I Introduction
With the explosive growth of mobile devices and bandwidthhungry applications such as video streaming and multimedia file sharing, user demands for mobile broadband are undergoing an unprecedented rise, which pushes the limits of current 4G LTE systems [2]. To improve spectrum efficiency and user experience, devicetodevice (D2D) communications underlaying LTE networks have been proposed as a promising approach to facilitate high data rate services in a short range and boost the performance of LTE systems [2, 4, 3] for future 5G communications and beyond. D2D communications enable mobile devices in proximity to establish a direct link without traversing the base station (BS), and reuse the spectrum with the LTE system by the control of the BS, which enjoy the benefits of fast access to the radio spectrum in terms of proximity gain, reuse gain, and paring gain [6, 7, 8, 5].
Recently, the operators expand LTE services to unlicensed spectrum to alleviate congestion. Mobile traffic offloading is a conventional method to utilize the unlicensed spectrum, in which the data is offloaded to WiFi networks [9, 10, 11, 12]. However, the offloading schemes commonly suffer from low efficiency and poor guarantee of qualityofservice (QoS) due to the inferior performance of WiFi and the lack of coordination between cellular and WiFi systems [13]. In light of these issues, the 3rd Generation Partnership Project (3GPP) has initialed the research on licensed assisted access (LAA) to integrate the unlicensed carriers with the licensed ones for data transmission [14]. Based on the LAA scheme, the LTEunlicensed (LTEU) technology is proposed to extend LTE to the unlicensed spectrum by the existing carrier aggregation (CA) technology [15, 16, 17, 18, 19, 20].
As LTEU technology shows satisfying performance, D2D communications underlaying LTE networks in unlicensed spectrum becomes a natural solution to further improve system throughput, in particular hotspot areas with large number of D2D links. However, due to the mutual interference among LTEU network, D2D users, and the opportunistic feature of unlicensed channel access in existing WiFi systems, D2DUnlicensed (D2DU) communication turns out to be much complicated. In this paper, we investigate the underlaid D2D communications in unlicensed spectrum. Note that different from most previous peertopeer communication technologies in unlicensed spectrum such as WiFi Direct [21, 22, 23], which builds the network upon the IEEE 802.11 infrastructure mode and allows users to negotiate with each other in an APlike method, D2DU requires assist and control from the central BS. With the involvement of BS, D2D users can work as an underlay of LTE system in both licensed and unlicensed spectra.
As aforementioned, the major challenges of implementing D2DU are (1) the opportunistic feature of unlicensed channel access due to current 802.11 mechanism adopted by WiFi systems; and (2) the interference management issue among the three types of systems, i.e., the access and transmission of D2DU users do not cause significant interference to the existing WiFi system as well as the LTEU system. To cope with the first challenge and be compatible with current LTE standards^{1}^{1}1The work in [24] proposed an access protocol based on listenbeforetalk (LBT) mechanism to mitigate collision with the ongoing WiFi transmissions. However, LBT requires changes to the LTE specifications., we design a duty cycle based protocol [25, 26, 27], in which the BS schedules transmissions according to the data demand. To tackle the second challenge, unlike the work in [28]
which only maximizes the total sumrate, we investigate the subchannel allocation problem to leverage the maximization of the sumrate of LTEU and D2DU users and the protection of WiFi performance. This subchannel allocation problem is originally a mixedinteger nonlinear programming (MINLP) problem, which is generally NPhard. For this reason, we reformulate it as a manytomany game with externalities
[30, 29, 31, 33, 32], and solve it with low computational complexity by designing an iterative usersubchannel swapmatching algorithm.The major contributions of this paper are summarized as follows.

We propose a feasible duty cycle based protocol for the LTEU and D2DU users to utilize the unlicensed spectrum.

An approximated model is elaborated to evaluate the interference to WiFi networks introduced by LTEU and D2DU users.

We investigate the subchannel allocation problem by a manytomany matching game with externality, and analyze its stability, convergence, complexity, and optimality.
The rest of the paper is organized as follows. In Section II, we first introduce the system model for the coexistence among LTE, D2D, and WiFi users, as well as their PHY/MAC features, and then discuss the interference issues. In Section III, a duty cycle based protocol is elaborated to support LTEU and D2DU users accessing the unlicensed spectrum. Then we formulate the optimization problem for subchannel allocation as a manytomany matching game with externalities in Section IV. In Section V, we develop an iterative algorithm to solve the manytomany matching game with its property analysis. In Section VI, the system performance is discussed. Numerical results in Section VII evaluate the proposed algorithm and the performance of the D2DU. Finally, conclusion remarks are drawn in Section VIII.
Ii System Model
In this section, we present the coexistence scenario of LTE, D2D, and WiFi systems in both licensed and unlicensed spectra. Then, the characteristics of LTE, D2D, and WiFi systems in the MAC and PHY layers are elaborated respectively. Furthermore, we discuss the interference issue within the coexistence network at the end.
Iia Scenario Description
As shown in Fig. 1, we consider an uplink scenario in an LTE network with one BS and WiFi access points (APs), denoted by . There exist LTE users, denoted by CU, and D2D users, denoted by , where D and D represent the transmitter and receiver of D2D user D, respectively. The system owns licensed subchannels with uniform bandwidth to support orthogonal frequency division multiple access (OFDMA) transmissions, denoted by .
For the WiFi system, we assume that within the coverage of AP, there exist WiFi users marked by WU. Besides, we assume that there are unlicensed channels to support different APs, e.g., there are 23 channels for IEEE 802.11n in the 5GHz band, and BS will select one of them to support LTEU and D2DU users. Since the bandwidth of a unlicensed channel is much wider than one licensed subchannel in LTE system, each LTE or D2D user only requires a fraction of the unlicensed channel. Thus, to reuse the unlicensed channel more efficiently, the unlicensed channel is divided to unlicensed subchannels with bandwidth [34], marked by , so that multiple LTEU users and D2D pairs can transmit on the unlicensed channel concurrently.
IiB Characteristics of LTEU, D2D, and WiFi Networks
In this part, we sequentially elaborate the PHY/MAC characteristics of the coexisting systems, i.e., LTE and D2D users in the licensed/unlicensed bands, and existing WiFi characteristics in the unlicensed band.
We assume that all devices transmit with fixed power in this work. Specifically, the transmit power of an LTE and D2D transmitter on any subchannel is fixed on and , respectively; and the transmit power of the APs as well as the WiFi users over the whole unlicensed channel is fixed on . The free space propagation pathloss model with Rayleigh fading is adopted to model the channel gain between any two devices in the network, i.e., for the link from device to device , the received power can be expressed as
(1) 
where represents the transmit power of user , is the distance between devices and , is the pathloss exponent, is the constant power gains factor introduced by amplifier and antenna, is a complex Gaussian variable representing Rayleigh fading, and
follows lognormal distribution representing shadowing fading. Besides, we assume that the thermal noise at each device satisfies independent Gaussian distribution with zero mean and the same variance
.IiB1 LTE network and underlaid D2D users
In PHY layer, LTE and D2D users can utilize both the licensed and unlicensed spectra orthogonally. We assume that each user is able to occupy multiple subchannels. In addition, to guarantee reliable transmission of the control signaling, an active LTE/D2D user must hold at least one licensed subchannel [16]. Similar to the channel sharing in the licensed spectrum, D2DU users can work as an underlay of the LTEU users. In other words, D2D users can utilize the licensed/unlicensed subchannels concurrently occupied by some LTE users.
As for the MAC layer, the LTE/D2D systems adopt a centralized MAC protocol. The BS controls the access of both types of users and decides the subchannel allocation in a centralized manner to mitigate mutual interference or maximize the system sumrate.
IiB2 WiFi systems
The WiFi systems operate only in the unlicensed spectrum. Different from the OFDMAbased channel utilization in LTE systems, the WiFi transmission covers the whole unlicensed channel. Thus, WiFi systems only allow one user to occupy the channel at a time.
For the MAC layer, without a central controller, the WiFi systems adopt a sensing and contentionbased MAC protocol, i.e., carrier sense multiple access with collision avoidance (CSMA/CA) [35]. Specifically, before transmission, a WiFi user first listens to the intended channel. If the channel is unoccupied, the WiFi user begins backoff process to avoid collision. Otherwise, the WiFi user keeps sensing until the channel is judged idle.
IiC Evaluation of Interference to WiFi Systems
When LTEU and D2DU users occupy the unlicensed channel, the nearby WiFi users cannot access, and thus the performance of WiFi system would be severely degraded. To quantify the performance degradation brought by LTEU and D2DU users, we introduce the definition of interference range
on the WiFi network. Within the interference range, each WiFi user is able to detect the channel unavailable, and then suspend their transmission attempts. In practice, the fading value varies between subframes, and thus, it is difficult for BS to detect the interference range of each LTEU or D2DU user at the beginning of a subframe. To better model the interference, the interference range of a LTEU/D2DU user is defined as the area where the expectation of the received power from this user exceeds the power threshold. Therefore, the interference range to the WiFi network is a circle centered at the transmitter, whose radius is positively related to the transmit power. Intuitively, the users with large interference range has low probability of utilizing unlicensed subchannels, because large numbers of WiFi users will be interfered by this user. On contrary, the users with small interference range are more likely to utilize unlicensed subchannels due to their limited interference to the WiFi network.
However, when multiple LTEU and D2DU users transmit on the unlicensed spectrum concurrently, their individual interference circles may overlap, which is hard to derive the closed form expression for the area of the total interference range. For this reason, in the following of this part, we present an approximated model of the interference range to evaluate the performance degradation in WiFi system. Intuitively speaking, a smaller interference range can be obtained if the BS allocates the unlicensed subchannels to those adjacent users rather than those whose interference ranges do not overlap. Inspired by this observation, we use the minimum distance between a new LTEU/D2DU user to others to approximate the additional interference range introduced by this addon user. Let and denote the radii of individual interference circles of LTEU and D2DU users, respectively, where . And we define as the user set in which users utilize unlicensed subchannels. With these notations, the weight functions for LTEU and D2DU users are given as below.
IiC1 LTEU user
When LTEU user CU is allocated to the unlicensed subchannel the first time, and CU is an LTEU user, the increased area of interference range is related to the distance between CU and CU, denoted by ^{2}^{2}2LTE users CU and CU will inform the BS of their locations, and then the BS can calculate the distance between these two users.. When the interference ranges of these two LTE users overlaps, i.e., , as CU and CU in Fig. 2, we assume that the weight is proportional to the distance . As CU and CU illustrated in Fig. 2, their interference ranges do not overlap, i.e., , the increased area will not grow as the distance . Therefore, the weight function for the increased area can be expressed as
(2) 
On the other hand, given D2DU user D, the increased area of interference range is also related to the distance between CU and D, . Note that , when the interference range of D is contained by that of CU as CU and D in Fig. 2, i.e., , the increased area is proportional to the increased diameter . When the interference range of CU overlaps with but does not contain that of D as D and CU, that is, , the weight is proportional to the distance as well. Besides, when the interference ranges of D and CU do not overlap as CU and D, i.e., , the increased area is a constant. Therefore, the weight function for the increased area is written by
(3) 
The weight of LTEU user CU is the minimum increased interference range between CU and any user allocated to unlicensed subchannels., that is,
(4) 
IiC2 D2DU user
Similar to the LTEU users, if D2DU user D is the first time to utilize unlicensed subchannels, and D is a D2DU user, the increased range is related to the distance between D and D. As illustrated in Fig. 2, the increased range can also be calculated under two conditions: (1) the interference ranges of these two D2D users overlap; (2) their interference ranges do not overlap. Thus, the weight function between D and D is provided by
(5) 
In addition, if there already exists LTEU user CU, the increased area is also related to the distance between CU and D, which can be given by
(6) 
Therefore, the weight of D is
(7) 
IiD Interference in the LTEU/D2D network
The mutual interference between the LTE and D2D users is analyzed in this subsection. We assume that one subchannel can be allocated to a maximum of one LTE user, and a subchannel can be allocated to at most users for the sake of QoS. Besides, we also assume that a user can utilize at most subchannels including licensed and unlicensed ones for the sake of fairness.
First, some notations regarding the subchannel allocation are listed as follows. The subchannel allocation matrix for LTE and D2D users is denoted by
(8) 
where , and stand for the subchannel allocation matrices for the LTE and D2D users, respectively. The values of and are defined as
(9) 
and
(10) 
Besides, we define the access indicators and to respectively represent whether the LTE and D2D users can access the unlicensed channel. If the LTE user CU can access the unlicensed channel, ; otherwise, . And it is the same for D2D users. We also define to represent the set of LTE and D2D users to which subchannel SC is allocated.
IiD1 Interference analysis in the licensed spectrum
In the licensed subchannels, under the assumption that a subchannel can be allocated to a maximum of one LTE user, the LTE users can only receive the cochannel interference from the underlaid D2D users, while the interference received by D2D users might be from LTE users and other cochannel D2D users. The SINR at the receiver of BS from CU over licensed subchannel SC can be given by
(11) 
where and represent the channel gains from CU and D to the BS, respectively. The SINR at D over licensed subchannel SC can be expressed as
(12) 
where and are the channel gains from D and CU to D, respectively. The data rates of CU and D over licensed subchannel SC are respectively given by
(13) 
IiD2 Interference analysis in the unlicensed spectrum
In the unlicensed subchannels, the D2DU and LTEU users will not only receive the mutual interference from D2DU and LTEU users as in the licensed subchannels, but also the interference from the WiFi users. Therefore, the SINR at the receiver of BS from CU over unlicensed subchannel SC is
(14) 
where is the total interference from WiFi system to BS. Here, the interference can be calculated as , where is the channel gain from the transmitting WiFi user WU to the BS. Similarly, the SINR at D over unlicensed subchannel SC can be written as
(15) 
where is the interference from WiFi system to D, whose value is , with representing the channel gain from the active WU to D.
The data rates of CU and D over unlicensed subchannel SC are respectively given by
(16) 
Iii Duty Cycle Based D2DU Protocol
In this section, we propose a duty cycle [25] based protocol for the LTEU and D2DU users to share the unlicensed spectrum with WiFi systems. The basic principle of the protocol is to allow the LTEU and D2DU users to access the unlicensed spectrum while protecting the incumbent WiFi performance.
Iiia Overview of the Proposed Protocol
As illustrated in Fig. 3, similar to the LTE standard, the timeline is slotted into subframes with length (e.g., 1ms in the LTE standard). There are three types of subframes, namely sensing subframes (SSs), transmission subframes, and reserved WiFi subframes. The SSs are inserted before the LTEU and D2D users attempt to select channel and initiate transmission to avoid collision with the ongoing WiFi transmissions. In transmission subframes, the LTEU and D2DU users perform transmission as in the conventional LTE standard. Then, to further protect the WiFi performance, we reserve several subframes for WiFi transmission, during which the LTEU users are not allowed to utilize the unlicensed spectrum.
IiiB Coexistence Mechanism
Without modifying current LTE PHY/MAC standards, two mechanism are used to safeguard that LTEU/D2DU users do not bring severe interference to their neighboring users in unlicensed spectrum. First, channel selection is performed to choose the cleanest channel avoiding the collision between the ongoing WiFi users and LTEU/D2DU users. In the event that no clean channel is available, channel sensing transmission is used to support transmission for D2DU/LTEU users.
IiiB1 Channel selection
In SS, LTEU/D2DU users will scan the unlicensed spectrum and identify the cleanest channel from the unlicensed channels for the uplink transmission. For an LTEU/D2DU user, the transmitter will perform energy detection, and measure the interference level. If the interference is sensed less than the predefined threshold for a sensing duration , the channel will be regarded as clean for this LTEU/D2DU user. Then, LTEU/D2DU users will inform the BS whether they collide with WiFi users according to the measured result. If in the operating channel, the number of interfered users is larger than a given threshold, and there is another cleaner channel available, i.e., the number of interfered users in this channel is less than that in the operating channel, the transmission will be switched to the new channel.
Some technologies are also used to improve detection sensitivity. For example, WiFi preambles are used to estimate the number of neighboring WiFi APs in a given channel. In addition, deviceassisted enhancements, such as 802.11k, in which the transmitter sends the request signals and the receiver replies the acknowledgment signal when the request signal is well received, can be used to address the hidden node effect, and thus help to select a better channel.
IiiB2 Channel sensing transmission
For most deployments, the channel selection is usually sufficient to meet the coexistence requirements. While in hyperdense deployment of 5G system, there is a probability that no clean channel can be found. For LTEU users, carriersensing adaptive transmission (CSAT) algorithm [25] is used to support the coexistence of LTEU and WiFi users. In the CSAT scheme, LTEU and WiFi users coexist in a TDM fashion. In particular, a duty cycle is defined where LTEU users transmit in a fraction of the cycle and gates off in the remaining time to hand over the unlicensed channel to WiFi users.
However, due to the short transmission range and low transmission power of D2D communications, it is possible to share the unlicensed spectrum with WiFi users during the full duty cycle. After the SS, the unlicensed channel can be still utilized by WiFi users to resume the ongoing data transmission. In these reserved subframes, those D2DU users which have sensed that the channel idle in SS are active in the unlicensed spectrum for data transmission, while other D2D and LTE users are only allowed to utilize licensed subchannels. When the reserved subframes for WiFi transmission expires, all the LTEU/D2DU users are activated in the unlicensed subchannels. At the begin of each subframe, the BS allocates licensed and unlicensed subchannels to the LTE/D2D users, in particular, only active LTEU/D2DU users have possibility to utilize unlicensed subchannels, which is elaborated in Section V.
IiiC Analysis of the protocol
IiiC1 Compatibility analysis
For LTE system, unlike the LBT based protocol for D2DU in [24] which requires LBT waveform and transmission modification of current LTE standard, our proposed protocol follows the current LTE PHY/MAC standards, such as frame structures, resource scheduling, and signaling. Thus, it can be directly implemented to current LTE network. And as for WiFi system, D2DU/LTEU users also perform energy detection to avoid the collision with WiFi users. Therefore, the LTEU and D2DU networks can be a good neighbor of WiFi network.
IiiC2 Signaling analysis
To describe the signaling cost over the control channels for the proposed protocol, we assume that messages are required to inform the BS the channel information sensed by a D2DU/LTEU user, messages are required for a user to report its location and subchannel estimation results, and messages are needed for the BS to notify a user about the allocated subchannels. In sensing subframe, each D2DU/LTEU user needs to report the sensing result over each channel. Therefore, at most messages are required in the SS. And before each subframe, each LTEU/D2DU user needs to report their locations and the subchannel estimation results for subchannel allocation, which requires messages. Then, the BS will perform resource allocation process with extra information, and notify each user by sending messages.
Note that in one duty cycle, each LTEU/D2DU user only performs one energy detection over one channel, thus, the signaling cost is under a tolerable level. In addition, the signaling cost of resource allocation is positively proportional to the number of D2DU/LTEU users, which is constrained by the limited subchannel resources. In each subframe, the signaling cost of the resource allocation can be also restricted to a tolerable level. Therefore, the signaling cost of the proposed duty cycle based protocol is acceptable for a practical system.
Iv Problem Formulation
In this section, we first formulate subchannel allocation problem considering both the performance of WiFi and the total sumrate of LTE and D2D users, and then reformulate this problem into a manytomany matching problem in consideration of its computation complexity.
Iva Sumrate Maximization Problem Formulation
Our objective is to maximize the total sumrate of LTE and D2D users while keeping the interference ranges under a tolerant level by setting the subchannel allocation variables in each subframe.
Since the BS does not hold the information of the interference from LTEU and D2DU users to WiFi systems, we use the approximate model described in the Subsection IVA to evaluate the performance degradation of WiFi system, and add it in the objective function as a penalty term. Assuming that the WiFi users are uniformly located in this plane, the number of interfered WiFi users is therefore positively proportional to the interference range. This can be used as an indicator of the performance degradation of WiFi system. Besides, provided that at least one unlicensed subchannels is allocated to CU or D, the WiFi users in its interference range cannot perform data transmission, and thus, the penalty term is in the form of sign function. Specifically, the penalty items and for CU and D can be respectively given by
(17) 
where is the sign function.
Taking the penalty into consideration, the subchannel allocation can be formulated as the following optimization problem:
(18a)  
(18b)  
(18c)  
(18d)  
(18e)  
(18f) 
where is the sensitivity factor for WiFi systems. Constraint (18b) is given under the assumption that one subchannel can be utilized by at the maximum of one cellular user. Constraints (18c) and (18d) imply that a user utilize at most subchannels, and a subchannel can be allocated to a maximum of users. According to the CA property, each LTEU or D2DU user needs to occupy at least one licensed subchannel for control signals, and thus constraint (18e) needs to be satisfied. Constraint (18f) is the sensing constraint, only the LTEU and D2DU users which have sensed that the channel is idle can access the unlicensed channel.
Note that the aforementioned problem is a MINLP problem, which is NPhard [36]. Considering the computational complexity, we reformulate the subchannel allocation as a manytomany twosided matching problem, which can be efficiently solved by utilizing the matching games.
IvB Matching Formulation
We consider the set of LTE and D2D users, , and the set of subchannels including licensed and unlicensed, , as two disjoint sets of selfish players aiming to maximize their own benefits. Each player can exchange information with one another without extra signaling cost^{3}^{3}3The BS is assumed to have the full knowledge of the channel side information (CSI), and performs subchannel allocation based on the obtained CSI., that is, the players have complete information about others. In this manytomany matching model, if subchannel SC is assigned to LTE user CU, then LTE user CU is said to be matched with subchannel SC, and form a matching pair, marked by .
A matching is an assignment of subchannels in to users in , which can be defined as:
Definition 1.
Given two disjoint sets, of the users, and of the subchannels, a manytomany matching is a mapping from the set to the set of all subsets of such that for every user CU or D, and subchannel SC:

, ;

;

;

;

;

, ;

, .
Conditions (1) and (2) state that each LTE or D2D user is matched with a subset of subchannels, and each subchannel is matched with a subset of users. Conditions (3) and (4) show the utilization constraints for a user and a subchannel. Due to the CA requirement, the users need to occupy at least one licensed subchannel, as expressed in condition (5). Condition (6) implies that only those users sensed idle unlicensed subchannel can utilize the unlicensed subchannels.
Considering mutual interference items in (12) and (15), any D2D user’s sumrate over its allocated subchannels SC is related to the set of other LTE users and D2D pairs sharing this subchannel. Besides, the penalty term in (18a) indicates that the objective of the LTEU and D2D users is relevant to other users operating in the unlicensed spectrum as well. Thus, each user cares about not only which subchannel it is matched with, but also the set of users matching with the same subchannel. For this reason, the aforementioned matching game is a manytomany matching game with externalities [32] or peer effects [33].
Affected by the peer effects, the outcome of this matching game greatly depends on the dynamic interactions among the users sharing the subchannels. To better describe the selection behavior and decision process of each player, we introduce a concept of preference relation for both users and subchannels. For any two subchannels , and any two matchings :
(19) 
where is related to the current subchannel allocation results. If D2D user D has not been allocated to unlicensed subchannels, the data rate needs to deduct the penalty item. This implies that the D2D user D prefers SC in to SC in if D can have a higher data rate over SC than SC. The same process will be done for an LTE user CU. LTEU user CU will also prefer the subchannel which can achieve higher data rate.
As for any subchannel SC, its preference relation over the set of users can be given in an uniform method. For any two subsets of users , and any two matchings :
(20) 
where also includes the penalty items. This indicates that subchannel SC prefers the set of users to only when SC can get a higher data rate from . is also used to indicate that subchannel SC likes the set of users at least as well as .
Different from traditional manytomany matchings in which the players’ preferences are substitutable, subchannels’ preferences do not satisfy substitutability. Specifically, given a subchannel SC, let represent its most preferred user set that containing two D2D pairs D and D. Besides, the data rate of D is higher than of D when they utilize subchannel SC independently. If D, then it is not necessary that D. Due to the mutual interference, the data rate may have changed after D is removed from , and thus, SC may not prefer D any more.
Due to the externalities, the manytomany matching model in this work is more complicated than the conventional twosided matching models. Under traditional definition of stable matching^{4}^{4}4Traditional stable matching refers to a matching in which there do not exist two players from opposite sets prefer each other to at least one of their current matches such that they form a new matching pair together for the sake of their interests, that is, there are no blocking pairs in a stable matching. in [32], there is no guarantee that a stable matching exists even in manytoone matchings. Because of the lack of substitutability, traditional deferred acceptance algorithm [32] cannot be applied to this model any more. To solve this matching problem, we introduce the swap matching [33] and propose a matching algorithm in Section V.
V ManytoMany MatchingBased Subchannel Allocation
In this section, we propose a matching algorithm to solve the problem formulated in Section IVB. We first introduce the notations and definitions of swap matching and stability into our manytomany matching model, and then elaborate on the swap matching algorithm.
Va Notations and Definitions
The concepts of swap matching and swapblocking pair are defined as below.
Definition 2.
Given a matching , two matching pairs and with SC, SC, SC, and SC, a swap matching is defined as:
(21) 
A swap matching is generated via swap operations, which is the twosided version of the exchange operation [37, 38]. In the swap operation, a pair of players exchange their matches while all other matchings remain unchanged. Different from conventional strategy change in onetoone matching performed by the individual player, the swap operation needs to be approved by both involved players. In the following, we provide the conditions in which the swap operations can be approved by introducing the concepts of swappable set and swapblocking pair.
Definition 3.
For LTEU user CU or D2D user D, its swappable set
is defined as a subchannel subset in which the user can swap for subchannel via swap matching. Specifically, if the sensing vector
, the swappable set of D is subchannels set including licensed and unlicensed ones; otherwise, its swappable set is licensed subchannels set . And it does the same for LTE user CU.Note that only those users which have sensed that the channel is idle can access the unlicensed subchannel, the users failed to sense idle unlicensed subchannels can only swap for licensed subchannels in the swappable sets.
Definition 4.
Provided a matching and a pair , and are matched in , and let and respectively represent the swappable sets of and . If there exist subchannels SC, SC, and SC such that:

,

,

,
then the swap matching is approved, and the pair is called a swapblocking pair in the matching .
The third condition in Definition 4 is to satisfy the CA requirement in which each user needs to utilize at least one licensed subchannel. The definition implies that once a swap matching is approved, at least one player’s data rate will increase, which leads to the increase in the total data rate.
Definition 5.
A matching is twosided exchangestable (2ES) if and only if there does not exist a swapblocking pair.
Intuitively speaking, from the perspective of network, a matching is said to be 2ES implies that there is not any user or subchannel SC, in which prefers another subchannel SC to its match SC, or SC likes another user rather than its match . Such a networkwide stable can be achieved by guaranteeing the involved players are beneficial from the swap operations, given the externalities in current matching .
VB Algorithm Description
With the notations of swap matching and the definition of stability, we propose a usersubchannel matching algorithm (Algorithm 1) to obtain a 2ES matching. This algorithm is a extension of the manytoone matching algorithm proposed in [39] with constraints that , , and .
As a part of Algorithm 1, each LTE user or D2D user needs to maintain a preference list. The preference list is established according to the following principles:

The subchannels in the preference list need to be contained in the swappable set.

The matched subchannel is removed from the preference list for each user.

The subchannels which have matched with users is removed from the preference list.

If the user is unmatched, i.e., the user has not been allocated to any subchannels, the licensed subchannels have priorities over the unlicensed ones.

The preference list is established based on the data rate over each subchannel.
In Algorithm 1, each user will send a proposal to the BS. According to definition 3, the proposed subchannel needs to be contained by swappable set. For each user, removing the matched subchannel is to avoid multiple proposals for the same subchannel. In addition, under the utilization constraints for a subchannel, the users can only send proposal to the available subchannels. The forth principle is designed in accordance with the CA requirement. This implies that if the user cannot compete for a licensed subchannel, the user needs to be silent. And according to the definition of preference relation in (19), the preference list is maintained based on the data rate. Due to the externalities, the preference list is dynamic in the swap matching process. Thus, in each iteration, the preference list will be updated based on the current matching.
The key idea of Algorithm 1 is to consider approving swap matchings among the players so as to obtain a 2ES matching. The algorithm is composed of two phases: initialization phase and swap matching phase. In the initialization phase, the BS will evaluate the channel gains for all users and interference from WiFi system. The swap matching phase contains multiple iterations in which the BS keeps executing the swap matching if there exist swapblocking pairs, and updates the current matching. Note that the higher a user’s data rate is, the higher probability it has to be accepted by the subchannel. In each iteration, the user updates its preference list, and sends a proposal to the subchannel SC ranked the first in the preference list unless it has been matched with subchannels. The acceptance can be regarded as a swap operation , where the element denotes a virtual user or subchannel. If this swap matching is approved, the proposed user is accepted by the subchannel SC, and the matching is updated. Then, the BS will search other swapblocking pairs and execute the swap matching to renew the current matching. The iterations stop until current matching is the same as the matching in the last iteration, and a final matching is determined.
Vi Performance Analysis
In this section, we analyze the effectiveness and efficiency of the proposed algorithm, and remark some key properties of the LTEU/D2D network. In the first part, the effectiveness and efficiency of the proposed algorithm is proved. Then, we discuss how the sensitivity factor impacts the subchannel allocation strategy.
Via Stability, Convergence, Complexity, and Optimality
Given the proposed Algorithm 1, we give remarks on the stability, convergence, complexity, and optimality.
ViA1 Stability and Convergence
We first provide the stability and convergence of Algorithm 1.
Lemma 1.
Phase II in Algorithm 1 converges after a limited number of swap operations.
Proof:
In each iteration of Algorithm 1, the matching is updated after a swap operation. Without loss of generality, we assume that after swap operation , the matching result is updated by swap matching . According to definition 4, after swap operation , the sumrates of subchannel SC and SC satisfy and , and these two equations cannot hold at the same time, while the sumrates of other subchannels remain unchanged. Therefore, the total sumrate over all subchannels strictly increases.
Note that the number of potential swapblocking pairs is finite since the number of users is limited, and the total sumrate has an upper bound due to limited subchannels. Therefore, there exists a swap operation after which no swapblocking pairs can be found and the total sumrate stops increasing. Then Algorithm 1 converges. ∎
Proposition 1.
Upon the convergence of Phase II, Algorithm 1 reaches a 2ES matching.
Proof:
The proof follows from these two considerations. First, the swap operations only occur when the players’ data rate strictly increases. Second, due to the convergency of Phase II, for any user , it cannot find another user to form a swapblocking pair with their matches when Algorithm 1 terminates. The matches of user must be the best choice in current matching. Hence, the terminal matching obtained by Algorithm 1 is 2ES. ∎
ViA2 Complexity
Having proved the convergence of Algorithm 1, we then discuss its computational complexity.
Note that in the swap matching phase, a number of iterations are performed to reach the 2ES matching. In every iteration, the BS needs to search for swapblocking pairs and all the approved swap operations are executed. Thus, the complexity of the swap matching phase lies in the number of both iterations and potential swap matchings in each iteration.
Proposition 2.
In the th iteration of Algorithm 1, at most swap matchings need to be considered.
Proof:
In each iteration of Algorithm 1, at most users send proposal to the subchannels which rank first in their preference lists. Therefore, in this step, at most swap matchings need to be considered.
If the proposals from users are accepted by subchannels, they might execute swap matchings with the existing matches. For user , it sends proposal to subchannel SC and is accepted. According to definition 2, this match can only execute swap matchings with matches which do not contain and SC. In each iteration, at most 1 match can be added to the current matching for each user. Therefore, for match pair , there are at most potential swap matchings in the th iteration. In the worst case, all the proposal for users are accepted by subchannels, and thus, there are a maximum of potential swap matchings.
Above all, at most swap matchings need to be considered in the th iteration. In practice, one iteration requires a significantly low number of swap operations, since only a small number of proposals from users can be accepted. ∎
ViA3 Optimality
We show whether Algorithm 1 can achieve an optimal matching.
Proposition 3.
All local maxima of total sumrate corresponds to a 2ES matching.
Proof:
Suppose the total sumrate of matching is a local maximum value. If is not a 2ES matching, then there exists at least one swapblocking pair, and any swap matching strictly increases data rates according to definition 2. However, this is in contradiction to the assumption that is a local maximum value. Therefore, must be 2ES. ∎
However, not all 2ES matchings obtained from Algorithm 1 are local maxima of total data rates. For example, there exists possibility that a user does not approve a swap matching since its data rate will decrease, but the other user will benefit from this swap matching, and the sumrates of SC and SC increase as well. The total sumrates will increase at the expense of stability if the swap operation is forced to execute.
To obtain a global optimum matching, we utilize a algorithm (GO Algorithm) proposed in [33]
by utilizing a Markov chain Monte Carlo heat bath method. In GO Algorithm, the swap matching does not need to be approved any more, instead, a swap matching
is executed with a probability which depends on the total sumrate as shown below:(22) 
where is a probability parameter. The algorithm keeps track of the optimum matching found so far, even when it moves to worse matchings. After sufficiently large amount of iterations, the matching moves towards the global optimal one [40].
ViB Selection of the Sensitivity Factor
Let be the maximum rate for a D2DU user over an unlicensed subchannel, be the maximum rate for an LTEU user over one unlicensed subchannel, and generally . How the value of the sensitivity factor tunes the performance can be analyzed in the following cases.
ViB1
This case implies that neither D2D nor LTE users can get access to an unlicensed subchannel, the value of penalty terms is sufficiently large that cannot satisfy all the conditions of swap matching. Therefore, in this case, any D2D or LTE users cannot utilize unlicensed subchannels, the LTE and D2D users can only utilize the licensed spectrum for total sumrate maximization.
ViB2
This case implies that D2D users can get access to unlicensed subchannels. In this case, any LTE users cannot utilize unlicensed subchannels. As for the D2D users, they can utilize both licensed and unlicensed subchannels. If one D2D user get access to a unlicensed subchannel, those D2D users whose interference ranges overlap with the accessed one will become more easier to get access to the unlicensed subchannels because the increased interference ranges will be less than that when this D2D user is the first one to get access to the unlicensed subchannels. Thus, in the view of geography, those D2D users allocated to unlicensed subchannels trend to form several clusters.
ViB3
This case means that both LTE and D2D users can get access to unlicensed subchannels. Similar to case 2, the users accessed to unlicensed subchannels also form several clusters. In addition, the accessed LTE users will decrease with the value of grows because of the large interference range, that is, more unlicensed subchannels are allocated to D2D users and the LTE users are allocated to more licensed subchannels.
ViB4
This case is the same as resource allocation problem in licensed scenario. In this case, both licensed and unlicensed subchannels are uniform, D2D and LTE users will make full use of the whole spectrum to maximize the total sumrate.
Vii Simulation Results
LTEU and D2DU Parameters  Values 

Cell radius  500 m 
D2D communication radius  20 m 
LTE’s transmit power  17 dBm 
D2D’s transmit power  10 dBm 
Subchannel bandwidth  180 kHz 
Number of subchannels  10 
Carrier frequency  1.9 GHz 
Noise figure  5 dB 
Decay factor of the path loss  2.2 
Power gains factor  33.58 dB 
Shadow fading standard deviation 
4 dB 
Maximum number of subchannels  4 
Maximum number of users  4 
WiFi Parameters  Values 
Number of subchannels  20 
Subchannel bandwidth  180 kHz 
WiFi user’s transmit power  23 dBm 
Number of APs  3 
LTEU interference radius  50 m 
LTEU interference radius  23 m 
In this section, we present the simulation results of Algorithm 1, in comparison to the GO Algorithm, a greedy algorithm, and the scenario without D2D, where all the users are LTE ones. In the greedy algorithm, the users will maintain a static preference list, and send proposal to the subchannels according to the preference list. We set the number of iterations as , in GO Algorithm such that the outcome of GO Algorithm can be regarded as the upper bound of the data sumrate. Note that the upper bound is unrealistic since the computational complexity is rather high. And the subchannels in the scenario without D2D are also allocated by Algorithm 1. In this simulation, we consider a single cell layout, where the LTE and D2D users are distributed randomly, and the communication distance of D2D users cannot exceed a predefined value. The simulation parameters based on existing LTEAdvanced specifications [41] are given in Table I. Note that the transmission power of WiFi user is over the whole unlicensed channel, while the transmission power of LTE or D2D user is over one subchannel.
Fig. 4 shows the data sumrate vs. the number of active D2D users , with the number of LTE users and the sensitivity factor . We observe that the sumrate increases with . This is because the number of concurrent transmission links grows while any two links which bring severe mutual interference are not allowed to exist in the same subchannel. However, it also shows that the sumrate becomes saturated when , as the number of subchannels is not sufficient to support more D2D users. In addition, it can be observed that the sumrate obtained by Algorithm 1 is 10.6% higher than the greedy algorithm, and 317.3% higher than the scenario without D2D, while it only has 3.7% gap to the upper bound when . This further implies that the BS can make full use of the unlicensed spectrum resources via D2D communications. The simulation results correspond to analysis in Section VIA.
Fig. 5 shows the sumrate v.s. the number of active D2D users with the number of LTE users . It can be easily observed that the total sumrate obtained by the same algorithm will decrease as the sensitivity factor increases. According to the discussions in Section VIB, means D2D and LTE users can use unlicensed spectrum. Since the subchannels are sufficient, the system sumrate is the maximum. means that almost all LTE and D2D users cannot use unlicensed spectrum, and thus the system sumrate is the minimum. Because of less available subchannels, the data sumrate decreases as the sensitivity factor increases. In addition, we can also find out that the data sumrate obtained by Algorithm 1 is always higher than that obtained by the greedy algorithm with the same sensitivity factor . In particular, the data sumrate is 100 bit/s/Hz higher when , which implies that Algorithm 1 can utilize the spectrum more efficiently.
In Fig. 6, we provide the interference ranges vs. the number of active D2D users with the number of LTE users . We use a uniform sampling and judge whether this sampling point is in the interference range of any LTE or D2D user using the unlicensed spectrum. The percentage of sampling points in the interference ranges is regarded as the interference ranges. From Fig. 6, it can be observed that the interference ranges obtained by the same algorithm will decrease as the sensitivity factor grows. However, we can find out that the decrease in interference ranges is at the expense of the data sumrate from Fig. 5. Therefore, we can utilize the unlicensed spectrum according to different traffic requirements by properly setting the value of . In addition, we can also observe the interference range obtained by Algorithm 1 is lower than that obtained by the greedy algorithm with the same , except , where LTE and D2D users cannot utilize the unlicensed spectrum. This implies that Algorithm 1 outperforms the greedy algorithm not only on the data sumrate, but also on the interference ranges.
Fig. 7 provides that the number of subchannels in both licensed and unlicensed spectra vs. the sensitivity factor with the number of LTE users and the number of D2D pairs . Note that each LTE user needs to occupy at least one licensed subchannel, and two LTE users are not allowed to transmit on the same subchannel. Thus, each LTE user will only utilize one licensed subchannel. Based on the constraint that a user can utilize at most subchannels, the total subchannels of D2D users cannot exceed 60 subchannels, and the total subchannels for LTE users cannot exceed 40 subchannels. It can be also observed that when the value of increases, the unlicensed subchannels for LTE and D2D users will decrease for the protection of WiFi system. However, due to the smaller interference ranges of D2D communications, the proportional reduction in unlicensed subchannels for D2D users is lower than that for LTE users. This is consistent with the discussions in Section VIB.
Fig. 8 shows the data sumrate vs. the D2D radius with the number of LTE users , the number of D2D users , and the sensitivity factor . In Fig. 8, we can learn that the data sumrate will decrease as the D2D radius grows. This is because the received power of D2D users downgrades as the transmission radius of D2D users upgrades. It can be also observed that Algorithm 1 always outperforms the greedy algorithm, and approaches to the upper bound with different values of D2D radius.
Viii Conclusions
In this paper, we investigate the D2DU technology, in which the D2D users operate as an underlay to the LTE system in both licensed and unlicensed spectra. A duty cycle based protocol is designed for LTEU and D2DU users while protecting the existing WiFi systems. Considering the complicated mutual interference between LTE, D2D, and WiFi systems, we study the subchannel allocation problem for D2D and LTE users sharing both licensed and unlicensed subchannels to leverage the performance degradation in WiFi systems and the maximization of the sumrate in LTE/D2D networks. Specifically, we formulate the allocation problem as a manytomany matching game with externalities, and develop a lowcomplexity usersubchannel swap matching algorithm. In addition, power control can be done in parallel with subchannel assignment. Analytical and simulation results show that enabling D2DU communications can significantly improve the system sumrate. Besides, the subchannel allocation strategy for LTEU and D2D users is closely related to how the BS adjusts the interference to WiFi systems. In an aggressive strategy where the WiFi performance degradation is not considered seriously, the BS allows more D2D and LTE users to transmit on the unlicensed spectrum. On contrary, in a WiFi friendly strategy, the BS tends to permit only a small fraction of D2D users to transmit on the unlicensed spectrum.
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