Opportunstc Interference Algnment wth -Bt Feedbac n 3-Cell Interference Channels Zhnan Xu, Mngmng Gan, Thomas Zemen,2 FTW (Telecommuncatons Research Center Venna), Venna, Austra 2 AIT Austran Insttute of Technology, Venna, Austra Contact: xu@ftw.at Abstract Opportunstc nterference algnment (OIA) explots channel randomness and multuser dversty by user selecton. The transmtter needs channel state nformaton (CSI), whch s usually measured on the recever sde and sent to the transmtter sde va a feedbac channel. Lee and Cho show that d degrees of freedom (DoF) per transmtter are achevable n a 3-cell MIMO nterference channel assumng a fully nformed networ, where every user feeds bac a real-valued varable to ther own transmtter. Ths paper nvestgates the achevable DoF usng only -bt feedbac per user. We prove that -bt feedbac s suffcent to acheve the optmal DoF d. Most mportantly, the requred number of users for OIA wth -bt feedbac remans the same as wth real-valued feedbac. Moreover, for a gven system confguraton, we provde an optmal choce of the -bt quantzer, whch captures most of the capacty provded by a system wth real-valued feedbac. I. INTRODUCTION Interference s a crucal lmtaton n next generaton cellular systems. To address ths problem, nterference algnment (IA) has attracted much attenton and has been extensvely studed lately. IA s able to acheve the optmal degrees of freedom (DoF) at hgh sgnal-to-nose ratos (SNR) resultng n a rate of M/2 log(snr)+o(log(snr)) for the M cell nterference channel. For IA, a closed-form soluton of the precodng vectors for sngle antenna nodes wth symbol extenson s nown []. owever, ths codng scheme s based on the assumpton that global channel state nformaton (CSI) s avalable at all nodes, whch s extremely hard to acheve and maybe even mpossble. For the sae of complexty reducton, opportunstc nterference algnment (OIA) has been studed lately [2] [7]. The ey dea of OIA s to explot the channel randomness and multuser dversty by proper user selecton. In [2] [7], sgnal subspace dmensons are used to algn the nterference sgnals. Each transmtter opportunstcally selects and serves the user whose nterference channels are most algned to each other. The degree of algnment s quantfed by a metrc. To facltate a user selecton algorthm, all potental users assocated wth the transmtter are requred to calculate and feedbac the metrc value based on the local CSI. Perfect IA can be acheved asymptotcally f the number of users scales fast enough wth SNR. The correspondng user scalng law to obtan the optmal DoF s characterzed for multple access channels n [2], [3] and for downln nterference channels n [5] [7]. The wor n [5] decouples a multple-nput multple-output (MIMO) nterference channel nto multple SIMO nterference channels and guarantees one spatal stream for each selected user. Snce each stream s assocated to one metrc value, multple metrc values have to be fed bac to allow for multple streams at the transmtters. The wor of [6] reduces the number of users to acheve the optmal DoF at the expense of ncreased feedbac nformaton from each user. In [6], each user has to feed bac a metrc value and a channel vector to cancel ntra-cell nterference. To enable multple spatal streams for each selected user, the authors of [7] nvestgate the requred user scalng n 3-cell MIMO nterference channels and show that the optmal DoF d s acheved f the number of users s scaled as SNR d2. Therefore, at hgher SNR, a larger number of users s requred to acheve the optmal DoF. Clearly, the level of requred total CSI feedbac also ncreases proportonally to the number of users. owever, n practcal systems, the feedbac s costly and the bandwdth of the feedbac channel s lmted. As a result, the feedbac rate should be ept as small as possble. For opportunstc transmsson n pont-to-pont systems, the problem of feedbac reducton s tacled n [8] [] by selectve feedbac. The soluton s to let the users threshold ther receve SNRs and notfy the transmtter only f ther SNRs exceed a predetermned threshold. The wor n [8], [9] reduces the number of real-valued varables that must be fed bac to the transmtter n SISO and MIMO multuser channels respectvely. But [8], [9] do not drectly address the queston of feedbac rate snce transmsson of real-valued varables requres nfnte rate. The wor n [] nvestgates the performance of opportunstc multuser systems usng lmted feedbac and proves that -bt feedbac per user can capture a double-logarthmc capacty growth wth the number of users. Note that [8] [] consder nterference-free pontto-pont transmssons. Unle pont-to-pont systems where the mperfect CSI causes only an SNR offset n the capacty, the accuracy of the CSI n nterference channels affects the slope of the rate curve,.e., the DOF. Thus, for OIA, a relaton to the DoF usng selectve feedbac s crtcal. Can we reduce the amount of feedbac and stll preserve the optmal DoF? Ths s addressed n our paper [] usng real-valued feedbac. It shows that the amount of feedbac can be dramatcally reduced by more than one order of magntude whle stll preservng the essental DoF promsed by conventonal OIA wth full feedbac. owever, to the best of our nowledge, the achevablty of the optmal
Fg.. cell Tx Tx2 Tx3 2 3 Intended ln Interferng ln User User User User User User Three-cell MIMO nterference channel wth canddates n each DoF wth lmted feedbac s stll unnown. In ths paper, we address ths problem by -bt feedbac for 3-cell MIMO nterference channels. Contrbutons of ths paper: We prove that the feedbac of only bt per user s suffcent to acheve the full DoF (wthout requrng more users) f the -bt quantzer s chosen judcously. We derve the schedulng outage probablty accordng to the metrc dstrbuton for -bt feedbac. We provde an optmal choce of the -bt quantzer to acheve the DoF of, whch captures most of the capacty provded by a system wth full feedbac. To acheve a DoF >, an asymptotc threshold choce s gven. We generalze the desgn of the DoF achevable threshold choces and provde the mathematcal expresson. Notatons: We denote a scalar by a, a column vector by a and a matrx by A. The superscrpt T and stand for transpose and ermtan transpose, respectvely. E[ ] denotes the expectaton operaton. I N s the N N dentty matrx. For a gven functon f(n), we wrte g(n) = O(f(N)) f and only f lm N g(n)/f(n) s bounded. II. SYSTEM MODEL Let us consder the system model for the 3-cell MIMO nterference channel, as shown n Fg.. It conssts of 3 transmtters wth N T antennas, each servng users wth N R antennas. The channel matrx from transmtter j to recever n cell s denoted by,j CNR NT,, j {, 2, 3} and {,..., }. Every element of,j s assumed as an ndependent dentcally dstrbuted (..d.) symmetrc complex Gaussan random varable wth zero mean and unt varance. For a gven transmtter, ts sgnal s only ntended to be receved and decoded by a sngle user for a gven sgnalng nterval. The sgnal receved at recever {,..., } n cell at a gven tme nstant s the superposton of the sgnals We are nterested n lmted feedbac for the metrc value. The wor of [6] addresses lmted feedbac to quantze a channel vector, whch s not relevant to our wor. transmtted by all three transmtters, whch can be wrtten as x =,s +,js j + n, () j=,j where vector s j C d denotes d transmtted symbols from transmtter j wth power constrant E{s j s j } = P d I d. The addtve complex symmetrc Gaussan nose n CN (, I N R ) has zero mean and unt varance. Thus, the SNR becomes SNR = P. In ths paper, we confne ourselves to the nontrval case of N R = 2d and N T = d. Ths s nterestng because t s the mnmum setup to acheve the full DoF d at each recever. In case the number of receve antennas N R > 2d, N R 2d DoF can be obtaned wth probablty one even wthout nterference management because uncoordnated nterference sgnals wll span a subspace wth a maxmum of 2d dmensons n the space C NR. On the other hand f N R < 2d, the full DoF d s not achevable because the nterference sgnals wll span at least a d dmensonal subspace even when they are perfectly algned. Defnng U CNR d as the postflterng matrx at recever n cell, the receved sgnal of user n cell becomes y =U x =U, s + j=,j U,j s j + n (2) where n = U n denotes the effectve spatally whte nose vector. The achevable nstantaneous rate for user n cell becomes ( R =log 2 det I d + P d U,, U ( P d j=,j ) ) U,j,j U + I d. (3) III. CONVENTIONAL OIA Wthout requrng global channel nowledge, OIA s able to acheve the same DoF as IA wth only local CSI feedbac wthn a cell. In ths secton, we descrbe the selecton crtera and the desgn of the postflter for the conventonal OIA algorthm. The ey dea of OIA [7] s to explot the channel randomness and the mult-user dversty, usng the followng procedure: Each transmtter sends out a reference sgnal. Each user equpment measures the channel qualty usng a specfc metrc. Every user feeds bac the value of the metrc to ts own transmtter. The transmtter selects a user n ts own cell for communcaton accordng to the feedbac values. The transmtters am at choosng a user, who observes most algned nterference sgnals from the other transmtters. The degree of algnment s quantfed by a subspace dstance measure, named chordal dstance. It s generally defned as d c (A, B) = / 2 AA BB F (4)
where A, B C NR d are the orthonormal bases of two subspaces and d 2 c (A, B) d. For OIA, each user fnds an orthonormal bass Q of the column space spanned by the two nterference channels respectvely,.e., Q p span( p ) and Q q span( q ) where p = ( + mod 3) and q = ( + 2 mod 3). Then the users calculate the dstance between two nterference subspaces usng the obtaned orthonormal bass, yeldng D = d 2 c(q p, Q q), (5) where D s the dstance measured at user n cell. For conventonal OIA, all users feed bac the dstance measure to ther own transmtter and the user selected by transmtter s gven by = arg mn D. (6) Therefore, the metrc value of the selected user becomes D. Defnng the receved nterference covarance matrx of the selected user as R = p p + q q, (7) the postflter appled at the selected user becomes U = [ u d+ (R ),, u NR (R )] (8) where u n (R) represent the sngular vector correspondng to the n-th largest sngular value of R. IV. TE ACIEVABLE DOF OF OIA WIT -BIT FEEDBAC In ths secton, we ntroduce the concept of -bt feedbac for OIA. The achevablty of the DoF s proved for d = frst, where a closed-form soluton exsts. We generalze the result to all d > based on asymptotc analyss. A. Prelmnares of the Chordal Dstance Measure As shown n [2], for quantzng a source A arbtrarly dstrbuted on the Grassmannan manfold G NR,d(C) usng a random codeboo C rnd wth codewords, the second moment of the chordal dstance can be bounded as [ ] Q() = E mn d 2 c(a, C ) (9) C C rnd Γ( d(n R d) ) d(n R d) (c N R,d) d(n R d) () where Γ( ) denotes the Gamma functon and the random codeboo C rnd G NR,d(C). The constant c NR,d s the ball volume on the Grassmannan manfold G NR,d(C),.e. c NR,d = Γ(d(N R d) + ) d = Γ(N R + ) Γ(d + ). () The problem of selectng the best user out of users s equvalent to quantzng an arbtrary subspace wth random subspaces on the Grassmannan manfold G NR,d(C) [7, Lemma 4]. Therefore, we have E [ ] D = Q() and E [ ] D = Q(). We brefly revst the results obtaned n [7], whch wll be used for comparson wth our -bt feedbac OIA. A fnte number of users results n resdual nterference. Let us defne the rate loss term due to resdual nterference as R loss = log 2 det j=,j U,j,j U + I d. (2) When the cell has users, the average rate loss at the selected user can be bounded ( as E[R loss ] d log 2 + P ) d E[D ] (3) ( =d log 2 + P ) d Q(), (4) where (3) s obtaned due to [7, Lemma 6]. The achevable DoF of transmtter usng OIA can be E[R expressed by d lm loss ]. The DoF d s acheved f the number of users s scaled as [7, Theorem 2] B. One-Bt Feedbac by Thresholdng P dd. (5) For conventonal OIA, the user selected for transmsson s the one wth the smallest chordal dstance measure. Ths requres that the transmtter collects the perfect real-valued chordal dstance measures from all the users. owever, the feedbac of real values requre nfnte bandwdth. The queston of how to effcently feedbac the requred CSI s stll not solved for OIA. To address ths problem, we propose a threshold-based -bt feedbac strategy where each user compares the locally measured chordal dstance to a predefned threshold x th and reports -bt nformaton to the transmtter about the comparson. In such a way, the transmtter can partton all the users nto two groups and schedule a user from the favorable group for transmsson. Therefore, we propose the followng steps for OIA usng -bt feedbac: Each transmtter sends out a reference sgnal. Each user equpment measures the channel qualty usng the chordal dstance measure. Each user compares the locally measured chordal dstance to a threshold. In case the measured value s smaller than the threshold, a wll be fed bac; otherwse a wll be fed bac. The transmtter wll randomly select a random user whose feedbac value s for transmsson. A schedulng outage occurs f all users send to the transmtter. In such an event, a random user among all users wll be selected for transmsson. To fnd the schedulng outage probablty P out, we frst denote the cumulatve densty functon (CDF) of D by F D (x), whch s defned as F D (x) = Pr(D x) (6) = Pr(d 2 c(a, C ) x) (7), x < c NR,d x d(nr d), x ˆx (8), x > ˆx
where ˆx satsfes c NR,d ˆx d(nr d) = and ˆx d. If d =, the CDF of (8) becomes exact. If d >, the CDF n (8) s exact when x. When < x < d, the CDF provded by (8) devates from the true CDF [2]. owever, we are manly nterested n small x < for the purpose of feedbac reducton by thresholdng. Therefore, the schedulng outage probablty corresponds to the event where all users exceed x, whch s denoted by P out = Pr(mn D x) (9) = Pr( mn d 2 c(a, C ) x) (2) C C rnd = ( F D (x th )). (2) We defne the probablty densty functons (PDFs) of D as f D (x), where x f D(x)dx = F D (x). In order to dstngush from the prevous conventonal OIA, we employ as the ndex of the selected user wth -bt feedbac. The expected metrc value of the selected user can be expressed as E[D ] = ( P out ) xth f D (x)x d F D (x th ) dx + P f D (x)x out x th F D (x th ) dx, (22) f where D (x) F D (x th ) and f D (x) F D (x th ) are the normalzed truncated PDFs of D n the correspondng ntervals [, x th) and [x th, d], satsfyng x th f D (x)dx F D (x th ) = and d f D (x)dx x th F D (x th ) =. The frst term n (22) represents the event where at least one user falls below the threshold and reports to the transmtter. The second term denotes a schedulng outage, where all the users exceed the threshold and report. C. Achevable DoF and User Scalng Law When d = For a gven, P out s unquely determned by the choce of the threshold x th. We ntend to fnd the optmal x th, such that (22) s mnmzed. The functon s convex n the range of [, ]. Thus, E[D ] has an unque mnmum wthn the nterval [, ]. To fnd the mnmum value and the correspondng threshold, ] we need to solve the equaton E[D x th =. For d =, accordng to (8) we have F D (x) = x and f D (x) = n the nterval [, ]. The expected metrc value E[D ] n (22) can be smplfed as D (x th ) = E[D ] = ( P out ) xth xdx x th x + P xdx th out x th = ( ( x th ) ) x th 2 + ( x th) ( + x th ). 2 (23) The optmal x th whch mnmzes E[D ] can be found by solvng D(x th) x th =,.e. ( x th ) + =. Thus we have the optmal threshold ˆx th = ( ). (24) Applyng ˆx th to (23), the mnmum of D (x th ) can be wrtten as a functon of as D (ˆx th ) = ( ) ( ) + 2 2 2. (25) Ths leads us to the followng lemma, whch wll then be used for the proof of the achevable DoF. Lemma. When the number of users goes to nfnty,.e., D (ˆx th ) s asymptotcally equvalent to log() 2, such that lm D (ˆx th ) lm =. (26) log 2 Proof. Accrodng to (25), the left hand sde of (26) can be wrtten as ( ) ( M ( log ) + ) ( ) + log M M (log M + ) log M + log M + M M M lm M = (27) (28) (29) (3) where (29) s obtaned by lettng M = / and applyng the L ôptal s rule. Thus, the proof s complete. Theorem. For d =, f the threshold s optmally chosen accordng to (24), -bt feedbac per user s able to acheve a DoF d [, ] per transmtter f the number of users s scaled as P d. (3) Proof. The achevable DoF of transmtter usng OIA can be expressed as d loss. If P d, the DoF loss term can be wrtten as d loss P E[R loss ] log 2 P (32) log lm 2 ( + P D (ˆx th )) (33) log 2 (P D (ˆx th )) (34) ( ) log 2 P log 2 (35) = ( d ) + lm P logp + O() (36) = ( d ). (37) The nequalty (33) s obtaned by usng the upper bound n (3) and nvong (25). Equalty (35) s due to the asymptotc equvalence n Lemma. Equalty (36) s obtaned usng the
Threshold value 2 Optmal threshold Asmpototcal optmal threshold log/ 3 2 4 6 8 Number of users Fg. 2. Comparson among the closed-form optmal threshold and the asymptotcally optmal threshold for d =. Sum rate [bps/z] 4 35 3 25 2 5 5 OIA wth bt feedbac OIA wth full feedbac IA DoF 2 3 4 SNR[dB] Fg. 3. Achevable sum rate for N R = 2, d =. The number of users = P for OIA. relatonshp P d and the L ôptal s rule. Therefore, the DoF d s obtaned at each transmtter. Remar. Compared to conventonal OIA n [7], the user scalng law achevng DoF d remans the same. The second term n (36) does not exst for conventonal OIA. owever, t goes to when P, and thus does not change the DoF. Therefore, -bt feedbac nether degrades the performance n terms of DoF nor requres more users to acheve the same DoF. D. Achevable DoF and User Scalng Law When d > For d >, due to the fact that an explct expresson for F D (x) s unnown for x (, d], we frst derve an upper bound for (22). owever, an explct soluton of x th whch mnmzes the upper bound s stll ntractable. For ths reason, we employ asymptotc analyss and fnd a soluton of x th, whch approaches the optmal value when. Therefore, we arrve at the followng theorem. Theorem 2. If the number of users s scaled as P dd, the feedbac of only -bt per user s able to acheve the DoF d [, d] per transmtter f the threshold ˆx th s chosen such that cˆx d2 th = (A log + B). (38) Proof. Due to the space lmt, the proof s gven n [3]. Remar 2. The result above s also applcable to the case of d =. For d =, the optmal threshold obtaned n (24) s asymptotcally equvalent to the above result. V. SIMULATION RESULTS In ths secton, we provde numercal results of the thresholds and sum rate of OIA usng -bt feedbac. Fg. 2 compares the threshold as a functon of the number of users for N R = 2, d =. The thresholds are obtaned by (24) and the asymptotc expresson log as mentoned n Remar 2. It can be seen that these thresholds are very close to each other, even for a small number of users. Therefore, ths result valdates the calculatons of our closed-form threshold and the asymptotc optmal threshold. Fg. 3 shows the achevable sum rate versus SNR of OIA wth full feedbac and OIA wth -bt feedbac, for N R = 2, d = and the number of users = P. We nclude also the sum rate acheved by closed-form IA n 3-user 2 2 MIMO channels. The threshold of our feedbac scheme s calculated accordng to (24). We can see that OIA wth -bt feedbac acheves slghtly lower rate than OIA wth full feedbac. At 3 db SNR, t can acheve 85% of the sum rate obtaned by full feedbac OIA. Importantly, OIA wth -bt feedbac s able to capture the slope and acheve the DoF d = (see the reference lne n Fg. 3). VI. CONCLUSION We analyzed the achevable DoF usng a -bt quantzer for OIA. We proved that -bt feedbac s suffcent to acheve the optmal DoF of d n 3-cell MIMO nterference channels. Most mportantly, the requred user scalng law remans the same as for OIA wth full feedbac. We derved a closedform threshold for d =. In the case of d >, an asymptotc threshold choce was gven, whch s optmal when the number of users. REFERENCES [] V. Cadambe and S. Jafar, Interference Algnment and Degrees of Freedom of the -User Interference Channel, IEEE Trans. Inf. Theory, vol. 54, no. 8, pp. 3425 344, Aug. 28. [2]. J. Yang, W.-y. Shn, B. C. Jung, and A. Paulraj, Opportunstc nterference algnment for MIMO nterferng multple-access channels, IEEE Trans. Wrel. Commun., vol. 2, no. 5, pp. 28 292, May 23. [3] B. C. Jung, D. Par, and W. Shn, Opportunstc nterference mtgaton acheves optmal degrees-of-freedom n wreless mult-cell upln networs, IEEE Trans. Commun., vol. 6, no. 7, pp. 935 944, Jul. 22. [4] T. Gou, T. oe-ano, and P. Orl, Improved and opportunstc nterference algnment schemes for mult-cell nterference channels, n Proc. IEEE Veh. Technol. Conf., May 22, pp. 5. [5] J.. Lee, W. Cho, and B. D. Rao, Multuser dversty n nterferng broadcast channels: achevable degrees of freedom and user scalng law, IEEE Trans. Wrel. Commun., vol. 2, no., pp. 5837 5849, Nov. 23. [6]. J. Yang, W.-y. Shn, B. C. Jung, C. Suh, and A. Paulraj, Opportunstc downln nterference algnment, n Proc. IEEE Int. Symp. Inf. Theory, vol. 9435, 24, pp. 588 592. [7] J.. Lee and W. Cho, On the achevable DoF and user scalng law of opportunstc nterference algnment n 3-transmtter MIMO nterference channels, IEEE Trans. Wrel. Commun., vol. 2, no. 6, pp. 2743 2753, Jun. 23. [8] D. Gesbert and M. Aloun, ow much feedbac s mult-user dversty really worth? n Proc. IEEE Int. Conf. Commun., 24, pp. 234 238. [9] M. Sharf and B. assb, On the capacty of MIMO broadcast channels wth partal sde nformaton, IEEE Trans. Inf. Theory, vol. 5, no. 2, pp. 56 522, Feb. 25. [] S. Sanaye and A. Nosratna, Opportunstc Downln Transmsson Wth Lmted Feedbac, IEEE Trans. Inf. Theory, vol. 53, no., pp. 4363 4372, Nov. 27. [] Z. Xu, M. Gan, and T. Zemen, Threshold-based selectve feedbac for opportunstc nterference algnment, n Proc. IEEE Wrel. Commun. Netw. Conf., 25, pp. 276 28. [2] W. Da, Y. E. Lu, and B. Rder, Quantzaton bounds on Grassmann manfolds and applcatons to MIMO communcatons, IEEE Trans. Inf. Theory, vol. 54, no. 3, pp. 8 23, Mar. 28. [3] Z. Xu, M. Gan, and T. Zemen, On the degrees of freedom for opportunstc nterference algnment wth -bt feedbac: The 3 cell case. [Onlne]. Avalable: http://arxv.org/abs/5.432