Previous work showed that the X network with M transmitters, N receivers has MN/(M+N-1) degrees of freedom. In this work we study the degrees of freedom of the X network with secrecy constraints, i.e. the X network where some/all messages are confide
ntial. We consider the $M times N$ network where all messages are secured and show that N(M-1)/(M+N-1) degrees of freedom can be achieved. Secondly, we show that if messages from only M-1 transmitters are confidential, then MN/(M+N-1) degrees of freedom can be achieved meaning that there is no loss of degrees of freedom because of secrecy constraints. We also consider the achievable secure degrees of freedom under a more conservative secrecy constraint. We require that messages from any subset of transmitters are secure even if other transmitters are compromised, i.e., messages from the compromised transmitter are revealed to the unintended receivers. We also study the achievable secure degrees of freedom of the K user Gaussian interference channel under two different secrecy constraints where 1/2 secure degrees of freedom per message can be achieved. The achievable scheme in all cases is based on random binning combined with interference alignment.
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-n
oise ratio, ${INR}={SNR}^alpha$. We find that the number of generalized degrees of freedom per user, $d(alpha)$, 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 $alpha=1$ where $d(1)=frac{1}{K}$. 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.
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 un
known. 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.
In this paper, we explore the benefits, in the sense of total (sum rate) degrees of freedom (DOF), of cooperation and cognitive message sharing for a two-user multiple-input-multiple-output (MIMO) Gaussian interference channel with $M_1$, $M_2$ anten
nas at transmitters and $N_1$, $N_2$ antennas at receivers. For the case of cooperation (including cooperation at transmitters only, at receivers only, and at transmitters as well as receivers), the DOF is $min {M_1+M_2, N_1+N_2, max(M_1, N_2)), max(M_2, N_1)}$, which is the same as the DOF of the channel without cooperation. For the case of cognitive message sharing, the DOF is $min {M_1+M_2, N_1+N_2, (1-1_{T2})((1-1_{R2}) max(M_1, N_2) + 1_{R2} (M_1+N_2)), (1-1_{T1})((1-1_{R1}) max(M_2, N_1) + 1_{R1} (M_2+N_1)) }$ where $1_{Ti} = 1$ $(0)$ when transmitter $i$ is (is not) a cognitive transmitter and $1_{Ri}$ is defined in the same fashion. Our results show that while both techniques may increase the sum rate capacity of the MIMO interference channel, only cognitive message sharing can increase the DOF. We also find that it may be more beneficial for a user to have a cognitive transmitter than to have a cognitive receiver.
It is known that the capacity of parallel (multi-carrier) Gaussian point-to-point, multiple access and broadcast channels can be achieved by separate encoding for each subchannel (carrier) subject to a power allocation across carriers. In this paper
we show that such a separation does not apply to parallel Gaussian interference channels in general. A counter-example is provided in the form of a 3 user interference channel where separate encoding can only achieve a sum capacity of $log({SNR})+o(log({SNR}))$ per carrier while the actual capacity, achieved only by joint-encoding across carriers, is $3/2log({SNR}))+o(log({SNR}))$ per carrier. As a byproduct of our analysis, we propose a class of multiple-access-outerbounds on the capacity of the 3 user interference channel.
Recent work has characterized the sum capacity of time-varying/frequency-selective wireless interference networks and $X$ networks within $o(log({SNR}))$, i.e., with an accuracy approaching 100% at high SNR (signal to noise power ratio). In this pape
r, we seek similar capacity characterizations for wireless networks with relays, feedback, full duplex operation, and transmitter/receiver cooperation through noisy channels. First, we consider 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. The sum capacity of this network is characterized as $frac{SD}{S+D-1}log({SNR})+o(log({SNR}))$. The implication of the result is that the capacity benefits of relays, causal feedback, transmitter/receiver cooperation through physical channels and full duplex operation become a negligible fraction of the network capacity at high SNR. Some exceptions to this result are also pointed out in the paper. Second, we consider a network with $K$ full duplex nodes with an independent message from every node to every other node in the network. We find that the sum capacity of this network is bounded below by $frac{K(K-1)}{2K-2}+o(log({SNR}))$ and bounded above by $frac{K(K-1)}{2K-3}+o(log({SNR}))$.
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 con
nected Gaussian $K$ user interference network with single antenna nodes, real, non-zero and contant 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.
