A Semi-Orthogonal Distributed Alamouti Space-Time Codes Design

Abstract

In this paper, a novel semi-orthogonal distributed Alamouti space-time codes transmission protocol is proposed for a four nodes cooperative communication system consisting of one source, one destination and two semi blind relays over block fading channels. In particular, by semi-orthogonal we mean two orthogonal frequency bands are invoked, one of which is for the transmission by the source node, while the other one is shared simultaneously by the two relay-destination links. Moreover, analytical performances of the proposed semi-orthogonal scheme are investigated in this paper. Specifically, the theoretical expressions of the exact SER and diversity order are presented. Our proposed scheme is capable of achieving higher spectral efficiency and remaining the same diversity order compared to the existing orthogonal one, while attaining better symbol error rate performance and higher diversity order against the non-orthogonal design. Finally, simulation results prove the correctness of the above conclusions and also verify our derivation for the analytical performances.

Appendices

Appendix 1

1.1 The Derivation of MGF $M_\gamma (t)$

First, for flat Rayleigh fading channels, the channel gain $\left| {h_{xy} } \right| ^{2}$ of the link $\text{ X }\rightarrow \text{ Y } (\text{ X }\in \{\text{ S, } \text{ R }_{1} ,\text{ R }_{2} \},\, \text{ Y }\in \{\text{ R }_{1} ,\text{ R }_{2} , \text{ D }\})$ is exponential distribution [24], and the probability density function (PDF) of $\left| {h_{xy} } \right| ^{2}$ can be expressed as

$$\begin{aligned} f_{\left| {h_{xy} } \right| ^{2}} \left( \left| {h_{xy} } \right| ^{2}\right) =\frac{1}{\Omega _{xy} }\exp \left( -\left| {h_{xy} } \right| ^{2}/\Omega _{xy} \right) . \end{aligned}$$

(26)

Based on (26), the MGF $M_\gamma \left( t|\left| {h_{r_i d} } \right| ^{2}\right)$ of $\gamma$ conditioned on $\left| {h_{r_i d} } \right| ^{2}$ can be written as

$$\begin{aligned}&M_\gamma (t|\left| {h_{r_i d} } \right| ^{2}) \nonumber \\&\quad =\int \limits _0^\infty {\exp }(\gamma t)f(\gamma )d_\gamma \nonumber \\&\quad =\int \limits _0^\infty {e^{t\left( \gamma _0 \left| {h_{sd1} } \right| ^{2}+\gamma _0 \left| {h_{sd2} } \right| ^{2}\right) }f_{\left| {h_{sd} } \right| ^{2}} } \left( \left| {h_{sd} } \right| ^{2}\right) d_{\left| {h_{sd} } \right| ^{2}}\nonumber \\&\quad \quad \times \int \limits _0^\infty {\int \limits _0^\infty {e^{t\gamma _0 \frac{\beta _{1}^{2} \left| {h_{sr_1 } } \right| ^{2}\left| {h_{r_1 d} } \right| ^{2}+\beta _{2}^{2} \left| {h_{sr_2 } } \right| ^{2}\left| {h_{r_2 d} } \right| ^{2}}{\beta _{1}^{2} \left| {h_{r_1 d} } \right| ^{2}+\beta _{2}^{2} \left| {h_{r_2 d} } \right| ^{2}+1}}f_{\left| {h_{sr_i } } \right| ^{2}} \left( \left| {h_{sr_i } } \right| ^{2}\right) d_{\left| {h_{sr_i } } \right| ^{2}} } }\nonumber \\&\quad =\frac{1}{\left( {1-t\Omega _0 \gamma _0 } \right) ^{2}}\prod _{i=1}^2 {\left( 1-\frac{t\left| {h_{r_i d} } \right| ^{2}\Omega _1 }{\left( \beta _1^2 \left| {h_{r_1 d} } \right| ^{2}+\beta _2^2 \left| {h_{r_2 d} } \right| ^{2}\right) /\beta _i^2 \gamma _0 +1/\beta _i^2 \gamma _0 }\right) ^{-1}}. \end{aligned}$$

(27)

Based on (27), we can obtain $M_\gamma (t)$ by averaging $M_\gamma (t|\left| {h_{r_i d} } \right| ^{2})$ on $\left| {h_{r_i d} } \right| ^{2}$, which can be expressed as [10]

$$\begin{aligned} M_\gamma (t)&= \frac{1}{\left( {1-t\Omega _0 \gamma _0 } \right) ^{2}}\int \limits _0^\infty {\int \limits _0^\infty {\prod _{i=1}^2 \left( 1-\frac{t\left| {h_{r_i d} } \right| ^{2}\Omega _1 }{\frac{\beta _1^2 \left| {h_{r_1 d} } \right| ^{2}+\beta _2^2 \left| {h_{r_2 d} } \right| ^{2}\text{+1 }}{\beta _i^2 \gamma _0 }}\right) }} ^{-1}\nonumber \\&\quad \times \frac{1}{\Omega _2^2 }e^{-\frac{\left| {h_{r_1 d} } \right| ^{2}+\left| {h_{r_2 d} } \right| ^{2}}{\Omega _2 }}d_{\left| {h_{r_1 d} } \right| ^{2}} d_{\left| {h_{r_2 d} } \right| ^{2}} \end{aligned}$$

(28)

Since $\beta _1^2 =\beta _2^2 =\beta ^{2}$, (28) can be simplified as:

$$\begin{aligned} M_\gamma (t)&= \frac{1}{\left( {1-t\Omega _0 \gamma _0 } \right) ^{2}}\int \limits _0^\infty {\int \limits _0^\infty {\prod _{i=1}^2 {\left( 1-\frac{t\left| {h_{r_i d} } \right| ^{2}\Omega _1 }{\frac{\left| {h_{r_1 d} } \right| ^{2}+\left| {h_{r_2 d} } \right| ^{2}}{\gamma _0 }+\frac{1}{\beta ^{2}\gamma _0 }}\right) }}} ^{-1}\nonumber \\&\quad \times \frac{1}{\Omega _2^2 }e^{-\frac{\left| {h_{r_1 d} } \right| ^{2}+\left| {h_{r_2 d} } \right| ^{2}}{\Omega _2 }}d_{\left| {h_{r_1 d} } \right| ^{2}} d_{\left| {h_{r_2 d} } \right| ^{2}}. \end{aligned}$$

(29)

Let $\tau =\left| {h_{r_1 d} } \right| ^{2}+\left| {h_{r_2 d} } \right| ^{2}$, (29) can be written as [10]

$$\begin{aligned} M_\gamma (t)&= \frac{1}{\left( {1-t\Omega _0 \gamma _0 } \right) ^{2}}\int \limits _0^\infty {\int \limits _0^\infty {\prod _{i=1}^2 {\left( 1-\frac{t\Omega _1 \left| {h_{r_i d} } \right| ^{2}}{\frac{\tau }{\gamma _0 }+\frac{1}{\beta ^{2}\gamma _0 }}\right) ^{-1}} } } \frac{1}{\Omega _2^2 }e^{-\frac{\tau }{\Omega _2 }}d_{\left| {h_{r_1 d} } \right| ^{2}} d_{\left| {h_{r_2 d} } \right| ^{2}} \nonumber \\&= \frac{1}{\left( {1-t\Omega _0 \gamma _0 } \right) ^{2}}\int \limits _0^\infty {H(\tau )} \frac{1}{\Omega _2^2 }e^{-\frac{\tau }{\Omega _2 }}d_\tau , \end{aligned}$$

(30)

where $H(\tau )=\int _0^\tau {\frac{1}{1-\alpha _1 \left| {h_{r_1 d} } \right| ^{2}}\frac{1}{1-\alpha _1 (\tau -\left| {h_{r_1 d} } \right| ^{2})}} d_{\left| {h_{r_1 d} } \right| ^{2}} ,\alpha _1 =\frac{t\Omega _1 }{\tau /\gamma _0 +1/\beta ^{2}\gamma _0 }$.

Using the method of partial fraction expansion, $H(\tau )$ can be expressed as [10]

$$\begin{aligned} H(\tau )&= \frac{1}{2-\alpha _1 \tau }\int \limits _0^\tau {\left( \frac{1}{1-\alpha _1 \left| {h_{r_1 d} } \right| ^{2}}+\frac{1}{1-\alpha _1 \left( \tau -\left| {h_{r_1 d} } \right| ^{2}\right) }\right) } d_{\left| {h_{r_1 d} } \right| ^{2}}\nonumber \\&= \frac{-2\ln (1-\alpha _1 \tau )}{\alpha _1 (2-\alpha _1 \tau )}. \end{aligned}$$

(31)

Finally, the MGF $M_\gamma (t)$ of $\gamma$ can be written as

$$\begin{aligned} M_{\gamma _q } (t)=\frac{1}{\left( {1-t\Omega _0 \gamma _0 } \right) ^{2}}\int \limits _0^\infty {\frac{-2\ln (1-\alpha _1 \tau )}{\alpha _1 (2-\alpha _1 \tau )}} \frac{1}{\Omega _2^2 }e^{-\frac{\tau }{\Omega _2 }}d_\tau . \end{aligned}$$

(32)

Appendix 2

1.1 Proof of Theorem 1

First, based on Eq. (3.352.4) in [23], we have the following equation:

$$\begin{aligned} -e^{\mu (t)\theta }\text{ Ei }(-\mu (t)\theta )=\int \limits _0^\infty {\frac{1}{t+\theta }} e^{-\mu (t)t}dt. \end{aligned}$$

(33)

where $\text{ Ei }(\,)$ is the exponential integral function [23].

Let $t=\hat{{t}}\theta$, (33) can be rewritten as

$$\begin{aligned} -e^{\mu (t)\theta }\text{ Ei }(-\mu (t)\theta )&= \int \limits _0^\infty {\frac{1}{t+\theta }} e^{-\mu (t)t}dt \nonumber \\&= \int \limits _0^\infty {\frac{1}{\theta }\frac{1}{1+\hat{{t}}}} e^{-\mu (\hat{{t}}\theta )\hat{{t}}\theta }d\left( {\hat{{t}}\theta } \right) \nonumber \\&= \int \limits _0^\infty {\frac{1}{1+\hat{{t}}}} e^{-\mu (\hat{{t}}\theta )\hat{{t}}\theta }d\hat{{t}}. \end{aligned}$$

(34)

Then let $\hat{{t}}=t$, (34) can be further expressed as

$$\begin{aligned} -e^{\mu (t)\theta }\text{ Ei }(-\mu (t)\theta )=\int \limits _0^\infty {\frac{1}{1+t}} e^{-\mu (t\theta )t\theta }dt. \end{aligned}$$

(35)

On the other hand, from Eq. (9.211.4) in [23], the confluent hypergeometric function $\Psi (\alpha ,r;z)$can be expressed as

$$\begin{aligned} \Psi (\alpha ,r;z)=\frac{1}{\Gamma (\alpha )}\int \limits _0^\infty {e^{-zt}t^{\alpha -1}(1+t)^{r-\alpha -1}dt} , \end{aligned}$$

(36)

where $\Gamma (\alpha )=\int _0^\infty {e^{-t}t^{\alpha -1}} d_t$ is the gamma function [23].

Let $\alpha =1, r=1$ and $z=\mu (t)\theta$, based on (36), we can obtain

$$\begin{aligned} \Psi (1,1;\mu (t)\theta )&= \frac{1}{\Gamma (1)}\int \limits _0^\infty {e^{-\mu (t)\theta t}(1+t)^{-1}dt} \nonumber \\&= \int \limits _0^\infty {e^{-t}} d_t \int \limits _0^\infty {e^{-\mu (t)\theta t}(1+t)^{-1}dt} \nonumber \\&= \int \limits _0^\infty {\frac{1}{1+t}e^{-\mu (t)t\theta }dt} \end{aligned}$$

(37)

From (35) and (37), we can conclude that $-e^{\mu (t)\theta }\text{ Ei }(-\mu (t)\theta )=\Psi (1,1;\mu (t)\theta )$.

Appendix 3

1.1 The Derivation of the MGF $M_{\gamma _{a1} } (t)$ and $M_{\gamma _{a2} } (t)$

According to the Lemma 1 in [19], we can obtain the following expression as

$$\begin{aligned} M_{\gamma _{a1} } (t)&= \frac{1}{\left( 1-t\gamma _0 \frac{\Omega _{sr_1 } \Omega _{sr_2 } }{\Omega _{sr_2 } +\Omega _{sr_2 } }\right) ^{2}\left( \beta _1^2 \Omega _{r_1 d} -\beta _2^2 \Omega _{r_2 d} \right) }\left[ \left( 1-t\gamma _0 \frac{\Omega _{sr_1 } \Omega _{sr_2 } }{\Omega _{sr_2 } +\Omega _{sr_2 } }\right) \left( \beta _1^2 \Omega _{r_1 d} -\beta _2^2 \Omega _{r_2 d} \right) \right. \nonumber \\&\quad \left. -t\gamma _0 \frac{\Omega _{sr_1 } \Omega _{sr_2 } }{\Omega _{sr_2 } +\Omega _{sr_2 } }e^{\frac{1}{\beta _1^2 \Omega _{r_1 d} \left( 1-t\gamma _0 \frac{\Omega _{sr_1 } \Omega _{sr_2 } }{\Omega _{sr_2 } +\Omega _{sr_2 } }\right) }}\text{ Ei }\left( \frac{1}{\beta _1^2 \Omega _{r_1 d} \left( 1-t\gamma _0 \frac{\Omega _{sr_1 } \Omega _{sr_2 } }{\Omega _{sr_2 } +\Omega _{sr_2 } }\right) }\right) \nonumber \right. \\&\quad \left. +t\gamma _0 \frac{\Omega _{sr_1 } \Omega _{sr_2 } }{\Omega _{sr_2 } +\Omega _{sr_2 } }e^{\frac{1}{\beta _2^2 \Omega _{r_2 d} \left( 1-t\gamma _0 \frac{\Omega _{sr_1 } \Omega _{sr_2 } }{\Omega _{sr_2 } +\Omega _{sr_2 } }\right) }}\text{ Ei }\left( \frac{1}{\beta _2^2 \Omega _{r_2 d} \left( 1-t\gamma _0 \frac{\Omega _{sr_1 } \Omega _{sr_2 } }{\Omega _{sr_2 } +\Omega _{sr_2 } }\right) }\right) \right] ,\end{aligned}$$

(38)

$$\begin{aligned} M_{\gamma _{a2} } (t)&= \frac{\Omega _{sr_1 } }{\left( {\Omega _{sr_1 } +\Omega _{sr_2 } } \right) ^{2}}\left[ \Omega _{sr_1 }^2 t\gamma _0 M\left( \frac{\beta _2^2 \Omega _{r_2 d} }{\beta _1^2 \Omega _{r_1 d} },-\Omega _{sr_1 } t\gamma _0 ,\frac{1}{\beta _1^2 \Omega _{r_1 d} }\right) +\Omega _{sr_1 } \right. \nonumber \\&\quad \left. +\Omega _{sr_2 } +\Omega _{sr_2 }^2 t\gamma _0 M\left( \frac{\beta _2^2 \Omega _{r_2 d} }{\beta _1^2 \Omega _{r_1 d} },-\Omega _{sr_2 } t\gamma _0 ,\frac{1}{\beta _1^2 \Omega _{r_1 d} }\right) \right] \nonumber \\&\quad +\frac{\Omega _{sr_2 } }{\left( {\Omega _{sr_1 } +\Omega _{sr_2 } } \right) ^{2}}\left[ \Omega _{sr_1 } +\Omega _{sr_2 } +\Omega _{sr_1 }^2 t\gamma _0 M\left( \frac{\beta _1^2 \Omega _{r_1 d} }{\beta _2^2 \Omega _{r_2 d} },-\Omega _{sr_1 } t\gamma _0 ,\frac{1}{\beta _2^2 \Omega _{r_2 d} }\right) \right. \nonumber \\&\quad \left. +\Omega _{sr_2 }^2 t\gamma _0 M\left( \frac{\beta _1^2 \Omega _{r_1 d} }{\beta _1^2 \Omega _{r_2 d} },-\Omega _{sr_2 } t\gamma _0 ,\frac{1}{\beta _2^2 \Omega _{r_2 d} }\right) \right] , \end{aligned}$$

(39)

where $M(x,y,z)$ is a function given in [19].

By using Theorem 1, (39) can be rewritten as:

$$\begin{aligned} M_{\gamma _{a1} } (t)&= \frac{1}{1-t\gamma _0 \frac{\Omega _{sr_1 } \Omega _{sr_2 } }{\Omega _{sr_2 } +\Omega _{sr_2 } }}+\frac{t\gamma _0 \frac{\Omega _{sr_1 } \Omega _{sr_2 } }{\Omega _{sr_2 } +\Omega _{sr_2 } }}{\left( 1-t\gamma _0 \frac{\Omega _{sr_1 } \Omega _{sr_2 } }{\Omega _{sr_2 } +\Omega _{sr_2 } }\right) ^{2}\left( \beta _1^2 \Omega _{r_1 d} -\beta _2^2 \Omega _{r_2 d} \right) } \nonumber \\&\quad \times \left[ e^{\frac{2}{\beta _1^2 \Omega _{r_1 d} \left( 1-t\gamma _0 \frac{\Omega _{sr_1 } \Omega _{sr_2 } }{\Omega _{sr_2 } +\Omega _{sr_2 } }\right) }}\Psi (1,1;u_1 (t))\right. \nonumber \\&\quad \quad \left. +e^{\frac{2}{\beta _2^2 \Omega _{r_2 d} \left( 1-t\gamma _0 \frac{\Omega _{sr_1 } \Omega _{sr_2 } }{\Omega _{sr_2 } +\Omega _{sr_2 } }\right) }}\Psi (1,1;u_2 (t))\right] , \end{aligned}$$

(40)

where $u_1 (t)=\frac{-1}{\beta _1^2 \left( 1-t\gamma _0 \frac{1}{\Omega _{r_2 d} }\frac{\Omega _{sr_1 } \Omega _{sr_2 } }{\Omega _{sr_2 } +\Omega _{sr_2 } }\right) }\frac{1}{\Omega _{r_1 d} }$ and $u_2 (t)=\frac{-1}{\beta _2^2 \left( 1-t\gamma _0 \frac{1}{\Omega _{r_2 d} }\frac{\Omega _{sr_1 } \Omega _{sr_2 } }{\Omega _{sr_2 } +\Omega _{sr_2 } }\right) }\frac{1}{\Omega _{r_2 d} }$.