I Introduction
Massive multipleinput multipleoutput (MIMO) is an efficient technique to boost the spectral efficiency. However, as elaborated in the existing literature, the issue of pilot contamination [1], which refers to the undiminished intercell interference caused by pilot reuse, may severely degrade its performance. Although there have been significant research efforts devoted to address the pilot contamination issue [2, 3] and some techniques proposed to mitigate the pilot contamination [4, 5], most of the works handle the intercell interference from the conventional cellular network point of view, instead of novel network topologies.
Recently, the deployment of wireless cache nodes in cellular networks has drawn significant research attentions [6, 7], where the devicetodevice (D2D) links has to be introduced into cellular network. The cochannel deployment of D2D communications and cellular networks was studied in some literature [8, 9]
, where it is shown that the D2D transmission reusing the cellular spectrum may cause severe interference. To alleviate the interference, D2D underlay massive MIMO cellular networks has been proposed by exploiting spatial degrees of freedom at the base station (BS). For example, in
[10], the authors proposed a pilot reuse strategy for D2D receivers and a novel interferenceaided minimum mean square error (MMSE) detector to suppress the D2Dtocellular interference. In [11], a novel revised graph coloringbased pilot allocation (RGCPA) algorithm was proposed for pilot allocation, and an iterative scheme was adopted to minimize the transmission power of D2D links. In order to evaluate the overall effect of D2D links on cellular network, a systemlevel performance analysis is necessary. The interplay between massive MIMO uplink transmission and cochannel D2D transmission has been studied in [12]. However, the conclusion of its analysis cannot be directly applied in downlink. Moreover, the authors assume that the antenna number at the BS is infinity, which is not practical^{1}^{1}1In massive MIMO systems, the typical number of antennas at the BS is up to a few hundred, which is large but not infinity.. As a result, performance analysis of massive MIMO downlink transmission with cochannel D2D links is still open.In this paper, we would like to shed some light on the above issue by analysing the performance tradeoff between massive MIMO downlink and cochannel D2D transmissions. Specifically, we consider a massive MIMO network with typical hexagonal cell structure and random distribution users and D2D links. The D2D links may refer to the direct transmission from wireless cache nodes to cell users, which offload the traffic from service BSs. In the analysis, we first obtain the asymptotic expressions of downlink and D2D signaltointerference ratios (SIRs) for sufficiently large (but finite) number of antennas at the BS. These expressions depend on the pathloss and smallscale fading of interference channels, which may be unknown to the service BS or D2D transmitters. For example, the interference to downlink users comes from neighbouring BSs and nearby D2D transmitters, and the channel condition of those interference is not easy to obtain at the service BS. When the randomness of user locations and smallscale fading are considered, it is possible that the transmission data rate is greater than the channel capacity, leading to the packet outage. Hence We use the average goodput [13], which measures the average number of bits successfully delivered to the receiver, as performance metric. We derive the approximated expressions of the downlink and D2D goodput, based on which the performance tradeoff between massive MIMO downlink and D2D can be evaluated. It is shown by numerical simulations that the analytical results matches the actual performance very well.
Ii System Model
Iia Network Model
We consider a typical cellular network with hexagonal cells where the radius of each cell is , as illustrated in Figure. 1. Each cell consists of a base station equipped with antennas, active singleantenna downlink users and active D2D transmitters. BSs and D2D transmitters transmit with constant powers and , respectively. The D2D links may refer to the data delivery between the downlink users and wireless cache nodes. For example, the desired data of downlink users is found in one cache node nearby, and then a direct D2D communication link is established. Since the focus of this paper is on the physicallayer SIR and throughput analysis, the establishment of D2D links is outside the scope of this work. Since massive MIMO technology is considered, is sufficiently large, e.g., a few hundred. The downlink users and D2D transmitters are uniformly and independently distributed. Without loss of generality, we investigate the performance of the first cell while other cells are all interfering cells. The th downlink user of the th cell is referred to as the th downlink user.
The massive MIMO network is working in timedivision duplex (TDD) mode. Thus it is assumed that the downlink channel of downlink users is estimated from their uplink pilot transmission within the same coherent fading block. Moreover, in order to improve the overall spectrum efficiency, we consider the cochannel deployment of D2D and downlink transmission, i.e., the D2D transmitters use the same spectrum as the cellular network. Note that the coexistence issue of massive MIMO uplink and D2D communications has been investigated in
[12], the focus of this paper is put on sharing the downlink transmission opportunities with D2D links. All D2D transmitters and receivers are equipped with single antenna. The th D2D link (transmitter or receiver) of the th cell is referred to as the th D2D link (transmitter or receiver). The notations of downlink and D2D transmissions are summarized below.
represent the downlink channel vectors from the
th BS to the th downlink user and th D2D receiver in the th cell, respectively. Each component of andis complex Gaussian with mean zero and variance
and respectively. and are the pathloss from the th BS to the th downlink user and th D2D receiver, where and are the distances from the th BS to the th downlink user and th D2D receiver. is the pathloss exponent between BSs and users. and are the aggregation of and within one cell. 
represents the downlink channel vector from the th D2D transmitters to the th D2D receiver and th downlink user in the th cell, respectively. and are complex Gaussian with mean zero and variance and , respectively. and are the pathloss from the th D2D transmitter to the th D2D receiver and th downlink user, where and are the distances from the th D2D transmitter to the th D2D receiver and th downlink user. is the pathloss exponent between users. and are aggregation of and .

is the pilot sequence of the th downlink user. We have and where is the transmission power of each mobile user and represents the pilot length in the uplink. is the aggregation of pilot sequences from active downlink users in the th cell. .
Remark 1.
The coexistence SIR analysis of massive MIMO uplink transmission and D2D transmission in [12] cannot be applied in downlink scenario, as the source of interference is completely different. In this paper, we also propose a new analytical framework to evaluate the asymptotic goodput performance with sufficiently large (but finite) number of antennas at BS . Note that the approach introduced in [12]
is for infinite M, which may not be accurate when M is only a few hundred. Moreover, we use Gaussian approximation to obtain a simple closedform expression of the cumulative distribution function (CDF) of SIR, based on which we also derive the average goodput as the performance metric so that the potential packet outage can be counted. These results cannot be obtained with the approach in
[12].IiB Channel Model
Since D2D links share the downlink transmission opportunities, they are silent in the uplink subframe. Thus in the channel estimation phase (as illustrated in Fig. 2), the received signal of the th BS is given by
With match filter, the estimated uplink channel can be written as
(1)  
where is the channel estimate error of the th BS. The first term is the desired CSI and the second term is the intercell interference due to pilot reuse (pilot contamination).
In the downlink transmission, the aggregated received signal of the users in the th cell (as illustrated in Fig. 1) is
(2) 
where is due to the interference from D2D transmitters. and are the downlink signals for the th downlink user and the th D2D receiver in the th cell. and are the aggregations of downlink data blocks for downlink and D2D receivers in the th cell with average power and .
Moreover, the aggregated received signal of the D2D receivers in the th cell during the downlink subframe can be written as
(3) 
We neglect the effect of noise because the noise is much smaller than the interference.
Since the channel is estimated at the BS before downlink transmission, the zeroforcing precoder based on imperfect channel state information (CSI) is designed as
Denoting the aggregated downlink data block before precoding as , the downlink signal for downlink users in the th cell is
(4)  
and the SIR of the th downlink user can be given by
(5) 
where is the channel estimate error of . Similarly, the downlink signal for D2D receivers in the th cell can be rewritten as
(6) 
It is assumed that the D2D transmitter has no knowledge of CSI, but the receiver has perfect CSI. Hence the SIR of the th D2D receiver can be given by
(7) 
Note that the D2D transmitter does not know , it has to determine the data rate according to the statistics of . Thus goodput is introduced in the following section as the performance metric.
IiC Average Goodput
It can be observed from (5) that the downlink SIR is determined by channel estimation error and CSI of interference channel , which may be unknown to the service BS. Hence the link capacity becomes random to the service BS, and it is possible that the scheduled downlink data rate may be larger than the channel capacity, which leads to the packet outage. In order to take the potential packet loss into consideration, we use the average goodput as the performance metric [13]. Given a scheduled downlink data rate for the th downlink user, the goodput is defined as
(8) 
where is is an indicator function with value if the event is true and otherwise. The the average goodput spanning all possible channel realization is given by
(9) 
In this paper, we consider the downlink transmission with a target outage probability
, thusDefine the SIR Threshold determined by as , where
(10) 
The average goodput of the th downlink user becomes
(11) 
where is the number of total downlink symbols within one subband of a frame.
Similarly, for the th D2D receiver, given the outage probability , the average goodput becomes
(12) 
where satisfies
(13) 
Iii Goodput Performance analysis
In this section, we first derive the asymptotic SIR expression for both downlink users and D2D receivers, and then provide the approximated expressions of goodput. First of all, we have the following lemma on asymptotic SIR.
Lemma 1 (Aysmptotic Downlink SIR).
When the number of BS antennas is sufficiently large, the asymptotic SIR of the th downlink user is given by
(14) 
and the asymptotic SIR of the th D2D receiver is
(15) 
Proof.
Please refer to the Appendix A. ∎
From equations (14) and (15), we can observe that the downlink performance is related to downlink users’ and D2D transmitters’ locations as well as the channel variation. Some of them are unknown to the service BS or D2D transmitters. In this case, the service BS and D2D transmitter cannot ensure that the data rate is below the channel capacity. However, if we know the distribution of the SIR, the outage probability can be controlled by setting the data rate appropriately. Therefore, we derive the CDF of downlink SIR for both downlink users and D2D receivers in the following.
Lemma 2 (CDF of SIRs).
Given that the number of BS antennas M and the number of interfering downlink users and D2D transmitters are sufficiently large, the CDF of the th downlink user’s downlink SIR can be approximated as
(16) 
where the
function is the tail probability of stantard normal distribution.
and are denoted as the expectation of and , while and are their variances.The CDF of the th D2D receiver’s downlink SIR can be approximated as
(17) 
where
function represents the tail probability of exponential distribution with expectation
and is the expectation of .Proof.
Please refer to the Appendix B. ∎
Remark 2.
Based on the uniform distribution of cellular users and D2D transmitters, and can be calculated as follows
and can be obtained Similarly. Note that the distance between D2D transmitters and receivers is much smaller than the cell radius, the distribution of D2D users can also be considered uniform, thus , and , can be calculated.
As a result, we have the following theorem on average goodput of downlink users and D2D receivers.
Theorem 1 (Average Goodput).
With outage probability , the average goodput of the th downlink user is given as
(18) 
where
Similarly, for the th D2D receiver the goodput data rate can be presented as
(19) 
where . Then the overall average goodput of the st cell is given as
(20) 
where and are expectations based on the distribution of locations for single downlink user’s average goodput and single D2D receiver’s average goodput.
Proof.
Theorem 1 is directly derived from lemma 2. ∎
The interference brought by D2D communication degrades the general performance of both downlink and D2D receivers, as revealed in (14) and (15). However, both downlink and D2D receivers with different locations may have different performance loss. Moreover, as the number of D2D links increases, the average goodput of both downlink users and D2D receivers degrades. However, the D2D number increase can improve the cell overall average goodput as revealed in (20). Thus the cell overall average goodput is determined by the tradeoff between the D2D number increase and the downlink users’ and D2D receivers’ average goodput degradation. As it may be quite complicated to figure out the tradeoff by numerical simulation, we show the analytical results in the next section.
Iv Simulation Results
In the simulation, we consider a network with 19 hexagonal cells each with radius . The number of downlink users in each cell is 10, and the antennas number of BS is 250. The length of pilot sequence is and the length of downlink symbols within one frame is . The pathloss exponents are and . The transmitting power of each BS and each D2D transmitter is 46 dBm and 23 dBm.
In Fig. 3, there are 10 D2D transmitters in each cell and the transmission distance of each D2D link is . For both downlink users and D2D receivers, we give the numerical results as well as the asymptotic results. It can be observed that the average goodput of downlink users decreases with respect to its distance to the service BS. This is because of stronger intercell interference. On the other hand, the average goodput of D2D links increases with respect to its distance to BS. This demonstrates the impact of interference from BS. Even there are a large number of antennas at the BS, its interference to D2D receiver is still strong and dominant.
Fig. 3 shows that the analytical results fit the actual performance quite well. Thus we can study the effect of D2D number with analytical expressions instead of complicated simulations. In Fig. 4, the effect of D2D number is demonstrated through analytical results, which show that the cell overall average goodput increases almost linearly with the D2D number, while the average goodput of all downlink users declines. Thus we reveal the tradeoff between cell overall average goodput and downlink users’ average goodput. Therefore, to achieve decent cell overall average goodput and fair average goodput for downlink users simultaneously, the number of D2D links per cell can be determined through our analytical results.
V Conclusion
In this paper, we focus on the average goodput performance analysis of a D2D underlay downlink massive MIMO system. It is assumed that the cell coverage is hexagonal and the distribution of downlink users and D2D transmitters is independent and uniform. The asymptotic SIR expressions for both downlink transmission and D2D links are firstly derived. To take the potential packet outage, we continue to derive the approximated expressions of average goodput for both downlink and D2D links, which measures the number of information bits successfully delivered to the receiver. Based on it, the tradeoff between the two types of links can be studied. Through simulation we show that the analytical results fit the numerical results quite well.
Appendix A Proof of Lemma 1
In massive MIMO systems, when is sufficiently large, we usually utilize asymptotic orthogonality of channel as
It is also the same for . Thus the interference term caused by neighboring BSs in Equation (4) can be simplified as
The first term remains because . The second term remains because . Similarly, for D2D receivers in the th cell, the interference from neighboring BSs can be simplified as
Therefore, the received signal of the th downlink user can be presented as
where the intracell interference for downlink users is neglected for it is much smaller than intercell interference. Similarly, the received signal of the th D2D receiver is
Appendix B Proof of lemma2
We first consider the downlink users. The asymptotic SIR expression is presented in (14). In the dominator, are independent variables with expectation and variance . Meanwhile, are also independent variables with expectation and variance . When the number of interfering cellular users and D2D transmitters is sufficiently large, we have and
, which satisfies requirements of the LindebergFeller Central Limit Theorem
[14]. According to the theorem, converges to a Gaussian variable when is sufficiently large. Thus, we apply Gaussian approximation on . Then the probability that the downlink SIR of the th cellular user is less than (coverage outage probability) can be written asThe probability that the SIR of the th D2D link is less than can be presented as
where , . Due to D2D receivers are quite close to their associated transmitters, the variation of the interference is relatively much smaller than the signal variation. Thus we can regard the interference as a constant. Since the downlink channel is a complex Gaussian variable, the signal power is an exponential variable, where the coverage outage probability can be simplified as
Thus (2) can be easily obtained. This finishes the proof.
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