FdICIC: Inter-cell Interference Coordination for Full-Duplex Cellular Systems

Abstract

This paper investigates the inter-cell interference coordination (ICIC) for heterogeneous networks (HetNets) with full-duplex (FD) small cell base stations to maximize the overall system throughput. To be compatible with the enhanced ICIC (eICIC) and the further enhanced ICIC (FeICIC) of existing LTE standards, the proposed FdICIC technique adopts cell range expansion and almost blank subframe to deal with the complicated interference scenario in such FD HetNets. In detail, the FdICIC includes four steps, namely, choosing the serving base station, pairing users for FD communication, resource block (RB) allocation, and power control of FD users. Through numerical simulation, we demonstrate that: (1) the overall system throughput can be improved by 10–25% with four FD small cell base stations when the residual self-interference is between − 130 to − 110 dB, (2) by carefully selecting the CRE bias, the overall system throughput can be further improved by about 30%, (3) the proposed user pairing, RB allocation, and power control steps have significant contributions to the performance improvement.

Appendices

Appendix 1: Proof of Theorem 1

Setting the first derivative of ${{\tilde{z}}^{NBS}}$ in (19) on $\beta _m$ to 0, we can derive that

$$\begin{aligned} {\beta _{m}}&= l - {Q_l} + Q_m^F, \end{aligned}$$

(31)

$$\begin{aligned} \phi _{j}&= l - {Q_l} + Q_j^H. \end{aligned}$$

(32)

Then we have

$$\begin{aligned} \beta _{m} = \dfrac{1}{2}\left[ {K -Q_l- \sum \limits _{{m_1 = 1,m_1} \ne {m}}^M {{ \beta _{m_1}}}-\sum \limits _{j = 1}^J {\phi _j+Q_m^F} } \right] , \quad\,\forall m \in M . \end{aligned}$$

(33)

Similarly, the optimal $\phi _j$ has a closed-form expression of

$$\begin{aligned} \phi _{j} = \dfrac{1}{2}\left[ {K - Q_l -\sum \limits _{{j_1 = 1,j_1} \ne j}^J {{\phi _{j_1}}} - \sum \limits _{m = 1}^M {\beta _m + Q_j^H} } \right] , \quad \forall j \in J . \end{aligned}$$

(34)

Then, summating both sides of (33) from $m = 1$ to M and (34) from $j = 1$ to J, we can get

$$\begin{aligned} (M + 1)\sum \limits _{m = 1}^M {{\beta _m}}&= M\left[ {K - \sum \limits _{j = 1}^J {{\phi _j}} - {Q_l}} \right] + \sum \limits _{m = 1}^M {Q_m^F}, \end{aligned}$$

(35)

$$\begin{aligned} (J + 1)\sum \limits _{j = 1}^J {{\phi _j}}&= J\left[ {K - \sum \limits _{m = 1}^M {{\beta _m}} - {Q_l}} \right] + \sum \limits _{j = 1}^J {Q_j^H}. \end{aligned}$$

(36)

Therefore, we can finally get the solution of l as

$$\begin{aligned} l^* = \dfrac{{K + (M + J){Q_l} - \sum \nolimits _{j = 1}^J {Q_j^H - \sum \nolimits _{m = 1}^M {Q_m^F} } }}{{M + J + 1}}. \end{aligned}$$

(37)

Substituting l to $\beta _m$ and $\phi _j$, we have

$$\begin{aligned} {\beta _m^*}&= \dfrac{1}{{M + J + 1}}\left( K - {Q_l} - \sum \limits _{m = 1}^M {Q_m^F - \sum \limits _{j = 1}^J {Q_j^H}}\right) + Q_m^F, \end{aligned}$$

(38)

$$\begin{aligned} {\phi _j^*}&= \dfrac{1}{{M + J + 1}}\left( K - {Q_l} - \sum \limits _{m = 1}^M {Q_m^F - \sum \limits _{j = 1}^J {Q_j^H } }\right) + Q_j^H. \end{aligned}$$

(39)

In a similar way, we can respectively get the optimal RB allocation results for the FD CRE user pair m and HD CRE user j, as

$$\begin{aligned} {\beta ^*_{m,CRE}}&= \dfrac{1}{{M_0 + J_0 }}\left( l^*- \sum \limits _{m = 1}^{M_0} {Q_m^F - \sum \limits _{j = 1}^{J_0} {Q_j^H}} \right) + Q_m^F, \end{aligned}$$

(40)

$$\begin{aligned} {\phi ^*_{j,CRE}}&= \dfrac{1}{{M_0 + J_0 }}\left( l^* - \sum \limits _{m = 1}^{M_0} {Q_m^F - \sum \limits _{j = 1}^{J_0} {Q_j^H }}\right) + Q_j^H. \end{aligned}$$

(41)

Fig. 8

Optimal power allocation for FD user pair

Appendix 2: Proof of Theorem 2

The aggregate data rate of a FD user pair (i, j) on the kth RB can be rewritten as

$$\begin{aligned} R_{i,j,k}=\log _2\left( 1+\xi ^{\mathrm{U}}_{i,k}\right) +\log _2\left( 1+\xi ^{\mathrm{D}}_{j,k}\right) , \end{aligned}$$

(42)

where $\xi ^{\mathrm{U}}_{i,k}$ and $\xi ^{\mathrm{D}}_{j,k}$ are the SINR for uplink user i and downlink user j on the kth RB, respectively. Now we take into account the QoS of each uplink user and downlink user, which can be respectively expressed as

$$\begin{aligned} \xi ^{\mathrm{U}}_{i,k}&= \dfrac{{p_{i,k}^{\mathrm{U}}h_i^{\mathrm{U}}}}{{\sigma _0^2+ p_{j,k}^{\mathrm{D}}{\eta } }} \ge \xi ^{\mathrm{U}}_{i,\mathrm{min}} , \end{aligned}$$

(43)

$$\begin{aligned} \xi ^{\mathrm{D}}_{j,k}&= \dfrac{{p_{j,k}^{\mathrm{D}}h_j^{\mathrm{D}}}}{{\sigma _0^2 + p_{i,k}^{\mathrm{U}}{h_{i,j}} }} \ge \xi ^{\mathrm{D}}_{j,\mathrm{min}} . \end{aligned}$$

(44)

According to [26], the feasible power region can be illustrated in Fig. 8, where lines $l_U$ and $l_D$ represent (43) and (44), respectively. There are only three situations for all possible FD user pairs, as in Fig. 8I–III. From Fig. 8, it is necessary that the slope of $l_D$ must be larger than that of $l_U$, which leads to the following admission control criterion

$$\begin{aligned} \dfrac{{\xi _{i,\mathrm{min}}^{\mathrm{U}}\eta }}{{h_i^{\mathrm{U}}}} < \frac{{h_j^{\mathrm{D}}}}{{\xi _{j,\mathrm{min}}^{\mathrm{D}}{h_{i,j}}}}. \end{aligned}$$

(45)

Then we analyze the optimal power allocation of each FD user pair that satisfies the admission control criterion.

We first prove that for any given power pair $(p^{\mathrm{U}}_{i,k}, p^{\mathrm{D}}_{j,k} )$ in the interior of the feasible power region, there always exists another power pair $(\alpha p^{\mathrm{U}}_{i.k}, \alpha p^{\mathrm{D}}_{j,k} )$ ($\alpha >1$) in the admissible area so that the throughput can be further improved, since

$$\begin{aligned} \begin{aligned}&R\left( \alpha p_{i,k}^{\mathrm{U}},\alpha p_{j,k}^{\mathrm{D}}\right) = {\log _2}\left( 1 + \dfrac{{\alpha p_{i,k}^{\mathrm{U}}h_i^{\mathrm{U}}}}{{{\sigma _0^2} + \alpha p_{j,k}^{\mathrm{D}}\eta }}\right) + {\log _2}\left( 1 + \dfrac{{\alpha p_{j,k}^{\mathrm{D}}h_j^{\mathrm{D}}}}{{{\sigma _0^2} + \alpha p_{i,k}^{\mathrm{U}}{h_{i,j}}}}\right) \\&\quad = {\log _2}\left( 1 + \dfrac{{p_{i,k}^{\mathrm{U}}h_i^{\mathrm{U}}}}{{{\sigma _0^2}/\alpha + p_{j,k}^{\mathrm{D}}\eta }}\right) + {\log _2}\left( 1 + \dfrac{{p_{j,k}^{\mathrm{D}}h_j^{\mathrm{D}}}}{{{\sigma _0^2}/\alpha + p_{i,k}^{\mathrm{U}}{h_{i,j}}}}\right) \\&\quad > {\log _2}\left( 1 + \dfrac{{p_{i,k}^{\mathrm{U}}h_i^{\mathrm{U}}}}{{{\sigma _0^2} + p_{j,k}^{\mathrm{D}}\eta }}\right) + {\log _2}\left( 1 + \dfrac{{p_{j,k}^{\mathrm{D}}h_j^{\mathrm{D}}}}{{{\sigma _0^2} + p_{i,k}^{\mathrm{U}}{h_{i,j}}}}\right) \\&\quad = R\left( p_{i,k}^{\mathrm{U}},p_{j,k}^{\mathrm{D}}\right) . \end{aligned} \end{aligned}$$

(46)

This implies that the optimal power pair $(p_{i,k}^{U*},p_{j,k}^{D*})$ must lay at the boundary of the feasible power area, i.e., $p_{i,k}^{U*}=P^{\mathrm{U}}_{MAX}$ or $p_{j,k}^{D*}=P^{\mathrm{D}}_{MAX}$.

For situation I, the power pair $(P^{\mathrm{U}}_{MAX},P^{\mathrm{D}}_{MAX})$ does not fall in the feasible power region, and

$$\begin{aligned} \dfrac{{P_{MAX}^{\mathrm{U}}h_i^{\mathrm{U}}}}{{{\sigma _0^2} + P_{MAX}^{\mathrm{D}}\eta }} \le \xi ^{\mathrm{U}}_{i,\mathrm{min}}. \end{aligned}$$

(47)

From the above conclusion, the optimal power pair must lay at line BC. According to [27], it is proved that $R_{i,j,k}$ is a convex function on $(p_{i,k}^{\mathrm{U}},p_{j,k}^{\mathrm{D}} )$ when the other variable, $p_{j,k}^{\mathrm{D}}$ or $p_{i,k}^{\mathrm{U}}$, is fixed at its maximum value. Therefore, the optimal power pair must lay at the end point B or C. Then the optimal power allocation for situation I can be expressed as

$$\begin{aligned} \left( p_{i,k}^{U * },p_{j,k}^{D * }\right) = \begin{array}{l} \arg \mathop {\max }\limits _{\left( p_{i,k}^{\mathrm{U}},p_{j,k}^{\mathrm{D}}\right) \in {\psi _1}} {\mathrm{}}f\left( p_{i,k}^{\mathrm{U}},p_{j,k}^{\mathrm{D}}\right) \end{array} , \end{aligned}$$

(48)

where ${\psi _1}$ is given in (30).

For situation II, the power pair $(P^{\mathrm{U}}_{MAX},P^{\mathrm{D}}_{MAX})$ also does not fall in the feasible power region, and

$$\begin{aligned} \dfrac{{P_{MAX}^{\mathrm{U}}h_i^{\mathrm{U}}}}{{{\sigma _0^2} + P_{MAX}^{\mathrm{D}}\eta }} > \xi ^{\mathrm{U}}_{i,\mathrm{min}}, \quad \dfrac{{P_{MAX}^{\mathrm{D}}h_j^{\mathrm{D}}}}{{{\sigma _0^2} + P_{MAX}^{\mathrm{U}}{h_{i,j}}}} \le \xi ^{\mathrm{D}}_{j,\mathrm{min}}. \end{aligned}$$

(49)

Therefore, the optimal pair must lay at the end point E or F. Then the optimal power allocation for this situation can be expressed as

$$\begin{aligned} \arg \mathop {\max }\limits _{\left( p_{i,k}^{\mathrm{U}},p_{j,k}^{\mathrm{D}}\right) \in {\psi _2}} {\mathrm{}}f\left( p_{i,k}^{\mathrm{U}},p_{j,k}^{\mathrm{D}}\right) , \end{aligned}$$

(50)

where ${\psi _2}$ is given in (30).

For situation III, the power pair $(P^{\mathrm{U}}_{MAX},P^{\mathrm{D}}_{MAX})$ sits in the feasible power region, i.e.,

$$\begin{aligned} \dfrac{{P_{MAX}^{\mathrm{U}}h_i^{\mathrm{U}}}}{{{\sigma _0^2} + P_{MAX}^{\mathrm{D}}\eta }}> \xi ^{\mathrm{U}}_{i,\mathrm{min}}, \quad \dfrac{{P_{MAX}^{\mathrm{D}}h_j^{\mathrm{D}}}}{{{\sigma _0^2} + P_{MAX}^{\mathrm{U}}{h_{i,j}}}} >\xi ^{\mathrm{D}}_{j,\mathrm{min}}. \end{aligned}$$

(51)

Therefore, the optimal pair must lay at the end point B, E, or F. Then the optimal power allocation for situation III can be expressed as

$$\begin{aligned} \arg \mathop {\max }\limits _{\left( p_{i,k}^{\mathrm{U}},p_{j,k}^{\mathrm{D}}\right) \in {\psi _3}} {\mathrm{}}f\left( p_{i,k}^{\mathrm{U}},p_{j,k}^{\mathrm{D}}\right) , \end{aligned}$$

(52)

where ${\psi _3}$ is given in (30).