Performance analysis of a multihop relay network using distributed Alamouti code

Abstract

In this paper, we apply the classical Alamouti coding technique in a distributed and cascaded fashion to multihop relay networks where relays perform the decode-and-forward (DF) protocol at each relay stage. The considered system consists of two antennas at the source, two single-antenna relays at each relay stage and one antenna at the destination. The source transmits at each coding step two symbols according to the Alamouti code, to the first relay stage where each relay decodes the two symbols independently. The two relays of each relay stage behave like the two antennas of the source and transmit the two estimated symbols to the next relay stage until the destination using distributed Alamouti code. We obtain a closed form expression for the outage probability of the N-hop relay network and derive a tight analytical upper bound for the bit error probability at the destination, by considering the two-hop case firstly and then generalizing for the N-hop networks. We also show that DF protocol provides better bit error performance compared to the amplify-and-forward protocol previously considered in the literature for N-hop relay networks.

Appendix

The diversity order is obtained by [22]

$$\begin{aligned} D = \lim \limits _{\overline{\gamma } \rightarrow \infty } -\frac{log(P_{out} (\gamma _{out}))}{log(\overline{\gamma })} \end{aligned}$$

(54)

where the outage probability is given by

$$\begin{aligned} P_{out}(\gamma _{out})=1-\left( \left(1+\frac{\gamma _{out}}{\overline{\gamma }}\right) e^{-\gamma _{out}/\overline{\gamma }}\right) ^{2N-1}. \end{aligned}$$

(55)

After some manipulation, we get

$$\begin{aligned} D = \lim \limits _{\overline{\gamma } \rightarrow \infty } - \frac{log\left( e^{\gamma _{out} (2N-1)/\overline{\gamma }} -\left( 1+\frac{\gamma _{out}}{\overline{\gamma }} \right) ^{2N-1}\right) }{log(\overline{\gamma })}. \end{aligned}$$

(56)

We cannot find the limit directly, so we need to use Taylor series expansion to have a proper expression. We know that the Taylor series expansions are given by

$$\begin{aligned} e^{\gamma _{out} (2N-1)/\overline{\gamma }}&= 1+\frac{(2N-1)\gamma _{out}}{ \overline{\gamma }} +\frac{1}{2}\left( \frac{(2N-1)\gamma _{out}}{\overline{\gamma }}\right) ^2 +O\left( \frac{1}{\overline{\gamma }^3}\right) , \\ \left( 1+\frac{\gamma _{out}}{\overline{\gamma }}\right) ^{(2N-1)}&= 1+\frac{(2N-1)\gamma _{out}}{\overline{\gamma }} +\frac{(2N-1)(2N-2)\gamma _ {out}}{2\overline{\gamma }^2} +O\left( \frac{1}{\overline{\gamma }^3}\right) , \end{aligned}$$

(57)

where $O(\cdot)$ represents omitted higher-order terms in power series. When we use (57) in (56) we have

$$\begin{aligned} D = \lim \limits _{\overline{\gamma } \rightarrow \infty } - \frac{log\left( \frac{1}{\overline{\gamma }^2} \left( C_1+O\left( \frac{1}{\overline{\gamma }}\right) \right) \right) }{log(\overline{\gamma })}, \end{aligned}$$

(58)

where $C_1$ is a constant. Then we have

$$\begin{aligned} D&= \lim \limits _{\overline{\gamma } \rightarrow \infty } - \frac{-2 log\left( {\overline{\gamma }}\right) + log\left( C_1+O\left( \frac{1}{\overline{\gamma }}\right) \right) }{log(\overline{\gamma })} \\&= 2 + \lim \limits _{\overline{\gamma } \rightarrow \infty } - \frac{log\left( C_1+O\left( \frac{1}{\overline{\gamma }}\right) \right) }{log (\overline{\gamma })} \\&= 2. \end{aligned}$$

(59)

So it is proved that the diversity order of the system is 2. $\square$