Subspace Alignment Chains and the Degrees of Freedom of the
ThreeUser MIMO Interference Channel
Abstract
We show that the 3 user MIMO interference channel where each transmitter is equipped with antennas and each receiver is equipped with antennas has degrees of freedom (DoF) normalized by time, frequency, and space dimensions, where . While the DoF outer bound of is established for every value, the achievability of DoF is established in general subject to a normalization with respect to spatialextensions, i.e., the scaling of the number of antennas at all nodes. Specifically, we show that DoF are achievable for the MIMO 3user interference channel, for some positive integer which may be seen as a spatialextension factor. is the scaling factor needed to make the value an integer. Given spatialextensions, the achievability relies only on linear beamforming based interference alignment schemes and requires neither channel extensions nor channel variations in time or frequency. In the absence of spatial extensions, it is shown through examples how essentially the same interference alignment scheme may be applied over timeextensions over either constant or timevarying channels. The central new insight to emerge from this work is the notion of subspace alignment chains as DoF bottlenecks. The subspace alignment chains are instrumental both in identifying the extra dimensions to be provided by a genie to a receiver for the DoF outer bound, as well as in the construction of the optimal interference alignment schemes.
The DoF value is a piecewise linear function of , with either or being the bottleneck within each linear segment while the other value contains some redundancy, i.e., it can be reduced without reducing the DoF. The corner points of these piecewise linear segments correspond to two sets, and . The set contains all those values of and only those values of for which there is redundancy in both and , i.e., either can be reduced without reducing DoF. The set contains all those values of and only those values of for which there is no redundancy in either or , i.e., neither can be reduced without reducing DoF. Because and represent settings with maximum and minimum redundancy, essentially they are the basis for the DoF outer bounds and inner bounds, respectively.
Our results settle the question of feasibility of linear interference alignment, introduced previously by Cenk et al., for the user MIMO interference channel, completely for all values of . Specifically, we show that the linear interference alignment problem (as defined in previous work by Cenk et al.) is feasible if and only if . With the exception of the values , and only with that exception, we show that for every value there are proper systems (as defined by Cenk et al.) that are not feasible. Evidently the redundancy contained in all other values of manifests itself as superfluous variables that are not discounted in the definition of proper systems, thus creating a discrepancy between proper and feasible systems.
Our results show that are the only values for which there is no DoF benefit of joint processing among colocated antennas at the transmitters or receivers. This may also be seen as a consequence of the maximum redundancy in the settings.
1 Introduction
The number of degrees of freedom (DoF) of a communication network is a metric of great significance as it provides a lens into the most essential aspects of the communication problem. DoF investigations have motivated many fundamental ideas such as interference alignment [4, 7], deterministic channel models [15], rational dimensions [5, 6], information dimensions [19], aligned interference neutralization [1] and manageable interference [2]. The DoF metric is especially valuable as a first order capacity approximation. Since even the smallest gap between the best available DoF inner and outer bounds translates into an unbounded gap in the best available capacity approximations, communication networks where DoF values are not known are some of the least understood problems, and hence, also the most promising research avenues for significant and fundamental discoveries.
DoF characterizations have recently been obtained for a wide variety of wireless networks. Since in this work we are interested primarily in interference channels, it is notable that the DoF of interference channels are known when all nodes are equipped with the same number of antennas, for almost all channel realizations, and regardless of whether the channels are timevarying, frequencyselective or constant. However, if each node may have an arbitrary number of antennas, then a general DoF characterization is not available beyond the 2 user interference channel. One obvious reason is the explosion of the number of parameters in considering arbitrary antenna configurations. However, as we show in this work, the problem involves fundamental challenges even when the number of parameters is restricted by symmetry. In particular, the difficulty of this problem has to do with the new notion of “depth” of overlap between vector subspaces that comes into play on the one hand, and of translating this notion into information theoretic bounds on the other. Specifically, in this work we explore what is perhaps the simplest setting for MIMO interference channels where the DoF remain unknown, and obtain its DoF characterization, while highlighting the challenging nature of the general problem in the process. The assumptions that define our primary focus in this work are:

3user symmetric MIMO interference channel: The number of users is set to , the number of antennas at every transmitter is set to the same value, , and the number of antennas at every receiver is set to the same value, . Thus, the problem space is parameterized by only two variables .

Spatiallynormalized DoF for almostall channel realizations: The DoF characterizations that we seek are intended in the “almostsurely” sense, with channel coefficients drawn from continuous distributions. Further, the DoF are normalized not only with respect to time and frequency dimensions, but also with respect to the spatial dimension, i.e., we allow channel extensions not only in time and frequency dimensions, but also in the spatial dimension (i.e., through a scaling of antennas).
As the smallest, and therefore the most elementary interference channel setting where interference alignment becomes relevant, the 3 user interference channel has special significance. The assumption of global channel knowledge, as well as the implicit assumption of comparable signal strengths from all transmitters at all receivers that follows from the definition of the DoF metric (as opposed to Generalized DoF), is most relevant to small clusters of, e.g., no more than 3, mutually interfering users.
The rationale for the remaining assumptions is our interest primarily in generic insights rather than the peculiarities and caveats associated with special structures. While the restriction to almostall channel realizations is by now a standard assumption for DoF studies, the normalization with respect to spatial dimension is less common. Spatial extension, i.e., proportional scaling of the number of antennas at each node and a corresponding normalization of DoF by the same factor, is appealing in that it allows us to deal with generic channels, thereby revealing generic insights into the geometry of alignments and relative signal space dimensions without having to deal with the added complexity of diagonal or block diagonal channel structures that would result from channel extensions over constant or timevarying channels, or the rational dimension arguments that are often invoked in the absence of sufficient channel diversity. As a further justification for the normalization with respect to spatial dimension, we note that so far, for every wireless network, with or without multiple antennas, where the DoF characterizations are available for almostall channel realizations, the DoF characterizations are unaffected by spatial normalization. Indeed, we conjecture that this should be the case in general, i.e., much like time, and frequency dimensions, the DoF of a network (for almost all channel realizations and with global channel knowledge as is assumed here) should also scale with the proportional scaling of the spatial dimension. Notably, the general question of the scaling of DoF with spatial dimension appears as an open problem in [20]. Also notable is the use of spatial normalization to characterize the DoF region for the general MIMO 2 user channel setting in one of the earliest works on interference alignment [3]. Finally, as explained towards the end of this paper through several examples, the interference alignment solutions developed in this work under spatial extensions may be directly applied to time and/or frequency extensions as well to obtain the same DoF characterizations but without relying on spatial extensions.
2 Background
We are interested in both the information theoretic DoF of MIMO interference channels, as well as the feasibility of linear interference alignment schemes. We start with a summary of relevant work on these topics.
2.1 DoF of MIMO Interference Channels
The two user () MIMO interference channel, where User 1 has transmit and receive antennas, and User 2 has transmit and receive antennas, is shown by Jafar and Fakhereddin in [11] to have DoF. For this result, the achievability is based on linear zero forcing beamforming schemes, and the converse is based on DoF outer bounds for the multiple access channel obtained by providing enough antennas to a receiver so that after decoding and subtracting its own signal, it is able to also decode the interfering signal.
In [7], Cadambe and Jafar introduced an asymptotic interference alignment scheme, referred to as the [CJ08] scheme (see [20] for an intuitive description of the [CJ08] scheme), leading to the result that in the user MIMO interference channel, each user can access halfthecake in terms of DoF^{1}^{1}1The “cake” refers to the maximum DoF accessible by a user when all interfering users are absent., for a total DoF value of DoF, almost surely, when all nodes are equipped with the same number of antennas and when channels are timevarying or frequencyselective. In [22], Motahari et al. introduced the rational dimensions framework based on diophantine approximation theory wherein the [CJ08] scheme is again applied to establish the same DoF result, but with constant channels and without time/frequency extensions. For the 3user MIMO interference channel setting with , Cadambe and Jafar also present a closed form linear beamforming solution in [7] that requires no timeextensions for even and 2 timeextensions for odd , to achieve the DoF outer bound value of per user, without the need for channel variations in time/frequency. The DoF outer bound in each of these cases is based on the pairwise outer bounds for any two users, as established previously for the single antenna setting, , by HostMadsen and Nosratinia in [21] and for the multiple antenna setting, , by Jafar and Fakhereddin in [11].
In [9], Gou and Jafar studied the DoF of the user MIMO interference channel under the assumption that is an integer and showed that each user has a fraction of the cake^{2}^{2}2Here, the “cake” corresponds to DoF., for a total DoF value of , almost surely, when the number of users .
Example: The 3user MIMO interference channel, i.e., the interference channel with 3 users where each transmitter has antenna and each receiver has antennas, has DoF per user, as does the 3user MIMO interference channel. The results holds for more than users as well, i.e., the user and interference channels have DoF per user, for all .
The results of [9], established originally over timevarying/frequencyselective channels, are extended to constant channels without the need for channel extensions, by Motahari et al. in [6], by employing the rational dimensions framework. Further, Motahari et al. show that each user in the user MIMO interference channel, has a fraction of the cake, even when is not an integer, provided the number of users . Interestingly, the achievability of DoF per user, or equivalently DoF per user, follows from the application of the [CJ08] scheme for every value, and for any number of users , and requires no joint signal processing between the multiple receive (transmit) antennas at any receiver (transmitter). However, the optimality of these achievable DoF has been shown only when either is an integer and or when for any . The outer bounds in each of these cases are based on allowing full cooperation among groups of transmitters/receivers and applying the DoF outer bound for the resulting user MIMO interference channel previously derived by Jafar and Fakhereddin in [11].
Example: Consider the 5user MIMO interference channel. Allowing full cooperation between users 1, 2, 3 and allowing full cooperation between users 4,5, we have a resulting two user MIMO interference channel where the effective User 1 has 6 transmit and receive antennas, and the effective User 2 has transmit and receive antennas. According to the DoF result for 2user MIMO interference channel shown by Jafar and Fakhereddin in [11], this channel has 6 DoF. Since cooperation cannot reduce the total DoF, it follows that the original 5 user interference channel has no more than DoF per user. Since DoF are always achievable per user, is the optimal value of DoF for this channel. Further, since DoF per user cannot increase with the number of users, is the optimal value of DoF per user in the user MIMO interference channel for all . The same conclusion applies in the reciprocal MIMO setting as well.
Note that the DoF value of the or the MIMO interference channel is not known if the number of users, , is 3 or 4. As a special case of our results in this work, we will show that is the optimal DoF value per user in the or MIMO interference channel for all , thereby resolving the DoF value for all in the and settings. Since outer bounds based on full cooperation are not enough, the challenge in this case will be to identify the genie signals that will lead us to this conclusion.
2.2 Feasibility of Linear Interference Alignment
While the DoF of MIMO interference channels are of fundamental interest, the achievable schemes are often built upon theoretical constructs such as the rational dimensions framework or Renyi information dimension, whose physical significance and robustness is not yet clear. On the other hand, linear beamforming schemes are well understood based on the abundance of MIMO literature. As a consequence, there is much interest in the DoF achievable through linear beamforming schemes, i.e., through linear interference alignment schemes. A central question in this research avenue is the feasibility (almost surely) of linear interference alignment based on only spatial beamforming, i.e., without the need for channel extensions or variations in time/frequency/space. The feasibility problem is introduced by Gomadam et al. in [23], where iterative algorithms were proposed to test the feasibility of desired alignments. Recognizing the feasibility problem as equivalent to the solvability of a system of polynomial equations, Cenk et al. in [12] draw upon classical results in algebraic geometry about the solvability of generic polynomial equations, to classify an alignment problem as proper if and only if the number of independent variables in every set of equations is at least as large as the number of equations in that set. While the polynomial equations involved in the feasibility of interference alignment are not strictly generic, Cenk et al. appeal to the intuition that proper systems are likely to be feasible and improper systems to be infeasible. For a user MIMO interference channel where each user desires DoF, denoted as the setting, Cenk et al. identified the system as proper if and only if
(1) 
The conjectured correspondence between proper/improper systems and feasible/infeasible systems is settled completely in one direction, and partially in the other direction, in recent works by Bresler et al. in [13] and Razaviyayn et al. in [14], who show that:

If are divisible by then proper systems are feasible [14].

For square channels, , proper systems are feasible [13].
While the properness of a system seems to work fairly well as an indicator of the feasibility of linear interference alignment in most cases studied so far, it is also remarkable that based on existing results it is possible to find examples of proper systems that are not feasible. For example. consider the 3 user interference channel where and where each user desires DoF. According to (1) this is a proper system, and according to [9] it is infeasible because the information theoretic DoF outer bound value for this channel is only per user. The DoF outer bound is easily found by allowing two of the users to cooperate fully, so that the resulting 2 user MIMO interference channel with has a total DoF value of according to [11]. Since cooperation does not hurt, and linear schemes (or any other scheme for that matter) cannot beat an informationtheoretic outer bound, it is clear that the linear interference alignment problem is infeasible for , and in particular for .
The observation that some proper systems are not feasible is also made by Cenk et al. in [12] who suggest including known information theoretic DoF outer bounds to further expand the set of infeasible systems. Interestingly, so far, all known DoF outer bounds for user MIMO interference channels come directly from the DoF result for the 2 user MIMO interference channel [11], applied after allowing various subsets of users to cooperate, while eliminating other users to create a 2 user interference channel (as also illustrated by the preceding example). As we show in this work, these DoF outer bounds do not suffice, even for the symmetric 3 user MIMO interference channel for all values. Thus, the feasibility of proper systems remains an open problem in general, even if restricted to the 3 user setting.
In this work, for the 3 user MIMO interference channel, we settle the issue of feasibility of linear interference alignment. Somewhat surprisingly within this setting, especially considering systems near the threshold of proper/improper distinction, we show that most proper systems are infeasible.
3 System Model and Metrics
We consider a fully connected threeuser MIMO interference channel where there are and antennas at each transmitter and each receiver, respectively. As shown in Fig. 1, each transmitter sends one independent message to its own desired receiver. Denote by the channel matrix from transmitter to receiver where . We assume that the channel coefficients are independently drawn from continuous distributions. While our results are valid regardless of whether the channel coefficients are constant or varying in time/frequency, we will assume the channels are constant through most of the paper. Global channel knowledge is assumed to be available at all nodes.
At time index , Transmitter sends a complexvalued signal vector , which satisfies an average power constraint for channel uses. At the receiver side, User receives an signal vector at time index , which is given by:
where an column vector representing the i.i.d. circularly symmetric complex additive white Gaussian noise (AWGN) at Receiver , each entry of which is an i.i.d. Gaussian random variable with zeromean and unitvariance.
Let denote the achievable rate of User where is also referred to as the SignaltoNoise Ratio (SNR). The capacity region of this network is the set of achievable rate tuples , such that each user can simultaneously decode its desired message with arbitrarily small error probability. The maximum sum rate of this channel is defined as , and denotes the maximum rate normalized per user. The sum DoF are defined as , and DoF stands for the normalized DoF per user. Throughout this paper, we also write DoF as for simplicity. Moreover, we use to denote the number of DoF associated with User , and the user index is interpreted modulo so that, e.g., User 0 is the same as User 3. Furthermore, we define the degrees of freedom normalized by the spatial dimension, as
and is similarly defined. The dependence on may be dropped for compact notation when no ambiguity would be caused, i.e., instead of we may write just .
Notation: For the matrix , and denote its row and column vector, respectively; denotes a row vector obtained from the row and the to columns of the matrix ; denotes a matrix obtained from the to rows of the matrix . We use the notation to represent any function such that . Further, we define .
4 Main Results
In order to present the DoF results for the 3user MIMO interference channel in a compact form, let us define the quantity DoF as follows.
Definition 1
(2) 
where .
The quantity denotes the length of the subspace alignment chain, to be described in the next section. Clearly, as and become approximately equal, i.e., becomes large, DoF converges to the “half the cake” value, .
To arrive at the main result of this paper, we proceed through two intermediate results, presented here as lemmas.
Lemma 1 (Outer Bound)
For the 3user MIMO interference channel, the DoF value per user is bounded above as:
(3) 
Remark: Note that the outer bound holds for arbitrary values of without any spatial normalization. However, also note that the outer bound does scale with spatial dimension, e.g., if we double the number of antennas at each node the value DoF would be doubled as well. In other words, the outer bound holds both with and without spatial normalization.
Remark: Unlike all previously used DoF outer bounds for MIMO interference channels, this outer bound does not follow from allowing cooperation among groups of users. Instead, what is needed is a careful choice of genie signals that provide a receiver access to parts of the signal space originating at interfering transmitters. Precisely which parts of a signal space can be provided as side information to produce the tight DoF outer bounds, is perhaps the most challenging technical aspect that we deal with in this paper. As such, the outer bounds represent the most significant contribution and a majority of this paper is devoted to their derivation.
Lemma 2 (Inner Bound)
For the 3user MIMO interference channel, the DoF per user value is achievable with linear beamforming over constant channels without the need for symbol extensions in time, frequency or space.
Remark: Note that the achievability result stated in Lemma 2 is limited to integer values of DoF. Typically, the issue of achievability for noninteger values of DoF is resolved by using symbol extensions over time or frequency dimensions, often with the need for timevarying/frequencyselective channels to create sufficient diversity. Time/frequency extensions can be used here as well, as will be discussed in Section 8.3. However, since our primary interest is in generic channels rather than the structured (blockdiagonal) channels that result from time/frequency extensions, we will focus on spatial extensions instead.
Theorem 1 (SpatiallyNormalized DoF)
For the 3user MIMO interference channel, the spatiallynormalized degrees of freedom value per user is given by:
(4) 
Proof: The proof of Theorem 1 follows directly from Lemma 1 and Lemma 2. Clearly, whenever DoF is an integer, we have an exact DoF characterization, DoF = DoF, without the need for any spatial extensions. For the cases where DoF is not an integer, let us express it in its rational form . Then, scaling the number of antennas by , we have a 3 user MIMO interference channel, for which the value DoF is both achievable and an outer bound, i.e., it is optimal. Since the normalized DoF outer bound is not affected by spatial scaling, no other spatial extension can improve the spatiallynormalized DoF, and we have the result of Theorem 1.
To understand the result of Theorem 1, the following equivalent, but more explicit representation of will be useful:
(5) 
or in compact form:
(6) 
While Theorem 1 allows spatial extensions to avoid dealing with structured channels resulting from symbol extensions in time/frequency, our results are not limited to settings with spatial extensions. As we illustrate through numerous examples, the achievable schemes translate directly from spatial extensions to channel extensions over time/frequency dimensions instead. With few exceptions, these channel extensions do not require channel variations in time/frequency. A sample of our DoF results, without spatial extensions appears in Fig. 2.
Lastly, we consider the most restricted setting where only linear alignment schemes are allowed, and no channel extensions are allowed in time/frequency/space. The next theorem settles the issue of feasibility of linear interference alignment for the 3 user MIMO interference channel.
Theorem 2 (Feasibility of Linear Interference Alignment)
For the 3user MIMO interference channel, the DoF demand per user, , is feasible with linear interference alignment if and only if .
Remark: Note that the feasibility of linear interference alignment is intended here in the same sense as studied previously by Cenk et al. in [12], Bresler et al. in [13] and Rezaviyayn et al. in [14], i.e., without symbol extensions in time/frequency/space.
Proof: The proof of Theorem 2 also follows directly from Lemma 1 and Lemma 2. Since the feasibility of linear interference alignment only concerns integer values of DoF per user, where the inner and outer bounds are tight, and the inner bound is achievable with linear interference alignment without the need for symbol extensions in time, frequency or space, the result of Theorem 2 follows immediately, and will be presented in Section 8.4 in detail.
5 Understanding the Result
In this section we provide an intuitive understanding of the main results based on linear dimension counting arguments, specifically by introducing the notion of subspace alignment chains, and highlight the key observations that follow from the main results.
5.1 The Simple Cases: and
If , then the DoF are already known, corresponding to the halfthecake value reported in [7].
If , i.e., , then interference alignment is not required. Consider first the setting , i.e., . Here, if then each receiver has enough antennas to zero force all interference at no cost to desired signals, and if then each transmitter has enough antennas to zero force all unintended receivers. Therefore, in this case every user achieves all of the cake, i.e., his interferencefree DoF, .
Next, consider the case , i.e., . Allowing any two users to cooperate, and using the DoF result for the resulting 2 user MIMO interference channel, we find the total DoF outer bound value of . This value is easily seen to be achievable with only zero forcing at the receivers () or at the transmitters (), e.g., with users 1 and 2 achieving DoF each and the third user achieving DoF. Note that the DoF assigned to each user can always be made equal by equal timesharing between all permutations of the users for a given DoF allocation.
For the remaining cases, i.e., , it turns out we need interference alignment. The challenge lies not only in constructing interference alignment schemes for this setting, but also, and to a greater extent, in finding the required DoF outer bounds. A new notion that emerges in this setting and that plays a central role in limiting the DoF values, is the notion of subspace alignment chains, which is introduced next.
5.2 Subspace Alignment Chains
The main DoF outer bound, presented in Lemma 1, consists of two outer bounds.
DoF  (7)  
DoF  (8) 
We will refer to these bounds as the bound and the bound, respectively. The parameter that appears in both bounds, is the length of the subspace alignment chain, a notion to be introduced in this section. Note that the first outer bound value is limited by and is an increasing function of while the second outer bound value is limited by and is a decreasing function of . The significance of this will become clear in the following description.
5.2.1 The bound: DoF
Consider the first outer bound, DoF . Since linear dimension counting arguments are identical for reciprocal networks [23], without loss of generality let us focus on the setting , so that . Now, since each receiver has at least as many antennas as any transmitter, zeroforcing of signals by the transmitters is not possible. Since interference cannot be eliminated, the next best thing is to align interference. Ideally, since each transmitter causes interference at two receivers, it should align with another interference vector at each of those undesired receivers. For example, let us consider a vector sent by Transmitter 1 that causes interference at Receiver 2. This vector should align with a vector sent by Transmitter 3, which is also undesired at Receiver 2. Now, the vector also causes interference at Receiver 1, so it should align there with a vector sent by Transmitter 2. The vector in turn also causes interference to Receiver 3, so it should align with a vector sent from Transmitter 1. Continuing like this, we create a chain of desired alignments:
The main question is whether this ideal scenario is possible, i.e., can we extend this subspace alignment chain indefinitely? Consider, for example the setting that is previously solved by Cadambe and Jafar in [7]. Cadambe and Jafar create this infinite chain of alignments using the asymptotic alignment scheme for and implicitly create an infinite alignment chain in the nonasymptotic solution for, e.g., , as the chain closes upon itself to form a loop, i.e., . The chain closes upon itself mainly because in this setting (as well as all cases where ) the optimal signal vectors are eigenvectors of the cumulative channel encountered in traversing the alignment chain starting from any transmitter and continuing until we return to the same transmitter. Thus, as shown by Cadambe and Jafar [7], the ideal solution of perfect alignment, achieved by an infinite (or closedloop) alignment chain, is possible when .
When , it turns out that the subspace alignment chain can neither be continued indefinitely, nor be made to close upon itself. The length of the subspace alignment chain is therefore limited to a finite value that is a function of and . The limited length of the subspace alignment chain creates the bottleneck on the extent to which interference can be aligned, and is ultimately the main factor limiting the DoF value.
Consider the setting , as shown in Fig. 3LABEL:, from a linear dimension counting perspective. Users 1, 2, and 3 are shown in green, yellow and blue in the figure. Let us start our subspace alignment chain with a signal space at Transmitter 1, shown in the figure as a black oval with the number 1. Consider the alignment that can occur at Receiver 3 with a corresponding signal space (marked with the number 2 to indicate the second signal in the alignment chain) originating at Transmitter 2. Note that interference can be aligned at each receiver only within that subspace which is accessible from both interfering transmitters. Since each transmitter can access only a dimensional subspace of the dimensional signal space available to Receiver 3, and because generic subspaces do not overlap any more than they have to, it follows that the size of the signal space accessible from both Transmitters 1 and 2, where interference alignment can take place, is no more than dimensional. Since , we note that is a positive number, i.e., such a space exists. However, the accounting for aligned dimensions does not stop here. Let us continue the alignment chain to see if further alignment is possible. The dimensional signal space 2 originating at Transmitter 2, can align with a corresponding signal space from Transmitter 3 at Receiver 1, in no more than dimensions. However, since , we note that is not a positive number, i.e., such a space does not exist. Thus, the alignment chain must stop here, and the maximum length of the alignment chain is , representing the number of transmitted signal spaces that participate in the alignment chain. Now, to find corresponding DoF outer bound, let us account for the signal space dimensions. Fig. 3LABEL: may be helpful here. Assuming each transmitted signal space is dimensional, the total number of dimensions transmitted is , and the number of dimensions occupied by interference at all three receivers is . Since the desired signal spaces must remain distinct from interference, the sum of desired signal dimensions at all receivers is . Thus, the total number of receive dimensions needed to accommodate both the desired signals and the interference is at least . Since the total number of receive dimensions available is we must have , i.e., . Since the total number of transmitted dimensions per user is , we have the corresponding outer bound value, DoF per user, which corresponds to , as expected.
As the next example, consider the setting , shown in Fig. 3LABEL:. Again, we start our alignment chain with the subspace numbered at Transmitter 1. As seen from Receiver 3, this space can align with a corresponding signal space (numbered 2) originating at Transmitter 2, in no more than dimensions. Continuing the chain, the next alignment must occur at Receiver 1, where the dimensional space (numbered 2) originating at Transmitter 2, can align with a corresponding signal space (numbered 3) originating at Transmitter 3, in no more than dimensions, which is also a positive number since , i.e., such a space exists. At this point the alignment chain has length 3. Continuing it further we note that the next alignment must occur at Receiver 2, where the dimensional space (numbered 3) originating at Transmitter 3 can align with a corresponding signal space (numbered 4) originating at Transmitter 1, in no more than dimensions, which is also a positive number since , i.e., such a space exists. Now the length of the alignment chain is 4. In order to continue the alignment chain further, we next consider Receiver , where the dimensional space (numbered 4) originating at Transmitter 1 can align with a corresponding space originating at Transmitter 2 in no more than dimensions, which is not a positive number since , i.e., such a space does not exist. Thus, the alignment chain ends here, and the maximum length of the alignment chain is for this example. Now, lets compute the implied DoF outer bound. Assuming each transmitted signal space is dimensional, the total number of dimensions transmitted is , and the number of dimensions occupied by interference at all three receivers is . Including the desired signal dimensions which must remain distinct from interference, the total number of receive dimensions needed is at least . Since the total number of receive dimensions available is we must have , i.e., . Since the total number of transmitted dimensions per user is , we have the corresponding outer bound, DoF per user, which again corresponds to , as expected.
These examples can be generalized in a straightforward manner, to verify that for the setting , the length of the subspace alignment chain , which can also be expressed as . The corresponding outer bound value is calculated as follows. A total of dimensions are transmitted, creating interference spaces of total dimension . The desired signal and interference together need a total of dimensions. Since only dimensions are available, we must have . Now, since the number of dimensions transmitted per user is , we have the bound, DoF .
We make an interesting comparison between the subspace alignment chains introduced above and the work of Bresler and Tse in [8]. Bresler and Tse considered linear interference alignment problem for a threeuser SISO Gaussian interference channel with timevarying/frequencyselective channel coefficients when the maximum diversity order that the channel can provide, e.g., the number of subcarriers, is limited to . They show that the maximum DoF achievable per user through linear interference alignment is a strictly increasing function of that monotonically converges to the information theoretic outer bound value of 1/2 as approaches infinity. Finite length chains of aligned vectors appear in the derivation of Bresler and Tse’s result as well. However, in spite of this superficial similarity, the alignment chains used by Bresler and Tse are fundamentally different from the subspace alignment chains in this work. First, the alignment chains of Bresler and Tse identify the limitations of DoF achievable through linear interference alignment schemes, but it is known that these limitations are surpassed by rational alignment schemes [6] that can achieve the information theoretic outer bound value of without the need for any channel variations. On the other hand, the subspace alignment chains presented in this work are much more fundamental in that they identify information theoretic outer bounds, i.e., these bounds cannot be beaten by linear alignment, rational alignment or any other scheme to be invented in the future. Second, the alignment chains in Bresler and Tse’s work reach their maximum length when it becomes impossible to separate the desired signal from interference. Alignment of interference, per se, is not a challenge in their setting. On the other hand, the subspace alignment chains discussed above reach their maximum length when it becomes impossible to align interference any further. Keeping the desired signal separate from interference is not the main concern in our setting.
5.2.2 The bound: DoF
Here we explain the second outer bound, DoF , from linear dimension counting arguments. Since linear dimension counting arguments are identical for reciprocal networks [23], without loss of generality let us focus on the setting , so that . Now, since each transmitter has more antennas than any receiver, zeroforcing of signals by the transmitters is possible. Ideally, we would like to zero force all interference. However, since , it is not possible for any transmitter to simultaneously zeroforce its transmitted signal to both unintended receivers. The next best thing is to zeroforce interference to the extent possible, and then align the remaining interference that cannot be zeroforced. This observation gives rise to a slightly different type of subspace alignment chains. The two ends of the alignment chain correspond to transmitted signals that are zero forced at one unintended receiver and cause interference at the other unintended receiver. These two nonzeroforceable interference terms are connected through a subspace alignment chain to alleviate the impact of the nonzeroforceable interference. Since zeroforcing is preferable to interference alignment, smaller interference alignment chains are preferable. Therefore, in this case, we will be limited by how quickly the semizeroforced signals can be connected through a subspace alignment chain. Therefore, this perspective gives rise to an outer bound which is a decreasing function of the length of the subspace alignment chain.
Consider the setting , illustrated in Fig. 4LABEL:. We start the alignment chain at Transmitter 1, where the number of transmitted dimensions that can be zeroforced at Receiver 2 is no more than . Since , these dimensions cannot be simultaneously zeroforced at Receiver 3. This creates the nonzeroforceable interference at Receiver 3 which initiates the subspace alignment chain. Recall that we would like the alignment chain to be as short as possible. This means that ideally, we would like the next signal in the chain (marked as number 2 in the figure), which must originate at Transmitter 2, to simultaneously accomplish the following objectives:

Signal space 2 should align with preexisting interference at Receiver 3.

Signal space 2 should be zeroforced at Receiver 1.
Note that if it is possible to accomplish these objectives, signal would not increase the interference space at any undesired receiver, which is why this is the preferred goal in choosing signal space 2. Let us see if this is possible. At Receiver 3, interference already spans dimensions, leaving an interference free space of dimensions. For signal space 2 to align with preexisting interference at Receiver 3, Transmitter 2 must zeroforce these interferencefree dimensions for Receiver 3. In addition, if signal space 2 is to be zeroforced at Receiver 1, then another dimensions must be zero forced. With generic signal spaces, the total number of dimensions to be zero forced is . Since Transmitter 2 has only antennas, it leaves dimensions within which both objectives can be accomplished. Since , we note that , i.e., such a space exists and it is possible to terminate the alignment chain. The resulting alignment chain length is . Now, let us compute the implied DoF bound. With dimensions assigned to each transmitted signal space, the total number of transmitted dimensions is and the total number of interference dimensions is . Adding up the desired signal dimensions and the interference dimensions (because the two must not overlap) we need a total of dimensions at the three receivers. The total number of dimensions at the three receivers is , which gives us the outer bound , or . Since the number of transmitted dimensions per user is we have the outer bound, DoF per user, which coincides with , as expected.
Next, let us consider the setting , illustrated in Fig. 4LABEL:. Once again, we start the alignment chain at Transmitter 1 and continue up to signal space 2. However, note that this time, because , there does not exist a signal space 2 which can be simultaneously aligned with interference at Receiver 3 and be zeroforced at Receiver 1. Thus, the alignment chain cannot be terminated at length 2. The next best thing is to only achieve interference alignment with signal space 2 and extend the subspace alignment chain. Recall that in order to align interference at Receiver 3, Transmitter 2 must zero force the dimensional nullspace of interference at Receiver 3, leaving exactly dimensions at Transmitter 2 that satisfy the interference alignment requirement. Since , note that the aligned dimensions cannot be zero forced at Receiver 1. Thus, the corresponding interference space at Receiver 1 is dimensional. Now, we extend the subspace alignment chain to signal space 3, which originates at Transmitter 3 (see Fig. 4LABEL:), and again we check if the alignment chain can be terminated. For the alignment chain to end with signal space 3, it must simultaneously satisfy the following two objectives:

Signal space 3 should align with preexisting interference at Receiver 1.

Signal space 3 should be zeroforced at Receiver 2.
Let us see if this is possible. At Receiver 1, the space accessible by interference produced so far occupies no more than dimensions. Equivalently, an interferencefree space of at least dimensions must be maintained at Receiver 1. For signal space 3 to align with preexisting interference at Receiver 1, Transmitter 3 must avoid the interferencefree dimensions available to Receiver 1, i.e., it needs to zeroforce these interferencefree dimensions. In addition, if signal space 3 is to be zeroforced at Receiver 2, then another dimensions must be zero forced. With generic signal spaces, the total number of dimensions to be zero forced is . Since Transmitter 3 has only antennas, it leaves dimensions within which both objectives can be accomplished. Since , , i.e., such a space exists and it is possible to terminate the alignment chain. The resulting alignment chain length is . Now, let us compute the implied DoF bound. With dimensions assigned to each transmitted signal space, the total number of transmitted dimensions is and the total number of interference dimensions is . Adding up the desired signal dimensions and the interference dimensions (because the two must not overlap) we need a total of dimensions at the three receivers. The total number of dimensions at the three receivers is , which gives us the outer bound , or . Since the number of transmitted dimensions per user is we have the outer bound, DoF per user, which coincides with , as expected.
Continuing along the same lines, these examples can also be generalized in a straightforward manner to verify that for the setting , the length of the subspace alignment chain , which can also be expressed as . The corresponding outer bound value is calculated as follows. A total of dimensions are transmitted, creating interference spaces of total dimension . The desired signal and interference together need a total of dimensions at the receivers. Since only dimensions are available at the receivers, we must have . Now, since the number of dimensions transmitted per user is , we have the outer bound, DoF .
We conclude this discussion with two plots of the DoF characterization presented in Theorem 1, shown in Fig. 5 and Fig. 6. Both figures are plotted as a function of the ratio . Clearly, as become less disparate, increases, and the length of the subspace alignment chain, increases as well, approaching infinity as . While the two figures are equivalent, the normalization with respect to in Fig. 5 and the normalization with respect to in Fig. 6 highlight the role of interferencealignment and zeroforcing, respectively. As increases from 0 to 1, subspace alignment chains become longer, a desirable outcome for interference alignment, which is reflected in Fig. 5. On the other hand, as increases from 0 to 1, alignment chains become longer, an undesirable outcome for zero forcing, which is reflected in Fig. 6. The interplay between the two bounds is evident in the piecewise analytical nature of the DoF function, with either or being the bottleneck within each analytical segment. A number of interesting observations can be made from the DoF result. These observations are the topic of the remainder of this section.
5.3 Other Key Observations
For these observations, we will refer to the characterization in Theorem 1 and its depiction as the piecewise linear curve in Fig. 5.
5.3.1 Redundant Dimensions
A particularly interesting observation from (6) and Fig. 5 is that within each piecewise linear interval, the value depends only on either or . This makes the other parameter somewhat redundant. In other words, within each piecewise linear section of the curve, either the number of dimensions at the transmitter or receiver can be increased/decreased without changing the value. For example, consider the user MIMO interference channel. This channel lies in the range , where the value depends only on . Therefore, there is some redundancy in . Since the redundant dimensions are fractional, they are much more explicitly seen in a larger space. So let us consider the user MIMO interference channel, which is simply a spatially scaled version of the original setting. Now, we know that the setting has exactly DoF (no spatial scaling required, since this is an integer value). However, note that the setting also has only DoF. Incidentally, the setting achieves the DoF with only linear beamforming based interference alignment, i.e., without the need for symbol extensions in time/frequency/space. Therefore, clearly, the receive antenna is redundant from a DoF perspective.
The previous example illustrates the situation for values that fall strictly inside the piecewise linear intervals. Now let us consider values that fall at the boundary points of the piecewise linear intervals. For example, consider the user MIMO interference channel, which corresponds to the value , i.e., a corner point, and has DoF value . However, note that the user MIMO interference channel also has DoF. Thus, the transmit antenna is redundant from a DoF perspective. Alternatively, consider the user MIMO interference channel, which also has DoF. Thus, evidently, the receive antenna is redundant in the setting. Thus, one can lose either a transmit antenna or a receive antenna (but not both) without losing DoF in the user MIMO interference channel. This is because the setting, which corresponds to , sits at the boundary of two piecewise linear segments where the redundancies in and the redundancies in meet. Therefore it contains both redundancies. The same observation is true for corner points .
Now consider the other set of corner values . As it turns out, these are the only values where neither the transmitter, nor the receiver has any redundant dimensions. We summarize the observations below:

For , the value of is the bottleneck, but the value of includes redundant dimensions that can be sacrificed without losing DoF.

For , the value of is the bottleneck, but the value of includes redundant dimensions that can be sacrificed without losing DoF.

For }, and only for these values, both and include redundant dimensions, either of which can be sacrificed without losing DoF.

For }, and only for these values, neither nor contains any redundant dimensions, i.e., reducing either will lead to loss of DoF.
Thus, the sets and represent maximally and minimally redundant settings. The redundancy, and the lack thereof, has interesting implications. For instance, the set corresponds to maximal redundancy and is also the precise set of values for which no joint processing is needed among the colocated antennas at any transmitter or receiver. On the other hand the set corresponds to noredundancy, and is also the precise set of values for which all proper systems are feasible from a linear interference alignment perspective. Next we elaborate upon these observations.
5.3.2 The DoF Benefit of MIMO Processing
In the user MIMO interference channel setting with , i.e., equal number of antennas at every node, Cadambe and Jafar have shown [7] that there is no DoF benefit of joint processing among multiple antennas, because the network has DoF even if each user is split into users, each with a single transmit and single receive antenna and with only independent messages originating at each transmitter. For the user MIMO interference channel setting, Ghasemi et al. have shown in [10] that DoF are achievable even if transmitter is split into singleantenna transmitters, each receiver is split into single antenna receivers, and there are no common messages. Note that while in general this achievable DoF value is not optimal, Ghasemi et al. have shown that is the optimal DoF value per user for the user MIMO interference channel when the number of users . Here we make related observations for our setting.

The user MIMO interference channel has DoF whenever , for some . Note that this statement is for any number of users (greater than 2), and not just for 3 users. Thus, we have a DoF characterization for any number of users whenever . The outer bound holds for any number of users because the DoF per user cannot increase with the number of users. The achievability is already established for any number of users. Since the two match, we know the DoF for any number of users. The result significantly strengthens the previous DoF characterization by Ghasemi et al. in [10]. As an example, consider the setting. Ghasemi et al. show that the this setting has DoF per user if the number of users is or higher. However, our result shows that this setting has DoF if the number of users is or higher. The improvement is even more stark for larger values of . For example, Ghasemi et al. show that the setting has exactly DoF per user if the number of users is 19 or higher, whereas our result shows that this setting has exactly DoF if the number of users is or higher.

There are no DoF benefits of MIMO processing whenever , for . This can be seen as follows. For our result illustrated in Figure 5, the piecewise linear curve is bounded below by the smooth curve shown with a light solid line plotting the value . Since , note that corresponds to DoF, i.e., the achievable DoF without any joint processing across multiple antennas, or the DoF achievable with independent coding/decoding at each transmit/receive antenna. Since the curve touches the curve whenever , there is no DoF benefit of MIMO processing in these settings.

Conversely, MIMO processing has DoF benefits whenever , for some . This is because for all these values, the curve is strictly above the value.

MIMO processing can enable linear achievable schemes when otherwise asymptotic alignment would be needed. We illustrate this with an example. Consider the 3 user interference channel (where ), which we now know, has DoF per user. We also know from the work of Ghasemi et al. that this DoF value can be achieved without any MIMO processing, but it requires the [CJ08] asymptotic interference alignment scheme [24] either in the rational dimensions framework or over timevarying channels. On the other hand, as we will show in this work, the DoF per user can be achieved purely through linear interference alignment based on beamforming without requiring any symbol extensions in space, time or frequency, by exploiting the MIMO benefit of joint processing among antennas located on the same node.
5.4 Infeasibility of Proper Systems
Our DoF results settle the issue of feasibility/infeasibility of linear interference alignment for the user MIMO interference channel. Prior work on this topic is summarized in Section 2. As stated earlier, it is known that improper systems are feasible, and also under certain conditions proper systems are feasible. The feasibility of proper systems is believed to be much more widely true. Somewhat surprisingly, our results show that for the user MIMO interference channel, the relationship between proper systems and feasibility of linear interference alignment is very weak, in the sense that most proper systems are infeasible.
Specializing the characterization of proper systems by Cenk et al. [12] to the user MIMO interference channel, the system is proper if and only if the desired DoF per user
(9) 
We will say that a system is strictly proper if we have equality in (9). In Figure 5, the curve is bounded above by a solid red line that plots the value . Note that this line only starts from because for (denoted as the dashed red line) the DoF per user is bounded above by the single user bound. This solid red line differentiates proper systems from improper systems. All systems above the curve are improper while those below the curve are proper. However, the DoF outer bound lies strictly below the curve except when for . Therefore, many proper systems and most strictly proper systems are infeasible. Remarkably, these systems are not only infeasible in terms of the restricted goal of achieving the desired DoF values through interference alignment with linear beamforming and without relying on channel extensions in time/frequency, but also they are infeasible in terms of the much more relaxed goal of achieving the desired DoF through any possible achievable scheme, linear or nonlinear, using interferencealignment or otherwise, using timevarying or constant channels, and using scalar or vector coding. This is because the solid blue curve is the informationtheoretic DoF outer bound.
Let us make this observation explicit with a few examples. Consider the setting (where ), with desired DoF value per user. This is a strictly proper system because . But from Lemma 1 we know that the information theoretic DoF per user for this channel is bounded above by . Therefore is clearly infeasible. Similarly, for any value of , i.e., that is not of the form , one can create an infeasible proper system. For example, suppose . Equivalently, . Choosing , we arrive at the desired DoF value per user which would make the system strictly proper. However, we know from Lemma 1 that the information theoretic DoF (per user) outer bound for is . Thus, once again the strictly proper system is infeasible. As yet another example consider the choice , i.e., . Choose the setting for which the informationtheoretic DoF per user is bounded above by . Therefore, if the desired DoF value per user is , the system is strictly proper and infeasible, if the desired DoF value per user is 86, the system is proper and infeasible. Proceeding similarly, we arrive at the following conclusions.

For every value of except , we can find proper systems that are infeasible, not only in terms of linear interference alignment, but also information theoretically infeasible.

For the values , proper systems are always feasible, i.e., the DoF demand per user, which is also the information theoretic outer bound, is actually achievable with only linear interference alignment, without the need for symbol extensions in time/frequency/space.
These observations, especially the infeasibility of proper systems for , can be understood in terms of the redundant dimensions explained in the previous section. Recall that Cenk et al. [12] make the distinction between proper/improper systems based on the number of variables involved in the system of polynomial equations. From the observations on redundant dimensions, we know that except for , every other setting contains redundant dimensions either in or . Evidently these redundant dimensions contribute superfluous variables which inflate the variable count, thereby qualifying a system as proper even when it is not feasible. We suspect that this observation may have significant implications in algebraic geometry where the solvability of systems of polynomial equations remains an unsolved problem.
5.5 Degrees of Freedom without Spatial Normalization
From our results it is clear that for all settings where DoF takes an integer value, we have a precise characterization of DoF (without the need for spatial normalizations). Precise DoF characterizations are also available for settings where for some , because in all these settings the outer bound of coincides with the achievability result of Ghasemi et al. [10], and the DoF value is . For the remaining cases, we propose a linear beamforming construction for achieving the DoF outer bound without relying on spatial extensions. Because the DoF outer bound is a fractional value, symbol extensions over time/frequency are needed to make the DoF value a whole number over the extended channel. Interestingly, the construction is always nonasymptotic, as in, the number of symbol extensions needed is only enough to make the DoF value an integer. We show analytically that while in many cases symbol extensions in time over constant channels are sufficient to achieve the information theoretic DoF outer bound, there are also cases where timevariations/frequencyselectivity of the channel is needed to achieve the DoF outer bound with linear interference alignment schemes. The feasibility of interference alignment can be settled in every case through a simple numerical test. We carry out this test to establish the DoF values for all values upto . In general, we end with the conjecture that in all cases, the DoF outer bound value is tight, even with constant channels, and may be achieved with nonlinear interference alignment schemes, e.g., exploiting the rational dimensions framework or the Renyi information dimensions framework.
6 DoF Outer Bounds: Preliminaries
Recall from the discussion on redundant dimensions, that the settings corresponding to are the ones that contain the most redundant dimensions, i.e., for these settings, and only for these settings, it is possible to reduce either or without losing DoF. It follows then, that in order to prove the strongest DoF outer bounds, it is these settings that must be considered. Indeed, as we will see in the informationtheoretic derivations of the DoF outer bounds, essentially the outer bounds must be shown for cases corresponding to and then with only a little additional effort, these outer bounds can be extended to all values.
A key step for the informationtheoretic DoF outer bound proof is to first perform a change of basis operation, corresponding to invertible linear transformations at both the transmitters and receivers. While invertible linear transformations at the transmitters and/or receivers do not affect the DoF, the change of basis identifies the subspace alignments which in turn helps identify the side information to be provided by a genie for the DoF outer bound. Next, we will present the invertible linear transformations for the case of . Since the change of basis operations are linear transformations, they are also directly applicable to the reciprocal channel according to the dual nature of reciprocal channels [23].
6.1 Change of Basis for
We begin with the simplest case, i.e., the MIMO interference channel. The linear transformations are carried out by multiplying an invertible square matrix at each transmitter and each receiver. The procedure of designing these transformation matrices is illustrated as follows. For brevity, we use to denote the invertible square matrix at Transmitter , and to denote the transformation matrix at Receiver .
Step 1: Consider the square matrix which has three rows. First, we determine its first row and the last row, corresponding to the first and the third antennas at Receiver . The linear transformation is designed in such a manner that the first antenna of Receiver does not hear Transmitter and the third antenna of Receiver does not hear Transmitter . This is illustrated in Fig.7(a). This operation is guaranteed by the fact that and are both matrices, and the left null space of each of them has one dimension. Therefore, the first row of lies in the left null space of , and the last row lies in the left null space of . That is,
(10)  
(11) 
If we choose the first entry of and the last entry of to be one, then their remaining entries can be solved through following equations.
(12)  
(13) 
Step 2: After the first and the third rows of are specified, we switch to the transmitter side and determine the two columns of , which correspond to the two antennas at Transmitter . Our goal is to ensure that the first antenna of Transmitter is not heard by the last antenna of Receiver but is heard by first antenna of Receiver , while the last antenna of Transmitter is not heard by first antenna of Receiver but is heard by the last antenna of Receiver . This is illustrated in Fig. 7(b). As a result, the first column of is chosen in the null space of the channel associated with the last antenna of Receiver (because the channels are generic, this is not in the null space of the first antenna of Receiver ). Similarly, the last column of is chosen in the null space of the channel associated with the first antenna of Receiver . Mathematically,
(14)  
(15) 
Since and are vectors, the null space of each of them has one dimension. Because the channel coefficients are generic, they are linearly independent almost surely. Therefore, if we choose the first entry of and last entry of to be one, then their remaining entries can be solved from following equations: