A New Opportunstc Interference Algnment Scheme and Performance Comparson of MIMO Interference Algnment wth Lmted Feedback Johann Lethon, Chau Yuen, Hmal A. Suraweera and Hu Gao Sngapore Unversty of Technology and Desgn, Dover Drve, Sngapore 88 E-mal: {johann lethon, yuenchau, hmalsuraweera, hu gao}@sutd.edu.sg Abstract We nvestgate the performance of several nterference algnment IA schemes for a three-user MIMO channel wth lmted feedback. In our study, we consder tradtonal IA wth quantzed channel state nformaton at each transmtter, and opportunstc IA OIA that explots the dea of opportunstc selecton to select users whose nterferng channels are most algned. The performance of the former s determned by the quantzaton resoluton, whereas the performance of the later depends on the number of users avalable for selecton. We show that OIA only requres a small number of users to acheve a comparable performance to tradtonal IA schemes wth quantzed feedback. Motvated by ths fact, we propose a new user selecton crteron that outperforms the exstng state of the art. Smulatons show the superor performance of our OIA scheme under dfferent strong/mld nterference condtons. Index Terms MIMO nterference channel, nterference algnment, lmted feedback, opportunstc user selecton. I. INTRODUCTION Capacty of a wreless network s heavly lmted by nterference due to smultaneous transmssons n the same frequency band or tme slot [], []. Tradtonal nterference management technques have reled on technques such as frequency reuse, codng, power control and multplexng and are consdered to be neffectve to handle nterference n emergng hgh data rate servces. Recently, a new nterference handlng scheme coned as nterference algnment IA has receved consderable attenton [], []. IA s a lnear precodng technque that ams to confne the total nterference at each recever n a space wth mnmum dmensons and leave the rest of the nterferencefree dmensons for sgnal decodng. In order to mplement IA, for e.g. n a K-user multple-nput multple-output MIMO network, certan feasblty condtons that depend on the number of transmt/receve antennas, number of streams and K must be met []. Moreover, a caveat wth IA s that such algnment requres perfect and global channel state nformaton CSI. As such the estmated channels at the recevers must be nformed to the transmtters. Ths requrement s consdered as a heavy burden for mplementaton as CSI has to be fed back often usng a fnte bandwdth reverse channel n practce []. To overcome the ssue, several lmted feedback schemes have been proposed: analog feedback [], quantzaton on the Grassman manfold Ths research s partly supported by the Sngapore Unversty Technology and Desgn Grant No. SUTD-ZJU/RES//. [8] and opportunstc IA OIA [9] []. In [9], an IA scheme wth opportunstc user selecton for MIMO cogntve rado networks was presented. The OIA concept was appled to maxmze the sum rate of a three-transmtter MIMO nterference channel by opportunstcally selectng the recever accordng to a chordal dstance based crteron n [] []. Global CSI knowledge requrement of IA can be relaxed wth OIA and leakage nterference mnmzaton s acheved by proper user selecton and post-processng matrx desgn. User selecton and post processng matrx computaton s performed only at the selected recevers []. As a result OIA enjoys a sgnfcantly reduced mplementaton complexty. To the best of authors knowledge, so far, there s no work that comprehensvely deals wth the comparson of IA wth lmted feedback schemes; and ths motvates us to study ther relatve performance. We show that OIA can acheve comparable performance to tradtonal IA wth quantzed CSI even wth a very small number of users. Inspred by ths result, we desgn a new OIA scheme that outperforms the current soluton reported n []. Ths new OIA selecton scheme exhbts a hgher sum rate than the prevous OIA schemes even when consdered n realstc asymmetrc network settngs wth dstance-based channel modelng. More specfcally, n ths paper our contrbuton s twofold: We compare the sum rate performance between tradtonal IA wth lmted feedback quantzed CSI and OIA schemes. We propose a new OIA scheme that outperforms prevous OIA scheme n []. Usng smulatons we show that the new OIA scheme s robust towards dfferent channel models. In certan stuatons, e.g., low quantzaton resoluton, t can be more practcal to desgn a post-processng matrx and/or explotng user dversty rather than employng the tradtonal IA approach wth quantzed CSI at the transmtters. Notaton: Complex matrces and vectors are denoted usng bold face upper case and lower case letters respectvely, A denotes the conjugate transpose of A, tra, λ n A and v n A denote the trace of A, ts nth largest egenvalue and the egenvector correspondng to λ n A, respectvely. Fnally I d represents the d d dentty matrx.
Group T T Group T Group Fg.. System model. The dots represent each N number of recevers served by a sngle transmtter. II. SYSTEM AND CHANNEL MODEL A. System Descrpton and Sgnalng We consder a N t N r,d K MIMO system as shown n Fg. where N t,n r, d are the number of antennas at each transmtter and at each recever, and the number of streams. The number of transmtters/recevers n IA s K and n OIA K refers to the number transmtters and groups. Each recever served by a sngle transmtter s selected from a group of N users n OIA. Each th transmtter, = {,, }, sends the symbol x usng a random untary beamformng matrx B C Nt d, such that B B = I d. In the tradtonal IA scheme, B s not random but desgned, e.g., accordng to the dstrbuted approach explaned n [, Algorthm ]. Each stream s allocated wth the same power, that s: E{x x } = P d I d, where P accounts for the total power. The sgnal receved by the user k n the th group can be expressed as y k = H,k B x + γ j=,j H j,k B j x j + n k, where H j,k C Nr Nt denotes channel matrx from transmtter j to the user k n group, n k s the Gaussan nose that follows a complex normal dstrbuton n k CN, I Nr and γ s a parameter related to the relatve propagaton path loss of the nterferng channels []. Selected users compute the post-processng untary matrx denoted as R k C d Nr, such that: R k R k = I d. Notce that R k s computed jontly wth B n the tradtonal IA scheme. The post-processed sgnal can be wrtten as R k y k = R k H,k x + γ j=j R k H j,k x j + R k n k, where H j,k = H j,k B j for {, j} = {,, } {,, }. From, we can express the nstantaneous sgnal-to-nose plus nterference rato SINR for the user k n the th group as Γ k = P d R k H,k H,k R P k I α d + R k d H j,k H j,k R k. j Therefore, the nstantaneous rate of the k user, s gven by I d + R P k d H,k H,k + j P α d H j,k H j,k R k C k =log, I d + P α d j R k H j,k H j,k R k log γp where α = log P, s the relatve propagaton path loss of the nterferng lnks. B. Channel Models Model I: Each element of H j,k, where j {,, } s the transmtter ndex and k {,...,N} s an arbtrary user n group {,, }, s drawn from an ndependent and dentcally dstrbuted..d. crcularly symmetrc complex normal dstrbuton wth zero mean and unt varance. Cross lnk channels H j,k wth j are scaled by the relatve propagaton path loss parameter γ whch allows to consder varyng nterference strengths. However, ths..d. model also consdered n [] may not be applcable n some practcal cases, f users are located at dfferent dstances from the transmtter. Model II: We also consder a more realstc channel model for the comparson of OIA and IA wth lmted feedback schemes. In ths model, channels are scaled by the factor r, where r s the dstance between transmtter and recever drawn from a unform dstrbuton between and the radus of the crcular coverage area for a gven transmtter. Transmtters are spread randomly n a squared area keepng a mnmum dstance between them. Ths dstance assumed to be same for all transmtters n the coverage area s set to twce the radus of coverage n our smulatons. III. LIMITED FEEDBACK IA SCHEMES AND CONVENTIONAL OPPORTUNISTIC USER SELECTION In ths secton, the lmted feedback schemes nvestgated n ths paper are descrbed, namely: tradtonal IA wth quantzed feedback, conventonal opportunstc user selecton schemes and OIA. A. Tradtonal IA wth Quantzed Feedback We now summarze the scheme descrbed n [] for the sake of completeness. In ths scheme, CSI at the th recever s quantzed usng the concept of aggregated channel matrx constructed as follows: recever feeds back the cross channels to the th transmtter by means of the aggregated channel matrx Φ C NtNr K defned as Φ = [ h, h, h,+ h,k ], where h, s a unt norm vector obtaned stackng the columns of the channel matrx from the th transmtter to the jth recever H,j as follows: ] [ h,j h hn t,j,j = [ ], h,j hn t,j
n where s the Eucldean norm and h,j s the nth column of the channel matrx H,j. Φ s quantzed usng the concept of composte Grassmann manfold wth a codebook C of sze B, and B s the number of bts needed to dentfy each codeword n C. Each codeword n C s generated through random vector quantzaton RVQ. As the quantzaton error depends on the chordal dstance between the actual channel Φ and ts quantzaton,.e., codeword C m C, a prequantzaton matrx G s used at the th recever n order to reduce the quantzaton error []. In ths paper, we consder both quantzaton schemes: tradtonal or conventonal scheme wthout pre-quantzaton matrx G as well as the optmzed scheme n [] wth pre-quantzaton matrx G. B. Maxmum SINR User Selecton MAX-SINR MAX-SINR s the optmal selecton crteron because the user that provdes the hghest nstantaneous achevable rate s chosen. Moreover, the post-processng matrx R k s computed so that t maxmzes the nstantaneous rate gven n. However, a closed form soluton s only avalable for the d =case []. The optmal combnng matrx for MAX- SINR scheme n the case of d =can be wrtten as [] L R MAX SINR k = L k v k P Ak L k, where L k s Cholesky decomposton of I NR + P α j H j,k H j,k and A k = H,k H,k. Each user computes ts optmal combnng matrx R MAX SINR k and feedbacks the correspondng nstantaneous rate. The user k MAX SINR wth the largest nstantaneous rate s chosen as follows: k MAX SINR =argmaxlog +λ R MAX SINR k. 8 k N C. Maxmum SNR User Selecton MAX-SNR Ths crteron seeks to maxmze the achevable rate gnorng the nterference. R k s chosen n order to maxmze the term I d + P d R k H,k H,k R k. Optmal R MAX SNR k s gven by ] R MAX SNR k = [v H,k H,k,..., v d H,k H,k. 9 Each user computes the nstantaneous rate wth the correspondng combnng matrx and feeds t back so that the transmtter chooses the user k MAX SNR that provdes the largest nstantaneous rate gven by k MAX SNR = arg max k N d m= + Pd λm H,k H,k. D. Mnmum INR User Selecton MIN-INR The post-processng matrx R k s chosen to mnmze the rate loss n,.e., the term I d + R k j P α d H j,k H j,k R k and s gven by R MIN INR k =[v d+ M k,...,v d M k ], where M k = j H j,k H j,k. The user k MIN INR s chosen f t mnmzes the rate loss term as follows: d k MIN INR =argmn + P α k N d λ m M k. m=d+ E. Lee-Cho OIA LC-OIA The OIA scheme proposed n [] s applcable for the -user MIMO nterference channel. The post-processng matrx R k s chosen to mnmze the rate loss term I d + P α d j R k H j,k H j,k R k n, and t s computed only by the selected user. The best user k s selected f t has the mnmum chordal dstance between the spaces spanned by ts nterferng channels. Wthout loss of generalty, for the sgnals from Transmtter to any user k n the Group, the chordal dstance between the two spaces spanned by the nterferng channels due to Transmtter and s gven by d d c H,k, H,k = λ m H,k H,k + H,k H,k, m=d+ where H,k s constructed from the egenvectors correspondng to the non-zero egenvalues of H,k H,k such that H,k H,k = H,k Λ,k H,k and Λ,k s a dagonal matrx comprsng unordered egenvalues of H,k H,k. Fnally, the preferred user s chosen accordng to the chordal dstance measure between the nterferng spaces and s gven by k =argmnd c H,k, H,k. k N IV. NEW OIA SCHEME Inspred by the sum rate expresson n, we propose a new metrc for opportunstc user selecton smlar to the LC-OIA scheme n []. The LC-OIA scheme has a much lower performance compared to the MAX-SINR scheme and ts sum rate performance shows degradaton n dstance-based channel models. In order to overcome these ssues, we propose a new selecton crteron to further optmze the performance of OIA. Frst, to avod the computaton of the nstantaneous rate and the combnng matrx at each recever, we base our selecton crteron on an analog value that s computed from the chordal dstances between spaces spanned by nterferng and non-nterferng channels. Unlke conventonal MAX-SINR and MIN-INR that requre extensve computatons at each recever, our new metrc only requres computaton of the post-processng matrx at the selected user, as n the LC-OIA scheme, and exhbts a superor performance as the smulaton results ndcate n Secton V. We propose a combnaton of the MAX-SNR and MIN- INR approaches whereby we seek to maxmze the SINR n wthout computng the post-processng matrx. Our crteron seeks to mnmze the chordal dstance between spaces spanned by nterferng channels and at the same tme seeks to maxmze the chordal dstance between the sgnal space of nterest and the nterferng spaces. Therefore, for the -transmtter nterference channel, three chordal dstances have
to be computed. Wthout loss of generalty, let us consder the data from the Transmtter as the sgnal space whereas, Transmtters and represent nterference. Our crteron seeks to mnmze the chordal dstance between the spaces spanned by sgnals from Transmtters and whle maxmzng the chordal dstance between the space spanned by the frst transmtter sgnal and both sgnals from Transmtters and. The metrcs we adopt to measure the dstance between nterferng spaces and and the sgnal space s the sum of the parwse chordal dstances, therefore our selecton metrc become: d c H,k, H,k = arg mn k N d c H,k, H,k k N OIA + d c H,k, H,k. Total Achevable Rate bps/hz Quantz Conventonal Quantz Enhanced Mn INR Un quantzed SNR db Ths selecton crteron s showed to perform better than LC- OIA. Moreover, we notce that the sgnal power has not yet been taken nto account n the selecton metrc proposed above. Hence, we further seek an mprovement by addng an addtonal element that accounts for the sgnal power as n the MAX-SNR crteron. We am to maxmze, by proper user selecton, the term I d + P d R k H,k H,k R k n wthout computng R k because t would ncrease the complexty of our scheme. In order to do ths, we consder the product H,k H,k whch dagonal contans the magntude values of the channel, then, whle observng, we seek to maxmze tr H,k H,k. Combnng these two metrcs leads to the next crteron: H,k, H,k k NOIA =argmn k N w j= d c d c H,k, H j,k + w tr H,k H,k, where w and w are weghtng factors for desgn purposes. Extensve smulatons showed that the optmal values are w = w =.. Notce that no CSI has to be fed back, and users are selected accordng to analog values that they compute. Each user trggers a tmer that lasts proportonally to ts analog value. Best user s the one that frst tme-out and nform ts transmtter. V. SIMULATION RESULTS In ths secton, we present smulaton results to evaluate the schemes descrbed n Secton III and Secton IV, respectvely. In order to carry out a far comparson between IA wth lmted feedback and OIA schemes, and to meet the IA feasblty condtons, we set the system parameters as follows: N t =, N r =, d =, K =so that we have a, system whch s feasble and meets the OIA requrements, e.g., N t d and N r =d []. Notce that, under perfect CSI, a, MIMO network attans the maxmum degrees of freedom because N t +N r K +d =[]. The new OIA scheme uses metrc n. Fg. shows a comparson of tradtonal IA wth fve bts quantzed feedback and OIA schemes. It can be seen, that OIA schemes outperform the tradtonal IA scheme wth quantzed feedback. Notce that selecton s not carred out Fg.. Total achevable rate of a, MIMO nterference channel wth B =, and OIA schemes wth N =. Channel model as n [] wth α =. Achevable rate total bps/hz 9 8 Quantz. Conventonal Quantz. Enhanced Mn INR Un quantz. SNR db Fg.. Total achevable rate of a, MIMO nterference channel wth B =, and OIA schemes wth N =. Channel model as n [] wth α =. at all snce sngle user groups are consdered. Ths shows how OIA approaches wthout any feedback can outperform feedback IA systems wth fve bts quantzed CSI. Fg. shows comparsons under a dfferent scenaro, e.g., quantzed feedback wth bts and OIA wth three users per group. We see that only three users per group are requred to match the performance of tradtonal IA wth enhanced bts quantzed CSI. We have assumed α = n Fgs. -, whch represents the worst scenaro for the OIA schemes,.e., nterference and sgnal space have the same power. As expected MAX- SINR exhbts the best performance, however, nvolves a hgher complexty. Moreover, these results show that n the low-snr regon, OIA schemes can outperform tradtonal IA even wth perfect CSI at the transmtter. Fg. shows results wth α =.,.e., a weak nterference scenaro. In all scenaros the proposed new OIA scheme outperforms the current approaches. Fg.. shows results for the channel model II wth range of coverage of m and mnmum dstance of m between transmtters. It can be seen that the new OIA crteron performs
Achevable rate per user bps/hz Mn INR Achevable rate per user bps/hz Mn INR SNR db 8 9 SNR db Fg.. Achevable rate per user of a, MIMO nterference channel, dfferent OIA selecton crtera wth N =. Channel model as n [], wth α =. Achevable rate per user bps/hz 8 8 Mn INR SNR db Fg.. Achevable rate per user of a, MIMO nterference channel, dfferent OIA selecton crtera wth N =. Channel model as n [], wth α =.. better than most of the schemes besdes MAX-SINR, whle the new OIA scheme has a much lower complexty than MAX- SINR. In the smulaton setup of Fg., as the nterference strength s altered by the dstance from the nterferng transmtters to the selected user, SNR-based crteron outperforms nterference-based MIN-INR and LC-OIA schemes. VI. CONCLUSIONS We have presented a performance comparson of several lmted feedback schemes; OIA, MIN-INR, MAX-SNR, MAX-SINR, and tradtonal IA quantzaton on the Grassmann manfold of aggregated channel matrces n a three-user MIMO nterference channel. It was shown that n the low-snr regme, proper post-processng matrx desgn outperforms tradtonal IA even wth perfect channel knowledge. Moreover, when the number of bts avalable for quantzaton s small, e.g., fve bts, then proper combnng matrx desgn should be preferred. Ths approach leads to a better performance and does not requre large codebooks and exhaustve comparsons n terms of chordal dstances. Furthermore, we proposed a new OIA crteron that outperforms the exstng OIA scheme and Fg.. Achevable rate per user of a, MIMO nterference channel, dfferent OIA selecton crtera wth N =and α =. s more robust towards realstc channel models. Smulatons ndcated that even for the case of bts of feedback, we only need three users per group n the new OIA scheme to match the performance. REFERENCES [] R. H. Etkn, D. N. C. Tse and H. Wang, Gaussan nterference channel capacty to wthn one bt, IEEE Trans. Inf. Theory, vol., pp. -, Dec. 8. [] S. A. Jafar and M. J. Fakhereddn, Degrees of freedom for the MIMO nterference channel, IEEE Trans. Inf. Theory, vol., pp. -, July. [] S. A. Jafar and S. Shama, Degrees of freedom regon of the MIMO X channel, IEEE Trans. Inf. Theory, vol., pp. -, Jan. 8. [] V. R. Cadambe and S. A. Jafar, Interference algnment and degrees of freedom of the K-user nterference channel, IEEE Trans. Inf. Theory, vol., pp. -, Aug. 8. [] C. Yets, T. Gou, S. Jafar, and A. Kayran, On feasblty of nterference algnment n MIMO nterference networks, IEEE Trans. Sgnal Process., vol. 8, pp. -8, Sept.. [] R. Tresch and M. Gullaud, Cellular nterference algnment wth mperfect channel knowledge, n Proc. IEEE ICC 9, Dresden, Germany, June 9, pp. -. [] O. E. Ayach and R. W. Heath, Interference algnment wth analog channel state feedback, IEEE Trans. Wreless Commun., pp. -, Feb.. [8] R. T. Krshnamachar and M. K. Varanas, Interference algnment under lmted feedback for MIMO nterference channels, arxv:9.9, Nov. 9. [9] S. M. Perlaza, N. Fawas, S. Lasaulce, and M. Debbah, From spectrum poolng to space poolng: Opportunstc nterference algnment n MIMO cogntve networks, IEEE Trans. Sgnal Process., vol. 8, pp. 8-, July. [] J. H. Lee and W. Cho, Opportunstc nterference algned user selecton n multuser MIMO nterference channels, n Proc. IEEE GLOBECOM, Mam, FL, Dec., pp. -. [] J. H. Lee and W. Cho, Interference algnment by opportunstc user selecton n -User MIMO nterference channels, n Proc IEEE ICC, Kyoto, Japan, June, pp. -. [] J. H. Lee, W. Cho and D. J. Love, On the optmalty of opportunstc nterference algnment n -transmtter MIMO nterference channels, arxv:9.. [] J. S. Km, S. H. Moon, S. R. Lee and I. Lee, A new channel quantzaton strategy for MIMO nterference algnment wth lmted feedback, IEEE. Trans. Wreless Commun., vol., pp. 8-, Jan.. [] K. Gomadam, V. R. Cadambe, and S. A. Jafar, A dstrbuted numercal approach to nterference algnment and applcatons to wreless nterference networks, IEEE Trans. Inf. Theory, vol., pp. 9-, June.