Degrees of Freedom of FullDuplex Multiantenna Cellular Networks
Abstract
We study the degrees of freedom (DoF) of cellular networks in which a full duplex (FD) base station (BS) equipped with multiple transmit and receive antennas communicates with multiple mobile users. We consider two different scenarios. In the first scenario, we study the case when half duplex (HD) users, partitioned to either the uplink (UL) set or the downlink (DL) set, simultaneously communicate with the FD BS. In the second scenario, we study the case when FD users simultaneously communicate UL and DL data with the FD BS. Unlike conventional HD only systems, interuser interference (within the cell) may severely limit the DoF, and must be carefully taken into account. With the goal of providing theoretical guidelines for designing such FD systems, we completely characterize the sum DoF of each of the two different FD cellular networks by developing an achievable scheme and obtaining a matching upper bound. The key idea of the proposed scheme is to carefully allocate UL and DL information streams using interference alignment and beamforming techniques. By comparing the DoFs of the considered FD systems with those of the conventional HD systems, we establish the DoF gain by enabling FD operation in various configurations. As a consequence of the result, we show that the DoF can approach the twofold gain over the HD systems when the number of users becomes large enough as compared to the number of antennas at the BS.
I Introduction
Current cellular communication systems operate in halfduplex (HD) mode by transmitting and receiving either at different time slots or over different frequency bands. The system is designed in such a way that the downlink (DL) and uplink (UL) traffics are structurally separated by time division duplexing (TDD) or frequency division duplexing (FDD). The advantage of such design principle is that it avoids the highpowered selfinterference that is generated during simultaneous transmission and reception. Recent results [1, 2, 3, 4, 5, 6], however, have demonstrated the feasibility of fullduplex (FD) wireless communication by suppressing or cancelling selfinterference in the RF and baseband level. Various practical designs to realize selfinterference cancellation have been proposed in the literature, including adding additional antennas [2], adding auxiliary transmit RF chains [3] or auxiliary receive RF chains [4], using polarization [4, 3], employing balun circuits [5], and many more. For more details, see [6, 7] and the references therein.
By enabling simultaneous transmission and reception, FD radio is expected to double the spectral efficiency of current HD systems [7], and is considered as one of the key technologies for next generation communication systems. Evidently, in situations where the base station (BS) and the user simultaneously transmit bidirectionally as shown in Figure 1(a), enabling FD doubles the overall spectral efficiency. This pointtopoint bidirectional communication example, however, is just one instance of how a FD cellular system will function.
In some practical cases, the system may have to support HD users which do not have FD radio due to extra hardware burden on mobile devices. In such case, the FD BS can simultaneously communicate with two sets of users, one receiving DL data from the BS and the other transmitting UL data to the BS (Figure 1(b)). In another configuration shown in Figure 1(c), for instance, when the BS has many more antennas compared to each user, the FD BS may wish to simultaneously communicate with multiple FD users using multiuser multipleinput and multipleoutput (MIMO) techniques. Since the BS is simultaneously transmitting and receiving, there is potential to double the overall spectral efficiency compared to the conventional HD only systems. However, the configurations shown in Figures 1(b) and 1(c) induce a new source of interference that does not arise in HD only networks. In Figure 1(b), since user 1 is transmitting to the BS while user 2 is receiving from the BS, the transmission from user 1 causes interference to user 2. Similarly, in Figure 1(c), the UL transmission of the users causes interference to the DL reception to each other. In cases where this type of interference is strong and proper interference mitigation techniques are not applied, the gain of having FD radios can be severely limited even when selfinterference is completely removed.
To manage interuser interference and fully utilize wireless spectrum with FD operation, in this paper we employ signal space interference alignment (IA) schemes optimized for FD networks including the cases in Figure 1. Initially proposed by the seminar works in [8, 9, 10], IA is a coding technique that efficiently deals with interference and is known to achieve the optimal DoF for various interference networks [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Especially, it is shown that IA can be successfully applied to mitigate interference in various cellular networks, such as twocell cellular networks [11, 12] and multiantenna UL–DL cellular networks [20]. Furthermore, the idea of IA can also be applied to the (multiuser) bidirectional cellular network with ergodic phase fading [21], in which the achievable scheme is based on the ergodic IA scheme proposed in [22].
Motivated by the aforementioned previous works related to IA, we propose the optimal transmission schemes that attain the optimal sum DoFs for two configurations: 1) a cellular network with a multiantenna FD BS and HD users (Figure 1(b)); 2) a cellular network with a multiantenna FD BS and FD users (Figure 1(c)). The key idea of the proposed schemes is to carefully allocate the UL and DL information streams using IA and beamforming techniques. The UL data is sent to the BS using IA such that the interuser interference is confined within a tolerated number of signal dimensions, while the BS transmits in the remaining signal dimensions via zeroforcing beamforming for the DL transmission.
With the proposed schemes, our primary goal is to answer whether if FD operation can still double the overall spectral efficiency even in the presence of interuser interference. We answer this question by providing matching upper bounds with the proposed achievable schemes, completely characterising the sum DoFs of the considered networks. As a consequence of the result, even in the presence of interuser interference, we show that the overall DoF can approach the twofold gain over HD only networks when the number of users becomes large as compared to the number of antennas at the BS. We further provide the DoF gain of the FD systems by considering various configurations (see Sections III and VI.).
Ia Previous Works
In [10], Cadambe and Jafar proposed a novel interference management technique called interference alignment (IA), which achieves the optimal sum DoF of for the user interference channel (IC) with timevarying channel coefficients. In addition, for the case in which all channel coefficients are constant, Motahari et al. [23, 24] proposed a different type of IA scheme based on numbertheoretic properties of rational and irrational numbers and showed that the optimal DoF of is also achievable. Later, alternative methods of aligning interference in the finite signaltonoise regime has been also proposed in [22, 25, 26, 27]. The concept of IA has been successfully adapted to various network environments, e.g., see [14, 15, 13, 16, 17, 18, 19] and the references therein.
The DoF of cellular networks has been first studied by Suh and Tse for both UL and DL environments, where intercell interference exists [11, 12]. It was shown that, for twocell networks having users in each cell, the sum DoF of is achievable for both UL and DL. Thus, multiple users at each cell are beneficial for improving the DoF of cellular networks. These models were further extended to more general cases in terms of the number of users and the number of antennas at each BS [28, 29, 30, 31, 32, 33]. In addition, recently, the DoF of the multiantenna UL–DL cellular network consisting of DL and UL cells has been studied in [20, 34]. For a cellular network with FD operation in the absence of selfinterference, the DoF of the (multiuser) bidirectional case has been studied in [21] for ergodic phase fading setting.
IB Paper Organization
The rest of this paper is organized as follows. In Section II, we describe the network model and the sum DoF metric considered in this paper. In Section III, we present the main results of the paper and intuitively explain how FD operation can increase the DoF. In Sections IV and V, we provide the achievability and converse proofs of the main theorems, respectively. In Section VI, we discuss the impacts of selfinterference and scheduling on the DoF. Finally, we conclude in Section VII.
Notations: We will use boldface lowercase letters to denote vectors and boldface uppercase letters to denote matrices. Throughout the paper, denotes , denotes the allzero vector, and denotes the identity matrix. For a real value , denotes . For a set of vectors , denotes the vector space spanned by the vectors in . For a vector , means that is orthogonal with all vectors in . For a set of matrices , denotes the block diagonal matrix consisting of .
Ii Problem Formulation
For a comprehensive understanding of the DoF improvement by incorporating FD operation, we consider two types of network models: the first network model consists of a single FD BS which simultaneously transmits to a set of DL users (in HD mode) and receives from a set of UL users (in HD mode); the second model consists of a single FD BS communicating with a set of FD users. Unless otherwise specified, we simply denote BS for FD BS in the rest of this paper.
Iia Network Model
In this subsection, we formally define the network models for the two cases mentioned above.
FDBS–HDuser cellular networks
This network model consists of a mixture of a FD BS and HD users. The HD users are partitioned into two sets, in which one set of users are transmitting to the BS, and the other set of users are receiving from the BS simultaneously. This cellular network is depicted in Figure 2. We assume that the FD BS is equipped with transmit antennas and receive antennas. On the user side, we assume that there are DL users and UL users, each equipped with a single antenna. Here, each user is assumed to operate in HD mode. The BS wishes to send a set of independent messages to the DL users and at the same time wishes to receive a set of independent messages from the UL users.
For , the received signal of DL user at time , denoted by , is given by
(1) 
and the received signal vector of the BS at time , denoted by , is given by
(2) 
where is the transmit signal vector of the BS at time , is the transmit signal of UL user at time , is the channel vector from the BS to DL user at time , is the scalar channel from UL user to DL user at time , and is the channel vector from UL user to the BS. The additive noises and are assumed to be independent of each other and also independent over time, and is distributed as and .
We assume that channel coefficients are drawn i.i.d. from a continuous distribution and vary independently over time. It is further assumed that global channel state information (CSI) is available at the BS and each UL and DL user. The BS and each UL user is assumed to satisfy an average transmit power constraint, i.e., and for all .
In the rest of the paper, we denote this network as a FDBS–HDuser cellular network.
FDBS–FDuser cellular networks
In this model, we consider the case where both the BS and users have FD capability (depicted in Figure 3). As before, we assume that the BS is equipped with transmit antennas and receive antennas. However, unlike the FDBS–HDuser cellular network, there is a single set of FD users, each equipped with a single transmit and a single receive antenna, that simultaneously transmits to and receives from the BS. The BS wishes to send a set of independent messages to the users and at the same time wishes to receive a set of independent messages from the same users.
For , the received signal of user at time is given by
(3) 
and the received signal vector of the BS at time is given by
(4) 
As before, we assume that selfinterference at the BS and each user is completely suppressed, which is reflected in the input–output relations in (3) and (4). The rest of the assumptions are the same as those of the FDBS–HDuser cellular network.
In the rest of the paper, we denote this network as a FDBS–FDuser cellular network.
IiB Degrees of Freedom
For each network model, we define a set of length block codes and its achievable DoF.
FDBS–HDuser cellular networks
Let and be chosen uniformly at random from and respectively, where and . Then a code consists of the following set of encoding and decoding functions:

Encoding: For , the encoding function of the BS at time is given by
For , the encoding function of UL user at time is given by
where .

Decoding: Upon receiving to , the decoding function of the BS is given by
Upon receiving to , the decoding function of DL user is given by
where .
A rate tuple is said to be achievable for the FDBS–HDuser cellular network if there exists a sequence of codes such that and as increases for all and . Then the achievable DoF tuple is given by
(5) 
We further denote the maximum achievable sum DoF of the FDBS–HDuser cellular network by , i.e.,
(6) 
where denotes the DoF region of the FDBS–HDuser cellular network.
FDBS–FDuser cellular networks
Similar to the FDBS–HDuser cellular network, we can define an achievable DoF tuple of the FDBS–FDuser cellular network. The key difference is that each user also operates in FD mode for this second model. Specifically, the encoding function of user at time is given by and the decoding function of user is given by , where . Then the definition of an achievable DoF tuple is the same as that of the FDBS–HDuser cellular network. Similarly, we denote the maximum achievable sum DoF of the FDBS–FDuser cellular network by .
Iii Main Results
In this section, we state the main results of this paper. We completely characterize the sum DoFs of both the FDBS–HDuser cellular network and the FDBS–FDuser cellular network.
Theorem 1
For the FDBS–HDuser cellular network,
(7) 
We demonstrate the utility of Theorem 1 by the following example.
Example 1 (Symmetric FDBS–HDuser cellular networks)
Consider the FDBS–HDuser cellular network, i.e., and . For this symmetric case, from Theorem 1. On the other hand, if the BS operates in HD mode, we can easily see that the sum DoF is limited by . By comparing the sum DoFs, we can see that there is a twofold gain by operating the BS in FD mode when we have enough number of users in the network, i.e., . Figure 4 plots with respect to when . As shown in the figure, FD operation at the BS improves the sum DoF as increases and eventually the sum DoF is doubled compared to HD BS for large enough .
For the FDBS–FDuser cellular network, we have the following theorem.
Theorem 2
For the FDBS–FDuser cellular network,
(8) 
From the network model and the DoF definition in Section II, any achievable sum DoF in the FDBS–HDuser cellular network is also achievable for the FDBS–FDuser cellular network. In particular, the encoding functions at the BS are the same for both network models, and the BS also receives the same signal as shown in (1) and (3). Comparing the user encoders, we can see that the user encoding function for the FDBS–FDuser cellular network is more general than the encoding function for the FDBS–HDuser cellular network. Furthermore, we can easily see that the received signal (4) is “better” than the received signal for the FDBS–HDuser cellular network (3), in that it has less interference (selfinterference is suppressed for the FD user case). Hence, from Theorem 1, the sum DoF of is achievable for the FDBS–FDuser cellular network, which coincides with in (8). The converse proof is given in Section V.
We demonstrate the utility of Theorem 2 by the following example.
Example 2 (Symmetric FDBS–FDuser cellular networks)
To be fair, the FDBS–FDuser cellular network in Example 2 has been considered in [21] under the ergodic fading setting assuming that the phase of each channel coefficient in is drawn independently from a uniform phase distribution. For this case, it has been shown in [21, Theorem 1] that the achievable DoF tuple satisfies:
(9) 
where (III) characterises the sum DoF. This result in [21] is general in that it provides a general achievable DoF region, while our result in Theorem 2 generalizes the sum DoF result in [21] by considering arbitrary number of transmit and receive antennas at the BS, and also extends to any i.i.d. generic channel setting including the ergodic fading setting.
In Section VI, we discuss in detail regarding the DoF improvement by enabling FD operation, and also the effect of imperfect selfinterference suppression.
Iv Achievability
In this section, we prove that the sum DoF in Theorem 1 is achievable. To better illustrate the main insight of the coding scheme, we first consider the achievablity of Theorem 1 for the case in Section IVA. The main component of the scheme utilizes IA via transmit beamforming with a finite symbol extension. For general , interference from multiple UL users should be simultaneously aligned at multiple DL users, which requires asymptotic IA, i.e., an arbitrarily large symbol extension. In Section IVB, we introduce transmit beamforming adopting such asymptotic IA for the general network configuration.
Iva The Case
For the FDBS–HDuser cellular network,
(10) 
from Theorem 1. For the proof on how (10) can be evaluated from (7) for the case , we refer to the proof in Lemma 1. In the following, we show that in (10) is achievable by considering two cases, and . For the first case , we can easily achieve by simply utilizing only the UL transmission, i.e., the BS receives from the UL users with receive antennas. Now consider the second case where , which we explain with the help of Figure 7.
For this case, communication takes place via transmit beamforming over a block of time slots, i.e., symbol extension. Denote
(11) 
where . The BS sends information symbols to the DL user via the beamforming vectors . On the other hand, UL user sends information symbols to the BS via the beamforming vectors .
We first construct as a set of linearly independent random vectors. Next, we construct linearly independent such that for each , all the information symbols that are indexed with are aligned at the DL user, i.e., satisfying the IA condition for all . Specifically, we first construct as a set of linearly independent random vectors. Then, for a given , we construct for all . By such construction, the resulting are linearly independent almost surely.
We now move on to the decoding step at the DL user. Due to the previous IA procedure of the UL users, the number of dimensions occupied by the interuser interference signals is given by . Furthermore, the DL signals sent by the BS occupy dimensions and are linearly independent of the interuser interference signals almost surely. Hence, the DL user is able to decode its intended information symbols achieving one DoF each. Next, consider decoding at the BS. Since are linearly independent, are also linearly independent almost surely. Hence, the BS is able to decode the information symbols. Finally, from the fact that a total of information symbols are communicated over time slots, is achievable for the case .
IvB General Case
Following the intuition in the previous subsection, with IA, we would like to confine the interference signals transmitted from multiple UL users into a preserved signal subspace at each DL user, leaving the rest of subspace for the intended signals sent from the BS. For general , this requires arbitrarily large number of symbol extensions [10].
For this purpose, a recently developed IA technique in [20] for the multiantenna UL–DL cellular network can be applied for the FDBS–HDuser cellular network. To show how the scheme in [20] fits into our problem, we begin with a brief overview of their network model. In [20], the authors consider a UL–DL cellular network (Figure 6), where two cells coexist (each cell consists of one BS and a set of users). In one cell, a BS with antennas transmits to a set of DL users, while in the other cell a set of UL users transmit to a BS with antennas. Thus, the network models the case when it can schedule each cell in DL or UL phase separately. The structural similarity with our FDBS–HDuser cellular network is apparent, and the key difference between them is that there is no intercell interference between the DL BS and UL BS (since in the FDBS–HDuser cellular network, UL and DL is performed with a single FD BS). Accordingly, the transmit signal vector of the DL BS in the UL–DL model (Figure 6) can also be used as the transmit signal vector of the FD BS in the FDBS–HDuser cellular network (Figure 2), and the transmit signal of each UL user in the UL–DL model (Figure 6) can also be used by each UL user in the FDBS–HDuser cellular network (Figure 2). Therefore, the IA scheme stated in [20, Section IVE] is applicable to the FDBS–HDuser cellular network. However, due to the selfinterference suppression capability in the FD BS case, the performance resulting from this scheme will be different for the two networks, and our contribution for achievability lies in the analysis of the sum DoF of the scheme for the FDBS–HDuser cellular network.
For completeness and better understanding, we briefly summarize how the IA scheme in [20, Section IVE] can be adapted to the FDBS–HDuser cellular network. We then give the analysis of its achievable sum DoF.
DL interference nulling and UL interference alignment
Communication takes place over a block of time slots, i.e., symbol extension. Denote
(12) 
for and . Each information symbol is transmitted through a length timeextended beamforming vector. Figure 7 is a conceptual illustration for this transmit beamforming. We refer to [20, Section IVE] for the detailed construction of beamforming vectors. Suppose that and as increases. For , the BS sends information symbols to DL user using the set of timeextended beamforming vectors . Similarly, UL user sends information symbols to the BS using the set of timeextended beamforming vectors , where .
As seen in Figure 7, the set of beamforming vectors transmitted from each UL user is set to align its interference at each DL user. More specifically, by applying asymptotic IA for , we can guarantee that occupies at most dimensional subspace in dimensional signal space for all almost surely in the limit of large , where as increases, see also [20, Lemma 2]. Then the set of beamforming vectors transmitted from the BS is set to null out its interference at each DL user. More specifically, is set to satisfy for all satisfying and , i.e., zeroforcing is performed using transmit antennas. In order to apply such DL interference nulling,
(13) 
should be satisfied. Again, as seen in Figure 7, for reliable decoding at each DL user achieving one DoF for each information symbol,
(14) 
should be satisfied. Similarly, for reliable decoding at the BS achieving one DoF for each information symbol,
(15) 
should be satisfied. Therefore, the proposed scheme is able to deliver information symbols over time slots under the constraints (13) to (15). Finally, from the fact that as increases, its achievable sum DoF is represented by the following optimization problem:
(16) 
Achievable sum DoF
In the following, we prove that the sum DoF attained by solving (16) is given as stated in Theorem 1. The linear program in (16) is divided into five cases depending on the feasible region of as depicted in Figure 8. Obviously, one of the corner points, which are marked as points in Figure 8, provides the maximum sum DoF. Hence, the maximum sum DoF attained from (16) is given by
(17) 
The following lemma then shows that (17) is represented as in Theorem 1, which completes the achievability proof of Theorem 1.
Lemma 1
The sum DoF in (17) is represented as
(18) 
For notational simplicity, denote
(19) 
Then denote . In the following, we show that for each of the five cases in (17), is represented as in the corresponding DoF expression in (17).

Case I (): Obviously, . For , . For , . Hence . In conclusion, for Case I.

Case II (): First consider the case where . Then . Also . Since , . Hence . Next consider the case where . Then and . Also . Hence . Finally, from the relation that for and for , for Case II.

Case III (): From the symmetric relation with Case II, for Case III.

Case IV (): The condition means that and . Hence and , which show . For , and . Similarly, and for . Hence . In conclusion, for Case IV.

Case V (): For , and then . Similarly, for . Hence