The Capacity Region of the MIMO Interference Channel and its Reciprocity to Within a Constant Gap

The capacity region of the 2-user multi-input multi-output (MIMO) Gaussian interference channel (IC) is characterized to within a constant gap that is independent of the channel matrices for the general case of the MIMO IC with an arbitrary number of antennas at each node. An achievable rate region and an outer bound to the capacity region of a class of interference channels were obtained in previous work by Telatar and Tse as unions over all possible input distributions. In contrast to that previous work on the MIMO IC, a simple and an explicit achievable coding scheme are obtained here and shown to have the constant-gap-to-capacity property and in which the sub-rates of the common and private messages of each user are explicitly specified for each achievable rate pair. The constant-gap-to-capacity results are thus proved in this work by first establishing explicit upper and lower bounds to the capacity region. A reciprocity result is also proved which is that the capacity of the reciprocal MIMO IC is within a constant gap of the capacity region of the forward MIMO IC.Comment: 22 pages, 5 figures, accepted in Trans. on Inform. T

The Diversity Multiplexing Tradeoff of the MIMO Half-Duplex Relay Channel

The fundamental diversity-multiplexing tradeoff of the three-node, multi-input, multi-output (MIMO), quasi-static, Rayleigh faded, half-duplex relay channel is characterized for an arbitrary number of antennas at each node and in which opportunistic scheduling (or dynamic operation) of the relay is allowed, i.e., the relay can switch between receive and transmit modes at a channel dependent time. In this most general case, the diversity-multiplexing tradeoff is characterized as a solution to a simple, two-variable optimization problem. This problem is then solved in closed form for special classes of channels defined by certain restrictions on the numbers of antennas at the three nodes. The key mathematical tool developed here that enables the explicit characterization of the diversity-multiplexing tradeoff is the joint eigenvalue distribution of three mutually correlated random Wishart matrices. Previously, without actually characterizing the diversity-multiplexing tradeoff, the optimality in this tradeoff metric of the dynamic compress-and-forward (DCF) protocol based on the classical compress-and-forward scheme of Cover and El Gamal was shown by Yuksel and Erkip. However, this scheme requires global channel state information (CSI) at the relay. In this work, the so-called quantize-map and forward (QMF) coding scheme due to Avestimehr {\em et} {\em al} is adopted as the achievability scheme with the added benefit that it achieves optimal tradeoff with only the knowledge of the (channel dependent) switching time at the relay node. Moreover, in special classes of the MIMO half-duplex relay channel, the optimal tradeoff is shown to be attainable even without this knowledge. Such a result was previously known only for the half-duplex relay channel with a single antenna at each node, also via the QMF scheme.Comment: 42 pages, 7 figures; submitted to the Trans. of I

Degrees of Freedom of the MIMO 2x2 Interference Network with General Message Sets

We establish the DoF region for the MIMO 2x2 interference network with a general message set, consisting of nine messages, one for each pair of a subset of transmitters at which that message is known and a subset of receivers where that message is desired. An outer bound on the general nine-message network is obtained and then it is shown to be tight, establishing the DoF region for the most general antenna setting wherein all four nodes have an arbitrary number of antennas each. The DoF-optimal scheme is applicable to the MIMO 2x2 network with constant channel coefficients, and hence, a fortiori, to time/frequency varying channel scenarios. In particular, a linear precoding scheme is proposed that can achieve all the DoF tuples in the DoF region. In it, the precise roles played by transmit zero-forcing, interference alignment, random beamforming, symbol extensions and asymmetric complex signaling are delineated. For instance, we identify a class of antenna settings in which ACS is required to achieve the fractional-valued corner points. Evidently, the DoF regions of all previously unknown cases of the 2x2 interference network with a subset of the nine-messages are established as special cases of the general result of this paper. In particular, the DoF region of the well-known four-message (and even three-message) MIMO X channel is established. This problem had remained open despite previous studies which had found inner and outer bounds that were not tight in general. Hence, the DoF regions of all special cases obtained from the general DoF region of the nine-message 2x2 interference network of this work that include at least three of the four X channel messages are new, among many others. Our work sheds light on how the same physical 2x2 network could be used by a suitable choice of message sets to take most advantage of the channel resource in a flexible and efficient manner.Comment: submitted to T-IT on Mar 4th, 201

The Diversity-Multiplexing Tradeoff of the Dynamic Decode-and-Forward Protocol on a MIMO Half-Duplex Relay Channel

The diversity-multiplexing tradeoff of the dynamic decode-and-forward protocol is characterized for the half-duplex three-terminal (m,k,n)-relay channel where the source, relay and the destination terminals have m, k and n antennas, respectively. It is obtained as a solution to a simple, two-variable, convex optimization problem and this problem is solved in closed form for special classes of relay channels, namely, the (1,k,1) relay channel, the (n,1,n) relay channel and the (2,k,2) relay channel. Moreover, the tradeoff curves for a certain class of relay channels, such as the (m,k,n>k) channels, are identical to those for the decode-and-forward protocol for the full duplex channel while for other classes of channels they are marginally lower at high multiplexing gains. Our results also show that for some classes of relay channels and at low multiplexing gains the diversity orders of the dynamic decode-and-forward protocol protocol are greater than those of the static compress-and-forward protocol which in turn is known to be tradeoff optimal over all {\em static} half duplex protocols. In general, the dynamic decode-and-forward protocol has a performance that is comparable to that of the static compress-and-forward protocol which, unlike the dynamic decode-and-forward protocol, requires global channel state information at the relay node. Its performance is also close to that of the decode-and-forward protocol over the full-duplex relay channel thereby indicating that the half-duplex constraint can be compensated for by the dynamic operation of the relay wherein the relay switches from the receive to the transmit mode based on the source-relay channel quality.Comment: 37 pages and 7 figure

Capacity Results for the K-User Broadcast Channel with Two Nested Multicast Messages

The K-user discrete memoryless (DM) broadcast channel (BC) with two nested multicast messages is studied in which one common message is to be multicast to all receivers and the second private message to a subset of receivers. The receivers that must decode both messages are referred to as private receivers and the others that must decode only the common message as common receivers. For two nested multicast messages, we establish the capacity region for several classes of DM BCs characterized by the respective associated sets of pair-wise relationships between and among the common and private receivers, each described by the well-known more capable or less noisy conditions. For three classes of DM BCs, the capacity region is simply achieved by superposition coding and the proofs of the converses rely on a recently found information inequality. The achievable rate region is then enhanced through the addition of a splitting of the private message into as many parts as there are common receivers and indirect decoding. A closed-form two-dimensional polyhedral description is obtained for it for a given coding distribution. Through a converse result that relies on the well-known Csiszar sum lemma and the information inequality, a specialization of this region that involves splitting the private message into just two sub-messages is proved to be the capacity region for several classes of DM BCs, beyond those for which superposition coding alone is capacity optimal, thereby underscoring the benefit of rate-splitting. All previously known capacity results for DM BCs with two nested multicast messages for the two and three-receiver DM BCs as well as DM BCs with one private or one common receiver are included in the general framework presented in this work.Comment: 24 pages, 6 figures. This manuscript is under review for possible publication in the IEEE Transactions on Information Theory. It was presented in part at 2017 IEEE International Symposium on Information Theory, Aachen, German

The Generalized Degrees of Freedom Region of the MIMO Interference Channel

The generalized degrees of freedom (GDoF) region of the MIMO Gaussian interference channel (IC) is obtained for the general case of an arbitrary number of antennas at each node and where the signal-to-noise ratios (SNR) and interference-to-noise ratios (INR) vary with arbitrary exponents to a nominal SNR. The GDoF region reveals various insights through the joint dependence of optimal interference management techniques (at high SNR) on the SNR exponents that determine the relative strengths of direct-link SNRs and cross-link INRs and the numbers of antennas at the four terminals. For instance, it permits an in-depth look at the issue of rate-splitting and partial decoding and it reveals that, unlike in the scalar IC, treating interference as noise is not always GDoF-optimal even in the very weak interference regime. Moreover, while the DoF-optimal strategy that relies just on transmit/receive zero-forcing beamforming and time-sharing is not GDoF optimal (and thus has an unbounded gap to capacity), the precise characterization of the very strong interference regime -- where single-user DoF performance can be achieved simultaneously for both users-- depends on the relative numbers of antennas at the four terminals and thus deviates from what it is in the SISO case. For asymmetric numbers of antennas at the four nodes the shape of the symmetric GDoF curve can be a "distorted W" curve to the extent that for certain MIMO ICs it is a "V" curve.Comment: 38 pages, 14 figures. Submitted to Trans. of I

A Unified Theory of Multiple-Access and Interference Channels via Approximate Capacity Regions for the MAC-IC-MAC

Approximate capacity regions are established for a class of interfering multiple access channels consisting of two multiple-access channels (MACs), each with an arbitrary number of transmitters, with one transmitter in each MAC causing interference to the receiver of the other MAC, a channel we refer to henceforth as the MAC-IC-MAC. For the discrete memoryless (DM) MAC-IC-MAC, two inner bounds are obtained that are generalizations of prior inner bounds for the two-user DM interference channel (IC) due to Chong {\em et al}. For the semi-deterministic MAC-IC-MAC, it is shown that single-user coding at the non-interfering transmitters and superposition coding at the interfering transmitter of each MAC achieves a rate region that is within a quantifiable gap of the capacity region, thereby extending such a result for the two-user semi-deterministic IC by Telatar and Tse. For the Gaussian MAC-IC-MAC, an approximate capacity region that is within a constant gap of the capacity region is obtained, generalizing such a result for the two-user Gaussian IC by Etkin {\em et al}. Contrary to the aforementioned approximate capacity results for the two-user IC whose achievability requires the union of all admissible input distributions, our gap results on the semi-deterministic and the Gaussian MAC-IC-MAC are achievable by only a subset and one of all admissible coding distributions, respectively. The symmetric generalized degrees of freedom (GDoF) of the symmetric Gaussian MAC-IC-MAC with more than one user per cell, which is a function of the interference strength (the ratio of INR to SNR at high SNR, both expressed in dB) and the numbers of users in each cell, is V-shaped with flat shoulders. ..

The diversity-multiplexing tradeoff of the MIMO Z interference channel

The fundamental generalized diversity-multiplexing tradeoff (GDMT) of the quasi-static fading MIMO Z interference channel (Z-IC) is established for the general Z-IC with an arbitrary number of antennas at each node under the assumptions of full channel state information at the transmitters (CSIT) and a short-term average power constraint. In the GDMT framework, the direct link signal-to-noise ratios (SNR) and cross-link interference-to-noise ratio (INR) are allowed to vary so that their ratios relative to a nominal SNR in the dB scale, i.e., the SNR/INR exponents, are fixed. It is shown that a simple Han-Kobayashi message-splitting/partial interference decoding scheme that uses only partial CSIT -- in which the second transmitter's signal depends only on its cross-link channel matrix and the first user's transmit signal doesn't need any CSIT whatsoever -- can achieve the full-CSIT GDMT of the MIMO Z-IC. The GDMT of the MIMO Z-IC under the No-CSIT assumption is also obtained for some range of multiplexing gains. The size of this range depends on the numbers of antennas at the four nodes and the SNR and INR exponents of the direct and cross links, respectively. For certain classes of channels including those in which the interfered receiver has more antennas than do the other nodes, or when the INR exponent is greater than a certain threshold, the GDMT of the MIMO Z-IC under the No-CSIT assumption is completely characterized.Comment: Submitted to the Transactions of Information Theory, 34 pages, 6 figure

The Generalized Degrees of Freedom Region of the MIMO Z-Interference Channel with Delayed CSIT

The generalized degrees of freedom (GDoF) region of the multiple-input multiple-output (MIMO) Gaussian Z-interference channel with an arbitrary number of antennas at each node is established under the assumption of delayed channel state information at transmitters (CSIT). The GDoF region is parameterized by $\alpha$, which links the interference-to-noise ratio (INR) to the signal-to-noise ratio (SNR) via $INR=SNR^{\alpha}$. A new outer bound for the GDoF region is established by maximizing a bound on the weighted sum-rate of the two users, which in turn is obtained by using a combination of genie-aided side-information and an extremal inequality. The maximum weighted sum-rate in the high SNR regime is shown to occur when the transmission covariance matrix of the interfering transmitter has full rank. An achievability scheme based on block-Markov encoding and backward decoding is developed which uses interference quantization and digital multicasting to take advantage of the channel statistics of the cross-link, and the scheme is separately shown to be GDoF-optimal in both the weak ($\alpha\leq1$) and strong ($\alpha>1$) interference regimes. This is the first complete characterization of the GDoF region of any interference network with delayed CSIT, as well as the first such GDoF characterization of a MIMO network with delayed CSIT and arbitrary number of antennas at each node. For all antenna tuples, the GDoF region is shown to be equal to or larger than the degrees of freedom (DoF) region over the entire range of $\alpha$, which leads to a V-shaped maximum sum-GDoF as a function of $\alpha$, with the minimum occurring at $\alpha=1$. The delayed CSIT GDoF region and the sum-DoF are compared with their counterparts under perfect CSIT, thereby characterizing all antenna tuples and ranges of $\alpha$ for which delayed CSIT is sufficient to achieve the perfect CSIT GDoF region or sum-DoF.Comment: submitted, IEEE Transactions on Information Theor

The Degrees of Freedom Region of the MIMO Interference Channel with Shannon Feedback

The two-user multiple-input multiple-output (MIMO) fast-fading interference channel (IC) with an arbitrary number of antennas at each of the four terminals is studied under the settings of Shannon feedback, limited Shannon feedback, and output feedback, wherein all or certain channel matrices and outputs, or just the channel outputs, respectively, are available to the transmitters with a finite delay. While for most numbers of antennas at the four terminals, it is shown that the DoF regions with Shannon feedback and for the limited Shannon feedback settings considered here are identical, and equal to the DoF region with just delayed channel state information (CSIT), it is shown that this is not always the case. For a specific class of MIMO ICs characterized by a certain relationship between the numbers of antennas at the four nodes, the DoF regions with Shannon and the limited Shannon feedback settings, while again being identical, are strictly bigger than the DoF region with just delayed CSIT. To realize these DoF gains with Shannon or limited Shannon feedback, a new retrospective interference alignment scheme is developed wherein transmitter cooperation made possible by output feedback in addition to delayed CSIT is employed to effect a more efficient form of interference alignment than is feasible with previously known schemes that use just delayed CSIT. The DoF region for just output feedback, in which each transmitter has delayed knowledge of only the receivers' outputs, is also obtained for all but a class of MIMO ICs that satisfy one of two inequalities involving the numbers of antennas.Comment: 30 pages, 3 tables, 9 figures. This paper was submitted to the IEEE Trans. Inform. Th. Oct. 2011. It was presented in part at the 49th Annual Allerton Conference on Communications, Control and Computing in Sept. 201