COGNITIVE RADIO NETWORKS BASED ON OPPORTUNISTIC BEAMFORMING WITH QUANTIZED FEEDBACK

COGNITIVE RADIO NETWORKS BASED ON OPPORTUNISTIC BEAMFORMING WITH QUANTIZED FEEDBACK Ayman MASSAOUDI, Noura SELLAMI 2, Mohamed SIALA MEDIATRON Lab., Sup Com Unversty of Carthage 283 El Ghazala Arana, Tunsa 2 LETI Lab., ENIS Unversty of Sfax 338 Sfax, Tunsa ABSTRACT In ths paper, we consder an opportunstc beamformng schedulng scheme of secondary users SUs whch can share the spectrum wth a prmary user PU n an underlay cogntve rado networ. In the schedulng process, the cogntve base staton CBS havng mult-antennas, generates orthogonal beams whch nsure the mnmum nterference to the PU. Then, each SU feeds bac ts maxmum sgnal to nterference and nose rato SINR and the correspondng beam ndex to the CBS. The CBS selects the users havng the largest SINRs for transmsson. The am of our wor s to study the effect of SINR feedbac quantzaton on the throughput of the secondary system. To do ths, we derve an accurate statstcal characterzaton of ordered beams SINR and then we derve the closed-form expresson of the system throughput wth SINR feedbac quantzaton based on Lloyd-Max algorthm. Index Terms Cogntve rado, opportunstc beamformng, feedbac quantzaton, Lloyd-Max quantzer practce, the SINRs are quantzed before beng fed bac to the base staton n order to mae more effcent use of the lmted resources bandwdth and power. The SINR quantzaton for OB non-cogntve system has been recently studed n the lterature [5 7]. In these studes, the mpact of the feedbac quantzaton on the throughput of OB system s analyzed. The context of ths wor s dfferent snce t deals wth cogntve rado networ. Indeed, we consder the cogntve users schedulng scheme of [3] and we propose to quantze the SINR feedbac usng the mnmum mean squared error MMSE optmal quantzer.e. the Lloyd-Max quantzer [8]. We propose to dentfy the optmal set of quantzaton thresholds and to study analytcally the mpact of quantzaton on the throughput of the secondary system. To do ths, we study analytcally the statstcs of the ordered beam SINRs for a partcular user. The dffculty here comes from the fact that these SINRs are correlated random varables. 2. SYSTEM MODEL. INTRODUCTION Cogntve rado CR s a novel approach for mprovng the utlzaton of the rado electromagnetc spectrum []. It allows unlcensed secondary users to share the spectrum wth lcensed prmary users wthout sgnfcantly mpactng ther communcaton [2]. In [3], we proposed an opportunstc beamformng schedulng scheme of secondary users SUs whch can share the spectrum wth a prmary user PU n an underlay cogntve rado networ. We assumed that the cogntve base staton CBS does not have full channel state nformaton CSI from SUs whle t has an mperfect CSI from the PU and we proposed a two-steps schedulng algorthm based on opportunstc beamformng OB [4]. In the frst step, orthogonal beams are generated by the CBS to mnmze the nterference to the PU. In the second step, each SU calculates the sgnal to nterference plus nose ratos SINRs for each beam and feeds bac ts maxmum SINR and the correspondng beam ndex to the CBS. The CBS selects for transmsson the users wth the hghest SINRs and assgns to each of these users the beam correspondng to the hghest SINR. In Fg.. System model We consder the system model llustrated n Fgure and descrbed n [3], where a cogntve rado networ coexsts wth a prmary networ. The prmary networ conssts of a prmary base staton PBS wth a sngle transmttng antenna and one PU wth a sngle recevng antenna. The cogntve networ comprses K SUs, wth a sngle recevng antenna each, and a CBS wth M transmttng antennas. Through- 978--9928626-3-3/5/$3. 25 IEEE 222

out ths paper, we assume that M K and the frequency dvson duplex FDD mode s used for both prmary and secondary lns. We consder the downln of the cogntve rado networ n whch the CBS transmts ndependent sgnals to scheduled secondary users, M the schedulng wll be explaned n the followng. We denote by S the set of the selected cogntve users. Snce the same carrer frequency s used wthn the prmary and the secondary networs, the receved sgnals at the SUs are corrupted by the sgnal transmtted by the PBS. Let h = [h,, h,2,, h,m ], where h,t s the channel tap gan between the t-th transmt antenna of the CBS and the -th secondary user, for t M and K. Let g = [g, g 2,, g K ], where g, for K, denotes the channel tap gan between the transmt antenna at the PBS and the -th cogntve user receve antenna. The entres of channel vectors h and g are ndependent and dentcally dstrbuted..d. complex Gaussan samples of a random varable wth zero mean and unt varance. We assume that the channels are constant durng the transmsson of a burst of T symbols and vary ndependently from burst to burst. The receved sgnal at the -th cogntve user, for K, can be wrtten as: y = P s h w x + P pu g x pu + n, S where P s and P pu denote the transmtted power for each selected cogntve user and for the prmary user, respectvely. In ths wor, fxed power allocaton for all selected users s adopted. The quanttes x pu and x denote the transmtted data from the PBS to the PU and from the CBS to the -th SU, respectvely, n denotes the nose at the -th cogntve user whch s a zero-mean Gaussan random varable wth varance σ 2. We assume that the varances σ2 for K are equal to σ 2. The weghtng vector w of sze M denotes the beamformng weght vector for the -th selected secondary user. 3. OPPORTUNISTIC BEAMFORMING SCHEDULING WITH SINR QUANTIZATION In our wor, we consder the two steps SUs schedulng method proposed n [3]. We assume that the CBS has an mperfect estmate of the nterference channel h pu channel between the CBS and the PU and has only partal channels nowledge about the secondary lns channels between the CBS and the SUs. In order to reduce the nterference to the PU, the CBS generates, n the frst step, orthogonal beams to the nterference channel estmate ĥ pu usng the Gram- Schmdt algorthm. In the second step, the CBS selects a set S of secondary users by applyng the opportunstc beamformng approach proposed n [4]. Thus, the cogntve base staton transmts the generated beams to all SUs. Then, by usng, each SU calculates the followng SINRs by assumng that x j, j, s the desred sgnal and the others x, j,, are nterferng sgnals as: SINR,j = Ns h w j 2 P s =, j h w 2 P s + g 2 P pu + σ 2. 2 In practce, each SU feeds bac to the CBS a quantzed verson of ts maxmum SINR.Thus, the value range of SINRs s dvded nto Q = 2 b ntervals, wth boundares values gven as: b < b <... < b Q. 3 If the largest SINR value of user, denoted by γ, s n the q-th nterval, where q Q,.e. < γ b q, then the -th user wll feedbac the ndex q of that nterval together wth the ndex of ts best beam. The CBS allocates the beams to selected users based on the feedbac nformaton. Specfcally, a beam wll be assgned to the user who has the largest SINR nterval ndex, among all the users requestng that beam. Notce that f many users feed bac the same quantzaton nterval ndex for the same beam, one of these users s selected at random. In addton, t may happen that no user requests one or several beams. In ths case, the CBS wll assgn that beam to a randomly chosen SU. We propose to use n ths paper the Lloyd-Max quantzer [8]. Then, we study the mpact of feedbac quantzaton on the secondary system sum rate. In order to dentfy the optmal set of quantzaton thresholds and to compute the sum rate, we derve the statstcs of the ordered beam SINR for a gven user. 4. LLOYD-MAX QUANTIZATION In ths secton, we consder the Lloyd-Max quantzaton [8]. Let Γ q, for q Q, be the reconstructon levels of the Q-level quantzer Ω. defned as: Ω γ = Γq f < γ b q 4 The quantzer s desgned to mnmze the average dstorton D Q gven by: D Q = Q q= γ Γ q 2 f γ γ dγ 5 where E [.] denotes the statstcal expectaton and f γ γ s the pdf of the largest SINR per user whch s ndependent of as wll be seen n secton 6. The expresson of f γ γ wll be gven n 8.The necessary condtons to mnmze D Q are: { DQ b q = D Q 6 Γ q = Solvng 6 usng the Lloyd-Max algorthm, we obtan the necessary condtons for mnmzaton as: b Γ q = q γf γ γ dγ 7 f γ γ dγ 223

b q = Γ q + Γ q+ 8 2 Mathematcally, the decson and the reconstructon levels are solutons of the above set of nonlnear equatons. In general, closed form solutons to equatons 7 and 8 do not exst and they can be solved by numercal technques n an teratve way by frst assumng an ntal set of values for the decson levels b q. For smplcty, one can start wth decson levels correspondng to unform quantzaton, where decson levels are equally spaced. Based on the ntal set of decson levels, the reconstructon levels can be computed usng equaton 7. These reconstructon levels are used n equaton 8 to obtan the updated values of b q. Solutons of equatons 7 and 8 are teratvely repeated untl convergence s acheved. In the next secton, we propose to study the mpact of SINR quantzaton on the secondary system sum rate. 5. SUM RATE OF THE SECONDARY SYSTEM In ths secton, we study the throughput of the secondary system. The loss n throughput due to the quantzaton s equal to: R loss = R A R Q 9 where R A respectvely R Q s the throughput of the secondary system wth analog respectvely quantzed feedbac. The throughput of the secondary system wth analog feedbac s expressed as [3]: + R A = F Sγ K dγ, + γ where F S x denotes the cdf of SINR,j n 2 and s gven n [3]: Ns a j exp F Sx = x / + ans exp x /, x + j x + j= where a j, for j, are constants. The exact sum rate expresson for the secondary system wth quantzed feedbac can be calculated as [ [9], Eq. 7.7]: K K K Ns R Q = = Q Fγ b q F γ b q q= F γ b q F γ Ns + K Ns u=2 log 2 + γ f γ γ dγ log 2 + γ f γ u γ dγ 2 where f γ u γ s the pdf of the u-th largest SINR per user and wll be gven n 5. In the next secton, we derve the statstcs of the ordered beam SINR for a gven user needed n 7, 8 and 2. 6. STATISTICS OF THE ORDERED BEAM SINRS FOR A PARTICULAR USER Because the SINR values for a partcular user are not ndependent, the order statstcs used n [3] cannot be appled. Indeed, the beam SINRs for the same user,.e. SINR,j n 2, for j, are correlated random varables as they nvolve the same channel vector h. The ordered SINRs for a gven user are denoted by γ Ns γ u γ. Snce for a user, the channels h and the noses n have statstcs whch are ndependent of, we omt the ndex n the followng. We show that the cdf of γ u, for 2 u, s gven by: F γ u γ = where = Ps σ and 2 Ns uy u γ y +y 2 +z+ f X u,y,y +,Z x, y, y 2, z dxdy dy 2dz. 3 f X u,y,y +,Z x, y, y 2, z = s u = Ns! y u xu 2 u! u! exp x + y + y 2 + z U y u x u! u 2! u y 2 x Ns u U y 2 x x >, y > u x, y 2 < ux, z > 4 where U. denotes the unt step functon and = Ppu P s. Proof: See the Appendx. By tang dervatve of 3 wth respect to γ, the pdf of γ u s gven by f γ u γ = f X u,y,y +,Z Ns uy u y + y 2 + z + γ y + y 2 + z +, y, y 2, z dy dy 2 dz. 5 We also show that the cdf of the largest SINR for a partcular user, denoted by γ, s gven by F γ γ = where f X,Y,Z x, y, z = γ y+z+ 2! f X,Y,Z x, y, z dxdydz. s = 6 y x Ns 2 exp x + y + z U y x 7 We omt here the proof of 7 due to the lac of space. After tang dervatve wth respect to γ, by applyng the bnomal theorem and usng equaton 2.323 n [], we can 224

obtan the closed-form expresson for the pdf of the largest SINR per user, f γ γ, gven by: f γ γ = γj γ + 2! j+ exp γ d γ s = where Σ,j,d γ = d! d l= Σ 2,j,d γ = d+ l= Ns s 2 Ns 2 j= γ Ns 2 j γ + Ns 2 j + γ d d= Ns 2 j d Σ,j,d γ + Σ2,j,d γ U γ d+!j! j+! γ+ + j! γ γ d l!x+ l+ d+ l γ γ d+ l!γ+ l+ 7. SIMULATION RESULTS. d l j and 8 In ths secton, we present smulaton results of our proposed schedulng algorthm based on quantzed SINR feedbac. We compare the performance of the secondary system sum rate for the Lloyd-Max quantzaton and the unform quantzaton. Sum rate bps/hz 7 6.5 6 5.5 5 4.5 Unform quant., b=2, smulaton 4 Unform quant., b=2, analyss Lloyd Max quant., b=2, smulaton 3.5 Lloyd Max quant., b=2, analyss Full feedbac, smulaton Full feedbac, analyss 3 2 3 4 5 6 7 8 9 K: Number of secondary users Fg. 2. Sum rate versus the number K of secondary users for dfferent quantzaton schemes for b = 2, M = 4, = 3 and = 5 db Fgure 2 shows the sum rate versus the number K of secondary users for b = 2, M = 4, = 3 and = 5 db for the Lloyd-Max quantzaton and the unform quantzaton. It also shows the sum rate for the deal scheme [3] wth analog best beam SINR feedbac full feedbac. In fgure 2, the curves obtaned by usng the numercal results n and 2 red curves are compared to the smulaton results blue sold curves for the deal scheme and the two quantzaton schemes. The fgure shows that the analytcal curves are nlne wth the curves obtaned by smulatons. We notce that the sum rate ncreases wth the total number of cogntve users snce the mult-user dversty ncreases [4]. Moreover, t can be seen from the fgure that the use of dfferent quantzaton schemes leads to a loss n terms of throughput. However, the Lloyd-Max quantzer clearly outperforms the unform quantzer as expected. R loss bps/hz 3 2.5 2.5.5 Unform quant., smulaton Unform quant., analyss Lloyd Max quant., smulaton Lloyd Max quant., analyss.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Number of quantzaton bts Fg. 3. Sum rate loss versus the number b of quantzaton bts for dfferent quantzaton schemes for K = 5, M = 4, = 3 and = 5 db Fgure 3 shows the sum rate loss versus the number b of quantzaton bts for the Lloyd-Max quantzer and the unform quantzer for K = 5, M = 4, = 3 and = 5 db. We notce that the curves obtaned by usng the numercal results n 9 red curves are nlne wth the smulaton results blue sold curves for the two quantzaton schemes consdered n ths fgure. We remar that the gap between the Lloyd-Max quantzer and the unform quantzer n terms of sum rate decreases as b ncreases. Moreover, for both quantzers, the sum rates converge to the one obtaned wth full feedbac as b ncreases. 8. CONCLUSIONS In ths paper, we consdered a SUs schedulng scheme based on opportunstc beamformng. We analyzed the mpact of SINR feedbac quantzaton on the throughput of the secondary system. In partcular, we derved the accurate statstcal characterzatons of ordered beams SINRs and we gave the exact analytcal expresson of the sum rate for the schedulng scheme based on quantzed SINR feedbac. We consdered the Lloyd-Max quantzer and we compared ts performance wth that of the unform quantzer. 9. APPENDIX Let X,j = h w j 2 for K and j, ordered as X > X2 >... > XNs, where Xu denotes the u-th largest value 2 u among the values X,j for j. We notce that SINR,j s the u-th largest beam SINR, for user, f and only f X,j = X u. Based on ths observaton, the u-th largest SINR of user, denoted γ u, 225

can be wrtten as: REFERENCES γ u = X u Y + Y + + Z + 9 where Y = u X =, Y + = X =u+, Z = g 2, = Ppu P s and = Ps σ. Snce for a user, the channels h 2 and the noses n have statstcs whch are ndependent of, we omt the ndex n the followng. The cdf of γ u can be calculated n terms of the jont pdf of X u, Y, Y + and Z, denoted by f Xu,Y,Y +,Z x, y, y 2, z, as F γ u γ = Ns uy u γ y +y 2 +z+ f X u,y,y +,Z x, y, y 2, z dxdy dy 2dz. 2 After tang dervatve wth respect to γ, the pdf of γ u s gven by Ns uy u f γ u γ = y + y 2 + z + f X u,y,y +,Z γ y + y 2 + z +, y, y 2, z dy dy 2 dz. 2 Applyng Bayesan rules, the jont pdf f X u,y,y +,Z x, y, y 2, z, can be obtaned as follows [9] f X u,y,y +,Z x, y, y 2, z = f X u x f Y y f Y + y 2 f Z z 22 where x >, y > u x, y 2 < ux, z >. Snce the random varables X,j, for a gven and j, are..d. random varables and have a gamma dstrbuton Γ,, f X u x, u, can be obtaned as [ [], Eq.2..6]: N f X u s! x = u! u! exp xns u exp ux 23 The pdfs f Y y and f Y + y 2 can be obtaned, respectvely, as [ [2], Eqs. 26, 27]: f Y y = y u x u 2 exp y + u x u 2! f Y + y 2 = u! s u = u y2 xns u exp y 2 exp x Ns u U y 2 x, 25 where U. denotes the unt step functon. The random varable Z follows the gamma dstrbuton Γ, and the PDF f Z z s gven by: f Z z = exp z. 26 Fnally, after proper substtuton, we can obtan the analytcal expresson gven n 4. [] J. Mtola and G. Q. Magure, Cogntve rado: Mang software rados more personal, IEEE Personal Communcatons, vol. 6, pp. 3 8, August 999. [2] A. Goldsmth, S. A. Jafar, I. Marc, and S. Srnvasa, Breang spectrum grdloc wth cogntve rados: An nformaton theoretc perspectve, Proceedngs of the IEEE, vol. 97, pp. 894 94, May 29. [3] A. Massaoud, N. Sellam, and M. Sala, Schedulng scheme for cogntve rado networs wth mperfect channel nowledge, PIMRC, pp. 345 349, 23. [4] M. Sharf and B. Hassb, On the capacty of MIMO broadcast channels wth partal sde nformaton, IEEE Transactons on Informaton Theory, vol. 5, pp. 56 522, February 25. [5] C. Anton-Haro, On the mpact of pdf-matched quantzaton on orthogonal random beamformng, IEEE Communcatons Letters, vol., no. 4, pp. 328 33, Aprl 27. [6] H.-C. Yang, P. Lu, H.-K. Sung, and Y. cha Ko, Exact sum-rate analyss of MIMO broadcast channels wth random untary beamformng based on quantzed snr feedbac, ICC, pp. 3669 3673, May 28. [7] O. Ozdemr and M. Torla, Optmum feedbac quantzaton n an opportunstc beamformng scheme, IEEE Transactons on Wreless Communcatons, vol. 9, no. 5, pp. 584 593, May 2. [8] S. Lloyd, Least squares quantzaton n PCM, IEEE Transactons on Informaton Theory, vol. 28, no. 2, pp. 29 37, Mar 982. [9] H. Yang and M. Aloun, Order Statstcs n Wreless Communcatons: Dversty, Adaptaton, and Schedulng n MIMO and OFDM Systems. Cambrdge Unversty Press, 2. U y u x, [] I. S. Gradshteyn and I. M. Ryzh, Table Of Integrals, 24 Seres And Products. 7th Edton, Elsever, 27. [] H. A. Davd, Order Statstcs, N. J. W.. Sons, Ed. New Yor, 98. [2] Y.-C. Ko, H.-C. Yang, S.-S. Eom, and M.-S. Aloun, Adaptve modulaton wth dversty combnng based on output-threshold MRC, Wreless Communcatons, IEEE Transactons on, vol. 6, no., pp. 3728 3737, October 27. 226