RECURSIVE STREAM SELECTION FOR SUM-RATE MAXIMIZATION ON THEINTERFERENCE CHANNEL

Mustapha Amara, Didier Le Ruyet, Mylene Pischella

CNAM CEDRIC/LAETITIA 292 rue Saint Martin 75003 Paris, Francemustapha.amara@eurecom.fr, {didier.le ruyet,mylene.pischella}@cnam.fr

ABSTRACT

This paper presents a successive stream selection algo-rithm than can be used for the interference channel. Theproposed solution is a stream-wise construction of theprecoding and decoding vector sets. These beams are se-lected to minimize the interference between the selectedstreams based on the projections of the nominal direc-tions derived via SVD. The projections are made suchthat the obtained new stream is in both null spaces ofthe receiving and transmitting directions of all previouslyselected streams. This means that it interferes the leastwith the other selected streams and is also weakly inter-fered by them.

1. INTRODUCTION

The degrees of freedom of a wireless interference chan-nel is the number of possible simultaneous streams con-nected between the communicating nodes. In the caseof a non degenerated channel, it is possible to optimallyexploit these degrees of freedom based on linear precod-ing and decoding. In fact, by properly constructing thebeams in such a way that they interfere the least withthe direction used by another couple of communicatingnodes, both couples can communicate properly. Apply-ing this to all K couples, induces a concentration of allthe interference in the non used direction. Thus it hasbeen called Interference Alignment [1] as all the interfer-ence to k is confined at maximum in NRk

−1 dimensions,leaving at least 1 dimension free of interference.The idea of interference alignment evolved starting fromthe 2-user X channel [2, 3] and extended to more andmore complex varieties of communication channels. Thepotential of the overlapping interference spaces was firstpointed out in [4, 5] where, the authors proposed someiterative schemes to compute the precoders and receiversfor the different couple of users.The common formulation of linear interference approachadopted is the one introduced by Jafar in [2]. In the bo-

This work was partially supported by french program FUI Mi-mologic SAMSUFI.

radcast channel (BC), the concept of Interference Align-ment has also been discovered by Weingarten, as men-tioned in [6].Some more recent results about interference alignmentalgorithms are the iterative algorithm proposed by Ne-gro et al. in [7]. This solution is based on a combinationof the WMMSE technique used in the BC and proposedby [8] and the iterative algorithm presented in [9] con-sisting in an alternating optimization procedure to block-diagonalize the interference channel matrix. Another op-timization approach has been proposed by Utschick etal. in [10]. In this paper, the authors propose for thesingle stream per user case a pricing method in orderto enhance the alignment. This pricing methodology al-lows the algorithm to converge to a solution even if thestarting point is not a proper configuration. The pric-ing is performed on the interference parts while gradu-ally increasing the powers. This method has similaritieswith the deterministic annealing solution presented in[11] where the power is incrementally increased and theoptimal solution is computed at each iteration via theWMMSE criteria to follow the homotopic point throughthe crystallization process. Nevertheless, all precoder de-sign techniques for interference alignment, either assumea predefined number of streams per user or require it-erative algorithms for stream allocation such in [7] thatallocates streams while optimizing the precoders and de-coders. Here we propose a recursive algorithm capableof selecting the best number of streams and optimallydistributing it among the considered couples.In this paper, we present in a first part, an unfair streamselection solution and precoder design based on a recur-sive procedure where the least interfering streams aresuccessively selected. In a second part, this procedure isameliorated and formulated in a more fair manner basedon successive projections and null space constructions.The article is organized as follows. The considered sys-tem model is briefly described in Section 2 followed bythe description of the recursive proposed algorithms inSection 3. And finally some simulation results are pre-sented in Section 4.

2012 IEEE 13th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

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2. SYSTEM MODEL

In this paper, we consider an interference channel con-sisting of K couple of users with NTk

transmitting anten-nas and NRk

receiving antennas for couple k. Assumingflat fading environment, the transmission channel is thusa NRk

× NTkmatrix noted Hk,k. The interference chan-

nel from user j to user k is noted Hk,j and is of sizeNRk

× NTj. Such a channel is depicted in Figure 1.

Fig. 1: System model.

The j-th transmitter generates interference to all k � jother receivers. Assuming a constant coefficient channel,the received signal yk at the k-th receiver is

yk = Hk,kxk +K∑

j=1j�k

Hk,jxj +nk (1)

where xj is the NTj× 1 symbol vector transmitted by

the j-th user and the NRk× 1 vector nk is the thermal

additive white Gaussian noise with zero mean and vari-ance σ2 present at receiver k.Each element of the channel matrices are considered ascompletely decorrelated complex random variables.Let Tk of size NTk

× qk denote the beamforming ma-trix of the k-th transmitter. The k-th user is capable oftransmitting qk = min(NRk

,NTk) independent streams.

The xk transmitted symbols can thus be computed as

xk = Tkdk (2)

where dk,l; l ∈ [1, . . . , qk] represents the data symbols tobe transmitted by user l. We assume that the dk,l havea Gaussian distribution with N (0,Iqk

). At the k-th re-ceiver, Dk is a qk ×NRk

decoding matrix applied to yk inorder to suppress the interference. The obtained decodeddata symbols are then the qk × 1 estimates of dk

d̂k = DkHk,kxk +K∑

j=1j�k

DkHk,jxj +Dknk (3)

The objective is to find the best distribution of streamsamong the available ones that maximizes the total sum-rate of the system. The number of streams per user isthus automatically defined by the proposed algorithmswithout any considerations to fairness between users.In the rest of the paper use the following notations. Cap-ital Greek and calligraphic letters denote sets. Upperscript in matrices denote the column vectors of the ma-trix. rank(A) denotes the rank of the matrix A, andx∗ represents the optimal solution for variable x throughthe argmax operator. Finally, P⊥

A denotes the projectionmatrix orthogonally to the space spanned by A

P⊥A = I − AH

(AAH

)−1A (4)

3. RECURSIVE SELECTION PROCEDURE

In this section, the proposed recursive selection proce-dure is explained. The objective here is to recursivelyfind the best combinations of beams. We thus selectthe least interfering links according to a given thresholdε. The idea is first of all to identify the main possiblestreams for couple k. The best stream is identified asthe one presenting the highest singular value, i.e. thebest channel. This initialization is obviously subopti-mal. Nevertheless, the sub-optimality can be minimizedby considering a set of possible initializations.Based on the SVD decomposition Hk,k = UkSkVH

k forall channels, the best stream (k∗, l∗) is identified. Thevirtual channel of the selected stream is

sl∗k∗ = Ul∗H

k∗ Hk∗,k∗Vl∗k∗ (5)

where Vl∗k∗ and Ul∗H

k∗ respectively denote the l∗ columnresp. row of Vk∗ resp. UH

k∗ . The precoder T and decoderD to use for this stream are

T = Vl∗k∗ (6)

D = Ul∗Hk∗ . (7)

We also define the Virtual Receiving Channel (VRC),representing the channel as it is seen by the transmitter,as the cascade of the channel of user k∗ and the corre-sponding decoder

V RCk∗ = Ul∗Hk∗ Hk∗,k∗ (8)

Similarly we define the Virtual Transmitting Channel(VTC), representing the channel seen from the receiver,as the cascade of the channel of user k∗ and the corre-sponding precoder,

V TCk∗ = Hk∗,k∗Vl∗k∗ (9)

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The first algorithm Algorithm 1 considers a predefinedset of streams with fixed directions. The direction ofthe beams are given by the SVD decomposition of allchannel matrices. Based on these SVDs, the initial set Γof potential streams is constructed.

Γ = {(k, l)} ;k ∈ [1 . . .K] ; l ∈ [1 . . .rank(Hk,k)

](10)

The set contains all potential streams. The streams aredenoted by the pair (k, l) where k is the couple of userand l is the position of the stream according to the per-formed SVD. The idea here is to find the ε-least interfer-ing streams in set Γ therefore, each time a stream is se-lected, an estimation of the interference generated to theremaining potential streams and the interference fromthe remaining streams are computed. This is used to re-duce the set of acceptable streams.

Algorithm 1 Successive Stream Selection

0. Compute the SVD of all couplesHk,k = UkSkVH

kConstruct

Γ = {(k, l)} ;k ∈ [1 . . .K] ; l ∈ [1 . . .rank(Hk,k)

]Set Ψ = ∅; i = 1;

1. While Γ � ∅1.1 Find the strongest stream

(k∗, l∗) = arg max(k,l)∈Γ

{sl

k

}1.2 Update

Ψi = (k∗, l∗) Γ = Γ− {(k∗, l∗)}Ti = Vl∗

k∗ Di = Ul∗k∗

1.3 Exclude interfering streams

Γ = Γ−{

arg(j,l)∈Γ,(j,l)�(k∗,l∗)

DHi Hj,k∗Vl

j > ε

}

1.4 Exclude interfered streams

Γ = Γ−{

arg(j,l)∈Γ,(j,l)�(k∗,l∗)

UljHk∗,jTi > ε

}

1.5 Increment i = i+1

end while

A stream is considered as acceptable and is kept in theset Γ if and only if the interference generated to the pre-viously selected streams is lower than the threshold εand the interference gathered from all previously selectedstreams is also lower than ε. Otherwise, the stream isdeleted from set Γ.Practically, the algorithm selects, at each recursion, thenew stream as the stream in set Γ presenting the largestsingular value. Each stream selected at recursion i is

saved in a new set noted Ψ as well as the correspondingprecoder Ti = Vl∗

k∗ and decoder Di = Ul∗k∗ .

It then eliminates from Γ the streams that are either in-terfering or interfered by the newly selected one. Theselected stream (k∗, l∗) is also deleted from Γ. An in-terfering stream (j,m) is detected as the stream witha beamforming precoder in the same direction as thereceiving virtual channel connecting the newly selectedstream (k∗, l∗) and (j,m). The interference is then

Ul∗Hk∗ Hk∗,jVm

j (11)

Identically, an interfered stream (j,m) is detected as thestream with a beamforming decoder in the same direc-tion as the transmitting virtual channel connecting thenewly selected stream (k∗, l∗) and (j,m). The interfer-ence level is then

UmHj Hj,k∗Vl∗

k∗ (12)

The algorithm is run until the set of potential candidatestreams is empty Γ = ∅. It is worth mentioning here thatthe algorithm always converges to a solution for all sys-tem parameters K,NT and NR without having to forcethem to respect a feasible condition as it is done in othersolutions proposed for interference alignment except forthe WMMSE iterative procedure proposed in [7].The second algorithm Algorithm 2 presented in this pa-per is an amelioration of the first one. In fact, as the firstalgorithm is based on a predefined set of streams derivedfrom the SVD decompositions, the probability of findinga new stream allocated to a second couple at the secondrecursion is very small. Thus the algorithm tends to se-lect streams from the same couple almost surely whenε is small enough. This affects a lot the fairness of theselection and is not realistic. Indeed, on the interferencechannel, a couple of user can not be completely deacti-vated and will always generate some interference.To solve these problems, we propose the following modi-fications: streams are selected recursively, the impact ofthe previously selected streams is taken into considera-tion by optimally reorienting the potential beamformers.This can be done by projecting the space spanned bythe remaining beamformers of user j orthogonally to thelast selected stream. This projection produces matricesnoted H⊥

k,k in the null space of the preciously selectedstreams. In fact, when projecting the matrix of user kthe space spanned looses one dimension and its’ largestsingular value decreases. This results in an increasingpenalty with the number of streams allocated to a givencouple k that enhances fairness. On the opposite tothe BC case, two protections are necessary. The firstone reduces the interference generated to the remainingstreams and consists in projecting the channel matricesH⊥

k,k generated at recursion i−1 orthogonally to the vir-

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tual transmitting channel H⊥k,k∗Vl∗

k∗ :{

H⊥k,k

}i= P⊥

Vl∗k∗ Hk,k∗

{H⊥

k,k

}i−1

(13)

The second projection reduces the interference gatheredfrom the remaining streams and consists in projectingthe channel matrices H⊥

k,k generated by the first projec-tion (13) orthogonally to the virtual receiving channelUl∗

k∗Hk∗,k {H⊥

k,k

}i=

{H⊥

k,k

}iP⊥

Ul∗k∗ Hk∗,k

(14)

Algorithm 2 Successive Null Space Stream Selection

0. Set Ψ = ∅; i = 1; H⊥k,k = Hk,k;∀k ∈ [1 . . .K]

Compute the SVD of all couplesH⊥

k,k = UkSkVHk

ConstructΞ =

{sl

k

};k ∈ [1 . . .K] ; l ∈

[1 . . .rank(H⊥

k,k)]

1. While Ξ � ∅1.1 Find the strongest stream

(k∗, l∗) = argmaxΞ(k,l)∈[1...K]×

[1...rank(H⊥

k,k)]

1.2 UpdateTi = Vl∗

k∗ ; DHi = Ul∗

k∗ ; Ψi = (k∗, l∗)1.3 Project orthogonally to the VRC

H⊥k,k = H⊥

k,kP⊥DiHk∗,k

;∀k ∈ [1 . . .K]

1.4 Project orthogonally to the VTCH⊥

k,k = P⊥TiHk,k∗ H⊥

k,k;∀k ∈ [1 . . .K]

1.5 Compute the SVD of all couplesH⊥

k,k = UkSkVHk ;∀k ∈ [1 . . .K]

1.6 Reconstruct ΞΞ =

{sl

k

};k ∈ [1 . . .K] ; l ∈

[1 . . .rank(H⊥

k,k)]

1.7 Increment i = i+1

end while

The SVD is then performed on the projected channelmatrices and the stream corresponding to the largest sin-gular value is selected. The corresponding precoder anddecoder are recursively saved.This proposed solution, and thanks to the successive pro-jection procedure performed on both the VTC and VRC,offers an interference free allocation of streams to the dif-ferent users by confining the interference in the null spaceof the directions selected for streams. Thus, the methodpermits to generate an interference alignment solutionwithout having to consider an a priory verification of the

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Fig. 3: Sumrate as a function of SNR, Algorithm 1.

feasibility conditions as it is usually done for interfer-ence alignment. The algorithm converges automaticallyto the best number of possible streams respecting theinterference alignment solution and enhancing fairness.

4. SIMULATIONS AND RESULTS

In all presented simulations, we consider the same num-ber of receiving antennas for all users NRk

= 3, NTk= 3

and K = 3. Channel coefficients are Rayleigh i.i.d. dis-tributed with E‖hk

i,j‖2=1. The SNR is taken as PT /σ2

where PT represents the total per-user transmit power.The throughput is averaged over channel realizations.Figures 2 and 3 depict the performances of Algorithm1. The first figure plots the achievable throughput infunction on ε. We see through the evolution of the per-formances that for a fixed set of directions, the perfor-mances are best obtained at very small ε. In fact withvery small ε, only the streams of the best user are se-

272

lected, as shown on Figure 2. The system becomes equiv-alent to the single user MIMO. At moderate values of εaround 0.25, some streams of a second user appear inthe selected set but the performances become very weakas plotted in Figure 3. This can easily be explained byhigher levels of interference between streams. Finally,the more ε is relaxed, the more streams are allowed inthe set but also the worse performances get.Figure 3 also presents a throughput comparison of theperformances of the first and second algorithm. It isclearly shown that reorienting the set of potential streamsin the null space of the previously selected ones highlyimproves the performances of the algorithm.Figure 4 represents the sum-rate of Algorithm 2 as afunction of the number of selected streams. These curvesshow that the proposed algorithm is capable of selectingthe optimal number of served streams and being fair atthe same time. The selection is done in the optic of sum-rate maximization.

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Fig. 4: Throughputs for Algorithm 2 in function ofnumber of imposed streams.

Finally analyzing the histogram (not plotted due to spaceconstraint) of the number of selected streams over theSNR range, show that in most of the cases, the algo-rithm selects one stream per couple of users especiallyat low and moderated SNRs. This corresponds to a de-gree of diversity of 3. Nevertheless, at very hight SNRs,the algorithm selects a smaller number of streams, i.e.smaller number of couple of users as the obtained peakis at 2 streams (i.e. 2 selected couples). This is dueto the high interference levels generating an interferencelimited system.

5. CONCLUSION

In this paper, we presented two algorithms for successivestream selection based on the SVD decomposition and

stream selection from the null space of the previouslyselected streams. Simulation results showed that it isimportant to consider a dynamic set of potential streamstaking into consideration the previously selected streamsto best enhance both performances and fairness amongusers.

6. REFERENCES

[1] V.R. Cadambe and S.A. Jafar, “Interference alignmentand spatial degrees of freedom for the k user interferencechannel,” in Proc. IEEE International Conference onCommunications, may 2008, pp. 971 –975.

[2] S.A. Jafar and S. Shamai, “Degrees of freedom region ofthe mimo x channel,” Information Theory, IEEE Trans-actions on, vol. 54, no. 1, pp. 151 –170, jannuary 2008.

[3] M.A. Maddah-Ali, A.S. Motahari, and A.K. Khandani,“Communication over mimo x channels: Interferencealignment, decomposition, and performance analysis,”Information Theory, Transactions on, vol. 54, no. 8, pp.3457 –3470, aug. 2008.

[4] M.A. Maddah-Ali, A.K. Khandani, A.S. Motahari, Uni-versity Waterloo. Dept. Electrical, and Computer Engi-neering, Communication over X channel: Signalling andmultiplexing gain, Dept. of Electrical and Computer En-gineering, University of Waterloo, 2006.

[5] M.A. Maddah-Ali, A.S. Motahari, and A.K. Khandani,“Signaling over mimo multi-base systems: Combinationof multi-access and broadcast schemes,” in Porc. IEEEInternational Symposium on Information Theory, july2006, pp. 2104 –2108.

[6] Hanan Weingarten, “On the compound mimo broadcastchannel,” in in Proc. Annual Information Theory andApplications Workshop UCSD, january-february 2007.

[7] Francesco Negro, S.P. Shenoy, Irfan Ghauri, andD. Slock, “Weighted sum rate maximization in theMIMO Interference Channel,” in Proc. IEEE 21st In-ternational Symposium on Personal Indoor and MobileRadio Communications, September 2010, pp. 684–689.

[8] S.S. Christensen, R. Agarwal, E. de Carvalho, andJ.M. Cioffi, “Weighted sum-rate maximization usingweighted mmse for MIMO-BC beamforming design,” inProc. IEEE International Conference on Communica-tions, June 2009, pp. 1–6.

[9] S.W. Peters and R.W. Heath, “Interference alignmentvia alternating minimization,” in Proc. InternationalConference on Acoustics, Speech and Signal Processing,April 2009, pp. 2445–2448.

[10] D.A. Schmidt and Wolfgang Utschick, “Algorithms forimproper single-stream MIMO interference networks,”in Proc. IEEE 8th International Symposium on WirelessCommunication Systems, November 2011, pp. 251–255.

[11] Francesco Negro, Irfan Ghauri, and D. Slock, “De-terministic annealing design and analysis of the NoisyMIMO Interference Channel,” in Proc. Information The-ory and Applications Workshop (ITA), February 2011,pp. 1–10.

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