Interference Coordination for FD-MIMO Cellular Network with D2D Communications Underlaying

Xiao Li＊, Nana Qin, Tingting Sun

National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China

Abstract: In this paper, we investigate the interference coordination for downlink full-dimension multiple-input multiple-output(FD-MIMO) systems with device-to-device(D2D) communications underlaying. With three-dimensional (3D) beamforming transmission applied for cellular users (CUEs), an approximation of the interference to signal ratio for CUEs is derived, and a coordination strategy is proposed to mitigate the interference from D2D pairs to CUEs. Based on the lower bound of the interference to signal ratio for D2D pairs, we propose coordination strategies for D2D pairs to mitigate the interference caused by base station (BS) and the interference between D2D pairs. The proposed strategies require only some statistical channel state information (CSI) of each user and the reduced-dimensional effective CSI of a few CUEs and D2D pairs. Simulation results show that the proposed coordination strategy performs well in terms of achieving good tradeoff between the achievable rate of CUEs and D2D pairs.

Keywords: D2D; FD-MIMO; 3D beamforming; interference coordination

I. INTRODUCTION

With the rapid development of wireless communication technology, smart mobile devices have experienced explosive growth, which leads to great demand for data rate and better user experience [1]. Massive multiple-input multiple-output (MIMO), where base station(BS) is equipped with large numbers of antennas, has been widely recognized as one of the key techniques to meet such demand [2-4].There remains several challenges for massive MIMO. One of the main challenges for massive MIMO is that the number of antennas can be deployed is limited by BS factor [5, 6]. This has motivated full-dimension MIMO (FD-MIMO) systems [5]. By placing antennas in two-dimensional (2D) grid, more antennas can be adopted and vertical dimension can be exploited in FD-MIMO systems. The large overhead of the channel state information (CSI)feedback in frequency-division-duplex (FDD)system is another challenge, since most cellular systems today employ FDD transmission algorithms. To overcome this challenge, transmission algorithm exploiting statistical CSI and instantaneous CSI of effective channel for FD-MIMO system has been proposed in [7].To further reduce the CSI feedback overhead,low-complexity 3D beamforming transmission algorithms for large-scale FD-MIMO systems was proposed in [8,9] with only statistical CSI at BS.

In this paper, the authors derived an approximation of the interference to signal ratio for cellular users, and proposed a coordination strategy exploiting only some statistical CSI to mitigate the interference from D2D pairs to cellular users.

Device-to-device (D2D) communication[10,11] is another technique to enhance the overall network performance. It was proposed to meet the continuously increasing demand for direct communications between wireless devices. However, resource sharing among D2D communications and macro network causes both inter-tier and intra-tier interference. To manage the interference, some power control strategies [12-17] have been provided.Most of them exploit full instantaneous CSI to coordinate the inter-tier and intra-tier interference [9], [12-14]. This brings significant CSI feedback and data exchange overhead, especially when the numbers of antennas and users become large. To reduce the CSI required,[15-17] investigated the interference coordination strategies exploiting statistical CSI as well as some instantaneous CSI for uplink transmission systems. Due to their benefits,the combination of massive MIMO and D2D is also attracting more and more attention[18-20]. In [19], an interference limited area control scheme was proposed for proposes an interference limited area scheme for an uplink D2D-underlaid MIMO cellular network with one D2D pair and multiple cellular users(CUEs). For multi-CUEs and multi-D2D pairs underlaying communications, downlink precoder schemes using instantaneous CSI were proposed in [20].

Motivated by the above observations, in this paper, we investigate the interference coordination for downlink FD-MIMO systems with D2D communications underlaying. To reduce the CSI feedback and data exchange,only each user’s statistical CSI and the reduced-dimensional effective CSI of a small fraction of users are available at the BS. With three-dimensional (3D) beamforming downlink transmission applied for cellular users, we derive an approximation of the interference to signal ratio for CUEs, and propose a coordination strategy exploiting only statistical CSI to mitigate the interference from D2D pairs to CUEs. Based on the lower bound of the interference to signal ratio for D2D pairs, interference coordination strategies exploiting statistical CSI and some effective CSI are proposed to mitigate the interference caused by CUEs to D2D pairs and the interference between D2D pairs. Simulation results show that the proposed coordination strategy performs well in terms of achieving good tradeoff between the throughput of CUEs and D2D pairs.

The rest of the paper is organized as follows. Section II introduces the system model.Section III describes the downlink transmission strategy, and proposes the interference coordination strategy. Simulation results are presented in Section IV and we conclude the paper in Section V.

We adopt the following notation: The superscripts (.)T, (.)*and (.)†indicate the matrix transpose, conjugate and conjugate-transpose operation, respectively. The complex number field is represented by C, and E[·] evaluates the expectation.

II. SYSTEM MODEL

In this paper, we consider a single-cell FD-MIMO downlink cellular system with multiple D2D communications underlaying. Assume that the D2D communications reuse the downlink resources of cellular links. There are L single-antenna CUEs to be scheduled by the BS, denoted as CUE1, CUE2, …, CUEL. Meanwhile, there are S coexisting D2D pairs, denoted as D2D1, D2D2, …, D2DS. Each D2D pair is comprised of one single-antenna transmitter and one single-antenna receiver, denoted as TUE and RUE. The TUE and RUE of D2Dsare denoted as TUEsand RUEs, respectively.

2.1 Macro-cell setup

We assume that the BS is deployed with a large-scale antenna array placed in a two-dimensional (2D) grid which has N antennas in each row and M rows in the vertical dimension. The distances between two adjacent antenna elements in each row and each column are both λ/2, where λ is the carrier wavelength. The BS can serve at most K≤MN users simultaneously in each time-frequency re-source block. The channel vector between the BS and CUEk, denoted as(gc,k∈CMN×1),can be expressed as

where βc,kis the large-scale fading coefficient between the BS and CUEk, and hc,kis the small-scale fading vector. According to [7],hc,kcan be written as

the (i,j)-th element of Hc,kis the small-scale fading coefficient between the antenna element on the i-th row and j-th column of BS antenna array and CUEk, Hc,w,kis a M×N random matrix with zero mean and unit variance i.i.d.complex Gaussian elements, Rc,V,k∈CM×Mand Rc,H,k∈CN×Nare the vertical and horizontal channel correlation matrices of CUEk. As in[22], we use the one-ring model to determine the channel correlation matrix Rc,V,kand Rc,H,k.

2.2 D2D pair setup

We define a binary vector to indicate the status of the D2D pairs:

where αs∈{0,1} indicates the status of D2Ds,i.e., αs=1 indicates that the transmission of D2Dsis admitted, while αs=0 indicates the transmission of D2Dsis turned off.

The channel gain between TUEiand RUEjis given as

where βdi,djis the large-scale fading coefficient, and hdi,dj～CN(0,1) is the small-scale fading coefficient. Similar to (5), the channel gain between TUEsand CUEkis modeled as

where βds,kis the large-scale fading coefficient between TUEsand CUEk, and hds,k～CN(0,1) is the small-scale fading coefficient.

Moreover, similar to equation (1), the channel vector between the BS and RUEsis expressed as

where βc,dsis the large-scale fading coefficient between the BS and RUEs, Hd,w,sis a M×N random matrix with zero mean and unit variance i.i.d. complex Gaussian elements,Rd,V,s∈CM×Mand Rd,H,s∈CN×Nare the vertical and horizontal channel correlation matrices of RUEs.

2.3 Signal model

Assume that, after user scheduling, CUEki,i=1,…,U, are selected to be served simultaneously, where U≤K. For the considered transmission system, the signal received by CUEkican be written as

where xkuis the data symbol intended for CUEkuwithis the beamforming vector of CUEku, Pc,kuis the transmit power for CUEkuwith total power constraintis the data symbol intended for RUEjwiththe transmit power of TUEj, nc,ki～ CN(0,σk2i)is the noise. In this paper, we assume equal power allocation among the scheduled users,i.e., Pc,ku= P/ U , u =1,…,U.

For the D2D pairs, the received signal of RUEscan be written as

III. DOWNLINK TRANSMISSION AND INTERFERENCE COORDINATION

In this section, we provide a downlink transmission scheme for CUEs and propose an interference coordination strategy for the considered network.

3.1 Downlink transmission for CUE

Let us define

where FM∈CM×Mand FN∈CN×Nare unitary discrete Fourier transform (DFT) matricesis the (m,n)-th element of FM,and the (m,n)-th element of FNisAssume that the largest diagonal elements of Λc,H,kiare the hki-th diagonal element of Λc,H,kiand the vki-th diagonal element of Λc,V,ki. According to [7], for uniform linear array (ULA) of large dimension, the eigenvectors of its correlation matrix can be approximated by DFT matrix. Therefore, when M→∞ and N→∞, Λc,H,kiand Λc,V,kibecome diagonal matrices.

Based on this property, a 3D beamforming downlink transmission scheme was proposed in [8]. In this paper, we adopt a downlink transmission scheme similar to the one in [8].The details of the transmission scheme are listed in Algorithm 1.

It can be seen that, to perform this downlink transmission scheme, the BS only needs to knowof each user and their corresponding indexes.

Algorithm 1. Downlink transmission for CUEs.1: Initialize sets: C=∅, where C is the set of scheduled users in macro cell;2: Choose N′+1 integers n~ n~0,…,N′ and M ′+1 integer m~ m~0,…,M′, so that M′ . For simplicity, we assume that M ′N′ =K;3: Divide the users into MN′′ groups, so that any user ki in group (m,n) satisfies n h n 0=＜＜＜n n n~ ~ ~0 1 … N′=N and 0= ＜ ＜m~ m~0 1…＜m~ =M~n−1+1≤ ≤~k n i and m v m~m−1+1≤ ≤~;k m i 4: Choose one user in each group, so that the user is the one with the largest λ λ i among the users in its cluster. Add these users into C;5: For CUEki∈C, calculate the beamforming vector:b F F cVk cHk max max,, ,,i k N M i=( ) ⊗hki( *)vki, (12)where i=1,…,U, (A)i denotes the i-th column of matrix A;6: Perform beamforming transmission for CUEs in C.

3.2 Interference coordination

As D2D communications and CUE communications share the same spectrum resources,both inter-tier and intra-tier interferences exist.To guarantee the quality of service (QoS) of both CUEs and D2D pairs, we first consider the interference coordination for CUEs, and then the interference coordination for D2D pairs.

3.2.1 Interference Coordination for CUEs

From (8), it can be obtained that the useful signal power CUEkireceived from BS isand the interference it received from TUEsisorder to maintain the QoS of CUEki, we would like that the ratiois below a certain threshold, i.e.,

where δcis a preset threshold. So that the total interference caused by D2D communications,is tolerable. Note that the interference coordination based onrequires full instantaneous CSI. This requires large amount of information feedback and data exchange. The following Theorem 1 provides an approximation of the expectation ofwhich depends only on statistical CSI.

Theorem 1. When MN→∞, S=μMN, and μ∈(0,∞), we have

and →represents convergence in probability.

Proof: See the Appendix.

It can be seen from Theorem 1 thatcontains only statistical CSI, so that we couldto approximatewhen MN＞＞1, and design interference coordination strategies based on it. Therefore, we modify the metric (13) to. There are two methods to satisfy this constraint, one is to turn off the communications of D2D pairs which do not satisfy the constraint, the other is to turn down the transmit power of these D2D pairs to the maximum value that satisfywhich is

However, turning off the communication of D2D pairs decreases the total throughput of the cell, while reducing the transmit power of D2D pairs degrades the QoS of these corresponding D2D communications. In this paper,to balance the total throughput and the QoS of D2D pairs, we set a minimum transmit power Pdminfor each D2D pair. At the beginning of every scheduling slot, we initialize each D2D pair to be active, i.e., αs=1, s=1,…,S,and transmit at its maximum transmit power Pdmax. If D2Dsdoes not satisfy the constraintcalculatethen setelse, set αs=0 and turn off the communication of D2Ds. The main procedure to coordinate the interference from D2D to CUEs is given in Algorithm 2.

It can be seen that, to perform this strategy to coordinate the interference from D2Dsto CUEs, some additional statistical CSI of each scheduled CUE is required, i.e., the large-scale fading coefficients between TUEsand each scheduled CUE, the large-scale fading coefficients between the BS and each scheduled CUE. These statistical CSI changes relatively slow, and can be obtained through long term feedback and exchange. Moreover, to perform this strategy, onlyfor each scheduled CUE corresponding to each D2D pair andfor some D2D pairs need to be calculated from(15) and (16). These parameters depend only on statistical CSI, and can be computed easily.

According to [23], due to the characteristics of short communication distance and low transmit power of D2D communication, the interference caused by D2D pairs is quite limited by the distance. Therefore, to further reduce the CSI feedback and complexity of the coordination scheme, we can only feedback the large scale fading coefficients of several D2D pairs close to the CUE, and perform interference coordination only for these D2D pairs.

3.2.2 Interference Coordination for D2D pairs

The interference received at each RUE consists of two parts. One is the interference from other TUEs, the other is the interference from the BS.

First, we consider the interference coordination between different D2D pairs. From(9), it can be obtained that the useful signal power RUEsreceived isand the interference it received from TUEpIn order to maintain the QoS of D2Ds, similar to (13), we would like that the ratiois below a certain threshold, i.e.,

Algorithm 2. Interference coordination for CUE.1: Initialize sets: αs=1,P P,= max;2: for CUEki, i=1,…,U do 3: for D2Ds, s=1,…,S do 4: Computes ds d, according to (15);5: ifω δ ωks(c)i,＞ , then 6: Computes(c)ks c i i according to (16);7: if Pds(k),i≥Pdmin, then 8: Set Pds(k),Pd,s=Pds(k),i;9: else 10: Set αs=0 and turn off D2Ds;11: end if 12: end if 13: end for 14: end for

Note that the interference coordination based onrequires full instantaneous CSI.In order to reduce the feedback overhead of the system, we also make use of a lower bound of the expectation of (16), which depends only on statistical CSI. Substituting (6) into (17),and using the Mullen’s inequality [25], i.e.,E [X/Y ]≥ E [X ]/ E[Y] if X and Y are independent random variables, the expectation ofbe lower bounded as

Thus, we change the metric (17) toIf this constraint cannot be satisfied,an interference coordination is needed. Here,we adopt an “ON/OFF” strategy [15, 16]: turn off the communication of the D2D pair which has lower expected signal power. It can be obtained that the expected signal power received at RUEsand RUEpcan be computed asspectively. Then, we summarize the strategy to coordinate the interference received by D2Dsfrom other D2D pairs in Algorithm 3.

Algorithm 3. Inter-tier coordination for D2D.1: for D2Ds, D2Dp, s=1,…,S, p=1,…,S do 2: while αs =1, αp =1 do 3: if p ≠ s do 4: Compute ,d according to (18);5: ifω δ ωsp()p≤ d do 6: Compare the expected signal power of D2Ds and D2Dp, and let i P d s(),=argmin{ }r∈{s,p} dr d d,βr ,r ;7: Turn off the communication of D2Di, and set αi = 0;8: else 9: End the interference coordination for D2Ds;10: end if 11: end if 12: end while 13: end for

To perform this proposed strategy coordinating the interference received by D2Dsfrom other D2D pairs, only some statistical CSI is needed: the transmit power of each D2D pair, the large-scale fading coefficient between the TUE and RUE of each D2D pair,and the large-scale fading coefficients between the TUEs of other D2D pairs to RUEs.Moreover, onlyneed to be computed to coordinate the interference from D2Dpto D2Ds. Note that due to the distance-related path-loss, each D2D pair can only cause strong interference to the D2D pairs which is located close to it. Therefore,to reduce the feedback overhead and the computation complexity, each D2D pair can only feedback the large scale fading coefficients correspondingly to the D2D pairs close to it,and only performs the interference coordination to these D2D pairs. Furthermore, since part of the D2D pairs are turned off during the interference coordination for CUEs and inter-tier coordination for D2D pairs, the interference coordination for D2D pairs do not need to go through all D2D pairs.

Next, we consider the interference from the BS to D2D pairs. From (9), it can be obtained that the interference power RUEsreceived due to CUEkiisUsing the similar method as the derivation of Algorithm 3, the expectation ofcan be lower bounded as

From (19), it can be seen that the average interference caused by CUEkidepends onAccording to [7], only a few adjacent diagonal elements of Λd,V,sand Λd,H,sare non-zero when M→∞ and N→∞. Therefore,CUEkican cause strong interference to RUEs,only when hkiis close to hsand vkiis close toHowever,according to the grouping method in Section 3.1, hkiand vkiof different scheduled CUEs are not continuous integers. It is probably that only one or two scheduled CUEs can have hkiand vkiclose to hsand vs. This means that, in most cases, there are only one or two scheduled CUEs that have strong interference to D2Ds. Therefore, to reduce the complexity of the interference coordination strategy, we only consider the strongest interference for D2Ds.Assume that the CUE that causes the strongest interference to D2Dsis CUEks. Then we com-no coordination is needed. If else, interference coordination for D2Dsis needed.

To make it more fair, so that both CUEksand D2Dscan maintain their communications,a two-stage precoding method is applied. In this method, a two-stage precoder

is adopted for CUEks, instead of the previous beamforming vector (12) given in Section 3.1,to eliminate the interference it caused to D2Ds.The outer precoder Bksdepends only on the statistical CSI, and is designed as

The inner precoder wksis designed to depend on the effective CSI Hks, and is chosen as the first column of the zero-forcing (ZF)matrix Qkscorresponding to Hks, i.e.,

Based on the above analysis, we summarize the procedure to coordinate the interference from BS to D2Ds. The main procedure is given in Algorithm 4.

To perform the steps in line 4 of Algorithm 4, only some statistical CSI is needed, i.e., the transmit power of TUEs, the transmit power of the BS, the large-scale fading coefficient between the TUE and RUE of D2Ds, and the large-scale fading coefficients between the BS and RUEs, the diagonal elements of Λd,V,sand Λ, and the indexes of

Algorithm 4. Intra-tier coordination for D2D.1: for D2Ds, s=1,…,S do 2: if αs =1 do 3: for CUEki, i=1,…,U do 4: Calculate, ;5: if θks(c), according to (18), and letks ks=argmax{ c}i kC i∈ θ()i,＞γ do 6: Compute the outer precoder Bks and the zero-forcing matrix Qks according to (24), (25) and (27);7: Get inner precoder wks according to (26);8: Calculate and update bks;9: else 10: End the interference coordination for D2Ds;11: end if 12: end for 13: end if 14: end for θks d(c)s

d,H,sof each scheduled CUE. These statistical CSI can be obtained through long term feedback and exchange. If steps in line 6-8 are needed,the indexes of the second largest diagonal elements of Λc,H,ksand Λc,V,ks, and some reduced-dimensional effective CSI Hksare required. The computation complexity of Algorithm 4 mainly depends on the computation of the matrix inversion in (27). Note that the effective channel matrix Hksis a 2×4 matrix, the matrix inversion that needs to be calculated in (27) is only the inversion of a 2×2 matrix.Thus, the computational complexity of this two-stage precoding method is greatly reduced compared to ZF precoding based on the full instantaneous CSI gc,ksand gc,ds. Moreover, to further reduce the CSI feedback and complexity of the coordination scheme, we can only feedback several non-zero diagonal elements of D2D pairs, and perform Algorithm 4 only for the CUE which is close to them.

Fig. 1. Throughput tradeoff curves for different coordination strategies.

Fig. 2. Average number of active D2D pairs under different coordination strategies.

IV. SIMULATION

In this section, we present the numerical results to validate the performance of the proposed interference coordination strategy.Through all the simulations, we set N=64,M=16, L=300, S=200, Pc=46 dBm=20 dBm,=14 dBm, δc=0.1, δd=0.1,γd=0.05, the noise level of all users are the same, i.e.,= σ2, and the cell edge SNR is 10 dB. The horizontal and vertical angle of the users relative to BS are both distributed uniformly in (−90°,90°). For largescale fading, as in [24], we only consider the distance-dependent path-loss, which is modeled as β=(1+(d/d0)γ)−1, where d denotes the distance (in kilometers) between the transmitter and receiver, d0is the cutoff distance,γ is the path-loss exponent. Throughout the simulation, we assume that d0=0.05, γ=3.76,the distance between CUE and the BS is uniformly distributed in [50 m, 1000 m], and the distance between the TUE and RUE of each D2D pair is uniformly distributed in [3 m, 20 m]. Additionally, both the horizontal and vertical angular spread of each user (both CUE and RUE) relative to macro BS are uniformly distributed in (5°,15°).

In the simulation results, we also show the performance of the system with no coordination, as well as other two strategies, denoted as ON/OFF strategy and Min-Power strategy.According to the methods in [15] and [16],in the ON/OFF strategy, we turn off the D2D pairs and CUEs that cause strong interference to other users. According to the coordination methods in [17], we turn down the transmit power of the D2D pairs to the minimum transmit power in Min-Power strategy.

Figure 1 shows the tradeoff between the average sum rate of CUEs and D2D pairs, while figure 2 compares the average number of active D2D pairs under different coordination strategies in a scheduling slot. These curves are both obtained by varying the parameter K=2kfrom k=1 to k=6. From figure 1, it can be observed that the increase of K results in the increase of cellular cell throughput but the reduction of D2D pairs throughput. This is because the interference coordination strategies improve the throughput of CUEs at the expense of some D2D pairs throughput by turning off some D2D pairs which can be seen from figure 2, and turning down the transmit power of some D2D pairs. Moreover, it can be obtained from figure 1 that the ON/OFF strategy performs best in increasing the throughput of CUEs but the worst in achieving high throughput of D2D pairs. This is because the active number of D2D pairs of the ON/OFF strategy is the smallest one among the three interference coordination strategies (see in figure 2). The proposed strategy performs better than the proposed strategy in D2D throughput but worse in cellular throughput. By allowing each D2D pair to transmit signal at its maximum power which satisfies the QoS of CUEs,the proposed strategy improves the throughput of CUEs as well as maintain the QoS of D2D pairs. In contrast, our proposed strategy provides a good tradeoff between the throughput of CUEs and D2D pairs.

Figure 3 and figure 4 show the cumulative distribution function (CDF) of the rate and the average rate of CUEs under different coordination strategies. In figure 3, K=64. It can be seen that the proposed interference coordination strategy can greatly improve the celledge performance of CUEs, while the ON/OFF strategy and Min-Power strategy perform slightly better than the proposed strategy. It can also be observed that the ON/OFF strategy performs best both in terms of the cell-edge rate and the average rate of CUEs, since it turns off all the users (both CUEs and TUEs)that cause strong interference to other users.Meanwhile, the proposed strategy has a suboptimal performance.

Fig. 3. CDF of the rate of CUEs under different coordination strategies, when K=64.

Fig. 4. Average rate of CUEs under different coordination strategies.

Fig. 5. CDF of the rate of D2D users under different coordination strategies, when K=64.

Fig. 6. Average rate of D2D pairs under different coordination strategies.

Figure 5 and figure 6 show the CDF of the rate and the average rate of D2D pairs respectively. From figure 5, it can be seen that the proposed strategy performs best in terms of cell-edge rate of D2D pairs, since it allows each active D2D pair transmits signal at its maximum transmit power which satisfies the QoS of CUEs. In the high rate region, the ON/OFF strategy has the best performance as the CUEs and D2D pairs that caused strong interference are turned off. For the average rate of D2D pairs, it can be observed from figure 6 that the ON/OFF strategy performs best while the proposed strategy also improve the average rate of D2D pairs a lot comparing to no coordination case. The average rate of D2D pairs under Min-power strategy and no coordination case decrease as K increases. This is because,under no coordination case, the number of active D2D pairs does not change as K increases,while under Min-power strategy, the number of D2D pairs with minimum transmit power increases as K increases. Besides, the average rate of D2D pairs under the proposed strategy and ON/OFF strategy first decrease and then increase as K increases. This is because, when K ≤ 8, the number of active D2D pairs under these strategies does not decrease much as K increases. Therefore, the interference D2D pairs received due to CUEs increases while the interference they received from other D2D pairs decreases dramatically, so that the interference between D2D pairs decreases significantly. And when K ≥ 16, the number of active D2D pairs decreases sharply as K increases.

In figure 7, we compare the performance of the proposed coordination strategy based on statistical CSI and the coordination strategy with pure instantaneous CSI. We assume 3D beamforming downlink transmission for macro users in both cases, since the performance of the 3D beamforming and beamforming based on pure instantaneous CSI for macro users have already been compared in[8], and it showed that the performance of 3D beamforming is comparable to MF and ZF precoding. The coordination strategy with pure instantaneous CSI is similar to our proposed coordination strategy, the difference is that it use instantaneous CSI instead of statistical CSI. Similar to [8], we assume that the instantaneous CSI obtained is imperfect due to channel estimation error, feedback latency, limited feedback, etc. The instantaneous channel vector obtained for user k iswhere gkis the real small-scale fading vector, gˆkrepresents the channel estimation error and the error in-curred by feedback, and α∈[0,1] indicates the accuracy of the channel vector obtained at the BS. It can be seen that the CSI obtained by the BS is perfect when α=0, and the CSI becomes increasingly more imprecise when α increases.Note that, in this comparison, the effective CSI used in Algorithm 4 of our proposed algorithm is also imperfect. In this figure, K=64,and the performance of the coordination strategies under α2=1/6 and α2=1/2 are shown.Note that this figure does not take into account the overhead of CSI feedback and information change.

From figure 7, it can be seen that the ergodic CUE sum rate performance of the proposed strategy is slightly inferior to the coordination strategy based on instantaneous CSI when α2=1/6. However, when the quality of the instantaneous CSI at the BS gets worse, the ergodic CUE sum rate performance of the proposed strategy remains almost the same, while the coordination strategy based on pure instantaneous CSI degrades a lot. The proposed strategy even outperforms the coordination strategy based on instantaneous CSI when α2=1/2. For the ergodic D2D sum rate performance, the coordination strategy based on instantaneous CSI outperforms the proposed statistical CSI based strategy even when the CSI quality is not good. The coordination strategy based on statistical CSI has about 10% performance loss. However, this result does not take into account the overhead in acquiring the CSI at the BS. The performance of the coordination strategy with instantaneous CSI will get worse when we take into account this overhead. Therefore, it can be seen that it is attractive and effective to design interference coordination strategy exploiting statistical CSI. It requires much less CSI and is of low complexity, while can still achieve a considerable throughput and is more robust to the CSI quality.

V. CONCLUSION

Fig. 7. Performance comparison of the coordination strategies based on instantaneous CSI and statistical CSI when K=64.

In this paper, we investigated the interference coordination for downlink FD-MIMO systems with D2D communications underlaying. We derived an approximation of the interference to signal ratio for cellular users, and proposed a coordination strategy exploiting only some statistical CSI to mitigate the interference from D2D pairs to cellular users. Based on the lower bound of the interference to signal ratio for D2D pairs, interference coordination strategies exploiting some statistical CSI and very little effective CSI were proposed to mitigate the interference caused by CUEs to D2D pairs and the interference between D2D pairs. Simulation results showed that the proposed coordination strategy performs well in terms of achieving good tradeoff between the achievable rate of CUEs and D2D pairs.

ACKNOWLEDGEMENT

This work was supported in part by the National Natural Science Foundation of China(Grants No. 61831013 and No. 61571112),the Foundation for the Author of National Excellent Doctoral Dissertation of PR China(FANEDD) (Grant No. 201446).

Appendix

Proof of Theorem 1

Substituting (1), (8) and (12) into (13), we can get that

Substituting (28) and (15) into (14), we can get that

First, let us consider a1. Since hds,ki, s=1,…,S are i.i.d. complex Gaussian variables with zero mean and unit variance, it can be obtainedS are uncorrelated,and

Then, according to Chebysheve Theorem[26], we can get that when S→∞,

Then, let us consider a2. From (2) and (3),it can be obtained that

where

For the simplicity of representation, let us define

Note that hc,w,kiis a random vector with zero mean and unit variance i.i.d. complex Gaussian elements, andFrom[27, Lemma 3], we have that

where C2is a constant,andis the j-th element of hc,w,ki. Note thatTherefore,

Substituting (41) and (42) into (39), we can get that

Furthermore, from [28, Theorem 3.5], we have that, for any ε ＞ 0,

where P(.) represents the probability. Substituting (43) into (44), we can get that

for any ε ＞ 0. Therefore, we get that [27], X →0 when MN→∞. Note thatsince most small cells are located far away and the correspondingare negligible, andWe can further obtain that

when MN→∞.

Finally, combining (29), (35), and (47), we get Theorem 1.