An Adaptive Power Allocation Scheme for Space-Time Block Coded MIMO Systems

Transcription

1 An Adaptive Power Allocation Scheme for Space-Time Block Coded IO Systems LiangXianandHapingLi School of Electrical Engineering and Compter Science Oregon State University Corvallis, OR 9733 USA , fax) Abstract Receive diversity yields a higher signal-to-noise ratio SNR) than transmit diversity when the total transmitted power and diversity order are the same. However, if the transmitter has complete or partial knowledge of channel, the SNR gap between these two schemes can be redced. This paper introdces an adaptive power allocation scheme for space-time block coded STBC) mltiple-antenna systems to improve system performance. For any set of channel fading coefficients, the transmit power scaling factors are controlled by a single design parameter. The proposed adaptive power allocation scheme improves the instantaneos SNR at the receiver. Special choices of reslt in some existing STBC schemes. Performance gain of the proposed scheme over the conventional eqal-power scheme nder the condition of perfect and imperfect feedback is stdied. The maximm achievable SNR gain limit over the conventional scheme is also derived. I. INTRODUCTION Space-time block codes STBC) provide transmit diversity over fading channels. In a commonly sed STBC, transmit power is eqally divided among all transmit antennas. However, if the transmitter has fll or partial knowledge of the channel, adaptive transmit power allocation that allocates more power to the transmit antenna with a better fading condition will improve the received signal-to-noise ratio SNR). In 4, several adaptive power allocation methods for systems with two transmit antennas were introdced. These schemes can be considered as a variation of the Alamoti scheme 5. In 6, a method to transmit the Alamoti block code based on selecting two ot of three transmit antennas was proposed. When the transmitter does not have perfect knowledge of the fading coefficients, none of the methods mentioned above can garantee the maximm SNR at the receiver. In this paper, we derive the maximm SNR gain limit achievable by adaptive power allocation for STBC designed for mltiple-inpt mltiple-otpt IO) systems when perfect feedback is available. Then, an adaptive power allocation scheme with imperfect feedback is proposed and analyzed. A design parameter is introdced to control the power scaling factors. SNR gain of the proposed scheme over the conventional scheme in which power is eqally distribted among all transmit antennas is provided. The conventional STBC scheme and the adaptive scheme analyzed in are special cases of the proposed scheme with specific choices of a design parameter. II. SYSTE ODEL Consider a wireless commnications system with transmit antennas and N receive antennas, denoted as an,n) system in this paper. Each receive antenna responds to each transmit antenna throgh a statistically independent fading coefficient. The received signals are frther corrpted by additive white Gassian noise that is statistically independent among different receive antennas and different symbol periods. Let the P transmission matrix be g, g, g, g, g, g, G = ) g P, g P, g P, and the transmitted symbol vector be s =s,s,,s K T, where T denotes transpose. Each element of G is a linear combination of symbols s,s,,s K and their complex conjgates. The p, m)th entry of G, g p,m, will be transmitted at time slot p from transmit antenna m. The code rate, as defined in 7, is given as K/P, where P is the nmber of time slots sed to transmit K symbols. The total average transmit power is normalized to. Average energy of each symbol is E s. Ths, the transmitted signal at time slot p from transmit antenna m is expressed as x p,m = α m Es g p,m, where α m is a real power scaling factor determined by feedback information. In order to maintain the same total average power after power scaling, it is reqired that αm =. ) For the conventional STBC scheme, α m = /, m =,,. The channel is assmed to be qasi-static, allowing it to be constant over a frame of symbols and change independently from one frame to another. Let h m,n denote the fading coefficient from the mth transmit antenna to the nth receive antenna of an,n) system. Rayleigh fading is considered so that h m,n is a zero-mean complex Gassian random variable. IEEE Commnications Society / WCNC /05/$ IEEE

2 The average power of the channel is also normalized so that h m,n has a nit variance. The received signal at time p by receive antenna n, r p,n,is given as r p,n = h m,n x p,m + ν p,n 3) where ν p,n is the additive zero-mean white Gassian noise component with variance. The maximm likelihood L) decoder calclates the following decision metric P N d = r p,n α m Es ĝ p,m 4) p= n= and the codeword ŝ,, ŝ K ) that minimizes d will be the decoder otpt. III. SNR ANALYSIS Assming a fll-diversity system coded with orthogonal space-time block codes OSTBC), L decoding can be achieved sing linear operations on r p,n, α m, and h m,n.in a system with a rate- transmission matrix or with a rate-3/4 transmission matrix, the decision variable for the kth element of s, ŝ k, is expressed as 5, 7 9 ŝ k = N E s αm h m,n s k + ξ k 5) n= where ξ k is the complex zero-mean Gassian noise component N whose variance is given as σξ k = αm h m,n.as n= an example, in a,) system =,N = ) with the Alamoti code 5, the received signals are expressed as r, = E s α h, s + α h, s )+ν, r, = E s α h, s + α h, s )+ν, and the decision variables are given as ŝ =α h,r, + α h, r, 6a) 6b) = E s α h, +α h, ) s +α h,ν, +α h, ν, ŝ =α h,r, α h, r, = E s α h, +α h, ) s +α h,ν, α h, ν,. In a system with rate-/ transmission matrix for complex signals, the decision variable is given as 8 ŝ k = N E s αm h m,n s k + η k 8) n= where η k is the complex zero-mean Gassian noise component whose variance is given as ση k = N n= α m h m,n. Obviosly, the SNR for rate-/ codes is dobled compared with rate- and rate-3/4 codes. With adaptive power allocation, however, the SNR gain will be the same for codes of rate, 3/4, and /. Specifically, let SNR a be the SNR with adaptive power allocation and SNR c be the SNR with the conventional eqal-power scheme. The ratio SNRa SNR c will be the same for codes of rate, 3/4, and /. Ths, in the following discssion, we will only focs on rate and rate 3/4 codes. The received instantaneos SNR is obtained as γ = E N s αm h m,n. 9) n= IV. ADAPTIVE POWER ALLOCATION A. inimm Feedback Allocation Scheme Antenna Selection) Let m = N n= h m,n. Withot loss of generality, we assme that.... Ths, we can write = + δ, = + δ + δ,..., = + δ δ, where i and δ j are nonnegative real nmbers. The instantaneos SNR is then expressed as γ = E s + δ αi + δ αi δ α. i= i= 0) Obviosly, when α =note that α m =), γ is maximized to be Es. This means that if... holds, the system shold allocate all its power to transmit antenna for best performance. The feedback reqired for this scheme is minimm; only log ) bits for each transmission, where denotes the nearest integer towards infinity. For simplicity, we will refer to this scheme as the minimm-feedback-allocation scheme FAS). Note that this scheme reslts in antenna selection one ot of ). Other advantages of the FAS inclde that there are no qantization errors for the feedback. Becase there is no inter-symbol interference, it is easy to realize a rate- transmission for complex signals with fll diversity, which is a challenging isse for IO systems with STBCs. However, this scheme, as will be seen from simlation reslts in Section V, is more sensitive to feedback errors than other power allocation schemes. B. A New Adaptive Power Allocation Scheme In practice when feedbacks are imperfect channel coefficients obtained by the transmitter throgh feedback contain errors), a very simple scheme with α > > α will improve the system performance if > >. In this case there are variables, α,,α α = α m ), to be solved, and it is rather difficlt to determine which set of combinations of α m give the best performance. Ths, we propose a new scheme with only one parameter that can be easily controlled to maximize SNR at the receiver. Additionally, this scheme is robst to feedback errors. In the proposed adaptive power allocation scheme, the real scaling factor for the mth transmit antenna is given as α m = m ) IEEE Commnications Society / WCNC /05/$ IEEE

3 where for a given set of channel coefficients, h m,n, parameter controls the power scaling factor α m. It is easy to verify that α m given in ) satisfies the reqirement given in ). It is worth of mentioning two special cases, = 0 and =, which correspond to, respectively, the conventional STBC scheme in which power is eqally distribted among all transmit antennas and the adaptive scheme proposed in for a system with two transmit antennas. By applying the power scaling factor α m given in ) to the instantaneos SNR given in 9), we obtain γ = E s + m. ) It will be interesting to examine the relationship between SNR and parameter for the adaptive power allocation scheme. The difference between γ + and γ is obtained to be γ + γ = Es = Es = Es + m + m i= i i= j= ) + m ) j= + ) ) j= + j j ) ) i= i j= + j i j i j ) i<j. 3) i + j It can be seen from Eq. 3) that γ + γ is always greater than or eqal to 0 with eqality only if = = =. If this condition does not hold, which is tre for any practical scenario, SNR increases monotonically with parameter note that does not necessarily need to be an integer). However, performance improvement with the proposed adaptive power allocation scheme will satrate as increases. This is proved as follows. Withot loss of generality, we assme that = = = w = max{,, }, where w<. The ratio γ + /γ can be written as γ + = γ = w + ɛ w + ɛ. It can be easily determined that which implies ) + + ) ) lim ɛ = lim ɛ =0, lim γ +/γ =. 4) Additionally, let s consider the limit of γ : lim γ = E s = E s lim lim ) ) + ) ) = E s. 5) Eq. 5) gives the ltimately achievable maximm SNR at receiver with the proposed adaptive power allocation scheme, which is the same as the SNR achieved by antenna selection. Based on Eq. 5) and the fact that γ is a continos fnction of, an appropriate cold reslts in the maximm achievable SNR. This redces the mltidimensional problem to a onedimensional problem. We define the average SNR gain as the ratio of the average SNR with the adaptive power allocation scheme to the average SNR with the eqal-power scheme. This ratio is expressed as 0 log E{γ} 0 E{γ 0} db, where E{ } denotes expectation. Recall that the average SNR for the traditional eqal-power scheme is given as E{γ 0 } = E s E { N } h m,n n= = NE s. 6) The maximm average SNR gain in db can be obtained as ) ) E{γ+ } E{max,, )} 0 log 0 = 0 log E{γ 0 } 0 N 7) where i, i =,,, are central chi-sqare-distribted random variables with freedom N in a Rayleigh fading environment. The cmlative distribtion fnction CDF) of i can be fond in closed form as 0 N F Y y) = e y/σ k=0 y ) k, y 0 8) k! σ where σ = /. TheCDFofmax,, ) is given as N FY max y ) k y) = e y/σ k! σ k=0 N = e y k! yk, y 0. 9) k=0 The probability density fnction PDF) of max,, ), p max Y y), can be calclated by differentiating FY max y). The expected vale of max,, ) is obtained as E{max,, )} = As an example, let s consider a,) system: 0 yp max Y y)dy. 0) F Y y) = e y, y 0 F max Y y) = e y ), y 0 p max Y y) = e y e y ) E{max, )} = 0 ye y e y )dy = 3. Therefore, the maximm average SNR gain for a,) system is 0 log 0 3 )=.76 db. Vales of the maximm average SNR gains for varios combinations of and N of a IO system are evalated nmerically and smmarized in Table. IEEE Commnications Society / WCNC /05/$ IEEE

4 Gain db) = =3 =4 =5 =6 N = N = N = N = N = Table : The maximm gains in the average SNR for IO systems. Examining Table, we find that the maximm gain in the average SNR de to the proposed adaptive power allocation increases as the nmber of transmit antennas increases, and decreases as the nmber of receive antenna increases. This can be intitively explained as follows. As increases with N fixed, E{max,, )} has more dimensions to provide a gain. On the other hand, when N increases with fixed, the difference between max,, ) and the average vale of i decreases. C. The New Scheme with Imperfect Feedback In a practical system, channel coefficients will not be perfectly known. Even if channel coefficients were perfectly known, there will be qantization errors in the feedback. In order to resolve the problem of imperfect feedback and lower the nmber of feedback bits reqired, we pre-determine a finite set of vales for α m. The receiver only needs to inform the transmitter that which pre-determined power scaling factor shold be assigned to antenna m. For example, in a system with two transmit antennas, we pre-determine a fixed set of vales for α m as α m 0.8, 0.6). If the receiver finds ot that >, it then needs only bit to instrct the transmitter to allocate 0.8 to antenna. For a general system with transmit antennas, log!) feedback bits are needed. For simplicity, we assme that the feedback system is a SISO system with the same constellation as the information channel. The average energy of feedback symbols is also E s. The pre-determined power scaling factors α m,m =,, for a particlar choice of parameter can be determined sing the method as follows. As defined earlier, m is a fnction of fading coefficients h m,,,h m,n.for each realization of the channel coefficients, let max = max{,, }. The pre-determined largest { power scaling factor α max can be set as α max = E max }. In the same manner, let sec be the second largest vale among,,, for each realization of the channel. The second largest { power scaling } factor α sec is calclated to be α sec = E sec. This method can be contined ntil the smallest scaling factor α min is determined as α min = α max αsec. As an example, if is chosen to be =for a system with =3, then the pre-determined power scaling factors can be calclated to be α,α,α 3 ) , 0.558, 0.360). If is chosen to be =, then α,α,α 3 ) 0.860, 0.444, 0.973). For 3 transmit antennas, we have to se log 3!) =3bits to feed back 3! = 6 possible grops α max, α sec, α min ). However, the 3 bits cold represent 8 niqe grops, yielding two invalid grops. The transmission power allocation strategy for this case is that if feedback symbols are erroneosly decoded as one of two invalid grops, eqal power allocation will be sed in next transmission. V. NUERICAL EXAPLES AND DISCUSSION Simlated reslts demonstrating the performance of the proposed adaptive power allocation schemes are obtained in this section. Fig. shows the error probability of different systems as a fnction of parameter. It is fond that for a ISO, BPSK, Es/No = 5dB IO, QPSK, Es/No = db Fig.. verss parameter =,N =, ), ) system with BPSK modlation operating at E s / = 5dB, the optimm vale of is 0.6. The corresponding power scaling factors for the proposed adaptive scheme with imperfect feedbacks can be determined to be α max,α min ) 0.896, ).Fora, ) system with QPSK modlation operating at E s / = db, the optimm vale of is fond to be. The corresponding power scaling factors for the two transmit antennas are determined to be α max, α min ) , ). It is sally not easy to determine the optimm vale of by an analytical approach since it depends on E s / in the information channel, the power of feedback symbols, the nmber of transmit and receiver antennas, N), and the modlation scheme. With the feedback model and PSK modlation, the optimm in the sense of minimizing error probabilities can be calclated nmerically sing the procedre described below. According to Eq. 5), orthogonal space-time block codes in an, N) system have the same performance as a, N) system a diversity-reception only system) sing maximal ratio combining, provided that the transmit power per antenna is the same in both systems to make the comparison fair. Therefore, the optimm vale of can be fond by sing the exact IEEE Commnications Society / WCNC /05/$ IEEE

5 FAS with perfect feedback SNR limit) new scheme with =0.6, imperfect feedback eqal power FAS, imperfect feedback Eb/No Fig.. verss E b / crves for different schemes =,N =, BPSK) 0 4 FAS with perfect feedback SNR limit) new scheme with =, imperfect feedback eqal power FAS, imperfect feedback Es/No Fig. 3. verss E s/ crves for different schemes =,N =, QPSK) error probability for mltichannel PSK signals given in 0, Appendix C. As an example, let s consider the Alamoti scheme sing BPSK in a, ) system. We can easily compte the error probability in the feedback channel P feedback and the error probability in the information channel for eqal P i,eqal power allocation P i,eqal no adaptive power allocation), where = f Es ) is a fnction of the received signal-tonoise ratio. Additionally, we have E{ max } =.5 from Table. Ths, E{ min } =.5 = 0.5. The average error probability for the information channel with adaptive power allocation nder imperfect feedback is given by P i,adp = P feedback )f.5αmax Es )+P feedback f0.5 αmax) Es ), where α max 0, ) is a variable that depends on. Ifwe fix Es, then P i,adp is a fnction of. The optimal vale of can be fond by minimizing P i,adp. Figs. and 3 compare the error performances of the FAS, the eqal-power scheme, and the adaptive power allocation scheme which applies the optimm. Althogh the optimm depends on E s /, for simplicity the vales of obtained in Fig. are sed for any E s / in Figs. and 3. It is fond that when perfect feedback symbols are assmed, the antennaselection scheme works the best. However, when there are feedback symbol errors, the antenna-selection scheme sffers from diversity loss. VI. CONCLUSION We have proposed a new power allocation scheme for spacetime block coded IO systems. If the channel coefficients are known, the power scaling factors for all transmit antennas are controlled by a single parameter which, for some special cases, can be predetermined nmerically. Different choices of parameter yields different SNR gains. The maximm achiev- able SNR gain can be achieved by choosing an appropriate vale of. Some special choices of parameter with the proposed adaptive power allocation scheme redce to some existing STBC power allocation schemes i.e., 5). A mch simpler power allocation scheme single antenna selection) that needs significantly less nmber of feedback bits is also proposed. Performance gains of the proposed schemes over the conventional eqal-power STBC scheme are simlated for systems with different nmber of antennas and modlation schemes. REFERENCES J. H. Horng, L. Li, and J. Zhang, Adaptive space-time transmit diversity for IO systems, Proc. of IEEE VTC 03, Apr. 003, pp T. Lo, Adaptive space-time transmission with side information, Proc. of IEEE WCNC 03, ar. 003, pp Seo and S. W. Kim, Power adaptation in space-time block code, Proc. IEEE Globecom 0, Nov. 00, pp G. Ganesan, P. Stoica, and E. G. Larsson, Diagonally weighted orthogonal space-time block codes, Proc. of Asilomar Conf. On Signals, Systems and Compters, Nov. 00, pp S.. Alamoti, Simple transmit diversity techniqe for wireless commnications, IEEE Jornal on Selected Areas in Commnications vol. 6, pp , Oct W. H. Wong and E. G. Larsson, Orthogonal space-time block coding with antenna selection and power allocation, Electron. Lett., vol. 39, no. 4, pp , Feb V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space-time block codes from orthogonal designs, IEEE Trans. on Information Theory, vol. 45, pp , Jly V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space-time block coding for wireless commnications: performance reslts, IEEE Jornal on Selected Areas in Commnications, vol. 7, pp , ar X. Li, T. Lo, G. Ye and C. Yin, A sqaring method to simplify the decoding of orthogonal space-time block codes, IEEE Trans. on Commnications, vol. 49, pp , Oct J. G. Proakis, Digital Commnications, 4th ed, cgraw-hill, 00. IEEE Commnications Society / WCNC /05/$ IEEE