For the fully connected K user wireless interference channel where the channel coefficients are time-varying and are drawn from a continuous distribution, the sum capacity is characterized as C(SNR) = K 2 log(SNR) +o(log(SNR)). Thus, the K user time-varying interference channel almost surely has K=2 degrees of freedom. Achievability is based on the idea of interference alignment. Examples are also provided of fully connected K user interference channels with constant (not time-varying) coefficients where the capacity is exactly achieved by interference alignment at all SNR values.

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hop, is especially interesting, as the intermediate hop takes place over an interference channel with single antenna nodes. While the two user interference channel with single antenna nodes has only one degree of freedom by itself, it is able to deliver degrees of freedom when used as an intermediate stage between a antenna source and a antenna destination [5]. The key is an amplify and forward scheme where the relay nodes, instead of trying to decode the messages, simply scale and forward their received signals. [1]–[3] consider end to end channel orthogonalization with distributed sources, relays and destination nodes and determine the capacity scaling behavior with the number of relay nodes. It is shown that distributed orthogonalization can be obtained even with synchronization errors if a minimum amount of coherence at the relays can be sustained. Degrees of freedom for linear interference networks with local side-information are explored in [22] and cognitive message sharing is found to improve the degrees of freedom for certain structured channel matrices. The MIMO MAC and BC channels show that there is no loss in degrees of freedom even if antennas are distributed among users at one end (either transmitters or receivers) making joint signal processing infeasible, as long as joint signal processing is possible at the other end of the communication link. The multiple hop example of [5], described above, shows that there is no loss of degrees of freedom even with distributed antennas at both ends of a communication hop (an interference channel) as long as the distributed antenna stages are only intermediate

We provide achievability as well as converse results for the degrees of freedom region of a multiple-input multipleoutput (MIMO) X channel, i.e., a system with two transmitters, two receivers, each equipped with multiple antennas, where independent messages need to be conveyed over fixed channels from each transmitter to each receiver. The inner and outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message. With M =1antennas at each node, we find that the total (sum rate) degrees of freedom are bounded above and below as 1? 4 X.IfM>1 and channel

We explore the degrees of freedom of M × N user wireless X networks, i.e. networks of M transmitters and N receivers where every transmitter has an independent message for every receiver. We derive a general outerbound on the degrees of freedom region of these networks. When all nodes have a single antenna and all channel coefficients vary in time or frequency, we show that the total number of degrees of freedom of the X network is equal to MN M+N−1 per orthogonal time and frequency dimension. Achievability is proved by constructing interference alignment schemes for X networks that can come arbitrarily close to the outerbound on degrees of freedom. For the case where either M = 2 or N = 2 we find that the degrees of freedom characterization also provides a capacity approximation that is accurate to within O(1). For these cases the degrees of freedom outerbound is exactly achievable. There is increasing interest in approximate capacity characterizations of wireless networks as a means to under-standing their performance limits. In particular, the high SNR regime- where the local additive white Gaussian noise (AWGN) at each node is de-emphasized relative to signal and interference powers- offers fundamental insights into optimal interference management schemes. The degrees-of-freedom approach provides a capacity

It has been conjectured by Høst-Madsen and Nosratinia that complex Gaussian interference channels with constant channel coefficients have only one degree-of-freedom regardless of the number of users. While several examples are known of constant channels that achieve more than 1 degree of freedom, these special cases only span a subset of measure zero. In other words, for almost all channel coefficient values, it is not known if more than 1 degree-of-freedom is achievable. In this paper, we settle the Høst-Madsen-Nosratinia conjecture in the negative. We show that at least 1.2 degrees-of-freedom are achievable for all values of complex channel coefficients except for a subset of measure zero. For the class of linear beamforming and interference alignment schemes considered in this paper, it is also shown that 1.2 is the maximum number of degrees of freedom achievable on the complex Gaussian 3 user interference channel with constant channel coefficients, for almost all values of channel coefficients. To establish the achievability of 1.2 degrees of freedom we introduce the novel idea of asymmetric complex signaling- i.e., the inputs are chosen to be complex but not circularly symmetric. It is shown that unlike Gaussian point-to-point, multiple-access and broadcast channels where circularly

Previous work showed that the 2×2×2 interference channel, i.e., the multihop interference network formed by concatenation of two 2-user interference channels, achieves the min-cut outer bound value of 2 DoF. This work studies the 2×2×2 interference channel with one additional assumption that two relays interfere with each other. It is shown that even in the presence of the interfering links between two relays, the min-cut outer bound of 2 DoF can still be achieved for almost all values of channel coefficients, for both fixed or time-varying channel coefficients. The achievable scheme relies on the idea of aligned interference neutralization as well as exploiting memory at source and relay nodes.

We explore similarities and differences in recent works on blind interference alignment un-der different models such as staggered block fading model and the delayed CSIT model. In particular we explore the possibility of achieving interference alignment with delayed CSIT when the transmitters are distributed. Our main contribution is an interference alignment scheme, called retrospective interference alignment in this work, that is specialized to settings with distributed transmitters. With this scheme we show that the 2 user X channel with only delayed channel state information at the transmitters can achieve 8/7 DoF, while the interfer-ence channel with 3 users is able to achieve 9/8 DoF. We also consider another setting where delayed channel output feedback is available to transmitters. In this setting the X channel and the 3 user interference channel are shown to achieve 4/3 and 6/5 DoF, respectively. 1

While the best known outerbound for the K user interference channel states that there cannot be more than K/2 degrees of freedom, it has been conjectured that in general the constant interference channel with any number of users has only one degree of freedom. In this paper, we explore the spatial degrees of freedom per orthogonal time and frequency dimension for the K user wireless interference channel where the channel coefficients take distinct values across frequency slots but are fixed in time. We answer five closely related questions. First, we show that K/2 degrees of freedom can be achieved by channel design, i.e. if the nodes are allowed to choose the best constant, finite and nonzero channel coefficient values. Second, we show that if channel coefficients can not be controlled by the nodes but are selected by nature, i.e., randomly drawn from a continuous distribution, the total number of spatial degrees of freedom for the K user interference channel is almost surely K/2 per orthogonal time and frequency dimension. Thus, only half the spatial degrees of freedom are lost due to distributed processing of transmitted and received signals on the interference channel. Third, we show that interference alignment and zero forcing suffice to achieve all the degrees of freedom in all cases. Fourth, we show that the degrees of freedom D directly lead to an O(1) capacity characterization of the form C(SNR) = D log(1 + SNR) +O(1) for the multiple access channel, the broadcast channel, the 2 user interference channel, the 2 user MIMO X channel and the 3 user interference

Abstract — Previous work showed that the X network with M transmitters, N receivers has MN degrees of freedom. In this