Analysis of NOMA based UAV assisted short-packet communication system and blocklength minimization for IoT applications

Abstract

Recently, academia and industry have shown keen interest in achieving ultra-reliable and low latency communication (URLLC) through short-packet communication to meet the strict demands concerning high reliability and latency for 5G and beyond applications. Un-manned aerial vehicle (UAV) has caught attention recently because of its cost, air time, and mobility, whereas, the non-orthogonal multiple access (NOMA) technique has proven effective in dense user network. Hence, a UAV based system with NOMA has been studied in this paper for remote coverage. In this system, devices communicate to the base station (BS) through a UAV relay where direct link from devices to BS is absent. As UAV has direct line of sight (LOS) to the remote devices, hence, all the analysis is done over a Rician fading channel. In this work, a closed-form expression of average block error rate (BLER) has been formulated for the given system model, which is used as performance metric to analyse the system performance. Moreover, exact BLER expression facilitates in optimization with partial channel state information (CSI) only, which reduces the latency and complexity. Furthermore, we derive an asymptotic expression for BLER in high signal to noise (SNR) regime. Also, a blocklength minimization problem is formulated and optimized with reliability constraints. Simulation results are presented to verify analytical work, as well as comparison of the results with orthogonal multiple access scheme are also shown.

Appendices

Appendix 1: Derivation of instantaneous BLER

Let $\theta _i^O$ denote the event that block destined to device i, where $i\epsilon \{1,2\}$ is decoded in error at location O, where $O\epsilon \{uav,D1,D2\}$. In addition, ${\overline{\theta }}_i^O$ denote the complement of $\theta _i^O$, i.e. ${\overline{\theta }}_i^O=1-\theta _i^O$. The error probability is given by Eq. (3) as

$$\begin{aligned} P(\theta _i^O)={\epsilon r}_i^O= Q\left( ln2\sqrt{\frac{m_p}{v}}\left( \log _2{\left( 1+\gamma _i^O\right) }-\frac{N_i}{m_p}\right) \right) \end{aligned}$$

(34)

where $N_i$ denotes the number of message bits destined to device Di, and $m_p$ denotes the pth phase blocklength, where $p\epsilon \{1,2\}$. Instantaneous probability of error in decoding of message $x_1$ at the uav is given by

$$\begin{aligned} P(\theta _1^{uav})={\epsilon r}_1^{uav}= Q\left( ln2\sqrt{\frac{m_1}{v}}\left( \log _2{\left( 1+\gamma _1^{uav}\right) }-\frac{N_1}{m_1}\right) \right) \end{aligned}$$

(35)

where SINR $\gamma _1^{uav}$ is given by Eq. (7a). Further, UAV performs SIC and decodes second message $x_2$. Since, decoding of message $x_2$ correctly depends on wether the message $x_1$ was decoded correctly or not, hence, the total instantaneous probability of decoding message $x_2$ at UAV in error, is given by

$$\begin{aligned} P\left( \theta _2^{uav}\right) =P\left( \theta _2^{uav}\mid \theta _1^{uav}\right) \ {P(\theta }_1^{uav})+P\left( \theta _2^{uav}\mid {\overline{\theta }}_1^{uav}\ \right) P({\overline{\theta }}_1^{uav}) \end{aligned}$$

(36)

Due to SIC, it is impossible to decode $x_2$ correctly when $x_1$ was decoded in error. Therefore, $P\left( \theta _2^{uav}\mid \theta _1^{uav}\right)$ is assumed to be 1. So from Eq. (34), the Eq. (36) can be expressed as

$$\begin{aligned} \epsilon _2^{uav}\approx 1\times {\epsilon r}_1^{uav}+ {\epsilon r}_2^{uav} (1-{\epsilon r}_1^{uav}) \end{aligned}$$

(37)

since under URLLC scenario errors are very small to the order of $10^{-5}$, hence term ${\epsilon r}_1^{uav}{\epsilon r}_2^{uav}$ can be neglected without compromising the Eq. (37) and total instantaneous probability of $x_2$ at UAV ($\epsilon _2^{uav}$) can be expressed as:

$$\begin{aligned} \epsilon _2^{uav}\approx \epsilon r_1^{uav}+\epsilon r_2^{uav} \end{aligned}$$

(38)

By using Eq. (3), $\epsilon _1^{uav}$ and $\epsilon _2^{uav}$can be expressed as

$$\begin{aligned} \epsilon_{1}^{{uav}} = & Q\left( {F\left( {\gamma _{1}^{{uav}} ,m_{1} \;,N_{1} \;} \right)} \right) \\ = & Q\left[ {ln2\sqrt {\frac{{m_{1} }}{{v_{1}^{{uav}} }}} }{\left( {\left( {\log _{2} 1 + \gamma _{1}^{{uav}} } \right) - \frac{{N_{1} }}{{m_{1} }}} \right)} \right] \\ \end{aligned}$$

(39)

and,

$$\begin{aligned} \epsilon _2^{uav}\approx \ Q\left( F\left( \gamma _1^{uav},m_1,N_1\ \right) \right) +Q\left( F\left( \gamma _2^{uav},m_1,N_2\ \right) \right) \end{aligned}$$

i.e.

$$ \begin{aligned}\epsilon _2^{uav}&\approx Q\left[ ln2\sqrt{\frac{m_1}{v_1^{uav}}}\left( \log _2({1+\gamma _1^{uav}})-\frac{N_1}{m_1}\right) \right] \\&\quad +\ Q\left[ ln2\sqrt{\frac{m_1}{v_2^{uav}}} \left( \log _2({1+\gamma _2^{uav}}) -\frac{N_2}{m_1}\right) \right] \end{aligned}$$

(40)

Appendix 2: Proof of Eq. (24)

The proof of Eq. (24) is as follow:

By Eq. (23)

$$\begin{aligned} \ F_{\gamma _1^{uav}}\left( z\right) =e^{-K_{b}}\ \ \sum _{j=0}^{\infty }{\frac{1}{j!}\left( {K_{b}}\right) ^j\frac{\eth\left( 1+j,\frac{\left( B^{uav}\right) z}{c_1-c_2z}\right) }{\Gamma \left( 1+j\right) }} \end{aligned}$$

(41)

and by Eq. (14)

$$\begin{aligned} {E}{\left( \epsilon ^{D1}\right) }={E}\left( \ \epsilon r_1^{uav}\right) +{E}\left( \epsilon r_1^{D1}\right) \end{aligned}$$

(42)

By Eqs. (41) and (18), (42) can be expressed as:

$$ \begin{aligned}{E}\left( \ \epsilon ^{D1}\right)& = {v_1^{uav}\int _{\phi _1^{uav}}^{\Delta _1^{uav}}{e^{-K_{b}}\ \ \sum _{j=0}^{\infty }{\frac{1}{j!}\left( {K_{b}}\right) ^j\frac{\eth\left( 1+j,\frac{\left( B^{uav}\right) z}{c_1-c_2z}\right) }{\Gamma \left( 1+j\right) }}}dz}\\&\quad + {v_1^{D1}\int _{\phi _1^{D1}}^{\Delta _1^{D1}}{e^{-K_{D1}}\ \ \sum _{j=0}^{\infty }{\frac{1}{j!}\left( {K_{b}}\right) ^j\frac{\eth\left( 1+j,\frac{\left( B^{D1}\right) z}{g_1-g_2z}\right) }{\Gamma \left( 1+j\right) }}}}\\ \end{aligned}$$

(43)

Both terms in Eq. (43) can be separately simplified. First term of the above expression indicating decoding error of $x_1$ at UAV. Hence, ${E}\left( \ \epsilon r_1^{uav}\right)$ can be evaluated as follows:

$$\begin{aligned}&{E}\left( \ \epsilon r_1^{uav}\right) = {v_1^{uav}\int _{\phi _1^{uav}}^{\Delta _1^{uav}}{e^{-K_{b}}\ \ \sum _{j=0}^{\infty }{\frac{1}{j!}\left( {K_{b}}\right) ^j\frac{\eth\left( 1+j,\frac{\left( B^{uav}\right) z}{c_1-c_2z}\right) }{\Gamma \left( 1+j\right) }}}dz} \end{aligned}$$

(44)

$$\begin{aligned}&\quad = {v_1^{uav}{e^{-K_{b}}\ \ \sum _{j=0}^{\infty }{\frac{1}{j!}\left( {K_{b}}\right) ^j\frac{\int _{\phi _1^{uav}}^{\Delta _1^{uav}}{\eth\left( 1+j,\frac{\left( B^{uav}\right) z}{c_1-c_2z}\right) dz}}{\Gamma \left( 1+j\right) }}}} \end{aligned}$$

(45)

Let $I_1$ denote the integral in Eq. (45). Then

$$\begin{aligned} I_1= {\int _{\phi _1^{uav}}^{\Delta _1^{uav}}{\eth\left( 1+j,\frac{\left( B^{uav}\right) z}{c_1-c_2z}\right) dz}} \end{aligned}$$

(46)

performing the following substitution $\ t=\frac{\left( B^{uav}\right) z}{c_1-c_2z}$, integral’s new limits are evaluated as:

$$\begin{aligned}l_1=\frac{B^{uav}\phi _1^{uav}}{c_1-c_2\phi _1^{uav}}\ \ \ \ \ and\ \ \ \ l_2=\frac{B^{uav}\Delta _1^{uav}}{c_1-c_2\Delta _1^{uav}}\end{aligned}$$

where $B^{uav}=\frac{K_{b}+1}{\beta _ub}$.Therefore, the Eq. (46) is written as:

$$\begin{aligned} I_1=B^{uav} c_1 {\int _{l_1}^{l_2}{\frac{\eth\left( 1+j,t\right) }{\left( B^{uav}+c_2t\right) ^2}\ dt}} \end{aligned}$$

(47)

Using integral by parts Eq. (47) can be written as

$$\begin{aligned} I_1=\frac{{B^{uav}c}_1}{c_2}\left\{ \frac{\eth\left( 1+j,l_1\right) }{B^{uav}+c_2l_1}-\frac{\eth\left( 1+j,l_2\right) }{B^{uav}+c_2l_2}+\int _{l_1}^{l_2}{\frac{\left( t\right) ^je^{-t}}{B^{uav}+c_2t}\ dt\ }\right\} \end{aligned}$$

(48)

Performing the following substitution and applying binomial expansion, $B_{uav}+c_2t=x$,

$$\begin{aligned}I_1&=\frac{{B^{uav}c}_1}{c_2}\left\{ \frac{\eth\left( 1+j,l_1\right) }{B^{uav}+c_2l_1}-\frac{\eth\left( 1+j,l_2\right) }{B^{uav}+c_2l_2}+\frac{e^\frac{B^{uav}}{c_2}}{c_2^{1+j}}\right. \nonumber \\&\quad \left[ \sum _{m=0}^{j}\left( {\begin{array}{c}j\\ m\end{array}}\right) \left( -B^{uav}\right) ^{j-m}\ c_2^m\right. \nonumber \\&\left. \left. \qquad \left( \int _{B^{uav}+c_2l_1}^{B^{uav}+c_2l_2}{e^{-\frac{x}{c_2}}{\ x}^{m-1}}dx\right) \right] \right\} \end{aligned}$$

(49)

by using equation (3.381) of [61] Eq. (49) can be evaluated as:

$$\begin{aligned}I_1&=\frac{{B^{uav}c}_1}{c_2}\left\{ \frac{\eth\left( 1+j,l_1\right) }{B^{uav}+c_2l_1}-\frac{\eth\left( 1+j,l_2\right) }{B^{uav}+c_2l_2}+\frac{e^\frac{B^{uav}}{c_2}}{c_2^{1+j}}\right. \\&\quad \left[ \sum _{m=0}^{j}\left( {\begin{array}{c}j\\ m\end{array}}\right) \left( -B^{uav}\right) ^{j-m} c_2^m \left( \Gamma \left( m,l_1+\frac{B^{uav}}{c_2}\right) \right. \right. \nonumber \\&\left. \left. \left. \quad -\Gamma \left( m,l_2+\frac{B^{uav}}{c_2}\right) \right) \right] \right\} \end{aligned}$$

(50)

Here $\Gamma \left( c,d\right)$ denotes the upper incomplete Gamma function, substituting $l_1$ and $l_2$ in (49), and writing equation in terms of lower gamma function yields

$$ \begin{aligned}I_1&=B^{uav}c_1\left[ \left( {\frac{1}{c_2B^{uav}}}-{\frac{\phi _1^{uav}}{c_1B^{uav}}}\right) \eth\left( 1+j,\frac{B^{uav}\phi _1^{uav}}{c_1-c_2\phi _1^{uav}}\right) \right. \\&\quad \left. -\left( {\frac{1}{c_2B^{uav}}} -{\frac{\Delta _1^{uav}}{c_1B^{uav}}}\right) \right. \\&\quad \left. \eth\left( 1+j,\frac{B^{uav}\Delta _1^{uav}}{c_1-c_2\Delta _1^{uav}}\right) +{\frac{e^\frac{B^{uav}}{c_2}}{c_2^{j+2}}}\sum _{m=0}^{j} \left( ^j_m\right) {\left( -B^{uav}\right) }^{j-m}{c_2^m}\right. \\&\quad \left. \left( \eth\left( m,\frac{B^{uav}c_1}{c_2(c_1-{c_2}{\Delta _1^{uav}})}\right) -\eth\left( m,\frac{B^{uav}c_1}{c_2(c_1-{c_2}{\phi _1^{uav}})}\right) \right) \right] \end{aligned}$$

(51)

Substituting $I_1$ in (45) and following the same procedure for the derivation of second term in Eq. (43) completes the proof.

Appendix 3: Proof of high SNR approximation given by Eq. (26)

The derivation of the expression is as follows: by Eq. (23), the CDF is given as

$$\begin{aligned} F_{\gamma _1^{uav}}\left( z\right) =e^{-K_{b}}\ \ \sum _{j=0}^{\infty }{\frac{1}{j!}\left( K_{b}\right) ^j\frac{\eth\left( 1+j,\frac{\left( B^{uav}\right) z}{c_1-c_2z}\right) }{\mathrm {\Gamma }\left( 1+j\right) }} \end{aligned}$$

(52)

using equation (8.354.1) of [61], series expansion of incomplete lower gamma function can be written as

$$\begin{aligned} \eth\left( a,t\right) =\sum _{n}^{\infty }\frac{\left( -1\right) ^nt^{a+n}}{n!\left( a+n\right) } \end{aligned}$$

(53)

If $t\ll 1$, only the first term dominates

$$\begin{aligned} \eth\left( a,t\right) \approx \frac{t^a}{a} \end{aligned}$$

(54)

Hence, for D1 $z<\Delta _1^{D1}$,

$$\begin{aligned}&{E}\left( \epsilon ^{D1}\right) =v_1^{uav}\int _{\phi _1^{uav}}^{\Delta _1^{uav}}F_{\gamma _1^{uav}}(z)dz + v_1^{D1}\int _{\phi _1^{D1}}^{\Delta _1^{D1}}F_{\gamma _1^{D1}}(z)dz \end{aligned}$$

(55)

$$\begin{aligned} &=e^{-K_{b}}v_1^{uav}\sum _{j=0}^{\infty }{\frac{1}{j!}\left( K_{b}\right) ^j}\frac{\int _{\phi _1^{uav}}^{\Delta _1^{uav}}\eth\left( 1+j,\frac{\left( B^{uav}\right) z}{c_1-c_2z}\right) }{\Gamma (1+j)}dz \\ &\quad+ e^{-K_{D1}}v_1^{D1}\sum _{j=0}^{\infty }{\frac{1}{j!}\left( K_{D1}\right) ^j}\frac{\int _{\phi _1^{D1}}^{\Delta _1^{D1}}\eth\left( 1+j,\frac{\left( B^{D1}\right) z}{g_1-g_2z}\right) }{\Gamma (1+j)}dz \end{aligned}$$

(56)

From Eq. (53), ${E}\left( \epsilon ^{D1}\right)$ is expressed as:

$$ \begin{aligned} {E}\left( \epsilon ^{D1}\right)& =e^{-K_{b}}v_1^{uav}\sum _{j=0}^{\infty }{\frac{1}{j!}\left( K_{b}\right) ^j}\frac{\int _{\phi _1^{uav}}^{\Delta _1^{uav}}\left( \frac{\left( B^{uav}\right) z}{c_1-c_2z}\right) ^{j+1}}{\Gamma (1+j)(j+1)}dz\\ &\quad+ e^{-K_{D1}}v_1^{D1}\sum _{j=0}^{\infty }{\frac{1}{j!}\left( K_{D1}\right) ^j}\frac{\int _{\phi _1^{D1}}^{\Delta _1^{D1}}\left( \frac{\left( B^{D1}\right) z}{g_1-g_2z}\right) ^{(j+1)}}{\Gamma (1+j)(j+1)}dz \end{aligned}$$

(57)

employing equation (3.194.1) from [61], the proof is complete.