Effective capacity of cooperative relaying systems with non-orthogonal multiple access

Abstract

In this paper, we present analytical results on the effective capacity (EC) for a dual-hop cooperative network employing non-orthogonal multiple access and the decode-and-forward half-duplex relaying protocol. Assuming that wireless propagation is modelled by the Nakagami-m distribution, novel analytical results for the EC are deduced. These analytical results are further extended to the generalized fading channels, modeled by a mixture gamma distribution. Our proposed analysis is validated by numerical results and Monte-Carlo simulations. Besides, the numerically evaluated results have demonstrated the impact of various system-level parameters on the performance of the considered system, including the impact of fading, the power allocation coefficient and the delay constraints.

Appendices

Appendices

Appendix A. Proof of Proposition 1

The EC of $s_1$, i.e., $R_{s_1 } \left( \theta \right)$, can be expressed as

$$\begin{aligned} \!\!R_{s_1 } \left( \theta \right) =\!-\frac{1}{\theta } \ln \left( \int _0^\infty \!\!\!\left( \frac{1 \!+\!\rho z_1}{1\!+\!\rho a_2 z_1}\right) ^{\!-\!\frac{\alpha }{2 \ln 2}} f_{z_1}(z_1) \textrm{d}z_1 \right) . \end{aligned}$$

(17)

In the generalized composite fading environment, the CDF of $|h_\ell |^2, \ell \in \{SR,SD,RD\}$, is given as [20],

$$\begin{aligned} F_{|h_\ell |^2}\left( x\right) =\sum \limits _{i=1}^{N_\ell } \alpha _i^l (\xi _i^l)^{-\beta _i} \gamma \left( \beta _i^l, \xi _i^l x \right) . \end{aligned}$$

(18)

From (18), the CDFs of $Z_1$ and $Z_2$ can be obtained as follows,

$$\begin{aligned} F_{Z_1}(Z_1)&=1-(1-F_{\lambda _{SR}}(Z_1))(1-F_{\lambda _{SD}}(Z_1))\nonumber \\&=\sum \limits _{t\in \{SR,SD\}}\sum \limits _{i=1}^{N_t}\alpha _i^t(\xi _i^t)^{-\beta _i^t}\gamma \left( \beta _i^t, \xi _i^tZ_1\right) \nonumber \\&-\prod \limits _{t\in \{SR,SD\}}\sum \limits _{i=1}^{N_t}\alpha _i^t\left( \xi _i^t\right) ^{-\beta _i^t}\gamma \left( \beta _i^t, \xi _i^tZ_1\right) {,} \end{aligned}$$

(19)

and

$$\begin{aligned} F_{Z_2}(z)&=\sum \limits _{t\in \{SR,SD\}}\sum \limits _{i=1}^{N_t}\alpha _i^t(\xi _i^t)^{-\beta _i^t}\gamma \left( \beta _i^t, \frac{\xi _i^tz}{a}\right) \nonumber \\&-\prod \limits _{t\in \{SR,SD\}}\sum \limits _{i=1}^{N_t}\alpha _i^t\left( \xi _i^t\right) ^{-\beta _i^t}\gamma \left( \beta _i^t, \xi _i^tz\right) {,} \end{aligned}$$

(20)

respectivly. Using $\gamma \left( n,x \right) =\int _0^x e^{-t} t^{n-1} \textrm{d}t$ and $\Gamma \left( n,x \right) =(n-1)! e^{-x} \sum \limits _{m=0}^{n\!-\!1} \frac{x^m}{m!}$, from (19), the PDF of $Z_1$ can be computed as

$$\begin{aligned}&f_{Z_1}(z)=\sum \limits _{t\in \{SR,SD\}}\sum _{i=1}^{N_t}\alpha _i^t z^{\beta _i^t-1}e^{-z\xi _i^t}\nonumber \\&\!\!-\!\!\!\!\!\prod \limits _{t\in \{SR,SD\}}\!\!\left( \sum \limits _{i=1}^{N_t}\sum \limits _{j=1}^{N_{\bar{t}}}\sum \limits _{r=0}^{\infty }\frac{(-1)^r\alpha _i^t\alpha _j^{{\bar{t}}} \left( \xi _j^{{\bar{t}}}\right) ^r z^{\beta _i^t+\beta _j^{\bar{t}}+r-1}e^{-z\xi _i^t}}{r!\left( r+\beta _j^{{\bar{t}}}\right) } \right) . \end{aligned}$$

(21)

By substituting (21) into (17) and performing some algebraic manipulations, integrals of the form

$$\begin{aligned}&\mathcal {I} =\int _0^\infty \left( 1+\rho z_1 \right) ^{ - \frac{\alpha }{{2\ln 2}}} \left( 1+\rho a_2 z_1 \right) ^{n} \nonumber \\&\quad \times \left( 1+\rho a_2 z_1 \right) ^{-n + {\frac{\alpha }{2\ln 2}}} z_1^{{i} + {j}} e^{-\omega _\Sigma z_1} \textrm{d}{z_1}, \end{aligned}$$

(22)

appears and needs to be solved. Expressing the exponential and power functions in terms of G-functions, using [23, Eq. (8.4.2.5)] and [23, Eq. (8.4.3.1)], as

$$\begin{aligned} e^{-\omega {z_1}} =G_{0,1}^{1,0}\left( {\omega {z_1}}{\mathop {\big |}\nolimits ^{\cdot }_{0}} \right) , \end{aligned}$$

$$\begin{aligned} \left( 1+\rho z_1 \right) ^{ - \frac{\alpha }{{2\ln 2}}} = \frac{1}{\Gamma \left( {\frac{\alpha }{{2\ln 2}}} \right) } G_{1,1}^{1,1}\left( {\rho {z_1} \left| \begin{array}{l} 1 - \frac{\alpha }{{2\ln 2}}\\ 0 \end{array} \right. } \right) , \end{aligned}$$

$$\begin{aligned} \left( 1+\rho a_2 z_1 \right) ^{-n + {\frac{\alpha }{2\ln 2}}}&=\frac{ G_{1,1}^{1,1}\left( {\rho {a_2}{z_1} \left| \begin{array}{l} 1 -n + \frac{\alpha }{{2\ln 2}}\\ 0 \end{array} \right. } \right) }{\Gamma \left( n- {\frac{\alpha }{{2\ln 2}}} \right) }, \end{aligned}$$

and with the help of binomial theorem as well as [24, Eq. (20)], (22) can be evaluated. Then, (14) can be proved.

The PDF of $Z_2$ is similar to (21), by substituting $f_{Z_2}(z_2)$ into (23), the follow integral $\mathcal {K}$ appears as

$$\begin{aligned} \mathcal {K}&=\mathop \sum \limits _{{{i}} = 1}^{{N_{SR}}} \alpha _{{{i}}}^{SR}{\left( {\frac{1}{{{a_2}}}} \right) ^{\beta _{{{i}}}^{SR}}}\int _0^\infty {{{\left( {1 + \rho {z_2}} \right) }^{ - \frac{\alpha }{{2\ln 2}}}}} {Z_2}^{ - 1 + \beta _{{{i}}}^{SR}}\\&\times {\mathrm{{e}}^{ - {Z_2}\frac{{\xi _{{{i}}}^{SR}}}{{{a_2}}}}}d{z_2}\\&=\sum \limits _{{i}=1}^{N_{SR}} \alpha ^{SR}_{{i}} \left( \frac{1}{a_2} \right) ^{ \beta ^{SR}_{{i}}} u\left( \frac{ \xi ^{SR}_{{i}}}{a_2},-1+ \beta ^{SR}_{{i}} \right) . \end{aligned}$$

So (15) can be proved. Thus, Proposition 4 can be proved.

Appendix B. Proof of Proposition 3

The EC of $s_2$ can be expressed as

$$\begin{aligned} R_{s_2 } \left( \theta \right)&=-\frac{1}{\theta } \ln \left( \int _0^\infty \left( 1+\rho z_2 \right) ^{-\frac{\alpha }{2 \ln 2}} f_{Z_2}(z_2) \textrm{d}z_2 \right) . \end{aligned}$$

(23)

By substituting $f_{Z_2}(z_2)$ into (23), the follow integrals $\mathcal {J}$ and $\mathcal {K}^{'}$ appear as

$$\begin{aligned} \mathcal {J}&= \int _0^\infty \left( 1+\rho z_2 \right) ^{-\frac{\alpha }{2 \ln 2}} \omega ' e^{-\omega ' z_2} dz_2 \\&= \frac{\omega '}{\rho } e^{\frac{\omega '}{\rho }} \in {i}^\infty t^{-\frac{\alpha }{2 \ln 2}} e^{\frac{-\omega ' t}{\rho }} dt, \end{aligned}$$

and

$$\begin{aligned} \mathcal {K^{'}}&=\sum \limits _{{i}=1}^{m_{SR}\!-\!1}\!\sum \limits _{{j}=1}^{m_{RD}\!-\!1} f({i},{j}) \omega ' \\&\quad \times \int _0^\infty \left( 1+\rho z_2 \right) ^{-\frac{\alpha }{2 \ln 2}} z^{t_3 + t_4} e^{-\omega ' z_2} \textrm{d}z_2. \end{aligned}$$

Utilizing the definition of $E_n(x)$, we obtain

$$\begin{aligned} \mathcal {J} =\frac{\omega '}{\rho } {e^{\frac{\omega '}{\rho }}} {E_{\frac{\alpha }{2\ln 2}}} \left[ {\frac{\omega '}{\rho }} \right] , \end{aligned}$$

(24)

and

$$\begin{aligned} \mathcal {K} =\sum \limits _{{i}=1}^{m_{SR}-1}\sum \limits _{{j}=1}^{m_{RD}-1} f({i},{j}) \omega ' u\left( \rho ,{{i}}+{{j}} \right) . \end{aligned}$$

(25)

With the help of (23), (24) and (25), Proposition 3 is proved.