A reliable relay selection scheme for SSK modulation in cooperative communication systems

Abstract

Space shift keying (SSK) modulation is a scheme in which the indices of active transmit antennas play an essential role in carrying information bits. In SSK modulation, the communication channel acts as modulating unit. Therefore, using the conventional relay selection schemes based on channel coefficients are impossible in cooperative systems using SSK modulation. In this paper, we propose a novel relay selection scheme that is applicable in cooperative systems using SSK modulation. The proposed scheme is based on calculating and measuring the Euclidean distances between received signal and all active or inactive channel coefficients. The proposed relay selection scheme is applied for a cooperative communication system with multiple relay nodes and a direct link between source and destination. The analytical expressions are derived to calculate the average bit error rate over Rayleigh fading channels. The analytical and simulation results show that this relay selection scheme can provide a full cooperative diversity order, for SSK modulation. In addition, we show that this scheme can be used at destination as a signal selection method, to select the best received signal from different paths. The analytical results are validated using Monte Carlo simulation studies.

Appendices

Appendix 1

In this appendix, it is proved that, when $N_{t}>2$, one of the $x_{min,1}$ or $x_{min,2}$ equals $\left| n \right| ^{2}$ with high probability, where n is the noise of the received signal. Assume that $x_{min}$ is a vector with the elements of $x_{min,q}$ which are the values of $d_{i}$’s in increasing order, where $q=1,2,\ldots ,N_{t}$. The probability that $x_{min,1}=\left| n \right| ^{2}$ is equal to the correct detection probability by optimum detector and is obtained as

$$\begin{aligned} Pr\left( x_{min,1}=\left| n \right| ^{2} \right) =1-\left( 1+\frac{2\sigma _{h}^{2}\rho _{s}}{N_{0}\left( N_{t}-1 \right) } \right) ^{-1}. \end{aligned}$$

(28)

Using the probability theory, the probability that $x_{min,2}=\left| n \right| ^{2}$ is calculated by

$$\begin{aligned} Pr\left( x_{min,2}=\left| n \right| ^{2} \right)&= \left( 1+\frac{2\sigma _{h}^{2}\rho _{s}}{N_{0}\left( N_{t}-1 \right) } \right) ^{-1}\nonumber \\&\quad \times \left[ 1-\left( 1+\frac{2\sigma _{h}^{2}\rho _{s}}{N_{0}\left( N_{t}-2 \right) } \right) ^{-1} \right] . \end{aligned}$$

(29)

Therefore, using the probability theory, the general expression of $Pr\left( x_{min,q}=\left| n \right| ^{2} \right)$ is derived as

$$\begin{aligned} Pr\left( x_{min,q}=\left| n \right| ^{2} \right)&= \left[ 1-\left( 1+\frac{2\sigma _{h}^{2}\rho _{s}}{N_{0}\left( N_{t}-q \right) } \right) ^{-1} \right] \nonumber \\&\quad \times \prod _{j=1}^{q-1} \left( 1+\frac{2\sigma _{h}^{2}\rho _{s}}{N_{0}\left( N_{t}-j \right) } \right) ^{-1} \end{aligned}$$

(30)

It should be noted that the probability that $x_{min,1}=\left| n \right| ^{2}$ or $x_{min,2}=\left| n \right| ^{2}$ is equal to $Pr\left( x_{min,1}=\left| n \right| ^{2} \right) +Pr\left( x_{min,2}=\left| n \right| ^{2} \right)$.

It is obvious from (30) that $Pr\left( x_{min,q}=\left| n \right| ^{2} \right)$ decreases more with increasing q. For $q>2$ and $\rho _{s}> 10$ dB, the value of $Pr\left( x_{min,q}=\left| n \right| ^{2} \right)$ is negligible in (30), in comparison to the value of $Pr\left( x_{min,1}=\left| n \right| ^{2} \right) +Pr\left( x_{min,2}=\left| n \right| ^{2} \right)$. Therefore, with high probability one of the $x_{min,1}$ or $x_{min,2}$ is equal to $\left| n \right| ^{2}$.

Appendix 2

In this appendix, we want to derive the error probability presented in (24). Using optimum detection, when $\left| n_{sd} \right| ^{2}+\left| n_{r_{j}d} \right| ^{2}>\mathop {\min }\nolimits _{i=1,\ldots ,N_{t} } \left\{ \left| y_{sd}-\sqrt{\rho }g_{i} \right| ^{2}+\left| y_{r_{j}d}-\sqrt{\rho }f_{j,i} \right| ^{2} \right\}$, an error will take place at the destination. We define w as $w=\left| n_{sd} \right| ^{2}+\left| n_{r_{j}d} \right| ^{2}$. The random variable w is sum of two independent exponential random variables with variances of $N_{0}$. The PDF of w is obtained as

$$\begin{aligned} f_{W}\left( w \right) =\left( \frac{1}{N_{0}} \right) ^{2}w~exp\left( \frac{-w}{N_{0}} \right) . \end{aligned}$$

(31)

As mentioned before, for large values of $\rho$, the value of $\mathop {\min }\nolimits _{i=1,\ldots ,N_{t} } \left\{ \left| y_{sd}-\sqrt{\rho }g_{i} \right| ^{2}+\left| y_{r_{j}d}-\sqrt{\rho }f_{j,i} \right| ^{2} \right\}$, is approximated as $\rho \mathop {\min }\nolimits _{i=1,\ldots ,N_{t} } \left\{ \left| g_{t}-g_{i} \right| ^{2}+\left| f_{j,t}-f_{j,i} \right| ^{2} \right\} =\rho \mathop {\min } \left\{ r_{i} \right\}$, where $r_{i}=\left| g_{t}-g_{i} \right| ^{2}+\left| f_{j,t}-f_{j,i} \right| ^{2}$ is sum of two independent exponential random variables, with variances of $2\sigma _{g}^{2}$ and $2\sigma _{f}^{2}$, respectively. The PDFs of $r_{i}$’s are obtained by

$$\begin{aligned} f_{R_{i}}\left( r_{i} \right) = \left\{ \begin{array}{ll} \left( \frac{1}{2\sigma _{g}^{2}} \right) ^{2}r_{i}~exp\left( \frac{-r_{i}}{2\sigma _{g}^{2}} \right) &{}\quad 2\sigma _{g}^{2} = 2\sigma _{f}^{2} \\ \frac{exp\left( \frac{-r_{i}}{2\sigma _{f}^{2}} \right) -exp\left( \frac{-r_{i}}{2\sigma _{g}^{2}} \right) }{2\sigma _{f}^{2}-2\sigma _{g}^{2}} &{}\quad 2\sigma _{g}^{2} \ne 2\sigma _{f}^{2}. \end{array}\right. \end{aligned}$$

(32)

The parameter v is defined as $v=\mathop {\min }\nolimits _{i=1,\ldots ,N_{t} } \left\{ r_{i} \right\}$. Then, the CDF of v is obtained by

$$\begin{aligned} F_{V}\left( v \right)&= Pr\left( \mathop {\min }\limits _{i=1,\ldots ,N_{t} } \left\{ r_{i} \right\} \leqslant v \right) \nonumber \\&= 1-\left[ \int _{v}^{\infty }f_{R_{i}}\left( r_{i} \right) dr_{i} \right] ^{N_{t}-1}. \end{aligned}$$

(33)

By obtaining the CDF of v, the PDF $f_{V}\left( v \right)$ is calculated as

$$\begin{aligned} f_{V}\left( v \right)&= \frac{dF_{V}\left( v \right) }{dv} \nonumber \\&= \left( N_{t}-1 \right) \left[ \int _{v}^{\infty }f_{R_{i}}\left( r_{i} \right) dr_{i} \right] ^{N_{t}-2} f_{R_{i}}\left( v \right) . \end{aligned}$$

(34)

Then, substituting (32) into (34) yields (25) and (26) for $2\sigma _{g}^{2}\ne 2\sigma _{f}^{2}$ and $2\sigma _{g}^{2}=2\sigma _{f}^{2}$, respectively. Finally, having $f_{W}\left( w \right)$ from (31) and $f_{V}\left( v \right)$ from (34), $Pr\left( E_{d} \right| C_{r})$ in (24) is completely calculated.