Abstract-

We provide innerbound and outerbound for the total number of degrees of freedom of the K user multiple input multiple output (MIMO) Gaussian interference channel with M antennas at each transmitter and N antennas at each receiver if the channel coefficients are time-varying and drawn from a continuous distribution. The bounds are tight when the ratio max(M,N ) min(M,N ) = R is equal to an integer. For this case, we show that the total number of degrees of freedom is equal to min(M,Achievability is based on interference alignment. We also provide examples where using interference alignment combined with zero forcing can achieve more degrees of freedom than merely zero forcing for some MIMO interference channels with constant channel coefficients.

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.