An Improved Interference Mitigation Scheme Basedon Interference Subspace Alignment

Shiwen HeSchool of Information Science

and Engineering, Southeast University,Nanjing 210096, China

Email: hesw01@seu.edu.cn

Yongming HuangSchool of Information Science

and Engineering, Southeast University,Nanjing 210096, China

Email: huangym@seu.edu.cn

Luxi YangSchool of Information Science

and Engineering, Southeast University,Nanjing 210096, China

Email: lxyang@seu.edu.cn

Abstract—This paper proposes an improved interference sub-space alignment scheme for the K-user interference channels,which combines the conventional interference subspace alignmentalgorithm and the power allocation algorithm between datastreams. More specifically, in the proposed scheme a power allo-cation scheme is developed to mitigate the leftover interferencefrom the conventional interference subspace alignment algorithm,achieving a performance improvement in terms of sum rate. Theadvantage of the proposed scheme is confirmed by computersimulation results.

Keywords—Interference Alignment, Power Allocation, SignalSubspace, Interference Subspace

I. INTRODUCTION

In interference channels, it is a general way to regardinterference signal as a “harmful” signal and strive to avoidthem by various strategies, such as the interference avoidance[1] and iterative waterfilling algorithms. Recently, interferencealignment (IA) which does not regard interference signal as a“harmful” signal has become an important tool to study thedegree of freedom of interference channel [2], [3], [4], [5], [6],[7]. It is shown that the number of degrees of freedom achievedby IA in interference channels scales linearly with the numberof users[4], [8], namely, C= K

2 log (SINR) + o(log (SINR)),K denotes the number of pairs between the transmittersand receivers, the o(log (SINR)) term becomes negligiblecompared to log (SINR) at high SNRs. The idea of the IA is tomake use of the symbol extension in time or frequency as longas the channel is varying across time or frequency. As multipleantenna techniques have been extensively studied recently [9],[10], [11], [12], [13], [14], it is natural to also extend symbolover antennas in IA design for MIMO interference channels. Inthis regard, some iterative IA algorithms based on interferenceand signal subspace decomposition were proposed [15], [16],[17], [18], [19]. However, these algorithms only considerhow to ensure that all interference signal at the receiver fall

This work was supported by the National Natural Science Foundationof China under Grants 60902012 and 61071113, the National Science andTechnology Major Project of China under Grants 2011ZX03003-001-02and 2011ZX03003-003-03, the Ph.D. Programs Foundation of Ministry ofEducation of China under Grants 20090092120013 and 20100092110010, theNatural Science Foundation of Jiangsu Province under Grants BK2011598and BK2011019.

into the interference subspace while the desired signal at thereceiver fall into the signal subspace. Due to limited iterationor numerical rounding errors, it is inevitable that a portion ofinterference signal leaks into the signal subspace, resulting inperformance degradation.

To address this issue, in this paper we proposed an improvedinterference subspace alignment algorithm by jointly usinginterference subspace alignment strategy and power allocation.The leftover interference from the conventional subspace-based IA scheme is mitigated by a post-processing power al-location between data streams. The advantage of the proposedscheme is validated via simulation results.

The rest of this paper is organized as follows: we describethe system model in section II. In section III, we illustratethe subspace IA algorithm and the improved subspace IAalgorithm. The simulation results are given in section IV, andconclusion is finally drawn in section V.

II. SYSTEM MODEL

We consider a K-user pairs MIMO interference channelsystem where the kth transmitter and receiver are equippedwith Mk and Nk antennas, respectively, as shown in Fig.1.

Fig. 1. System Model

The receive signal of the kth user is written as:

yk =K∑i=1

Hk,ixi + nk, k = 1, · · · ,K (1)

where nk denotes zero mean unit variance circularly sym-metric additive white Gaussian noise vector at receiver k, xk

denotes the Mk×1 transmitted signal vector at the transmitter

978-1-4577-1010-0/11/$26.00 ©2011 IEEE

k and the Nk ×Mi matrix Hk,i models the channel betweentransmitter i and receiver k. The transmit power at transmitteri denoted as E[∥xi∥2] = pi.

Given the channel matrix Hk,i, k, i = 1, · · · ,K, the basicidea of the IA is to design a precoding matrix and aninterference suppression matrix at each transmitter and eachreceiver, respectively, to satisfy the following conditions givenby

UHk Hk,iVi = 0, ∀i ̸= k

rank(UHk Hk,kVk) = dk

(2)

where Vk is a Mk × dk precoding matrix at transmitter kand Uk is a Nk × dk interference suppression at receiverk, dk ≤ min(Mk, Nk) denotes the degree of freedom foruser k, and the superscript H denotes conjugate transpose.The solution to the feasibility condition Eq.(2) is not knownin general. In other words, given a set of randomly gen-erated channel matrices and a degree-of-freedom allocation(d1, · · · , dK), it is not known that whether one can alwaysfind the transmit precoding matrices and the interferencesuppression matrices that will satisfy the conditions of Eq.(2).In this paper, we factorize the interference suppression matrixUk into an interference subspace matrix and a filtering matrix,namely

Uk = WHk GH

k (3)

Without ambiguity, the matrix Wk is called interferencesuppression matrix and the matrix Gk denotes as the linearreceived filter. Combining the precoding matrices, the Eq.(1)could be rewritten as follows:

yk = Hk,kVKsk +K∑i=1i̸=k

Hk,iVisi + nk, k = 1, · · · ,K (4)

where sk is the symbol vector intended for users k, assumingE(sksHk ) = I . The second term in the above equation iscalled coordinated interference term, since it is caused by othertransmitters and can be coordinated to minimize its effect onthe desired signal.

III. IMPROVED SUBSPACE IA ALGORITHM

From the above analysis, we can learn that the technique ofthe interference alignment separates the received signal spaceinto two linearly independent subspaces, denoted as signalsubspace Sk and interference subspace Ck, respectively. Theexisting interference alignment techniques strive to ensure allinterference signal at the receiver fall into the interferencesubspace, and the desired signal fall into the signal subspace.To achieve this aim, a typical solution is to perform alternatingoptimization between each transmitter and receiver [16], whichis implemented in an iterative manner. It should be noted thatthis strategy only strives to restrict the interference signal intothe interference subspace by iterative optimization. It doesnot consider the impact of the leftover interference signalon the desired signal. Often, the iterative optimization IAalgorithm can not restrict the interference signal perfectly into

the interference space. To address this problem, in this sectionwe present an improved subspace IA algorithm that combinessubspace IA algorithm and power allocation algorithm.

A. Subspace IA AlgorithmDefinition 1. The distance between two orthonormal matricesA and B (or equivalently between two subspaces) is definedas

∥A, B∥M = ∥A−BBHA∥F (5)

where ∥ • ∥F denotes the Frobenius norm.

We first review the subspace IA algorithm which wasformulated in the paper [16]. Given the precoding matrix{Vk}Kk=1 , let Ck denote the orthonormal basis matrix ofthe interference subspace Ck. In order to restrict as more aspossible the interference signal from other transmitters i, i ̸= kinto the interference subspace Ck, the optimization problem isformulated using the following criterion,

Ck = argmin︸ ︷︷ ︸CK

k CK=I

K∑i=1i̸=k

∥Hk,iVi, Ck∥2M

= argmin︸ ︷︷ ︸CK

k CK=I

K∑i=1i̸=k

∥Hk,iVi −CkCHk Hk,iVi∥2

(6)

Using the basic properties of linear algebra tr(AB) =tr(BA) and ∥A∥2F = tr(AAH), we have

Ck = argmin︸ ︷︷ ︸CK

k CK=I

K∑i=1i̸=k

∥Hk,iVi −CkCHk Hk,iVi∥2

= argmax︸ ︷︷ ︸CK

k CK=I

CHk tr

( K∑i=1i̸=k

Hk,iViVHi HH

k,i

)Ck

(7)

It is easy to see that the solution of Ck is the eigenvec-tors corresponding to the largest Nk − dk eigenvalues of∑K

i=1,i̸=k Hk,iViVHi HH

k,i. When the interference subspacebasis {Ci}Ki=1 has been given, the optimal precoding Vk canbe determined from the following optimization problem

Vk = argmin︸ ︷︷ ︸V K

k VK=I

K∑i=1i̸=k

∥Hi,kVk, Ci∥2M

= argmin︸ ︷︷ ︸V K

k VK=I

K∑i=1i̸=k

∥Hi,kVk −CiCHi Hi,kVk∥2

(8)

Using a similar method, we can get that the solution of Vk isthe eigenvectors corresponding to the smallest dk eigenvaluesof

∑Ki=1,i̸=k H

Hi,k(I − CiC

Hi )HH

i,k. Based on these results,the alternating minimization method can be used to solve thesubspace IA problem formulated as [16]

min︸︷︷︸V H

i Vi=I,∀i,CH

k Ck=I,∀k

K∑k=1

K∑i=1i̸=k

∥Hk,iVi −CkCHk Hk,iVi∥2F , (9)

where the interference suppression matrix at each receiver isformualted as Wk = I −CkC

Hk .

B. Power Allocation For Minimization Leakage InterferencePower

The goal of the above subspace IA algorithm is to ensureall interference signal at the receiver fall into the interferencesubspace, which is implemented in an iterative manner. How-ever, due to the limited iterative times, it is highly possible thatthe interference signal can not fully fall into the interferencesubspace. This means that there are some leftover interferencesignal falling into the desired signal subspace, resulting in aperformance loss. To address this problem, we propose to use apower allocation algorithm to reduce the impact of interferencesignal on the desired signal.

It is known from the above subspace IA algorithm thatthe columns of the precoding matrix Vk are the least dkdominant eigenvectors of

∑Ki=1i̸=k

HHi,k(I − CiC

Hi )HH

i,k. This

means that each data stream employs identical transmit power.A proper power allocation between data streams has a potentialof further improving the system performance. Motivated bythis, we formulate the following power allocation problem forthe subspace IA scheme,

Pk = argmin︸ ︷︷ ︸P

tr(PHV H

k

( K∑i=1i̸=k

HHi,k(I −CiC

Hi )Hi,k

)VkP

)

s.t.

dk∑i=1

pi,i = pk

(10)

where, P = diag(√p1,1, · · · ,

√pdk,dk

) denotes the powerallocation matrix at transmitter k for each transmitted datastream. Next we will develop the following theorem to solvethis problem.

Theorem 1. Let A and P denote two diagonal matri-ces respectively, i.e., A = diag(λ1, · · · , λN ) and P =diag(p1, · · · , pN ), satisfying λ1 ≥ λ2 ≥ · · · ≥ λN > 0 and∑N

i=1 pi = p, pi ≥ 0, i = 1, · · · , N . Then, the solution to thefollowing optimization problem:

argmin︸ ︷︷ ︸P

tr(AP ), s.t.

N∑i=1

pi = p (11)

is given bypi =

p1λi

∑Ni=1

1λi

(12)

Proof: It is easy to prove the following inequality:

tr

λ1 · · · 0

.... . .

...0 · · · λN

p1 · · · 0

.... . .

...0 · · · pN

=N∑i=1

λipi ≥ NN

√√√√ N∏i=1

λipi

(13)

Based on which, one can see that the lower bound can beobtained by setting λ1p1 = · · · = λNpN . Thus, under thecondition

∑Ni=1 pi = p, we can get the equation (12).

Following this theorem, the solution to the optimizationproblem (10) for the power allocation at transmitter k iswritten as:

pi,i =pk

1λMk−dk+i

∑dk

l=11

λMk−dk+l

, i = 1, · · · , dk (14)

where, {λMk−dk+i}dki=1 is the least significant dk eigenvalues

of the∑K

i=1,i̸=k HHi,k(I − CiC

Hi )HH

i,k, ∀i. By getting thepower allocation matrix Pk for data streams at transmitterk, we can combine the power allocation matrix Pk with theprecoding matrix Vk to precode the transmit symbol vectorsk, namely xk = VkPksk. Eq.(1) can be rewritten as:

yk =

K∑i=1

Hk,iViPisi + nk, k = 1, · · · ,K (15)

Then, the coordinated interference could be canceled with leftmultiplication of Wk at receiver k.

Note that each receiver must still separate the desired spatialstreams after the coordinated interference has been canceledthrough left multiplication of Wk. For example, we can designthe MMSE filter for the receiver filter Gk to separate thedesired signal streams. In this paper, we do not explicit thedesign of the receive filter.

IV. SIMULATION RESULTS

The general setup for our simulation is as follows. Wedenote the K user MIMO interference channel as (Nk ×Mk, dk)

k[20]. Once the precoding matrices and the powerallocation matrices of the IA algorithm are designed, the sumrate can be computed as following

R =

K∑k=1

log

∣∣∣∣I +

(Rk +

K∑i=1i̸=k

Hk,iViPiPHi V H

i HHk,i

)−1

Hk,kVkPkPHk V H

k HHk,k

∣∣∣∣(16)

where, Rk = E(nknHk ) = I . We regard the sum rate as

performance metric to compare the performance of differentIA algorithms because of the ability of the sum rate capturethe total network throughput in a single scalar. Note that thesum rate is obtained after the ideal non-linear decoding of thesignal at the receivers. We assume that each transmitter hasthe same maximum transmit power P .

Fig.2 and Fig.3 show the sum rate performance of the im-proved subspace IA algorithm, subspace IA algorithm, hybridinterference and signal IA algorithm and maximum SINRalgorithm for the interference channel of (5×5, 2)3,(6×6, 2)4

respectively. The latter three algorithms adopt uniform powerallocation between data streams. The simulation results showthat the improved subspace IA algorithm gets a better per-formance than that of the hybrid interference and signal IA

algorithm and that of the subspace IA algorithm, but has a littlegap from the performance of the maximum SINR algorithm. Itshows that through allocating the transmit power for differentdata streams can further reduce the power of interferencesignals at receivers.

Fig. 2. The sum rate curve of (5× 5, 2)3 interference channel

Fig. 3. The sum rate curve of (6× 6, 2)4 interference channel

V. CONCLUSION

The conventional subspace IA algorithm only aims to ensurethat all interference signal at the receivers falls into the respec-tive interference subspaces, while the desired signal falls intothe signal subspaces. But in real application it is impossible toput all interference signal exactly into each limited dimensionsubspace at each receiver. This leads to that some interferencesignal may leak into the desired signal subspace. In orderto further reduce the impact of interference signal on the

desired signal, we have designed a new effective algorith-m of combining the subspace IA algorithm and the powerallocation algorithm. The simulation results demonstrate thatthe sum rate of the proposed scheme is improved contrast tothe conventional subspace IA algorithm. Through the workof this paper, we illustrate the importance of adopting thepower allocation in the interference suppression after applyinginterference alignment. However, how to allocate differentnumber of data streams for different transmitter is still an openproblem, which we will leave as our future work.

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