Viveck R. Cadambe

Interference Alignment and Degrees of Freedom of the

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/2log(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.

K

Recent results establish the optimality of interference alignment to approach the Shannon capacity of interference networks at high SNR. However, the extent to which interference can be aligned over a finite number of signalling dimensions remains unknown. Another important concern for interference alignment schemes is the requirement of global channel knowledge. In this work we provide examples of iterative algorithms that utilize the reciprocity of wireless networks to achieve interference alignment with only local channel knowledge at each node. These algorithms also provide numerical insights into the feasibility of interference alignment that are not yet available in theory.

-User Interference Channel

Recent results establish the optimality of interference alignment to approach the Shannon capacity of interference networks at high SNR. However, the extent to which interference can be aligned over a finite number of signalling dimensions remains unknown. Another important concern for interference alignment schemes is the requirement of global channel knowledge. In this work, we provide examples of iterative algorithms that utilize the reciprocity of wireless networks to achieve interference alignment with only local channel knowledge at each node. These algorithms also provide numerical insights into the feasibility of interference alignment that are not yet available in theory.

Approaching the Capacity of Wireless Networks through Distributed Interference Alignment

We show that the sum capacity of the K user frequency selective (or time-varying) interference channel is C(SNR) = (K/2) log(SNR) +o(log(SNR)) meaning that the channel has a total of K/2 degrees of freedom per orthogonal time and frequency dimension. Linear schemes of interference alignment and zero forcing suffice to achieve all the degrees of freedom and multi-user detection is not required.

A Distributed Numerical Approach to Interference Alignment and Applications to Wireless Interference Networks

We explore the degrees of freedom of M times 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 outer bound 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 outer bound 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 outer bound is exactly achievable.

Interference Alignment and Spatial Degrees of Freedom for the K User Interference Channel

It has been conjectured by Hø-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 use 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 degrees-of-freedom, i.e., the assumption of time-variations/frequency-selectivity used in prior work to establish the same result, is not needed.

Interference Alignment and the Degrees of Freedom of Wireless

We find the degrees of freedom of a network with S source nodes, R relay nodes, and D destination nodes, with random time-varying/frequency-selective channel coefficients and global channel knowledge at all nodes. We allow full-duplex operation at all nodes, as well as causal noise-free feedback of all received signals to all source and relay nodes. An outer bound to the capacity region of this network is obtained. Combining the outer bound with previous interference alignment based achievability results, we conclude that the techniques of relays, feedback, full-duplex operation and noisy cooperation do not increase the degrees of freedom of interference and X networks. As a second contribution, we show that for a network with K full-duplex nodes and K(K-1) independent messages with one message from every node to each of the other K-1 nodes, the total degrees of freedom are bounded above and below by [( K(K-1))/( (2K-2))] and [( K(K-1))/( (2K-3))], respectively.

X

An interference alignment example is constructed for the deterministic channel model of the K user interference channel. The deterministic channel example is then translated into the Gaussian setting, creating the first known example of a fully connected Gaussian K user interference network with single antenna nodes, real, non-zero and constant channel coefficients, and no propagation delays where the degrees of freedom outerbound is achieved. An analogy is drawn between the propagation delay based interference alignment examples and the deterministic channel model which also allows similar constructions for the 2 user X channel as well.

Networks

The high repair bandwidth cost of (<i>n</i>,<i>k</i>) maximum distance separable (MDS) erasure codes has motivated a new class of codes that can reduce repair bandwidth over that of conventional MDS codes. In this paper, we address (<i>n</i>,<i>k</i>,<i>d</i>) exact repair MDS codes, which allow for any single failed node to be repaired exactly with access to any arbitrary set of <i>d</i> survivor nodes. We show the existence of exact repair MDS codes that achieve minimum repair bandwidth (matching the cut-set lower bound) for arbitrary admissible (<i>n</i>,<i>k</i>,<i>d</i>), i.e., <i>k</i> ≤ <i>d</i> ≤ <i>n</i>-1. Moreover, we extend our results to show the optimality of our codes for multiple-node failure scenarios in which an arbitrary set of <i>r</i> ≤ <i>n</i>-<i>k</i> failed nodes needs to repaired. Our approach is based on asymptotic interference alignment proposed by Cadambe and Jafar. As a byproduct, we also characterize the capacity of a class of multisource nonmulticast networks.

Interference Alignment With Asymmetric Complex Signaling—Settling the Høst-Madsen–Nosratinia Conjecture

Author(s): Maleki, H; Cadambe, VR; Jafar, SA | Abstract: The index coding problem is studied from an interference alignment perspective providing new results as well as new insights into, and generalizations of, previously known results. An equivalence is established between the capacity of multiple unicast index coding (where each message is desired by exactly one receiver), and groupcast index coding (where a message can be desired by multiple receivers), which settles the heretofore open question of insufficiency of linear codes for the multiple unicast index coding problem by equivalence with groupcast settings, where this question has previously been answered. Necessary and sufficient conditions for the achievability of rate half per message in the index coding problem are shown to be a natural consequence of interference alignment constraints, and generalizations to feasibility of rate 1/(L+1)per message when each destination desires at least L messages, are similarly obtained. Finally, capacity optimal solutions are presented to a series of symmetric index coding problems inspired by the local connectivity and local interference characteristics of wireless networks. The solutions are based on vector linear coding.