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# Real Interference Alignment 

Abolfazl Seyed Motahari $^{†}$, Shahab Oveis-Gharan $^{†}$, 

Mohammad-Ali Maddah-Ali $^{††}$, and Amir Keyvan Khandani† 

† Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON, Canada N2L3G1 { abolfazl,shahab,khandani } @cst.uwaterloo.ca 

†† Department of Electrical Engineering and Computer Sciences, University of California-Berkeley, Berkeley, CA, USA maddah-a@eecs.berkeley.edu 

Abstract —In this paper, we show that the total Degrees-Of- Freedoms (DOF) of the K -user Gaussian Interference Chan- nel (GIC) can be achieved by incorporating a new alignment technique known as real interference alignment . This technique compared to its ancestor vector interference alignment performs on a single real line and exploits the properties of real numbers to provide optimal signaling. The real interference alignment relies on a new coding scheme in which several data streams having fractional multiplexing gains are sent by transmitters and interfering streams are aligned at receivers. The coding scheme is backed up by a recent result in the field of Diophantine approximation, which states that the convergence part of the Khintchine-Groshev theorem holds for points on non-degenerate manifolds. 

Index Terms —Interference channels, interference alignment, number theory, Diophantine approximation. 

A CHIEVING the optimum throughput of a system re- quires efficient interference management. Interference alignment is a type of interference management that exploits spatial Degrees-Of-Freedoms (DOF) available at transmitters and receivers. In [1], Maddah-Ali, Motahari, and Khandani introduced the concept of interference alignment and showed its capability in achieving the full Degrees-Of-Freedom (DOF) for certain classes of two-user X channels. 

Interference alignment in n -dimensional Euclidean spaces for n ≥ 2 is studied by several researchers, c.f. [1]–[3]. In this method, at each receiver a subspace is dedicated to interfer- ence, then the signaling is designed such that all the interfering signals are squeezed in the interference subspace. Such an approach saves some dimensions for communicating desired signal, while keeping it completely free from the interference. Using this method, Cadambe and Jafar showed that, contrary to the popular belief, a K -user Gaussian interference channel with varying channel gains can achieve its total DOF, which is K 2 . Later, in [4], it is shown that the same result can be achieved using a simple approach based on a particular pairing of the channel matrices. The assumption of varying channel gains, particularly noting that all the gains should be known at the transmitters, is unrealistic, which limits the application of these important theoretical results in practice. This paper aims to remove this shortcoming by proving that the same result can be achieved by relying on a new alignment technique. 

Some techniques are proposed for alignment in real do- mains. In [5] interference alignment is applied in single 

## I. INTRODUCTION 

antenna systems when all receivers are interference free but one. In [6] and [7], the results from the field of Diophan- tine approximation in Number Theory are used to show that interference can be aligned using properties of rational and irrational numbers and their relations. However, in their examples there is no need for signaling design. The first example of interference alignment in one-dimensional spaces, which requires signaling design, is presented in [8]. Using irra- tional numbers as transmit directions and applying Khintchine- Groshev theorem, [8] shows the two-user X channel achieves its total DOF. In this paper, we extend the result of [8] and prove the following theorem 

Theorem 1: The total DOF of the K -user GIC with real and time invariant channel coefficients is K 2 for almost all channel realizations. 

Remark 1: In [9], the total DOF of the K -user MIMO GIC is addressed. The application of real interference alignment to channels with complex coefficient gain is addressed in [10]. 

The ingredients of real interference alignment technique are 

1) Decoding based on Diophantine approximation. 

2) Alignment based on structural encoding. 

3) Achieving asymptotically perfect alignment based on partial interference alignment. 

We provide three examples to clarify the impact of the preceding items. We first look at a three-user multiple acces s channel modeled by y = x$_{1}$ + ax$_{2}$ + bx$_{3}$ + z. Let us assume that all three users communicate with the receiver using a single data stream. The data streams are modulated by the constellation U = A ( − Q,Q )$_{Z}$ where A is a factor controlling the minimum distance of the received constellation. ( a,b )$_{Z}$ denotes the set of integers between a and b 

The received constellation consists of points representable by A ( u$_{1}$ + au$_{2}$ + bu$_{3}$ ) where u$_{i}$ s are integer. Let us choose two distinct points v$_{1}$ = A ( u$_{1}$ + au$_{2}$ + bu$_{3}$ ) and v$_{2}$ = A ( u ′ $_{1}$+ au ′ $_{2}$+ bu ′ $_{3}$) in the received constellation. The distance between these two points is d = A | ( u$_{1}$ − u ′ $_{1}$)+ a ( u$_{2}$ − u ′ $_{2}$)+ b ( u$_{3}$ − u ′ $_{3}$) | . The following theorem due to Groshev can be used to lower bound the minimum distance of the received constellation. 

Theorem 2 (Khintchine-Groshev): The set of m -tuple real numbers satisfying 

## II. REAL INTERFERENCE ALIGNMENT 

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