On Feasibility of Interference Alignment in MIMO Interference Networks
We explore the feasibility of interference alignment in signal vector space – based only on beamforming – for MIMO interference channels. Our main contribution is to relate the feasibility issue to the problem of determining the solvability of a multivariate polynomial system, considered extensively in algebraic geometry. It is well known, e.g. from Bezout’s theorem, that generic polynomial systems are solvable if and only if the number of equations does not exceed the number of variables. Following this intuition, we classify signal space interference alignment problems as either proper or improper based on the number of equations and variables. Rigorous connections between feasible and proper systems are made through Bernshtein’s theorem for the case where each transmitter uses only one beamforming vector. The multi-beam case introduces dependencies among the coefficients of a polynomial system so that the system is no longer generic in the sense required by both theorems. In this case, we show that the connection between feasible and proper systems can be further strengthened (since the equivalency between feasible and proper systems does not always hold) by including standard information theoretic outer bounds in the feasibility analysis.
Degrees of freedom, interference alignment, interference channel, MIMO, Newton polytopes, mixed volume
The degrees of freedom (DoF) of wireless interference networks represent the number of interference-free signaling-dimensions in the network. In a network with transmitters and receivers and non-degenerate channel conditions, it is well known that non-interfering spatial signaling dimensions can be created if the transmitters or the receivers are able to jointly process their signals. Until recently it was believed that with distributed processing at transmitters and receivers, it is not possible to resolve these signaling dimensions so that only one degree of freedom is available. However, the discovery of a new idea called interference alignment has shown that the DoF of wireless interference networks can be much higher .
I-a Evolution of Interference Alignment
Interference alignment refers to the consolidation of multiple interfering signals into a small subspace at each receiver so that the number of interference-free dimensions remaining for the desired signal can be maximized. The idea evolved out of the DoF studies for the 2-user X channel [88, 113] and has since been applied to a variety of networks in increasingly sophisticated forms. The majority of interference alignment schemes proposed so far, fall into one of two categories – (1) signal space alignment and (2) signal level alignment.
I-A1 Interference Alignment in Signal Vector Space
The potential for overlapping interference spaces was first pointed out by Maddah-Ali et. al. in [123, 111] where iterative schemes were formulated for optimizing transmitters and receivers in conjunction with dirty paper coding/successive decoding schemes. The idea of interference alignment was crystallized in a report by Jafar  where the first explicit (closed form, non-iterative) and linear (no successive-decoding or dirty paper coding) interference alignment scheme in signal vector space was presented. The explicit linear approach introduced by Jafar in  was adopted by Maddah-Ali et. al. in their subsequent report and journal paper [112, 113], while  developed into the journal paper by Jafar and Shamai . Interference alignment was also independently discovered by Weingarten et. al.  in the context of the compound multiple input single output (MISO) broadcast channel (BC).
Following the early success on the X channel and the compound MISO BC, signal space interference alignment schemes were introduced for the interference channel with equal (unequal) number of antennas at all transmitters and receivers by Cadambe and Jafar (Gou and Jafar) in  (), for X networks with arbitrary number of users by Cadambe and Jafar in , for cellular networks by Suh and Tse in , for MIMO bidirectional relay networks (Y channel) by Lee and Lim in , for ergodic fading interference networks by Nazer et. al. in , and for interference networks with secrecy constraints in . Interference networks with constant channel coefficients posed a barrier for signal space interference alignment schemes because they did not provide distinct rotations of vector spaces on each link that were needed for linear interference alignment. The problem was circumvented to a certain extent for complex interference channels in , where phase rotations were exploited in a similar manner through the use of asymmetric complex signaling. However, for constant and real channel coefficients, these linear alignment schemes were not sufficient and a different class of alignment schemes based on structured (e.g. lattice) codes that align interference in signal scale were introduced.
I-A2 Interference Alignment in Signal Scale
The first interference alignment scheme in signal scale was introduced for the many-to-one interference channel by Bresler et. al. in  and for fully connected interference networks by Cadambe et. al. in . Unlike random codes for which decoding the sum of interfering signals is equivalent to decoding each of the interfering signals, these schemes rely on codewords with a lattice structure, which opens the possibility that the sum of interfering signals can be decoded even when the individual interfering signals are not decodable. This is because the sum of lattice points is another lattice point, and hence may be decoded as a valid codeword. Lattice based alignment schemes were further investigated for interference networks by Sridharan et. al. in [102, 103] and for networks with secrecy constraints by He and Yener in . An interesting interference alignment scheme in signal scale was introduced by Etkin and Ordentlich in . This work used fundamental results from diophantine approximation theory to prove that the rational and irrational scaled versions of a lattice “stood apart” from each other, and thus could be separated. The result was extended to almost all irrational numbers by Maddah-Ali et. al. in  by translating the notion of linear independence (exploited in linear interference alignment schemes) into the notion of rational independence in signal scale. With this new insight, the asymptotic alignment scheme of Cadambe and Jafar from  was essentially adopted in  to achieve interference alignment in signal scale and following the approach in , was shown to approach the DoF outer bound.
I-B The Feasibility Question - Examples
I-B1 Symmetric Systems
Let denote the MIMO interference network, where every transmitter has antennas, every receiver has antennas and each user wishes to achieve DoF. We call such a system a symmetric system. Consider the following examples.
- It is shown in  that for the 3-user interference network with 2 antennas at each node, each user can achieve 1 DoF by presenting a closed form solution for linear interference alignment, i.e., by linear beamforming at the transmitters and linear combining at the receivers. However, is there a way to analytically determine the feasibility of this system without finding a closed form solution?
- Consider the 4-user interference network with 5 antennas at each user and we wish to achieve 2 DoF per user for a total of 8 DoF. A theoretical solution to this problem is not known but numerical evidence in  clearly indicates that a linear interference alignment solution exists. Numerical algorithms are one way to determine the feasibility of linear interference alignment. However, is there a way to analytically determine the feasibility of alignment without running the numerical simulation?
I-B2 Asymmetric Systems
Let us introduce the notation to denote the MIMO interference network, where the transmitter and receiver have and antennas, respectively and the user demands DoF. We call such a system an asymmetric system. Consider the following examples.
Consider the simple system , which is clearly feasible (proper) because simple zero-forcing is enough for achievability. However, now consider the system, where the same total number of DoF is desired. Although these systems have the same number of total antennas, is the latter system still achievable?
Consider the 2-user interference network , where a total of 2 DoF is desired. The achievable scheme for this system is presented in . Now, consider the same scheme with increased number of users; that is, the 4-user interference network , where a total of 4 DoF is desired. Is this system still achievable, where DoF is doubled by simply going from two users to four users?
In this paper, we address all these questions. Our approach is to consider the signal space interference alignment problem as the solvability of a multivariate polynomial system, and then place it into perspective with classical results in algebraic geometry where these problems are extensively studied.
Ii-a System Model
We consider the same MIMO interference network as considered in . The received signal at the channel use can be written as follows:
. Here, are the received signal vector and the zero mean unit variance circularly symmetric additive white Gaussian noise vector (AWGN) at the receiver, respectively. is the signal vector transmitted from the transmitter and is the matrix of channel coefficients between the transmitter and the receiver. is the transmit power of the transmitter. Hereafter, we omit the channel use index for the sake of simplicity. The DoF for the user’s message is denoted by .
As defined earlier, denotes the symmetric MIMO interference network, where each transmitter and receiver has and antennas, respectively and each user demands DoF; therefore, the total DoF demand is . In general, let denote the asymmetric MIMO interference network, where the transmitter and receiver have and antennas, respectively and the user demands DoF. Some sample symmetric and asymmetric systems are shown in Fig. 1.
Ii-B Interference Alignment in Signal Space - Beamforming and Zero Forcing Formulation
In interference alignment precoding, the transmitted signal from the user is , where is a vector that denotes the independently encoded streams transmitted from the user. The precoding (beamforming) filters are designed to maximize the overlap of interference signal subspaces at each receiver while ensuring that the desired signal vectors at each receiver are linearly independent of the interference subspace. Therefore, each receiver can zero-force all the interference signals without zero-forcing any of the desired signals. The zero-forcing filters at the receiver are denoted by . In , it is shown that an interference alignment solution requires the simultaneous satisfiability of the following conditions:
where denotes the conjugate transpose operator. Very importantly,  explains how the condition (2) is automatically satisfied almost surely if the channel matrices do not have any special structure, and are designed to satisfy (1), which is independent of all direct channels . We assume that general MIMO channels have no structure and we force the transmit and receive filters to achieve the required ranks by design. Thus, (2) is automatically satisfied for us as well.
Iii Proper System
Based on classical results in algebraic geometry, like Bezout’s theorem, it is well known that a generic system of multivariate polynomial equations is solvable if and only if the number of equations does not exceed the number of variables. While the qualification “generic system of polynomials” is intended in a precise sense and limits the scope of settings where the result can be rigorously applied, the intuition behind this statement is believed to be much more widely true. This conventional wisdom forms the starting point for our work. By accurately accounting for the number of equations, , and the number of variables, , we classify a signal space interference alignment problem as either improper or proper, depending on whether or not the number of equations exceeds the number of variables.
Iii-a Counting the Total Number of Equations and Variables
We rewrite the conditions in (1) as follows:
where and are the transmit and receive beamforming vectors (columns of precoding and interference suppression filters, respectively).
is directly obtained from (3) as follows:
However, calculating the number of variables is less straightforward. In particular, we have to be careful not to count any superfluous variables that do not help with interference alignment.
At the transmitter, the number of transmit beamforming vectors to be designed is . Therefore, at first sight, it may seem that the precoding filter of the transmitter, , has variables. However, as we argue next, we can eliminate of these variables without loss of generality.
The linearly independent columns of transmit precoding matrix span the transmitted signal space
Thus, the columns of are the basis for the transmitted signal space. However, the basis representation is not unique for a given subspace. In particular, consider any full rank matrix . Then, continuing from the last step of the above equations,
Thus, post-multiplication of the transmit precoding matrix with any invertible matrix on the right does not change the transmitted signal subspace. Suppose that we choose to be the matrix that is obtained by deleting the bottom rows of . Then, we have , which is a matrix with the following structure:
where is the identity matrix and are vectors. It is easy to argue that there is no other basis representation for the transmitted signal space with fewer variables.
Therefore, by eliminating superfluous variables for the interference alignment problem, the number of variables to be designed for the precoding filter of the transmitter, , is . Likewise, the actual number of variables to be designed for the interference suppression filter of the receiver, , is . As a result, the total number of variables in the network to be designed is:
Iii-B Proper System Characterization
To formalize the definition of a proper system, we first introduce some notation. We use the notation to represent the equation
The set of variables involved in an equation is indicated by the function . Clearly
where is the cardinality of a set.
Using this notation, we denote the set of equations as follows:
This leads us to the formal definition of a proper system.
A system is proper if and only if
In other words, for all subsets of equations, the number of variables involved must be at least as large as the number of equations in that subset.
The condition to identify a proper system can be computationally cumbersome because we have to test all subsets of equations. However, several simplifications are possible in this regard. We start with symmetric systems.
Iii-C Symmetric Systems
For symmetric systems, simply comparing the total number of equations and the total number of variables suffices to determine whether the system is proper or improper.
A symmetric system is proper if and only if
Because of the symmetry, each equation involves the same number of variables and any deficiency in the number of variables shows up in the comparison of the total number of variables versus the total number of equations. Plugging in the values of and computed earlier, we have the result of Theorem 1. ∎
Consider the system. For this system, so that this system is proper.