In a K-pair-user two-way interference channel (TWIC), 2K messages and 2K transmitters receivers form a K-user IC in the forward direction (K messages) and another K-user IC in the backward direction which operate in full-duplex mode. All nodes may interact, or adapt inputs to past received signals. We derive a new outer bound to demonstrate that the optimal degrees of freedom (DoF, also known as the multiplexing gain) is K: full-duplex operation doubles the DoF, but interaction does not further increase the DoF. We next characterize the DoF of the K-pair-user TWIC with a MIMO, full-duplex relay. If the relay is non-causal instantaneous (at time k forwards a function of its received signals up to time k) and has 2K antennas, we demonstrate a one-shot scheme where the relay mitigates all interference to achieve the interference-free 2K DoF. In contrast, if the relay is causal (at time k forwards a function of its received signals up to time k 1), we show that a full-duplex MIMO relay cannot increase the DoF of the K-pair-user TWIC beyond K, as if no relay or interaction is present. We comment on reducing the number of antennas at the instantaneous relay.
The @math -user interference channel, as an extension of the 2-user interference channel, information theoretically models wireless communications in networks involving more than two-pairs of users. Using the idea of interference alignment @cite_13 @cite_33 @cite_20 , the DoF of the @math -user (one-way) IC for both time-varying channels and (almost all) The precise definition of almost all'' may be found in @cite_36 . constant channels has been shown to be @math in @cite_7 and @cite_36 respectively. The generalized DoF of the @math -user IC without and with feedback have been characterized in @cite_25 and @cite_27 (full feedback from receiver @math to transmitter @math ) respectively. Authors in @cite_3 showed that for almost all constant channel coefficients of fully connected two-hop wireless networks with @math sources, @math relays and @math destinations (source nodes are not destination nodes as they are here, i.e. the network is one-way), the DoF is @math .
@cite_33 @cite_7 @cite_36 @cite_3 @cite_27 @cite_13 @cite_25 @cite_20
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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 @math directly lead to an @math capacity characterization of the form @math for the multiple access channel, the broadcast channel, the 2 user interference channel, the 2 user MIMO X channel and the 3 user interference channel with M>1 antennas at each node. Fifth, we characterize the degree of freedom benefits from cognitive sharing of messages on the 3 user interference channel. In this paper, we develop the machinery of real interference alignment. This machinery is extremely powerful in achieving the sum degrees of freedom (DoF) of single antenna systems. The scheme of real interference alignment is based on designing single-layer and multilayer constellations used for modulating information messages at the transmitters. We show that constellations can be aligned in a similar fashion as that of vectors in multiple antenna systems and space can be broken up into fractional dimensions. The performance analysis of the signaling scheme makes use of a recent result in the field of Diophantine approximation, which states that the convergence part of the Khintchine-Groshev theorem holds for points on nondegenerate manifolds. Using real interference alignment, we obtain the sum DoF of two model channels, namely the Gaussian interference channel (IC) and the X channel. It is proved that the sum DoF of the K-user IC is (K 2) for almost all channel parameters. We also prove that the sum DoF of the X-channel with K transmitters and M receivers is (K M K + M - 1) for almost all channel parameters. We show that fully connected two-hop wireless networks with K sources, K relays and K destinations have K degrees of freedom for almost all values of constant channel coefficients. Our main contribution is a new interference-alignment-based achievability scheme which we call aligned network diagonalization. This scheme allows the data streams transmitted by the sources to undergo a diagonal linear transformation from the sources to the destinations, thus being received free of interference by their intended destination. The symmetric K user interference channel with fully connected topology is considered, in which (a) each receiver suffers interference from all other K − 1 transmitters, and (b) each transmitter has causal and noiseless feedback from its respective receiver. The number of generalized degrees of freedom (GDoF) is characterized in terms of α, where the interference-to-noise ratio (INR) is given by INR = SNRα. It is shown that the number of per-user GDoF of this network is the same as that of the 2-user interference channel with feedback, except for α = 1, for which existence of feedback does not help in terms of GDoF. The coding scheme proposed for this network, termed cooperative interference alignment, is based on two key ingredients, namely, interference alignment and interference decoding. We characterize the generalized degrees of freedom of the K user symmetric Gaussian interference channel where all desired links have the same signal-to-noise ratio (SNR) and all undesired links carrying interference have the same interference-to-noise ratio, INR = SNRα. We find that the number of generalized degrees of freedom per user, d(α), does not depend on the number of users, so that the characterization is identical to the 2 user interference channel with the exception of a singularity at α = 1 where d(1) = 1 K. The achievable schemes use multilevel coding with a nested lattice structure that opens the possibility that the sum of interfering signals can be decoded at a receiver even though the messages carried by the interfering signals are not decodable.