Two-unicast layered interference networks under the Shannon feedback and limited Shannon feedback settings are studied from a degrees of freedom (DoF) perspective. The main focus is on the archetypal two-hop multiple-input, multiple-output (MIMO) (M,N), × (M,N) × (M,N) network, denoted here as the (M,N)3network, which is a layered two-unicast network with two transmitters, two relays, and two receivers, with the first hop network between the transmitters and the relays, and the second hop network between the relays and the receivers, both being MIMO Gaussian interference channels. In particular, the first transmitter-receiver pair and the first relay have M antennas each and the second transmitter-receiver pair and the second relay having N antennas each. The (full) Shannon feedback setting for the (M,N)3network is one in which the transmitters have delayed knowledge of the first and second hop channel coefficients and the relay and destination outputs and the relays have delayed knowledge of the second hop channel coefficients and destination outputs. The DoF region under this setting is established. A key result in this paper shows that this Shannon feedback DoF region can in fact be achieved with much less side information-under what we refer to as the limited Shannon feedback setting-wherein the transmitters have no channel state or output feedback whatsoever and only the M-antenna relay (assuming M ≥ N) has delayed knowledge of the coefficients of the second-hop channel and of only the received signal of the N-antenna receiver. For this limited Shannon feedback setting, the DoFs region of the (M,N)3network is established by introducing a retro-cooperative interference alignment scheme. These DoF region results for the (M,N)3network with feedback are also extended to more general layered interference networks including the two-unicast l-hop layered networks as well as layered networks with more general numbers of antennas at the various terminals.

The degrees of freedom (DoF) region of the two-user multiple-input multiple-output (MIMO) interference channel is established under hybrid CSIT models in which there is some combination of instantaneous, delayed or no channel state information at the transmitters (CSIT) of each of the four channel matrices. In a standard version of this hybrid CSIT model, there is delayed CSIT at one transmitter and instantaneous CSIT at the other transmitter of incoming channel matrices at their respective unpaired receivers. The DoF regions for this standard model is established and, moreover, it is shown to be applicable to a large class of hybrid CSIT models. In so doing, a new DoF-optimal achievable scheme based on a combination of transmit beamforming and interference alignment is developed. Conditions are obtained on the numbers of antennas at each of the four terminals such that the DoF region under hybrid CSIT is equal to that under (a) instantaneous CSIT and (b) delayed CSIT. It is also demonstrated that by alternating between constituent hybrid CSIT states it is possible to obtain synergistic DoF benefits.

The question of whether the degrees of freedom (DoF) of multiuser networks can be enhanced even under isotropic fading and no channel state information or output feedback at the transmitters (CSIT) is investigated. Toward this end, the two-user multiple-input multiple-output (MIMO) broadcast and interference channels are studied with no side-information at the transmitters and with receivers equipped with full-duplex radios. The full-duplex feature allows for receiver cooperation because each receiver, in addition to receiving the signals sent by the transmitters, can also simultaneously transmit a signal in the same band to the other receiver. Unlike the case of MIMO networks with CSIT and full-duplex receivers, for which DoF are known, it is shown that for MIMO networks with no CSIT, full-duplex receiver cooperation is beneficial to such an extent that even the DoF region is enhanced. Indeed, for important classes of two-user MIMO broadcast and interference channels, defined by certain relationships on numbers of antennas at different terminals, the exact DoF regions are established. The key to achieving DoF-optimal performance for such networks are new retro-cooperative interference alignment schemes. Their optimality is established via the DoF analysis of certain genie-aided or enhanced version of those networks. Taken together, the results of this paper show that full-duplex receiver cooperation can recover the DoF lost due to lack of CSIT. In particular, even without CSI and/or output feedback, it has the potential of yielding most, if not all, the gains promised by delayed CSIT, or even Shannon, feedback.

The interference channel with common information (IC-CI) consists of two transmit-receive pairs that communicate over a common noisy medium. Each transmitter has an individual message for its paired receiver, and additionally, both transmitters have a common message to deliver to both receivers. In this paper, through explicit inner and outer bounds on the capacity region, we establish the capacity region of the Gaussian IC-CI to within a bounded gap of one bit, independently of the values of all channel parameters. Using this constant-gap characterization, the generalized degrees of freedom (GDoF) region is determined. It is shown that the introduction of the common message leads to an increase in the GDoF over that achievable over the Gaussian interference channel without a common message, and hence, to an unbounded improvement in the achievable rate. A surprising feature of the capacity-within-one-bit result is that most of the available benefit (i.e., to within one bit of capacity) due to the common message is achieved through a simple and explicit coding scheme that involves independent signaling at the two transmitters so that, in effect, this scheme forgoes the opportunity for transmitter cooperation that is inherently available due to shared knowledge of the common message at both transmitters.

The layered two-hop, two-unicast multi-input, multi-output (MIMO) interference network consists of two transmitters, two relays, and two receivers with the first and the second hop networks between transmitters and relays, and between relays and receivers, respectively, both being Gaussian MIMO interference channels. The degrees of freedom (DoF) region is established in the general case in which there are an arbitrary numbers of antennas at each of the six terminals. It is shown that the DoF region coincides with the min-cut outer bound for both time-varying or fixed-channel coefficients. The min-cut bound is shown to be achievable via a concatenated communication scheme that consists of linear vector space joint beamforming (that includes zero forcing and signal alignment) at all terminals as the inner precoding scheme and point-to-point coding and aligned interference neutralization as the outer precoding scheme.

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 and output feedback, wherein channel matrices and outputs, or just the channel outputs, respectively, are available to the transmitters with a delay. While for most numbers of antennas at the four terminals, the degree-of-freedom (DoF) regions with Shannon feedback are equal to that with just delayed channel state information (CSIT), it is shown that for the rest of the MIMO ICs, the DoF region with Shannon feedback is strictly larger than the DoF region with just delayed CSIT. To realize these DoF gains with Shannon feedback, a new interference alignment scheme is obtained 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 output-only feedback is also obtained for all but a class of MIMO ICs that satisfy one of two inequalities involving the numbers of antennas. Moreover, the DoF region for Shannon feedback is shown to be applicable to two limited Shannon feedback settings where the transmitters have knowledge only of certain channel matrices and certain outputs with delay.

The degrees of freedom (DoF) region of the two-user multiple-input multiple-output (MIMO) interference channel (IC) is studied under the assumptions of fast fading and delayed channel state information (CSI) at the transmitters (CSIT). Under our fast fading assumption, the channel matrices vary independently across time, so that delayed CSIT is equivalent to outdated CSIT. In particular, the DoF region under delayed CSIT is established for the general MIMO IC with an arbitrary numbers of antennas at each of the four terminals. Toward this end, a set of outer bounds to the DoF region of the general MIMO IC is derived. These bounds are then shown to be tight by developing DoF-region-optimal interference alignment schemes. A comparison of the DoF region of the MIMO IC under the delayed CSIT assumption with those under the two extremes of instantaneous CSIT and no CSIT assumptions is made. This comparison reveals that there are nonempty classes of MIMO ICs, defined by certain relationships between the numbers of antennas at the four terminals, that correspond to each of the following four scenarios: the no CSIT DoF region is strictly contained by, or is equal to, the delayed CSIT DoF region, which in turn is strictly contained by, or is equal to, the instantaneous CSIT DoF region. It is notable that within the class of MIMO ICs for which the delayed CSIT DoF region is strictly larger than the no CSIT DoF region, there is a subclass for which the interference alignment scheme which uses just delayed CSIT achieves the entire DoF region previously known to be achievable only with instantaneous CSIT.

The degree-of-freedom (DoF) regions are characterized for the multiple-input multiple-output (MIMO) broadcast channel (BC), interference channels (ICs), including X and multihop ICs, and the cognitive radio channel (CRC), when there is no channel state information at the transmitter(s) (CSIT) and for fading distributions in which transmit directions are statistically indistinguishable. For the K-user MIMO BC, the exact DoF region is obtained, which shows that time division is DoF-region optimal. For the two-user MIMO IC and CRC, inner and outer bounds are obtained that coincide for a vast majority of the relative numbers of antennas at the four terminals. Finally, the DoF of the K-user MIMO IC, the CRC, and X networks are obtained for certain classes of these networks. The results herein are derived for fading distributions and additive noises that are more general than those considered in other simultaneous related works. The DoF with and without CSIT are compared and conditions under which a lack of CSIT does, or does not, result in the loss of DoF are identified, thereby 1) providing robust no-CSIT schemes that have the same DoF as their previously found CSIT counterparts and 2) identifying situations where CSI feedback to transmitters would provide gains that are significant enough that even the DoF could be improved.

The two-user multiple-input multiple-output (MIMO) interference and cognitive radio channels are considered, where theith transmitter and theith receiver haveMiandNiantennas, respectively. In particular, the degrees of freedom (DoF) regions of these channels are studied under the assumption of having no channel state information at the transmitters. Making certain assumptions about the distributions of channel matrices of increasing generality respectively, Huang, Zhu and Guo, and the authors of this paper characterized the DoF region of the MIMO interference channel for all values of the four-tuple (M1,M2,N1,N2), except when min(M1,N1) >;N2>;M2or min(M2,N2) >;N1>;M1. More recently, for isotropic fading distributions, Zhu and Guo have solved this latter case by providing a tight outer bound to the DoF region. Here, a simpler and more widely applicable proof of that outer bound is given based on the idea of interference localization. Using the same idea, under Rayleigh fading, the DoF region is then also established for the MIMO cognitive radio channel (when the second transmitter is cognitive) with min(M1+M2,N1) >;N2>;M2-the only class for which the inner and outer bounds previously reported by the authors were not tight-thereby completing the DoF region characterization of this channel under Rayleigh fading for all values of (M1,M2,N1,N2).

The generalized degrees of freedom region (GDoF) of the multiple-input multiple-output (MIMO) interference channel (IC) is studied under the “no CSIT” assumption under which there is perfect channel state information (CSI) at the receivers and no CSI at the transmitters (CSIT). In the very weak interference regime, where the ratio of channel gains (in dB) of the interfering and direct links, α, is ≤ 0.5, the GDoF regions are characterized for the two classes of the MIMO ICs defined by (a) M1= N1>; M2≥ N2and (b) M1= N1>; N2>; M2(where Miis the number of antennas at transmitter i and Niis the number of antennas at receiver i, i ∈ {1, 2}). In particular, inner-bounds are obtained by developing CSI-independent coding schemes using which it is shown that for each of the two classes a significant portion of the perfect-CSIT GDoF region can be achieved even without CSIT. Furthermore, tight outer-bounds to the no-CSIT GDoF regions are obtained that simultaneously account for the interference encountered by both the receivers. These bounds are thus fundamentally different from those derived in earlier works which deal with the case of α = 1, i.e., the degrees of freedom (DoF) regions. Interestingly, it is found that the loss of DoFs due to lack of CSIT is much less pronounced for the α ≤ 1 over 2 than it is for α = 1.

The degrees of freedom (DoF) region of the K-user MIMO (multiple-input multiple-output) Gaussian broadcast channel (BC) is studied under i.i.d. fading when there is delayed channel state information at the transmitter (CSIT). The general case of the MIMO BC is considered where each terminal has an arbitrary number of antennas. The delayed CSIT assumption is that the transmitter has perfect knowledge of `stale' channel states (i.e., with some delay) but no knowledge of current CSI. An outer-bound to the DoF region is derived. This bound is shown to be tight in the 2-user case via an interference alignment scheme that optimally accounts for multiple and possibly distinct number of antennas at the two receivers.

The two-user multi-input, multi-output (MIMO) interference and cognitive radio channels are studied under the assumption of no channel state information at the transmitter (CSIT) from the degrees of freedom (DoF) region perspective. With Miand Nidenoting the number of antennas at transmitter i and receiver i respectively, the DoF region of the MIMO interference channel was recently characterized by Huang et al., Zhu and Guo, and by the authors of this paper for all values of the 4-tuple (M1,M2,N1,N2), except when min(M1,N1) >; N2>; M2(or min(M2,N2) >; N1>; M1). This latter case was solved more recently by Zhu and Guo, who provided a tight outer-bound. Here, a simpler and more widely applicable proof of that outer-bound is given based on the idea of interference localization. Using it, the DoF region is also established for the class of MIMO cognitive radio channels (under certain restrictions on the fading distributions) when min(M1+ M2,N1) >; N2>; M2(with the second transmitter cognitive) - the only class for which the inner and outer bounds previously obtained by the authors were not tight.

The interference channel with common information (IC-CI) is studied. In an IC-CI, each transmitter has an individual message for its paired receiver, and additionally, both transmitters collaboratively must deliver a common message to both the receivers. For the symmetric Gaussian IC-CI, the capacity region is characterized to within a gap of 3 bits independently of the values of the channel parameters. Further, using this constant-gap characterization, the generalized degrees of freedom region is also determined. Insight is provided on how the common message helps in markedly improving the achievable rates over that in an IC without a common message.

The degrees of freedom (DoF) region of the 2-user MIMO interference channel (IC) is studied under fast fading and the assumption of delayed channel state information at transmitters (CSIT) wherein the receivers know all channel matrices perfectly and the transmitters acquire this knowledge with a certain delay. An outer bound to the DoF region is first derived. This bound is then shown to be tight for all possible values of the number of antennas at the four terminals. This is done by developing interference alignment based achievability schemes. A comparison of the DoF region under the delayed CSIT assumption is made with that of the idealistic perfect CSIT assumption where perfect and global channel knowledge (without delay) is available at the transmitters (and receivers) on the one hand and with the DoF region of the conservative no CSI assumption on the other, where CSI is available at the receivers but not at all at the transmitters.

The cognitive interference channel (C-IC) is defined as the interference channel that consists of two transmitter-receiver pairs and in which any one or more of the four terminals is cognitive. The degrees of freedom (dof) region of the C-IC is studied for the case in which there is perfect and no channel state information at the receivers and the transmitters, respectively. The proposed inner and outer-bounds yield the precise characterization of dof region, except for a few cases of cognition in which for certain values of the number of antennas at the four terminals, the bounds are not tight. Finally, an example on the feasibility of interference alignment without transmitter knowledge of the channel realizations is shown.

The operation of the multiple input multiple output (MIMO) Gaussian broadcast channel (BC) with K transmit antennas and K over r users with r receive antennas each is analyzed when channel state information (CSI) is obtained at the transmitter from the ithuser via finite-rate feedback link of capacity Bibits. It is proved that if Biis scaled at the rate of αir(K - r) log2P, where P is the transmit power and α ∈ [0, 1], then using dirty paper coding (DPC), the multiplexing gain of αir can be achieved for user i. This result on the feedback scaling rate rests on an elegant achievability scheme, the construction of which brings to light several new insights on the manner in which DPC achieves the complete multiplexing gain in the perfect-CSIT case.

A dirty paper coding (DPC) based transmission scheme for the Gaussian multiple-input multiple-output (MIMO) cognitive radio channel (CRC) is studied when there is imperfect and perfect channel knowledge at the transmitters (CSIT) and the receivers, respectively. In particular, the problem of optimizing the sum-rate of the MIMO CRC over the transmit covariance matrices is dealt with. Such an optimization, under the DPC-based transmission strategy, needs to be performed jointly with an optimization over the inflation factor. To this end, first the problem of determination of inflation factor over the MIMO channel Y = H1X + H2S + Z with imperfect CSIT is investigated. For this problem, two iterative algorithms, which generalize the corresponding algorithms proposed for the channel Y = H(X + S) + Z, are developed. Later, the necessary conditions for maximizing the sum-rate of the MIMO CRC over the transmit covariances for a given choice of inflation factor are derived. Using these necessary conditions and the algorithms for the determination of the inflation factor, an iterative, numerical algorithm for the joint optimization is proposed. Some interesting observations are made from the numerical results obtained from the algorithm. Furthermore, the high-SNR sum-rate scaling factor achievable over the CRC with imperfect CSIT is obtained.