Unified Performance Analysis of Multi-antenna Techniques in Dual Hop Networks over Nakagami-m Fading Channels

Abstract

In this paper, various multi-antenna dual hop relay networks are studied over flat Nakagami-m fading channels. The studied scenarios include transmit diversity techniques such as orthogonal space-time block coding, maximal-ratio transmission, and transmit antenna selection techniques and receive diversity techniques such as selection combining and maximal-ratio combining, and hybrid versions of these transmit and receive diversity techniques. In order to study and analyze the performance of the multi-antenna relay networks, closed-form and exact expressions for the moment generating function, the signal-to-noise ratio moments, the outage probability, and the symbol error probability are derived. Besides, to provide further insight about the performance of the studied scenarios asymptotic expressions revealing the array and diversity gain are also derived. Theoretical results are approved by the Monte Carlo simulations.

Appendices

Appendix 1

The CDF of end-to-end SNR for the CSI-based gain is obtained as follows:

$$\begin{aligned} F_{\gamma _{\mathrm {csi}}}\left( x\right)&=\Pr \left( \frac{\gamma _1\gamma _2}{\gamma _1+\gamma _2}\le x\right) , \nonumber \\&= \int _{0}^{\infty } \int _{0}^{\frac{zx}{z-x}} f_{\gamma _1,\gamma _2}\left( y,z\right) dy dz, \nonumber \\&= \int _{0}^{\infty } \left[ \int _{0}^{\frac{zx}{z-x}} f_{\gamma _1}\left( y\right) dy\right] f_{\gamma _2}\left( z\right) dz, \nonumber \\&=\int _{0}^{x} f_{\gamma _2}\left( z\right) dz + \int _{x}^{\infty } F_{\gamma _1}\left( \frac{zx}{z-x}\right) f_{\gamma _2}\left( z\right) dz, \nonumber \\&= 1 - \int _{x}^{\infty } \bar{F}_{\gamma _1}\left( \frac{zx}{z-x}\right) f_{\gamma _2}\left( z\right) dz, \nonumber \\&= 1 - \int _{0}^{\infty } \bar{F}_{\gamma _1}\left( x+\frac{x^2}{w}\right) f_{\gamma _2}\left( x+w\right) dw, \end{aligned}$$

(23)

where $\bar{F}_{\gamma _1}\left( \cdot \right)$ and $f_{\gamma _2}\left( \cdot \right)$ are the complementary CDF of $\gamma _1$ and the PDF of $\gamma _2$, respectively. In order to derive the CDF of the end-to-end SNR for the CSI-based gain, $\bar{F}_{\gamma _1}\left( \cdot \right)$ and $f_{\gamma _2}\left( \cdot \right)$ have to be derived. We expressed $\gamma _i,\, i=1,2$ in (4). Let $Y_{ij}=\sum \nolimits _{k=1}^{B_i} X_{ijk}$ be defined, the CDF of $Y_{ij}$ can be expressed as $F_{Y_{ij}}(x)=\psi \left( mB_i,x\lambda _i\right) /{\Gamma }\left( mB_i\right)$, where $\lambda _i=\frac{m_i}{\Xi _i\bar{\gamma }_i}$. Since $\gamma _i=\max \nolimits _{1\le {j}\le {A_i}}\left\{ Y_{ij}\right\}$, CDF of $\gamma _i$ can be derived by using the highest order statistics [40], as $F_{\gamma _i}\left( x\right) =\left[ F_{Y_{ij}}\left( x\right) \right] ^{A_i}$. Then by substituting $F_{Y_{ij}}\left( x\right)$, CDF of $\gamma _{i}$ can be obtained as

$$\begin{aligned} F_{\gamma _i}\left( x\right) =\left[ \frac{\psi \left( m_iB_i,x\lambda _i\right) }{{\Gamma }\left( m_iB_i\right) }\right] ^{A_i}, \end{aligned}$$

(24)

and the PDF of $\gamma _i$ is obtained as

$$\begin{aligned} f_{\gamma _i}\left( x\right) =\frac{A_i x^{m_iB_i-1}}{{\Gamma }\left( m_iB_i\right) } \lambda _i^{m_iB_i} e^{-x\lambda _i} \left[ \frac{\psi \left( m_iB_i,x\lambda _i\right) }{{\Gamma }\left( m_iB_i\right) }\right] ^{A_i-1}. \end{aligned}$$

(25)

Series expansion of the lower incomplete gamma function given in [38, eq. (8.352.6)] is substituted, then by using binomial expansion the following expression can be obtained

$$\begin{aligned} F_{\gamma _i}\left( x\right) =1+\sum _{\ell _i=1}^{A_i} \sum _{n_i=0}^{\ell _i\left( m_iB_i-1\right) } C_{A_i}^{\ell _i} (-1)^{\ell _i} \mu _{n_i} \left( \ell _i\right) \left( x\lambda _i\right) ^{n_i} e^{-\ell _ix\lambda _i}, \end{aligned}$$

(26)

where $C_j^k=j!/[(j-k)!k!]$ denotes the coefficients of binomial expansion and $\mu _{\ell }(k)$ are the coefficients of multinomial expansion given in [38, eq. (0.314)], and defined as $\mu _{\ell }(k)=\frac{1}{a_0\ell }\sum _{\tau =1}^{\ell }(k\tau -\ell +\tau )\,a_{\tau }\, \mu _{\ell -\tau }(k),\, \ell \ge 1$, where $a_{\tau }={1}/{\tau !}$ and $\mu _0(k)=1$. By using the same way to obtain the CDF of $\gamma _i$, the PDF of $\gamma _i$ is obtained as

$$\begin{aligned} f_{\gamma _i}\left( x\right)&=\frac{A_i}{\Gamma \left( m_iB_i\right) } \sum _{\ell _i=0}^{A_i-1} \sum _{n_i=0}^{\ell _i\left( m_iB_i-1\right) } {C_{A_i-1}^{\ell _i} \lambda _i^{m_iB_i+n_i} \mu _{n_i}\left( \ell _i\right) (-1)^{\ell _i}} \nonumber \\&\quad\times\, x^{n_i+m_iB_i-1}e^{-x\left( \ell _i+1\right) \lambda _i}. \end{aligned}$$

(27)

By using the binomial expansion the first integrand in (23) is obtained as

$$\begin{aligned} \bar{F}_{\gamma _1}\left( x+\frac{x^2}{w}\right)&= \sum _{\ell _1=1}^{A_1} \sum _{n_1=0}^{\ell _1\left( m_1B_1-1\right) } \sum _{p_1=0}^{n_1} C_{A_1}^{\ell _1} C_{n_1}^{p_1} \lambda _1^{n_1} \mu _{n_1}\left( \ell _1\right) (-1)^{\ell _1+1} \nonumber \\&\quad \times\, w^{-p_1} x^{n_1+p_1} e^{-\ell _1\lambda _1\left( x+\frac{x^2}{w}\right) }. \end{aligned}$$

(28)

Similarly, by using the binomial expansion the second integrand in (23) is derived as

$$\begin{aligned} f_{\gamma _2}(x+w)&= \sum _{\ell _2=0}^{A_2-1} \sum _{n_2=0}^{\ell _2\left( m_2B_2-1\right) } \sum _{p_2=0}^{n_2+m_2B_2-1} \frac{A_2 C_{A_2}^{\ell _2} C_{n_2+m_2B_2-1}^{p_2} \lambda _2^{m_2B_2+n_2} \mu _{n_2}\left( \ell _2\right) }{\Gamma \left( m_2B_2\right) (-1)^{\ell _2} e^{(x+w)\left( \ell _2+1\right) \lambda _2}} \nonumber \\&\quad \times\, x^{n_2+m_2B_2-p_2-1} w^{p_2}. \end{aligned}$$

(29)

When (28) and (29) are substituted into (23),

$$\begin{aligned} F_{\gamma _{\mathrm {csi}}}(x)&= 1- \sum _{\varvec{\ell },\varvec{n},\varvec{p}} \frac{A_2C_{A_1}^{\ell _1}C_{n_1}^{p_1}C_{A_2}^{\ell _2}C_{n_2+m_2B_2-1}^{p_2}}{\Gamma \left( m_2B_2\right) (-1)^{\ell _1+\ell _2+1}} \mu _{n_1}\left( \ell _1\right) \mu _{n_2}\left( \ell _2\right) \lambda _1^{n_1} \lambda _2^{m_2B_2+n_2} \nonumber \\&\quad \times\, \frac{x^{n_1+n_2+m_2B_2+p_1-p_2-1}}{\exp \left[ x\left( \ell _1\lambda _1+\lambda _1\left( \ell _2+1\right) \right) \right] } \int _{0}^{\infty } w^{p_2-p_1} e^{-w\lambda _2\left( \ell _2+1\right) } e^{-\ell _1\lambda _1{x^2}/{w}} dw \end{aligned}$$

(30)

is obtained. The integral in (30) is taken with the help of [38, eq. (3.471.9)], then the CDF of $\gamma _{\mathrm {csi}}$ is derived as given in (7).

Appendix 2

The CDF of the end-to-end SNR for the fixed relay gain can be obtained as follows:

$$\begin{aligned} F_{\gamma _{\mathrm {fix}}}\left( x\right)&=\Pr \left( \frac{\gamma _1\gamma _2}{\gamma _2+Z}\le x\right) , \nonumber \\&= \int _{0}^{\infty } \int _{0}^{x+\frac{xZ}{y}} f_{\gamma _1,\gamma _2}\left( w,y\right) dw dy, \nonumber \\&=\int _{0}^{\infty } F_{\gamma _1}\left( x+\frac{xZ}{y}\right) f_{\gamma _2}\left( y\right) dy. \end{aligned}$$

(31)

The first integrand can be obtained by using binomial expansion as

$$\begin{aligned} F_{\gamma _1}\left( x+\frac{xZ}{y}\right)&= 1+ \sum _{\ell _1=1}^{A_1} \sum _{n_1=0}^{\ell _1\left( m_1B_1-1\right) } \sum _{p_1=0}^{n_1} C_{A_1}^{\ell _1} C_{n_1}^{p_1} Z^{p_1} \mu _{n_1}\left( \ell _1\right) (-1)^{\ell _1} \nonumber \\&\quad \times\, w^{-p_1} \left( x\lambda _1\right) ^{n_1} e^{-\ell _1\lambda _1\left( x+\frac{xZ}{y}\right) }. \end{aligned}$$

(32)

When (27) and (32) are substituted into (31)

$$\begin{aligned} F_{\gamma _{\mathrm {fix}}}(x)&=1+\frac{A_2}{\Gamma \left( m_2B_2\right) } \sum _{\varvec{\ell },\varvec{n},p_1} C_{n_1}^{p_1} C_{A_1}^{\ell _1} C_{A_2}^{\ell _2} \mu _{n_1}\left( \ell _1\right) \mu _{n_2}\left( \ell _2\right) (-1)^{\ell _1+\ell _2} Z^{p_1} e^{-x\ell _1\lambda _1} \nonumber \\&\quad\times\, \left( x \lambda _1\right) ^{n_1} \left( \lambda _2\right) ^{m_2B_2+n_2} \int _{0}^{\infty } y^{n_2+m_2B_2-p_1-1} e^{-y\lambda _2\left( \ell _2+1\right) } e^{-\ell _1 \lambda _1 \left( \frac{xZ}{y}\right) } dy \end{aligned}$$

(33)

is obtained. The integral in (33) is taken with the help of [38, eq. (3.471.9)], then the CDF of $\gamma _{\mathrm {fix}}$ is derived as given in (8).

Appendix 3

The received SNR for the CSI-based gain is expressed with a tight approximation as $\gamma _{\mathrm {csi}} \le \gamma _{\mathrm {acsi}}=\min \left\{ \gamma _1, \gamma _2\right\}$ [41]. CDF of $\gamma _{\mathrm {acsi}}$ can be expressed as $F_{\gamma _{\mathrm {acsi}}}\left( x\right) =1-\bar{F}_{\gamma _1}\left( x\right) \bar{F}_{\gamma _2}\left( x\right)$, and by taking the first derivative of $F_{\gamma _{\mathrm {acsi}}}\left( x\right)$ PDF of $\gamma _{\mathrm {acsi}}$ is derived as

$$\begin{aligned} f_{\gamma _{\mathrm {acsi}}}(x)= f_{\gamma _1}(x)+ f_{\gamma _2}(x)- f_{\gamma _1}(x)F_{\gamma _2}(x)- f_{\gamma _2}(x) F_{\gamma _1}(x). \end{aligned}$$

(34)

By the asymptotic characteristic of the lower incomplete gamma function, which is $\psi \left( v,x\rightarrow \infty \right) \approx {x^v/v}$ given in [42, eq. (45:9:1)], $F_{\gamma _i}\left( x\right)$ and $f_{\gamma _i}\left( x\right) ,\, i=1,2$ are expressed, respectively as

$$\begin{aligned}&F_{\gamma _i}(x)\approx \beta _i\left( \frac{x}{\bar{\gamma }}\right) ^{t_i}, \end{aligned}$$

(35)

$$\begin{aligned}&\quad f_{\gamma _i}(x)\approx t_i \beta _i\left( \frac{x}{\bar{\gamma }}\right) ^{t_i-1}. \end{aligned}$$

(36)

In (35) and (36), $t_i=m_iA_iB_i,\, \beta _i=\frac{1}{\left[ \Gamma \left( m_iB_i+1\right) \right] ^{A_i}}\left( \frac{m_i}{\Xi _i}\right) ^{t_i},\, i=1,2$ and $\bar{\gamma }=\bar{\gamma }_1=\kappa \bar{\gamma }_2$, where $\kappa$ is a positive constant, are defined. When (35) and (36) are substituted into (34) the following expression is obtained

$$\begin{aligned} f_{\gamma _{\mathrm {acsi}}}(x)&\approx t_1\beta _1\left( \frac{x}{\bar{\gamma }}\right) ^{t_1-1} + t_2\beta _2\left( \frac{x\kappa }{\bar{\gamma }}\right) ^{t_2-1} \nonumber \\&-t_1\beta _1\beta _2\kappa ^{t_2} \left( \frac{x}{\bar{\gamma }}\right) ^{t_1+t_2-1} - t_2\beta _1\beta _2\kappa ^{t_2-1}\left( \frac{x}{\bar{\gamma }}\right) ^{t_1+t_2-1}. \end{aligned}$$

(37)

The first non-zero derivative of (37) is obtained for $(t-1)$-th derivative, where $t=\min \left\{ t_1,t_2\right\}$. By this way asymptotic PDF of $\gamma _{\mathrm {csi}}$ is derived as

$$\begin{aligned} f_{\gamma _{\mathrm {csi}}}(x)\approx \alpha _{\mathrm {csi}}\left( \frac{x}{\bar{\gamma }}\right) ^{t-1}, \end{aligned}$$

(38)

where $\alpha _{\mathrm {csi}}$ is given in (19).

Similarly, for the fixed relay gain the end-to-end SNR is expressed with a tight approximation as $\gamma _{\mathrm {fix}} \le \gamma _{\mathrm {afix}}=\min \left\{ \gamma _1, \gamma _1\gamma _2/Z\right\}$ [43]. CDF of $\gamma _{\mathrm {afix}}$ can be expressed as

$$\begin{aligned} F_{\gamma _{\mathrm {afix}}}(x)=F_{\gamma _1}(x)+\Pr \left( x<\gamma _1<\frac{xZ}{\gamma _2}\right) . \end{aligned}$$

(39)

The asymptotic expression for $F_{\gamma _1}$ is as given in (35). On the other hand the second term can be derived as follows:

$$\begin{aligned} \Pr \left( x<\gamma _1<\frac{xZ}{\gamma _2}\right)&=\int _x^{\infty }f_{\gamma _1}(u)\int _0^{\frac{xZ}{u}}f_{\gamma _2}(v)dvdu \nonumber \\&=\int _x^{\infty }f_{\gamma _1}(u)F_{\gamma _2}\left( \frac{xZ}{u}\right) du. \end{aligned}$$

(40)

$f_{\gamma _1}(x)=\frac{t_1\beta _1x^{t_1-1}}{\left( \bar{\gamma }_1\right) ^{t_1}}e^{-x\lambda _1}$ and (35) are substituted for the first and the second integrand, respectively we obtain

$$\begin{aligned} \Pr \left( x<\gamma _1<\frac{xZ}{\gamma _2}\right)&=\frac{t_1\beta _1\beta _2\kappa ^{t_2}Z^{t_2}}{\bar{\gamma }^{t_1+t_2}}\int _x^{\infty } u^{t_1-t_2-1} e^{-u\lambda _1}du \nonumber \\&=\frac{t_1\beta _1\beta _2\kappa ^{t_2}Z^{t_2}e^{-x\lambda _1}x^{t_1-t_2-1}}{\bar{\gamma }^{t_1+t_2}}\int _0^{\infty } \left( 1+\frac{y}{x}\right) ^{t_1-t_2-1} e^{-y\lambda _1}dy \end{aligned}$$

(41)

With the help of [43, eq. (11)] and by utilizing $Z=1+\bar{\gamma }_1\rightarrow \bar{\gamma }_1$ and $e^{-x\lambda _1}\rightarrow 1$ for the high SNR regime (i.e., $\bar{\gamma }\rightarrow \infty$) into (41) we obtain

$$\begin{aligned} \Pr \left( x<\gamma _1<\frac{xZ}{\gamma _2}\right) = {\left\{ \begin{array}{ll} {\frac{t_1\beta _1\beta _2}{\left( t_2-t_1\right) }\left( \frac{x}{\bar{\gamma }}\right) ^{t_1}}, & t_1<t_2, \\ {t_1\beta _1\beta _2\ln \left( \frac{1}{\lambda _1}\right) \left( \frac{x}{\bar{\gamma }}\right) ^t}, & t_1=t_2=t, \\ {\frac{t_1\beta _1\beta _2\Gamma \left( t_1-t_2\right) }{\left( m_1/\Xi _1\right) ^{t_1-t_2}}\left( \frac{x}{\bar{\gamma }}\right) ^{t_2}}, & t_1>t_2. \end{array}\right. } \end{aligned}$$

(42)

After (35) and (42) are substituted into (39), by having the first derivative the asymptotic PDF of $\gamma _{\mathrm {fix}}$ is obtained as $f_{\gamma _{\mathrm {fix}}}(x)\approx \alpha _{\mathrm {fix}}\left( x/\bar{\gamma }\right) ^{t-1}$, where $\alpha _{\mathrm {fix}}$ is given in (20).