Degrees of Freedom of the MIMO Interference Network with General Message Sets
We establish the degrees of freedom (DoF) region for the multiple-input multiple-output (MIMO) two-transmitter, two-receiver () 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 interference network is obtained and then it is shown to be tight, establishing the DoF region for the most general antenna setting wherein all four nodes have an arbitrary number of antennas each. The DoF-optimal scheme is applicable to the MIMO 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 interference network with a subset of the nine-messages are established as special cases of the general result of this paper. In particular, the DoF region of the well-known four-message (and even three-message) MIMO channel is established. This problem had remained open despite previous studies which had found inner and outer bounds that were not tight in general. Hence, the DoF regions of all special cases obtained from the general DoF region of the nine-message 22 interference network of this work that include at least three of the four channel messages are new, among many others. Our work sheds light on how the same physical 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.
In order to design communication systems that can flexibly and efficiently handle the complex signaling requirements of modern applications, such as in the delivery phase of caching systems over wireless interference channels, it may be necessary to offer multiple physical layer modes that allow for the transmission of some or all of multiple unicast, multiple multicast, multiple broadcast (i.e., -channel), and/or cooperative/cognitive/common messages. In this paper, rather than considering each such transmission mode in isolation, we study the unified setting in which any subset (including all) such messages can be transmitted simultaneously over the MIMO interference network. For this simple network, depending on the subset of the two transmitters at which a message is known, and the subset of the two receivers where it is desired, there are nine possible messages in the general message set. For this fully general message set, the associated nine-dimensional DoF region of the MIMO interference network is established herein.
The most studied and also the best understood setting of the interference network is the two-unicast setting, referred to in the literature as the interference channel , in which each transmitter has a private message for its single distinct intended receiver (cf. [3, 4, 5] and the references therein). In particular, the DoF region of the two-user MIMO interference channel was found in  and more refined characterizations in terms of generalized degrees of freedom and constant bit-gap to capacity were found in  and , respectively.
The four private message case, which can be thought of as a two-broadcast network, more commonly known as the channel, allows for the transmission of a private message to each of the two receivers from each transmitter. The now well-known, and more broadly applicable, linear precoding technique known as interference alignment is needed to achieve the DoF in some cases. With its use, the MIMO channel was shown in [6, 7] to achieve higher sum DoF than the MIMO interference channel. For example, when all transmitters and receivers are equipped with the same number, , of antennas, the two-user MIMO interference channel has a sum DoF of , while the MIMO channel can achieve a sum DoF of for , achievable with interference alignment. The key idea (when is a multiple of 3) is that by aligning undesired signals (i.e., interference) from the two transmitters into the same subspace at a receiver, one can maximize the desired signal dimensions at that receiver. In , an outer bound on the DoF region of the MIMO channel is given based on the sum rate outer bound of the embedded MAC, BC and channels in the channel. Moreover,  gives an achievability scheme based on interference alignment and presents an achievable DoF region that is given as the convex hull of all integer-valued degrees of freedom within that outer bound region. But these inner and the outer bounds of  are not identical. However, using interference alignment over multi-letter extensions of the MIMO channel, it was shown that the outer bound is tight (including non-integer corner points) when all nodes have equal number of antennas , when . In the context of the general MIMO channel with an arbitrary numbers of antennas at the four terminals,  claims that the DoF outer bound region obtained therein is tight in “most cases”, but a precise statement and proof of this claim is not provided. Later, the authors of  introduced a novel technique named asymmetric complex signaling (ACS). By allowing the inputs to be complex but not circularly symmetric and using an alternative representation of the channel models in terms of only real quantities, the problem is transformed to delivering real messages over channels with real-valued coefficients. Consequently, it was shown that the 2-user single-input, single-output (SISO) channel with constant channel coefficients achieves the outer bound of DoF. However, it remained an open problem as to whether the outer bound of  is tight for any of the multiple antenna cases. For instance, the problem remained open as to whether there are other scenarios in which ACS is required in addition to multi-letter extensions, as did the problem of identifying cases in which just multi-letter extensions suffice to achieve the outer bound. More recently, it was shown in  that the outer bound on the sum DoF for the MIMO channel (with generic channel coefficients) derived in  is tight for any antenna configuration. The work in  proposes a linear precoding method based on the generalized singular value decomposition (GSVD), and with the aid of computational experiments, the authors of  offer a conjecture that the outer bound region obtained in  is also tight. The general DoF region result of this paper for the MIMO interference network with nine distinct messages, when specialized to the four private-message MIMO channel, settles this conjecture in the affirmative. It therefore also expands on, and makes precise, the claim in . The outer bound on the DoF region of  is indeed tight.
Besides the aforementioned MIMO interference and channels (and its embedded MAC and/or BC), in which only private messages are considered (see also [4, 5, 10]), the interference network can work in various other modes if common messages, multicast messages and/or transmitter cognition are allowed. For example, if both transmitters share the same three messages, and each receiver demands one of the first two messages while both demand the third, we have what is known as the broadcast channel with private and common messages (BC-CM) [11, 12]. On the other hand, if each transmitter has a private message, and both receivers demand both of the messages, the system works as a compound multiple access channel (C-MAC) . If there are two private messages as in the interference channel and there is a common message known by both transmitters and also demanded by both receivers, the network is known as the interference channel with common message (IC-CM) [14, 15]. The network is referred to as a cognitive channel in  if there are four independent messages to be sent as in the channel, but with one of the four messages known at both transmitters. A new three-message setting could be defined in which one transmitter has 2 messages, each intended for a distinct receiver, and a third shared message that is known to both transmitters and desired at one of the receivers. Interpreting the second transmitter as a relay, such a setting could be described as a broadcast channel with a partially cognitive relay (BC-PCR). A six-message cognitive channel could be defined as having the four private messages as in the channel as well as two more messages that are known to both transmitters with each desired at a distinct receiver. Evidently, based on different message sets, the interference network can represent many different settings and potential applications.
Notation: co(A) is the convex hull of set A, and denote the set of non-negative -tuples of real numbers and integers, respectively. represents the larger of the two numbers, and . denotes the Kronecker product of matrix and . means the horizontal concatenation of matrix and , and means the vertical concatenation of matrix and . and denote the real part and imaginary part of complex matrix , respectively. denotes the null space of the linear transformation . denotes the subspace spanned by the column vectors of matrix .
Ii System Model
We consider the complex Gaussian network with two transmitters and two receivers, as it is shown in Figure 1. The two transmitters are equipped with , antennas respectively, and the two receivers are equipped with , antennas respectively. We denote the channel between transmitter and receiver as the complex matrix and assume all channels to be generic, i.e., all the channel coefficient values are drawn independently from a continuous probability distribution. The channel is assumed to be constant over the duration of communication and all channel coefficients are perfectly known at all transmitters and receivers. The received signal at receiver is given by , where is the input vector at transmitter , is the additive white Gaussian noise (AWGN) vector at receiver r.
General message sets are considered in this paper. For interference network, there are at most nine possible messages classified by different sources and destinations. We index them as , , , , , , , and , as shown in Figure 1. is a private message sent from transmitter to receiver ; is a common message transmitted cooperatively from both transmitters to receiver ; is a multicast message transmitted from transmitter and demanded by both receivers simultaneously; is a common multicast message transmitted cooperatively from both transmitters and demanded by both receivers.
Assume the total power across all transmitters to be equal to and indicate the message set size by . For codewords occupying channel uses, the rates are achievable if the probability of error for all nine messages can simultaneously be made arbitrarily small by choosing appropriately large . The capacity region of the MIMO interference network with general message sets is the set of all achievable rate-tuples =. Define the degrees of freedom region for MIMO interference network with general message sets as
This definition is the general message set counterpart of the one provided in  for the MIMO X channel. Note that is a closed convex set.
In the following section, we consider first the previously studied MIMO channel, for which the best inner and outer bounds of [7, 8, 9] known to date are not coincident in general. The MIMO channel provides the context in which to introduce the notation used in this paper and all the relevant linear precoding techniques, namely, zero-forcing, interference alignment, symbol extension, and ACS. We provide a class of antenna configurations for which, among linear schemes, ACS is required and is sufficient, along with multi-letter extensions and the other linear precoding techniques, to achieve all fractional DoF corner points for those antenna configurations. More generally, we show that the use of linear precoding techniques including symbol extensions and ACS, whether ACS is required or not, are sufficient to achieve any corner point of the DoF region regardless of the antenna configuration. The DoF region of the general nine-message problem is established in IV.
Iii The MIMO Channel
The MIMO channel is an important special case of the interference network in which only the four private messages, namely, , , , , are present. Hence the message index set in this case is . Each of these four messages is intended for one of the two receivers and is a source of interference to the other receiver.
We start by stating the DoF region of the MIMO channel.
The DoF region of the MIMO channel with constant generic channel coefficients is (with probability one)
That the above DoF region is an outer bound for the DoF region of the MIMO channel is proved in Theorem 2 of . The outer bounding inequalities result, respectively, from the embedded multiple-access channel, broadcast channel and channels, in the MIMO channel. The readers can refer to  for details. Moreover, these outer bounds are generalized to the general nine-message problem in Section V.
The authors of  also provide a constructive achievability proof to show that the convex hull of all the integer-valued DoF-tuples in is achievable. The techniques used in the achievable scheme are zero-forcing, interference alignment and random beamforming. Since these techniques are among the techniques used in our interference network with general message sets problem, we provide a succinct account of them in Section III-A, describing in the process, the notation used in this paper as well. The techniques of symbol extension and ACS are described in Sections III-B and III-C to follow.
Iii-a Zero-forcing and interference alignment
Consider message as an example. If , the null space of channel is not empty. By transmitting some symbols of message using the beamformers chosen from the null space , we can zero-force these symbols at receiver and thus introduce no interference to it. The maximum number of such symbols that can be zero-forced is , which is equal to the rank of . Similarly, we can transmit, at most, symbols of message via the nullspace of channel and zero-force them all at their unintended receiver, . Note that the null space and are both subspaces of the null space of the concatenated channel . The remaining dimension of is equal to . By choosing beamformers for message and jointly from the rest of the subspace of , it is possible to align this part of message and into the same subspace and thus reserve more dimensions for the desired messages at receiver , and the maximum number of such pairs of streams is equal to . If there are more symbols of message left, they can be transmitted using random beamforming, which would create unavoidable interference at its unintended receiver.
Since the technique of zero-forcing is the more efficient in terms of reducing interference than interference alignment, it is given the highest priority when constructing precoding beamformers. Following that, interference alignment is used to the extent possible, and following which all of the remaining symbols are sent using random beamforming. The beamformers for each private message is hence divided into three linearly independent parts based on the precoding technique used. Here we use superscript ’Z’ to indicate a message is zero-forced at its unintended receiver, ’A’ to indicate a message is aligned with another interference at their commonly unintended receiver, and ’R’ to indicate the remainder of a certain message that is transmitted using random beamforming. Hence a message for (recall for the MIMO channel) is split in general into three components or sub-messages, denoted and , with the number of symbols (dimensions) in each denoted as , and , respectively. In general, we use the notation and with and for the component messages and dimensions, respectively. Similarly, the precoding matrix for any sub-message is denoted as . Thus we have that and let denote the horizontally concatenated matrix , where . It was shown in  that any integer-valued DoF-tuple within the outer bound can be divided into three such parts within the decoding ability of the channels. It is thus achievable.
Iii-B Symbol extensions
When a corner point of is not integer-valued, it is rational-valued. It is therefore natural to consider a multi-letter extension of the channels to obtain a larger but equivalent system with the corresponding corner point of the DoF region being integer-valued. The length of symbol extensions can be chosen to be the least common multiple of the denominators of all the fractional values. To this time-extended channel, the techniques of zero-forcing, alignment and random beamforming can be applied as described in the previous section. This was proposed in .
Consider symbol extensions of the channel with complex and constant (across time) channel coefficients. We have the equivalent channel matrix , in which , , , and
Hence, we effectively have an channel with antennas at the th transmitter and antennas at the th receiver and channel matrices . To achieve a degrees of freedom tuple for the original system, we need to achieve for this equivalent system, and we can use the exact same precoding scheme designed for integer-valued corner points.
However, the equivalent channel matrices after symbol-extension are unlike those for their original counterparts (with ) in that they are block-diagonal. The primary question that arises is whether the the channel matrices of the time-extended channel continue to yield the linear independence results of the single-letter generic unstructured channels in spite of their special non-generic structure. If they do, then it can be asserted that multi-letter extensions are sufficient to achieve all fractional DoF tuples of .
However, this is not the case in general. Indeed, as it was observed in  the symbol extension technique is not sufficient even for the SISO channel. Interestingly, on the other hand, it is shown in  that, in the symmetric MIMO case, where all nodes have equal number, , of antennas, and , the same idea works.
Nevertheless, the authors of  claim, based on a few examples, that the DoF outer bound region obtained therein is tight in “most cases”, and give the SISO case as an exception. But it is not clear if there are other cases that are also such exceptions, and if so, whether they can indeed be seen as exceptions, i.e., it is unclear as to how commonly these exceptions arise, in which just symbol extensions are not enough to achieve all the fractional corner points of . This brings us to the next section.
Iii-C Asymmetric Complex Signaling
As stated previously, since the equivalent channel matrices after symbol-extension will be block-diagonal, many nice properties of the original generic channels can be lost. It is shown in  that, in the SISO case, the precoding scheme provided previously (with three symbol extensions) fails to achieve the important integer-valued corner point which achieves sum-DoF, because of the block diagonal structure in the extended channel matrices.
In response to this phenomenon, the authors of  introduced a new technique named asymmetric complex signaling (ACS). The key idea of ACS is to allow the inputs to be complex but not circularly symmetric and use an alternative representation of the channel models in terms of only real quantities. All dimensions of the new system will be doubled and all channel coefficients, beamformers, inputs and outputs will be real-valued. Let be the original complex channel matrices, their alternative real representations will have the following forms
In order to transmit complex-valued streams over the original system, we need to transmit real-valued streams over the equivalent real channels.
It is shown in  that using ACS, the outer bound of degrees of freedom is achievable for the SISO channel. In particular, with a three-symbol extension and ACS, all equivalent channel matrices are of size , and using the same precoding scheme as used in the other MIMO cases, two real-valued symbols can be transmitted via the real channels. The missing independence requirement in the previous complex-valued transmission disappears almost surely in this new model. Thus the sum-DoF of is achievable (and hence also the DoF region). The readers are referred to  for further details.
Iii-D Closing the gap
The important question as to whether there are MIMO antenna configurations for which, among linear schemes including symbol extensions, ACS is necessary, remains open. The question is also open about whether ACS, along with the other linear techniques, is sufficient for MIMO antenna configurations to achieve all fractional DoF-tuples in . If so, for what antennas configurations is it sufficient? Are there DoF-tuples and antenna configurations for which linear precoding schemes including time extensions and ACS are not sufficient?
In this section, all of the above questions are definitively answered. In particular, a class of antenna configurations (that includes the SISO case) are identified that require ACS among linear schemes; i.e., in which just employing symbol extensions alone doesn’t suffice. More generally, it is shown that ACS along with the other linear schemes is sufficient to achieve any fractional corner points of the DoF region of the MIMO channel for any antenna configuration.
In the case that and , if interference alignment is needed to achieve any fractional DoF-tuple in , then the achievability scheme in III-A, applied to the -symbol extended interference network, fails to make the corresponding symbols distinguishable at the receiver where they are desired. In particular, if or , then ; if or , then .
We give the proof of Lemma 1 in the case that or , and the validity for the case that or follows in the same way.
First, consider the situation when , and we have that . Consequently, zero-forcing any symbol of message and at receiver is not possible, i.e., . However, since , there exists a one dimensional null space of the concatenated channel . Thus, it is possible to align one symbol of message with one symbol of message at receiver . When channel extensions are used, the available dimension for interference alignment is equal to . Suppose the basis vector of the null space of is given by111The dimensions of matrices will be specified in a subscript when such dimensions have to be emphasized or defined for the first time.
Then one set of basis vectors of -dimensional subspace after symbol extension will be the column vectors of matrix
All beamformers generated from this basis should be of the form
where are random scalars. Since , and will be scalars. and will have the following form
Hence is also aligned with at their commonly desired destination, i.e., . This makes and indistinguishable at receiver .
Next, consider the situation when . In this case, and . In other words, the null space of channel does not exist, and the null space of channel may exist. Recall that we only do interference alignment after zero-forcing of more symbols is not possible. Thus, when interference alignment is used, streams of the message have already been zero-forced at receiver . Since , it is still possible to transmit another symbol of . The dimension of the null-space of the concatenated channel is equal to . Letting vector be in the subspace of null space , we have that will belong to the null space of . In other words, dimensions of the null space are already occupied when doing zero-forcing of message . The remaining dimension of null space is equal to . Thus, 1 dimension of interference alignment is possible at receiver for messages and . When channel extensions are applied, the available dimension for interference alignment is equal to , and the dimension of zero-forcing subspace is equal to . The beamformers and are also in the form of (3)-(5). However, since can be greater than 1 in this case, and are no longer scalars, and we don’t have the desirable result that is aligned with any more.
To prove that , we instead prove that , where are any pair of alignment vectors drawn from the same beamformer from the null space of . Then, we will have that , which is the desired result. Let and , we have that
Let column vectors of be a basis of the nullspace . We have that
Note that the dimensional space at receiver can be partitioned into linearly independent subspaces according to different symbol extension slot index. In order to prove that , it is sufficient to prove that for all . Since here are all scalars, we only need to show that .
Because is generated from the nullspace of channel , it is independent with channel matrix . Consequently, will almost surely reserve the column rank of , since is a generic full matrix whose rank is greater than ’s. In other words, almost surely. Since is linearly independent with the column vectors of , will also be linear independent with the column vectors of almost surely. Thus, . In other words, the column vectors of would span the entire -dimensional subspace at receiver . Since vector also belongs to the same subspace, we have that , which leads to that . Thus, message and are indistinguishable at receiver .∎
By using the technique of ACS together with symbol extensions, the problem of unexpected alignment of desired messages is avoided.
The equivalent channel matrices, when doing -symbol extension and ACS, are given as where is given in equation 2. We need to transmit real-valued streams over the equivalent real channels.
Consider again the independence of and for the cases in Lemma 1. If , when doing asymmetric complex signaling, the dimension of and in (3) will be and , respectively. and will instead have the following form
where are random real scalars. Now, and are both real matrices rather than scalars as in (6) and (7). Each diagonal block of or works as if it is to rotate a random real vector with a certain degree. However, the randomness of makes the projections in different symbol extension slots independent with each other. Thus, and will be linearly independent almost surely.
Consider again the case that , the column rank of will be equal to . In other words, there is still 1 dimension left in the receiver subspace. Thus, is independent with almost surely. In the situation that the column dimension of and are , which is greater than 1, i.e., there are multiple pairs of symbols to be aligned, the column size of will be equal to and thus the columns of are linear dependent with columns of . However, since the coefficients required for dependence for are different almost surely in different time slots, the dimensional will still be linearly independent with almost surely, so long as .
In summary, the desired messages are still linearly independent with each other at both receiver.∎
The authors of  introduced ACS in the context of the SISO channel and showed that the total DoF of can be achieved in that channel. In this paper, we provided a new and simplified perspective on how ACS works. In particular, it transforms the previous scalar multiplication to a local vector rotation, thus obviating the unexpected linear dependences among all the beamformers. Using this we broaden its applicability to MIMO channel, and more generally, ia a later section, to the nine-message MIMO 22 interference network.
For all the other antenna settings not included in the cases given in Lemma 1, there is no unexpected loss of independence of desired messages when doing symbol extensions. Thus, ACS is not necessary in those cases.
Iii-E Further results on the MIMO channel
In this section, we discuss several other observations/results about the MIMO channel.
In the symmetric antenna setting, the maximum sum DoF of the MIMO channel is given by
Lemma 3 is a special case of Theorem 1. In terms of sum-DoF performance, there are hence redundant antennas at the transmitters if or , and there are redundant antennas at the receivers if or . In the case that , the redundancy exists both at the transmitters and at the receivers. For example, the three antenna settings of , and all have the same maximum sum-DoF of 4.
Interestingly, for the equal-antenna case of , one can easily achieve the DoF-tuple of by turning off one antenna at each receiver and then transmitting all four symbols of the private messages using zero-forcing beamforming in each of the one-dimensional null space of the remaining channel matrices. No explicit interference alignment is actually needed to achieve the optimal sum-DoF. Given that explicit interference alignment was first discovered in the context of the symmetric three-antenna MIMO channel as being the key ingredient  needed to achieve DoF-optimality, this observation is surprising. To the best of the authors’ knowledge, this is the first time this simple result has been noted. Shutting down the redundant antenna at each receiver could however be seen as implicitly aligning interference in a subspace that would only be seen by that antenna and then discarding that subspace.
For the special cases given in Lemma 1, in which ACS is required to achieve the maximum sum-DoF, the maximum sum-DoF is equal to , where . The DoF tuple to achieve the maximum sum-DoF is given by .
We give the proof for the case that here. The other cases follow in the same way.
Since and , we have that , , and . Adding the first 4 inequalities in together, we have that
Thus, the sum-DoF is bounded by . It is easy to verify that the DoF tuple achieves the optimal sum DoF and is within the DoF region . Thus, the maximum sum DoF is equal to .
A symbol extension of length 3, together with ACS, is required to achieve this corner point.∎
In the case that only three private messages are transmitted in the channel, all the corner points will be integer-valued. Thus, neither symbol extension nor ACS is necessary.
Since the channel is isotropic with respect to any message, we can assume without loss of generality that the three private messages are , and . By deleting from and removing the redundant inequalities, we obtain the following 3-dimensional DoF region.
Each corner point of this 3-D region will be the intersection of three of the nine facets describing the polytope. Observing the constraints, it is easy to verify that the only possible combination of facets that can have a fractional intersection are (9), (10) and (11), and the corresponding vertex is
These three values will be all integers or all non-integer fractions which are an odd-multiple of .
From constraint (8), we have that
Otherwise, this corner point will be outside the DoF region. Consequently, one of and will be . If it is less than zero, this corner point is outside the DoF region and therefore irrelevant; if it is equal to 0, then the other two values will be integers.
The intersection of all other combinations of facets will be integer-valued, thus, all the corner points of are integer-valued, and neither symbol extension nor ACS are necessary to achieve them.∎
For the MIMO channel of an arbitrary antenna setting, if there are fractional-valued corner points and symbol extension is required to achieve this corner point, the length of symbol extension will be at most 3.
Again, each corner point of the 4-dimensional DoF region is the intersection of four of the facets describing the polytope. Since the coefficients of any facet are either 0 or 1, any selected 4-by-4 coefficient matrix will be a binary matrix. According to the Hadamard maximal determinant problem , the determinant of an order 4 binary matrix can at most be 3. Consequently, the inverse of any 4-by-4 coefficient matrix, if it exists, can at most have a denominator of 3. Thus, for any non-integer valued corner points, the denominator will be at most 3. Thus, the length of symbol extension will be at most 3.
More specifically in this problem, it is shown that there is only one corner point whose denominator is 3, and this corner point is the intersection of the four facets corresponding to the first four constraints in . ∎
Iii-F Cognitive MIMO channel
If one of the four private messages in the MIMO channel, for example , is made available non-causally at the other transmitter, the channel is named cognitive MIMO channel. It is shown in  that the sum DoF of the cognitive MIMO channel with equal number, , of antennas at each terminal is equal to , which is greater than the sum DoF of of the symmetric channel. So, cognitive message sharing helps increase sum DoF in this case. We discuss more general properties of the cognitive MIMO channel here.
The degrees of freedom region of the cognitive MIMO channel with message , , and is given by
Theorem 2 follows directly from our main result of the 9-dimensional DoF region of the MIMO Gaussian interference network with general message sets given in Section IV. When the above DoF region is specialized to the symmetric, equal-antenna case, all but the first three bounds are redundant, and it is easy to see that the DoF-tuple , the achievability of which was shown in  for (using two-symbol extensions), is a maximum sum-DoF corner point of for any .
More generally, the DoF region of cognitive MIMO channel is in general greater than that of the MIMO channel. For example, consider the case of , , , . When , and