Interference Channel with Delay: Noisy interference Anas Chaaban and Aydin Sezgin Emmy-Noether Research Group on Wireless Networks Institute of Telecommunications and Applied Information Theory Ulm University, 89081, Ulm, Germany Email: anas.chaaban@uni-ulm.de, aydin.sezgin@uni-ulm.de Abstract—We consider the interference channel with two transmit-receive pairs when there is a delay in receiving the desired message compared to interference. We study this channel in the weak interference regime (noisy interference), and show that delay does not affect the capacity of the interference channel in this region. This also holds for an interference channel with a more general delay, where receiving the desired message is delayed compared to the interference, or vice versa. This result is then applied to a previous results on the interference MIMO relay channel where a scheme is used to transform it to an interference channel with delay. We show that using a relay can be helpful when interference is treated as noise in the interference relay channel. I. I NTRODUCTION The capacity of wireless networks has been the topic of intensive research for decades. For the simple case of two transmit-receive pairs, the capacity region was characterized to within one bit in [1]. However, it remains an open problem for the general case where more transmit-receive pairs need to communicate. There has been important steps towards solving this problem, especially with the introduction of interference alignment schemes [2], [3]. In [4]–[6], it was shown that the capacity of the interference channel (IC) is achieved using the simple scheme of treating interference as noise (TIN), as long as the cross channels h 12 and h 21 and the transmit powers P 1 and P 2 at the first and the second user respectively satisfy |h 12 ( 1+ h 2 21 P 2 ) | + |h 21 ( 1+ h 2 12 P 1 ) |≤ 1. Extensions of this work to the multiple antenna case were considered in [7], [8]. TIN is a simple practical technique that does not require complicated processing. It is always interesting to analyze such simple suboptimal schemes from a practical point of view. For example, the rate region of a K-user interference channel is analyzed for the case in which the interference is treated as noise in [9]. Another area of interest in wireless networks is to charac- terize the interference relay channel (IC-R). The IC-R is of practical importance where a relay is used to aid the com- munication between transmit-receive pairs. The relay might have access to abundant power and might be capable of This work is supported by the German Research Foundation, Deutsche Forschungsgemeinschaft (DFG), Germany, under grant SE 1697/3. complicated processing. Several schemes were proposed for the IC-R, e.g. [10]–[12], including block Markov encoding [13], dirty paper coding [14], or the Han-Kobayashi scheme [15]. For the interference MIMO relay channel (IC-MR), a scheme was proposed in [16], where block Markov encoding, beamforming, and Willems’ backward decoding [17] were used. The low transmit power performance of this scheme was analyzed in [18]. With this scheme, an important part of interference is eliminated, and the remaining interference is treated as noise. Interestingly, this scheme transforms the IC-MR to an IC with delay (IC-D), which we study in this paper. In [19], it was shown that delay makes a difference in interference networks with more than two users, where it increases the degrees of freedom. The delay is exploited in this case and used for interference alignment. Delay is inevitable in practice, and hence it would be useful to exploit it. We consider here the IC with two users, and show that, at least for the TIN region, delay does not affect the capacity. This holds as long as the duration of the communication is much longer than the delay. The optimality of TIN in the IC- D is analyzed, i.e. we derive a condition on the IC-D that allows us to optimally treat interference as noise. In fact, we show that the same condition holds as for the IC. Then, we use this condition to establish conclusions on the IC-MR with the transmission scheme proposed in [16], that transforms the IC-MR to an IC-D with channel parameters given in [18]. The IC-MR is studied for the case of low transmit power at the sources, with possible higher power at the relay, which is of practical relevance. The model of the IC-D is given in II, the TIN region of the IC-D is derived in section III. Then we study the IC-MR and the impact of the technique in [16] on the TIN region in section IV. A numerical example is included in section V. Finally, we conclude with section VI. II. IC-D CHANNEL MODEL We consider a symmetric interference channel with a delay of one block on the direct channel, as shown in figure 1. The input-output equations are given by: Y n(b) i = X n(b−1) i + hX n(b) j + Z n i (1) where in block b, X n(b) i and Y n(b) i are the n-symbol transmit and received signals of receiver and transmitter i respectively, 2010 International ITG Workshop on Smart Antennas (WSA 2010) 978-1-4244-6072-4/10/$26.00 ©2010 IEEE 8