Degrees Of Freedom Region Research Articles

MISO Networks With Imperfect CSIT: A Topological Rate-Splitting Approach

Recently, the Degrees-of-Freedom (DoF) region of multiple-input-single-output (MISO) networks with imperfect channel state information at the transmitter (CSIT) has attracted significant attentions. An achievable scheme is known as rate-splitting (RS) that integrates common-message-multicasting and private-message-unicasting. In this paper, focusing on the general $K$-cell MISO IC where the CSIT of each interference link has an arbitrary quality of imperfectness, we firstly identify the DoF region achieved by RS. Secondly, we introduce a novel scheme, so called Topological RS (TRS), whose novelties compared to RS lie in a multi-layer structure and transmitting multiple common messages to be decoded by groups of users rather than all users. The design of TRS is motivated by a novel interpretation of the $K$-cell IC with imperfect CSIT as a weighted-sum of a series of partially connected networks. We show that the DoF region achieved by TRS covers that achieved by RS. Also, we find the maximal sum DoF achieved by TRS via hypergraph fractional packing, which yields the best sum DoF so far. Lastly, for a realistic scenario where each user is connected to three dominant transmitters, we identify the sufficient condition where TRS strictly outperforms conventional schemes.

Optimal DoF Region for the Asymmetric Two-Pair MIMO Two-Way Relay Channel

In this paper, we study the optimal degrees of freedom (DoF) region for the two-pair MIMO two-way relay channel (TWRC) with asymmetric antenna setting, where two pairs of users exchange information with the help of a common relay. Each user $i$ is equipped with $M_i$ antennas, for $i=1,2,3,4$, and the relay is equipped with $N$ antennas. First, we derive an outer bound of the DoF region by using the cut-set theorem and the genie-message approach. Then, we propose a new transmission scheme to achieve the outer bound of the DoF region. Due to the asymmetric data exchange, where the two users in each pair can communicate a different number of data streams, we not only need to form the network-coded symbols but also need to process the additional asymmetric data streams at the relay. This is realized through the joint design of relay compression matrix and source precoding matrices. After obtaining the optimal DoF region, we study the optimal sum DoF by solving a linear programming problem. From the optimal DoF region of this channel, we show that in the asymmetric antenna setting, some antennas at certain source nodes are redundant and cannot contribute to enlarge the DoF region. We also show that there is no loss of optimality in terms of the sum DoF by enforcing symmetric data exchange, where the two users in each pair are restricted to communicate the same number of data streams.

Degrees of Freedom for the MIMO Multi-Way Relay Channel With Common and Private Messages

In this paper, we study the general multiple input multiple output (MIMO) multi-way relay channel, i.e., MIMO Y channel, with common and private messages. In this channel, $K$ users exchange messages through a common relay. Each user transmits a private message to each user in addition to a common message to all the other users. The $i$ th user and the relay are equipped with $M_{i}$ and $N$ antennas, respectively. First, we derive the degrees of freedom (DoF) region of the symmetric three-user MIMO Y channel, where $M_{i} = M$ and $i \in \{1,2,3\}$ . Due to the symmetry of the network, we focus on the case where the DoF of all private messages are equal and the DoF of all common messages are equal. In this case, the DoF region has two dimensions: the DoF of private messages and the DoF of common messages. We develop an outer bound on the DoF region based by using cut-set and one-sided genie bounds. We prove the achievability of the outer bound on the DoF region by using linear beamforming and signal space alignment (SSA) schemes. Second, based on our study of the DoF region of the symmetric channel, we define a weighted sum DoF metric that integrates all the network messages and weights the common messages by a factor of $\alpha$ . We study the weighted sum DoF maximization problem and show that sending common messages only is optimal when $\alpha$ exceeds $\frac {4}{3}$ . Next, we focus on the weighted sum DoF with $\alpha = 2$ that represents the total number of received interference-free streams at the users. First, we show that the weighted sum DoF, with $\alpha = 2$ , of the MIMO Y channel with an arbitrary number of antennas is given by $\min \{3N,2N+M_{3},N+M_{2}+M_{3},2M_{2}+2M_{3},M_{1}+M_{2}+M_{3}\}$ . Second, we study the weighted sum DoF, with $\alpha = 2$ , of the $K$ –user case in the symmetric setting. We derive an outer bound on the weighted sum DoF using cut-set bounds, and show that the network has $K\min \{N,M\}$ weighted DoF. The achievability results are obtained by using SSA for network coding in the multiple access phase, and zero-forcing precoding in the broadcast phase.

Degrees of Freedom of the MIMO $2 \times 2$ Interference Network with General Message Sets

We establish the degrees of freedom (DoF) region for the multiple-input multiple-output (MIMO) two-transmitter, and two-receiver ( $2 \times 2$ ) 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 $2 \times 2$ interference network is obtained and it is shown to be tight, thereby establishing the DoF region for the most general antenna setting wherein the four nodes have an arbitrary number of antennas each. The DoF-optimal scheme is applicable to the MIMO $2 \times 2$ interference 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 (ACS) 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 $2 \times 2$ interference network with a subset of the nine-messages are newly established as special cases of the general result of this paper. For instance, the DoF region of the well-known four-message (and even three-message) MIMO $X$ channel is newly 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 $2\times 2$ interference network of this paper that include at least three of the four $X$ channel messages are new, among many others. This paper sheds light on how the same physical $2 \times 2$ interference 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.

Degrees of Freedom Region of the MIMO Z-Interference Channel With Mixed CSIT

The degrees of freedom (DoF) region is established for the two-user multiple-input multiple-output Z-interference channel (IC) under the assumption of mixed channel state information at transmitters (CSIT), in which each transmitter obtains accurate delayed CSIT as well as an imperfect estimate of the current channel state, with the quality of the estimate determined by a parameter, $\alpha $ . A block-Markov achievability scheme is designed that uses transmit beamforming, interference quantization, and multicasting along with backward decoding, and is shown to be DoF optimal. When applied to a two-user IC with mixed CSIT and one cross-link statistically weaker than the rest, this scheme can not only improve over the DoF region otherwise achievable by using the DoF-optimal scheme for the IC (which treats all links as comparable in strength) but also dispenses with the requirement of obtaining delayed feedback or current channel estimate for the weaker link.

MISO Broadcast Channel With Hybrid CSIT: Beyond Two Users

We study the impact of heterogeneity of channel-state-information available at the transmitters (CSIT) on the capacity of broadcast channels with a multiple-antenna transmitter and $k$ single-antenna receivers (MISO BC). In particular, we consider the $k$-user MISO BC, where the CSIT with respect to each receiver can be either instantaneous/perfect, delayed, or not available; and we study the impact of this heterogeneity of CSIT on the degrees-of-freedom (DoF) of such network. We first focus on the $3$-user MISO BC; and we completely characterize the DoF region for all possible heterogeneous CSIT configurations, assuming linear encoding strategies at the transmitters. The result shows that the state-of-the-art achievable schemes in the literature are indeed sum-DoF optimal, when restricted to linear encoding schemes. To prove the result, we develop a novel bound, called Interference Decomposition Bound, which provides a lower bound on the interference dimension at a receiver which supplies delayed CSIT based on the average dimension of constituents of that interference, thereby decomposing the interference into its individual components. Furthermore, we extend our outer bound on the DoF region to the general $k$-user MISO BC, and demonstrate that it leads to an approximate characterization of linear sum-DoF to within an additive gap of $0.5$ for a broad range of CSIT configurations. Moreover, for the special case where only one receiver supplies delayed CSIT, we completely characterize the linear sum-DoF.

Optimal Degrees of Freedom Region for the Asymmetric MIMO Y Channel

This letter studies the optimal degrees of freedom (DoF) region for the asymmetric three-user MIMO Y channel with antenna configuration $(M_1,M_2,M_3,N)$, where $M_i$ is the number of antennas at user $i$ and $N$ is the number of antennas at the relay node. The converse is proved by using the cut-set theorem and the genie-message approach. To prove the achievability, we divide the DoF tuples in the outer bound into two cases. For each case, we show that the DoF tuples are achievable by collectively utilizing antenna deactivation, pairwise signal alignment and cyclic signal alignment techniques. This work not only offers a complete characterization of DoF region for the considered channel model, but also provides a new and elegant achievability proof.

Degrees of Freedom Region of Wireless X Networks Based on Real Interference Alignment

We first consider a single hop wireless X network with $K$ transmitters and $J$ receivers, all with a single antenna. Each transmitter conveys an independent message for each receiver. The channel is assumed to have constant coefficients. We develop an interference alignment scheme for this setup and derive an achievable degrees of freedom (DoF) region. We show that in some cases, the derived region meets a previous outer bound, and hence, is the (exact) DoF region. For our achievability schemes, we divide each message into streams and use real interference alignment on the streams. Several previous results on the DoF region and total DoF for several special cases can be recovered from our result. Next, we consider a network model, such that each transmitter emits an arbitrary number of messages and each receiver can request an arbitrary subset of all emitted messages. We term this network the X network with multicast. We derive an achievability result for the DoF region of such networks, as well as an outer bound. Our achievability scheme is based on a novel extension of real interference alignment that enables us to achieve more points of the DoF region. This approach allows multiple messages from a single user to be aligned with other users’ messages. Finally, we discuss some points on the outer bounds that are not achievable with any of the alignment schemes that we present.

DoF Analysis of the MIMO Broadcast Channel With Alternating/Hybrid CSIT

We consider a K-user multiple-input singleoutput (MISO) broadcast channel (BC) where the channel state information (CSI) of user i(i = 1,2, .. ., K) may be instantaneously perfect (P), delayed (D), or not known (N) at the transmitter with probabilities λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sup> , λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sup> , and λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sup> , respectively. In this setting, according to the three possible CSI at the transmitter (CSIT) for each user, knowledge of the joint CSIT of the K users could have at most 3K states. In this paper, given the marginal probabilities of CSIT (i.e., λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sup> , λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sup> , and λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sup> ), we derive an outer bound for the degrees of freedom (DoF) region of the K-user MISO BC. Subsequently, we tighten this outer bound by considering a set of inequalities that capture some of the 3K states of the joint CSIT. One of the consequences of this set of inequalities is that for K ≥ 3, it is shown that the DoF region is not completely characterized by the marginal probabilities in contrast to the two-user case. Afterwards, the tightness of these bounds is investigated through the discussion on the achievability. Finally, a two user multiple-input multipleoutput BC having CSIT among P and N is considered in which an outer bound for the DoF region is provided, and it is shown that in some scenarios, it is tight.

The Degrees of Freedom Region of the Cognitive Interference Channel With Delayed Channel State Information Feedback

This paper studies two-user cognitive multiple-input multiple-output (MIMO) interference channel with delayed channel state information at transmitter (CSIT), where a transmitter possesses noncasual knowledge of data originating at the other user. We first derive an upper bound of degrees-of-freedom (DoF) region and then propose DoF bound achieving schemes for various antenna configurations. Since interference condition and interference suppression capability vary with antenna configurations, the proposed DoF optimal schemes differently utilize cognitive transmission based on delayed CSIT to suppress interference and thus differ in the required numbers of time slots and transmit symbols, although the key idea of swapping interfering signals between users is common. Comparing with the DoF regions of relevant channels, we identify and analyze the contributions of cognitive transmission and delay CSIT toward enlarging the DoF region. Our analysis specifies antenna configurations where delayed CSIT and cognitive transmission are useful, respectively.

Degrees of Freedom of the MIMO Z-Interference Channel With Delayed CSIT

The degrees of freedom region is established for the 2-user multiple-input multiple-output Z-interference channel with an arbitrary number of antennas at each node, under the assumption of delayed channel state information at the transmitters. A new interference alignment scheme is obtained and is shown to be optimal in the degrees of freedom (DoF) metric. When used over an interference channel with one cross-link being statistically weaker than the other three links, this scheme, by explicitly recognizing the relative weakness of that link, can lead to a DoF improvement over the DoF-optimal scheme designed for the fully connected interference channel, while also requiring no feedback of the weak link's channel state information.

Rank Matching for Multihop Multiflow

We study the degrees of freedom (DoF) of the layered $2 \times 2 \times 2$ multiple-input-multiple-output (MIMO) interference channel where each node is equipped with arbitrary number of antennas, the channels between the nodes have arbitrary rank constraints, and subject to the rank-constraints the channel coefficients can take arbitrary values. The DoF outer bounds reveal a fundamental rank-matching phenomenon, reminiscent of impedance matching in circuit theory. It is well known that the maximum power transfer in a circuit is achieved not for the maximum or minimum load impedance but for the load impedance that matches the source impedance. Similarly, the maximum DoF in the rank-constrained $2 \times 2 \times 2$ MIMO interference network is achieved not for the maximum or minimum ranks of the destination hop, but when the ranks of the destination hop match the ranks of the source hop. In fact, for mismatched settings of interest, the outer bounds identify a DoF loss penalty that is precisely equal to the rank-mismatch between the two hops. For symmetric settings, we also provide achievability results to show that along with the min-cut max-flow bounds, the rank-mismatch bounds are the best possible, i.e., they hold for all channels that satisfy the rank-constraints and are tight for almost all channels that satisfy the rank-constraints. Limited extensions—from sum-DoF to DoF region, from 2 unicasts to $X$ message sets, from 2 hops to more than 2 hops and from 2 nodes per layer to more than 2 nodes per layer—are considered to illustrate how the insights generalize beyond the elemental $2 \times 2 \times 2$ channel model.

On Maximum Linearly Achievable Degrees of Freedom Region of 2&lt;formula formulatype="inline"&gt;&lt;tex Notation="TeX"&gt;$\,\times\,$&lt;/tex&gt; &lt;/formula&gt;2&lt;formula formulatype="inline"&gt;&lt;tex Notation="TeX"&gt;$\,\times\,$&lt;/tex&gt; &lt;/formula&gt;2 Interference Network With Arbitrary Antenna Configurations

We obtain the exact maximum Degrees of Freedom (DoF) region achieved by linear transceivers without symbol extensions of the 2 $\,\times\,$ 2 $\,\times\,$ 2 multiple-input multiple-output (MIMO) interference network, which is comprised of two sources, two relays and two destinations, each with arbitrary number of antennas. We prove that the maximum achievable DoF region for the networks under some antenna configurations coincide with the cut-set bound of the networks, while there is only one DoF gap from the bound for the networks under other configurations, which can not be bridged by linear transceivers. The basic idea of the proof is to construct a two-tier transceiver, where the first tier decomposes the general networks to several basic networks with special configurations, and the second tier achieves the maximal DoF tuples for these basic networks. A mechanism for constructing the second tier transceiver is proposed, which is referred to as channel scaling assisted interference elimination. Combined with interference avoidance at sources and interference cancelation at destinations depending on the antenna configurations, interference-free transmission can be ensured.

The Degrees of Freedom of Two-Unicast Layered MIMO Interference Networks With Feedback

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)\, \times \, (M,N) \,\times \, (M,N) $ network, denoted here as the $(M,N)^{3}$ network, 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 $(M,N)^{3}$ network 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 $M \geq N$ ) 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 $(M,N)^{3}$ network is established by introducing a retro-cooperative interference alignment scheme. These DoF region results for the $(M,N)^{3}$ network 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.

A New Proof for the DoF Region of the MIMO Networks With No CSIT

In this paper, a new proof for the degrees of freedom (DoF) region of the K-user multiple-input multiple-output (MIMO) broadcast channel (BC) with no channel state information at the transmitter (CSIT) and perfect channel state information at the receivers (CSIR) is provided. Based on this proof, the capacity region of a certain class of MIMO BC with channel distribution information at the transmitter (CDIT) and perfect CSIR is derived. Finally, an outer bound for the DoF region of the K-user MIMO interference channel (IC) with no CSIT is provided.

The Degrees of Freedom Region of the MIMO Interference Channel With Hybrid CSIT

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.

On Degrees of Freedom Region of Three-User MIMO Interference Channels

Degrees of freedom (DoF) region of three-user interference channels is studied in this paper. The outer-bound of sum DoF of three-user interference channels was derived based on subspace alignment chain, but the complete DoF region remains unknown. In this paper, an outer-bound of DoF region of three-user interference channels is first derived. Then, a linear interference alignment scheme based on spatial beamforming is proposed that can achieve all integer DoF inside the region. As a result, the region can be seen as both the sufficient and necessary condition for the feasibility of linear interference alignment based on spatial beamforming in three-user interference channels. Moreover, the achievability of fractional DoF inside the outer bound DoF region is also addressed with the proposed beamforming scheme in combination with symbol extension.

The Degrees of Freedom Region of the &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$2 \times 2 \times 2$ &lt;/tex-math&gt;&lt;/inline-formula&gt; MIMO Interference Network

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.

A tight outer bound of the degrees of freedom for the MIMO interference channel with the delayed CSIT and partial local feedback

We have researched a new model for the general two-user MIMO interference channel. The model is constructed under the assumption that the instantaneous channel state information (CSI) is available to the receive nodes and the delayed CSI at transmitters (defined as CSIT) and partial local feedback are accessible to the transmit nodes. From this model, an outer bound of the degrees of freedom (DoF) region has been deduced for such channel. A special case of (6,1,5,2) interference channel is used to demonstrate the proposed interference cancellation scheme. This case shows that the limit DoF pairs (3,1) and (92,12) lied on the region boundary can be achieved. Based on this scheme, the general demonstration for the achievability of the DoF region is also presented. Our work shows that the derived outer bound is tight and under some cases, the DoF region with delayed CSIT and partial local feedback is comparable to that with instantaneous CSIT or delayed CSIT and global feedback.

The Degrees of Freedom of MIMO Networks With Full-Duplex Receiver Cooperation but no CSIT

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.