Y - Δ Product in 3-Way Δ and Y - Channels for Cyclic Interference and Signal Alignment Henning Maier ∗ , Anas Chaaban † and Rudolf Mathar ∗ ∗ Institute for Theoretical Information Technology, RWTH Aachen University, 52056 Aachen, Germany † Chair of Communication Systems, Ruhr-Universität Bochum, 44780 Bochum, Germany Email: {maier, mathar}@ti.rwth-aachen.de, {anas.chaaban}@rub.de Abstract—In a full-duplex 3-way Δ channel, three transceivers communicate to each other, so that a number of six messages is exchanged. In a Y -channel, however, these three transceivers are connected to an intermediate full-duplex relay. Loop-back self- interference is suppressed perfectly. The relay forwards network- coded messages to their dedicated users by means of interference alignment (IA) and signal alignment. A conceptual channel model with cyclic shifts described by a polynomial ring is considered for these two related channels. The maximally achievable rates in terms of the degrees of freedom measure are derived. We observe that the Y - channel and the 3-way Δ channel provide a Y -Δ product relationship. Moreover, we briefly discuss how this analysis relates to spatial IA and MIMO IA. I. I NTRODUCTION In two-way full-duplex communication systems, users op- erate both as transmitters and receivers, i. e., transceivers, and exchange messages with each other in a bidirectional manner. A generalization of the two-way channel is the K-user full- duplex interference channel. In [1], this channel is considered for time-varying channel coefficients. It is shown that a full- duplex channel can be equivalently represented by a fully- connected K- user interference channel with perfect feedback links between the transmitters and the receivers with the same index. To achieve the upper bounds on the degrees of freedom (DoF), the innovative concept of Interference Alignment (IA) [2] is applied. For K = 3 we call this a 3-way Δ channel. A Y - channel [3] is a related 3-way communication system but with one intermediate relay. Each transceiver sends two messages to the relay, and the relay forwards three network- coded messages back to the dedicated users. The DoF of the MIMO Y - channel with an arbitrary antenna configuration are provided in [4]. In [5], the capacity region of the related linear shift deterministic Y - channel is derived. A conceptual channel model based on polynomials and inspired by cyclic codes as introduced in [6] to investigate the impact of interference in multi-user networks. Therein, Cyclic IA on the X- channel and the K- user interference channel is considered. A Cyclic Interference Neutralization scheme on this channel model was investigated in [7]. The polynomial model is closely related to the finite-field model in [8] and to the linear shift deterministic channel model introduced in [9]. This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) within the project Power Adjustment and Constructive Interference Alignment for Wireless Networks (PACIA - Ma 1184/15-2) of the DFG program Communication in Interference Limited Networks (COIN) and fur- thermore by the UMIC Research Centre, RWTH Aachen University. Contributions. In 3-way Δ channels and Y - channels, each user intends to convey two messages, i.e., one message to each other. We derive optimal Cyclic IA schemes for both channels in terms of the conceptual polynomial channel model. We ob- serve that the provided schemes achieving the same proposed upper bounds and are essentially equivalent. The Y -channel is expressed by a Δ channel using a Y - Δ product relationship. This relationship is evidently motivated by elementary circuit theory. To the best of our knowledge, it has not been reported in terms of information theory of multi-user communications yet. Note that, in contrast to our previous works [6] and [7], the channel matrices are not subject to further conditions. Organization. In Sec. II we define the conceptual model of the polynomial representation for the 3-way Δ channel and the Y - channel. An upper bound on the DoF is provided in Sec. III. We propose corresponding Cyclic IA schemes for both channels in Sec. IV and V. The Y - Δ product of Cyclic IA is discussed in Sec. V-D. In Sec. VI, we briefly relate our results to IA schemes in [1], [4]. II. SYSTEM MODEL We adapt the notation used in [6] and [7]. The set of user- indices is defined by K ∶= {1, 2, 3}. In a full-duplex system, a user is a combined transmitter and receiver and denoted as a transceiver T i , i ∈K. There are 6 independent message vectors w ji , namely, w 12 , w 21 , w 13 , w 31 , w 23 and w 32 , dedicated to be conveyed from a transceiver T i to a transceiver T j , with i ≠ j ∈K, i. e., each transceiver broadcasts two message vec- tors. The message vectors w ji contain α ji ∈ N submessages W [∗] ji and are denoted by a vector w ji =(W [1] ji ,...,W [αji] ji ). A submessage is a string of t ∈ N bits W [∗] ji ∈ B ={0, 1} t . We interpret the different number of the submessages as individual rate demands per user-pair. The number of submessage per dedicated user-pair is expressed by the messaging matrix: M = ⎛ ⎜ ⎜ ⎝ 0 α 12 α 13 α 21 0 α 23 α 31 α 32 0 ⎞ ⎟ ⎟ ⎠ , (1) and the total number of submessages amounts to: M =∥M∥ 1 = α 12 + α 21 + α 13 + α 31 + α 23 + α 32 . We consider polynomial rings F(x)/(x n − 1) with the inde- terminate x. The channel access at each T i is partitioned into arXiv:1401.3995v1 [cs.IT] 16 Jan 2014