Performance Analysis of Relay Selection with Enhanced Dynamic Decode-and-Forward and Network Coding in Two-Way Relay Channels

Abstract

Relay selection (RS), enhanced dynamic decode-and-forward (EDDF), and network coding (NC) have been proven to effectively improve the performance of cooperative communications systems. This motivates us to combine these three techniques to see how they can improve the performance of cooperative communications systems. In this study, we adopt the RS protocol proposed by Bletsas, Khisti, Reed and Lippman in 2006. The protocol features an EDDF and NC system for two-hop two-way multirelay networks. All nodes are single-input single-output and half-duplex (i.e., they cannot transmit and receive data simultaneously). The outage probability and its asymptotic behavior at a high SNR, diversity gain, coding gain at a high SNR, multiplexing gain of the sum rate, and spectrum efficiency of the RS-DDF&NC and RS-EDDF&NC schemes were analyzed. In this paper, we present comparisons of the outage probability in various scenarios under the Rayleigh fading channel. Our results demonstrate that the RS protocol with the EDDF and NC scheme exhibits optimal performance regarding outage probability based on the considered DF relay scheme if there exist sufficient relays. In addition, the performance loss is large if a relay is selected randomly. This demonstrates the importance of RS strategies.

Appendices

Appendix 1: Proof of Lemma 1

Let $X_l = |h_{\mathrm{A},l}|^2$ and $Y_l = |h_{\mathrm{B},l}|^2$ for $l = 1, \ldots , L$. Then, $X_1, \ldots , X_L$, $Y_1, \ldots , Y_L$ are independent identically distributed (i.i.d.) exponential random variables with the parameter $\lambda = 1$. The PDFs of $X_l$ and $Y_l$ are

$$\begin{aligned} f_{X_l}(x_l) = {\left\{ \begin{array}{ll} e^{- x_l}, &{} x_l \ge 0 \\ 0, &{} {\text {otherwise}} \end{array}\right. } \end{aligned}$$

(63)

and

$$\begin{aligned} f_{Y_l}(y_l) = {\left\{ \begin{array}{ll} e^{- y_l}, &{} y_l \ge 0 \\ 0, &{} {\text {otherwise}}, \end{array}\right. } \end{aligned}$$

(64)

respectively.

Define $S_l = \min \{X_l, Y_l\}$ and $T_l = \max \{X_l, Y_l\}, \quad l = 1, \ldots , L.$ Then, the joint PDF of $S_l$ and $T_l$ is

$$\begin{aligned} f_{S_l,T_l}(s_l,t_l) = {\left\{ \begin{array}{ll} 2 e^{- s_l - t_l}, &{} 0 \le s_l \le t_l \\ 0, &{} {\text {otherwise}}. \end{array}\right. } \end{aligned}$$

(65)

The PDF of $S_l$ is

$$\begin{aligned} f_{S_l}(s_l) = {\left\{ \begin{array}{ll} 2 e^{- 2 s_l}, &{} s_l \ge 0 \\ 0, &{} {\text {otherwise}}. \end{array}\right. } \end{aligned}$$

(66)

The conditional PDF of $T_l$ given $S_l$ is

$$\begin{aligned} f_{T_l|S_l}(t_l|s_l) = {\left\{ \begin{array}{ll} e^{- (t_l - s_l)}, &{} t_l \ge s_l \\ 0, &{} {\text {otherwise}}. \end{array}\right. } \end{aligned}$$

(67)

Let

$$\begin{aligned} l_0 = \arg \max _{l} S_l. \end{aligned}$$

(68)

Define $Z_1 = X_{l_0}, \quad Z_2 = Y_{l_0}, \quad U = \min \{Z_1, Z_2\},$ and $V = \max \{Z_1, Z_2\}.$ Then,

$$\begin{aligned} U = \max \{ S_1, \ldots , S_L\}. \end{aligned}$$

(69)

The cumulative distribution function (CDF) of U is

$$\begin{aligned}&F_U(u) \nonumber \\&\quad = P[U \le u] \nonumber \\&\quad = P[S_1 \le u] \cdots P[S_L \le u] \nonumber \\&\quad = {\left\{ \begin{array}{ll} (1 - e^{- 2 u})^L, &{} u \ge 0 \\ 0, &{} {\text {otherwise}}. \end{array}\right. } \end{aligned}$$

(70)

The PDF of U is

$$\begin{aligned} f_U(u) = {\left\{ \begin{array}{ll} 2 L e^{- 2 u} (1 - e^{- 2 u})^{L - 1}, &{} u \ge 0 \\ 0, &{} {\text {otherwise}}. \end{array}\right. } \end{aligned}$$

(71)

If $u \ge v \ge 0$, then the joint PDF of U and V is

$$\begin{aligned}f_{U,V}(u,v)&= f_{V|U}(v|u) f_U(u) \nonumber \\&= e^{- (v - u)} 2 L e^{- 2 u} (1 - e^{- 2 u})^{L - 1} \nonumber \\&= 2 L e^{- (u + v)} (1 - e^{- 2 u})^{L - 1}, \end{aligned}$$

(72)

where $f_{V|U}(v|u)$ is the conditional PDF of V given U. We have

$$\begin{aligned} f_{U,V}(u,v) = {\left\{ \begin{array}{ll} 2 L e^{- (u + v)} (1 - e^{- 2 u})^{L - 1}, &{} 0 \le u \le v \\ 0, &{} {\text {otherwise}}. \end{array}\right. } \end{aligned}$$

(73)

We also have

$$\begin{aligned} f_{U,V}(u,v)= & {} {\left\{ \begin{array}{ll} f_{Z_1,Z_2}(u,v) + f_{Z_1,Z_2}(v,u),&{} 0\le u\le v\\ 0, &{}{\text {otherwise}}, \end{array}\right. } \end{aligned}$$

(74)

$$\begin{aligned} f_{Z_1,Z_2}(a,b)= & {} f_{Z_1,Z_2}(b,a). \end{aligned}$$

(75)

Therefore,

$$\begin{aligned} f_{Z_1,Z_2}(z_1,z_2) =&{\left\{ \begin{array}{ll} \frac{1}{2} f_{U,V}(z_1,z_2), &\quad{} 0\le z_1\le z_2\\ \frac{1}{2} f_{U,V}(z_2,z_1), &\quad{} 0\le z_2\le z_1\\ 0, &\quad{} {\text {otherwise}}. \end{array}\right. } \nonumber \\ =&{\left\{ \begin{array}{ll} L e^{-(z_1+z_2)} (1-e^{-2 z_1})^{L-1}, &\quad{} 0\le z_1\le z_2\\ L e^{-(z_1+z_2)} (1-e^{-2 z_2})^{L-1}, &\quad{} 0\le z_2\le z_1\\ 0, &\quad{} {\text {otherwise}}. \end{array}\right. } \end{aligned}$$

(76)

Appendix 2: Proof of Theorem 1

The probability of event $O_{\mathrm{A,B}}^{\mathrm{DDF \& NC}}$ is

$$ \begin{aligned} \mathrm{P}[O_{\mathrm{A,B}}^{\mathrm{DDF \& NC}}]&=\mathrm{P}\left[ \frac{2}{3}\log _2(1+\rho |h_{\mathrm{A},\mathrm{B}}|^2)<R\right] \nonumber \\&=\mathrm{P}\left[ |h_{\mathrm{A},\mathrm{B}}|^2<\frac{2^\frac{3R}{2}-1}{\rho }\right] \nonumber \\&{\mathop {=}\limits ^\mathrm{(b)}} 1 - \exp \left( - \frac{2^{\frac{3 R}{2}} - 1}{\rho } \right) \triangleq p_1. \end{aligned}$$

(77)

The equality (b) holds because $|h_{\mathrm{A},{\mathrm{B}}}|^2$ is an exponential random variable with a rate parameter of 1. Moreover,

$$ \begin{aligned} P_{\mathrm{out}}^{\mathrm{RS-DDF \& NC}}= p_1 \int \int _{{\mathcal {R}}} f_{Z_1,Z_2}(z_1,z_2) dz_1 dz_2, \end{aligned}$$

(78)

where the integration region ${{\mathcal {R}}}$ is

$$\begin{aligned} {{\mathcal {R}}} = \bigcup \limits _{i=1}^4 {{\mathcal {R}}}_{i}, \end{aligned}$$

(79)

where

$$\begin{aligned} {{\mathcal {R}}}_1= & {} \left\{ (z_1, z_2) \in {\mathbb {R}}_+^2 \left| \log _2(1 + \rho z_2) < \frac{J R}{J-1} \right. \right\} , \end{aligned}$$

(80)

$$\begin{aligned} {\mathbb {R}}_+^2= & {} \left\{ (z_1, z_2) \in {\mathbb {R}}^2 \left| z_1 \ge 0, z_2 \ge 0 \right. \right\} , \end{aligned}$$

(81)

$$\begin{aligned} {{\mathcal {R}}}_2= & {} \left\{ (z_1, z_2) \in {\mathbb {R}}_+^2 \left| \log _2(1 + \rho z_1) < \frac{J R}{J-1} \right. \right\} , \end{aligned}$$

(82)

$$\begin{aligned} {{\mathcal {R}}}_3= & {} \left\{ (z_1, z_2) \in {\mathbb {R}}_+^2 \left| \frac{2 J + J_1 - J_2}{3 J} \log _2(1 + \rho z_1) < R \right. \right\} , \end{aligned}$$

(83)

$$\begin{aligned} {{\mathcal {R}}}_4= & {} \left\{ (z_1, z_2) \in {\mathbb {R}}_+^2 \left| \frac{2 J + J_2 - J_1}{3 J} \log _2(1 + \rho z_2) < R \right. \right\} , \end{aligned}$$

(84)

$$\begin{aligned} J_1= & {} \min \left( J, \left\lceil \frac{J R}{\log _2(1 + \rho z_1)} \right\rceil \right) , \end{aligned}$$

(85)

and

$$\begin{aligned} J_2 = \min \left( J, \left\lceil \frac{J R}{\log _2(1 + \rho z_2)} \right\rceil \right) . \end{aligned}$$

(86)

Equations (85) and (86) come from (7) and (11), respectively, with the assumption $\rho _{\mathrm{A},l^*}= \rho _{\mathrm{B},l^*}= \rho$. From (79)–(86), we find that ${{\mathcal {R}}}$ is symmetric about the line $z_1 = z_2$ and restricted in the first quadrant on the $z_1$–$z_2$ plane. The integrand of (78), $f_{Z_1,Z_2}(z_1,z_2)$, is also a symmetric function of $z_1$ and $z_2$. Therefore,

$$\begin{aligned} \int \int _{{\mathcal {R}}} f_{Z_1,Z_2}(z_1,z_2) dz_1 dz_2 = 2 \int \int _{{{\mathcal {R}}} \cap {{\mathcal {R}}}_{\pi /4}} f_{Z_1,Z_2}(z_1,z_2) dz_1 dz_2, \end{aligned}$$

(87)

where

$$\begin{aligned} {{\mathcal {R}}}_{\pi /4}= \left\{ \left. (z_1, z_2) \in {\mathbb {R}}_+^2 \right| z_1 \ge z_2 \right\} . \end{aligned}$$

(88)

We can further derive ${{\mathcal {R}}} \cap {{\mathcal {R}}}_{\pi /4}$ as

$$\begin{aligned} {{\mathcal {R}}}_5&\triangleq {{\mathcal {R}}} \cap {{\mathcal {R}}}_{\pi /4}\nonumber \\&= ({{\mathcal {R}}}_1 \cup {{\mathcal {R}}}_4) \cap {{\mathcal {R}}}_{\pi /4}. \end{aligned}$$

(89)

First, we must find the intersection point $P_1$ of the boundary of ${{\mathcal {R}}}_4$ and the line $z_2 = z_1$. The boundary of ${{\mathcal {R}}}_4$ is

$$\begin{aligned} (2 J + J_2 - J_1) \log _2(1 + \rho z_1) = 3 J R. \end{aligned}$$

(90)

Substituting $z_2 = z_1$ into (90) yields

$$\begin{aligned} P_1 = (\rho ^{-1}(2^{3R/2} - 1), \rho ^{-1}(2^{3R/2} - 1)) \triangleq (z_{1, 1}, z_{2, 1}). \end{aligned}$$

(91)

The region ${{\mathcal {R}}}_5$ is illustrated in Fig. 16, where $N_s$ is the number of stairs. The value of $N_s$ can be easily found as in (33).

Fig. 16

Integration region ${{\mathcal {R}}}_5$

The boundary of ${{\mathcal {R}}}_1$ is

$$\begin{aligned} \log _2(1 + \rho z_2) = \frac{J R}{J - 1}. \end{aligned}$$

(92)

Solving (92) yields

$$\begin{aligned} z_2 = \rho ^{-1} \left[ 2^{J R / (J - 1)} - 1 \right] \triangleq c. \end{aligned}$$

(93)

The next step is to find $z_{1, i}$ and $z_{2, i}$ for $i \ge 2$. Noting that the edge of each stair is located at $z_1 = z_{1, i}$ for $i = 2, 3,\ldots , N_s+1$. Let the boundary of ${{\mathcal {R}}}_5$ be $f_5(z_1, z_2) = 0$. Then,

$$\begin{aligned} \frac{d f_5(z_1, z_2)}{d z_1} \end{aligned}$$

(94)

must have Dirac delta functions at $z_1 = z_{1, i}$ for $i = 2, 3,\ldots , N_s+1$. After some calculation, we find

$$\begin{aligned} z_{1, i} = \rho ^{-1} \left[ 2^{JR/(\lfloor 2 J / 3 \rfloor - i + 2)} - 1 \right] \end{aligned}$$

(95)

for $i \ge 2$. Substituting $z_{1, i}$ in $f_5(z_1, z_2) = 0$ can yield $z_{2, i}$ for $i \ge 2$, as shown in (31).

Third, the following must be computed:

$$\begin{aligned} \int \int _{{{\mathcal {R}}}_5} f_{Z_1,Z_2}(z_1,z_2) dz_1 dz_2. \end{aligned}$$

(96)

We first calculate the integration up to $z_1 = z_{1, 2}$. Let

$$\begin{aligned} {{\mathcal {S}}}_1 = \{ (z_1,z_2) \in {\mathbb {R}}_+^2 | z_2 \le z_1 \le z_{1, 2}, z_2 \le z_{2, 1} \}. \end{aligned}$$

(97)

Then,

$$\begin{aligned}&\int \int _{{{\mathcal {S}}}_1} f_{Z_1,Z_2}(z_1,z_2) dz_1 dz_2 \nonumber \\&\quad = \int _0^{z_{2, 1}} \int _{z_2}^{z_{1, 2}} L e^{-(z_1 + z_2)} (1 - e^{-2 z_2})^{L - 1} dz_1 dz_2 \nonumber \\&\quad = \frac{1}{2} (1 - e^{-2 z_{2, 1}})^L - L e^{-z_{1, 2}} f_1(z_{2, 1}). \end{aligned}$$

(98)

Because of space limitations, we do not detail the integration here. Next, consider the integration region

$$\begin{aligned} {{\mathcal {S}}}_2 = \{ (z_1,z_2) \in {\mathbb {R}}_+^2 | z_{1, 2} \le z_1 \le z_{1, 3}, 0 \le z_2 \le z_{2, 2} \}. \end{aligned}$$

(99)

Then,

$$\begin{aligned}&\int \int _{{{\mathcal {S}}}_2} f_{Z_1,Z_2}(z_1,z_2) dz_1 dz_2 \nonumber \\&\quad = \int _0^{z_{2, 2}} \int _{z_{1, 2}}^{z_{1, 3}} L e^{-(z_1 + z_2)} (1 - e^{-2 z_2})^{L - 1} dz_1 dz_2 \nonumber \\&\quad = L [e^{-z_{1, 2}} - e^{-z_{1, 3}}] f_1(z_{2, 2}). \end{aligned}$$

(100)

Similarly, for the integration region

$$\begin{aligned} {{\mathcal {S}}}_i = \{ (z_1,z_2) \in {\mathbb {R}}_+^2 | z_{1, i} \le z_1 \le z_{1, i + 1}, 0 \le z_2 \le z_{2, i} \}, \end{aligned}$$

(101)

we have

$$\begin{aligned} \int \int _{{{\mathcal {S}}}_i} f_{Z_1, Z_2}(z_1, z_2) dz_1 dz_2 = L [e^{-z_{1, i}} - e^{-z_{1, i + 1}}] f_1(z_{2, i}) \end{aligned}$$

(102)

for $i = 3, 4,\ldots , N_s$. For the integration region

$$\begin{aligned} {{\mathcal {S}}}_{N_s+1} = \{ (z_1,z_2) \in {\mathbb {R}}_+^2 | z_1 \ge z_{1, N_s + 1}, 0 \le z_2 \le c \}, \end{aligned}$$

(103)

we have

$$\begin{aligned} \int \int _{{{\mathcal {S}}}_{N_s+1}} f_{Z_1,Z_2}(z_1,z_2) dz_1 dz_2 = L e^{-z_{1, N_s + 1}} f_1(c). \end{aligned}$$

(104)

Summing up (98), (102), and (104) yields

$$\begin{aligned}&\int \int _{{{\mathcal {R}}}_5} f_{Z_1,Z_2}(z_1,z_2) dz_1 dz_2 \nonumber \\&\quad = \frac{1}{2} (1 - e^{-2 z_{2, 1}})^L - L e^{-z_{1, 2}} f_1(z_{2, 1}) + L \sum _{i = 2}^{N_s} (e^{-z_{1, i}} - e^{-z_{1, i + 1}}) f_1(z_{2, i}) \nonumber \\&\qquad + L e^{-z_{1, N_s + 1}} f_1(c). \end{aligned}$$

(105)

Finally,

$$ \begin{aligned} P_{\mathrm{out}}^{\mathrm{RS-DDF \& NC}}= 2 p_1 \int \int _{{{\mathcal {R}}}_5} f_{Z_1,Z_2}(z_1,z_2) dz_1 dz_2 \end{aligned}$$

(106)

which is equal to (28).

Appendix 3: Proof of Theorem 2

First, consider the first term of (28):

$$\begin{aligned}&2 p_1 \left[ \frac{1}{2} (1 - e^{-2 z_{2,1}})^L \right] \nonumber \\&\quad = \left[ 1 - \exp \left( - \frac{2^{3 R / 2} - 1}{\rho } \right) \right] \left\{ 1 - \exp \left[ - \frac{2 (2^{3 R / 2} - 1)}{\rho } \right] \right\} ^L \nonumber \\&\qquad \approx 2^L \left( \frac{2^{3 R / 2} - 1}{\rho } \right) ^{L + 1}, \end{aligned}$$

(107)

where the approximation results from the power series [37] expansion for the second line of (107) about the point $\rho = \infty$ to order $\rho ^1$.

Similarly, the second term of (28) can be approximated as

$$\begin{aligned} 2 p_1 L e^{- z_{1,2}} f_1(z_{2,1}) \approx 2^L \left( \frac{2^{3 R / 2} - 1}{\rho } \right) ^{L + 1}. \end{aligned}$$

(108)

To find an approximation of the third term of (28), we must first find an approximation of $z_{2,i}$ for $i \ge 2$. When $\rho$ is very large,

$$\begin{aligned} \frac{J R}{\log _2(1 + \rho x)} \end{aligned}$$

(109)

is very small. Therefore,

$$\begin{aligned} J > \left\lceil \frac{J R}{\log _2(1 + \rho x)} \right\rceil \end{aligned}$$

(110)

and

$$\begin{aligned} \min \left( J, \left\lceil \frac{J R}{\log _2(1 + \rho x)} \right\rceil \right)&= \left\lceil \frac{J R}{\log _2(1 + \rho x)} \right\rceil \nonumber \\&\approx \frac{J R}{\log _2(1 + \rho x)}. \end{aligned}$$

(111)

Similarly,

$$\begin{aligned} \min \left( J, \left\lceil \frac{J R}{\log _2(1 + \rho z_{1,i})} \right\rceil \right) \approx \frac{J R}{\log _2(1 + \rho z_{1,i})}. \end{aligned}$$

(112)

Substituting (111) and (112) into

$$\begin{aligned}&\left[ 2 J + \min \left( J, \left\lceil \frac{J R}{\log _2(1 + \rho x)} \right\rceil \right) - \min \left( J, \left\lceil \frac{J R}{\log _2(1 + \rho z_{1,i})} \right\rceil \right) \right] \nonumber \\&\cdot \log _2(1 + \rho x) - 3 J R = 0, \end{aligned}$$

(113)

which comes from (31), leads to

$$\begin{aligned} \left[ 2 J + \frac{J R}{\log _2(1 + \rho x)} - \frac{J R}{\log _2(1 + \rho z_{1,i})} \right] \log _2(1 + \rho x) - 3 J R \approx 0. \end{aligned}$$

(114)

Solving (114) yields

$$\begin{aligned} x \approx \rho ^{-1} \{ 2^{2 R / [ 2 - R / \log _2(1 + \rho z_{1,i} ) ]} - 1 \} \triangleq {\hat{z}}_{2,i}. \end{aligned}$$

(115)

Therefore, the third term of (28) can be approximated as

$$\begin{aligned}&2 p_1 L \sum _{i = 2}^{N_s} (e^{-z_{1, i}} - e^{-z_{1, i + 1}}) f_1(z_{2,i}) \nonumber \\&\quad \approx 2 p_1 L \sum _{i = 2}^{N_s} (e^{-z_{1, i}} - e^{-z_{1, i + 1}}) f_1({\hat{z}}_{2,i}) \nonumber \\&\quad \approx 2^L (2^{3 R / 2} - 1) \rho ^{-(L + 2)} \sum _{i = 2}^{N_s} \left( 2^{\frac{J R}{2 - i + \left\lfloor \frac{2 J}{3} \right\rfloor }} - 2^{\frac{J R}{1 - i + \left\lfloor \frac{2 J}{3} \right\rfloor }} \right) \nonumber \\&\quad \cdot \bigg ( 2^{\frac{2 R}{2- \frac{R}{\log _2(J R) - \left\lfloor \frac{2 J}{3} \right\rfloor + i - 2}}} - 1 \bigg )^L. \end{aligned}$$

(116)

Using the technique of power series expansion again, we approximate the fourth term of (28) as

$$\begin{aligned} 2 p_1 L e^{-z_{1, N_s + 1}} f_1(c) \approx 2^L (2^{3 R / 2} - 1) [2^{J R / (J - 1)} - 1]^L \rho ^{-(L + 1)}. \end{aligned}$$

(117)

Substituting (107), (108), (116), and (117) into (28) yields (35). Note that (116) can be ignored because it contains the term $\rho ^{-(L + 2)}$. When $\rho$ is very large, $\rho ^{-(L + 1)} \gg \rho ^{-(L + 2)}$.

Appendix 4: Proof of Theorem 3

The diversity gain [2] of the RS-DDF&NC scheme is

$$ \begin{aligned}&d^{\mathrm{RS-DDF \& NC}}(r) \nonumber \\&\quad = - \lim _{\rho \rightarrow \infty } \frac{\log _2 \left( P_{\mathrm{out}}^{\mathrm{RS-DDF \& NC}}\right) }{\log _2 \rho } \nonumber \\&\quad = - \lim _{\rho \rightarrow \infty } \frac{\log _2 \left\{ 2^L \left( 2^{3 R / 2} - 1 \right) [ 2^{J R / (J - 1)} - 1]^L \rho ^{-(L + 1)} \right\} }{\log _2 \rho } \nonumber \\&\quad = - \lim _{\rho \rightarrow \infty } \frac{\log _2 \left( 2^L \right) }{\log _2 \rho } - \lim _{\rho \rightarrow \infty } \frac{\log _2 \left( 2^{3 R / 2} - 1 \right) }{\log _2 \rho } \nonumber \\&\qquad - \lim _{\rho \rightarrow \infty } \frac{\log _2 \left\{ \left[ 2^{J R / (J - 1)} - 1 \right] ^L \right\} }{\log _2 \rho } - \lim _{\rho \rightarrow \infty } \frac{\log _2 \left[ \rho ^{-(L + 1)} \right] }{\log _2 \rho } \nonumber \\&\quad = - 0 - \lim _{\rho \rightarrow \infty } \frac{\log _2 \left[ 2^{3 r (\log _2 \rho ) / 2} \right] }{\log _2 \rho } \nonumber \\&\qquad - \lim _{\rho \rightarrow \infty } \frac{L \log _2 \left[ 2^{J r (\log _2 \rho ) / (J - 1)} \right] }{\log _2 \rho } + L + 1 \nonumber \\&\quad = (36). \end{aligned}$$

(118)

Appendix 5: Proof of Theorem 4

When we do not use coding (i.e., we do not use DDF or NC), only RS is performed. The system block diagram is shown in Figs. 17 and 18.

Fig. 17

Illustration of Phases 1 and 2 for the RS scheme

Fig. 18

Illustration of Phases 3 and 4 for the RS scheme

First, consider the outage event of User A. If the links from User B to User A and those from relay $l^*$ to User A are both in outage, then User A cannot receive the information of User B. Therefore, the outage event of User A can be expressed as follows:

$$\begin{aligned} O_{\mathrm{A}}^{\mathrm{RS}}= O_{\mathrm{B,A}}^{\mathrm{RS}}\cap O_{l^*,\mathrm{A}}^{\mathrm{RS}}. \end{aligned}$$

(119)

Similarly, User B cannot receive the information of User A when the links from User A to User B and those from relay $l^*$ to User B are both in outage. We express the outage event of User B as follows:

$$\begin{aligned} O_{\mathrm{B}}^{\mathrm{RS}}= O_{\mathrm{A,B}}^{\mathrm{RS}}\cap O_{l^*,\mathrm{B}}^{\mathrm{RS}}. \end{aligned}$$

(120)

In the next step, we analyze the outage events between User A, User B, and relay $l^*$. The outage event between Users A and B is

$$\begin{aligned} O_{\mathrm{A,B}}^{\mathrm{RS}}&= \{ J \log _2 (1 + \rho |h_{\mathrm{A},\mathrm{B}}|^2)< 2 J R \} \nonumber \\&= \{ \log _2 (1 + \rho |h_{\mathrm{A},\mathrm{B}}|^2) < 2 R \}. \end{aligned}$$

(121)

Clearly, $O_{\mathrm{B,A}}^{\mathrm{RS}}= O_{\mathrm{A,B}}^{\mathrm{RS}}$ because $h_{\mathrm{B},\mathrm{A}}= h_{\mathrm{A},\mathrm{B}}$.

Then, we consider the outage event between relay $l^*$ and User A:

$$\begin{aligned} O_{l^*,\mathrm{A}}^{\mathrm{RS}}&= \left\{ \frac{J}{2} \log _2 (1 + \rho |h_{l^*,\mathrm{A}}|^2)< 2 J R \right\} \nonumber \\&= \left\{ \log _2 (1 + \rho |h_{l^*,\mathrm{A}}|^2) < 4 R \right\} . \end{aligned}$$

(122)

Similarly,

$$\begin{aligned} O_{l^*,\mathrm{B}}^{\mathrm{RS}}= {\log _2 (1 + \rho |h_{l^*,\mathrm{B}}|^2) < 4 R}. \end{aligned}$$

(123)

The outage event of the overall system can be expressed as

$$\begin{aligned}O^{\mathrm{RS}}&= O_{\mathrm{A}}^{\mathrm{RS}}\cup O_{\mathrm{B}}^{\mathrm{RS}}\nonumber \\&= (O_{\mathrm{B,A}}^{\mathrm{RS}}\cap O_{l^*,\mathrm{A}}^{\mathrm{RS}}) \cup (O_{\mathrm{A,B}}^{\mathrm{RS}}\cap O_{l^*,\mathrm{A}}^{\mathrm{RS}}) \nonumber \\&= O_{\mathrm{B,A}}^{\mathrm{RS}}\cap (O_{l^*,\mathrm{A}}^{\mathrm{RS}}\cup O_{l^*,\mathrm{B}}^{\mathrm{RS}}). \end{aligned}$$

(124)

Therefore, the outage probability of RS can be expressed as

$$\begin{aligned} P_{\mathrm{out}}^{\mathrm{RS}}=&\int _{-\infty }^{\infty } \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } I(O^{\mathrm{RS}}|_{Z = z, Z_1 = z_1, Z_2 = z_2}) f_Z(z) f_{Z_1, Z_2}(z_1, z_2) dz dz_1 dz_2 \nonumber \\ =&\int _{-\infty }^{\infty } I(O_{\mathrm{B,A}}^{\mathrm{RS}}|_{Z = z}) dz \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } I(O_{l^*,\mathrm{A}}\cup O_{l^*,\mathrm{B}}|_{Z_1 = z_1, Z_2 = z_2}) \nonumber \\&\cdot \, f_{Z_1, Z_2}(z_1, z_2) dz_1 dz_2. \end{aligned}$$

(125)

Because of space limitations, we do not detail the derivation here. The first integral in (125) is equal to

$$\begin{aligned} \int _0^\infty I(\{ \log _2 (1 + \rho z) < 2 R \}) e^{-z} dz = 1 - \exp [ -\rho ^{-1} (2^{2 R} - 1 ) ]. \end{aligned}$$

(126)

The double integral in (125) is equal to

$$\begin{aligned}&\int _0^\infty \int _0^\infty I ( \{ \log _2 ( 1 + \rho z_1 )< 4 R \} \cup \{ \log _2 ( 1 + \rho z_2 ) < 4 R \} ) \nonumber \\&\quad \cdot \, f_{Z_1, Z_2}(z_1, z_2) dz_1 dz_2 \nonumber \\&\quad = \{ 1 - \exp [ - \rho ^{-1} ( 2^{4 R} - 1 ) ] \}^L. \end{aligned}$$

(127)

Substituting (126) and (127) into (125) yields

$$\begin{aligned} P_{\mathrm{out}}^{\mathrm{RS}}= \{ 1 - \exp [ -\rho ^{-1} ( 4^R - 1 ) ] \} \{ 1 - \exp [ - 2 \rho ^{-1} ( 16^R - 1 ) ] \}^L. \end{aligned}$$

(128)

When $\rho$ is large, using the Taylor series to expand (128) up to the first-order term results in

$$\begin{aligned} P_{\mathrm{out}}^{\mathrm{RS}}\approx \left( 4^R - 1 \right) \left[ 2 \left( 16^R - 1 \right) \right] ^L \rho ^{-(L + 1)}. \end{aligned}$$

(129)

From (35), we know

$$ \begin{aligned}P_{\mathrm{out}}^{\mathrm{RS-DDF \& NC}}& \approx 2^L \left( 2^{3 R / 2} - 1 \right) \left[ 2^{J R / (J - 1)} - 1 \right] ^L \rho ^{-(L + 1)} \nonumber \\& \Rightarrow \rho \approx \left( \frac{2^{3 R / 2} - 1}{P_{\mathrm{out}}^{\mathrm{RS-DDF \& NC}}} \right) ^\frac{1}{L + 1} \left\{ 2 \left[ 2^{J R / (J - 1)} - 1 \right] \right\} ^\frac{L}{L + 1}. \end{aligned}$$

(130)

From (129), we know

$$\begin{aligned} \rho \approx \left( \frac{4^R - 1}{P_{\mathrm{out}}^{\mathrm{RS}}} \right) ^\frac{1}{L + 1} \left[ 2 \left( 16^R - 1 \right) \right] ^\frac{L}{L + 1}. \end{aligned}$$

(131)

When $P_{\mathrm{out}}^{\mathrm{RS}}= P_{\mathrm{out}}^{\mathrm{RS-DDF \& NC}}= P_{\mathrm{out}}$, we let the SNR of RS be $\rho ^{\mathrm{RS}}$ and that of the RS-DDF&NC scheme be $\rho ^{\mathrm{RS-DDF \& NC}}$. Then, the coding gain of the RS-DDF&NC scheme is

$$ \begin{aligned} G^{\mathrm{RS-DDF \& NC}}= \frac{\rho ^{\mathrm{RS}}}{\rho ^{\mathrm{RS-DDF \& NC}}} \approx (37). \end{aligned}$$

(132)

Appendix 6: Proof of Theorem 5

Both Users A and B transmit at rate R; therefore, the sum rate is $R_{\Sigma } = 2R$. The multiplexing gain of $R_{\Sigma }$ is

$$\begin{aligned} r_{\Sigma } = \lim _{\rho \rightarrow \infty } \frac{R_{\Sigma }}{\log _2 \rho } = 2 r. \end{aligned}$$

(133)

According to Theorem 3, the diversity gain of the RS-DDF&NC scheme is

$$ \begin{aligned} d^{\mathrm{RS-DDF \& NC}}= - \left( \frac{L J}{J - 1} + \frac{3}{2} \right) r + L + 1. \end{aligned}$$

(134)

Equation (134) can be rewritten as

$$ \begin{aligned} r = \frac{L + 1 - d^{\mathrm{RS-DDF \& NC}}}{\frac{L J}{J - 1} + \frac{3}{2}}. \end{aligned}$$

(135)

Substituting (135) into (133) yields (38).

Appendix 7: Proof of Theorem 6

When the SNR is high, $\rho$ is very large. The multiplexing gain

$$\begin{aligned} r = \frac{R}{\log _2 \rho } \end{aligned}$$

(136)

is a very small positive number. Therefore, $r < 0.5$ and the EDDF and DDF protocols are identical ([6], Theorem 3). Thus, the RS-DDF&NC and RS-EDDF&NC schemes have the same asymptotic behavior.