Amel Tibhirt
email: amel.tibhirt@orange.com

Dirk Slock
email: dirk.slock@eurecom.fr

Yi Yuan-Wu

Distributed Beamforming Design in Reduced-Rank MIMO Interference Channels and Application to Dynamic TDD

Keywords: Dynamic TDD, MIMO, rank deficient, interference alignment

Employing dynamic Time Division Duplexing (DynTDD) can increase the system-wide spectral efficiency of scenarios with varying and unbalanced uplink (UL) and downlink (DL) data traffic requirements. However when using DynTDD, a different DL/UL slot configuration is likely to be selected by neighboring cells, leading to Cross Link Interference (CLI) between the Base Stations (BS), which is known as BS-to-BS or DLto-UL interference, and between User Equipments (UE) which is known as UE-to-UE or UL-to-DL interference. Rank deficient channels are frequently encountered in Multi-Input Multi-Output (MIMO) networks, due to poor scattering and keyhole effects, or when using Massive MIMO and moving to mmWave. While the implications of rank deficient channels are well understood for the single user point to point setting, less is known for interference networks. In this paper we study the Degrees of Freedom (DoF) focusing on the UE-to-UE interference, by establishing necessary conditions, for centralized and distributed designs. We discuss the variation of these conditions for different scenarios involving rank deficient channels. In particular we observe reduced DoF gaps between distributed and centralized techniques compared to existing results.

I. INTRODUCTION

Dynamic Time Division Duplexing (DynTDD) is one promising way to improve the spectrum efficiency of the wireless communication networks since flexible traffic adaptation can be achieved by dynamically changing uplink (UL) or downlink (DL) transmission direction. DynTDD performance has been analyzed in the literature. [START_REF] Ding | Performance Analysis of Dense Small Cell Networks With Dynamic TDD[END_REF] investigates the impact of synchronous DynTDD on the performance of the DL/UL in dense small cells networks (SCNs). The results show that DynTDD outperforms the static TDD in terms of average total Time Resource Allocation (TRA), and DL and UL Area Spectral Efficiency (ASE). However, DynTDD also brings some new challenges because of the introduction of crosslink interference (CLI), including DL-to-UL interference (e.g., gNB-to-gNB interference) and UL-to-DL interference (UE-to-UE interference). In this scope many techniques have been proposed for the cross-link interference mitigation. Some authors propose solutions based on an optimization problem, such as Mean Square Error (MSE) minimization with the constraint of the maximum transmit power for the DL BS and UL UE in [START_REF] De Olivindo Cavalcante | Bidirectional Sum-Power Minimization Beamforming in Dynamic TDD MIMO Networks[END_REF], and the minimization of the total transmit sum power while satisfying a minimum signal-to-interferenceplus-noise ratio (SINR) threshold for every UE in [START_REF] Yoon | MSE-Based Downlink and Uplink Joint Beamforming in Dynamic TDD System Based on Cloud-RAN[END_REF].

Degrees of Freedom (DoF) is one powerful metric, that allows an approximate characterization of rates at high Signal to Noise Ratio (SNR). In Multi-Input Multi-Output (MIMO) settings, with multiple antennas at transmitters (Tx) and receivers (Rx), DoF maximization requires (spatial) interference alignment (IA). Some authors works on interference alignment (IA) and interference neutralization methods. [START_REF] Na | Interference Alignment Based Dynamic TDD For Small Cells[END_REF] proposes an interference alignment based MIMO transmission scheme that effectively addresses the interference problem in DynTDD systems and improves the system capacity. The proposed technique in [START_REF] Ko | Distributed Interference Alignment for Multi-Antenna Cellular Networks With Dynamic Time Division Duplex[END_REF] is a distributed interference alignment (DIA) in a two-cell MIMO network with DynTDD mode under local channel state information (CSI) assumption, and [START_REF] Du | Interference Neutralization With Partial CSIT for Full-Duplex Cellular Networks[END_REF] proposes an interference neutralization scheme to eliminate the inter user interference with the help of partial channel state information at the transmitter (CSIT). CLI cancellation methods have been studied in [START_REF] Huo | Joint User Scheduling and Transceiver Design for Cross-link Interference Suppression in MU-MIMO Dynamic TDD Systems[END_REF], where a joint user scheduling and transceiver design based CLI suppression scheme is investigated in multi-cell multi-user MIMO DynTDD systems to eliminate UL-to-DL interference. An algorithm is designed to avoid scheduling DL UEs which will be interfered by neighboring UL UEs. To suppress DLto-UL interference, the DL-to-UL interference channel (IC) is divided into several interference sub-channels and a novel precoding and detection design is provided to make the wanted signal channel orthogonal to these interference sub-channels.

The feasibility conditions of IA have been analyzed in [START_REF] Ko | Distributed Interference Alignment for Multi-Antenna Cellular Networks With Dynamic Time Division Duplex[END_REF], [START_REF] Liu | On the Feasibility of Linear Interference Alignment for MIMO Interference Broadcast Channels With Constant Coefficients[END_REF], [START_REF] Jeon | The Feasibility of Interference Alignment for MIMO Interfering Broadcast-Multiple-Access Channels, journal=Trans. Wireless Communications[END_REF] [10] [START_REF] Negro | Interference alignment limits for K-user frequency-flat MIMO interference channels[END_REF], [START_REF] Negro | Interference alignment feasibility in constant coefficient MIMO interference channels[END_REF] and [START_REF] Chen | A General Model for DoF-based Interference Cancellation in MIMO Networks with Rank-deficient Channels[END_REF]. [START_REF] Ko | Distributed Interference Alignment for Multi-Antenna Cellular Networks With Dynamic Time Division Duplex[END_REF] also mathematically characterizes the achievable DoF of their proposed DIA technique for a given number of antennas at BS/MS. In [START_REF] Liu | On the Feasibility of Linear Interference Alignment for MIMO Interference Broadcast Channels With Constant Coefficients[END_REF] the authors analyze the feasibility of linear IA for the MIMO Interfering Broadcast Channel (IBC) with constant coefficients. They pose and prove the necessary conditions of linear IA feasibility for general MIMO-IBC. Except for the proper condition, they find another necessary condition to ensure a kind of irreducible interference to be eliminated. They then prove the necessary and sufficient conditions for a special class of MIMO-IBC, where the numbers of antennas are divisible by the number of data streams per user. [START_REF] Jeon | The Feasibility of Interference Alignment for MIMO Interfering Broadcast-Multiple-Access Channels, journal=Trans. Wireless Communications[END_REF] established a necessary and sufficient condition on IA feasibility for MIMO Interfering Broadcast Multiple Access Channel (IBMAC), which characterizes the optimal sum DoF for various practical network configurations.

[13] addresses (centralized) attainable DoF for general interference networks with general channel rank conditions. The multiple antennas give each node a certain zero-forcing (ZF) budget that for a given DoF distribution needs to be coordinated between all nodes to handle all interference. [START_REF] Chen | To Cancel or Not to Cancel: Exploiting Interference Signal Strength in the Eigenspace for Efficient MIMO DoF Utilization[END_REF] gives also an approach to find the spatial filter matrices that offer the desired DoF scheduling and reduce the unwanted interference signal strength to close to zero (rather than absolute zero).

For the work reported in this paper, the starting point was the recently introduced simplified framework from [START_REF] Chen | A General Model for DoF-based Interference Cancellation in MIMO Networks with Rank-deficient Channels[END_REF] to analyze DoF feasibility (sufficient conditions) in interference networks. This ZF constraint accounting framework is particularly convenient to also analyze MIMO channels with reduced rank. However, it turns out that the presumed sufficient conditions in [START_REF] Chen | A General Model for DoF-based Interference Cancellation in MIMO Networks with Rank-deficient Channels[END_REF] are actually necessary conditions, and correspond to a role distribution in the well-known proper condition. We also make a correction to the conditions obtained in [START_REF] Chen | A General Model for DoF-based Interference Cancellation in MIMO Networks with Rank-deficient Channels[END_REF]. We furthermore introduce a second set of conditions based on the Sylvester rank inequality, which may allow to construct sufficient conditions. We apply our thus revised framework to the UL-to-DL MIMO interference scenario arising in DynTDD, which allows to characterize the DoF region as a function of number of users, antennas and channel ranks. We provide analytical expressions for SumDoF for a number of specific scenarios, including the uniform asymmetric scenario (antennas, ranks and user DoFs identical between users on one side, but different between UL and DL). We observe that the results differ (sometimes substantially) from the classical MIMO interference channel (IC) scenario, as here in DynTDD the useful signal links are to nodes outside of the UL-to-DL interfering group. Apart from the centralized design considered so far, we also introduce a distributed design, for which we had analyzed necessary and sufficient DoF conditions before [START_REF] Lejosne | Decoupled, Rank Reduced, Massive and Frequency-Selective Aspects In MIMO Interfering Broadcast Channels[END_REF]. We actually generalize the distributed design of [START_REF] Lejosne | Decoupled, Rank Reduced, Massive and Frequency-Selective Aspects In MIMO Interfering Broadcast Channels[END_REF] to a finer granularity of ZF role distribution between Tx and Rx, and we consider another distributed design which consists of unilateral ZF at either UL or DL side. The combination of these options lead to an optimized distributed solution of which the nature varies as a function of the channel ranks involved. It also leads to a reduced DoF gap to centralized designs and there exists a low rank region in which no DoF loss is observed. All this enhances the attractivity of a distributed design, which are in any case far more feasible to implement in practice since they only require local CSI.

II. DYAMIC TDD SYSTEM MODEL

We consider a MIMO system with two cells, one operating in DL and the other one in UL. Each cell has one BS of M antennas, with K ul and K dl interfering/interfered users in the UL and DL cell respectively. The kth DL UE and the lth UL UE have N dl,k and N ul,l antennas respectively. This scenario brings the two types of interference, the BS-to-BS interference, and the UE-to-UE interference between the UEs that are particularly on the edge of the two cells as shown in Fig 1 . The channel between the lth user in the UL cell and the kth user in the DL cell is denoted as H k,l ∈ C N dl,k ×N ul,l with k ∈ [1, ..., K dl ] and l ∈ [1, ..., K ul ]. Denote d dl,k and d ul,l as the number of data stream from the DL BS to the kth DL UE and from the lth UL UE to the UL BS respectively. We denote the rank of the UE-to-UE interference channel (IC) as r k,l . We have r k,l distinguishable significant paths contribute to H k,l . Then we can factorize H k,l as:

H k,l = B k,l A H k,l (1)
with a full rank matrices B k,l ∈ C N dl,k ×r k,l and A k,l ∈ C N ul,l ×r k,l . We have r k,l distinguishable significant paths contribute to H k,l , where distinguishable means with linearly independent antenna array responses from other paths, at both the Tx side and the Rx side. and G l ∈ C N ul,l ×d ul,l as the Rx/Tx beamforming (BF) matrices at the kth DL and the lth UL users respectively. ZF from UL UE l to the DL UE k requires:

F H k H k,l G l = 0, ∀k ∈ {1, ..., K dl }, ∀l ∈ {1, ..., K ul } .
(2) For the ZF conditions in interference networks [START_REF] Chen | A General Model for DoF-based Interference Cancellation in MIMO Networks with Rank-deficient Channels[END_REF] introduces binary variables 1 T k,l and 1 R k,l with the following definition:

1 T k,l = 1 if Tx node l is active for ZF from l to k, 0 otherwise (3) 1 R k,l = 1 if Rx node k is active for ZF from l to k, 0 otherwise (4)
We denote by z R k,l (resp. z T k,l ) the number of ZF constraints satisfied by the Rx (resp. the Tx). To cancel all interference from the UL UEs to the DL UEs, according to Theorem 1 in [START_REF] Chen | A General Model for DoF-based Interference Cancellation in MIMO Networks with Rank-deficient Channels[END_REF], the following conditions should be satisfied (here formulated for the UL2DL interference in the DynTDD problem considered):

z R k,l 1 R k,l + z T k,l 1 T k,l = min(1 R k,l d ul,l + 1 T k,l d dl,k , r k,l ) (5a) (1 R k,l , 1 T k,l ) = (0, 0) (5b) d dl,k + l∈I dl,k z R k,l ≤ N dl,k (5c) d ul,l + k∈I ul,l z T k,l ≤ N ul,l (5d)
where I dl,k denotes the set of UL UEs for which the CLI is zero-forced at the kth DL UE, and I ul,l denotes the set of DL UEs for which the CLI is zero-forced at the lth UL UE. Equation (5c) means that DL UE k has N dl,k antennas to receive d dl,k streams while performing ZF to the CLI coming from a certain number of UL UEs. And similarly for an UL UE in (5d). (1 R k,l , 1 T k,l ) = (1, 0) or (0, 1) or (1, 1) means that in this link, the ZF is performed by the Rx or the Tx or by both in a shared fashion. [START_REF] Chen | A General Model for DoF-based Interference Cancellation in MIMO Networks with Rank-deficient Channels[END_REF] believes that these conditions are sufficient (i.e. correspond to a feasible design of Tx/Rx filters) but actually they are necessary conditions. Furthermore, the first term on the RHS of (5a) is suboptimal in the case

(1 R k,l , 1 T k,l ) = (1, 1)
, when the minimum corresponds to the first argument. The work in this paper is inspired by the ideas of [START_REF] Chen | A General Model for DoF-based Interference Cancellation in MIMO Networks with Rank-deficient Channels[END_REF], which at first sight appears to introduce an elegant and simplified approach to DoF analysis in general interference networks, furthermore applicable to MIMO channels with general rank conditions. However, [START_REF] Chen | A General Model for DoF-based Interference Cancellation in MIMO Networks with Rank-deficient Channels[END_REF] does not seem to be aware of the vast body of work on interference alignment, and believes that the state of the art corresponds to one-sided ZF. As a result, they e.g. believe that the DoF region in [13, Fig. 1] represents an improvement w.r.t. their assumed state of the art, but actually the point [START_REF] Liu | On the Feasibility of Linear Interference Alignment for MIMO Interference Broadcast Channels With Constant Coefficients[END_REF][START_REF] Liu | On the Feasibility of Linear Interference Alignment for MIMO Interference Broadcast Channels With Constant Coefficients[END_REF] is also achievable in the DoF region of [13, Fig. 1]. But due to the suboptimality of (5a), [START_REF] Chen | A General Model for DoF-based Interference Cancellation in MIMO Networks with Rank-deficient Channels[END_REF] does not capture this. On the other hand, in other scenarios, some DoF distributions that are assumed to be feasible by [START_REF] Chen | A General Model for DoF-based Interference Cancellation in MIMO Networks with Rank-deficient Channels[END_REF] will in reality not be.

IV. INTERFERENCE ALIGNMENT (IA) CONDITIONS FOR THE DYNTDD UE-TO-UE IC -CENTRALIZED CASE

In this section we analyze the overall UL UE to DL UE interference, in the centralized case in which a central design unit disposes of the knowledge of all channels involved. Spatial linear interference alignment (IA) corresponds to joint ZF by TX and RX of all units involved, as indicated in (2).

A. Proper (IA) Conditions

We start by establishing the proper conditions, which were introduced in [START_REF] Lejosne | Decoupled, Rank Reduced, Massive and Frequency-Selective Aspects In MIMO Interfering Broadcast Channels[END_REF] for rank deficient MIMO channels. The proper conditions express that in order for a set of variables to be able to satisfy a set of non-linear (here bilinear) equations, the number of variables involved needs to equal at least the number of equations (constraints).

• The total number of variables in G l is d ul,l (N ul,l -d ul,l ), since only the column space of G l counts. Hence for ZF, G l is determined up to a d ul,l × d ul,l mixture matrix. The global proper condition is then

K ul l=1 d ul,l (N ul,l -d ul,l ) + K dl k=1 d dl,k (N dl,k -d dl,k ) ≥ K ul l=1 K dl k=1 min(r k,l d dl,k , r k,l d ul,l , d ul,l d dl,k ) . (6)
The proper condition is a necessary condition for the feasibility of IA. Now, inspired by the work in [START_REF] Chen | A General Model for DoF-based Interference Cancellation in MIMO Networks with Rank-deficient Channels[END_REF], we shall investigate localized instances of the proper condition, or stated differently, we shall consider a distribution of the roles of the Tx/Rx variables in satisfying the ZF conditions. Actually, we had already considered such a role distribution perspective in [START_REF] Negro | Interference alignment feasibility in constant coefficient MIMO interference channels[END_REF], which in general can go beyond the global proper condition. But in the scenario considered here, in which each DL UE receives interference from each UL UE (at least if all channel ranks are positive), the ensemble of local proper conditions add up to the single global proper condition. In any case, for the local version, consider first a somewhat simplified scenario in which the cross link ZF in ( 2) is either handled completely by the corresponding Rx F k or completely by the corresponding Tx G l . For the links (k, l) handled by the Rx, i.e. Lemma 1]. These constraints can actually be interpreted as applying to each column of F k , which a Rx beamformer for the corresponding stream of user k. For the overall BF matrix F k , we account for the d dl,k streams and we get a total of d dl,k z R k,l = min(d dl,k d ul,l , d dl,k r k,l ) ZF constraints. If we take into account that Rx F k will handle the ZF for the links in I dl,k , then we get the following local proper condition for F k :

(1 R k,l , 1 T k,l ) = (1, 0), then (2) represents a linear ZF equation in F k which represents z R k,l = rank(H k,l G l ) = min(d ul,l , r k,l ) constraints [13,
d dl,k (N dl,k -d dl,k ) ≥ l∈I dl,k d dl,k z R k,l (7)
⇒ N dl,k -d dl,k ≥ l∈I dl,k z R k,l (8)
where the last equation corresponds exactly to (5c). It can be interpreted as a proper condition per stream, where the subtraction on the LHS makes sure that after the N dl,k antennas are used for ZF of interfering links, d dl,k dimensions are left for receiving that many streams.

A completely analogous reasoning can be made for the case (1 R k,l , 1 T k,l ) = (0, 1) in which the ZF conditions are handled by the Tx side G l , which will lead to (5d). The remaining case is (1 R k,l , 1 T k,l ) = (1, 1), which is not handled correctly in [START_REF] Chen | A General Model for DoF-based Interference Cancellation in MIMO Networks with Rank-deficient Channels[END_REF] or in (5b). The correct treatment actually corresponds to a finer split between the ZF roles at Tx and Rx sides of a UE-to-UE link at stream level. The resulting correct local shows a relatively limited DoF loss for the more realistic distributed schemes. We illustrated the DoF in a symmetric uniform scenario for a range of Tx/Rx design schemes, which have varying coordination and CSI needs.

Fig. 1 :

 1 Fig. 1: DynTDD system Model

  DoF per user as a function of the rank of any cross link channel for a uniform symmetric scenario with N = 15, K ul = K dl = 4.

•

  The total number of variables in F k is d dl,k (N dl,k -d dl,k ), since again only the column space of F k counts. F k is determined up to a d dl,k × d dl,k mixture matrix. • Equation (2) represents min(d dl,k r k,l , d ul,l r k,l , d ul,l d dl,k ) constraints for the cross (interfering) link from UL UE l to DL UE k [15]. • The total number of cross links is K dl K ul .

ACKNOWLEDGEMENTS

EURECOM's research is partially supported by its industrial members: ORANGE, BMW, SAP, iABG, Norton LifeLock, and by the projects EEMW4FIX (French ANR) and SPOT-LIGHT (EU ITN).

proper conditions are ∀k, l :

where 1 x = 1 if x > 0 and 1 x = 0 otherwise. One can check easily that the cases (1 R k,l , 1 T k,l ) = (1, 0), (0, 1) discussed above can be recovered from [START_REF] Jeon | The Feasibility of Interference Alignment for MIMO Interfering Broadcast-Multiple-Access Channels, journal=Trans. Wireless Communications[END_REF]. The proof of (9) appears in Appendix A. It turns out that summing up all local conditions in (9) leads to the global proper condition [START_REF] Du | Interference Neutralization With Partial CSIT for Full-Duplex Cellular Networks[END_REF].

B. Sylvester Rank Inequality

Whereas the proper conditions may be necessary and sufficient for MIMO channels that are (close to) square, they do not capture the complete picture in general. Other conditions can be obtained by considering Sylvester's law of nullity [START_REF] Horn | Matrix Analysis[END_REF], which states that the dimension of the null space of a product of two matrices cannot be larger than the sum of the null space dimensions of the two factors. This leads to a condition closely related to [11, (16) or ( 22)]. For singular (factorizable) matrices, one may consider an extension for the product of three matrices, which is provided by the Frobenius rank inequality [START_REF] Horn | Matrix Analysis[END_REF]: rank(ABC) ≥ rank(AB) + rank(BC)rank(B) [START_REF] Chen | On DoF-Based Interference Cancellation Under General Channel Rank Conditions[END_REF] of which Sylvester's rank inequality is a special case with

Then putting together (2) for all Tx, we get

For F k to be able to perform this ZF while receiving d dl,k streams, we require

On the other hand Sylvester's rank inequality implies rank(H k,: G) ≥ rank(H k,: ) + rank(G) -N ul = min(N dl,k , r k,: , N ul ) + d ul -N ul [START_REF] Chen | A General Model for DoF-based Interference Cancellation in MIMO Networks with Rank-deficient Channels[END_REF] where the statement about rank(H k,: ) should be interpreted w.p. 1 for a sufficiently random channel distribution. Putting together [START_REF] Negro | Interference alignment feasibility in constant coefficient MIMO interference channels[END_REF] and ( 13), we get

By interchanging the roles or Tx and Rx (UL/DL duality), we get similarly N ul,l -d ul,l ≥ min(N ul,l , r :,l , N dl ) + d dl -N dl , ∀l . [START_REF] Lejosne | Decoupled, Rank Reduced, Massive and Frequency-Selective Aspects In MIMO Interfering Broadcast Channels[END_REF] In the likely case that min(N dl,k , r k,: , N ul ) = N dl,k and min(N ul,l , r :,l , N dl ) = N ul,l , (14) becomes

and [START_REF] Lejosne | Decoupled, Rank Reduced, Massive and Frequency-Selective Aspects In MIMO Interfering Broadcast Channels[END_REF] becomes

which would not be very constraining.

V. INTERFERENCE ALIGNMENT (IA) CONDITIONS FOR THE DYNTDD UE-TO-UE IC -DISTRIBUTED CASE

The necessary conditions for the distributed solutions considered in this section are actually also sufficient conditions (the corresponding Tx/Rx designs are feasible) since they correspond to linear ZF equations.

A. Distributed Solutions Exploiting the Low Rank Channel Factorizations

As opposed to the centralized case, in the distributed case, each Tx/Rx disposes of at most local CSI, i.e. of the channels directly connected to it. In this case the global proper condition (6) reduces to

which corresponds to decoupling between the designs for the F k and the G l . The corresponding local proper conditions become

This represents a generalization of [START_REF] Lejosne | Decoupled, Rank Reduced, Massive and Frequency-Selective Aspects In MIMO Interfering Broadcast Channels[END_REF] in which we considered the special case in which either

In other words the interference of a particular UE-to-UE link is handled completely by either the Tx or the Rx. But the handling of all UE-to-UE links is still partitioned between UL and DL UEs. The ZF of any particular UE-to-UE link can also be shared between Tx and Rx, as considered here. The first line in (19) can be interpreted as

where the zero rows in the two factors are complementary so that (2) is satisfied.

B. Distributed Solutions Based on Fixed Tx/Rx Factors

When the selections (1 R k,l , 1 T k,l ) = (1, 0) or (0, 1) are applied to all links, then in general the design of Tx and Rx is coupled. This coupling can be broken if either

) is applied to all links. In this case we get ∀k, l :

where of course (5c), (5d) continue to apply. When the Rx handle all the ZF, then the Tx can be designed separately, e.g. based on the UE-BS channels, and vice versa.

VI. UNIFORM SCENARIOS

To have a simplified version of the conditions above we define a uniform asymmetric scenario, in which:

Then the centralized proper condition (6) becomes:

We define also a uniform symmetric case when For

Now, consider the local proper conditions and introduce

) denote the number of UL (resp. DL) UEs for which the cross link interference is cancelled by the kth DL UE (resp. the lth UL UE). For this ZF role distribution to ensure the cancellation of all CLI, we require

In the uniform case, n F,k = n F , ∀k ∈ [1, ..., K dl ] and n G,l = n G , ∀l ∈ [1, ..., K ul ], equation (24) becomes:

The optimization of n F , n G depends on the desired point (d dl , d ul ) in the DoF region. For a uniform asymmetric case, we get on the Rx side (

and on the Tx side ((1 R k,l , 1 T k,l ) = (0, 1)).

Exploring the case (1 R k,l , 1 T k,l ) = (1, 1) only leads to a finer granularity of ZF roles (at stream level instead of user level). For the symmetric case we have n F = n G = K 2 and we get back (23).

We can consider the distributed approaches with fixed Tx/Rx factors (symmetric uniform case):

• (1 R k,l , 1 T k,l ) = (1, 1): considering (5a) from [START_REF] Chen | A General Model for DoF-based Interference Cancellation in MIMO Networks with Rank-deficient Channels[END_REF], applied to all links, we can take

Which all three yield worse DoF than (23).

If we consider the distributed solution based on low rank channel factorizations, then for the uniform asymmetric case we obtain:

As in the centralized case, to ensure the cancellation of all the cross link interference equation ( 25) should be satisfied.

For the uniform symmetric case, we get n F = n G = K 2 , and (31a), (31b) become:

VII. DISCUSSION

From the proper conditions established in the previous sections for the DynTDD UE-to-UE IC, we notice for the uniform asymmetric scenario that:

• When min(d ul , r) = min(d dl , r) = r, the conditions in the centralized case ((26) and ( 27)) meet the conditions in the distributed case ((31a) and (31b)).

• At a stream level the bilateral solution (1 R k,l , 1 T k,l ) = (1, 1) leads to a distributed design for r ≤ min(d dl , d ul ) (and a half distributed solution when r is in between the two d's, see Appendix A), In Table I we evaluate the DoF for a number of dimensions considering the uniform symmetric case. The first line corresponds to the global proper (necessary) condition, which in this case also yield feasible DoF. The other approaches lead to reduced DoF. The distributed solution does not exhibit suboptimality w.r.t. the centralized optimal solution for a rank up to r = 5. From the analysis of the table and considering a distributed design we can conclude:

• For the uniform symmetric case the best distributed solution is given by the distributed approach (32) for r < d, and for r > d it is given by the unilateral cases (ZF at only Rx (28) or only Tx (29)).

VIII. CONCLUSIONS

In this work, we addressed the interference between users arising in DynTDD systems. Considering the centralized and distributed cases, we studied the proper and sufficient conditions for the interference cancellation. This analysis highlights the gap between centralized and distributed schemes, but APPENDIX A

Here we prove the constraints on z R k,l , z T k,l appearing in (9), esp. for the case

Case r ≤ min(d dl , d ul )

(2) can be rewritten as

An application of Sylvester's rank inequality to (33) yields

We can choose B H k F k,l to have z R k,l zero rows and A H k G k,l to have z t k,l zero rows so that z R k,l + z T k,l = r k,l . The optimized values for z R k,l , z T k,l depend on the other variables d dl , d ul , N ul , N dl , K ul and K dl . We see that when r ≤ min(d dl , d dl ), the case (1 R k,l , 1 T k,l ) = (1, 1) leads to a distributed design: the design of F k depends only on the factor B k,l in H k,l and not on G l , and similarly G l only depends on A k,l .

Case r ≥ max(d dl , d ul ) In this case F H k H k,l G l is a priori full rank, in the case of arbitrary Tx/Rx. Here we assume a uniformity of the number of zeros produced by the columns (beamformers) in F k and G l . Let F k now produce z R k,l zeros in each row of the matrix product F H k H k,l G l , i.e. in total it produces d dl,k z R k,l zeros (the position of the zeros in each row may be different so that the number of non-zeros per column is also equal between all columns). Then let G l produce z T k,l zeros in each column, with the constraint that we produce a total of d dl,k z R k,l +d ul,l z T k,l = d dl,k d ul,l zeros. In this case, the design of F k and G l is clearly coupled. Now, in the combination of the two cases above, care has to be taken with the limiting cases z R k,l = 0 or z T k,l = 0, corresponding to one-sided ZF. In that case we have linear ZF equations representing a number of ZF constraints equal to the rank of the matrix of coefficients, as mentioned in the discussion of the cases (1 z R k,l , 1 z T k,l ) = (1 R k,l , 1 T k,l ) = (1, 0) or (0, 1). This results in the first condition appearing in [START_REF] Jeon | The Feasibility of Interference Alignment for MIMO Interfering Broadcast-Multiple-Access Channels, journal=Trans. Wireless Communications[END_REF].