Retrospective Interference Alignment over Interference Networks Hamed Maleki * , Syed A. Jafar * and Shlomo Shamai (Shitz) † * CPCC, EECS Dept., University of California Irvine, Irvine, CA 92697, USA † EE Dept., Technion-Israel Institute of Technology, Technion City, Haifa 32000, Israel Email:{hmaleki,syed}@uci.edu, sshlomo@ee.technion.ac.il Abstract—Maddah-Ali and Tse recently introduced the idea of retrospective interference alignment, i.e., achieving interference alignment with only outdated (stale) channel state information at the transmitter (CSIT), in the context of the vector broadcast channel. Since the scheme relies on the centralized transmit- ter’s ability to reconstruct all the interference seen in previous symbols, it is not clear if retrospective interference alignment can be applied in interference networks consisting of distributed transmitters and receivers, where interference is contributed by multiple transmitters, each of whom can reconstruct only the part of the interference caused by themselves. In this work we prove that even in such settings, retrospective interference alignment is feasible. Specifically, we show that it is possible to achieve more than 1 DoF based on only delayed CSIT in the 3 user interference channel and the 2 user X channel consisting of only single antenna nodes. Retrospective interference alignment is also shown to be possible in other settings, such as the 2 user MIMO interference channel and with delayed channel output feedback. I. I NTRODUCTION There is much interest in studying the degrees of freedom (DoF) — and thereby exploring the potential for interference alignment (IA) — in wireless networks in the absence of instantaneous CSIT [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. On the one hand, there are pessmistic results that include [1], [2], [3], [4], [8], [9], [10] where the DoF are found to collapse due to the inability of the transmitters to resolve spatial dimensions. On the other hand, there are more recent optimistic results [5], [6], [7], [11], [12], [13] where the feasibility of interference alignment is demonstrated under various models of channel uncertainty at the transmitter(s). Closely related to this work are the papers on blind interference alignment [11], [12] and especially the recent work on interference alignment with delayed CSIT [13]. Reference [11] introduces the idea of blind interference alignment, i.e., the ability to achieve interference alignment without any knowledge (not even delayed knowledge) of chan- nel coefficient realizations at the transmitters, by exploiting only statistical knowledge in the form of heterogeneous chan- nel coherence structures associated with different users. The sensitivity of network DoF to channel coherence structures un- der the assumption of no CSIT and perfect CSIR goes against The work of Syed Jafar and Hamed Maleki was supported by ONR under grant N00014-12-1-0067 and by NSF CCF under grants 0546860 and 0830809. The work of S. Shamai was supported by the Israel Science Foundation (ISF). the conventional wisdom that because channel coherence inter- vals are only relevant in the training and channel estimation at the receivers, they cannot affect capacity when perfect CSIR is assumed (thereby taking training and channel estimation issues at the receiver out of the picture). Indeed in the single user setting channel capacity with no CSIT and perfect CSIR is independent of the channel coherence interval. It is therefore surprising that under no CSIT and perfect CSIR, not only the network capacity but also the network DoF depend on the channel coherence intervals, making a case against the commonly used i.i.d. fading model for studying the capacity of wireless networks. Blind interference alignment shows that heterogenous channel coherence structures, whether present naturally in the form of different channel coherence times and coherence bandwidths experienced by different users [11], or imposed artifically by antenna switching [12], create oppor- tunities for interference alignment. This is mainly because of the insight that because the receivers observe a linear combination of all transmitted symbols with the coefficients determined by the channel realizations, a simple repetition of information symbols from all transmitters will create linearly independent combinations of the transmitted symbols for the receivers that experience changing channel conditions, but will repeat the same linear combination of transmitted symbols for those receivers whose channel conditions remain fixed. Thus, each receiver’s desired symbols should be transmitted across those dimensions where its own channel changes while the other (undesired) receivers’ channels remain fixed. This allows each receiver to resolve its desired symbols which appear in linearly independent combinations each time, while all interference symbols (because they are observed in the same linear combination each time) are aligned along the all- ones vector — thereby achieving interference alignment. More recently, reference [13] has introduced the delayed CSIT model, that will also be the main focus of this paper. The delayed CSIT model assumes i.i.d. fading channel conditions, with no knowledge of current channel state at the transmitter. However, perfect knowledge of channel states is available to the transmitter with some delay. The surprising finding of [13], in the context of the vector broadcast (BC) channel, is that not only is CSIT helpful even when it is outdated, but also that it can have a very significant impact as it is capable of increasing the DoF. The delayed CSIT model studied in [13] is particularly relevant in practice where invariably a delay is involved in any feedback from the receivers to the