Interference is known as the major limiting factor in the performance of wireless communication. Theoretically, techniques such as interference alignment [1, 2, 3] promise to increase the throughput of the network significantly to its first order. However, implementation of such techniques faces serious practical challenges. Fortunately, in a major class of wireless networks, i.e. cellular networks, there exist some backhaul links, which provide the possibility of collaboration among the interfering links. Such backhaul resources can be used to manage interference and increase throughput in wireless links. The major question here is how much rate improvement is expected, for a given increase in the backhaul load?
In , the authors investigate the effect of receivers cooperation in a two–user interference channel, and characterize the capacity region versus backhaul load trade-off within a constant gap. The achievable scheme is based on a form of Han and Kobayashi method. In high regimes, the scheme of  reduces to a simple strategy. In particular, whenever cooperation is allowed, it is optimum to pass the received signal (indeed its quantized version) at receiver one to receiver two, perform the joint decoding at receiver two, and pass the decoded message of receiver one back again. For the case of no cooperation, orthogonal transmission (e.g. in time or frequency) is used.
Let and respectively denote, the optimum rate in wireless link and the backhaul load per user, normalized by , for large transmission power . Then from , we conclude that
From the results in , the same trade–off region can also be derived for two–user interference channel with transmitters cooperation.
The result in  has been extended in  to the cases where users are equipped with multiple antennas. To be more specific, the authors in  consider a multiple antenna two–user interference channel with receivers cooperation and provide an approximate capacity region assuming some fixed backhaul capacity. It is shown that the gap between the inner and outer bounds is a function of total number of antennas at the receivers and independent of the signal power, therefore, the trade–off between versus is fully characterized.
In , the authors characterize the trade–off between DoF versus receivers backhaul load for three user single antenna interference channel. The achievable scheme proposed in  is fundamentally different from the scheme of  and it has a new ingredient in the cooperation scheme, called cooperation alignment. Such approach outperforms the schemes that are based on collecting all received signals at one node and jointly perform the decoding at that node, by fifty percent and achieves the optimum trade–off. In cooperation alignment, some alignment techniques has been used in developing backhaul messages such that at each receiver the interfering terms in the signals received through the backhaul and through the wireless link are aligned, and these two together provides the means for canceling the interference and revealing the desired signal. As such alignment is not possible in one shot solutions, the idea in  is to divide the intended signals into many sub-signals. In such method, upon receiving backhaul messages, each receiver is able to cancel part of the interference and continuing the message passing phase, all the receivers are able to decode their intended signals. Similarly, for the case of no cooperation interference alignment is exploited. In  it is shown that versus trade–off follows the same formula as in (1).
The results of  and  suggest that for –user case, the versus follows (1) as well. The main objective of this paper is to show that such generalization is in fact wrong. Indeed the cases of two and three users interference channels are only exception rather than a rule.
In this paper, we consider a –user multiple antenna interference channel with backhaul cooperation. We assume fully connected backhaul network and non-degenerate wireless channels and characterize the full DoF versus backhaul load trade–off region, for both receivers cooperation and transmitters cooperation cases. To be more specific, we derive the full trade–off region for the cases of even number of users, while for the cases of odd number of users an achievable bound and a converse bound is provided. The gap between the achievable bound and the converse bound in case of odd number of users vanishes as the number of users increases, hence characteristic of the full trade–off region for large is presented by
where is the number of antennas at each individual node.
When no cooperation is allowed, the achievable scheme is based on interference alignment. On the other hand to achieve full degrees of freedom per user, we use a centralized scheme which is of great practical interest. For the case of receivers cooperation, we collect the quantized version of all the received signals at one of the receivers, where the joint decoding takes place. Then, the decoded signals are sent back to the corresponding receiver through the backhaul network. For the transmitters cooperation, we collect all the messages in one of the transmitters where an interference management scheme (such as zero forcing) takes place. Then, the computed signals are sent back to the corresponding transmitters.
In order to develop an outer bound, we propose a new converse based on dividing the set of cooperating nodes into two balanced sets and assuming full cooperation inside each set and only considering the cooperation load between the sets. For finite number of users, we show that our achievable scheme is optimal when the number of users are optimal. For odd number of users, we show that the gap between the achievable scheme and the converse is diminishing as the number of users increases.
the backhaul networks are fully connected, and the wireless channels from each transmitter to every receiver are degenerate with zero probability. However, these are not valid assumptions for a cellular network, specially when the number of usersincreases. Specifically, due to the power limitations, the signal sent by a transmitter can be detected by only a limited number of receivers. In addition, although there exist paths connecting every arbitrary pair of base stations, there are limitations in the backhaul network, and not all the base stations are directly connected to each other. As an example, in , the authors assume that the backhaul network follows linear Wyner model, and each transmitter only interferes on the two receivers closest to it.
, we assume the system follows general connectivity. In general connectivity, at the wireless side, the wireless channel between some of the transmitters and receivers are so low that we can consider them to be zero. Subsequently at the wireless side, with respect to the channel coefficient matrix between a transmitter and a receiver, either all elements are identically drawn from a continuous probability distribution or all of them are zero. In addition, in general, at the backhaul side, some of the direct links between cooperating nodes do not exist.
In the second part of this paper, we find conditions on the wireless and the backhaul connectivity, such that the centralized scheme remains optimum. However, it is important for the conditions to be tractable, since we might deal with very large networks. We have shown that the proposed conditions are tractable, i.e., verifiable in polynomial time with respect to the network size .
The rest of this paper is organized as follows. In Section II, we formulate our problem from the information theoretic view point and the main results of the paper are presented in Section III. Section IV contains the discussions on the two–user interference channels, forming the foundations of the proofs for the main results. The proofs of our main results for the case of full connectivity is presented in Section V, while the discussion on generalized configurations is presented in Section VI. Finally, in Section VII we present the complexity analysis of the conditions for the optimality of the central processing.
Ii Problem Formulation
Consider an Interference Channel, with receivers and a transmitter corresponding to each receiver. Each transmitter is equipped with antennas and each receiver is equipped with antennas. The set of all transmitters and the set of all receivers are denoted by and , respectively. Transmitter , intends to convey a message to its corresponding receiver. The channel is a Gaussian Interference Channel, which, in a narrow-band environment, is given by
In the above equation, is the received signal at receiver and is the transmitted signal from transmitter , is the additive circularly symmetric Gaussian noise at receiver with zero mean and identity
co-variance matrix, all at time. In addition, is the channel coefficient matrix from transmitter to receiver and is assumed to be fixed during the whole wireless transmission period. We assume sufficient distance among the antennas, consequently all the channel coefficients are independent, yet randomly chosen from an identical distribution. We further assume that the full Channel State Information (CSI) is available at all transmitters and receivers.
To be more precise, we assume that the channel coefficient matrix from receiver to receiver consists of the multiplication of a large scale factor and a small scale factor . The elements of the small scale matrices are independent and randomly chosen from a continuous distribution. This means that the channel coefficient matrix is either zero, or non-zero with probability one. We collect all the large scale channel coefficients in a binary matrix L, denoted by the adjacency matrix of the channel.
Representation by a Bipartite Graph
The channel has an equivalent bipartite graph , with bipartitions . In general, there exists a link among receiver and transmitter if the large scale channel coefficient is one.
In most parts of this paper, we assume that the equivalent bipartite graph is fully connected. To be more precise, this means that all the receivers are subject to the interference from all the transmitters. In the other parts, although we assume that small scale coefficients can be zero, we assume that the wireless direct channel coefficient matrices are full rank with probability one, as defined rigorously in Condition 1 (Direct Connectivity Condition).(Condition 1).
Condition 1 (Direct Connectivity Condition).
In the wireless channels, for every , is one and hence, the matrix is full rank with probability one.
The justification for Condition 1 (Direct Connectivity Condition).(Condition 1), is that in the scheduling stage, if a direct link is weaker than a given threshold, we do not assign the receiver to that transmitter.
We assume that there exists a backhaul network providing the ability of cooperation as depicted in Fig. 2. The backhaul network can be either at the receivers side (Fig. 1(a)), or at the transmitters side (Fig. 1(b)) as described in what follows. The backhaul network has an equivalent graph , where is the set of all cooperating nodes and represents the set of all available backhaul links. In most part of this paper, we assume that the graph of the backhual network is fully connected, i.e., there exists a link between every pair , of the cooperating nodes through which, they can directly pass backhaul messages. In the other parts, we assume that some of the direct links in the backhaul network are zero.
Ii-a Receivers Cooperation
In this scenario, receivers cooperate through the backhaul network as depicted in Fig. 1(a). Here, the set of vertices in the equivalent graph of the backhaul network is the set of all receivers, i.e., . In order for transmitter to create the signal , it encodes the message into a codeword , using a block code of length subject to the following average power constraint
Note that the outgoing signals of the transmitters are completely independent, since they are not connected and the messages and are independent, for all .
Transmitter and its corresponding receiver agree on the message sets . It uses the encoding function
Subsequently, the receivers participate in the message passing process. The backhaul message , sent from receiver to receiver at round of collaboration phase, is chosen from which is a finite set denoting the backhaul message alphabet. In order to construct the message , receiver employs all the signals previously received over its wireless terminal , as well as all previously received backhaul messages .
In particular, receiver uses the backhaul message generating function
in order to form a backhaul message to receiver at round of collaboration phase. Let be the collection of all received backhaul messages upto round of collaboration phase, at receiver , i.e.,
then, we have
Finally, for receivers to decode their intended signals, receiver chooses the decoding function
in order to decode its desired message, where
The corresponding probability of error can be calculated as
Ii-B Transmitters Cooperation
As shown in Fig. 1(b), in this scenario, transmitters cooperate through a backhaul network. Note that in this scenario, the set of vertices in the equivalent graph of the backhaul network is the set of all transmitters, i.e., . In order for transmitter to create the signal , it exploits the message and all the received backhaul messages at the cooperation phase. The resulting signal contains time slots and is subject to the average power constraint (4).
In this case, transmitter and its corresponding receiver, agree on the message set . Then the transmitters participate in the message passing process. The backhaul message , sent from transmitter to transmitter at round of collaboration phase, is chosen from which is a finite set denoting the backhaul message alphabet. In order to construct the message , transmitter uses its intended message , as well as all previously received backhaul messages .
In particular, transmitter uses the backhaul message generating function
to form a backhaul message to transmitter at round of collaboration phase, i.e.,
where is defined at (5).
In addition, transmitter uses an encoding function
in order to convey its message to the intended receiver through the interference channel
Finally, receiver uses a decoding function
in order to decode its desired message, i.e.,
where is the decoded message at receiver . The corresponding probability of error can be calculated accordingly, as in (6).
Ii-C Capacity and DoF Region
We define the rate of each backhaul link and the average (per user) cooperation rate, as in . Specifically, the rate of each backhaul link is defined as the average entropy of the messages passing through that link
and the average cooperation rate is also defined as the sum rate of all backhaul links, normalized by the number of users
For the achievablity of the rate vector, it is required that for every , there exists an integer such that , for every block length , while
Here, is a function of the average power constraint and indicates the capacity of the backhaul network.
The closure of all achievable rate vectors, subject to the average backhaul constraint (9), forms the capacity region . Considering only the interference effect on the capacity region, we omit the noise impact by solely focusing on high regimes.
In high regimes, we define the backhaul capacity of the link from cooperating nodes to cooperating node as,
and the average (per user) backhaul cooperation load as,
To be more specific, we define the backhaul capacity and the average backhaul load as the limit for the backhaul rate and the capacity of the backhaul network divided by the rate of a point to point Gaussian channel at high , respectively.
In the same way, we also define the achievable degrees of freedom for each individual user as,
and the average (per user) achievable degrees of freedom as,
The average DoF (per user) of the channel is denoted by and is defined as the supremum of , over all achievable schemes. We denote as the region, and characterizing the trade-off between the and the backhaul load , is one of the main focus points of this paper.
Definition 1 precisely defines a class of centralized schemes. This definition generalized the concept of centralized scheme and provides the proper tool for writing rigorous converse proofs.
The class of –centralized scheme consists of all the schemes, achieving degrees of freedom of per user with the backhaul load of per user, where each receiver decodes its own message and at least one of the cooperating nodes, say cooperating node , is able to decode , , , with vanishing probability of error. This means that for every , there exists an integer that for codes with the block lengths , we have , and . We denote cooperating node as the Central Processor.
Iii Main Results
In this section, we present our main results. For the rest of the paper we consider the cases where all transmitters and receivers have equal number of antennas, i.e., , unless otherwise stated.
Iii-a Fully Connected Wireless Network
In this subsection, we bound the achievable DoF per user with limited backhaul capacity, where both the wireless and the backhaul networks are fully connected. This means that at the backhaul network, for every arbitrary pair of cooperating nodes, there exists a two-way backhaul link. Also, in the wireless channel, all the entries of the adjacency matrix are one, i.e., the channel coefficients matrix , for all and , are full rank. We treat the problem separately for the cases of Even and Odd number of users.
In a –user interference channel where each node is equipped with antennas, for even values of and with the average backhaul load , we have
In a –user interference channel where each node is equipped with antennas, for odd values of and with the average backhaul load , we have
The results of Theorem III-A, are valid for the case of receivers cooperation (II-A) as well as the case of transmitters cooperation (II-B). Therefore, we have shown the reciprocity of DoF vs. Backhaul load trade–off for both receivers and transmitters cooperation. This was previously discussed in  and  for two–user interference channel for rate versus backhaul load trade-off. Here, we extend the result to –user interference channels in terms of DoF versus backhaul load. To be more specific, we show that the effect of transmitters cooperation and receivers cooperation on the DoF are similar and the gain from exploiting either of them is the same.
For the achievability, we use time-sharing between two corner points. In case of no collaboration (i.e., ), we use interference alignment to achieve DoF of per user. On the other hand, to eliminate the entire effect of interference, and approximately achieving the capacity of interference-free MIMO links, we follow centralized processing as follows:
Receivers Cooperation: A quantized version of the received signals at all the receivers are collected at one receiver. At that receiver, the decoding is done jointly, and the decoded messages are sent back to the corresponding receivers.
Transmitters Cooperation: The messages corresponding to all the transmitters are collected at one transmitter. That transmitter encodes each of the messages, and the encoded signals then go through to a linear transformation, by multiplying to the inverse of the channel coefficient matrix (i.e., zero forcing). The resulting signals are sent back to the corresponding transmitter, and subsequently, wireless transmission phase takes place. Thus, each receiver receives the interference free version of the encoded message of its corresponding transmitter.
Using the results of Theorem III-A, we settled down the problem raised in . To be more specific, we solve the problem of characterizing the DoF vs. backhaul load trade-off region for –user interference channel with receiver backhaul cooperation. In case of Even number of users, (10) characterizes full trade-off region. Moreover in case of Odd number of users, with finite number of antennas, the gap characterized in (11) vanishes for large network size, .
The authors in  show that in case of three–user Interference channel with receivers cooperation and single antenna users, it is possible to reduce the amount of required backhaul load by employing the concept of Cooperation Alignment and decentralized encoding and decoding. Note that the problem formulation admits the solutions in which interference management is performed through pairwise collaboration. However, our results show that the most straightforward scheme in which one node collects all the signals, perform interference management, and send back the computed signals to the corresponding nodes is (almost) optimal.
According to (11), the minimum required backhaul load for , is given by . This bound is tight for the case of three–user interference channel with single antenna users, which is consistent with the result of . It is worth noting that the difference between and the required backhaul load in the Centralized Scheme (briefly discussed in Remark 2) is , which for finite number of antennas shrinks as increases.
For the full DoF vs. backhaul load trade-off region of a –user Interference Channel with backhaul cooperation, we have
for large enough number of users .
Consider the scenario in which we have , receivers each equipped with antennas. Corresponding to each receiver there exist single antenna transmitters, i.e., total transmitters. In this scenario, the results of Theorem III-A hold.
Consider another scenario in which we have transmitters each equipped with antennas. Corresponding to each transmitter there exist single antenna receivers, i.e., total receivers. The results of Theorem III-A holds in this scenario.
Iii-B General Wireless Networks
In this subsection, we introduce a condition on wireless network connectivity such that in the presence of that condition, the class of –centralized schemes are optimum, for large values of . In a general wireless network, the elements of the adjacency matrix are allowed to become zero, i.e., the channel coefficient matrix , are either zero or full rank, for all and . Here, first we introduce Extended Hall Condition as follows.
Condition 2 (Extended Hall’s Condition).
Consider a -user interference channel with backhaul cooperation where the Condition 1 (Direct Connectivity Condition). (Condition 1) holds and all the receivers and transmitters are equipped with antennas. Whenever Condition 2 (Extended Hall’s Condition). (Condition 2) holds, the class of –centralized schemes is optimum, for large values of .
Centralized scheme is of great interest from the practical point of view. The above theorem deals with the cases where some of the wireless links between transmitters and receivers are zero. In this case, the question is whether the DoF of and the backhaul of form a point on the boundray of the region of backhual load versus DoF trade–off. The above theorem states some necessary condition for such optimality.
Note that Condition 2 (Extended Hall’s Condition). (Condition 2) deals with exponentially many subset of nodes. To be more precise, in order to check Condition 2 (Extended Hall’s Condition). exhaustively is exponentially hard in network size. Still, Theorem 3 shows that similar to the Condition 4 (Hall’s Condition)., we can verify Condition 2 (Extended Hall’s Condition). in polynomial time in our problem setting.
In the following section, we focus on two–user interference channel which forms the foundation of the proofs for Theorem III-A.
Iv Two–User Interference Channels
In this section, we focus on the case of two–user interference channels with backhaul cooperation. The problem setup is similar to Section II, except we assume that transmitter is equipped with antennas and receiver is equipped with antennas. Specifically, in this section we remove the assumption of equal number of antennas both for the transmitters and the receivers.
Lemma 1 characterizes the DoF region of the general two–user interference channel. This DoF regions forms the main ingredient for the proof of Theorem III-A (Section V), as well as proof of Theorem 4 (to be discussed later in this section).
For a two–user interference channel with backhaul cooperation, the DoF region is characterized by
This result holds for receivers cooperation scenario as well as transmitters cooperation scenario.
The authors in  have derived the same result (Theorem 2) for the case of receivers cooperation. However, here we prove the bounds for both transmitters cooperation and receivers cooperation. The challenge to prove Lemma 1 for transmitters cooperation case is that the transmitting signals are not independent, in spite of the fact that the messages are independent. This is due to the fact that the cooperation phase has taken place before the wireless transmission phase and the transmitters are, to some extend, aware of each others’ messages. Therefore, the method used in  is not applicable anymore. Our main contribution here, is to deal with such dependency and showing the reciprocity of transmitters and receivers cooperation case for general two–user interference channel with backhaul cooperation. To handle this case, we bound the effect of dependency between transmitting messages and show that in terms of degrees of freedom this does not affect the results. On the other hand, in  the power constraint is of the form , which is different from (4). This will also affect the proof, since (4) is more general.
Following the lines of the proof, one is able to verify that the only constraints for the results of Lemma 1 to hold, is that the channels coefficient matrices from transmitter to receivers and must be full rank, for and . This means that the direct and cross channel coefficient matrices must be full rank. Such observation is of great importance for the discussions in Section III-B.
Using the results from Lemma 1, we have the following Theorem for two–user interference channels.
For a two–user interference channel with backhaul cooperation and average per user backhaul load , where transmitter is equipped with antennas and receiver is equipped with antennas, we have
To prove Theorem 4, we first present the converse then the achievable scheme. Without loss of generality, we assume that . For the converse proof, we directly use the results of Lemma 1, where by setting we have
By adding the above two inequalities, we have
On the other hand, since there are antennas at each transmitter and each receiver , the maximum achievable DoF equals per user. This completes the converse proof.
For the achievable scheme, we use the time sharing between two corner points. The first corner point is where no cooperation is allowed and we simply turn the firss transmitter–receiver pair off and let the second pair work to achieve a DoF of per user. Moreover, the second corner point is obtained by eliminating the entire effect of interference, and approximately achieve DoF of per user. We use the central processing as introduced in Remark 2 while the cooperating node two handles the processes. This scheme requires backhaul messages to achieve DoF of and the backhaul load is per user. ∎
For a two–user interference channel where each transmitter and each receiver are equipped with antennas, the DoF versus backhaul load trade of region is given by
which is consistent with the results of Theorem III-A.
So far we have dealt with two–user multiple antenna interference channels. Now we are ready to move to more general case and prove Theorem III-A.
V Proof of Theorem Iii-A
In this section, we first provide a converse proof for Theorem III-A, and then we prove the achievability of the theorem.
V-a Converse Proof
First, we notice that since all the transmitters and all the receivers are equipped with antennas, we have , for all , and consequently . We call this as the single user bound.
For the rest of the proof we develop an upper–bound using the DoF region of a two–user interference channel with backhaul cooperation as stated in Lemma 1.
Let us partition the whole set of transmitter–receiver pairs, into two groups. In this partitioning, there exists pairs of transmitters-receivers in the first group and pairs in the second group. Note that such a partitioning is not unique and by putting different pairs of transmitter–receiver in different groups, we can form different realizations of such a partitioning. Let represent the set of indices of transmitter-receiver pairs in group , and denote the set of indices of all transmitter-receiver pairs. For any we define
Corresponding to each realization of the above partitioning, we form a two–user interference channel, in which the transmitters (receivers) in fully cooperate with each other at zero backhaul cost. In other words, the transmitters (receivers) in form the multiple antenna transmitter (receiver ). We also set the capacity of the backhaul link connecting cooperating node to cooperating node in the two–user interference channel equal to the sum of the capacities of the backhaul links connecting each of the cooperating nodes in group to each of the cooperating nodes in group of the original –user interference channel (For an example see Fig. 3).
For a given backhaul load limit, the total achievable DoF in each realization of the above partitioning, is upper bounded by the total achievable DoF of the two–user interference channel corresponding to that realization.
Any achievable solution on the original interference channel with backhaul cooperation can be used on the corresponding two–user interference channel. ∎
where is the number of antennas for each individual transmitter (receiver) and is defined according to (16). In addition, and , therefore equals the total achievable DoF, i.e.,
Part I (Converse for Theorem 1-A.)
In case of even number of users, we have for some . In this case we choose . By substituting and and adding both equations in (17) we have
for all realizations of and with .
In order to form one of such realizations, we have different options. Consider the realizations in which, pair is in one group and pair is in the other, for and . In such realizations, the capacity of the backhaul link connecting node to node , denoted by , and the capacity of the backhaul link connecting node to node denoted by , show up in (18). In order to form a realization, we need to choose pairs, out of remaining pairs. This means that there are different options.
Using the above argument and summing (18) over all options, we have
Since , we have
Part II (Converse for Theorem 1-B.)
In case of odd number of users, we have , for some . In this case we choose and . By substituting and and adding both equations in (17) we have
for any realization of and with and .
In order to form one of such realizations, we need to choose pairs, out of available pairs. This means that there are different options. Consider a realization in which, pair is in one group and pair is in the other, for some and . In this realization, the capacity of the backhaul link connecting node to node , denoted by , and the capacity of the backhaul link connecting node to node , denoted by , show up in (20). To form such a realization, we need to choose pairs, out of remaining pairs.
Using the above argument and summing (20) over all realizations we have
Since , we have,
which completes the proof.
Referring to Remark 14 and the fact that the converse proof is based on the results from two–user interference channel, one is able to verify that for the converse bounds (19) and (21) to hold, it is required that for all realizations the channel coefficient matrices from the transmitters in group to receiver in group and must be full rank, for and . More details are given in Lemma 3.
V-B Achievable Schemes
First consider the case where no cooperation is permitted, i.e., . This case is widely investigated in the literature and it is shown that a DoF of half per user is achievable using Interference Alignment [1, 2, 3]. Consequently, we have,
Let us now focus on the cases where each user is able to achieve full DoF of . We treat this part separately for the cases of receivers cooperation and transmitters cooperation.
V-B1 Receivers Cooperation
In this scenario, transmitter uses a Gaussian code book, carrying degrees of freedom and transmit the message . At the receivers side, one of the receivers, say receiver 1, is chosen to be the central processor. Then, all other receivers, quantize their signals with unit squared error distortion and send it to the central processor, i.e. receiver 1, using the backhal links.
The central processor, upon receiving all the backhal messages, is able to jointly process all received signals and decode all the messages. Exploiting the backhaul links again, central processor sends back the desired message of each receiver.
V-B2 Transmitters Cooperation
In this case each transmitter uses a Gaussian code book, carrying degrees of freedom. The cooperation phase takes place prior to transmission and one of the transmitters, say transmitter 1, is chosen be to the central processor. Then, all other transmitters send their intended messages to the central processor using the backhaul links. Upon receiving all the backhaul messages, the central processor is aware of all intended messages and it encodes each message separately for the corresponding receiver. With the encoded messages and the channels knowledge in hand, the central processor performs a linear beamforming. To be more specific, let denote the supper channel matrix, and denote the signal intended for receiver at time . The central processor performs a linear zero–forcing and forms the signal . Here, and and for .
Subsequently, it quantizes the signal u with unit squared error distortion, resulting in , and for all , and sends back to transmitter using the backhal links. Each transmitter consequently sends what it has received from the central processor through the channel. After the transmission phase, each receiver is able to decode its desired signal, since the received signals are interference-free.
Each of the above achievable schemes, so-called the Centralized Scheme, requires per user backhaul load and achieves full DoF of per user, i.e.,
Up to now we have shown the achievability of two corner points of the DoF versus backhaul load trade-off region, i.e., and . Using time-sharing, one is able to achieve every point connecting these two points in the trade-off region. Consequently, we have
Vi Generalized Configurations
Up to now, we assumed that the backhaul network is fully connected and all the large scale wireless channel coefficients are equal to one. To be more precise, in our problem setting, the equivalent graph of the backhaul network contains all possible links and the transmitted signal of each transmitter contributes to the signal received by all the receivers.
In the schemes introduced in Section V-B, we do not use all the links of the backhaul network. To be more specific, there are links available in a fully connected backhaul network, while our scheme only employs of them. Therefore, even in the absence of some of the backhaul links the class of –centralized scheme might be achievable.
On the other hand, with respect to the wireless links, there are large scale channel coefficients, all of which are one in the fully connected scenario. However, for the optimality of class of –centralized schemes, it is not required for all the large scale channel coefficients to be equal to one.
Thus, we expect the class of centralized schemes to be optimum in some networks with less connectivity, as well. In this section, we aim to explore such class of networks.
In general configurations, in the backhaul side, some of the direct links among the cooperating nodes do not exist and on the wireless side, the channel between some of the transmitters and receivers are so weak that we can consider them to be zero. To be more precise, we assume that the large scale channel coefficients are either zero or one. This means that the channel coefficient matrix is either zero (), or where all the elements of are drawn i.i.d from a continuous distribution. Following the definition of the adjacency matrix in Section II, for a group of transmitters and a group of receivers , the matrix is called the adjacency matrix of the set of transmitters and the set of receivers.
We assume that, Condition 1 (Direct Connectivity Condition). (Condition 1) always holds, i.e., , for all , while could be either zero or one for . Then, the question is whether the centralized scheme is still optimum in those configurations or not?
The objective in this section is to investigate the optimality and feasibility of the class of centralized schemes in more general configurations.
Vi-a Optimality Condition
As mentioned in Corollary 1, in a –user interference channel with fully connected backhaul network and wireless channel, the centralized scheme is optimal for large enough values of . To be more specific, there is no scheme that achieves DoF of per user, with a backhaul load less than per user.
In this sub-section, we assume the backhaul network remains fully connected, and identify some wireless configurations, where despite the fact that zero channels are allowed, the class of –centralized schemes, remains to be optimal. To identify this class of wireless networks, we find some sufficient condition on wireless channel connectivity such that the converse proof in Section V-A is still valid.
First, we have the following theorem about the connectivity of the wireless channel.
In the presence of Condition 1 (Direct Connectivity Condition). (Condition 1), the following conditions are equivalent:
(a) Condition 2 (Extended Hall’s Condition). (Condition 2)
Let be any arbitrary integer in . In the equivalent bipartite graph, for each group of arbitrary subset of transmitters,
(b) Full Rank Sub-Matrices Conditions
Let , be such that . For any arbitrary subset of transmitters with and any arbitrary subset of receivers with , is full rank, almost surely.
(c) Non Degenerate Direct and Cross Channels Conditions
Let . For each arbitrary set of users with , the channel coefficient matrices and are full rank with probability one, where .
We first show the equivalency of (a) and (b).
Assume (a) does not hold. Then, there exists at least one and a set , containing transmitters, with less than neighboring receivers, i.e., . Moreover, there exists a set , containing receivers, and , i.e., . Consider another set , containing transmitters, where .
There exists a subset of with nodes, which has less than neighbors in . According to Theorem 6, there is no perfect matching between and . This results in rank-deficiency of the coefficient matrix of the channel between and , due to Proposition 2. Therefore, (b) does not hold.
We provide the proof, for the cases of even and odd number of users separately.
Case 1: Even number of users ()
Here, it is only required to show that the channel coefficient matrix from any set containing transmitters to any set containing receivers, is full rank.
Assume (a) holds, i.e., for any and any arbitrary set , containing transmitters, there are at least neighboring receivers.
Since there are receivers in total, one concludes that in each arbitrary set , containing receivers, i.e., , there are at least neighbors of , i.e.,
According to Theorem 6, a perfect matching exists for each sub-graph of of the equivalent bipartite graph, where .
Exploiting Proposition 2, the existence of the perfect matching results in the rank-efficiency of the equivalent matrix, i.e., (b) holds.
Case 2: Odd number of users ()
Here, it is only required to show that the channel coefficient matrix from any set containing either or transmitters to any set containing receivers, is full rank. It is also required to show that the channel coefficient matrix form any set containing transmitters to any set containing either or receivers, is full rank.
Assume (a) holds, i.e., for any and any set , containing transmitters, there are at least neighboring receivers. Since there are receivers in total, one concludes that in each arbitrary set , containing receivers, i.e., , there are at least neighbors of , i.e., . Therefore, and according to Theorem 6, for each sub-graph of where and there exists a matching saturating all transmitters.
On the other hand, since (a) holds and according to Lemma 21, for any and any set , containing receivers, there are at least neighboring transmitters. With the same argument as above, since there are totally transmitters, one concludes that in each arbitrary set , containing transmitters, i.e., , there are at least neighbors of , i.e., . Therefore, and according to Theorem 6, for each sub-graph of where and there exists a matching saturating all receivers.
Exploiting Proposition 2, the existence of a matching results in the rank-efficiency of the equivalent matrix, i.e., (b) holds.
Now we provide the proof for the equivalency of (b) and (c). However, it is very easy to verify that if (b) holds, then (c) also holds.
Note that, since Condition 1 (Direct Connectivity Condition).(Condition 1) holds, according to Proposition 1, the direct channel coefficient matrices are always full rank. For the rest of proof we use contradiction. Consider a set , containing transmitters and another set , containing receivers, with rank–deficient channel coefficients matrix. According to Proposition 2, there is no perfect matching in the adjacency graph and consequently, due to the results from Theorem 6, there exists a set , containing transmitters with .