Syed Ali Jafar

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

We provide an overview of the extensive results on the Shannon capacity of single-user and multiuser multiple-input multiple-output (MIMO) channels. Although enormous capacity gains have been predicted for such channels, these predictions are based on somewhat unrealistic assumptions about the underlying time-varying channel model and how well it can be tracked at the receiver, as well as at the transmitter. More realistic assumptions can dramatically impact the potential capacity gains of MIMO techniques. For time-varying MIMO channels there are multiple Shannon theoretic capacity definitions and, for each definition, different correlation models and channel information assumptions that we consider. We first provide a comprehensive summary of ergodic and capacity versus outage results for single-user MIMO channels. These results indicate that the capacity gain obtained from multiple antennas heavily depends on the available channel information at either the receiver or transmitter, the channel signal-to-noise ratio, and the correlation between the channel gains on each antenna element. We then focus attention on the capacity region of the multiple-access channels (MACs) and the largest known achievable rate region for the broadcast channel. In contrast to single-user MIMO channels, capacity results for these multiuser MIMO channels are quite difficult to obtain, even for constant channels. We summarize results for the MIMO broadcast and MAC for channels that are either constant or fading with perfect instantaneous knowledge of the antenna gains at both transmitter(s) and receiver(s). We show that the capacity region of the MIMO multiple access and the largest known achievable rate region (called the dirty-paper region) for the MIMO broadcast channel are intimately related via a duality transformation. This transformation facilitates finding the transmission strategies that achieve a point on the boundary of the MIMO MAC capacity region in terms of the transmission strategies of the MIMO broadcast dirty-paper region and vice-versa. Finally, we discuss capacity results for multicell MIMO channels with base station cooperation. The base stations then act as a spatially diverse antenna array and transmission strategies that exploit this structure exhibit significant capacity gains. This section also provides a brief discussion of system level issues associated with MIMO cellular. Open problems in this field abound and are discussed throughout the paper.

-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.

Capacity limits of MIMO channels

We provide achievability as well as converse results for the degrees of freedom region of a multiple-input multiple-output (MIMO) X channel, i.e., a system with two transmitters, two receivers, each equipped with multiple antennas, where independent messages need to be conveyed over fixed channels from each transmitter to each receiver. The inner and outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message. With M = 1 antennas at each node, we find that the total (sum rate) degrees of freedom are bounded above and below as 1 les eta*x les 4/3. If M > 1 and channel matrices are nondegenerate then the precise degrees of freedom eta*x = (4/3)M. Thus, the MIMO X channel has noninteger degrees of freedom when M is not a multiple of 3. Simple zero forcing without dirty paper encoding or successive decoding, suffices to achieve the (4/3)M degrees of freedom. If the channels vary with time/frequency then the channel with single antennas (M = 1) at all nodes has exactly 4/3 degrees of freedom. The key idea for the achievability of the degrees of freedom is interference alignment-i.e., signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired. We also explore the increase in degrees of freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio.

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.

Degrees of Freedom Region of the MIMO

We explore the feasibility of interference alignment in signal vector space-based only on beamforming-for K-user MIMO interference channels. Our main contribution is to relate the feasibility issue to the problem of determining the solvability of a multivariate polynomial system which is considered extensively in algebraic geometry. It is well known, e.g., from Bezouts theorem, that generic polynomial systems are solvable if and only if the number of equations does not exceed the number of variables. Following this intuition, we classify signal space interference alignment problems as either proper or improper based on the number of equations and variables. Rigorous connections between feasible and proper systems are made through Bernshteins theorem for the case where each transmitter uses only one beamforming vector. The multibeam case introduces dependencies among the coefficients of a polynomial system so that the system is no longer generic in the sense required by both theorems. In this case, we show that the connection between feasible and proper systems can be further strengthened (since the equivalency between feasible and proper systems does not always hold) by including standard information theoretic outer bounds in the feasibility analysis.

X

In this correspondence, we consider the problem of maximizing sum rate of a multiple-antenna Gaussian broadcast channel (BC). It was recently found that dirty-paper coding is capacity achieving for this channel. In order to achieve capacity, the optimal transmission policy (i.e., the optimal transmit covariance structure) given the channel conditions and power constraint must be found. However, obtaining the optimal transmission policy when employing dirty-paper coding is a computationally complex nonconvex problem. We use duality to transform this problem into a well-structured convex multiple-access channel (MAC) problem. We exploit the structure of this problem and derive simple and fast iterative algorithms that provide the optimum transmission policies for the MAC, which can easily be mapped to the optimal BC policies.

Channel

We solve the transmitter optimization problem and determine a necessary and sufficient condition under which beamforming achieves Shannon capacity in a linear narrowband point-to-point communication system employing multiple transmit and receive antennas with additive Gaussian noise. We assume that the receiver has perfect channel knowledge while the transmitter has only knowledge of either the mean or the covariance of the channel coefficients. The channel is modeled at the transmitter as a matrix of complex jointly Gaussian random variables with either a zero mean and a known covariance matrix (covariance information), or a nonzero mean and a white covariance matrix (mean information). For both cases, we develop a necessary and sufficient condition for when the Shannon capacity is achieved through beamforming; i.e., the channel can be treated like a scalar channel and one-dimensional codes can be used to achieve capacity. We also provide a waterpouring interpretation of our results and find that less channel uncertainty not only increases the system capacity but may also allow this higher capacity to be achieved with scalar codes which involves significantly less complexity in practice than vector coding.

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

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.

On Feasibility of Interference Alignment in MIMO Interference Networks

We provide inner bound and outer bound 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,N)K if K ≤ R and min(M,N)[(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.