# On Achievable Schemes of Interference Alignment in Constant Channels via Finite Amplify-and-Forward Relays

###### Abstract

This paper elaborates on the achievable schemes of interference alignment in constant channels via finite amplify-and-forward (AF) relays. Consider sources communicating with destinations without direct links besides the relay connections. The total number of relays is finite. The objective is to achieve interference alignment for all user pairs to obtain half of their interference-free degrees of freedom. In general, two strategies are employed: coding at the edge and coding in the middle, in which relays show different roles. The contributions are that two fundamental and critical elements are captured to enable interference alignment in this network: channel randomness or relativity; subspace dimension suppression.

## I Introduction

Since interference alignment (IA) is a new multiplexing gain maximizing technique [1] and relay is considered a cost-effective solution for coverage extension and capacity enhancement, recently many researchers have been investigating schemes to combine these two techniques [2, 3]. Various scenarios have been intensively studied, e.g. decode-and-forward relays [4, 5], multi-user broadcast channel [6], clustered relays [7], two-way relay selection [8]. Among all the scenarios, the model of multi-user peer-peer two-hop interference channel with amplify-and-forward relays without direct links has received increasing attention due to its wide application in practice [9, 10, 11, 12], which show that relays help the system to obtain higher degrees of freedom (DoF) in high signal-to-noise ratio (SNR) regime.

While peer-peer multi-user interference channel has been traditionally regarded as a challenging scenario due to channel inseparability [13], not to mention with relays in between. Furthermore, current relay networks are categorized into two main sets: one is that relays are auxiliary links in addition to direct links between both ends [2, 14, 15, 16, 17]; the other is that relays are the only connection path for end nodes without direct links [3]. In general, relay networks with direct links are well structured to construct IA. However, this work considers relay networks without direct links where relay generated equivalent channels are quite complicated and poorly structured [3, 18, 19]. In that case, many approaches are proposed, e.g. [3] requires an infinite number of relays to eliminate interference; [18] exploits ergodic nature in fading channels; [19] illustrates interference neutralization in a special network; [9] uses mean squared error (MSE) numerical method to minimize interference.

Three important features are highlighted in the scenario of this work: the number of relays is finite so that it is impossible to have the unpractical solution in [3] to eliminate interference with infinite number of relays; each node could only have single antenna to achieve IA; time-invariant channels are also available to achieve IA even with single antenna nodes [1]. This work borrows the two strategies generalized in multiple unicast problem [20], which are coding at the edge and coding in the middle. In the first strategy of coding at the edge, relays randomly construct equivalent channels while end nodes proceed with conventional interference alignment schemes. Compared with other research, the most unfavorable conditions are set in this work: all end and relay nodes are single antenna; all channels are time-invariant; relay number is set to be finite, e.g one or two. Max-flow-min-cut theorem is not directly applicable in this scenario. In the second strategy of coding in the middle, optimization techniques are applied to numerically approach interference alignment with all nodes set to be multi-antenna.

Our contributions are in two folds:

A new fact is unveiled that when the network has only one relay, Cadambe-Jafar scheme is not applicable in signal vector alignment; Motahari-Khandani scheme is not applicable in signal level alignment; asymmetric complex signaling is not applicable in phase alignment. The reason is that relay-emulated channels lose randomness or relativity. Then the problem is settled by two ways. On one hand, at least two relays are necessary to emulate qualified channels to generate randomness. On the other hand, to generate randomness still in the one-relay channel, space-time type of precoding methods are applicable at the edge; the conceptual idea of blind interference alignment is also applicable to fluctuate channels with the only one relay. By all these analysis and schemes, a novel unique role of relays in the network is revealed to overcome the channel randomness issue.

A novel solution is proposed for IA via relay coding based on a non-trivial application of existing rank constrained rank minimization (RCRM) method [21, 22]. Conventional optimization and numerical algorithms are non-convex or hard for interference alignment [23, 9, 5]. Since RCRM method is originally not for relay networks, the convexity is proved for the new application. This novel solution has three advantages: a) it considers DoF at high SNR by subspace dimension minimization while other numerical methods could only consider sum rate and mean squared error (MSE); b) it is universal to be conveniently applied to a number of scenarios, e.g. MIMO amplify-and-forward(AF) relay channel, MIMO two-way AF relay channel, MIMO Y channel, and MIMO multi-hop relay channel. c) actually conventional analytical and numerical solutions are hard to obtain for the mentioned scenarios, while this novel method accomplishes the design and is robust to poor conditions. Numerical results show its effectiveness.

## Ii Problem Statement and Model Description

### Ii-a Basic Model

Consider sources and destinations connected by relays as shown in Fig. 1. Direct links between end users are not available. All nodes are single antenna, denote the channel from source to relay as and the channel from relay to destination as . Denote the sets , , so that , . All channels are quasi-static, i.e. and are scalar constants.

Time-extended MIMO scheme in [3] is used so that consecutive symbols form one signal and the relay coding matrix at -th relay is

(1) |

in which is the relay gain from the -th time slot to the -the time slot at the -th relay. Then the received signal at -th user can be written as

(2) |

(3) |

where is the input streams at -th source and is precoding matrix. is the noise at -th receiver, and . The equivalent channel matrix from the -th source to the -th destination is defined as for simplicity. The holistic system equation is also written as:

(4) |

### Ii-B General Model

For more general cases, the model is illustrated in Fig. 2. Denote nodes on one end as ; relay nodes in the middle as ; nodes on the other end as . Define sets , . All channels are time-invariant. Assume all nodes have all channel knowledge to cooperate.

In the case when all nodes are single antenna, denote , , as channels from to , to , respectively, where , , all channel coefficients are scalars. While in the case when all nodes are multiple antenna, denote , , , , as channels from to , to , to , to , respectively, where , , all channel coefficients are matrices.

## Iii Characteristics of Amplify-and-Forward and Finite Relays

Before detailed design and analysis, this relay scenario needs to be characterized and some critical features should be exposed. First, the applicability of max-flow-min-cut theorem in this network is actually in doubt along with the amplify-and-forward scheme. Second, the number of relays, achievable DoF, and coding strategy are correlated in a complicated form.

### Iii-a Amplify-and-Forward Scheme and Max-Flow-Min-Cut Theorem

A natural question arises here: Does max-flow-min-cut theorem still work in this IA relay network? Another similar question is: in a -pair network with finite relays, what does the achievable DoF equal to? The answers to these two questions are quite nuanced actually.

Look into the -pair network with relays. The first hop is a MIMO link and the second hop is a MIMO link. Conventionally, both the first and the second channel matrices only have a rank of , which means that for the whole -pair network, the minimum cut is . However, the degrees of freedom for the network is not necessarily . Consider the -pair network regardless of the middle connections, so that there are potentially DoF for this equivalent network as long as with joint processing across the transmitters or receivers. The critical difference is that the intermediate relay nodes do not necessarily process data flows directly, and could act only as equivalent channels. In addition, instead of joint processing, a distributed manner such as interference alignment could still possibly achieve DoF. However, it is also important to highlight that when there are finite single-antenna relays, the equivalent channel may be rank deficient, which is discussed detailedly in the following parts.

DoF as in the signal processing model is not simply equal to capacity flow links as in network coding model. In the flow network model, each node has separate inputs and outputs, and each edge represents a separate link. While in amplify-and-forward relay networks, arbitrary flows could overlap through same relays, and links in each hop does not represent separate flows.Max-flow min-cut theorem is originally applied to wired networks with single-letter characterizations. A recent model for wireless relay networks is known as the linear deterministic relay network model with a max-flow min-cut result pertaining to it [24]. The algorithmic framework is introduced by Avestimehr, Diggavi and Tse, incorporating the key features of broadcasting and superposition. The signals are elements of a finite field and the interactions between the signals are assumed to be linear. This model is based on linking systems and the max-flow min-cut theorem is applicable with matroid intersection or partition.

Moreover, [20] shows that in a network with 3 unicast sessions each with min-cut of 1, whenever network alignment can achieve rate of 1/2 per session, there exists an alternative approach including routing, packing butterflies, random linear network coding, or other network coding strategies instead of alignment. However when there are more than 3 sessions, alignment is required to obtain the maximum rate as 1/2 the min-cut, while no other method can achieve it.

In summary, the max-flow min-cut theorem is not exactly available for this amplify-and-forward system, and DoF is not determined directly based on the number of relays. So that it is non-trivial to investigate new schemes in the amplify-and-forward strategy. Before further understanding and analyzing the DoF of this network, different situations need to be classified as following.

### Iii-B Finite/Infinite Relays, DoF Limits, and Strategies

For the -pair amplify-and-forward relay network, all the cases are roughly classified into three categories: when infinite relays are provided, full DoF is achievable, by using relay coding; when finite relays are provided, only fractional DoF is achievable as the number of users grows, by using conventional MIMO precoding approach; when a specific range of finite relays are provided, DoF is possible to be obtained, by using asymptotic interference alignment precoding approach. The details are shown in Table I and discussed as following.

Case | Relay Number | Target DoF | Coding | Channel | IA Scheme |
---|---|---|---|---|---|

1 | Infinite | Relay Coding | Generic | N/A | |

2 | Finite | Precoding | |||

3 | Precoding | Diagonal | Asymptotic |

#### Iii-B1 Infinite Relays, Full DoF, and Relay Coding

Consider the single antenna -pair relay network in Fig. 1 with the system equation of (2) and the equivalent channel in (3). Let the interference at any receiver to be zero, and then the condition for -th user should satisfy the following:

(5) | ||||

Equation (5) represents matrix equations, equalling to linear equations. The matrices contain variables. Generally, the equations are solvable when: , i.e. . It could be also interpreted as for each set of elements of there are equations. Therefore in this case, the network achieves DoF by only using the linear relay coding.

#### Iii-B2 Finite Relays, Fractional DoF, and Precoding

Consider the same network in Fig. 1. However, there’re only finite relays in this case. Instead of relay coding as above, a scheme of precoding on the equivalent channels is proposed, i.e. following the coding at the edge strategy [20]. Relays generate equivalent channels, and the transmitters and receivers only see the equivalent channels regardless of relays. Then conventional interference alignment methods could be used directly for all users [1], [23].

The relay gain matrices of (1) are randomly chosen to be in a generic form (full elements), so that and corresponding are equivalent to general MIMO channels. The leakage minimization algorithm of [23] is applicable to approach interference alignment. However, according to the feasibility condition in [25] for symmetric MIMO channels, the equivalent system in (2) must satisfy

(6) |

So that each user obtains DoF bounded by . Numerical results are shown in Fig. 3 in which three cases denoted by ‘Generic’ use the leakage minimization algorithm in [23] to achieve IA for different numbers of relays: relay number , and no relay. Total user number , and the time extension length , and each user has streams.A MIMO network with 21 antennas for all nodes is set as a reference case. When the channels are generated by 1 relay or 5 relays, each user could obtain the same number of DoF as the reference MIMO channel case roughly as .

#### Iii-B3 Specific Finite Relays, Half DoF, and Precoding

Although the feasibility condition in [25] limits the DoF in the constant MIMO channels, however there is still chance to achieve DoF for the network actually. The reason is that the relays are capable of flexibly constructing desired channel structures such as time-variant channels.

Manipulate relay gain matrices to be of diagonal structures as following:

(7) |

where is diagonal function which place all inputs on diagonal line of a matrix output. Then the equivalent channels are constructed in diagonal structure as well:

(8) |

If we only deal with the relay constructed diagonal channels by using the conventional leakage minimization method as before, the DoF results would have no improvement as shown in Fig. 3. It shows three cases denoted by ‘Diagonal’, which indicate the same DoF curves as the equivalent generic MIMO channels where . Observe that the relay number does not affect the algorithmic IA result either. Theoretically, there has been no conclusions so far on algorithmic IA feasibility of diagonal channels [25].

However, if we deal with the relay constructed diagonal channels by using the analytical asymptotic design (Cadambe-Jafar scheme) for frequency selective channels [1], each user could approach DoF regardless of the number of users in the network. Take the 3-user network as an example to apply Cadambe-Jafar scheme in this relay network. The system equation (2) needs to be modified a little: let , , , , where is a positive integer. Thus DoF for the three users are not symmetric here and set as , , respectively. When is large enough, each user could approach DoF. Design the aligned interference subspace at each user as following:

(9) |

where . In this way, the three-user network could approach DoF eventually. It is important to highlight that this scheme could be extended to arbitrary -user case. Then (9) and (10) are upgraded to a more complicated form accordingly [1]. The achievable multiplexing gain of the network becomes , where , is non-negative integer. As a result each user could approach DoF.

Numerical results are shown in Fig. 3. In the 3-user network, let so that the time extension length . There are four cases: relay number , , , and no relay. In addition, a case of frequency selective channel with 21 dimensions for all nodes is introduced as a reference. The four cases using the asymptotic design are denoted by ‘Asymptotic’. When the channels are generated by 2 relays or 5 relays, each user could obtain the same DoF as the reference frequency selective channel case. However, when the channels are generated by only 1 relay, each user obtains zero rate, and zero DoF, i.e. the interference alignment scheme fails.

The reason for the failure of the case of is investigated as following. For relay, (3) becomes:

(11) |

(12) | ||||

where is scalar coefficient equal to 1, and is an identity matrix. Observe in (12) the three precoding matrices , , have only rank 1 because all are linear dependent to be eliminated, so that the asymptotic solution fails due to the lack of channel randomness or relativity in [1], which is the core idea of the IA mechanism. Similarly, in the -user case, the asymptotic solution also fails when there is only 1 relay. Meanwhile recall the numerical result by leakage minimization algorithm in equivalent MIMO scheme, the achieved DoF is low but non-zero.

Comparing with the case of , the relay generated diagonal channel in (8) becomes:

(13) |

It needs to be proved that are not linear dependent to lose channel randomness with the following lemma.

###### Lemma 1

and are linear independent almost surely for arbitrary non-identical and .

Proof: Assume

. , such that .

Expand and group by and to:

.

and are arbitrary generic diagonal matrices, therefore almost surely the coefficients are zeros, i.e.

.

Forming the ratio , so we have .

Since are all random independent scalars, the equality fails almost surely. So that , do not exist almost surely. Then and are linearly independent almost surely.

Notice ‘almost surely’ cases are recognized as successful interference alignment [1]. Lemma 1 shows that ,, and are not scaled identity matrices, therefore their total product, , is not a scaled identity matrix almost surely considering channel relativity of these three terms. As a result the asymptotic design in (10) still works here, rather than degenerating to the form (12). In summary, Lemma 1 reveals that two or more relays can generate channel randomness or channel relativity in the equivalent channels with problems in (11) and (12) successfully prevented. Then the asymptotic design in (10) can be applied successfully.

## Iv Role of Relay in Precoding Scheme in Constant Channels

The above 3-user case reveals a critical issue for the constant channel to implement interference alignment. Therefore it is important to look into general -user networks, as well as different class of IA schemes. In this section, for the general scenarios of relay networks, the model is shown in Fig. 2 : define nodes as sources and as destinations, whereas each communicates with each via relays . Relays are half-duplex so that the transmission procedure consists of two stages where relays forward data in the second stage. Interference alignment is designed with the following strategy: relays construct equivalent channels; source and destination nodes proceed precoding and zero-forcing, i.e. coding at the edge as [20] claimed. The objective is still to approach 1/2 due DoF (exclude duplex factor) for every user, which is quite difficult and non-trivial for a relay network under quasi-static channel condition. The work is published in [26] and submitted in [27].

### Iv-a Time-extended Signal Vector Alignment

The most common approach to implement IA in this network is the time-extended MIMO scheme as in [3], where consecutive symbols form one signal vector and IA is designed in the vector space. The above example of 3-user network as in equations (7) and (8) exactly constructs the relay gain matrix and equivalent channels with this time-extended MIMO approach. Therefore, it is important to extend the design to arbitrary -user networks, and look into the same issue caused by constant channels with single relay in the network.

and could be diagonal or generic. If they are generic, it is equivalent to MIMO channel constrained by feasibility conditions [25], so that each user could only approach DoF. If they are diagonal, the channel is equivalent to a frequency selective channel in [1, 28], then the asymptotic design of Cadambe-Jafar scheme could be applied. Notice [28] has the same core design structure as [1], so that the design of [28] is illustrated here for a general -pair relay network as in the following form:

(14) |

(15) | ||||

where is the precoding matrix for arbitrary source , which contains input streams. . is the all one vector, is integer. Notice and . has a dimension of ; has a dimension of , . So that each user could approach DoF when is large for arbitrary .

#### Iv-A1 The case when the network has single relay,

However, in the special case when , this scheme does not work. The reason is easily shown in the equations (3) and (14) as following:

(16) | ||||

#### Iv-A2 The case when the network has at least two relays,

While for the case of , asymptotic IA scheme could be much likely applied to obtain DoF of this network for arbitrary . As a primitive investigation, set . The key is to prove that the column subspace of in (15) almost surely maintains its rank. First, we need to look at the core elements which constitute . The objective is to prove all the terms are linear independent. Then the expanded form is as in equation (17) and proved through the following three lemmas:

(17) | ||||

###### Lemma 2

, , , are linear independent almost surely.

Proof: Let . denotes the function to generate a diagonal matrix with all elements in the set. Suppose the lemma is false, then , , , are linear dependent. So that there exist non-zero satisfying the element-wise equations of , . Since are random generated independent parameters, and rank-3 polynomials could not have non-identical roots almost surely, so it proves the lemma is true almost surely.

###### Lemma 3

and are linear independent almost surely for arbitrary non-identical , where .

Proof:

Suppose satisfying , where denotes an all-zero vector. Derived from (14), the following equation is obtained:

(18) |

(19) | ||||

In (19), since , , , are actually four independent -dimensional bases, then all their coefficients must be zero. We obtain another four linear equations about , , with coefficients composed of , which are all random independent scalar values of the channels. Therefore the solutions of are zero almost surely, which contradicts the initial non-zero assumption. So that it proves the lemma.

###### Lemma 4

All vectors in are linear independent almost surely, where .

Proof:

The procedure is similar to Lemma 3. Only notice that the set contains vectors. The new equation corresponding to (18) has degree of over . Expand to a new equation corresponding to (19), similarly there are totally -dimensional bases such as etc. Then there are linear equations about variables , which are forced to be zero. So that the independency is proved.

Finally, in addition, consider the linear independency of the exponentiation terms of in the precoders in (15). Because the signal dimension is large enough to afford the number of bases in space, so that all are possibly well-conditioned to implement interference alignment. However, it needs further rigorous proof in our future work.

In summary, the comparison of the cases and in signal vector space illustrates that in single antenna constant relay channels, single relay is not able to provide channel randomness or relativity which is the key to interference alignment, while at least two relays are necessary to provide this feature to approach 1/2 DoF (excluding duplex factor) for every user.

### Iv-B Number-Based Signal Level Alignment

Besides signal vector alignment for single antenna constant channels, there is another major class of schemes called signal level interference alignment. There are several approaches to investigate signal level alignment, including lattice coding, deterministic models, and number theory. Among these approaches, IA on the number domain is a very novel and canonical method. While our focus in the following work is to show, in relay connected constant channels, in a similar manner to vector alignment scheme, how level alignment would also face the applicability issue of existing designs when there is only one relay and the issue solved when there are at least two relays.

#### Iv-B1 Basic Concept and Design Procedure

At first, signal level could be viewed on the rational number scale which represents infinite fractional DoF [29]. Then Khintchine-Groshev theorem also reveals that the field of real numbers is rich enough to be equivalent to vector space to design IA. Furthermore, [30, Theorem 7] uses a generalized version of Khintchin-Groshev theorem to extend the designs to complex channels. In real channels, DoF is defined as while in complex channel DoF is defined as where is sum capacity and is transmission power.

By using this Motahari-Khandani scheme in [29, 30], -user single antenna constant channels could approach DoF. Since the objective in this work is to study the role of relays, the core structure and procedure of IA are briefly described in a complex channel setting.

Define as the number of datastreams sent from node to node , . , i.e. belongs to an integer constellation. Each datastream is multiplied by a number , which is called a modulation pseudo-vector serving as distinct directions. In order to satisfy the power constraint and control the minimum distance of the received constellation, transmission signals should be scaled with a constant .

In the meanwhile, each relay node generates a random gain to be a rational number and the equivalent channel from to is as following:

(20) |

Then the received signal at destination node is:

(21) |

According to [29, 30], the structure design of pseudo-vectors on number domain is similar to the design of in vector space of [28], however there exists notable difference as well. In this design, all belong to a set :

(22) |

is integer, so that the number of streams .

For destination node , the received signal space of (21) contains desired signal subspace formed by and interference subspace formed by . Observe all the interference subspaces overlap to the same set of :

(23) |

So that . Meanwhile the desired signal subspace and interference subspace are distinct. According to [29, Theorem 6], the total DoF is , which approaches when is large.

#### Iv-B2 Issue in the Three-User Special Case

In the three-user case, there is a special form of design in [29, Definition 1] presented as following:

(24) | ||||

So that user 1 has the transmit directions , user 2 has the directions of , and user 3 has the directions of .

(25) |

Then the encoding set used by all three users collapses. The scheme of [29, 30] is not applicable to this case to support normal transmission with interference alignment.

If the network has two relays, , IA scheme is applicable as proved by the following lemma:

###### Lemma 5

holds almost surely, for random independent parameters , , , , , , where .

Proof:

(26) | ||||