CADAMBE AND JAFAR: INTERFERENCE ALIGNMENT AND DEGREES OF FREEDOM OF THE-USER INTERFERENCE CHANNEL3435Fig. 3. Interference Alignment: Everyone gets half the cake.VI. CONCLUSION We have shown that with perfect channel knowledge the user interference channel has (almost surely) degrees of freedom when the channel coef cients are time-varying and are generated from a continuous distribution. The key idea is interference alignment which maximizes the overlap between the signal spaces of all interference signals at each receiver so that the size of the interference-free space is maximized for the desired signal. Due to relativity of alignment, it is possible that signals align at the receivers where they are not desired and remain distinguishable at the receivers where they are desired. Somewhat surprisingly it is shown that all the interference can be concentrated roughly into one half of the signal space at each receiver, leaving the other half available to the desired signal and free of interference. The alignment can be accomplished for any number of users but as the number of users increases a larger signal space is needed for each user to recover nearly half of it. While this work shows the potential bene ts of interference alignment, several challenges must be overcome before these bene ts translate into practice. One key issue is the assumption of global channel knowledge. While a node may acquire channel state information for its own channels, it is much harder to learn the channels between other pairs of nodes with which this node is not directly associated. On the other hand, global channel knowledge may not be necessary if there is a feedback channel through which the receivers can guide the transmitters into aligned con gurations in real time by applying incremental corrections. Also iterative algorithms based on channel reciprocity may be able to align interference in a distributed fashion [28]. The key insight of this paper is the role of interference alignment in a wireless network. From a capacity perspective the idea of interference alignment reaf rms the need for structured codes in wireless networks, also pointed out by [29]. For the single user point to point Gaussian channel it is well known that the capacity can be achieved through random (Gaussian) codebooks as well as through structured (lattice) codes. There is a growing realization that structured codes, optional for the single user case, may be necessary for approaching the capacityof networks. In an interference network when we design one user’s codebook we are also designing the interference/noise that will be seen by other users. Having structure in the interference may therefore be necessary. It is the structure imposed on the transmitted signals that facilitates interference alignment in this work. The intuition from this work is that since random codes will not automatically align themselves, structured codes will be necessary for wireless networks. Indeed interference alignment at the codeword level has been shown to be optimal in the capacity sense in [21] and in the degrees of freedom sense in [22] for some interesting cases. A combination of Han–Kobayashi [3] type achievable schemes and structured codes is a promising avenue in the quest for the capacity of wireless networks. APPENDIX I EXAMPLE: INTERFERENCE ALIGNMENT VIA DELAY OFFSETS Consider the -user interference network shown in Fig, 3, where there is a propagation delay associated with each channel. In particular, let us assume that the propagation delay is equal to one symbol duration for all desired signal paths and two symbol durations for all paths that carry interference signals. is de ned as The channel output at receiver (30) where during the th time slot (symbol duration) transmitter sends symbol and is i.i.d. zero mean unit variance Gaussian noise (AWGN). All inputs and outputs are complex. The transmit power at each transmitter is E . In the absence of interference, each user . Now, with all would achieve a capacity of the interferers present, suppose each transmitter transmits only ) and is silent during the during odd time slots (with power even time slots. Let us consider what happens at receiver 1. The symbols sent from its desired transmitter (transmitter 1) are received free from interference during the even time slots and all the undesired (interference) transmissions are received simultaneously during the odd time slots. Thus, each user is able to access the channel one-half of the time with no interference from other users. Each user achieves a rateAuthorized licensed use limited to: Harbin Institute of Technology. Downloaded on June 01,2010 at 01:21:44 UTC from IEEE Xplore. Restrictions apply.