A scheduling algorithm based on downlink performance for full-duplex MU-MIMO systems

A full-duplex (FD) multiuser multiple-input multiple-output (MU-MIMO) system, where a FD base station (BS) with multiple antennas serves multiple half-duplex (HD) user equipments (UEs) in both uplink (UL) and downlink (DL) via the same time-frequency resources, is considered. UE scheduling is in demand to manage the UL-to-DL interference (UDI) incurred by the FD operation. Existing scheduling algorithms require UDI channel state information between each pair of the candidate UEs, which incurs a signiﬁ-cant amount of overhead as the number of UEs grows. To reduce the overhead, a new UE scheduling algorithm that uses channel reciprocity and UL-DL duality to design a scheduling algorithm based on downlink performance was designed. By selecting UEs only based on their DL channels and received UDI strength, the proposed scheme no longer requires the massive UDI channel state information (CSI). Numerical results demonstrate the proposed algorithm can achieve a near optimal performance without knowing any UDI CSI.

and the influence on the selected downlink UEs. The paper [13] proposed an alternative UE scheduling algorithm. First, a downlink UE is selected based on signal-to-interference-plusnoise ratio (SINR), and then an uplink UE is selected based on the signal-to-leakage-and-noise ratio (SLNR), and so on alternately. However, the existing scheduling algorithms require FD BS to aggregate channel state information (CSI) of UDI channels between each pair of candidate UEs. Since each of the UDI channels needs to be estimated at the UL/DL UEs first and then feedback to the BS, the scheduling algorithms requiring UDI CSI will incur a massive overhead as the number of UEs increases [14]. Therefore, for the existing UE scheduling algorithm, obtaining the UDI channel estimation value will become a performance bottleneck, and the entire system is not scalable. How to schedule UEs in a FD cellular without knowing the UDI channels remains an open and interesting issue.
This paper proposes a DL-centric UE scheduling strategy which does not require CSI of the UDI channels by utilizing two fundamental properties of wireless communications: channel reciprocity and UL-DL duality [15]. The existing UE scheduling schemes require UDI CSI mainly because they rely on the CSI of the UDI channels to calculate the caused interference to other DL UEs when selecting an UL UE. An important fact that each UL UE will also play as a DL UE in another time slot or frequency band, and its caused interference to other UEs in the UL mode will be proportional to its received interference from those UEs in the DL mode due to channel reciprocity. Therefore, we propose to always schedule a new UE based on its DL performance. The DL performance is in turn determined by the DL channel and the received UDI power, which can be directly measured by the DL UE, which leads to reduction of CSI overhead of the UDI channels. Moreover, the UL-DL duality implies that the UL and the DL performance of a cell are proportional to each other. Thus the proposed DL-centric UE scheduling will also improve the UL performance to some extent. A specific DL-centric UE scheduling algorithm is proposed to illustrate the above idea, and numerical results verify that the proposed algorithm still achieves a near optimal performance without knowing the UDI channels.
The paper is organised as follows: Section 2 articulates the system model; Section 3 proposes a DL-centric UE scheduling algorithm. Section 4 analyzes the complexity of the existing algorithm and the proposed algorithm from three aspects: search complexity, channel estimation overhead and information feedback. Section 5 presents the simulation results, comparing the existing algorithm and the proposed algorithm in terms of complexity and system sum rate performance. Finally, Section 6 gives the conclusion.

SYSTEM MODEL
As shown in Figure 1, we consider a single-cell FD MU-MIMO system, where the FD BS is equipped with N t transmit antennas and N r receive antennas, and suppose that N t = N r = N . where a BS is equipped with FD MIMO radios with N antennas. The BS serves multiple single-antenna UEs with HD radios in both UL and DL channels simultaneously over the same frequency band. The set of candidate UEs within the cell is denoted by  = {1, 2, … , K }, with K being the total number of UEs, and suppose that K >2N . The HD UEs operate in TDD mode, where all scheduled UEs can be classified into two disjoint sets according to the assignment of TDD frames: where A j ∈ and B i ∈ represent the j -th UE in  A and the i-th UE in  B , respectively. The basic TDD frame structure for the UEs are depicted in Figure 2, where each frame is equally divided into two parts, namely T 1 and T 2 . During T 1 the UEs in  A and  B respectively operate in DL and UL, and vice versa during T 2 . Since both UL and DL transmissions are active concurrently, SI at the FD BS and UDI at the DL UEs are inevitable. Figure 1 demonstrates these interferences during T 1 for illustration purpose.
In this work, we model imperfect FD transmit radio chain as an equivalent additive white Gaussian "transmitter noise" according to the dynamic range model introduced in [16]. The variance of the transmitter noise is ( ≪1) times the power of the transmit signals, where is the dynamic range parameter. The transmitter noise will propagate through the SI channel and can not be mitigated by digital SI cancellation, thus it will become residual SI at the FD receiver. However, the transmitter noise that propagates over the DL channels will be attenuated by the path loss and can be ignored compared with the receiver noise [16].
During T 1 the UEs in  B are active in the DL transmission, where the data symbol x B i intended for UE B i is multiplied by a unit-norm beamforming vector v B i ∈ℂ N ×1 . Consequently, the received signal at UE B i is given by where h B i ∈ ℂ N ×1 denotes the channel vector between the BS and UE B i , g A j ,B i represents the channel coefficient from UE A j to UE B i , x A j is the transmitted data symbol of UE A j , and n B i is the thermal noise which follows  (0, 2 UE ). In addi-tion, the allocated power for UE B i is P DL term in (1) accounts for multiuser interference (MUI) and the third term represents the UDI between UEs in  A and  B . We exploit the zero-forcing (ZF) beamforming for the DL to mitigate the MUI [17]. As a result, the DL rate for the UEs in  B can be written as Moreover, during T 1 the UEs in  A is active in the UL transmission, and the received signal vector at the FD BS is given by where h A j ∈ ℂ N ×1 denotes the channel vector between UE A j and the FD BS, H SI ∈ ℂ N ×N is the SI channel, and e BS ∈ ℂ N ×1 represents the equivalent "transmit noise" caused by the imperfect FD transmit front-end chain [16], which follows  (0, P BS ∕N I N ) with P BS denoting the total transmit power of the FD BS. Finally, n BS ∼ (0, 2 BS I N ) indicates the receiver thermal noise at the BS. I N means the unit matrix with a dimension of N .
The FD BS performs SI cancellation based on the SI CSI and its own DL signal. Hence after the SI cancellation, the UL signal in (3) can be expressed as For the above UL signal, we adopt a minimum mean square error (MMSE) receiver at the BS [18], whereby the the UL rate of the UEs in  A can be written as During T 2 the UEs in  A and  B are respectively active in the DL and UL transmission, where the DL rate R DL A and UL rate R UL B can be respectively obtained similarly to (2) and (5) as Consequently, the sum rate of the FD system during T 1 and T 2 can be respectively given by

DOWNLINK-CENTRIC UE SCHEDULING ALGORITHM
Unlike existing UE scheduling algorithms [9,13] that aim to maximize R T 1 (or equivalently R T 2 ) without considering the structure of the TDD frames, that is In this letter we propose a DL-centric UE scheduling objective that can be mathematically expressed as where R UL A has been replaced by R DL A based on the following intuitions: firstly, when the residual SI and UDI are both small compared with the desired signal, the DL and UL of the FD system can achieve the same performance due to the UL-DL duality [15]; secondly, in high SINR regime MMSE receiver mimics ZF receiver [18], thus the sum rate of UL transmission with MMSE receiver R UL A will be close to that of DL transmission with ZF beamformer R DL A . Inspired by the semi-orthogonal UE selection (SUS) algorithm for HD MU-MIMO systems in [17], we propose a DLcentric UE scheduling algorithm to solve (10), which is summarized in Algorithm 1, as shown in Figure 3.
Algorithm 1 is the proposed DL-centric scheduling algorithm, where the ⌈N ∕S ⌉ layer outer loop means that the entire algorithm requires ⌈N ∕S ⌉ TDD frames to complete. The first S -layer inner loop means that S new UEs are selected for the set  B during T 1 of each TDD frame. Similarly, the second S -layer inner loop means selecting S new UEs for the set  A during T 2  Figure 4 shows a schematic diagram of the entire operation of Algorithm 1. In addition, the basis vectors {f (1) , … , f (m−1) } and {g (1) , … , g (m−1) } are the results of Gram-Schmidt orthogonalization of the selected UE downlink channel vectors in the set  A and set  B , respectively. Projecting the candidate UE channel vector to the zero space of the basis vector can obtain the effective channel gain of the selected UE during zero-forcing (ZF) beamforming, and then determine the equivalent SINR value of the candidate UE.
The number S of new UEs scheduled each time in Algorithm 1 is a key parameter, which not only determines the time overhead of the entire user scheduling, but also affects the sum rate performance of the scheduling result. For example, when S = 1, Algorithm 1 selects only one new UE each time, and a total of N TDD frames are required to complete N pairs of UEs scheduling. The scheduling process is similar to alternate scheduling in existing algorithms. The following simulation results show that the two have similar sum rate performance. When S = N , Algorithm 1 can complete the entire scheduling in a TDD frame, select N new UEs for the set  B at time T 1 of the frame, and then select N new UEs for the set  A at time T 2 of the frame. The scheduling process is similar to the two-step scheduling in the existing algorithm, and the subsequent simulation results also show that the two have similar sum rate performance. In summary, the value of 1 ≤ S ≤ N determines the trade-off between the scheduling time overhead and the fineness of each scheduling. During T 1 of each of these TDD frames, the candidate UEs in  B are in DL, thus can measure and feedback their interference-plus-noise (IN) metrics (the sum of the UDI power of the selected UE and the noise power of its own receiver). Then during T 2 of each TDD frame, the candidate UEs in  A are in DL, thus can measure and feedback their IN metrics. Therefore, one significant advantage of the proposed algorithm is that it does not require the CSI of the UDI channels between the K candidate UEs. The threshold parameter in Algorithm 1 is another key parameter. Assuming = 10 dB, it means that candidate UEs whose UDI is higher than the noise floor by more than 10 dB will be directly eliminated during the scheduling process. This makes the finally selected UEs not suffer excessive interference in the downlink, and does not cause excessive interference to other UEs in the uplink. If takes a larger value, it means that the hard limit on UDI in the scheduling process is relaxed. Usually in the conventional signal-to-noise ratio (SNR) range of 5 → 40 dB, the FD system needs to control the interferenceto-noise ratio (INR) within 10 dB to obtain a significant spectral efficiency gain. This article relaxes slightly in the following simulations, and = 15 dB is used unless otherwise specified. Algorithm 1 reduces the senseless operations in subsequent iterations by eliminating those UEs that suffer from high UDI, reduces the complexity of the algorithm, and greatly reduces the running time of the algorithm. The advantage in algorithmic complexity is further analyzed and demonstrated via comparison with existing schemes [9,13] in simulations.

Search complexity
Search complexity refers to the size of the search space in the scheduling process, that is, the total number of candidate combinations. Let us first analyze the search complexity of the optimal scheduling algorithm using exhaustive search. Assuming that the number of selected downlink UEs is m and the number of uplink UEs is n, the exhaustive search selects m+n UEs from K UEs, and there are   . This formula increases exponentially with the increase of the number K of candidate UEs, so exhaustive search is only applicable to the case where the number of candidate UEs is small.
The two-step scheduling algorithm and the alternate scheduling algorithm both schedule UEs one by one. When selecting the i-th UE, one needs to be selected from K − i + 1 candidate UEs, and finally a total of 2N UEs are scheduled at most. Therefore, the search complexity is 2N + 1). This formula increases linearly with the increase of the number K of candidate UEs, which is far lower than the above exhaustive search.
Algorithm 1 also schedules UEs one by one. In the worst case, the search complexity is the same as the previous two algorithms, that is, the upper limit is N (2K − 2N + 1). However, Algorithm 1 excludes those UEs whose UDI exceeds a certain threshold from the candidate set, and no longer considers them in subsequent loops, which will reduce the overall search complexity of the algorithm. The simulation results show that the search complexity of the algorithm is about half of the twostep user scheduling algorithm and the alternate user scheduling algorithm , which is about N (K − N ).

Channel estimation overhead
All user scheduling algorithms need to know the uplink and downlink channel information between BS and UEs, that is, the channel estimation overhead of various algorithms is the same, so we mainly compare the UDI channel gain estimation overhead of various algorithms. The optimal scheduling algorithm, two-step scheduling algorithm and alternate scheduling algorithm all need to use the UDI channel gain between candidate UEs, and a total of ( K 2 ) = K (K −1)∕2 UDI channel gains need to be estimated. However, there is no need to estimate ( K 2 ) UDI channels one by one, because when any candidate UE sends a pilot signal, other candidate UEs can estimate the UDI channel gain at the same time. Therefore, the two-step scheduling algorithm and the alternate scheduling algorithm UDI channel estimation overhead is equal to the number of candidate users K .
Although Algorithm 1 does not need to know the UDI channel gain between candidate UEs, in each outer loop of the algorithm, all candidate UEs need to estimate the sum of their received interference noise power IN k . Fortunately, all candidate UEs are in the downlink mode during each scheduling period, and they can estimate their IN at the same time. Therefore, the channel estimation overhead of Algorithm 1 is equal to the number ⌈N ∕S ⌉ of outer loops of the algorithm.

Information feedback amount
The optimal scheduling algorithm, the two-step scheduling algorithm and the alternate scheduling algorithm all need to feed back the UDI channel gains between K candidate UEs to the BS, so the channel information feedback overhead is In the running process of Algorithm 1, 2S new users are selected in each outer loop, and some users with too high IN will be excluded during the scheduling process. Therefore, in the i-th outer loop, the candidate sets  B and  A have at most K −(2i−2)S and K −(2i−1)S users need to feedback IN. Therefore, the upper bound of the information feedback amount of Algorithm 1 is: In short, in terms of search complexity, Algorithm 1 is half of the existing two-step scheduling algorithm and alternate scheduling algorithm; in terms of channel estimation overhead, the existing algorithm is directly proportional to the number K of candidate UEs, while Algorithm 1 is inversely proportional to S and has nothing to do with K ; In terms of information feedback, the existing algorithms increase with the square of K , and Algorithm 1 increases linearly. As the number K

SIMULATION RESULTS
In this section, we present numerical simulations under the 3GPP LTE specifications [19] for small cell deployments to evaluate the system performance. The K candidate UEs are uniformly and randomly dropped within a radius of 40 m around the BS, with a minimum UE-BS distance of 30 m. The largescale fading for BS-UE, UE-UE channels which include path loss and shadowing effect follow the 3GPP model in [19]. The SI channel model is based on the existing experiment data given in [20], where the dynamic range parameter is = −70 dB and the propagation loss of the SI channel is 40 dB. We set = 15 dB for the IN threshold. We run 2000 of random drops of UEs in each simulation. Detailed simulation parameters are the same as those given in [21].

Complexity comparison
As shown in Figure 5, under different N and K , we compare the alternate scheduling algorithm, the two-step scheduling algorithm, and the search complexity of Algorithm 1 when S = 1. This paper defines the search complexity as the sum of the number of candidate UEs in each cycle during the execution of the entire scheduling algorithm. Theoretical analysis shows that the search complexity of optimal UE scheduling and two-step scheduling are and N (2K −2N +1) respectively, where the former increases exponentially with K (hence not shown in Figure 5) and the latter is linear growth. The search complexity of the alternate scheduling algorithm is the same as that of the two-step scheduling algorithm. In addition, the search complexity of Algorithm 1 is (2K −2N +S )⌈N ∕S ⌉ as the upper limit, but there is no specific closed expression. The value in Figure 5 is obtained through numerical simulation. Figure 5 shows that the search complexity of Algorithm 1 also increases linearly with K , but it is significantly lower than the existing two algorithms. The reason is that Algorithm 1 excludes those UEs whose UDI exceeds a certain threshold from the candidate set, so they do not need to be considered in the subsequent cycle. Figure 6 shows how the UDI channel estimation overhead of the existing scheduling algorithm (including exhaustive search, alternate scheduling and two-step scheduling) and Algorithm 1 changes with K when the number of BS transceiver antennas N = 6. The data in this figure come directly from the previous theoretical analysis results, which more intuitively shows that in the scenario where the number of candidate UEs K >2N , Algorithm 1 has a significant advantage in terms of UDI channel estimation overhead. Figure 7 shows the variation of the channel information feedback amount of the existing scheduling algorithm and Algorithm 1 with K when the number of BS transceiver antennas N = 6. The data of the existing algorithms in this figure are directly derived from the previous theoretical analysis results, the data of Algorithm 1 are obtained from simulation. Figure 7 shows that as the number of users S increases from 1 to N each time, the amount of information feedback of Algorithm 1 gradually decreases, and all increase linearly with K . Even when S = 1, the channel information feedback of Algorithm 1 is significantly lower than that of the existing algorithm.

Sum rate performance comparison
As shown in Figure 8, in the scenario where the transmit power of BS and UE are 24 dBm and 23 dBm, the cell radius is 40 m, and the number of BS transceiver antennas N = 2, the average sum rate (R T 1 +R T 2 )∕2 obtained by different user scheduling algorithms is compared with the change of the number K of candidate UEs. Among them, the optimal UE scheduling for FD and HD systems is obtained through the exhaustive search method, and the optimal performance of the FD system simulated by the exhaustive search method is too long when K >30, so it is not shown in Figure 8. First, Figure 8 shows that the performance of Algorithm 1 when S = 1 is almost the same as that of the alternate scheduling algorithm, and both are very close to the optimal performance. The alternate scheduling algorithm selects a new UE for the downlink and then selects a new UE for the uplink, and then alternately loops until enough UEs are selected. The difference between the alternate scheduling algorithm and Algorithm 1 is that when it selects new uplink UEs, the metrics LN (leakageand-noise) used is the sum of the uplink UEs leakage interference and receiver noise power, not the IN metrics. When all UEs have the same transmit power, the LN index of a certain uplink UE is the same as the IN metrics after the UE enters the downlink mode. Therefore, in the same scenario, when S = 1, the scheduling results of the alternate scheduling algorithm and Algorithm 1 are similar, so their sum rate performance are relatively close. The huge advantage of Algorithm 1 over the former is that it does not need to know the UDI channel estimation value between candidate UEs, and has a smaller channel information estimation and feedback overhead.
Secondly, Figure 8 shows that when the number of candidate UEs K is small (for example, K = 10), the performance of Algorithm 1 is even slightly higher than that of alternate scheduling. This is because Algorithm 1 excludes candidate UEs with excessively high IN during the scheduling process, while alternate scheduling does not have this operation. Therefore, at the end of the alternate scheduling process, UEs with higher SINR (or SLNR) but stronger UDI may be selected, thereby dragging down the overall system performance.
Again, Figure 8 shows that the sum rate performance of Algorithm 1 when S = N = 2 is similar to but slightly lower than the two-step scheduling algorithm, and both are much worse than the performance of Algorithm 1 when S = 1. The reason is that the operation process of Algorithm 1 when S = N = 2 is similar to the two-step scheduling algorithm. When scheduling downlink UEs in the first step, the influence of UDI is not considered. The selected downlink UEs are randomly distributed in the cell, which may be unfavorable the second step uplink UE scheduling. In addition, the performance of Algorithm 1 is slightly lower than that of the two-step scheduling algorithm when S = N = 2, because the two-step scheduling algorithm directly uses the uplink and downlink sum rate of formula (9) as the performance criterion in the second step, while Algorithm 1 using formula (10) as the performance criterion, there is a certain estimation bias. Moreover, Figure 8 shows that FD with random UE scheduling achieves little performance gain over HD with random UE scheduling due to the UDI, and that FD with the proposed UE scheduling obtains a significant rate gain over HD with optimal UE scheduling.
In order to better observe the impact of the number S of scheduled users on the performance of Algorithm 1, in the scenario where the cell radius is 80 m and the number N = 6 of BS transceiver antennas, the change of the sum rate performance with the number K of candidate UEs is simulated. Due to the larger cell radius, the BS and UE transmit power increased to 34 dBm and 33 dBm respectively in the simulation.
As shown in Figure 9, the sum rate performance of Algorithm 1 under different values of = 15 dB and S = [1, 2, 3, 6] is shown. A similar phenomenon to Figure 8 is that the performance of Algorithm 1 when S = 1 is close to that of the alternate scheduling algorithm, and the performance of S = N = 6 is close to the two-step scheduling algorithm. When S increases from 1 to 6, the sum rate performance of Algorithm 1 gradually decreases. The reason is that as the number S of scheduled UEs increases, the fineness of Algorithm 1 gradually decreases and it gets closer and closer to the two-step scheduling algorithm.
To further observe the influence of parameter on the performance of Algorithm 1, based on the parameter setting in Figure 9, set = 100 dB, and the simulation result is shown in  Figure 10 shows that the performance of Algorithm 1 almost coincides with the alternate scheduling when S = 1. The reason is the same as the analysis of the similar phenomenon in Figure 8 above. Compared with the situation in Figure 9 = 15 dB, in Figure 10 = 100 dB, the sum rate performance of Algorithm 1 at S = [2, 3, 6] is significantly reduced, even lower than the two-step scheduling. The reason is that after the IN threshold limit is cancelled, there may be strong UDI in the scheduling result of Algorithm 1 when S = [2, 3, 6], which will drag down the overall sum rate performance. It can be seen that setting a reasonable IN threshold in Algorithm 1 can not only reduce the search complexity, but is also extremely necessary to ensure the scheduling result sum rate performance.

CONCLUSIONS
We have proposed a low-complexity DL-centric UE scheduling algorithm for FD MU-MIMO systems that only requires DL channels and received UDI strength, based on the reciprocity of the UDI channels to reduce the CSI overhead. Numerical results have verified that the proposed algorithm maintains a near optimal UE scheduling, which is close to the optimal scheduling algorithm using exhaustive search, and is almost twice that of the HD system using optimal UE scheduling. This shows the effectiveness of the proposed UE scheduling algorithm in suppressing UDI in the FD MU-MIMO network. In terms of search complexity, this algorithm is half lower than the existing two algorithms. In addition, the algorithm is lower than the existing two algorithms in terms of channel estimation overhead and information feedback, and as the number of candidate UEs increases, the relative advantage of the algorithm proposed is greater. The DL-centric scheduling ideas put forward can be extended to a more general multi-cell cellular system. In the multi-cell FD cellular system, each DL UE is not only subject to intra-cell UE-UE interference, but also inter-cell UE-UE interference. As a result, the traditional user scheduling algorithms will face greater channel estimation and feedback overhead, while the scheduling algorithm proposed will have greater advantages in multi-cell system because it does not depend on UDI channel gain. Therefore, the popularization of the DLcentric user scheduling algorithms in multi-cell cellular systems is a promising and interesting problem for future work.