Interactive Interference Alignment

Interactive Interference Alignment

Quan Geng, Sreeram Kannan, and Pramod Viswanath Coordinated Science Laboratory and Dept. of ECE
University of Illinois, Urbana-Champaign, IL 61801
Email: {geng5, kannan1, pramodv}@illinois.edu
Abstract

We study interference channels (IFC) where interaction among sources and destinations is enabled, e.g., both sources and destinations can talk to each other using full-duplex radios. The interaction can come in two ways: 1) In-band interaction: sources and destinations can transmit and listen in the same channel simultaneously, enabling interaction. 2) out-of-band interaction: destinations talk back to the sources on an out-of-band channel, possible from white-space channels. The flexibility afforded by interaction among sources and destinations allows for the derivation of interference alignment (IA) strategies that have desirable “engineering properties”: insensitivity to the rationality or irrationality of channel parameters, small block lengths and finite SNR operations. We show that for several classes of interference channels the interactive interference alignment scheme can achieve the optimal degrees of freedom. In particular, we show the first simple scheme (having finite block length, for channels having no diversity) for that can achieve the optimal degrees of freedom of even after accounting for the cost of interaction. We also give simulation results on the finite SNR performance of interactive alignment under some settings.

On the technical side, we show using a Gröbner basis argument that in a general network potentially utilizing cooperation and feedback, the optimal degrees of freedom under linear schemes of a fixed block length is the same for channel coefficients with probability . Furthermore, a numerical method to estimate this value is also presented. These tools have potentially wider utility in studying other wireless networks as well.

I Introduction

Due to the broadcast nature of wireless communication, an active transmitter will create interference to all other receivers using the same channel (in time and frequency). This is a basic challenge in supporting concurrent reliable wireless communications. Traditional methods have involved power control, scheduling and antenna beamforming/nulling to manage interference. Interference alignment is a novel interference management technique which makes concurrent wireless communications feasible [1, 2].

Since its introduction, substantial research has been conducted in understanding the gains of interference alignment. From a theoretical perspective, the focus has been mainly in understanding the degrees of freedom (first term in the high SNR approximation of the information theoretic capacity region). A seminal result, shown in [1], is that we can communicate at a total degrees of freedom for nearly all time-varying/frequency selective -user interference channel (this is also an upper bound). To get to degrees of freedom, the scheme employed (vector space interference alignment) requires the channel diversity to be unbounded; in fact, one needs the channel diversity to grow like [3]. The diversity order required is huge even for modest number of users (such as 3 and 4). This is definitely a huge impediment in a practical communication system, where there is hardly enough channel diversity to make such a vector space interference alignment scheme feasible.

A practically relevant problem is to understand the fundamental degrees of freedom for a fixed deterministic channel. For fully connected channel matrix , the total degrees of freedom are upper bounded by and [4] shows that this upper bound is achievable for almost all channel matrices using a coding scheme based on Diophantine approximation. However, this result is limited in two ways. First, the coding scheme is very sensitive to whether entries of are rational or irrational. Second, although it is provable degrees of freedom are achievable for almost all , for a given , in general it is not known what are the optimal achievable degrees of freedom. [5] shows that the result for almost all can be derived using Rényi information dimension. Again, the result is sensitive to whether entries of are rational or irrational, and for fixed channel matrix , in general the optimal achievable degrees of freedom is unknown. The recent works of [6, 7] address this issue to a good extent for the case of the two-user X channel and the symmetric -user interference channel respectively, but the engineering implication of the proposed coding schemes remains unclear.

Therefore, despite significant theoretical progress on the -user interference channel problem, it is still unclear how to make interference alignment practical. The drawbacks of existing schemes may be inherent to the channel model which assumes sources can only transmit and destinations can only listen, while in practice radios can both transmit and receive. We study new channel models where interaction among sources and destinations is enabled, e.g., both source and destination can talk to each other. The interaction can come in two ways: 1) In-band interaction: sources and destinations can transmit and listen in the same channel simultaneously, enabling interaction. 2) out-of-band interaction: destinations talk back to the sources on an out-of-band channel, possible from white-space channels.

Although [8] shows that for interference channel, relays, feedback, and full-duplex operation cannot improve the degrees of freedom beyond , we demonstrate that the interaction among sources and destinations enables flexibility in designing simple interference alignment scheme and in several cases achieves the optimal degrees of freedom. Both of these interaction methods are enabled by full-duplex radios, especially the in-band interaction requires high quality full-duplex systems with good self-interference suppression, which have attracted renewed attention in recent times [9] [10].

Our main contribution is to propose a simple interference alignment scheme by exploiting the interactions among sources and destinations, and prove that the scheme can achieve the optimal degrees of freedom for several classes of interference channels, including -user IFC with out-of-band interaction, -user and -user IFC with in-band interaction, and -user MIMO IFC with in-band interaction. One specific aspect of our model, namely, feedback using the reciprocal interference channel, has been considered in prior work [11]. In this work, we improve on this state-of-the-art in two ways: we prove new results for this specific model and also generalize this model to exploit more general modes of interaction, which admits the possibility of source-cooperation [12, 13], destination-cooperation [14, 15] and in-band feedback in a single setting, in order to achieve interference alignment. The general modes of interaction permit simpler schemes. In particular, for , we show surprisingly that in-band interaction permits the first practical scheme that can achieve degrees of freedom, even after accounting for interaction cost. In addition to these results, we do extensive numeric simulations and show the proposed interactive interactive alignment scheme also works for some other classes of IFC empirically. We use tools from algebraic geometry to show why success of numeric simulations can suggest that the scheme should work well for almost all channel parameters in a rigorous way.

Along the way, we present a mathematical method for understanding the degrees of freedom in a general network with cooperation, feedback and relaying, where the nodes are constrained to using linear schemes of a fixed block length. While in general, the degrees of freedom achievable in a network will depend on the channel realization, in the case of all known channels, the degrees of freedom is the same for a measure of the channel realizations; and hence this value is called the degrees of freedom of the network. We show that this remains true for general networks with linear schemes as well. Furthermore, once the block length is fixed we give a numerical way of estimating this number.

The paper is organized as follows. We describe the channel model in Section II and give some motivating examples in Section III. We present the interactive communication scheme for out-of-band interaction and write down the interference alignment conditions in Section IV, and our main technical results from algebraic geometry on the interference alignment feasibility in Section V. We also show how to use this method to infer properties of degrees of freedom of a general network with linear schemes in Section V. Section VI and Section VII study interactive interference alignment for -user interference channels with out-of-band interaction and in-band interaction, respectively. Simulation results on the finite SNR performance evaluation are presented in Section VIII. Section IX discusses a general multi-phase interactive communication scheme for -user interference channel with large , and lists several open problems. Section X concludes this paper.

Ii System Model

Consider a -user interference channel with radios, where radios are sources and radios are destinations. For each , source wants to send an independent message to destination .

Let denote the forward channel matrix from sources to destinations, and the input and output signals of the forward channel are related as

(1)

where is the time index, is the signal received by destination at time , is the signal sent out by source at time , is the channel noise, and is the channel coefficient from source to destination .

The above is a canonical channel model for -user interference channel.

In this work, we consider two channel models where interaction among sources and destinations can be enabled.

Ii-a Out-of-Band Interaction

The first model we consider is a simple model for out-of-band interaction. While the sources talk to the destinations on its channel, the destinations are assumed to talk back to the source in a different channel, which could come from white-space channel. Depending on how the out-of-band channel is obtained, one may or may not want to account for the cost of this channel. Here we will see that when , this mode is useful even in the absence of Let denote the feedback channel (or reverse channel) matrix from destinations to sources. Similarly, the input-output relation of the feedback channel is

(2)

where is the time index, is the signal sent out by destination at time , is the signal received by source at time , is the channel noise, and is channel coefficient from destination to source .

This mode can also be enabled by in-band interaction using half-duplex radios. If the destinations use in-band half-duplex transmissions, due to the reciprocity of wireless channels, we have

(3)

where denotes the transpose of .

Fig. 1: System Model For Out-of-Band Interaction: are sources and are destinations. is the forward channel matrix from sources to destinations, and is the feedback channel matrix from destinations to sources.

Ii-B In-Band Interaction

Full-duplex radios can both send and receive signals using the same channel simultaneously, and this capability naturally enables the interaction among all radios in the network. We assume that all sources and destinations have full-duplex antennas, and all nodes can transmit and receive signals using the same channel (in-band) simultaneously. Let be the channel matrix from sources to destinations, be the channel matrix among sources, and be the channel matrix among destinations. The input-output relation of this interference channel with in-band interaction is

(4)
(5)

where is the time index, and are the signals sent out by source and destination at time , respectively, and are the signal received by destination and source at time , and are the channel noise, is channel coefficient from to and also the channel coefficient from to due to channel reciprocity, is the channel coefficient from to , and is the channel coefficient from to .

Fig. 2: System Model For In-Band Interaction: are sources and are destinations. is the forward channel matrix from sources to destinations, is the channel matrix among sources, and is the channel matrix among destinations.

Iii Motivating Examples

The main theme of this paper is to show how interaction among destinations and sources can help do interference alignment/neutralization and thus make concurrent wireless communication feasible.

In this section, we start with a simple example of 3-user interference channel with out-of-band interaction, where communication is impossible without interaction from destinations, and describe a simple three-phase transmission scheme which makes concurrent communication feasible by exploiting the interaction from destinations to sources. Then, we generalize this three-phase transmission scheme to a class of -user interference channels where the forward channel matrix can be written as the sum of a diagonal matrix and a rank 1 matrix.

Iii-a A symmetric 3-user interference channel

Consider a 3-user interference channel with forward channel matrix

(6)

where all interference link gains are 1, and all direct link gains are 0.

Since the direct links have gains 0, it is trivial to see that without interaction from destinations, no reliable communication is possible. However, if we allow destinations to talk back to sources using the reciprocal channel, we can achieve degrees of freedom as seen below. For simplicity, let be the reverse channel; one way of achieving this is by using half-and-half interaction.

Consider the following three-phase interactive transmission scheme:

  • Phase 1: All sources send their independent symbols simultaneously. Then destinations receive

    (7)
    (8)
    (9)
  • Phase 2: After receiving signal from sources in Phase 1, all destinations simultaneously send out to sources using the reciprocal channel, and sources get

    (10)
    (11)
    (12)
  • Phase 3: Now each source has two sets of signals and , and each can simultaneously send out to destinations using the forward channel. destinations will get

    (13)
    (14)
    (15)

Therefore, after two forward transmissions and one reverse transmission, each destination will get the desired symbol from the corresponding source without any interference. If we do not count the reverse transmission, the total degrees of freedom achieved are . Even if we account for reverse transmission, we get a total degrees of freedom of , which cannot be achieved in the absence of

Iii-B A symmetric -user interference channel with special channel matrix

We can extend the above result to -user interference channel where the channel matrix is an all-one matrix except all diagonal entries being zero for any .

Consider a -user interference channel where the channel matrix satisfies

(16)
(17)

We can show that the following three-phase interactive transmission scheme with two forward transmissions and one reverse transmission can make each destination get the desired symbol from source for all .

  • Phase 1: All sources send their independent symbols simultaneously. Then each destination get

    (18)
  • Phase 2: After receiving signal from sources in Phase 1, all destinations simultaneously send out to sources using the reciprocal channel. Then each source gets

    (19)
  • Phase 3: Now each source has two sets of signals and , and each can simultaneously send out

    (20)

    to destinations using the forward channel. Then each destination will get

    (21)

For , . Therefore, each destination will get the desired symbol from the source without any interference for all .

In fact, we can show that if can be written as the sum of diagonal matrix and a rank 1 matrix, then in general two forward transmissions and one reverse transmission can make each destination get the desired symbol from source for all .

Theorem 1.

Consider a special -user interference channel with reciprocal feedback channel (), where the forward channel matrix can be written as the sum of a diagonal matrix and a rank 1 matrix, i.e.,

(22)

where is a diagonal matrix and are column vectors.

If is invertible, each component of and is nonzero, and

(23)

then there exists a three-phase interactive transmission scheme such that two forward transmissions and one reverse transmission can make each destination get the desired symbol from source without any interference for all .

Proof:

See Appendix A. ∎

Iv Communication Scheme and Interference alignment conditions

The three-phase interactive transmission scheme described in Section III is a natural scheme to exploit the interaction from destinations to sources to make concurrent wireless communication feasible. We can generalize the transmission scheme there to any interference channel with out-of-band interaction. We will do so in this section, by first describing the scheme with certain design parameters and then writing down the constraints required on the design parameters in order for interference alignment to be achieved.

  • Phase 1 (forward transmission): All sources send their independent symbols simultaneously. And destinations get , where is the additive noise of the channel.

  • Phase 2 (interaction from destinations): After receiving signals from sources in phase1, all destinations scale and send back to sources using the reverse channel. Sources get , where is the additive noise of the channel. Since each source and each destination only knows what signals they sent out and received, the coding matrix has to be diagonal.

  • Phase 3 (forward transmission): Now each source has two sets of signals and , and sources can send out a linear combination of and to destinations via the forward channel. Destinations get

    (24)
    (25)

    Again, the matrices are constrained to be diagonal.

Fig. 3: Phase 1: All sources send their independent symbols simultaneously, and destinations get .
Fig. 4: Phase 2: After receiving signals from sources in phase1, all destinations scale and send back to sources using the feedback channel. Sources get .
Fig. 5: Phase 3: Now each source has two sets of signals and , and sources can send out a linear combination of and to destinations via the forward channel. Destinations get .

In the above scheme, since each source and each destination only know what signals they sent out and received, the coding matrices and have to be diagonal, i.e., each node can only do coding over what signals it sent out and received. Note that since we are interested only in degrees of freedom calculations in this paper, we will not keep track of the particular structure of noise, as long as it has finite variance and is independent of everything else in the equation.

To make destinations be able to decode desired signals, one possibility is do interference neutralization, i.e., the interference term is zero for each of the receivers. This can be guaranteed if we choose diagonal matrices and such that is a diagonal matrix and every diagonal entry is nonzero, which makes each destination get the desired signal from without any interference. In fact we have used coding matrices with this property for the special class of interference channels presented in Section III. So a natural question to ask is whether there always exists such and .

Requiring that all off-diagonal entries be zero gives rise to polynomial equations, and each diagonal entry being nonzero leads to inequalities, whereas in total there are only variables, corresponding to the variables in each of the diagonal coding matrices and . One may therefore be led to conjecture that if the number of variables is more than the number of equations, a solution should exist that simultaneously satisfies all equations and inqualities. Unfortunately, this conjecture is not true: there are cases where even though the number of variables is more than equations, due to the coupled constraints on the equations and inequalities, all off-diagonal entries of being zero implies that at least one diagonal entry is zero. A concrete example is the 3-user interference channel with reciprocal reverse channel, which will be discussed in Section VI.

One way to overcome this problem is to do interference alignment. In this case, we want to align the interference seen by the receiver during the first and the third phase, i.e., the interference terms in (which the destinations received during the first phase) and (which destinations receive during the third phase) instead of relying purely on alone, which was the case in interference neutralization. Now, the destinations can cancel out all interference by taking a linear combination of and if the interference terms in are aligned with the interference terms in .

More precisely, let . Then for destination , in Phase 1 it receives

(26)

and in Phase 3 it receives

(27)

If for all , we have

(28)

or equivalently

(29)

for some constant , then interference terms of and for each destination are aligned with each other. So destination can compute and get

(30)

Therefore, if , or equivalently,

(31)

then destination can get the desired signal sent by source without any interference.

Without introducing the auxiliary variables , aligning interference terms for each destination corresponds to equations, and thus in total we have interference alignment equations. Preserving the desired signals after canceling all interference terms leads to inequalities. Therefore, to make the three-phase communication scheme work and thus achieve the optimal degrees of freedom, we need to solve equations with variables and check that whether the solution satisfies the inequalities.

In some cases, we can reduce the polynomial equations corresponding to interference alignment equality constraints to linear equations, the solution of which has a closed-form expression and thus it can be easily verified whether the inequality constraints are satisfied. However, in general the system of polynomial equations is nonlinear, and it may not have closed-form solution, making it hard to check whether the inequality constraint can be satisfied. Our main approach is to convert the system of polynomial equations and polynomial inequalities to a system of polynomial equations, and then use tools from algebraic geometry to check the existence of solution for the system of polynomial equations.

First, we show how to reduce the problem to checking the existence of solutions to a system of polynomial equations. Then we will present our main technique for solving the polynomial system using tools from algebraic geometry in Section V.

Suppose there are polynomial equation constraints , , and polynomial inequality constraints , , over variables . By introducing an auxiliary variable , we define a polynomial function as

(32)
Lemma 1.

There exists satisfying equality and inequality constraints

(33)
(34)

if and only if there exists satisfying

(35)
(36)
Proof:

If is a solution to (33) and (34), then is a solution to (35) and (36), where

(37)

On the other hand, if is a solution to (35) and (36), then satisfies both (33) and (34). ∎

Therefore, due to Lemma 1, the problem of checking the existence of solutions to a system of polynomial equations and inequalities can be reduced to the problem of checking existence of solutions of a system of polynomial equations, which is well studied for algebraically closed field in algebraic geometry [16].

V General Solution Methodology

In this section we present our main technical results on checking the existence of solutions to the interference alignment equations for generic channel parameters and using tools in algebraic geometry. For more details on algebraic geometry, we refer the reader to Appendix B and the excellent textbook [16].

We follow the standard notation in [16]. Let denote a field, and let denote the set of all polynomials in the variables with coefficients in .

Definition 1 (Definition 1 on page 5 of [16]).

Let be a field, and let be polynomials in . Then we call

(38)

the affine variety defined by .

As discussed in Section IV, our problem can be reduced to checking the existence of solutions to a system of polynomial equations. In the language of algebraic geometry, the problem is to check whether the affine variety defined by some polynomials is an empty set or not, which is well studied for algebraically closed field in algebraic geometry. In wireless communication, the channel coefficients are represented as complex numbers , which is an algebraically closed field.

The standard approach to checking whether an affine variety is an empty set is to use Buchberger’s algorithm to compute the Gröbner basis of the given polynomials, and from the Gröbner basis we can easily conclude whether the corresponding affine variety is empty or not [17].

One important implication of these results in algebraic geometry is that if the coefficients in the polynomial equations are rational functions of variables , then except a set of which satisfies a nontrivial polynomial equation on and thus has a measure 0, either for all numeric values of the polynomial equations have a solution, or for all numeric values of the polynomial equations do not have a solution.

More precisely,

Theorem 2.

Let be polynomials in , where and all coefficients of the polynomials are rational functions of variables . Then there exists a nontrivial polynomial equation on , denoted by , such that except the set of , either for all , , or for all , .

Proof:

See Appendix C. ∎

For the polynomial equations describing the interference alignment problem, the coefficients of the polynomials are rational functions of channel parameters and in symbolic form. Therefore, in the context of the interference alignment feasibility problem, this main result can be restated as follows. Either one of the following two statements hold:

  • For almost all111We emphasize that in our results “almost all” means for all numeric values except a set of parameters which satisfy a nontrivial polynomial equation. Therefore, in contrast to results in [5] and [4], our results are not sensitive to whether the channel parameters are rational or irrational. channel realizations of and , there exists solution to the system of interference alignment equations.

  • Or, for almost all channel realizations of and , there does not exist solution to the system of interference alignment equations.

Although in theory we can compute in finite number of steps the symbolic Gröbner basis for these polynomials with symbolic coefficients to check whether for almost all and there exists solutions, it turns out to be computationally infeasible to run the Buchberger’s algorithm for the symbolic polynomial equations for most of our interference alignment problems, due to the fact that the orders of intermediate symbolic coefficients can increase exponentially. However, it is much easier to compute a Gröbner basis numerically. Due to Theorem 2,

Corollary 1.

If we draw the channel parameters according to a continuous probability distribution, then with probability one the numeric polynomial equations have a solution if and only if for almost all channel realizations the polynomials equations have solution.

Hence, while we may not be able to prove that certain polynomial equations have solution for almost all channel parameters because of computational difficulty, numeric simulations can let us make claims with high credibility.

V-a Application to General Networks

In a general network with potentially feedback, cooperation and relaying, an important question to understand is when linear schemes can acheive a certain degrees of freedom. While this is a hard question in general, once we restrict to a fixed block length, we can start answering this question. Let us fix a block length of communication and let each user transmit a linear combination of the symbols that he received. At the end of this communication, the receivers apply a linear combination of all received inputs to construct the decoded vector. We want this decoded vector to equal the transmitted vector. We leave all the multiplication matrices to be design variables and ask when does this system of equations have a solution. The answer to this question, of course, depends on the realization of the channel. However, if we assume that that channel coefficients are drawn from a measure with a probability density, then we can invoke Theorem 2 and Corrolory 1 to show that there are only two possibilities.

  1. With probability over the channel measure, the particular degrees of freedom is achievable.

  2. With probability over the channel measure, the particular degrees of freedom is not achievable.

In order to test which of the two hypotheses is true, we can run simulations, where the channel is generated according to the measure. Once the channel is fixed, we can then run numerical Gröbner basis algorithm to determine whether there is a solution or not. If for almost all the simulations, the solution exists, then we can declare that the former case is true; if not, the latter case is true. While this step of going from the simulation to the conclusion has technical difficulties due to the fact that computer has numerical precision, we can make such claims with high confidence.

Vi Out-of-Band Interactive Interference Alignment

In this section, we study interactive interference alignment for -user interference channel with out-of-band interaction. We show results only for small values of , in particular only for .

Vi-a

We described in Section IV, a three-phase interactive communication scheme for out-of-band interference alignment and the required conditions on the design matrices. Recall that, for such a scheme work, we would like to find diagonal coding matrices , and to make be a diagonal matrix with each diagonal entry being nonzero, so that interference neutralization can be achieved in the third phase. For -user interference channel, the number of interference neutralization equations is , and the number of variables is . Even so, one may hope that the interference neutralization equations are solvable, due to the coupled constraints on the equations and inequalities, it can be quickly seen that all off-diagonal entries of being zero implies that at least one diagonal entry is zero. A simple counter example is the case when the reverse channel is reciprocal to the forward channel, i.e., . We did extensive numeric simulations, and in each numeric instance we find that there is no solution to solve both equations and inequalities by using the methodology (computing Gröbner basis) introduced in section V. Therefore, due to Corollary 1, we believe the following claim holds.

Claim 1.

For 3-user interference channel with reciprocal feedback channel (), for generic matrix222Generic matrix means all numeric matrices except a set of matrices the entries of which satisfy a nontrivial polynomial equation. , there does not exist and such that is a diagonal matrix and every diagonal entry is nonzero.

As discussed in Section IV, it is not necessary to do exact interference neutralization. Instead, interference alignment is also sufficient. When , the number of variables is , which is much more than the number of interference alignment equations , so it is not surprising that we can find and to solve the interference alignment equations and inequalities. [11] gives an analytical solution to do interference alignment for -user interference channel with feedback. Here, we present a different proof technique, which also will be applied in interference alignment with in-band interaction in Section VII.

First we state a lemma which will be used in the proof of Theorem 3.

Lemma 2.

Let be a linear subspace with dimension . For any linear subspaces with dimensions respectively, if

(39)

then is a nonempty set.

Proof:

Let be a set of bases of . Let such that every by submatrix of has full rank. It is easy to show the existence of with such property. Indeed, if every entry of is independently generated according to Gaussian distribution (or other continuous probability distributions), then with probability 1 every by submatrix of has full rank.

Define vectors by setting

(40)

for all . Since every by square submatrix of has full rank, every vectors of are linearly independent.

Suppose is the empty set, i.e., . Then . By pigeonhole principle, there exists such that contains at least vectors of . Since every vectors of are linearly independent, the dimension of is no less than , contradicting with the fact that .

Therefore, is a nonempty set.

Theorem 3.

For 3-user interference channel with reciprocal feedback channel (), and for generic channel matrix , there exists solutions to the interference alignment equations and inequalities.

Proof:

The equations are

(41)
(42)
(43)

and the inequalities are

(44)
(45)
(46)

Since the number of variables is much more than the number of equations, we do not need to use all the variables. In fact, we can set to be a simple diagonal matrix

(47)

Write and as

(48)
(49)

Then equations (41), (42) and (43) are homogeneous linear equations in terms of variables and . Let denote the set of solutions to linear equations (41), (42) and (43). It is easy to check that the corresponding coefficient matrix has a rank of . Therefore, is a linear subspace of dimension .

Consider the following equations corresponding to the inequality constraints

(50)
(51)
(52)

Let denote the set of solutions to (41), (42), (43) and (50), let denote the set of solutions to (41), (42), (43) and (51), and let denote the set of solutions to (41), (42), (43) and (52). We can check that all the coefficient matrices of the linear equations have a rank of , so all of and are linear subspaces of dimension .

Now consider the set . Any element in satisfies both the equations (41), (42), (43), and the inequality constraints (44), (45), (46).

Since is linear subspace of dimension , which is larger than the dimensions of and , is a nonempty set by Lemma 2. This completes the proof.

The same result continues to hold when the feedback channel matrix is also a generic matrix independent of .

It is worth pointing out that the 3-user interference channel under an alternate local feedback model, where each receiver sends feedback to its corresponding transmitter, was studied in [18] recently. In contrast to our model, where obtaining feedback using natural reciprocal channel takes up a single time slot, local feedback can take time slots to implement.

Vi-B

For 4-user interference channel, we have polynomial equation constraints and inequality constraints with variables. We cannot use the same technique as in the case of three-user interference channel by setting or to be a deterministic constant matrix and thus reducing the polynomial equations to linear equations. The reason is that if we do not use or as variables, the number of equations will be the same as the number of variables, and this will prevent us from finding a solution which also satisfies the inequality constraints. Instead, we use the tools introduced in Section V to study the feasibility of interactive interference alignment. More precisely, we first convert the system of polynomial equations and polynomial inequalities to a system of polynomial equations, and then we check whether it has solutions or note by computing the Gröbner basis of corresponding polynomials.

Due to computational difficulty, it is hard to compute the Gröbner basis for the symbolic polynomials. We do extensive numeric verifications: each time we generate random numeric channel matrices, and then apply Buchberger’s algorithm to compute a Gröbner basis. In all numeric instances, the Gröbner basis does not contain a nonzero scalar, which implies that solution to (28) and (31) exists. Therefore, due to Corollary 1, although we cannot compute a symbolic Gröbner basis explicitly, from numeric verifications we believe the following claims hold.

Claim 2.

For 4-user interference channel with reciprocal feedback channel (), for generic matrix , there exists and such that each row of is proportional to the corresponding row of except the diagonal entries.

Claim 3.

For 4-user interference channel with out-of-band feedback channel, for generic matrices and , there exists and such that each row of is proportional to the corresponding row of except the diagonal entries.

[11] formulates the interference alignment problem as a rank constrained nonconvex optimization problem, and reports that all numeric simulations confirm the existence of solutions to the interference alignment equations and inequalities.

In some cases when has special structures, we are able to compute the symbolic Gröbner basis of the interference alignment polynomials, and thus we can conclude whether for almost all with such structure there exists interference alignment solutions. One positive example is that if is a symmetric matrix and all diagonal entries are zero, then for almost all such , the interference alignment equations are solvable.

Theorem 4.

For 4-user interference channel with reciprocal feedback channel (), if is symmetric channel and all diagonal entries are zero, i.e., can be written as

(53)

then for generic , i.e., for all except a set of which satisfies a nontrivial polynomial equation, there exists and such that is proportional to the corresponding row of except the diagonal entries.

Proof:

We derive the interference alignment equations and then use commercial computer algebra system Maple to compute the corresponding symbolic Gröbner basis, which does not contain a nonzero scalar. Therefore, due to Corollary 4 in Appendix B, we conclude that for generic , there exists and such that is proportional to the corresponding row of except the diagonal entries. ∎

Vi-C and

Recall that the number of variables is , the number of equations is and the number of inequalities is . grows faster than as increases, and when , . We expect that when the number of variables is not more than the number of inequations, a valid solution may not exist. Indeed, we confirm this conjecture for and via computing Gröbner basis for random numeric and and finding that is in the Gröbner basis in all instances.

Claim 4.

For -user interference channel with reciprocal feedback channel () where or , for generic matrix , there does not exist and such that each row of is proportional to the corresponding row of except the diagonal entries.

Claim 5.

For -user interference channel with out-of-band feedback feedback channel where or , for generic matrices and , there does not exist and such that each row of is proportional to the corresponding row of except the diagonal entries.

Since one reverse transmission from destinations to sources is not enough, in Phase 2 (interaction phase) we can let destinations make more than one reverse transmissions to sources. In one extreme case, if each destination sends what it receives in Phase 1 to all sources sequentially, where in total we use reverse transmissions333In fact, it is easy to see that reverse transmissions are also sufficient., then each source can compute the symbol signals sent out by all sources in the first phase, and then each source can do precoding over all symbols in the second phase, which reduces to a MIMO broadcast channel.

It turns out that when or , two reverse transmissions in Phase 2 is sufficient to do interference alignment while still preserving the desired signals. A natural three-phase scheme is as follows:

  • Phase 1: All sources send their independent symbols simultaneously. And destinations get , where is the additive noise of the channel.

  • Phase 2-1: After receiving signals from sources in phase1, all destinations scale and send back to sources using the feedback channel. sources get .

  • Phase 2-2: All destinations scale using a different coding matrix and send back to sources. sources get .

  • Phase 3: Now each source has three sets of signals and , and sources can send out a linear combination of and to destinations via the forward channel. destinations get

    (54)
    (55)

With two reverse transmissions, the number of variables is increased to from , and is more than for . Extensive numeric simulations show that for generic matrix and , there exists diagonal coding matrices to align interferences while still preserving desired signals.

Claim 6.

For -user interference channel with reciprocal feedback channel () where or , for generic matrices , there exists and such that each row of is proportional to the corresponding row of except the diagonal entries.

Claim 7.

For -user interference channel with out-of-band feedback channel where or , for generic matrices and , there exists and such that each row of is proportional to the corresponding row of except the diagonal entries.

Vii In-Band Interactive Interference Alignment

In this section, we study how to exploit interaction in interference channel where all sources and destinations have full-duplex antennas so that all nodes can transmit and receive in the same band simultaneously. We call this model as in-band interactive alignment. This model captures the full range of possibilities of interaction, including source-cooperation, destination-cooperation and feedback, where these cooperation modes arise naturally in a fully-connected wireless network. We show a surprising result that the presence of these modes can highly simplify the nature of communication schemes. In particular, for the -user interference channel with in-band interaction, these modes, the optimal sum degrees of freedom of can be achieved using the proposed alignment schemes with a simple two-phase scheme even when the channel coefficients are fixed. We also show a similar result is true even when each user has antennas.

Vii-a Interference Channels with

Recall in the system model of IFC, is the channel matrix from sources to destination, is the channel matrix among sources, and is the channel matrix among destinations. Consider the following simple two-phase transmission scheme.

  • Phase 1: All sources send out signals simultaneously. After the transmission, sources get , and destinations get .

  • Phase 2: sources send out a linear combination of and , and destinations send out scaled version of . Therefore, destinations get , where and are diagonal coding matrices.

Note that in this transmission scheme no feedback channel from destinations to sources is required. Similarly, if each row of is proportional to the corresponding row of except all diagonal entries, then interferences at all destinations are aligned and all destinations can retrieve their designed signals from sources without any interference, and thus achieve the optimal degrees of freedom.

Our first result is that the above two-phase transmission scheme works for and .

Theorem 5.

For and , for generic channel matrices , and , there exists diagonal matrices and such that each row of is proportional to the corresponding row of except all diagonal entries.

Proof:

The interference alignment equations are linear in and , so as in the proof of Theorem 3 we can use dimension argument to show that there always exists a solution to the system of linear interference alignment equations and it also satisfies the inequality constraints. ∎

Using the same dimension argument, we can prove that the above scheme does not work for or bigger.

Vii-B Multi-antenna IFC with

Next we show that for four-user MIMO interference channel with in-band interaction can help design a simple transmission scheme to achieve the optimal degrees of freedom.

For -user MIMO interference channel where all sources and destinations are equipped with antennas, [1] shows that for , vector space interference alignment can achieve the optimal degrees of freedom and channel diversity is not required. [3] and [19] prove that in general vector space MIMO interference alignment can at most get degrees of freedom, which is strictly less than the optimal when . We show that for four-user MIMO interference channel with in-band interaction, a simple two-phase transmission scheme can achieve the optimal degrees of freedom at least for all .

Suppose source wants to send symbols to destination , for all . Let . Consider a natural two-phase transmission scheme for 4-user MIMO IFC with in-band interaction, described as follows.

  • Phase 1: All sources send out signals simultaneously, i.e., the th antenna sends for all . After the transmission, sources get , and destinations get .

  • Phase 2: sources do linear coding over and , and send , and destinations do linear coding over and send , where and are block diagonal matrices with diagonal block size of . Therefore, destinations get

    (56)

Note that since each node can only do coding over what signals it has or received from other nodes, the coding matrices and have to be block diagonal matrices with diagonal block size of .

A sufficient condition for all destinations to retrieve the desired symbols from the corresponding sources without interference is that at each antenna of destination , all interference symbols are aligned except the desired symbols , for all .

More precisely, let and denote the index set by . Then for the destination , in Phase 1 it receives

(57)

and in Phase 2 it receives

(58)

The first term on the RHS of (57) and (57) correspond to the desired signals, and the second term are interferences. If at each antenna of , all interferences are aligned, i.e., for all , there exists , such that

(59)

then by computing , can cancel all interferences

(60)
(61)

Now gets linear equations in variables