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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 symmetric complex Gaussian inputs are optimal, for interference channels optimal inputs are in general asymmetric. With asymmetric complex signaling, we also show that the 2 user complex Gaussian X channel with constant channel coefficients achieves the outer bound of 4/3 degreesof-freedom, i.e., the assumption of time-variations/frequency-selectivity used in prior work to establish the same result, is not needed. * The ordering of authors is alphabetical. arXiv:0904.0274v1 [cs.IT] 1 Apr 2009 1. Signal Vector Space Alignment Schemes -Linear transmitter precoding and receiver combining operations are used to transform the interference channel into multiple non-interfering

We show that the 2 × 2 × 2 interference network, 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, for almost all values of channel coefficients, for both time-varying or fixed channel coefficients. The key to this result is a new idea, called aligned interference neutralization, that provides a way to align interference terms over each hop in a manner that allows them to be cancelled over the air at the last hop. Interference Alignment ApproachAny approach that treats either hop (or both hops) of the 2 × 2 × 2 IC as an interference channel can only achieve a maximum of 1 DoF, because of the bottleneck created by the 2 user interference channel which has only 1 DoF [35]. Interestingly, the interference channel approach is highly suboptimal at high SNR. This is because the interference channel approach precludes interference alignment.Interference alignment refers to a consolidation of undesired signals into smaller dimensions so that the signaling dimensions available for desired signals at each receiver are maximized. Interference alignment was observed first by Birk and Kol [36] for the index coding problem, and then by Maddah-Ali et. al. for the X channel in [37], followed by Weingarten et. al. for the compound vector broadcast channel in [38]. The idea was crystallized as a general concept in [2,3] by Jafar and Shamai, and Cadambe and Jafar, respectively, and has since been applied in increasingly sophisticated forms [13,39,40,41,42,20,5,43,44,45] across a variety of communication networks -both wired and wireless -often leading to surprising new insights.2 Unlike the interference channel approach which can achieve no more than 1 DoF, Cadambe and Jafar show in [4] that the 2 × 2 × 2 IC can achieve 4 3 DoF almost surely. This is accomplished by a decode and forward approach that treats each hop as an X channel. Specifically, each transmitter divides its message into two independent parts, one intended for each relay. This creates a total of 4 messages over the first hop, one from each source to each relay node, i.e., the 2 × 2 X channel setting. After decoding the messages from each transmitter, each relay has a message for each destination node, which places the second hop into the X channel setting as well. It is known that the 2 × 2 X channel with single antenna nodes has 4 3 DoF. The result was shown first by Jafar and Shamai in [2] under the assumption that the channel coefficients are time-varying. By using a combination of linear beamforming, symbol extensions and asymmetric complex signaling, Cadambe et. al. showed in [40] that 4 3 DoF are achievable on the 2 × 2 X channel even if the channels are held constant for almost all values of channel coefficients. Motahari et. al. [46] proposed the framework of rational dimensions which allows 4 3 DoF to be achieved almost surely even if the channels are fixed and restricted to real values. Thus, regardless of whether the channels are time-varying or constan...

We show that the 3-user M T × M R MIMO interference channel where each transmitter is equipped with M T antennas and each receiver is equipped with M R antennas has d(M, N ) = min M 2−1/κ , N 2+1/κ degrees of freedom (DoF) normalized by time, frequency, and space dimensions, whereWhile the DoF outer bound of d(M, N ) is established for every M T , M R value, the achievability of d(M, N ) DoF is established in general subject to a normalization with respect to spatial-extensions, i.e., the scaling of the number of antennas at all nodes. Specifically, we show that qd(M, N ) DoF are achievable for the 3-user qM T × qM R MIMO interference channel, for some positive integer q which may be seen as a spatial-extension factor. q is the scaling factor needed to make the value qd(M, N ) an integer. Given spatial-extensions, the achievability relies only on linear beamforming based interference alignment schemes and requires neither channel extensions nor channel variations in time or frequency. In the absence of spatial extensions, it is shown through examples how essentially the same interference alignment scheme may be applied over time-extensions over either constant or time-varying channels. The central new insight to emerge from this work is the notion of subspace alignment chains as DoF bottlenecks. The subspace alignment chains are instrumental both in identifying the extra dimensions to be provided by a genie to a receiver for the DoF outer bound, as well as in the construction of the optimal interference alignment schemes.The DoF value d(M, N ) is a piecewise linear function of M, N , with either M or N being the bottleneck within each linear segment while the other value contains some redundancy, i.e., it can be reduced without reducing the DoF. The corner points of these piecewise linear segments correspond to two sets, A = {1/2, 2/3, 3/4, · · · } and B = {1/3, 3/5, 5/7, · · · }. The set A contains all those values of M/N and only those values of M/N for which there is redundancy in both M and N , i.e., either can be reduced without reducing DoF. The set B contains all those values of M/N and only those values of M/N for which there is no redundancy in either M or N , i.e., neither can be reduced without reducing DoF. Because A and B represent settings with maximum and minimum redundancy, essentially they are the basis for the DoF outer bounds and inner bounds, respectively.Our Our results show that M/N ∈ A are the only values for which there is no DoF benefit of joint processing among co-located antennas at the transmitters or receivers. This may also be seen as a consequence of the maximum redundancy in the M/N ∈ A settings.

We propose a blind interference alignment scheme for the vector broadcast channel where the transmitter is equipped with M antennas and there are K receivers, each equipped with a reconfigurable antenna capable of switching among M preset modes. Without any knowledge of the channel coefficient values at the transmitters and with only mild assumptions on the channel coherence structure we show that M K M +K−1 degrees of freedom are achievable. The key to the blind interference alignment scheme is the ability of the receivers to switch between reconfigurable antenna modes to create short term channel fluctuation patterns that are exploited by the transmitter. The achievable scheme does not require cooperation between transmit antennas and is therefore applicable to the M × K X network as well. Only finite symbol extensions are used, and no channel knowledge at the receivers is required to null the interference.