Transmit Beamforming and Iterative Water-Filling Based on SLNR for OFDMA Systems
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1 Transit Beaforing and Iterative Water-Filling Based on SLNR for OFDMA Systes Kazunori Hayashi, Megui Kaneko, Takeshi Fui, Hideaki Sakai Graduate School of Inforatics, Kyoto University, Yoshida Honachi Sakyo-ku, Kyoto, 66-85, JAPAN {kazunori, eg, takec 8, Yo Okada Suitoo Electric Industries, Ltd. Kitahaa, Chuo-ku, Osaka, 54-4, JAPAN Abstract This paper proposes a transit beaforing and subcarrier power allocation ethod for orthogonal frequency division ultiple access (OFDMA) systes based on signal-toleakage-plus-noise ratio (SLNR). As the transit beaforing vector control criterion, we eploy the axiization of the SLNR at each base station, while the subcarrier power allocation is perfored by iterative water-filling algorith using the SLNR of each subcarrier. The SLNR based approach enables us to obtain closed for beaforing vector, and achieve power allocation without sending any signal to the obile terinal. We also discuss the validity of the SLNR based beaforing vector in ters of Pareto optiality. Coputer siulations show the proising perforance of the proposed ethod and the validity of the analysis of the SLNR based beaforing vector. Index Ters OFDMA, transit beaforing, power allocation, SLNR, iterative water-filling, Pareto optial I. INTRODUCTION Much effort to apply block transission schees using cyclic prefix [] [3], such as orthogonal frequency division ultiplexing (OFDM), to obile counications systes has been ade as typified by WiMAX syste, where orthogonal frequency division ultiple access (OFDMA) [4] is adopted for the physical layer / ediu access control layer protocol. In cellular obile counications systes, co-channel interference is one of the ost serious probles, especially when frequency reuse factor is set to be in order to achieve high spectral efficiency. Receive beaforing using ultiple antenna eleents with the cobining vector based on axiu signal-to-interference-plus-noise ratio (SINR) criterion is effective for the interference suppression. However, eploying ultiple antenna eleents and calculating the vector at the obile terinal are undesirable fro view points of cost and power consuption. Therefore, for downlink counications in the cellular systes, transit beaforing is ore attractive. In this paper, we consider the downlink transit beaforing for OFDMA systes. The deterination of the transit beaforing vector based on the SINR is difficult copared as the receive beaforing, because the received SINR of This work was supported in part by the KMRC R&D Grant for Mobile Wireless fro Kinki Mobile Radio Center, Foundation, JAPAN, and by the Grant-in-Aid for Scientific Research, Grant no. 7689, fro the Ministry of Education, Science, Sports, and Culture of Japan. each user becoes a function of beaforing vectors of all the base stations. Moreover, SINRs at all the obile terinals have to be taken into consideration siultaneously for the calculation of the beaforing vector. Instead of the siultaneous axiization of received SINRs, we utilize socalled signal-to-leakage-plus-noise ratio (SLNR) [5] as the optiization etric of beaforing vectors. The SLNR of the base station is defined as the ratio of the received signal power fro the base station at the desired obile terinal to the received signal power at obile terinals in the other cells plus noise power. The sae idea as the SLNR is used in [6], where the beaforing vector is deterined by using the received SINR of virtual uplink. With the SLNR based criterion, each base station can obtain closed for beaforing vector based only on locally available inforation. We discuss the validity of the SLNR based beaforing vector in ters of the Pareto optiality. We also consider the power allocation proble over subcarriers cobined with the transit beaforing in order to further reduce the ipact of the co-channel interference. As the power allocation strategy, we utilize iterative water-filling [7] [9] for the SLNR instead of the SINR, which is coonly used in the iterative water-filling algorith. The eployent of the SLNR enables us to deterine transit power without sending any signal to obile terinals. Fro the coputer siulation results, we discuss achievable rate of the SLNR based beaforing vector coparing with the perforance of axiu-ratio-cobining (MRC) vector, zero-forcing (ZF) vector, and their linear cobinations [], []. Moreover, the su-rate perforance of the transit beaforing and power allocation ethod is evaluated with highlighting the difference between the SLNR and the SINR based iterative water-filling power allocation algoriths. II. SIGNAL MODEL Consider downlink of OFDMA systes with N pairs of base station and obile terinal. For each base station, one obile terinal out of the N obile terinals is a desired terinal, while the signals fro the rest of the N base stations are considered as interference for the obile terinal. Let s i = [s i,,sm i ] T denote the frequency doain transitted signal block fro the i-th base station to the i- th obile terinal, where M is the nuber of subcarriers and s i is the sybol on the -th subcarrier of the OFDMA
2 signal. Moreover, Pi denotes the transitted signal power of the i-th base station on the -th subcarrier, and we define wi = [w, i,,w,q i ] T as the transit beaforing weight vector on the -th subcarrier of the transitted signal fro the i-th base station, where Q is the nuber of antenna eleents at each base station and wi is assued to be unit nor, i.e., wi =. Furtherore, assuing the length of the guard interval is greater than or equal to the order of the channel ipulse response between the q-th antenna eleent of the i-th base station and the j-th obile terinal h q = [h q (),,hq (L )]T, the frequency response between the q-th antenna eleent of the i-th base station and the j-th obile terinal is given by λ,q.. λ M,q = D h q ], () (M L) where D is M M a discrete Fourier transfor (DFT) atrix, whose {, n} eleent is {D},n =/ M exp( j πn M ) and (M L) denotes a zero vector of (M L). Defining the frequency response vector on the -th subcarrier as λ =[λ,,,λ,q ] T, the received signal at the j-th obile terinal on the -th subcarrier is written as N rj = P i (wi ) H λ s i + n j, () i= where n j denotes a zero ean additive white Gaussian noise (AWGN) at the j-th terinal on the -th subcarrier with the variance of σn. For the j-th obile terinal, only the signal fro the j- th base station is the desired signal, and the signals fro the other base stations result in the interference. Therefore, the SINR at the j-th terinal on the -th subcarrier is given by P j Γ j = (w j )H λ jj(λ jj) H wj Ni=. (3) Pi (wi )H λ (λ ) H wi + σn i j Unlike the case of the receive beaforing, (3) includes beaforing vectors of all the base stations w,, wn and channel frequency responses between the j-th obile terinal and all the base stations λ j,, λ jn, therefore, SINRs of all the users Γ,...,Γ N have to be taken into the consideration siultaneously for the deterination of the beaforing vectors and the power allocation. In order to avoid the coplicated joint optiization proble and the huge overhead caused by exchanging the channel state inforation, we take a two-step suboptial approach for the transit beaforing and the power allocation based on SLNR rather than SINR as explained in the following sections. III. BEAMFORMING VECTOR CONTROL We consider the axiization of SLNR [5] at each base station with respect to the beaforing vector. The SLNR of the j-th base station is defined as the ratio of the received signal power fro the base station at the desired obile terinal (the j-th terinal) to the received signal power at [ all the obile terinals in the other cells plus noise power. This can be also considered as the ratio obtained fro (3) by replacing the interference power observed at the j-th obile terinal in the denoinator with the interference power caused by the j-th base station. The SLNR of the -th subcarrier at the j-th base station is written as Γ j = Ni= i j P j (w j )H λ jj(λ jj) H w j P j (w j )H λ ij (λ ij ) H w j + σ n. (4) Key issue here is that the SLNR is coposed only by locally available inforation, while the SINR (3) includes soe values, which are not directly observable for the j-th base station, like wi or λ for i j. Each transit beaforing vector is controlled so that the SLNR of each base station is axiized. For given transit power Pi, the beaforing vector, which axiizes (4) can be directly obtained as the eigenvector corresponding to the axiu eigenvalue of [ N i= Pj λ ij (λ ij ) H + i j σni Q ] [Pj λ jj(λ jj) H ], (w j ) SLNR = C j ( N ) Pi λ ij (λ ij ) H + σni Q Pj λ jj, i= where Cj is a scaling factor, which keeps (wj )SLNR unit nor, and I Q denotes Q Q identity atrix. In this way, by utilizing the SLNR, we can obtain closed for expression of the beaforing vector. The validity of the utilization of the SLNR based vector is discussed in Sec. V. IV. POWER ALLOCATION For fixed beaforing vectors wj, we consider the transit power allocation with total power constraint of P = M = (5) P j. (6) In order to axiize each user s achievable rate for given beaforing vectors, we can apply the idea of iterative water-filling [7] [9], which is based on the well-known waterfilling theore [] which says that ore power should be allocated on the subcarrier with better SINR. The iterative water-filling can be ipleented in a distributed anner, however, it requires actual signal transission and feedback of observed SINR at the obile terinal several ties for the deterination of the transit power. The proposed ethod utilizes the sae approach as the iterative water-filling but with the SLNR, where ore power is allocated to the subcarrier if the frequency response of the desired channel (w j )H λ jj is greater than that of the interference channel (w j )H λ ij, i j. The utilization of the SLNR enables us to deterine the transit power without sending any signal to the obile terinal, while the power allocation is perfored by an iterative anner. Note that, although the syste considered in the paper has ultiple antennas, the iterative water-filling
3 is perfored not in spatial doain as in [9] but in frequency doain. The transit power of the j-th base station on the -th subcarrier at the n-th iteration is described as Pj (n) hereafter. Defining the su of the leakage and noise power of the - th subcarrier at the n-th iteration noralized by the channel frequency response including the effect of the beaforing vector as Ni= P Xj j (n)(w j )H λ ij (λ ij ) H wj + σn i j (n) = (wj )H λ jj(λ jj) H wj, (7) the proposed algorith to update the transit power is suarized as follows: ) Initialize K =, R(n) = P + X j (n). M ) If R(n) > ax X j (n) then, Pj (n +) = R(n) Xj (n) and exit, otherwise go to 3 3) Define a set A = { R(n) Xj (n)} and K = Card(A), where Card denotes the nuber of eleents in the set. Modify R(n) as R(n) = P + / A X j (n), and if R(n) > ax M K / A X j (n) then go to 4, otherwise { go to 3. R(n) X 4) Pj (n +)= j (n) (/ A) ( A) V. PARETO OPTIMALITY OF SLNR BASED TRANSMIT BEAMFORMING VECTOR We focus on the transit beaforing on a certain subcarrier, so the superscript is dropped and the transit power Pj is set to be in the sequel for siplicity. The achievable rate on the subcarrier at the j-th obile terinal is given by wj R j (w,...,w N ) = log (+ Hλ ) jj i j wh i λ, (8) + σn where the bandwidth is noralized to. The achievable rate region is defined by a set of all rate points (R,...,R N ) achieved by all the possible transit beaforing vectors with unit nor. A rate point is Pareto optial if it is ipossible to increase one of the rate without decreasing at least one of the other rates []. For the special case of two pairs of the base station and the obile terinal, we show the following proposition regarding the Pareto optiality of the SLNR based beaforing vector. Proposition : For N =, transit beaforing vectors, which axiize SLNRs, achieve a Pareto optial rate point. Proof: Let {w SLNR, w SLNR } be the beaforing vectors, which axiize SLNRs, and consider to change the SLNR solution to arbitrary beaforing vectors {w, w }. With the change of vectors, we can write (w ) H λ = α (w SLNR ) H λ, (9) (w ) H λ = β (w SLNR ) H λ, () (w ) H λ = γ (w SLNR ) H λ, () (w ) H λ = δ (w SLNR ) H λ, () where α, β, γ, δ R, α>, β>, γ>, and δ>. Since {w SLNR, w SLNR } are the SLNR solution, SLNRs decrease with any change fro the beaforing vectors as (w SLNR ) H λ (w SLNR ) H λ + σn (w SLNR ) H λ (w SLNR ) H λ + σn α (wslnr ) H λ β (w SLNR, (3) ) H λ + σn γ (wslnr ) H λ δ (w SLNR. (4) ) H λ + σn The proposition is proved by contradiction. Suppose that the rate of user increases with the change of beaforing vectors, naely, the SINR of user increases with the change as (w SLNR ) H λ (w SLNR ) H λ + σ < α (wslnr ) H λ n δ (w SLNR. (5) ) H λ + σn Fro (4) and (5), we have γ( (w SLNR ) H λ + σn) δ (w SLNR ) H λ + σn <α( (w SLNR ) H λ + σn), and this eans γ<α. (6) Here, if we assue the rate of user does not decrease with the change of vectors, then we have (w SLNR ) H λ γ (w SLNR ) H λ (w SLNR ) H λ + σn β (w SLNR. (7) ) H λ + σn Fro (3) and (7), we have α( (w SLNR ) H λ + σn) β (w SLNR ) H λ + σn γ( (w SLNR ) H λ + σn), and this eans γ α, which is a contradiction. Thus, for any change of beaforing vectors, which increases user s rate, the rate of user decreases and vice versa. VI. NUMERICAL RESULTS In this section, the perforance of the transit beaforing and power allocation ethod shown in Sec. III and IV, and the validity of the analysis in Sec. V are exained via coputer siulations. Figs. and show exaples of the achievable rate region (shaded area) of two base stations and obile terinals with two antenna eleents (N =and Q =)for different channel realizations obtained by randoly generated beaforing vectors with nor less than. In the figures, SLNR denotes the rate point achieved by the beaforing vector of (5), while MRC and ZF stands for the rate point achieved by MRC vector and ZF vector, respectively. Since it has been proved that a point on the Pareto boundary is achieved by a linear cobination of the MRC vector and the ZF vector [], [], we have evaluated the achievable rate of the linear cobination by changing the weight of the two vectors (the dashed line connecting MRC and ZF). We can see that the SLNR based vector and the linear cobination with a certain cobination ratio can achieve rate points of Pareto boundary, while achieved rate points are different.
4 . Rate region SLNR Q= Achievable Rate of User MRC ZF Su-Rate / Subcarrier 5 DUR=9 [db] SINR SLNR (proposed) 5. DUR= [db] Achievable Rate of User Fig.. Exaple of rate region (N =and Q =) SNR [db] Fig. 3. Su-rate per subcarrier (Q =). Q= 4 Achievable Rate of User Rate region SLNR.8 MRC.6.4. ZF Achievable Rate of User Fig.. Exaple of rate region (N =and Q =) Su-Rate / Subcarrier SINR 5 DUR=9 [db] SLNR (proposed) 5 DUR= [db] SNR [db] Fig. 4. Su-rate per subcarrier (Q =4) Then, we have evaluated su-rate per subcarrier by the proposed transit beaforing and power allocation ethod. Here, the su-rate per subcarrier is defined as N N M j= = log ( + ˆΓ j ), (8) where ˆΓ j is the observed SINR of the j-th user on the -th subcarrier in the siulation. Figs. 3 and 4 show the su-rate perforance with the nuber of antenna eleents of Q = and Q = 4, respectively. In the figures, the nuber of base stations (or obile terinals) N is set to be 3, and DUR denotes the power ratio of the desired channel frequency response to the interference channel frequency response E[ λ,q jj ]/E[ λ,q ]. Note that the nuber of antenna eleents is less than the nuber of incoing signals for the case of Q =. The perforance of the proposed ethod using SINR in the iterative water-filling is also plotted in the sae figures. Fro the figures, we can see that the degradation of the perforance due to the eployent of the SLNR in the iterative water-filling is rather sall, especially for the case of Q =4. In the perforance evaluation, the overhead caused by the actual signal transission and the feedback of the SINR required by the SINR based iterative water-filling is not taken into consideration, therefore, the proposed ethod ay even outperfor the SINR based approach if all the overheads are counted. Finally, Figs. 5 8 show typical exaples of the achieved power allocation of each user by the SLNR and the SINR based approaches for the nuber of antenna eleents of Q = and Q = 4. The sae channel realization is used for the sae nuber of antenna eleents, and the SNR and the DUR are set to be db and db, respectively. Fro the figures, we can see that not only the SINR based but also the SLNR based iterative water-filling algoriths decrease the transit power of a subcarrier, if the other users have large power on the corresponding subcarrier. This eans that the autoatic interference avoiding subcarrier allocation aong the three users is achieved by the proposed SLNR based iterative water-filling algorith. For Q =4, the SLNR based iterative water-filling can achieve alost the sae power allocation as that of the SINR based ethod. This is because the interference is alost copletely suppressed for this case, and the SLNR coincides with the SINR in the absence of the interference. It should be also noted that, for Q =,the SLNR based approach results in copletely different power allocation copared to the SINR based approach, while the degradation of the achievable su-rate due to the eployent
5 Q= SLNR based allocation User User Q=4 SLNR based allocation User User Fig. 5. Exaple of power allocationslnr based (Q =) Fig. 7. Exaple of power allocationslnr based (Q =4) Q= SINR based allocation User User Q=4 SINR based allocation User User Fig. 6. Exaple of power allocationsinr based (Q =) Fig. 8. Exaple of power allocationsinr based (Q =4) of the SLNR is rather sall. VII. CONCLUSION We have considered transit beaforing and power allocation ethod for downlink OFDMA systes. The beaforing vector is deterined based on the axiization of the SLNR, while the transit power is allocated by the iterative water-filling based on the SLNR. We have also discussed the Pareto optiality of the SLNR based beaforing vector and proved its optiality for N =. Coputer siulation results show that the SLNR based beaforing vector and the linear cobination of the MRC and ZF vectors with a certain cobination ratio achieve different Pareto optial points. Moreover, the degradation of the achievable su-rate due to the eployent of the SLNR based power allocation is rather sall, and the difference is alost negligible for the case with a sufficient nuber of antenna eleents. Future work will include the analytical evaluation of the SLNR based beaforing vector for a larger nuber of users in ters of Pareto optiality. REFERENCES [] H. Sari, G. Kara and I. Jeanclaude, Transission techniques for digital terrestrial TV broadcasting, IEEE Coun. Mag., vol. 33, pp. 9, Feb [] D. Falconer, S. L. Ariyavisitakul, A. Benyain-Seeyar and B. Eidson, Frequency doain equalization for single-carrier broadband wireless syste, IEEE Coun. Mag., vol. 4, pp , Apr.. [3] Z. Wang and G. B. Giannakis, Wireless ulticarrier counications, IEEE Signal Process. Mag., vol. 7, pp. 9 48, May. [4] IEEE Std. 8.6e, Air interface for fixed and obile broadband wireless access systes aendent for physical and ediu access control layers for cobined fixed and obile operation in licensed bands, IEEE, 6. [5] M. Sadek, A. Tarighat, and A. H. Sayed, Active antenna selection in ulti-user MIMO counications, IEEE Trans. Signal Process., vol. 55, no. 4, pp , April 7. [6] F. R. Farrokhi, K. J. Ray Liu, and L. Tassiulas, Transit beaforing and power control for cellular wireless systes, IEEE J. Select. Areas Coun., vol. 6, no. 8, pp , Oct [7] W. Yu, G. Ginis, and J. M. Cioffi, Distributed ultiuser power control for digital subscriber lines, IEEE JSAC, vol., no. 5, pp. 5 5, June. [8] K. B. Song, S. T. Chung, G. Ginis, and J. M. Cioffi, Dynaic spectru anageent for next-generation DSL systes, IEEE Coun. Mag., vol. 4, no., pp. 9, Oct.. [9] W. Yu, W. Rhee, S. Boyd, and J. M. Cioffi, Iterative water-filling for Gaussian vector ultiple-access channels, IEEE Trans. Infor. Theory, vol. 5, no., pp. 45 5, Jan. 4. [] E. A. Jorswieck, E. G. Larsson, and D. Danev, Coplete characterization of the Pareto boundary for the MISO interference channel, IEEE Trans. Signal Process., vol. 56, no., pp , Oct. 8. [] Z. Ka, M. Ho, and D. Gesbert, Spectru sharing in ultiple-antenna channels: a distributed cooperative gae theoretic approach, in Proc. IEEE PIMRC 8, pp. 5, Sept. 8. [] Thoas M. Cover and Joy A. Thoas, Eleents of inforation theory, nd edition, Wiley-Interscience, 6.
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