Relay Selection in Non-coherent AF MIMO–OFDM Relay-Assisted Systems with OSTBC

Abstract

This paper investigates the relay selection in non-coherent amplify-and-forward relay-assisted MIMO–OFDM system with orthogonal space–time block coding transmission. We propose three relay selection schemes. The best relay is selected based on OFDM subcarrier, OFDM symbol, and block. In this analysis, the source–relay and relay–destination channels are considered as Rayleigh fading and Rician fading, respectively. Exact closed-form expressions for end-to-end outage probability of all proposed schemes are derived. Furthermore, these schemes are studied in high SNR and also the diversity order and power gain are obtained. In addition, we showed that the selecting best relay among the multiple relay causing performance improvement. Moreover, this performance improvement will be greater with increasing the number of relays.

Appendix: Proof of Diversity Order for Three Relay Selection Schemes

Since the relays are independent from the others, we have

$$\begin{aligned} P_{out}^{RSS-1} =\left[ {1-\prod _{k=1}^N {\left( {1-F\left( {\gamma _{th} } \right) } \right) } } \right] ^{L}\approx \left[ {\sum _{k=1}^N {F\left( {\gamma _{th} } \right) } } \right] ^{L} \end{aligned}$$

(26)

Considering $b = \beta \rho$ for high-SNR region [13, 16]. Since $\rho$ is large, the $F\left( {\gamma _{th} } \right)$ which is obtained in [13, Eq.(25)] is simplified to (28). Following consider the expression (see Appendix C.1 from [32])

$$\begin{aligned} K_\upsilon (x)\approx \frac{\Gamma (\upsilon )}{2}\left( {\frac{2}{x}} \right) ^{\upsilon } ,\hbox { for}\, x\rightarrow 0 \end{aligned}$$

(27)

Also, we have

$$\begin{aligned} F\left( {\gamma _{th} } \right)&= 1-2e^{-\lambda }\sum _{k=0}^{N_S -1} {\sum _{i=0}^k {\sum _{l=0}^\infty {\left( {\frac{\left( {_i^k } \right) \gamma _{th}^k \lambda ^{l}}{k!\alpha ^{k}\left( {\frac{\beta }{N_R }} \right) ^{i}\sigma ^{2i}\Gamma \left( {p+l} \right) \Gamma \left( {l+1} \right) }} \right) }} }\nonumber \\&\times \left( {\frac{\left( {{N_R \gamma _{th} }/{\beta \sigma ^{2}\alpha \rho }} \right) ^{{\left( {p+l-i} \right) }/2}}{\rho ^{k}}} \right) K_{p+l-i} \left( {2\sqrt{\frac{N_R \gamma _{th} }{\beta \sigma ^{2}\alpha \rho }}} \right) \end{aligned}$$

(28)

Using (27) we can simplify (28) as

$$\begin{aligned} F\left( {\gamma _{th} } \right) \approx 1- 2e^{-\lambda }\sum _{k=0}^{N_S -1} {\sum _{i=0}^k {\sum _{l=0}^\infty {\left( {\frac{\left( {_i^k } \right) \gamma _{th}^k \lambda ^{l}\Gamma \left( {p+l-i} \right) }{2\left( {k!} \right) \alpha ^{k}\left( {\frac{\beta }{N_R }} \right) ^{i}\sigma ^{2i}\Gamma \left( {p+l} \right) \Gamma \left( {l+1} \right) }} \right) } .} } \left( {\frac{1}{\rho }} \right) ^{2k} \end{aligned}$$

(29)

Therefore, we can write

$$\begin{aligned} F\left( {\gamma _{th} } \right) \approx 1-2e^{-\lambda }\sum _{k=0}^{N_S -1} {\left( {\frac{1}{\rho }} \right) ^{2k}\sum _{i=0}^k {\sum _{l=0}^\infty {\left( {\frac{\left( {_i^k } \right) \gamma _{th}^k \lambda ^{l}\Gamma \left( {p+l-i} \right) }{2\left( {k!} \right) \alpha ^{k}\left( {\frac{\beta }{N_R }} \right) ^{i}\sigma ^{2i}\Gamma \left( {p+l} \right) \Gamma \left( {l+1} \right) }} \right) } } }\nonumber \\ \end{aligned}$$

(30)

where, we define

$$\begin{aligned} \Upsilon \left( k \right) =-\sum _{i=0}^k {\sum _{l=0}^\infty {\left( {\frac{\left( {_i^k } \right) \gamma _{th}^k \lambda ^{l}\Gamma \left( {p+l-i} \right) }{2\left( {k!} \right) \alpha ^{k}\left( {\frac{\beta }{N_R }} \right) ^{i}\sigma ^{2i}\Gamma \left( {p+l} \right) \Gamma \left( {l+1} \right) }} \right) } } \end{aligned}$$

(31)

which $\Upsilon \left( k \right)$ is not function of $\rho$. Hence, for large $\rho$, we have

$$\begin{aligned} F\left( {\gamma _{th} } \right) \le \rho ^{-2N_S +2}\Upsilon \left( {N_S -1} \right) \end{aligned}$$

(32)

In high-SNR using (26) and (32), we can achieve

$$\begin{aligned} P_{out}^{RSS-1} \le \left( {N\Upsilon \left( {N_S -1} \right) } \right) ^{L}\rho ^{L\left( {-2N_S +2} \right) } \end{aligned}$$

(33)

Using the diversity order and power gain definition in (18) and (19), respectively, we can obtain $d^{RSS-1}=L\left( {2N_S -2} \right)$ and $G^{RSS-1}=\left( {N\Upsilon \left( {N_S -1} \right) } \right) ^{L}$, where $\Upsilon \left( {N_S -1} \right)$ is denoted by (34).

$$\begin{aligned} \Upsilon \left( {N_S -1} \right)&= -2e^{-\lambda }\left[ {\sum _{i=0}^{N_S -1} {\left( {\frac{\gamma _{th} }{\alpha }} \right) ^{N_S -1}} \left( {\frac{1}{2\left( {i!} \right) \left( {N_S -i-1} \right) !\left( {\frac{\beta }{N_R }} \right) ^{i}\sigma ^{2i}}} \right) } \right] \nonumber \\&\times \sum _{l=0}^\infty {\left( {\frac{\lambda ^{l}\left( {p+l-i-1} \right) !}{l!\left( {p+l-1} \right) !}} \right) } \end{aligned}$$

(34)

Also, for RSS-2 we have

$$\begin{aligned} P_{out}^{RSS-2} =1-\prod _{k=1}^N {\left[ {1-\prod _{i=1}^L {F\left( {\gamma _{th} } \right) } } \right] } \end{aligned}$$

(35)

Since the relays are independent from the others, we have

$$\begin{aligned} P_{out}^{RSS-2} =1-\prod _{k=1}^N {\left[ {1-\left[ {F\left( {\gamma _{th} } \right) } \right] ^{L}} \right] } \approx \sum _{k=1}^N {\left[ {F\left( {\gamma _{th} } \right) } \right] ^{L}} \end{aligned}$$

(36)

Similar to RSS-1, in the high SNR using (32) we can write

$$\begin{aligned} P_{out}^{RSS-2} \le N\left( {\Upsilon \left( {N_S -1} \right) } \right) ^{L}\rho ^{L\left( {-2N_S +2} \right) } \end{aligned}$$

(37)

Therefore, the diversity order and the power gain for RSS-2 are given as $d^{RSS-2}=L\left( {2N_S -2} \right)$ and $G^{RSS-2}=N\left( {\Upsilon \left( {N_S -1} \right) } \right) ^{L}$, respectively, hence, (20) and (22) are obtained.

Now, for RSS-3 we have

$$\begin{aligned} P_{out}^{RSS-3} =1-\prod _{b=1}^B {\left\{ {1-\prod _{i=1}^L {\left[ {1-\prod _{k=1}^{N_B } {\left( {1-F\left( {\gamma _{th} } \right) } \right) } } \right] } } \right\} } \end{aligned}$$

(38)

Since the relays are independent from the other, we have

$$\begin{aligned} P_{out}^{RSS-3}&= 1-\prod _{b=1}^B {\left\{ {1-\left[ {1-\prod _{k=1}^{N_B } {\left( {1-F\left( {\gamma _{th} } \right) } \right) } } \right] ^{L}} \right\} } \approx \sum _{b=1}^B {\left[ {1-\prod _{k=1}^{N_B } {\left( {1-F\left( {\gamma _{th} } \right) } \right) } } \right] ^{L}}\nonumber \\&\approx \sum _{b=1}^B {\left[ {\sum _{k=1}^{N_B } {F\left( {\gamma _{th} } \right) } } \right] ^{L}} \end{aligned}$$

(39)

Similar to RSS-1 and RSS-2, in the high SNR using (32) we can write

$$\begin{aligned} P_{out}^{RSS-3} \le B\left( {N_B \Upsilon \left( {N_S -1} \right) } \right) ^{L}\rho ^{L\left( {-2N_S +2} \right) }. \end{aligned}$$

(40)

Therefore, the diversity order and power gain for RSS-3 are $d^{RSS-3}=L\left( {2N_S -2} \right)$ and $G^{RSS-3}=B\left( {N_B \Upsilon \left( {N_S -1} \right) } \right) ^{L}$, respectively, and also (20) and (23) are provided.