Degrees of Freedom of Two-Hop Wireless Networks: “Everyone Gets the Entire Cake”
We show that fully connected two-hop wireless networks with sources, relays and destinations have degrees of freedom both in the case of time-varying channel coefficients and in the case of constant channel coefficients (in which case the result holds for almost all values of constant channel coefficients). Our main contribution is a new achievability scheme which we call Aligned Network Diagonalization. This scheme allows the data streams transmitted by the sources to undergo a diagonal linear transformation from the sources to the destinations, thus being received free of interference by their intended destination. In addition, we extend our scheme to multi-hop networks with fully connected hops, and multi-hop networks with MIMO nodes, for which the degrees of freedom are also fully characterized.
The conventional design of wireless networks is based on a centralized architecture where a base station, or an access point, directly exchanges data with the end users. Thus, communication is essentially restricted to the one-to-many (broadcast) and many-to-one (multiple-access) single-hop paradigms. However, as the number of users and the data demand increase, and we move quickly towards the future of wireless networks, multi-hop and multi-flow paradigms are expected to play a very important role by enabling a denser spatial reuse of the spectrum and adaptation to heterogeneous scenarios characterized by user-deployed and user-operated infrastructures.
A major challenge in multi-hop multi-flow wireless networks is that “interference management” and “relaying” are coupled with each other. In other words, wireless relay nodes must play a dual role: they serve as intermediate steps for multi-hop communication and as part of the mechanism that allows interference management schemes. Nonetheless, in the information theory literature, these two tasks have traditionally been addressed separately. The relaying problem is usually studied in the context of multi-hop single-flow wireless networks (or relay networks). For such networks, the capacity is shown in  to be within a constant gap to the cut-set bound, and several relaying strategies are known to achieve the capacity to within a constant gap (e.g., quantize-map-forward , lattice quantization followed by map-and-forward  and compress-and-forward ). On the other hand, the problem of interference management is mostly studied in the context of multi-flow single-hop wireless network (or interference channels). While the exact capacity and even a constant-gap capacity approximation for interference channels are still unknown (except in the two-user case [5, 6, 7, 8, 9, 10, 11, 12] and some special -user cases [12, 13, 14]), the total degrees of freedom of such networks are known to be half of the cut-set bound and are achievable by interference alignment techniques [15, 16].
As we move to the multi-hop multi-flow paradigm, a natural question is whether a decoupled approach for relaying and interference management is optimal. For example, if we consider a coarse metric such as degrees of freedom, do we need coupled strategies in order to perform optimally? To make this question clear, consider the wireless network, shown in Figure 1. One approach that decouples relaying and interference management would basically consist of viewing the wireless network as the concatenation of two -user interference channels. For the -user interference channel, it is known that, for almost all values of channel gains, degrees of freedom are achievable both when the channel gains are fixed and when they are time-varying [15, 16]. Therefore, by repeating the scheme described in [15, 16] at each hop, we can also achieve degrees of freedom on the wireless network for almost all values of the channel gains. Another similar decoupled approach consists of viewing each hop of the wireless network as a -user X-channel. This approach in fact achieves degrees of freedom , which is slightly better than . A strategy that couples relaying and interference management can be devised using the result from  that shows that, in an wireless network, a linear scheme can neutralize the interference at all destinations as long as . Thus, it is possible to achieve (roughly ) degrees of freedom on the wireless network, by using only a subset of source-destination pairs. As depicted in Fig. 2, this coupled scheme only outperforms the Interference Channel and -Channel approaches for . Another coupled strategy was recently proposed for the case in . The proposed scheme, named Aligned Interference Neutralization, manages to achieve the cut-set bound of two degrees of freedom, and outperforms all decoupled approaches. However, in general, for , all known schemes fall short of the cut-set outer bound of degrees of freedom. This makes the wireless network a canonical example of a multi-unicast network where the gap between the state-of-the-art inner bounds and the outer bounds is very significant, and an important step in understanding how suboptimal decoupled approaches can be in general.
In this work, we introduce a new achievability scheme called Aligned Network Diagonalization (AND), which handles relaying and interference management in a coupled manner, and manages to close the gap between inner and outer bounds. In particular, we show that the wireless network has degrees of freedom for time-varying channels and constant channels, in which case the result holds for almost all channel gain values. This result also implies that coupled strategies can significantly outperform a strategy that handles relaying and interference management separately.
The scheme takes two forms, depending on whether the channels are fixed or time-varying. In the case of time-varying channels, Aligned Network Diagonalization is in fact a linear scheme. By viewing multiple network uses as generating a single vector use of the network, we can interpret AND as a solution to a diagonalization problem: is it possible to choose linear transformations for the sources, relays and destinations such that the resulting end-to-end transformation is a diagonal transformation (with non-zero diagonal elements)? Our scheme shows that, with probability over the channel gain realizations, this diagonalization can indeed be obtained. This way, interference-free channels are effectively created between each source and its corresponding destination, allowing each user to achieve arbitrarily close to one degree of freedom, i.e., each user can get “the entire cake”.
Similar to the aligned interference neutralization scheme in , each source starts by encoding its message into several data streams, each one corresponding to a direction in a vector space. The key idea behind AND, which differentiates it from the scheme in , is in the goal of the operations performed by the relays. Each relay receives data streams along several directions, and performs carefully chosen linear operations in order to modify each of these directions. In particular, the new directions are chosen so that it looks like the transfer matrix of the first hop is the inverse of the transfer matrix of the second hop. This way, by forwarding these effectively received signals, the end-to-end transformation is diagonalized.
In the case of fixed channels, however, using the network multiple times does not provide us with the diversity we need to perform the end-to-end diagonalization. Therefore, in order to achieve the same degrees of freedom in this setting, each of the data streams is transmitted along distinct rational dimensions, using the real interference alignment framework from . Then, similar ideas to those used in the time-varying case can be used in order to modify the rational dimensions at the relays so that the transfer matrix of the first hop looks like the inverse of the transfer matrix of the second hop. Once again the result is that the signals transmitted at the sources essentially undergo a diagonal transformation until they reach the destinations.
Several interesting extensions of our main result are possible. In particular, for multi-hop layered networks with source-destination pairs, if all hops are fully connected, the number of degrees of freedom is the minimum between and the minimum number of relays in a single layer. The case of MIMO sources, relays and destinations is also addressed. Interestingly, our result implies that, from the point of view of degrees of freedom, the multiple antennas in a single MIMO relay can be equivalentely seen as separate relays, meaning that cooperation between relays in the same layer cannot increase the number of degrees of freedom.
Recently, a number of works have focused their attention to networks with source-destination pairs (two-unicast networks). For instance, the work in  provides constant-gap approximations to the capacity of ZZ and ZS networks. In , the focus are wireless networks. The authors investigated how the common information between the two relays can be exploited in the second hop and proposed relaying strategies based on distributed MIMO broadcast techniques. In , the authors also considered the wireless network under a degrees-of-freedom perspective. By introducing a new scheme called aligned interference neutralization, which applies ideas from interference alignment to a multi-hop scenario, they showed that these networks have two degrees of freedom both in the case of time-varying channels and in the case of fixed channels. General layered networks with two source-destination pairs were later considered in . In this work, two new notions were introduced. The first one is the idea of network condensation, by which a network with an arbitrary number of layers is reduced to a network with at most four layers with the same degrees of freedom. The second is a graph theoretic characterization of when the interference in a network is manageable, i.e., when all the interference can be simultaneously neutralized. This allowed the degrees of freedom of two-unicast layered networks with an arbitrary number of layers and arbitrary connectivity between adjacent layers to be completely characterized and shown to only attain the values , and . In , the authors revisited the setting with constant channel gains but under the constraint that the relays have to performing linear operations. They showed that the optimal degrees of freedom in this case are and can be achieved by a time-varying linear scheme.
When an arbitrary number of source-destination pairs is considered, the results are scarcer. One effort along this direction is found in , where the authors focus on two-hop networks structured as wireless networks where is very large (and edge effects can be neglected) and investigate communication strategies based on rate-splitting and successive interference cancellation at each hop. In , networks with source-destination pairs and hops with nodes each were considered under the fast fading scenario. The authors show that, under some assumptions on the joint distribution of the channel gains, degrees of freedom can be achieved. The main idea is to have the relays forward their received signals at carefully chosen times, so that the signals transmitted by the sources undergo an approximately diagonal end-to-end transformation.
Ii Problem Setup
The wireless network is made up of sources , relays , and destinations , organized as a two-hop layered network, as shown in Figure 1. We will consider two distinct scenarios.
Time-varying channels: We let the channel gain between source and relay at time be , and the channel gain between relay and destination at time be , for . We assume that and are mutually independent i.i.d random processes each obeying an absolutely continuous probability distribution with finite second moment.
Constant channels: We assume that and remain the same throughout the entire communication period.
In both cases we will assume that instantaneous channel state information is available at all nodes. To simplify our notation, we let be the channel state information available at time (which includes all past channel realizations as well as the current one).
Communication will take place over a block of discrete time steps. At each time , each node transmits a real-valued signal . The received signal at a relay and at a destination are respectively given by
where and , for , are sequences of i.i.d. noise terms distributed as . The noise terms are also assumed to be independent from all transmit signals and noise terms at different nodes.
A coding scheme with block length and rate tuple for the wireless network consists of:
Encoding functions for each source , , and for each time . For each message and channel state information , the codeword satisfies an average power constraint of .
Relaying functions , for , for each relay , , satisfying the average power constraint
for all and .
A decoding function for each destination , .
The error probability of a coding scheme (as defined in Definition 1), is given by
where we assume that each is chosen independently and uniformly at random from , that source transmits over the time-steps, and relay transmits at time , for .
A rate tuple is said to be achievable for the wireless network if there exists a sequence of coding schemes with rate tuple and blocklength , for which , as . The sequence of coding schemes , , is then said to achieve rate tuple .
The capacity region of a wireless network is the closure of the set of achievable rate tuples, and the sum-capacity is defined as
The degrees of freedom of a wireless network are defined as
Iii Main Results
Our main result settles the question of the number of degrees of freedom of a wireless network, in both the case of time-varying and constant channel coefficients.
For a wireless network with time-varying channels, .
For a wireless network with constant channels, for (Lebesgue) almost all values of the channel gains.
Since the cut-set outer bound trivially implies that, in both cases, , we only need to show that degrees of freedom are achievable. The achievability scheme we propose for both the time-varying channel case and the constant channel case are based on interference alignment techniques. Similar to the approach taken in , in the time-varying case our alignment is performed over time dimensions, while in the constant channel case, it is performed over rational dimensions. More precisely, when we have time-varying channels, the alignment is performed in the vector space created by multiple channel uses, using the framework introduced in . In this case, our construction results in a linear scheme, i.e., where relaying functions are restricted to linear transformations. When the channels are constant, on the other hand, alignment over time dimensions is not feasible, and we instead use the real interference alignment frameworks introduced in .
In both cases, each of the sources will transmit data streams, each one along a different transmit dimension (be it time or rational). These data streams are aligned at the relays, which allows each relay to decode approximately linear combinations of the data streams which can then be re-modulated using new transmit directions. These new transmit directions are chosen so that all the intereference is cancelled at each destination, and the data streams from each source arrive at their intended destination along independent rational dimensions, which allows perfect decoding with high probability. These operations guarantee that, with small probability of error, the data streams chosen at all sources are mapped to received directions at the destinations by a diagonal linear transformation. Hence, we call this scheme Aligned Network Diagonalization.
The result in Theorems 1 and 2 has important consequences. Consider a two-hop -unicast wireless network where, instead of having relays, we have relays; i.e., a wireless network. It is easy to see that the cut-set outer bound states that no more than degrees of freedom can be achieved. Now, if , we can simply ignore of the relays and use aligned network diagonalization to achieve degrees of freedom. Similarly, if , we can ignore source-destination pairs, and achieve degrees of freedom. A similar idea can be used in a multihop wireless network with layers, source-destination pairs and relays in the th layer (hence ). If we call such a network a wireless network, we have the following result.
For a wireless network, in the time-varying case and for almost all values of the channel gains in the constant channel case.
Iv Achievability Scheme
In this section we describe the Aligned Network Diagonalization scheme, which achieves degrees of freedom on a the wireless network. First, in Section IV-A, we give a high-level overview of the scheme and describe the intuition behind it. These ideas are then formalized in Sections IV-B and IV-C, where we consider, respectively, the time-varying case and the constant channel case, and describe the operations performed by the sources, relays and destinations.
Iv-a Scheme Overview and Intuition
In order to understand the main idea behind AND, we start by considering a different but related problem. Suppose we have a two-hop network with sources, destinations, and a single MIMO relay with (full-duplex) antennas. Equivalently, this setup, illustrated in Fig. 3, can be seen as our wireless network where the relay nodes are allowed to collaborate in the computation of their transmit signals. This new problem is clearly easier than our original problem, in the sense that any scheme for the wireless network can be used to achieve the same rates on the network with a single MIMO relay node.
Achieving degrees of freedom in the setting from Fig. 3 is not difficult. As illustrated in Fig. 4, a simple scalar linear scheme can be used to diagonalize the network. More precisely, if each source transmits a signal at time , , the received signal at the MIMO relay at time is a length- vector given by , where . Then, if we assume that the transfer matrices and are invertible (which is the case with probability under the distribution assumptions in Section II), the relay can build its transmit signal for time through the linear transformation . If we let be the vector of the received signals at the destinations, it is clear that , where is the vector of effective noises at the destinations. Therefore, each destination receives its desired source signal plus a Gaussian noise term, meaning that the relay operations essentially diagonalized the end-to-end transfer matrix of the network, since , where is the identity matrix. It is easy to see that a slight modification of this scheme can guarantee that the transmit power constraints are satisfied at the relays and can thus be used to show that degrees of freedom are achievable in this setup.
When we move back to our original problem with single-antenna relay nodes, we notice that the same scheme cannot be implemented because the relays are not allowed to cooperate in order to compute . Therefore, a natural question is whether it possible to apply the linear transformation distributedly. More precisely, can we find functions such that
for all ? In the case of general transfer matrices and , the answer is no. In fact, if is not diagonal, it is easy to see that at least one component of depends on multiple components of .
Therefore, in order to pursue our objective of diagonalizing the network with distributed relays, we must consider a more general question than the aforementioned one. In particular, we will reformulate the question of whether the network can be diagonalized by bringing in the channels’ time variation, and by including linear transformations at each source and at each destination. Since our channels are time-varying, we notice that, if each hop of the network is used for consecutive time steps, we can view both the transmit signals and the received signals of the network as length- vectors. The transfer matrix of the first hop is now given by
for . The transfer matrix of the second hop, can be similarly built. In this new setting, we have transfer matrices constituted of diagonal blocks, and we could restate the goal in (3) by having each be a length- column vector. In this new setting, by assuming that each relay applies a linear transformation to its vector of received signals, the diagonalization problem becomes the problem of finding block diagonal matrices (with blocks , for ), (with blocks , for ) and (with blocks , for ) such that
where , and correspond to the linear transformations applied by the sources, relays and destinations. Notice that the identity matrix is , and the parameter regulates how much information the sources are transmitting. Our goal is to solve the problem specified by (4) for large enough so that .
In this work, our main contribution is to show that the problem in (4), with probability over the channel realizations, indeed admits a sequence of solutions parameterized by , with the property that as . The scheme that provides this solution, which we call Aligned Network Diagonalization, can be roughly described as follows. The source matrices , , are all chosen to be the same matrix , whose columns are all of the form
for some nonnegative integers , , where is a column vector with all entries equal to . It is then not difficult to see that the result of
is a matrix with blocks whose columns are again of the form in (5). The key idea in the AND scheme is in the design of the relaying matrices . Once again, we will choose a single matrix and let = for , where is a matrix whose columns are the vectors of the form (5) that appear in any of the blocks in and is obtained from by replacing each column as given in (5) with the column
for diagonal matrices to be defined. The key observation is that the result of any vector , as given in (5), undergoing the transformation is
where is a standard basis vector with the at the entry corresponding to the position of the column in . Therefore, the transformation applied by each relay can be understood as replacing each “direction” with a new direction . Each matrix is chosen as what would have been if . This essentially makes it look like the first hop of the network is , rather than . More precisely, we have , where is obtained by replacing each column in one of the blocks of with . This reduces the end-to-end transformation in (4) to
Finally, since can be seen to admit a block diagonal left inverse, we can set to be this matrix and obtain our desired end-to-end diagonalization. In the next section, we describe this scheme in more detail. In particular, several issues such as power constraints and invertibility of the matrices are properly addressed, and the fact that we can choose and sufficiently large such that approaches is proved.
Iv-B Aligned Network Diagonalization for Time-Varying Channels
In order to use the Aligned Network Diagonalization in the time-varying scenario, sources and relays will choose their transmit directions based on the channel gain values at each time-step.
Encoding at the sources:
Each source starts by breaking its message into submessages. Each of the submessages will be encoded in a separate data stream, using Gaussian random codebooks with codewords of length and entries drawn as . We will let
and , and we define the set of transmit directions for the sources at time to be
for some arbitrary . This selection of directions is similar in flavor to the directions chosen in the Interference Alignment scheme introduced in . Notice that the number of transmit directions (which is also the number of data streams) is . To simplify the notation we will let be a vector of indices and write .
Communication will take place over a block of time-steps, where . The th symbol of the codeword associated to the submessage of stream of source will be written as , for . At time for and , source will thus transmit
The constant is chosen so that the transmit power
does not exceed . In (7), we used the fact that the were independently generated. Notice that does not depend on or and can be chosen strictly positive, since the fact that the channel gains are independent and have finite variances implies for all .
The received signal at relay at time can be written as
Even though writing the received signal as in (8) does not emphasize the alignment that occurs at the relays, it will still be a useful representation of the received signal. To capture the alignment, we consider rearranging the terms in the summation in (8) by viewing it as a polynomial on the variables , for , where the coefficients are given by sums of terms. It can then be seen that the actual set of received directions at each relay is a subset of , and the received signal at relay at time can be alternatively written as
where and we define if any component of is or . At the end of the th block of received signals (i.e., the block consisting of signals received at ), relay can form a -dimensional vector of received signals
for . Notice that, for each , is a distinct monomial on the variables for . The following lemma, whose proof is in Appendix A, will thus be useful.
Let , where each is a distinct monomial on the variables . Then, the determinant of the matrix
is a non-identically zero polynomial on the variables .
Let be the matrix whose columns are
for . From Lemma 1, we see that , seen as a polynomial on the variables for and , is not identically zero. Thus, since for and are all indepedent and drawn from absolutely continuous distributions, is invertible with probability . Moreover, if we fix some arbitrary , we can find such that with probability . At time , the relays will verify whether this is satisfied. In case , all the relays will simply remain silent at times . As we will see later, this is important to guarantee that the entries of are not too large, which could lead to a violation of the transmit power constraints at the relays. Otherwise, if , in order to build its transmit signals, each relay will construct the vector of estimates of the s
In order to build the transmit signal for time , each relay will compute the determinant of
Lemma 1 in this case implies that is a non-identically zero polynomial on the variables , , and we can find such that for with probability , for any fixed . Since the event is independent for each time , we will choose and the corresponding small enough so that
where is the same previously chosen parameter. If , all relays will simply stay silent at time . Otherwise, after obtaining , relay will encode all these symbols using new transmit directions. To describe the new set of transmit directions, we first define
Next, we let
and, similar to (6), we can define the set of transmit directions for the relays to be
Relay will encode the s by transmitting, at time ,
where is the effective noise term which results from the additive noise terms in the estimates s. The constant is chosen so that the transmit power
does not exceed . Since each can be written as a ratio between a polynomial on the variables , and , and , we see that for all . Moreover, the fact that , for each , and guarantees that the variance of is finite and independent of . Thus, for sufficiently large, can be chosen independent of and .
We then have the following claim.
The transmit signal of relay , given in (17), can be re-written as