arXiv subjects

Syed A. Jafar

Publications and source records attributed to Syed A. Jafar.

At least 109 records · Page 6Linked to original sources

arXiv2009

Degrees of Freedom of Multi-Source Relay Networks

We study a multi-source Gaussian relay network consisting of $K$ source--destination pairs having $K$ unicast sessions. We assume $M$ layers of relays between the sources and the destinations. We find achievable degrees of freedom of the network. Our schemes are based on interference alignment at the transmitters and symbol extension and opportunistic interference cancellation at the relays. For $K$-$L$-$K$ networks, i.e., 2-hop network with $L$ relays, we show $\min\{K,K/2+L/(2(K-1))\}$ degrees of freedom are achievable. For $K$-hop networks with $K$ relays in each layer, we show the full $K$ degrees of freedom are achievable provided that $K$ is even and the channel distribution satisfies a certain symmetry.

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arXiv2009

Exploiting Channel Correlations - Simple Interference Alignment Schemes with no CSIT

We explore 5 network communication problems where the possibility of interference alignment, and consequently the total number of degrees of freedom (DoF) with channel uncertainty at the transmitters are unknown. These problems share the common property that in each case the best known outer bounds are essentially robust to channel uncertainty and represent the outcome with interference alignment, but the best inner bounds -- in some cases conjectured to be optimal -- predict a total collapse of DoF, thus indicating the infeasibility of interference alignment under channel uncertainty at transmitters. Our main contribution is to show that even with no knowledge of channel coefficient values at the transmitters, the knowledge of the channels' correlation structure can be exploited to achieve interference alignment. In each case, we show that under a staggered block fading model, the transmitters are able to align interference without the knowledge of channel coefficient values. The alignment schemes are based on linear beamforming -- which can be seen as a repetition code over a small number of symbols -- and involve delays of only a few coherence intervals.

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arXiv2009

On the Degrees of Freedom of Finite State Compound Wireless Networks - Settling a Conjecture by Weingarten et. al

We explore the degrees of freedom (DoF) of three classes of finite state compound wireless networks in this paper. First, we study the multiple-input single-output (MISO) finite state compound broadcast channel (BC) with arbitrary number of users and antennas at the transmitter. In prior work, Weingarten et. al. have found inner and outer bounds on the DoF with 2 users. The bounds have a different character. While the inner bound collapses to unity as the number of states increases, the outer bound does not diminish with the increasing number of states beyond a threshold value. It has been conjectured that the outer bound is loose and the inner bound represents the actual DoF. In the complex setting (all signals, noise, and channel coefficients are complex variables) we solve a few cases to find that the outer bound -and not the inner bound- of Weingarten et. al. is tight. For the real setting (all signals, noise and channel coefficients are real variables) we completely characterize the DoF, once again proving that the outer bound of Weingarten et. al. is tight. We also extend the results to arbitrary number of users. Second, we characterize the DoF of finite state scalar (single antenna nodes) compound X networks with arbitrary number of users in the real setting. Third, we characterize the DoF of finite state scalar compound interference networks with arbitrary number of users in both the real and complex setting. The key finding is that scalar interference networks and (real) X networks do not lose any DoF due to channel uncertainty at the transmitter in the finite state compound setting. The finite state compound MISO BC does lose DoF relative to the perfect CSIT scenario. However, what is lost is only the DoF benefit of joint processing at transmit antennas, without which the MISO BC reduces to an X network.

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arXiv2009

On Degrees of Freedom Region of MIMO Networks without CSIT

In this paper, we study the effect of the absence of channel knowledge for multiple-input-multiple-output (MIMO) networks. Specifically, we assume perfect channel state information at the receivers, no channel state information at the transmitter(s), and independent identically distributed (i.i.d.) Rayleigh fading across antennas, users and time slots. We provide the characterization of the degrees of freedom (DoF) region for a 2-user MIMO broadcast channel. We then provide a DoF region outer bound for a 2-user MIMO interference channel. This bound is shown to be tight for all possible combinations of the number of antennas at each node except for one case. As a byproduct of this analysis we point out the potential of interference alignment in the 2-user MIMO interference channel with no CSIT.

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arXiv2009

On the Beamforming Design for Efficient Interference Alignment

An efficient interference alignment (IA) scheme is developed for $K$-user single-input single-output frequency selective fading interference channels. The main idea is to steer the transmit beamforming matrices such that at each receiver the subspace dimensions occupied by interference-free desired streams are asymptotically the same as those occupied by all interferences. Our proposed scheme achieves a higher multiplexing gain at any given number of channel realizations in comparison with the original IA scheme, which is known to achieve the optimal multiplexing gain asymptotically.

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arXiv2009

Capacity of a Class of Symmetric SIMO Gaussian Interference Channels within O(1)

The N+1 user, 1 x N single input multiple output (SIMO) Gaussian interference channel where each transmitter has a single antenna and each receiver has N antennas is studied. The symmetric capacity within O(1) is characterized for the symmetric case where all direct links have the same signal-to-noise ratio (SNR) and all undesired links have the same interference-to-noise ratio (INR). The gap to the exact capacity is a constant which is independent of SNR and INR. To get this result, we first generalize the deterministic interference channel introduced by El Gamal and Costa to model interference channels with multiple antennas. We derive the capacity region of this deterministic interference channel. Based on the insights provided by the deterministic channel, we characterize the generalized degrees of freedom (GDOF) of Gaussian case, which directly leads to the O(1) capacity approximation. On the achievability side, an interesting conclusion is that the generalized degrees of freedom (GDOF) regime where treating interference as noise is found to be optimal in the 2 user interference channel, does not appear in the N+1 user, 1 x N SIMO case. On the converse side, new multi-user outer bounds emerge out of this work that do not follow directly from the 2 user case. In addition to the GDOF region, the outer bounds identify a strong interference regime where the capacity region is established.

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arXiv2009

Feasibility Conditions for Interference Alignment

The degrees of freedom of MIMO interference networks with constant channel coefficients are not known in general. Determining the feasibility of a linear interference alignment solution is a key step toward solving this open problem. Our approach in this paper is to view the alignment problem as a system of bilinear equations and determine its solvability by comparing the number of equations and the number of variables. To this end, we divide interference alignment problems into two classes - proper and improper. An interference alignment problem is called proper if the number of equations does not exceed the number of variables. Otherwise, it is called improper. Examples are presented to support the intuition that for generic channel matrices, proper systems are almost surely feasible and improper systems are almost surely infeasible.

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arXiv2009

Interference Alignment with Asymmetric Complex Signaling - Settling the Host-Madsen-Nosratinia Conjecture

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

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arXiv2008

On the Capacity and Generalized Degrees of Freedom of the X Channel

We explore the capacity and generalized degrees of freedom of the two-user Gaussian X channel, i.e. a generalization of the 2 user interference channel where there is an independent message from each transmitter to each receiver. There are three main results in this paper. First, we characterize the sum capacity of the deterministic X channel model under a symmetric setting. Second, we characterize the generalized degrees of freedom of the Gaussian X channel under a similar symmetric model. Third, we extend the noisy interference capacity characterization previously obtained for the interference channel to the X channel. Specifically, we show that the X channel associated with noisy (very weak) interference channel has the same sum capacity as the noisy interference channel.

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arXiv2008

On the Secure Degrees of Freedom of Wireless X Networks

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 confidential. 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.

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arXiv2008

A Layered Lattice Coding Scheme for a Class of Three User Gaussian Interference Channels

The paper studies a class of three user Gaussian interference channels. A new layered lattice coding scheme is introduced as a transmission strategy. The use of lattice codes allows for an "alignment" of the interference observed at each receiver. The layered lattice coding is shown to achieve more than one degree of freedom for a class of interference channels and also achieves rates which are better than the rates obtained using the Han-Kobayashi coding scheme.

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arXiv2008

Degrees of Freedom of the $K$ User $M \times N$ MIMO Interference Channel

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 $\frac{\max(M,N)}{\min(M,N)}=R$ is equal to an integer. For this case, we show that the total number of degrees of freedom is equal to $\min(M,N)K$ if $K \leq R$ and $\min(M,N)\frac{R}{R+1}K$ if $K > R$. 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.

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arXiv2008

Capacity of Symmetric K-User Gaussian Very Strong Interference Channels

This paper studies a symmetric K user Gaussian interference channel with K transmitters and K receivers. A "very strong" interference regime is derived for this channel setup. A "very strong" interference regime is one where the capacity region of the interference channel is the same as the capacity region of the channel with no interference. In this regime, the interference can be perfectly canceled by all the receivers without incurring any rate penalties. A "very strong" interference condition for an example symmetric K user deterministic interference channel is also presented.

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arXiv2008

Generalized Degrees of Freedom of the Symmetric Gaussian $K$ User Interference Channel

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, ${INR}={SNR}^α$. 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 $d(1)=\frac{1}{K}$. 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.

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arXiv2008

Approaching the Capacity of Wireless Networks through Distributed Interference Alignment

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.

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arXiv2008

Degrees of Freedom of the MIMO Interference Channel with Cooperation and Cognition

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 $M_1$, $M_2$ antennas 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.

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arXiv2008

Multiple Access Outerbounds and the Inseparability of Parallel Interference Channels

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 $3/2\log({SNR}))+o(\log({SNR}))$ 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.

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arXiv2008

Capacity of Wireless Networks within o(log(SNR)) - the Impact of Relays, Feedback, Cooperation and Full-Duplex Operation

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 paper, 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 $\frac{K(K-1)}{2K-3}+o(\log({SNR}))$.

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