The Interference Channel Revisited: Eliminating Interference with a Two Antenna Receiver
Abstract
It is shown that a receiver equipped with two antennas may null an arbitrary large number of spatial directions to any desired accuracy, while maintaining the interferencefree signaltonoise ratio, by judiciously adjusting the distance between the antenna elements. The main theoretical result builds on ergodic theory. The practicality of the scheme in moderate signaltonoise systems is demonstrated for a scenario where each transmitter is equipped with a single antenna and each receiver has two receive chains where the desired spacing between antenna elements is achieved by selecting the appropriate antennas from a large linear antenna array. We then extend the proposed scheme to show that interference can be eliminated also in specular multipath channels as well as multipleinput multipleoutput interference channels. To demonstrate the performance of the scheme, we show significant gains at low and moderate signaltonoise ratios (020 dB) with 4 and 6 users scenarios. The robustness of the proposed technique to small direction estimation errors is also explored.
I Introduction
The informationtheoretic model of an interference channel is an abstraction that is motivated by the physical channel model of transmitterreceiver pairs that communicate over a shared wireless medium. While abstraction often leads to insights that may then be translated to more complicated reallife models, it is now recognized that the interference channel is an example that generalization also carries with it the risk of overabstraction, i.e., losing some key features of the true problem. Therefore, it is worthwhile to reexamine the problem formulation from time to time as has been demonstrated, e.g., in the case of magnetic recording channels; see e.g., [1] for an overview of the evolution of the physical models and its impact on the relevant informationtheoretic and coding techniques. Another example is the evolution that led to the V.90 voiceband modem [2, 3].
Indeed, works on interference alignment [4, 5, 6] reveal that a twouser model is nonrepresentative and further that the linear Gaussian model allows for elegant schemes that do not carry over to the general interference channel model. Nonetheless, interference alignment techniques have faced serious difficulties in translating the theoretical asymptotic gains to the operating conditions of communication systems.
In the present paper, we argue that taking a further step in bringing back into the model some simple considerations stemming from the physical propagation medium yields new insights on how near interferencefree transmission can be approached by an effective signal processing method, thereby resolving some of the drawbacks of interference alignment techniques. Specifically, the proposed method substitutes simultaneous alignment of interference by simultaneous near nulling. This difference is crucial as it eliminates the power penalty involved in interference alignment techniques and thus the associated gains are not restricted to the very high signaltonoise ratio (SNR) regime.
To convey the essence of the advocated approach, consider the standard interference nulling performed in multiantenna wireless communication. It is well known that given an adaptive array with receive antennas, one can null out (singleantenna) interferers and enjoy a full degreeoffreedom (DoF) for one (singleantenna) desired source. This leads to low utilization of the receive antennas, since only of the degrees of freedom convey useful information. The main advantages of receive beamforming are its ease of implementation and its robustness, since channel state information (CSI) feedback is not required.
Nevertheless, as mentioned, works on achievable rates for the interference channel [4, 5, 6] demonstrate that half of the degrees of freedom can be achieved, independent of the number of interferers, even when employing a single antenna at each node. While appealing from a theoretical point of view, interference alignment techniques face some major challenges in reallife applications; see, e.g., [7]. Beyond knowledge of full CSI of the complete interference network at all transmitters being required, the results are highly asymptotic. The SNR at which a tangible improvement over naive schemes is achieved is extremely high or requires very specific system configurations.
Most of the work on the interference channel concentrated on a simplified channel modelwhich assumes that the wireless channel is represented by an arbitrary matrix with random elements. However, as is recognized for many years in the communications theory literature, wireless channels are better represented as a combination of a small number of reflections with complex random coefficients caused by the small scale fading at the reflectors. Examples of such models include the well known Saleh and Valenzuela model [8], that is prevalent in recent applications of wireless communications (see e.g., [9] and the references therein), as well as raybased MIMO models [10, 11, 12, 13, 14]. These models are characterized by a finite (typically small) set of reflection clusters with well defined directionofdeparture (DoD) and directionofarrival angles (DoA) together with fading coefficients.
We consider the particular class of lineofsight (LOS) interference channels as well as specular multipath channels that are the basis for physical modeling of recent of wireless communication systems. We then extend the results to the MIMO interference channel.
Our main result is as follows: given a large spaced array with receive antennas and the possibility of selecting two antennas, one can approximately null out any number of sources in the plane, affording (with probability one) a full DoF to a single desired source. Accordingly, for a user interference channel with singleantenna transmitters, we achieve degrees of freedom, i.e., a utilzation factor of one half, similar to the best achievable DoF of interference alignment schemes. In contrast to the latter, the scheme only requires receiveside CSI. Moreover, it achieves substantial gain at practical values of SNR. The scheme can be implemented using a simple array and selection mechanism as depicted in Figure 1.
To gain insight into the proposed approach, consider a fouruser lineofsight interference channel where we focus on the receiver of user 1. The direction of the transmitters were with respect to user 1 array. Selecting antennas having a separation of , yields the beampattern depicted in Figure 2. The desired user’s gain is close to 2 which is the interferencefree gain, while the gains corresponding to the signals of all other users are almost completely suppressed. Theorem 1 proves that such a beam pattern is almost always achievable provided that the array is sufficiently large.
Ia Related Work
Handling interference efficiently is a major challenge in multiuser wireless communication. Recently, it has become clear that this challenge can sometimes be overcome via interference alignment [15, 16]. For instance, consider the user Gaussian interference channel, where transmitterreceiver pairs wish to communicate simultaneously. Through clever encoding strategies, the received signals can be aligned so that each receiver only observes its desired signal along with a single effective interferer. As a result, each user can achieve asymptotically roughly half the rate that would be available if there were no interference whatsoever; i.e., DoF are available. However, many schemes, such as the CadambeJafar framework [16] and ergodic interference alignment [17], require a large number of independent channel realizations to achieve nearperfect alignment and suffer from a significant SNR penalty due to channel inversion. In certain settings, this level of channel diversity may not be attainable; ideally, we would like to achieve alignment over a single channel realization.
The capacity region of the static Gaussian user interference channel [18] is unknown in general, although significant progress has been made recently, in part due to the discovery of interference alignment and the shift from exact capacity results to capacity approximations [19, 20]. Motahari et al. showed that DoF are achievable for almost all channel realizations [21] but thus far this result has not been translated into real gains outside of the very high SNR regime.
More recent work has concentrated on extending the basic zeroforcing approach suggested by Cadambe and Jafar [4] by obtaining more favorable channel conditioning through various methods; see, e.g., [22] and references therein. These are practical only for particular system configurations. In general, at present time, the gains of interference alignment have yet to be realized.
Apart from the obvious connection to works on the interference channel, the idea of altering the physical propagation channel bears some similarity to “mediabased modulation”, “spatial modulation” and “index modulation” schemes; see [23, 24, 25, 26] for an overview of these interrelated concepts. In all of these works, the physical medium is modulated based on the informationbearing signal. In contrast to these works, the present work only requires subsampling of the spatial channel at the receiver and can cope with an arbitrary number of interfering signals.
Ii The interference channel  revisited
We extend the standard interference channel model to a model where for each user the number of transmit/receive chains is not necessarily equal to the number of transmit/receive antennas, respectively. To that end, consider a Kuser interference channel where each transmitter has transmit chains and antennas and each receiver has receive chains and antennas; see Figure 1 depicting a link between one transmitter and one receiver. This configuration, where all transmitter/receiver pairs have the same parameters, is denoted as the symmetric interference channel. We note that this model is of significance for modern systems which utilize massive antenna arrays where the number of Tx/Rx chains differs from the number of antennas. We begin with the traditional case where and .
Consider an interference channel with transmitters and corresponding receivers. We assume for simplicity that all transmitters are equipped with the same number of antennas and all receivers are equipped with antennas. Denoting by , the channel matrix from transmitter to receiver , the received signal is given by
(1) 
where is i.i.d. (between users and over time) circularlysymmetric complex Gaussian noise.
Several variants of this problem have been addressed. For instance, the case of and real timevarying (which can be thought of as a diagonal matrix) coefficients has been studied in [4] where it was shown that for almost all channel coefficients, interference alignment attains half a DoF per user. A similar result was shown for scalar but timeinvariant channels in [5] through alignment on the signal scale using lattice codes.
Both of these approaches are very asymptotic in nature and require high resolution transmitside CSI as well as very high SNR conditions to start to play a beneficial role. Extensions to more general MIMO channels have subsequently revealed, e.g., [27, 28] that the DoF alignment gains are much more modest under more realistic assumptions. As a partial remedy, antenna switching has been proposed as a means for improving the channel coefficients to facilitate alignment [29].
We note that the interference channel is equivalent to requiring that for each user , the transmitter and receiver must employ linear frontend selection matrices , each having exactly and nonzero elements which are not in the same row or column, respectively. Applying selection matrices at both ends of the link of each user, (1) becomes
(2) 
From a practical perspective, implementing the selection mechanism yields a substantial reduction in hardware complexity.
Iii Lineofsight interference channels
In this section we describe a novel approach to the interference channel. The classical signal processing literature deals primarily with Nyquistresolution beamformers, where at least some antennas are separated by at most . In this case, the array has a single main lobe in the desired direction, and the resolution of the array is determined by the farthermost elements. This is because that when all distances between antennas are larger than , an ambiguous beam pattern occurs. An example of this phenomenon is depicted in Figure 2. Interestingly, an ambiguous beam pattern, can prove extremely advantageous when dealing with interference, since such patterns have multiple nulls. We will show that by judiciously designing the beam pattern, we can point multiple nulls at the multiple interferers simultaneously. In fact, with a highly undersampled array, any number of interferers in almost any set of directions can be suppressed. This follows from an ergodic theory argument.
In what follows we assume that and . The spacing is assumed to be and the receiver selects two out of antennas to be switched into the receive chains. We assume that the receiver has full directional CSI.
To develop the general framework for receiver antenna array design, we first introduce the lineofsight interference channel model. The use of highfrequency communication in general and mmwave and THz frequency communication has prompted recent interest in LOS communication channels [30, 31, 32]. Moreover, such channels form the basis for the more elaborate channel models described in Sections V and VI. Specifically, we make the following assumptions:

We assume a single transmit chain and transmit antenna per user. Hence is trivial for all .

We assume two receive chains per user, i.e. the matrices in (II) are reduced to LOS vectors .

The vectors consist of array manifold vectors which depend on the selection matrix, setting the spacing (in units of ) between the chosen antennas.

Let be the array response towards direction . Thus, the array response is given by
(3) 
We assume that each receiver has perfect CSI w.r.t. all channel gains corresponding to impinging signals.

Transmitters on the other hand need not have access to any CSI beyond the rate at which they should communicate with their respective receiver.

For simplicity, we assume a linear array and planar geometry where all sources are far field point sources.

Without loss of generality, we use the array manifold as the channel, since the signal attenuation can be absorbed in the power of the signal .

We assume that the locations of all transmitters and receivers are independently uniformly distributed in angle with respect to the origin.

We assume that the power of all transmitters is bounded by .
Note that by 9, the incidence angle of each received signal is uniformly distributed as well. Under this setting, it suffices to focus on the receiver of a single user as the operations at all receivers will be similar.
It follows that (1) becomes:
(4) 
Using a received beamforming vector , the received signal of the th user becomes:
(5) 
where
(6) 
Therefore,
(7) 
Straightforward algebraic simplification yields:
(8) 
In the next section we show that by properly selecting , we can obtain the following:
(9) 
where is Kronecker’s delta function.
Iv Eliminating Interference via Ergodic Nulling
We now show that by judiciously adjusting the distance between the receive antennas, we can (with probability 1) suppress all interferers to any desired level. Furthermore, we show that this can be achieved by a separation which is an integer multiple of the wavelength and hence, implementation via antenna selection applied to a linear array is possible.
Specifically, we demonstrate that for almost all angles of arrival, one can approach the interferencefree rate of any desired user. This is proved using the uniform distribution property of sequences modulo . Moreover, we show that without loss of generality, the beamforming vector can be chosen as .
The proof utilizes an integer antenna spacing (in terms of wavelength ). Hence, it can be translated to an implementation using antenna selection applied to an array of antennas with spacing.
Theorem 1 (Main Theorem).
Assume that the directions are such that are independent over . Then, for every , one can find a spacing such that applying receive beamforming with the vector yields:
(10) 
Before we prove the main theorem, note that it provides a full DoF per transmitter. This is the case since the gain in the desired direction can be made arbitrarily close to 1 while the total interference is suppressed to any desired level.
Proof.
To prove the main theorem, recall the following definition by Weyl (see [33]):
Definition 1.
A dimensional sequence of real vectors is uniformly distributed modulo if for every box
(11) 
Weyl [34] proved that whenever is a vector of irrational real numbers that are linearly independent over , the sequence is uniformly distributed modulo . In the present context, assume that are linearly independent over . Note that this holds with probability one. By Weyl’s theorem the sequence , , is uniformly distributed modulo 1. Define a box
(12) 
where
where Therefore, we can find a such that
(13)  
(14) 
By continuity of , as given in (8) , for a given , we can find an such that (10) is satisfied. ∎
V Multipath Channels and related interference channel models
We now show that the proposed approach generalizes to the case of multipath [35, 36] with a finite number of reflections.
We assume a physical channel model such that the carrier frequency is much larger than the signal bandwidth, which is typical in cellular and indoor wireless communications. We first show that in the case of a interference channel with intersymbol interference (ISI), we can approach the singleuser interferencefree rate.
Adhering to discrete time and allowing a different path loss for each reflection, the channel, as given in (II), now generalizes to
(15) 
for , where is the number of reflections of the ’th signal received by user , and is the complex path loss of the signal arriving from direction and is defined in (3). Following standard models, we can assume that is either Rician or Rayleigh, or even deterministic.
Let
(16) 
where for all :
and . As in the previous section, by invoking Weyl’s theorem, there exists a such that:
(17) 
for all and
(18) 
for all and . By continuity of , for a given , there exists an such that:
(19)  
(20) 
We conclude that one can suppress all specular multipath components of the interference signals to any desired level. Hence, the resulting received signal is given by:
(21) 
where
is composed of the receiver noise as well as the residual interference at receiver ,
Note that the power of the residual interference satisfies:
(22) 
By selecting sufficiently small, can be can be made arbitrarily small. Moreover, for all desired signal paths are (simultaneously) arbitrarily close to by a proper choice of . It follows that (21) amounts to a standard ISI channel, with coefficients arbitrarily close to the interferencefree ISI channel.
Vi Ergodic nulling for the MIMO interference channel
We now turn to analyze the MIMO interference channel where for simplicity we assume that the number of transmit and receive antenna elements as well as RF chains are the same for all transmitter and receiver pairs, i.e., of dimensions and .
Following the vast literature of physical spatial pointtopoint MIMO channel models, we note that the MIMO channel between the transmit antennas of user and the receive antennas of user can be described as
(23) 
where, are the DoD between transmit array and reflector and the DoA between receive array and reflection . Without loss of generality we also assume the are monotonically decreasing in . As is common in the MIMO literature, we assume that the scattering is sufficiently rich. In the present context, this requires that for all so that (almost surely) for all , we have . The following theorem holds:
Theorem 2.
Let be given and assume that , and . Further, assume that each receiver has directional CSI. Then, for any , there is a sufficiently large and a selection matrix , such for user any rate satisfying
(24) 
is achievable in the interference channel, where
and
Furthermore, if the transmitter has CSI, then the rate
is achievable where Q is a positive semidefinite matrix satisfying .
Proof.
Let the transmitter use an i.i.d. isoptropic Gaussian codebook of dimension and with power per dimension. Let denote the dimensional standard unit vectors. The receiver uses a selection matrix followed by a beamforming matrix where
Here, if and only if the antenna is selected such that the beamformer receive only direction (and approximately nulling all other directions, both from the desired user as well as from all others users). Recall that by Theorem 1, this is possible.
Thus, user obtains an equivalent MIMO channel
(25) 
where
and
is composed of the receiver noise as well as the residual interference at receiver . Note that the power of the residual interference can be made as small as desired.
By construction
and . Hence, can be made arbitrarily close to the channel .
Thus, an achievable rate for this channel is given by
The case of full CSIT follows by standard MIMO techniques. ∎
This should be compared against the isotropic transmission interferencefree benchmark of
For large , the two rates nearly coincide.
Example: Threeuser MIMO interference channel. Assuming a specular multipath model with at least two reflections for every desired user (at the respective receiver), we note that we can achieve a total of DoFs per user. This should be compared to the DoFs per user achieved (in the generic) MIMO interference channel [4, 27, 28].
Vii Optimizing the beamformer
We now discuss the practical implementation of the proposed method. While Theorems 1 and 2 guarantee that interference can be suppressed to any desired level, they do not exploit the full optimization parameter space on the one hand. On the other hand, in practice is fixed apriori. Ultimately, our goal is to maximize the signaltointerferenceplusnoise ratio by properly choosing and . To simplify notation, we focus on a single receiver since the design of all receivers is equivalent.
Explicitly, the desired solution for user is given by
(26) 
where are the receive power of the desired and interfering signals, and are the directions of the desired and interfering signals, respectively. While this equation is highly nonlinear, given the received signal and interference CSI, we need to optimize the receive array by enumerating over . This is a two dimensional search with a moderate complexity. Note that in the MIMO case, when we can further optimize the receive beamforming matrix, given the selection of the antennas using an MMSE criterion and allowing a full (combinatorial) search over the subsets of receive antennas.
Viii Simulations
To test the proposed ergodic interference nulling scheme we performed several simulations. In the first set of simulations, we tested the performance as a function of SNR and the sensitivity to directional errors. For SNR values ranging from dB to dB, we generated 100 LOS interference channels, with four users. We repeated the experiment for three values which is a reasonable number for practical massive MIMO scenarios. We tested the capacity of user with all transmitters randomly located at directions chosen between and degrees. All interferers were assumed to be received with the same power. As a benchmark for comparison we took nonnaive timedivision multiple access (TDMA), with two users transmitting per time slot assuming MMSE nulling of the undesired signal. We calculated the average achievable rate over all the channel realizations, optimized over using full search with resolution in and a uniform linear array. To test for robustness, we also tested the performance of a receiver suffering from i.i.d receiver directional errors with for respectively. The results are depicted in Figures 35. The interferencefree rate is clearly attained up to an SNR of roughly dB for , and even at dB when . The slowing of the growth of the attained rates of proposed nulling scheme is due to the limited size of the array. We can see that the scheme shows reasonable sensitivity to small errors in direction.
Finally, we have tested a 6user interference channel scenario, where . As expected, the gain over TDMA is smaller. This indicates that combining TDMA with ergodic nulling can be beneficial over pure ergodic nulling, in particular, at high SNR.
To test the dependence on we chose SNR dB, and computed the achievable rate as a function of . The results are depicted in figure 7. While attaining the interferencefree rate requires about separation, there is a very significant performance gain, compared to nonnaive TDMA, even at where the rate is higher.
Ix Discussion
In this paper we proposed a novel technique for dealing with the interference channel. The approach is based on judiciously setting the distance between the two receive antennas to attain a beamforming vector with approximate nulls in the direction of the interferers. The main theorem shows that we can achieve half the degrees of freedom afforded by the system. A significant advantage of the proposed approach compared to traditional interference alignment is the fact that we only require receiveside CSI. Thus, the scheme does not require CSI information at the transmitter beyond transmission rate. Since the scheme is applicable when there is a single transmitter per user, this allows us to operate in the regime where transmit ZF is impossible.
In practice, moving the antennas to set the desired separation may be difficult to implement. To overcome thi, one possibility is to use the standard approach taken in massive MIMO systems, where two antennas of a large array are switched into two receiver chains; see, e.g., [29].
In a practical implementation, it is preferable to limit the dimensions of the array. To that end, a receiver could divide the interferers into two groups, a small group of strong interferers for which approximate nulling is required, and a residual that is treated as noise. Moreover, from a system perspective, the users could be partitioned into orthogonal groups in which the number of strong interferers is limited.
Since the proposed approach is capable of suppressing any (finite) number of interferers, it is applicable also for non symmetric interference channels with configurations as long as all are large enough and for all .
Similarly, the results can be easily extended to configurations of the interference channel with as long as is sufficiently large and provided that the directional CSI is available at the transmitters.
Finally, we note that the advocated approach easily extends to the model of an interference multipleaccess channel. Namely, given receive chains, singleantenna users can be afforded a full DoF while suppressing an arbitrary number of interferers, thus yielding a DoF utilization factor of . Similar approach applies to the downlink.
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