \end{aligned}$$

(66)

$Pr(f=1)$ and $Pr(f < 1)$ can be obtained from (48) and (49) respectively. Hence, only $Pr\left( {\mathcal{O}}^{DDF_{hd}}_{MUE}|f<1\right)$ needs to be evaluated. Let

$$\begin{aligned} Pr\left( {\mathcal{O}}^{DDF_{hd}}_{MUE}|f<1\right)& \stackrel{.}{=} \gamma ^{-\tilde{d}^{DDF_{hd}}_{MUE}} \end{aligned}$$

(67)

As in the case of the FUE, the achievable DMT at the MUE can be written as in (13).

In (13), only $\tilde{d}^{DDF_{hd}}_{MUE}$ needs to be evaluated. It can be evaluated as

$$\begin{aligned} \tilde{d}^{DDF_{hd}}_{MUE}&= \min _f \min _{v_{22},v_{32}} \min _{v_{23}} \left\{ v_{22}+v_{32}+ v_{23}\right\} \nonumber \\&\text {subject to } v_{ij} > 0 \quad \forall \quad \left\{ i,j\right\} \in \left\{ 2,3\right\} \nonumber \\&f [1-v_{22}]^+ + (1-f) \max \left\{ [1-v_{22}]^+,[1-v_{32}]^+\right\} \le r_2 \nonumber \\ f&= \min \left\{ 1,\frac{ r_2}{\left[ 1-v_{23}\right] ^+}\right\} \end{aligned}$$

(68)

This problem can be equivalently written as

$$\begin{aligned} \tilde{d}^{DDF_{hd}}_{MUE}&= \min _f \left\{ d_{inner}(f)+\left( 1-\frac{r_2}{f}\right) \right\} \nonumber \\&\text {subject\, to } v_{ij} > 0 \; \; \forall \; \;\left\{ i,j\right\} \in \left\{ 2,3\right\} \nonumber \\&f [1-v_{22}]^+ + (1-f) \max \left\{ [1-v_{22}]^+,[1-v_{32}]^+\right\} \le r_2 \nonumber \\&f = \frac{ r_2}{\left[ 1-v_{23}\right] ^+} \end{aligned}$$

(69)

$$\begin{aligned}&\text {where, }d_{inner}(f) = \min _{v_{22},v_{32}}v_{22}+v_{32} \nonumber \\&\text { subject to: } v_{22}> 0 \;\; v_{32} > 0 \nonumber \\&f [1-v_{22}]^+ + (1-f) \max \left\{ [1-v_{22}]^+,[1-v_{32}]^+\right\} \le r_2 \end{aligned}$$

(70)

The inner optimization problem can be solved as

$$\begin{aligned} d_{inner}(f) &= {\left\{ \begin{array}{ll} 2[1-r_2]^+ \text { if } 0< f< \frac{1}{2} \\ \left[ \frac{1-r_2}{f}\right] ^+ \text { if } \left\{ \frac{1}{2},(1-r_2)\right\}< f< 1 \\ 1+\left[ \frac{1-r_2-f}{1-f}\right] ^+ \text { if } \frac{1}{2}< f < 1-r_2 \end{array}\right. } \end{aligned}$$

(71)

Let $d^{MUE}_{I}$, $d^{MUE}_{II}$ and $d^{MUE}_{III}$ be the achievable DMTs corresponding to the above three cases. Then $\tilde{d}^{DDF_{hd}}_{MUE}$ can be calculated as

$$\begin{aligned} \tilde{d}^{DDF_{hd}}_{MUE} = \min \left\{ d^{MUE}_{I},d^{MUE}_{II},d^{MUE}_{III}\right\} , \end{aligned}$$

(72)

where

$$\begin{aligned} d^{MUE}_{I}&= 2[1-r_2]^+ \text { if } r_2 < \frac{1}{2}, \end{aligned}$$

(73)

$$\begin{aligned} d^{MUE}_{II}&= {\left\{ \begin{array}{ll} [2-2r_2]^+ \text { if } r_2 \le \left\{ \frac{1}{2},1\right\} \\ \left[ \frac{1-r_2}{\max \left\{ \frac{1}{2},r_2\right\} }\right] ^+ + \left[ 1-\frac{r_2}{\max \left\{ \frac{1}{2},r_2\right\} }\right] ^+ \text { if } \frac{1}{2} \le r_2 < 1, \\ \end{array}\right. } \end{aligned}$$

(74)

$$\begin{aligned} d^{MUE}_{III}&= \min \left\{ {\mathcal{G}}_4\left( r_2,\max \left\{ \frac{1}{2},r_2\right\} \right) , {\mathcal{G}}_4 \left( r_2,1-r_2\right) \right\} \nonumber \\&\text { if } \max \left\{ \frac{1}{2},r_2\right\} < (1-r_2). \end{aligned}$$

(75)