Opportunistic Interference Alignment in MIMO Interference Channels

Opportunistic Interfer ence Alignment in MIMO Interfer ence Ch annels Samir Medina Perlaza 1 , M ´ erouane Debbah 2 , Samson Lasaulce 3 and Jean-Marie Chaufray 1 1 France T elecom R&D - Orange Labs Paris. France { Samir .Me dinaPerlaza, Jean Marie.Chaufray } @orange -ftgroup.com 2 Alcatel Luce nt Chair in Flexible Radio - SUP ELEC. France Merouane.De bbah@su pelec.fr 3 Laboratoire des Signaux et Syst ` emes (LSS) - CN RS, SUPELEC, Univ . Paris Sud. France Samson.L asaulce@lss .supelec.fr Abstract —W e present two interference a lignment tech- niques such that an opport unistic point- to-point multiple input multiple output (MIMO) link can reuse, without generating any additional interf erence, the same f requency band of a similar pre-existing primary link. I n this scenario, we explo it the fa ct that under power constra ints, although each radio maximizes independently its rate by water-filling on their channel t ransfer matrix singula r values, frequently , not all of t hem are used. Theref ore, by alig ning the interference o f the opportunistic radio it is possible to transmit at a significant rate while ins uring zero-interference on the pre-existing link. W e pr opose a linear pre-coder for a perfect interfe rence alignment and a power allocat ion scheme which maximizes the individual data rate of the secondary link. Our numerical results s how that significant data rates are achieved even for a reduced number of antennas. I . I N T RO D U C T I O N W e con sider the ca se of radio devices attempting to opportunistically exploit the same frequency b and being utilized by license d networks under the constraint that no additional interference must be generated . This can be implemented for exa mple by as suming that opp ortunistic users are c ogniti ve radios [1], [2]. T y pically , cognitiv e radios temporally exploit the unus ed frequency bands, named white-spa ces, to transmit their data. T his c learly improves the spectral efficiency sinc e more use rs are allowed to co-exist in the same bandwidth. However , a higher spec trum ef ficiency co uld be attained by simul- taneously allo wing the opportunistic radios to transmit with the licensees if no harmful interference is g ener- ated. In this c ase, interference alignment (IA) has been identified as a powerful tool to achieve such a goal. This tech nique, was initially introduced in [3], [4]. IA allows a giv en transmitter t o partially or completely “align” its interferenc e with unused d imensions of the primary terminals [5]. The concept of d imension cou ld T x 1 T x 2 Rx 1 Rx 2 Primary Link Secondary Link H 1 , 1 H 2 , 2 H 1 , 2 H 2 , 1 N t N t N r N r 2 1 2 1 2 1 2 1 Figure 1. T wo-user MIMO interference channel. be a ssociated with a s pecific sp atial direction, freque ncy carrier or time s lot [4], [6]. An extensiv e stud y has been condu cted [4], [6], [7] to es timate the number of interference-free d imensions a giv en radio might find to transmit when s ev eral ra dio sy stems co-exist. In [4] and [6] several IA schemes to exploit such d imensions are proposed . Similarly , in [8], a linear pre-code r based on V andermond e matrices allows an orthogon al frequency divi sion multiplexing (OFDM) radio to co-exist with similar pre-existing terminals without ge nerating any additional interference. The idea is to exploit the re- dundan cy o f the O FDM cyclic prefix and frequency selectivity of the c hannel. In this s tudy , we propose a novel interference align- ment tec hnique for the sec ondary users exploiting the fact tha t under a power-l imitation, a primary use r which maximizes its own rate b y water-fill ing on its MIMO channe l singu lar values, might leave some of them unused . i.e. no transmission takes plac e along the corre- sponding spatial directions. The se unused directions may be op portunistically utilized by a se conda ry transmitter , since its signal would not interfere with the signal sent by the primary transmitter . W e p resent a linear pre-coder which perfectly aligns the interference gen erated by the second ary transmitter with such unuse d spatial direc- tions. Similarly , we present a power alloca tion scheme based on the water- filling idea which maximizes the indi vidual data r ate of the opportunistic radio. Simulation results show that a significa nt da ta rate can be a chieved by the se condary link follo wing our approach. I I . S Y S T E M M O D E L Notation: In the following, matrices and vectors are denoted by boldface up per case symbols a nd boldface lower c ase symbols, respectively . Th e i th entry of the vector x is d enoted x ( i ) . The entry correspo nding to the i th row and j th column of the matrix X is d enoted by X ( i, j ) . Th e N -dimension ide ntity a nd nu ll matrix are represented by I N and 0 N , respectively . The Hermitian transpose is denoted ( · ) H , and the expected value is represented by the operator E [ . ] . W e consider two point-to-point unidirectional links simultaneously operating in the same frequency ban d and producing mutual interference a s shown in Fig. 1 [9]. Both transmitters are eq uipped with N t antennas while both receiv ers u se N r antennas . The first transmitter- receiv er pair , i.e. T x 1 and Rx 1 , is a primary l ink licensed to exclusiv ely exploit a given frequen cy band. The pair , T x 2 and Rx 2 is an opportunistic li nk exploiting the same frequency band sub ject to the con straint that no ad di- tional interference must be ge nerated over the primary system. Note that no coope ration be tween terminals is allo wed, i.e. transmitters do not share or excha nge any signal before transmitting. Therefore, the mu ltiple acces s interference (MAI) is considered as additi ve white Gaussian noise (A WGN). The chan nel transfer matrix from transmitter j ∈ { 1 , 2 } to receiver i ∈ { 1 , 2 } is de noted H i,j ∈ C N r × N t , where the entries of H i,j are indepe ndent and identically distrib uted (i.i.d) comp lex Gaussian circularly sy mmetric random variables. The cha nnel ma trices are suppos ed to be fixed for the whole transmission du ration. This correspond to assu ming (static) Gau ssian links. But our analysis readily extends to the case of slow- fading channe ls b y as suming the cha nnels to be cons tant over each data block. Regarding the chan nel state informa- tion (CSI) conditions, we assume the primary terminals (transmitter and recei ver) to only ha ve perfect knowledge of the matrix H 1 , 1 . On the othe r han d, the sec ondary terminals hav e perfect knowledge of a ll the channe l transfer matrices H i,j , for e very i and j ∈ { 1 , 2 } . Although unrealistic, this c ondition provides us with an upper bound on the achie vable rate of the second ary user . It can h owe ver be met in practice in the TDD (T ime Di vision D uplex) mode if the s econda ry user exploits opportunistically the training se quenc es a nd s ignaling communication between the primary devices. Follo wing a matrix notation, the primary and sec- ondary received signals ca n be written as  y 1 y 2  =  H 1 , 1 H 1 , 2 H 2 , 1 H 2 , 2   V 1 s 1 V 2 s 2  +  n 1 n 2  , (1) where the vectors s i ∈ C N t × 1 and n i ∈ C N r × 1 represent the transmitted symbols and an A WGN proces s with zero mean and covari ance matrix σ 2 I N r for the i th link. For all i ∈ { 1 , 2 } the matrices V i ∈ C N t × N t represent the linear pre-coders us ed for interferenc e a lignment. Furthermore, at each receiver , the inpu t signal is linearly process ed with the matrix F i ∈ C N r × N r . The sign al at the ou tput of the linear filter i is r i = F i y i . Both matrices V i and F i are de scribed later on. The po wer allocation matrices are defined as the inpu t covariance matrices P i = E  s i s H i  ∈ ( R + ) N t × N t , for the i th transmitter . The power c onstraints are ∀ i ∈ { 1 , 2 } , T race  V i P i V H i  6 p i, max , (2) where p i, max is the maximum transmit p ower lev el for the i th transmitter . W ithout los s o f gene rality , we as sume identical max imum transmit powers for all the terminals i.e. ∀ i ∈ { 1 , 2 } , p i, max = p max . I I I . I N T E R F E R E N C E A L I G N M E N T S T R A T E G Y In this se ction we foc us on the study of the p re-coding V i and p ost-processing F i matrices. Suppose that the primary terminals complete ly ignore the prese nce of the opportunistic transmitter . H ence, in order to maximize its own data rate, the primary transmitter follows a water - filling power a llocation as in the single-user case [10]. A. Primary link design Under the assumption that the channel ma trix H 1 , 1 is known at the rec eiv er a nd transmitter , the primary terminal c hoose s its pre-coding V 1 and post-process ing F 1 matrices in suc h a way that their channel transfer matrix is dia gonalized, i.e. V 1 and F 1 = U H 1 satisfy the singular value d ecompos ition H 1 , 1 = U 1 Λ 1 V H 1 , where U 1 ∈ C N r × N r and V 1 ∈ C N t × N t are unitary matrices and Λ 1 ∈ ( R + ) N r × N t is a diagona l matrix which contains min( N r , N t ) n on-zero singular values, λ 1 , . . . , λ min( N r ,N t ) . Thus, the receiv ed signal after linear process ing, r 1 , can be written as r 1 = F 1 y 1 = Λ 1 s 1 + n ′ 1 , (3) where n ′ 1 = U H n 1 is an A WGN process w ith zero mean an d covariance matrix σ 2 I N r . The n, the achiev able rate of the primary user is ma ximized by the power allocation ma trix P 1 which is a so lution to the following optimization problem maximize log 2   I N t + 1 σ 2 H 1 , 1 V 1 P 1 V H 1 H H 1 , 1   subject to T race ( P 1 ) 6 p max . (4) The solution to (4) is the classical water- filling a lgorithm [10]. Followi ng this approac h, the optimal power alloca- tion matrix is a diagona l matrix with entries ∀ n ∈ { 1 , . . . , N t } , P 1 ( n, n ) =  β − σ 2 λ 2 n  + , (5) with [ p ] + = max (0 , p ) . The c onstant β is a La grangian multiplier that is d etermined to sa tisfy N X j =1 P 1 ( j, j ) = p max . B. Seconda ry link d esign Depending on the channel singular v alues λ 1 , . . . , λ min( N r ,N t ) , the po wer allocation matrix P 1 might con tain zeros in its main d iagonal. A zero power allocation for a g i ven s ingular v alue means that no transmiss ion takes place along the co rresponding spatial direction. This means that the s econda ry terminal can a lign its transmitted signa l with the unuse d singular modes such that it does not interfere with the signal transmitted by the primary user . If o ne con verts the spatial problem into the freque ncy one, the result is similar to the cog niti ve s cenario where the sec ondary would o pportunistically use the unexploited frequency modes. The main difference here lies in the fact tha t in the spa tial d omain, there is no universal prec oder which diagonalizes the ba sis of all the devices wherea s this is the c ase in the frequency domain with the use o f the FFT . As a c onseq uence, in the spatial domain, the co r - responding orthogo nality co ndition (suc h that the sec- ondary user gene rates no interference on the primary link) is given by U H 1 H 1 , 2 V 2 = α ¯ P 1 , (6) where the ma trix ¯ P 1 is a diago nal matrix with entries ∀ n ∈ { 1 , . . . , N t } , ¯ P 1 ( n, n ) =  σ 2 λ 2 n − β  + , (7) such that the cond ition P 1 ¯ P 1 = 0 N r always holds. It can be e asily verified since both matrices are diag onal. Additionally , the co nstant α is ch osen to satisfy the power con straints (2) with i = 2 . Assuming tha t perfect estimates of H 1 , 1 and H 1 , 2 are available at the s econda ry transmitter , the secon dary precoder (when the in verse of H 1 , 2 exists) is giv en b y: V 2 = α H − 1 1 , 2 U 1 ¯ P 1 . (8) For the cas e where N r > N t , i.e. the receiv er ha s more antennas than the transmitter , it is s till p ossible to obtain the pre-coding matrix by using the Moore-Penrose pseudo -in verse of H 1 , 2 , V 2 = α  H H 1 , 2 H 1 , 2  − 1 H H 1 , 2 U 1 ¯ P 1 . (9) Once the pre-decod er V 2 has been a dapted to s atisfy (8) or (9) at the second ary transmitter , no a dditional interference impairs the primary user . Howe ver , the second ary rec eiv er still undergoes the interferenc e from the primary transmitter . T ypica lly , this effect is a colored noise with covariance Q ∈ C N r × N r due to the c hannel H 2 , 1 and the pre-cod er V 1 . Here, Q = H 2 , 1 V 1 P 1 V H 1 H H 2 , 1 + σ 2 I N r . (10) Hence, the receiv ed sign al y 2 can be whitened by u sing the matrix F 2 = Q − 1 2 , to ob tain r 2 = F 2 y 2 , suc h that r 2 = Q − 1 2 H 2 , 2 V 2 s 2 + n ′ 2 , (11) where n ′ 2 = Q − 1 2 ( H 2 , 1 V 1 s 1 + n 2 ) is an i.i.d. A WGN process with zero mean and a covariance ma trix p ropor- tional to the identity . Let S be the number of zeros on the ma in diag onal of P 1 . Th en, the matrix V 2 contains N r − S zero c olumns. Note that S = 0 implies that no transmission takes p lace in the sec ondary link. In the sequel, we always assume that S > 0 (which w ill be the case at low signal to noise ratio as shown in the simulations). I V . I N P U T C OV A R I A N C E M A T R I X O P T I M I Z A T I O N In the latter section the proposed p re-coding sch eme does no t ge nerate any inte rference on the primary use r but the transmission rate for the secon dary us er was not optimized. For this pu rpose, the cho ice of the power al- location of the seco ndary transmitter , i.e. the matrix P 2 , needs to b e optimized. F irst, we present the mos t simple case where un iform p ower allocation is performed. Sec- ond, we introduce a power alloca tion which maximizes the indi vidual transmission rate. In both ca ses we assume that the pre-code r has be en p re viously ada pted to sa tisfy the orthogonality cond itions (8) or (9). A. Uniform P owe r Alloca tion For the u niform power allocation sc heme the inp ut covari ance matrix is set to P 2 = I N t and the constan t α from (6) is tune d in orde r to mee t the condition T race  V 2 V H 2  = p max . Th e rate ach iev ed by the second ary user while generating z ero-interference to the primary receiver is R 2 = log 2    I N r + Q − 1 2 H 2 , 2 V 2 V H 2 H H 2 , 2 Q − 1 2    . (12) B. Optimal P owe r A llocation The transmission rate for the se condary link is maxi- mized by adopting a power allocation matrix P 2 which is a solution of the followi ng op timization problem, arg max P 2 R 2 ( P 2 ) s.t. T race  V 2 P 2 V H 2  6 p max , (13) where R 2 ( P 2 ) = log 2    I N + Q − 1 2 H 2 , 2 V 2 P 2 V H 2 H H 2 , 2 Q − 1 2    . (14) Note that so lving this optimization problem requ ires the knowledge of the covariance ma trix Q , whic h is calcu - lated at the seco ndary receiver based on the knowledge of the chan nel H 2 , 1 . This can be done if the seconda ry receiv er e stimates Q a nd feeds it back to the seco ndary transmitter . Here, we as sume a perfect kn owl edge of Q is av ailable a t the se conda ry transmitter . By definition (Eq. ( 13)), the matrix V 2 is not full rank. Therefore, the o ptimization problem (13) does not hav e a simple so lution. W e propos e a two-step optimization which lea ds to a water -filling s olution. First, w e defi ne a new inpu t covariance ˆ P 2 such that, ˆ P 2 =  V H 2 V 2  1 2 P 2  V H 2 V 2  1 2 . (15) By replacing t he expression (15) in (13), t he optimization problem become s arg max ˆ P 2 n log 2    I N r + G ˆ P 2 G H    o s.t T race  ˆ P 2  = p max , (16) where G = Q − 1 2 H 2 , 2 V 2  V H 2 V 2  − 1 2 ∈ C N r × N t . The idea here is to so lve the a priori non-trivial optimization problem defined by express ion (13) by introducing an equiv alent chann el matrix G to simplify the problem. Using G w e can then apply a s ingular value de composi- tion to the new chan nel such that G = E ∆ Z H , wh ere E ∈ C N r × N r and Z ∈ C N t × N t are un itary matrices , and the matrix ∆ ∈ ( R + ) N r × N t contains the s ingular values η 1 , . . . , η min { N r ,N t } of G . U nder thes e assumptions , the optimal solution P ∗ 2 = Z H ˆ P 2 Z is ∀ n ∈ { 1 , . . . , N t } , P ∗ 2 ( n, n ) =  ρ − 1 η 2 n  + , (17) where ρ is a Lagrang ian multiplier that is de termined to satisfy N X j =1 P ∗ 2 ( j, j ) = p max . Once P ∗ 2 has been obtained, then the op timal power alloca tion matrix [10] is P 2 =  V H 2 V 2  − 1 2 Z ˆ P ∗ 2 Z H  V H 2 V 2  − 1 2 (18) The cons tant α in (6) is tune d su ch that the condition (2) is met for i = 2 . V . N U M E R I C A L R E S U LT S In this section we s how n umerical examples to il- lustrate the performance of our interference alignment strategy . Con sidering the same number of antenna s at the receiver and the transmitter , we an alyze the numbe r of un used singular v alues or free dimensions available for the s econda ry link as well as its achieved data rate. Recall that the primary li nk is interference-free, therefore its d ata rate correspon ds to the sing le u ser c ase rate [10]. In Fig. 2 we show the number of unus ed singular values in the primary link as a func tion of the n umber of antennas an d S N R = p max σ 2 . No te that in low SNR regime, the transmitter attempts to conce ntrate all its power in the best singular values leaving all the oth- ers u nused. On the co ntrary , in high SNR regime the primary transmitter tends to spread its power amon g all its av ailable singular v alues. Thus , in the first case the oppo rtunistic link ha s plenty of free dimen sions, while in the sec ond one, it is effecti vely limited. This power a llocation be havior ha s been a lso repo rted in [11] and [12]. Similarly , it is observed that increasing the number o f antennas leads on av erage, to a linea r sc aling of the unu sed singular values. In Fig. 3, we show the achieved data rate of se conda ry link whe n optimal power allocation is implemented ( R 2 , optimal ) as a func tion o f the numbe r o f antennas and the SNR. Therein, it is shown that at very low an d very high SNRs the data rate approach es zero bits/sec. In lo w SNR regime this effect is na tural since de tection is difficult due to the nois e. Howe ver , at high SNRs it is d ue to the fact that the primary trans mitter does not leave any unused singular value. No netheless , at intermediate SNRs, significan t data rates are achieved by the s econd ary link. Note that increasing the number of a ntennas always leads to higher data rates for the s econda ry link. In Fig. 4, we plot the data rate of the primary R 1 and secondary link for the case of uniform R 2 , uniform and o ptimal R 2 , optimal power allocation. No te tha t in h igh SNR regime, the R 2 , uniform and R 2 , optimal performs similarly . It is due t o the fact that few or even none of the singular values are left unused by the primary link, the refore the uniform power allocation does n ot differ from the optimal in av erage. In contrast, at intermediate SNRs the dif ferenc e in performance is more significant for a lar ge nu mber of antenn as. V I . C O N C L U S I O N S W e provided a novel interference alignment sche me which a llo ws an opportunistic p oint-to-point MIMO link to co-exist with a s imilar pre-existing primary link on the same fully utilized band without generating any ad- ditional interference. The proposed scheme exploits the fact that t he licensed t ransmitter , while performi ng water - filling power allocation on it s MIMO channe l (in order to −20 −10 0 10 20 0 2 4 6 8 10 0 2 4 6 8 10 SNR [dB] Number of Antennas (N r = N t ) Free Dimensions Figure 2. A verage number of unused singular valu es in the primary link as a function of the total number of antennas and the S N R = p max σ 2 . The SNR and the number of antennas N r = N t are assumed the same for the primary and secondary links −20 −10 0 10 20 0 2 4 6 8 10 0 1 2 3 4 5 SNR [dB] Number of Antennas (N r = N t ) Data Rate [bits/sec] Figure 3. A verage data rate of t he secondary link when optimal po wer allocation is implemented as a function of the number of antennas N r = N t and S N R = p max σ 2 . The SNR and the number of antennas are assumed t he same for the primary and secondary link. maximize its single user rate) will leav e so me sing ular values unused when c onstrained by p ower limitati ons. Hence, no t ransmission takes place along the correspond- ing spatial directions. W e propos ed a linear pre-cod er for the opp ortunistic radio, wh ich perfectly aligns the transmitted signa l with s uch unused dimen sions. W e also provided a power a llocation sche me which maximizes the data rate of the opportunistic link. Numerica l results show that significant data rates are a chieved by the second ary link e ven for a reduced n umber of an tennas. Further stud ies w ill extend the novel approa ch to multi- user multi-carrier s ystems and the case of incomp lete CSI. 0 5 10 15 20 0 5 10 15 20 25 SNR [dB] Data Rate [bits/seg] Primary Link (N r = N t = 3) Secondary Link − Uniform Power (N r = N t = 3) Secondary Link − Optimal Power (N r = N t = 3) Primary Link (N r = N t = 20) Secondary Link − Uniform Power (N r = N t = 20) Secondary Link − Optimal Power (N r = N t = 20) Figure 4. 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