Probability Analysis of Capacity Gain by Cooperative Relaying Over Single-Hop OSTBC Transmission in Low SNR Regime

Abstract

In this paper, we investigate the capacity gain obtained by putting a multiple-antenna relay in an original single-hop multiple-input multiple-output system using an orthogonal space-time block code over uncorrelated Rayleigh fading channels. As relaying techniques, we consider decode-and-forward relaying as well as decouple-and-forward relaying in which no decoding is performed at the relay. Focusing on the low signal-to-noise ratio regime, closed-form expressions of the probability of the capacity gain are provided for the respective relaying schemes. The probability results in an explicit function of an average power ratio of the per-hop channel in dual-hop relaying to the single-hop channel. Numerical examples show the impact of the power ratio, the relaying strategy, the number of antennas and the relay location on the capacity gain, respectively.

Appendices

Appendix A: Derivation of $\mathcal P ^{DCF}_{A}(\lambda )$

Inserting (9) and (10) into (7), $\mathcal P ^{DCF}_{A}(\lambda )$ for $\lambda > 1/R_c$ is obtained by

$$\begin{aligned} \mathcal P ^{DCF}_{A}(\lambda )&= 1- \int \limits ^{\infty }_{0} F_{\gamma ^{DCF}_{R}} ( (R_c \lambda -1) \gamma ) f_{\gamma _0}(\gamma ) d\gamma \nonumber \\&= \frac{2}{(m_0-1)!(m_2-1)!} \left( \frac{z^S}{G_0}\right) ^{m_0} \left( \frac{z^R}{G_2}\right) ^{m_2} \sum ^{m_1-1}_{k=0}\frac{1}{k!} \nonumber \\&\quad \times \left( \frac{z^S z^R}{G_1 G_2}\right) ^{\frac{k}{2}} \sum ^{m_2+k-1}_{j=0} \left( \begin{array}{c} m_2+k-1 \\ j \\ \end{array} \right) \left( \frac{G_2 z^S}{G_1 z^R}\right) ^{\frac{j+1}{2}}\nonumber \\&\quad \times \left( R_c \lambda -1 \right) ^{m_2+k} \int \limits ^{\infty }_{0} e^{-\gamma \left( \frac{z^S}{G_0} + (R_c \lambda -1) \left( \frac{z^S}{G_1} +\frac{z^R}{G_2} \right) \right) } \nonumber \\&\quad \times \,\gamma ^{m_0+m_2+k-1} K_{j-k+1}\left( 2\gamma (R_c \lambda -1) \sqrt{\frac{z^S z^R}{G_1 G_2}} \right) d\gamma . \nonumber \\ \end{aligned}$$

(18)

Using [21, eq. (6.621.3)], the integral in (18) is solved as follows:

$$\begin{aligned}&\sqrt{\pi } \left( 4 \sqrt{\frac{z^S z^R}{G_1 G_2} }\right) ^{j-k+1} (R_c \lambda -1)^{-\xi -k} \left( \frac{z^S}{G_0} \right) ^{-\xi -j-1} \nonumber \\&\quad \times \,\omega _{+}^{-\xi -j-1} \frac{\Gamma {(\xi +j+1)} \Gamma {(\xi +2k-j-1)}}{\Gamma {(\xi +k+\frac{1}{2})}} \nonumber \\&\quad \times \, _{2}F_{1} \left( \xi +j+1,j-k+\frac{3}{2}; \xi +k+\frac{1}{2}; \frac{\omega _{-}}{\omega _{+}} \right) \!, \end{aligned}$$

(19)

where

$$\begin{aligned} \omega _{+} = \frac{1}{R_c \lambda -1} + \frac{G_0}{G_1} + \frac{G_0 z^R}{G_2 z^S} + 2 \sqrt{\frac{G_0^2 z^R}{G_1 G_2 z^S}},\end{aligned}$$

(20)

$$\begin{aligned} \omega _{-} = \frac{1}{R_c \lambda -1} + \frac{G_0}{G_1} + \frac{G_0 z^R}{G_2 z^S} - 2 \sqrt{\frac{G_0^2 z^R}{G_1 G_2 z^S}}, \end{aligned}$$

(21)

and $\xi =m_0+m_2$. Inserting (19) into (18) and using $L_1 = G_1/G_0$ and $L_2 = G_2/G_0$, (18) can be expressed as (11).

Appendix B: Derivation of $\mathcal P ^{DF}_A(\lambda )$

Let $I_i(\gamma ) = \sum ^{m_i-1}_{k=0} \frac{1}{k!} \left( \frac{z_i \gamma }{G_i} \right) ^k e^{-z_i \gamma / G_i}$ for $i=1,2$, where $z_1=z^S$ and $z_2=z^R$. Inserting (9) and (14) into (8), $\mathcal P ^{DF}_A(\lambda )$ for $\lambda > 1/R_c$ is then obtained by

$$\begin{aligned} \mathcal P ^{DF}_A(\lambda )&= \int \limits ^{\infty }_{0} \left\{ 1-F_{\gamma _1} ( R_c \lambda \gamma ) \right\} \times \left\{ 1-F_{\gamma _2} ( (R_c \lambda -1) \gamma ) \right\} f_{\gamma _0}(\gamma ) d\gamma \nonumber \\&= \int \limits ^{\infty }_{0} I_1(R_c \lambda \gamma ) I_2( (R_c \lambda -1) \gamma ) f_{\gamma _0}(\gamma ) d\gamma \nonumber \\&= \frac{1}{(m_0-1)!}\left( \frac{z^S}{G_0}\right) ^{m_0} \sum ^{m_1-1}_{k=0} \sum ^{m_2-1}_{j=0} \frac{1}{k!j!} \left( \frac{z^S R_c \lambda }{G_1}\right) ^k \nonumber \\&\quad \times \left( \frac{z^R (R_c \lambda -1)}{G_2}\right) ^j \int \limits ^{\infty }_{0} e^{-\gamma \left( \frac{z^S R_c \lambda }{G_1} + \frac{z^R (R_c \lambda -1)}{G_2} + \frac{z^S}{G_0} \right) } \gamma ^{k+j+m_0-1} d\gamma . \end{aligned}$$

(22)

Using [21, eq. (3.381.4)], (22) is obtained by

$$\begin{aligned} \mathcal P ^{DF}_A(\lambda )&= \frac{1}{(m_0-1)!}\left( \frac{z^S}{G_0}\right) ^{m_0} \sum ^{m_1-1}_{k=0} \sum ^{m_2-1}_{j=0} \frac{1}{k!j!} \nonumber \\&\quad \times \left( \frac{z^S R_c \lambda }{G_1}\right) ^k \left( \frac{z^R (R_c \lambda -1)}{G_2}\right) ^j \nonumber \\&\quad \times \left( \frac{z^S R_c \lambda }{G_1} + \frac{z^R (R_c \lambda -1)}{G_2} + \frac{z^S}{G_0} \right) ^{-k-j-m_0} \nonumber \\&\quad \times \, \Gamma (k+j+m_0). \end{aligned}$$

(23)

Using $L_1 = G_1/G_0$ and $L_2 = G_2/G_0$, (23) can be expressed as (15).

Substituting (9) and (14) into (8), $\mathcal P ^{DF}_A(\lambda )$ for $\lambda \le 1/R_c$ is obtained by

$$\begin{aligned} \mathcal P ^{DF}_A(\lambda )&= \int \limits ^{\infty }_{0} \left\{ 1-F_{\gamma _1} ( R_c \lambda \gamma ) \right\} f_{\gamma _0}(\gamma ) d\gamma \nonumber \\&= \frac{1}{(m_0-1)!}\left( \frac{z^S}{G_0}\right) ^{m_0} \sum ^{m_1-1}_{k=0} \frac{1}{k!} \left( \frac{z^S R_c \lambda }{G_1}\right) ^k \nonumber \\&\quad \times \int \limits ^{\infty }_{0} \gamma ^{k+m_0-1} e^{-\gamma \left( \frac{z^S R_c \lambda }{G_1} +\frac{z^S}{G_0} \right) } d\gamma . \end{aligned}$$

(24)

Using [21, eq. (3.381.4)], (24) is then obtained by

$$\begin{aligned} \mathcal P ^{DF}_A(\lambda )&= \frac{1}{(m_0-1)!}\left( \frac{z^S}{G_0}\right) ^{m_0} \sum ^{m_1-1}_{k=0} \frac{1}{k!} \left( \frac{z^S R_c \lambda }{G_1}\right) ^k \nonumber \\&\quad \times \left( \frac{z^S R_c \lambda }{G_1} + \frac{z^S}{G_0} \right) ^{-k-m_0} \Gamma (k+m_0). \end{aligned}$$

(25)

Using $L_1 = G_1/G_0$ and $L_2 = G_2/G_0$, (25) can be expressed as (16).