SCIENCE CHINA Information Sciences, Volume 61, Issue 2: 022302(2018) https://doi.org/10.1007/s11432-016-9042-8

## Fractional full duplex cellular network: a stochastic geometry approach

Wenping BI1,2,3, Limin XIAO1,2,3, Xin SU1,2,3, Shidong ZHOU1,2,3,*
• AcceptedFeb 28, 2017
• PublishedJul 28, 2017
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### Abstract

In-band full-duplex (FD) communication has been considered as a promising technology to enhance the spectral efficiency (SE) for the next generation of wireless system. However, the severe interference in FD cellular network may largely degrade the system performance, especially for cell edge users. In this paper, multi-cell cellular network consisting of BSs with FD capability and legacy half duplex (HD) users is studied. In order to relieve the received interference and enhance the system SE, a new resource allocation strategy named fractional FD (FFD) is proposed and its main idea and procedure are described as follows.Firstly, the frequency and time resources are partitioned into FD resource blocks (RBs) and HD RBs based on the network topology. Then all the users are classified into two groups named cell center users (CCUs) and cell edge users (CEUs) based on their channel conditions. At last, in each cell, each FD RB is allocated to a pair of CCUs with one in uplink and the other in downlink transmission direction, while only one CEU in the uplink (or downlink) transmission direction is scheduled over each HD RB. Tractable results of both the coverage probabilities and the ergodic rates of FFD, FD and HD systems are derived using stochastic geometry method. Numerical results show that, FFD can significantly improve the coverage probability of all users especially for CEUs compared with FD cellular system, and higher system SE is obtained compared with FD and HD cellular network. With proper design of the classification criterion and under the simulation settings of this paper, the SE of FFD system outperforms FD and HD system by $1.25$ and $1.38$ times, respectively.

### Acknowledgment

This work was supported by National Basic Research Program of China (Grant No. 2012CB316002), National Natural Science Foundation of China (Grant No. 61631013), National High Technology Research and Development Program of China (863 Program) (Grant No. 2015AA01A706), National Natural Science Foundation of China (Grant No. 61321061), Tsinghua University Initiative Scientific Research Program (Grant No. 2015Z02-3), National S&T Major Project (Grant No. 2014ZX03001011), Key Project of International Science and Technology Innovation Cooperation Between the Government (Grant No. 2016YFE0122900), and Huawei Technologies.

### Supplement

Appendix

Proof of Lemma sect. 3.1

Starting with the definition of laplace transform, we can get \begin{aligned} {L_{{I_{{\rm h}q}}}}\left( s \right) \mathop = \limits^{\left( {\rm a} \right)} {{{\rm E}}_{{\Phi _{{\rm h}q}}}}\left\{ {\prod\limits_{z \in {\Phi _{{\rm h}q}}\backslash c_o} {{{{\rm E}}_{{h_{z,d_o}}}}\exp \left( { - s{h_{z,d_o}}D_{z,d_o}^{ - {\alpha _{\rm bu}}}} \right)} } \right\}\mathop = \limits^{\left( {\rm b} \right)} \exp \left\{ { - 2\pi {\lambda _{\rm s}}\int\nolimits_R^\infty {\left\{ {1 - {{{\rm E}}_{{\rm h}}}\left[ {{{\mathop{\rm e}\nolimits} ^{\left( { - s{h}v^{ - {\alpha _{\rm bu}}}} \right)}}} \right]} \right\}v{\rm d}v} } \right\}, \end{aligned} \tag{23} where $c_o$ and $d_o$ are $b_o$ ($u_o$) and $u_o$ ($b_o$) if $q=d(u)$, respectively. (a) follows the fact that $\Phi~_{{\rm~h}q}$ and $h_{z,d_o}$ are independent from each other. Therefore, the expectation order can be exchanged. (b) follows from the probability generating function of PPP. Carrying on the proof in (23), the followings can be obtained \begin{align} &\exp \left\{ { - 2\pi {\lambda _{\rm s}}\int\nolimits_R^\infty {\left\{ {1 - {{{\rm E}}_{\rm h}}\left[ {\exp \left( { - sh{v^{ - {\alpha _{\rm bu}}}}} \right)} \right]} \right\}v{\rm d}v} } \right\} \\ &\mathop = \limits^{\left( {\rm c} \right)} \exp \left\{ - \frac{2}{{{\alpha _{\rm bu}}}}\pi {\lambda _{\rm s}}{{{\rm E}}_{\rm h}}\left\{ - {{\left( {sh} \right)}^{\frac{2}{{{\alpha _{\rm bu}}}}}}\Gamma \left( { - \frac{2}{{{\alpha _{\rm bu}}}}} \right)- \frac{{{\alpha _{\rm bu}}}}{2}{R^2} + {{\left( {hs} \right)}^{\frac{2}{{{\alpha _{\rm bu}}}}}}\Gamma \left( { - \frac{2}{{{\alpha _{\rm bu}}}},hs{R^{ - {\alpha _{\rm bu}}}}} \right) \right\} \right\}, \tag{24} \end{align} where (c) is proofed in [17]. Based on the Rayleigh fading assumption, $h\sim~\exp(1)$, we can obtain the expectations as \begin{eqnarray}&\displaystyle{{{\rm E}}_{\rm h}}\left\{ {{{ h }^{\frac{2}{{{\alpha _{\rm bu}}}}}}} \right\} = \int\nolimits_0^\infty {{{ x }^{\frac{2}{{{\alpha _{\rm bu}}}}}}{{\rm e}^{ - x}} {\rm d}x} \mathop = \limits^{\left( {\rm d} \right)} \Gamma \left( {\frac{2}{{{\alpha _{\rm bu}}}} + 1} \right) , \tag{25} \\ &\displaystyle{ {\rm E}_{\rm h}}\left\{ {{{ h}^{\frac{2}{{{\alpha _{\rm bu}}}}}}\Gamma \left( { - \frac{2}{{{\alpha _{\rm bu}}}},hs{R^{ - {\alpha _{\rm bu}}}}} \right)} \right\}= \int\nolimits_0^\infty {{x^{\frac{2}{{{\alpha _{\rm bu}}}}}}\Gamma \left( { - \frac{2}{{{\alpha _{\rm bu}}}},xs{R^{ - {\alpha _{\rm bu}}}}} \right){{\rm e}^{ - x}}{\rm d}x}\mathop = \limits^{\left( {\rm e} \right)} J\left( {\frac{2}{{{\alpha _{\rm bu}}}} + 1,1, - \frac{2}{{{\alpha _{\rm bu}}}},s{R^{ - {\alpha _{\rm bu}}}}} \right), \tag{26} \end{eqnarray} where $J(\mu~,\beta~,\nu~,\gamma~)$ is defined in (11) and proofed in [17]. (d) and (e) can be obtained by Eqs. (3.478) and (3.381) 3), respectively. Substitute (25) and (26) into (24) and we can get the result of Lemma sect. 3.1.

Gradshteyn I, Ryzhik I. Table of Integrals, Series, and Products. Manhattan: Academic Press, 2014. 346–370.

Proof of Lemma 3.2

Starting with the definition of laplace transform, we can get \begin{align} &{L_{{I_{{\rm f}q}}}}\left( s \right) \mathop = \limits^{\left({\rm f}\right)}{{{\rm E}}_{D,h,{\Phi _{\rm fd}}}}\left\{ {\prod\limits_{z \in {\Phi _{{\rm fd}\backslash b_o}}} {\exp \left( { - s{P_{\rm fd}}{h_{z,d_o}}{{k}_{z,d_o}}D_{z,d_o}^{ - {\alpha _{\rm d}}}} \right)} } \right\}{{{\rm E}}_{D,h,{\Phi _{\rm fu}}}}\left\{ {\prod\limits_{x \in {\Phi _{\rm fu}}\backslash c_o} {\exp \left( { - s{P_{\rm fu}}{h_{x,d_o}}{{k}_{x,d_o}}D_{x,d_o}^{ - {\alpha _{\rm u}}}} \right)} } \right\} \\ & \mathop = \limits^{\left({\rm g}\right)} \exp \left\{ { - 2\pi {\lambda _{\rm s}}{{\rm E}_{\rm h}}\int\nolimits_{R_{\rm d}}^\infty {\left\{ {1 - \left[ {\exp \left( { - {s_{\rm d}}h{v^{ - {\alpha _{\rm d}}}}} \right)} \right]} \right\}v{\rm d}v} } \right\} \exp \left\{ { - 2\pi {\lambda _{\rm s}}{{\rm E}_{\rm h}}\int\nolimits_{R_{\rm u}} ^\infty {\left\{ {1 - \left[ {\exp \left( { - {s_{\rm u}}h{v^{ - {\alpha _{\rm u}}}}} \right)} \right]} \right\}v{\rm d}v} } \right\}, \tag{27} \end{align} where $c_o$ and $d_o$ are $\emptyset$ ($u_o$) and $u_o$ ($b_o$) if $q=d({\rm~u})$, respectively. (f) follows the fact that $\Phi~_{\rm~fu}$ and $\Phi~_{\rm~fd}$ are assumed to be independent. In Step (g), when we focus on the downlink, the inter-user interference distance can be arbitrarily close, therefore the integration lower bound $R_{\rm~u}$ is $\eta~_{\rm~uu}R$ when taking the interference cancelation strategy into consideration. Based on the association principle, the interfering distance between focused downlink user and neighbor BSs is at least $R$, so $R_{\rm~d}~=~R$. While when considering uplink transmission, the inter-BS interference distance between focused BS and the neighbor BSs can also be arbitrarily close, as a consequence $R_{\rm~d}$ is $\eta~_{\rm~bb}R$ when SIC is applied. In the same way, the distance from the neighbor uplink users to the focused BS is at least $R$, which gives $R_{\rm~u}=~R$. Then following the proof in sect. 1, we can get the result in (13).

Proof of Lemma 3.3

Before solving joint LT of $I_{\rm~d}$ and $\hat~I_{\rm~d}$, some properties of the sum of two exponential distribution random variables are analysed.

We define $G$ as the sum of two exponential distribution random variables as follows: $$G = {s_2}{h_1} + {s_2}{h_2}, \tag{28}$$ where both $h_1$ and $h_2$ follow exponential distribution, i.e, $h_1~~\text{and}~~h_2\sim~{\rm~exp}(1)$. Then, when $s_1=s_2=s$, $G$ is the sum of two exponential random variables with same rate. As presented in [17], $G$ follows Erlang distribution. While, if $s_1~\ne~s_2$, $G$ follows the hypo-exponential distribution.

Based on the PDF of $G$, we can have {{{\rm E}}_G}\left( {{G^\delta }} \right) = \left \{ \begin{aligned} &{s^\delta }\Gamma \left( {\delta + 2} \right), \text{if s_1=s_2=s}, \\ &\frac{1}{{{s_2} - {s_1}}}\Gamma \left( {\delta + 1} \right)\left( {{s_2}^{\delta + 1} - {s_1}^{\delta + 1}} \right), \text{otherwise}, \end{aligned} \right. \tag{29} where (29) can be obtained by Eq. (3.478)$^{3)}$. Furthermore, according to Eq. (3.381)$^{3)}$, we can also obtain $$\begin{array}{l} {{\rm E}_G}\left( {{G^{\mu - 1}}\Gamma \left( {\nu ,\gamma G} \right)} \right) = \left\{ \begin{array}{l} {s}^2J\left( {\mu {\rm{ + }}1,{s}^{ - 1},\nu ,\gamma } \right), {\rm{ if }} {{\rm{s}}_{\rm{1}}}{\rm{ = }}{{\rm{s}}_{\rm{2}}}{\rm{ = s }}, \\ \frac{1}{{{s_2} - {s_1}}}\left( {J\left( {\mu ,\frac{1}{{{s_2}}},\nu ,\gamma } \right) - J\left( {\mu ,\frac{1}{{{s_1}}},\nu ,\gamma } \right)} \right), {\rm{ otherwise}}, \end{array} \right. \end{array} \tag{30}$$ where $J(~{\cdot,~\cdot,~\cdot,~\cdot~})$ is defined in Eq. (11). We also start with the definition of joint LT. \begin{align}&{\mathcal{L}_{{{\hat I}_{\rm d}},{I_{\rm d}}}}\left( {{s_{\rm h}},{s_{\rm f}}} \right) = {{{\rm E}}_{{{\hat I}_{\rm d}},{I_{\rm d}}}}\left( {{{\rm e}^{ - {s_{\rm h}}{{\hat I}_{\rm d}} - {s_{\rm f}}{I_{\rm d}}}}} \right) \\ &\mathop = \limits^{\left(k \right)}{\rm E}\left( \exp \left( - {s_{\rm hd}}\left( {\sum\limits_{z \in {\Phi _{\rm hd}}\backslash b_o} {{h_{z,u_o}}{D_{z,u_o}}^{ - {\alpha _{\rm bu}}}} } \right)- \left( {\sum\limits_{z \in {\Phi _{{\rm fd}\backslash b_o}}} {{s_{\rm fd}}{h_{z,u_o}}{{ {{D_{z,u_o}}} }^{ - {\alpha _{\rm bu}}}}} + \sum\limits_{x \in {\Phi _{\rm fu}}} {{s_{\rm fu}}{h_{x,u_o}}{{ {{D_{x,u_o}}} }^{ - {\alpha _{\rm uu}}}}} } \right) \right) \right) \\ &\mathop = \limits^{\left( l \right)} {\rm E}\left( \exp \left( - \left( {\sum\limits _{z \in {\Phi _{\rm fd}}\backslash b_o} {\left( {{s_{\rm hd}}{h_{z,u_o}} + {s_{\rm fd}}{h_{z,u_o}}} \right){D_{z,u_o}}^{ - {\alpha _{\rm bu}}}} } \right) - {s_{\rm fu}}\left( {\sum\limits_{x \in {\Phi _{\rm fu}}} {{h_{x,u_o}}{{\left( {{D_{x,u_o}}} \right)}^{ - {\alpha _{\rm uu}}}}} } \right) \right) \right) \\ &\mathop = \limits^{\left( m \right)} \exp \left\{ { - 2\pi {\lambda _{\rm s}}{\rm E}\int\nolimits_R^\infty {1 - {{\mathop{\rm e}\nolimits} ^{\left( { - \left( {{{s}_{\rm fd}}{h_{\rm fd}} + {{s}_{\rm hd}}{h_{\rm hd}}} \right){v^{ - {\alpha _{\rm bu}}}}} \right)}}v{\rm d}v} } \right\}\exp \left\{ { - 2\pi {\lambda _{\rm s}}{\rm E}\int\nolimits_{\eta _{\rm uu}R} ^\infty {1 - {{\mathop{\rm e}\nolimits} ^{\left( { - {{s}_{\rm fu}}{h_{\rm fu}}{v^{ - {\alpha _{\rm uu}}}}} \right)}}v{\rm d}v} } \right\} \\ &\mathop = \limits^{\left( n \right)}\exp \left\{ - 2\pi {\lambda _{\rm s}}{{\rm E}_{{G_{\rm u}}}}\left\{ - \frac{1}{{{\alpha _{\rm bu}}}}{G_{\rm u}}^{\frac{2}{{{\alpha _{\rm bu}}}}}\Gamma \left( - \frac{2}{{{\alpha _{\rm bu}}}}\right) - \frac{1}{2}{R^2}+ \frac{1}{{{\alpha _{\rm bu}}}}{G_{\rm u}}^{\frac{2}{{{\alpha _{\rm bu}}}}}\Gamma \left( - \frac{2}{{{\alpha _{\rm bu}}}},\frac{{{G_{\rm u}}}}{{{R^\alpha }}}\right) \right\} \right\}\omega \left( {{\alpha _{\rm uu}},s_{\rm fu}\eta _{\rm uu}R } \right), \tag{31} \end{align} where ${s}_{\rm~fd}={{s_{\rm~f}}}{{P_{\rm~fd}}}{{k}_{\rm~fd}},~~{s}_{\rm~hd}={{s_{\rm~h}}}{{P_{\rm~hd}}}{{k}_{\rm~hd}}$ and${s_{\rm~fu}}={{s_{\rm~f}}}{{P_{\rm~fu}}}{{k}_{\rm~fu}}$ in Step (k). In Step (l), the interference BS sets in HD RBs and FD RBs are the same, therefore, the interfering distances between the focused downlink user and the BSs after classifying the user as CEU is the same as the distances when the focused user is regarded as cell CCU. Hence, we can write them together as the first sum item of Step (l). $G_{\rm~u}={{{s}_{\rm~fd}}{h_{\rm~fd}}~+~{{s}_{\rm~hd}}{h_{\rm~hd}}}$ which is the same as (28) and its properties have already been given by (29) and (30). Then the conclusion of Lemma 3.3 is obtained.

Proof of Theorem sect. 4.1

The proof begins with the definition of the CEU and coverage probability, then we can obtain \begin{align} &F_{{\rm edge},q}\left( {T} \right) = \mathbb{P}\left( {{{{\rm SINR}}} > T|{\rm SINR} < {{\gamma}_q}} \right) \\ &\mathop = \limits^{\left({\rm h}\right)} \frac{{{\rm E}\left( {\mathbb{P}\left( {{{\rm SINR}} > T,{\rm SINR} < {{\gamma}_q}|R,{I_{{\rm f}q}},{I_{{\rm h}q}}} \right)} \right)}}{{{\rm E}\left( {\mathbb{P}\left( {{\rm SINR} < {{\gamma}_q}|R,{I_{{\rm f}q}}} \right)} \right)}}\mathop = \limits^{\left(I\right)} \frac{{{\rm E}\left( {{{\mathop{\rm e}\nolimits} ^{\left( { - {s_{\rm h}}\left( {{\sigma ^2} + {{\hat I}_{\rm e}}} \right)} \right)}}\left( {1 - {{\mathop{\rm e}\nolimits} ^{\left( { - {s_{\rm f}}\left( {{\sigma ^2} + {I_{\rm e}} + {\delta }{I_{\rm SI}}} \right)} \right)}}} \right)} \right)}}{{{\rm E}\left( {\left( {1 - {{\mathop{\rm e}\nolimits} ^{\left( { - {s_{\rm f}}\left( {{\sigma ^2} + {I_{\rm e}} + {\delta }{I_{\rm SI}}} \right)} \right)}}} \right)} \right)}}, \tag{32} \end{align} where SINR and $I_{\rm~e}$ denote the received SINR and interference when the user is regarded as CCU. While SINR$~and~$hat I_rm e$~represent~the~received~SINR~and~interference~after~the~user~is~classified~as~CEU.~SINR~is~defined~as~(\ref{equation2}).~Step~(h)~follows~Bayes~theorem.~Step~(I)~is~obtained~based~on~two~facts.~one~is~the~independency~of~the~small~scale~fading~over~FD~and~HD~RBs,~the~other~fact~is~that~the~small~scale~fading~is~assumed~to~be~Rayleigh,~i.e,~$h∼ rm exp1)$,~so~$mathbbPh<t)=1-exp (-t).~Then~carry~on~the~proof~in~Step~(I)~and~we~have \begin{align} \label{equation_coverage_edge} &F_{{\rm edge},q}\left( {T} \right) = \mathbb{P}\left( {{{\rm SINR}} > T|{\rm SINR} < {{\gamma}_q}} \right)= \frac{{\int\nolimits_{R = 0}^\infty {{\rm E}\left( {{{\mathop{\rm e}\nolimits} ^{\left( { - {s_{\rm h}}{\sigma ^2}} \right)}}\left( {{{\mathop{\rm e}\nolimits} ^{ - {s_{\rm h}}{{\hat I}_{\rm e}}}} - {{\mathop{\rm e}\nolimits} ^{\left( { - {s_{\rm f}} {{{\sigma }^2} + {s_{\rm h}}{\hat I_{\rm e}} + {s_{\rm f}}{I_{\rm e}}} } \right)}}} \right)} \right){f_R}\left( R \right){\rm d}R} }}{{\int\nolimits_{R = 0}^\infty {{\rm E}\left( {1 - {{\mathop{\rm e}\nolimits} ^{\left( { - {s_{\rm f}}\left( {{{\sigma }^2} + {I_{\rm e}}} \right)} \right)}}} \right){f_R}\left( R \right){\rm d}r} }}\nonumber\\ &\mathop = \limits^{\left({\rm j}\right)} \frac{{\int\nolimits_{R = 0}^\infty {\left( {{{\mathop{\rm e}\nolimits} ^{ - {s_{\rm h}}{\sigma ^2}}}\left( {{L_{{I_{{\rm h}q}}}}\left( {{s_{\rm h}}} \right) - {{\mathop{\rm e}\nolimits} ^{ - {s_{\rm f}}\sigma {^2}}}{L_U}\left( {{s_{\rm h}},{s_{\rm f}}} \right)} \right)} \right){f_R}\left( R \right){\rm d}R} }}{{\int\nolimits_{R = 0}^\infty {\left( {1 - {{\mathop{\rm e}\nolimits} ^{ - {s_{\rm f}}\sigma {^2}}}{L_{{I_{{\rm f}q}}}}\left( {{s_{\rm f}}} \right)} \right){f_R}\left( R \right){\rm d}R} }}, \end{align} where~f_R(R)$~is~defined~in~(\ref{distribution_r}).~We~can~obtain~Step~(j)~according~to~the~definition~of~LT.~$L_I_m$~is~the~LT~of~interference~$I_m$~($m ∈ łeft rm hq,rm fq right$~)~and~they~are~defined~in~(\ref{laplace_hd})~and~(\ref{laplace_fq}).~The~joint~LT~$L_U$~is~different~in~the~downlink~and~uplink~transmission.~In~downlink~transmission,~the~interfering~BSs~are~the~same~before~and~after~allocating~HD~RBs~to~the~CEUs,~which~results~in~$hat I_rm e$~and~$I_rm e$~are~correlated~with~each~other.~As~discussed~in~Lemma~\ref{Theorm_laplace_jiont},~$L_U$~is~defined~in~\ref{L_U}.~While~in~the~uplink,~the~uplink~interfering~user~sets~have~already~changed~when~the~serving~RBs~of~CEU~switch~from~FD~RBs~to~HD~RBs.~Therefore,~in~uplink~$L_U$~can~be~written~as~$L_rm hułeft(s_rm hright)L_rm fułeft(s_rm fright). Then the conclusion of Theorem sect. 4.1 is obtained.

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• Figure 1

An exemplary interference scenario of HD/FD cellular network. (a) Synchronous TDD cellular network; protectłinebreak (b) FD cellular network.

• Figure 2

Resource blocks allocation.

• Figure 9

SE vs. successive interference cancellation capability.

• Table 1   Simulation parameters
 RB band width Number of RB Thermal noise $P_b$ $P_{\rm~u}$ $\eta~$ $\lambda_{\rm~s}$ $\lambda_{\rm~u}$ $\gamma_{\rm~d}$ ${\left|~{{\varepsilon~}}~\right|^2}$ Path loss [25] 1 MHz 128 $-$174 dBm$\cdot$Hz$^{-1}$ 0.1 W 0.1 W 0 ${10^{{\rm{~-~}}3}}$ m$^{-2}$ ${1}$ m$^{-2}$ 0 dB $-110$ dB 140.7 + 36.7lg$R$ ($R$ in km)
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