An Adaptive Bit Loading Algorithm for MIMOOFDMA Systems with Fixed Rate

精品论文推荐An Adaptive Bit Loading Algorithm for MIMO-OFDMA Systems with Fixed RateCui QingDepartment of Communication and Information System, Beijing University of Posts andTelecommunications, Beijing (100044)E-mail: cuiqing122AbstractThis paper considers the optimal resource allocation for MIMO-OFDM systems in multi-user downlinkscenario. By using zero-forcing based SDMA scheme, the multi-user MIMO channels are converted to several SISO channels, which facilitate the proposed bit loading algorithm with the objective of minimizing the system bit error rate (BER). We provide the general mathematical formulation of the optimization problem which includes multi-user interference cancellation, multi-user diversity, bits and power allocation. We consider only the case that the total power and rate are defined by a system. Thus, our proposed algorithm can be expressed through a closed form equation, and it has very low complexity comparing with the iterative ones.Keywords: MIMO; OFDM; multi-user; Zero-forcing; adaptive bit and power allocation1. Intr oduct i on Multiple-input multiple-output (MIMO) systems have been recognized as a promising candidate for next-generation wireless communications due to their potential for dramatic gains in channel capacity 12. In a single-user context, the diversity gain over the single-input single output (SISO) is roughly the smaller number of antennas between the base station and the user terminal.Because of the size and cost constraints the number of the receive antennas are always the smaller one. It is obvious that significant capacity benefit cannot be obtained from the multiple transmit antennas at the base station. The solution to this problem is to serve multiple users simultaneously which can exploit multi-user diversity gain by taking advantage of the independence of the channel conditions of different users.On the other hand, there is growing interest in orthogonal frequency division modulation (OFDM), which has become a mature technique for providing high speed data services and is adopted in many existing systems, such as IEEE802.11a/g Wireless Local Area Networks, IEEE802.16 (known as WiMax), and also a strong candidate in ultra-wideband (UWB) standard. Since OFDM converts a frequency-selective fading channel into a set of parallel flat-fading channels, SDMA-related algorithms can be implemented on each sub-carrier. It is also found capable of providing further system capacity gain by exploiting multi-user diversity in OFDM systems 34, deducing a new multiple access technique so called orthogonal frequency division multiple access (OFDMA). By combining SDMA and OFDMA, multi-user diversity can be exploited both in the spatial domain and the frequency domain.For the above reasons, MIMO-OFDMA-SDMA system has considerable potential to increase degrees of freedom of the rich scattered channels and facilitates adaptive resource allocation which can enhance system throughput further more. A set of parallel, independent and spectrally flat sub-channels must be shaped before resource allocation scheme is applied. For downlink, zero-forcing (ZF) technique or block diagonalization (BD) has been proposed for SDMA to cancel the co-channel multi-user interference 56. It is feasible and desirable to move the most of the signal processing complexity from the user terminals to the base station by adopting preprocessing the inter-user interference at the base station, while subsequently a simple receive method which neglects the other users interference can be utilized at the user terminal.This paper considers the optimal resource allocation in ZF based MIMO-OFDMA-SDMAdownlink system for the purpose of minimizing the average BER. The MIMO-OFDMA-SDMA-10-system accommodates the users in both frequency domain and spatial domain. So the resource allocation not only refers to the power and bits allocation among the sub-carriers, but also user selection and optimal beam-former on each sub-carrier.The rest of this paper is organized as follows. In section 2, we describe the system model andpresent the ZF based optimal beam-forming which results in a set of independent parallels to accommodate multiple users data and power. And then the optimal resource allocation algorithm with discrete bit loading scheme are demonstrated. Section 3 presents the simulation results comparison. Finally, Section 4 concludes this paper.2. System Model an d Optim al Sch e me 2.1 Syste m Model In this paper, we consider a downlink adaptive multi-user MIMO-OFDM system equippedwith N sub-carriers,NT transmit antennas at a base station and K mobile users, each withNR receive antennas. And the MIMO channel on sub-carrier n between the base station and userk can be characterized by aNR NTmatrixH k , n whose elements can be viewed as thetransmission coefficient between a receive-transmit antennas pair assuming a flat-fading case. Sothe received signal for the k -th user on the n -th sub-carriery k ,n = Hk , n xn + nk , ny k ,n is(2-1)Where xnis the transmitted signal vector on sub-carrier n including the multiple usersdata, andnk ,n denotes aNR 1White Gaussian noise at the k -th user terminal with a variance 2 for each dimension. The system block diagram for each sub-carrier n is depicted in Figure1.d1, nM1,nW1,ny1,nd2, ndK ,n M 2, nM K , nW2,nWK ,n y 2,ny K , n Fig1 The multi-user MIMO-OFDMA downlink system on sub-carrier nTo avoid a cost and complex receiver design, we put the burden of interference elimination entirely on the base station transmitter. The multiple users can share the same time slot and the same frequency band resource by multiplying pre-coding matrixes containing a ZF-based processing for the inter-user interference cancellation. And each user can only use a simple MRCmethod to detect its own valid signal at the receiver 7.Denotingvector is thusM n , k as the pre-coding matrix for user k on sub-carrier n , the transmit signalKxn = M k , n dk , n .(2-2)k =1Wheredk ,n is a data vector of arbitrary dimensionm% n ,k for user k .The received signal atthe k -th user terminal is thus KKy k ,n = Hk , n Ml , n dl , n + nk ,n = H k , n M k , n dk , n + Hk , n Ml , n dl ,n + nk , n .(2-3) l =1l =1,l kThe first term in (2-3) is the valid signal, the second and the last one are, respectively indicated the interference from other users and the additive Gaussian noise.In some practical context, there is no need to reach the capacity of the transmission system, but to transmit constant data rate with the specified total power at the lowest error rate possible. Our objective here is to optimize such a multi-user downlink MIMO-OFDM system depicted infigure1 with fixed rate and fixed power. Specifically, we want to obtain several parallelindependent single user SISO channels by choosing an appropriate pre-coding matrixM k ,n foruser k on sub-carrier n to convert the original inter-user interference channels. And then a discrete bit loading scheme is applied among these equivalent independent channels, while meeting the constraint of constant rate and constant power. Thus, our target can be described asthe following constrained optimization problems:P1:K Hk , n Ml ,n = 0l =1,l k(2-4)P2:Minimize error probability(2-5) With the additional constraints:D Ri =RTi =1D Si =STi =1(2-5a).(2-5b)Where D is the total number of the valid independent converted channels, RiandSi arerespectively the assigned bits and power for i -th channel, RTsystem.and STare specified by a2.2 Co m b ined Opti m i za tion in Spatial Do m a in and F requency Do m a in This section addresses a solution to the optimization problems in (2-4) and (2-5). P1 should be considered first, and then P2 will be solved by applying a resource allocation algorithm based on the equivalent channels obtained from former processes.2.2.1 Inte r-user Inte r f ere nce C a n cel l a ti on To simplify the signal processing, a restriction is imposed that a same pre-coding scheme is adopted for all the sub-carriers. The channel state information (CSI) is estimated perfectly at the receiver and is feedback to the base station with no errors. According to (2-4), an equivalent form is expressed asHl ,n M k , n = 0, l k .(2-6)To achieve thisM k ,n should be chosen from the null space ofH% k , n , which is defined asTTTTTH% k , n = H1, nK H k 1, nHk +1,nK H K , n.(2-7)By using singular value decomposition (SVD),H% k , n can be rewritten asHk , n H%= U%V% (1) V% (0) .(2-8)k , nk,nk , n k , n k ,n V%Where(1) k , n holds the first L%n , k right singular vectors andV% (0) holds the last ( N L%n , k )V%ones, lettingL%n , k = Rank (H% k , n ) . Hence,(0) k , n forms an orthogonal basis for the null space ofH% k , n . Thus,Mk ,n can be generally presented ask , nk ,n k , n TM= V% (0) A.(2-9)Ak , n is a ( NT L%n , k ) m% n ,kmatrix , and will act in the next step referring to adaptiveresource allocation .Substituting (2-9) into (2-3), we obtain(y k ,n =(0) H V%)k ,n k , nAk , n dn + nk , n = Hk , n Ak , n dk , n + nk , n.(2-10)andSo the downlink system reduces to K parallel non-interfering single user MIMO channels,Hk , n denotes the equivalent independent channel.2.2.2 O p ti m a l Beam -form i n gSince the aim of optimization is to minimize the error probability, maximizing the receivedSNR will help in space domain processing. It is well known beam-forming is optimal in terms of maximizing the received signal-to-noise rate (SNR). After ZF-based preprocessing , the multipleusers channels can be viewed as several single user channels, andAk , n can be seen asbeam-forming matrix on the k -th user channels. Consider the situation that only one data streamcan be allowed to serve each user. Thus, a system can have no more thanNT Nindependentchannels, andAk , n is restricted to a ( NT L%n , k ) 1vector. Then a loading scheme can work onthese channels as it dose in single user system.Based on (2-10) and using Maximum ratio combining (MRC) at the receiver, thecorresponding optimum weighted combining vectorrk ,n is obtained:Hence, the combined received signal isrk ,n = (Hk ,n Ak ,n )H.(2-11)HHz k , n = (Hk , n Ak , n )Hk , n Ak ,n dk , n + ( Hk , n Ak ,n )nk , n .(2-12)SNRk ,n at the k -th receiver on the n -th sub-carrier isE HA)H (k ,nk ,n 2 HHk ,n Ak ,ndk ,n PSNR= =k ,n ( A)H (H)H HAHH.(2-13)k ,n (k ,nk ,n 2 E HA)Rnk ,n N 2k ,nk ,nk ,nk ,nWherePk,ndenotes assigned power for k -th user on n -th sub-carrier. So maximizingtheSNRk ,n is transformed to maximize the quantity of ( Ak ,n ) ( Hk ,n )Hk ,n Ak ,n .By eigenvalue decomposition, we can obtainHHk , n Hk , n = Uk,n k , n Uk,n .(2-14)The optimal beamforming vector is then chosen as 8A= umaxk ,n k ,nk ,n .(2-15)uWheremaxk ,n indicates the eigenvector corresponding to the maximum eigenvalue max .Then (2-13) becomesP maxSNR= k ,n k ,n (2-16)Rk ,n N 22.2.3 Po w er a nd bi t al l o cati on Thanks to the spatial preprocessing, a set of parallel independent channels with best channel quality index (CQI) are abstracted from multi-user MIMO channels. A discrete bit loading scheme can be utilized while ignoring these channels belong to different users. To achieve P2 described in (2-5), the rate and power are distributed according to minimize error probability, which also means maximizing the received SNR.On condition that square MQAM modulation is adopted, the approximate expression for theerror symbol rate with Gray bit mapping at i -th independent channel is 9P = K Q d 2 4 i .(2-17)2NiiiWhereQ ( x ) = 12 xexpt 22 dtis the complementary Gaussian integralfunction, anddi is the minimum Euclidean distance in the constellation. For conciseness Kiis selected as a constant numberK= 4 1 1 4.(2-18)iM Hence it is equal for all i and its influence can be neglected.The square constellation can be scaled with the half of minimum Euclidean distancedi , andthus the signal powerSi can be deducedS = 2 ( M 1) d 22 2Ri d 2(2-19)i3i3iWhere M is the number of signal constellation and is equal to 2Ri .Since the total error rate would dominate by the highest one, we must force all Pito a sameminimum valueP0 . Thus, P2 in (2-5) together with (2-17) and (2-19) is translated tod 2 4 i MaximizeSNR0 =Ni 2(2-20)Based on the constraints in (2-5a) and (2-5b), we obtainS = 1 N 2Ri SNRi3i0(2-21)Then (2-20) can be written as=SNR03ST Di max.(2-22)Ni =1 2RiUsing Lagrange optimization combined with (2-5a), (2-22) is equivalent toD DRiL = Ni 2 Ri RT min(2-23)i =1 i =1 iwhere is nonnegative Lagrange multiple. It is obvious that (2-23) is concave, and thenecessary condition for optimal isiiLRi= N 2Ri N 2Ri = , for all i(2-24)iwhich results in deriveN 2Ri should be constant. Taking a D times product overN 2Ri , we( Ni 2Ri )DD= Nii =1 2Ri = 2RTD Nii =1.(2-25)Then by a logarithm operation over the two sides in (2-25), Riis deduced2 DR = RT + 1 log Ni ND i =1 (2-26)iDDiIf Ri 0happens for some i D , the related channel must be in a terrible condition andnot be allowed to serve transmission. Among a smaller number of channels excluding theforbidden one, (2-26) can be applied once again. This is done iteratively until all ratesRi of theremaining channels are positive. The number of these channels is denoted by D and thecorresponding indices should be comprised in the set .Besides, the rates must be integers for MQAM modulation.Ri obtained in precedingprocedure have to be quantized to the nearest integer numberRQi .Substituting (2-22) into (2-21) with the knowledge ofRi ,Si is obtained byS N 2RiiS = T i .(2-27) Nii 2RiThus, bits and power allocated on each sub-channel can be solved successively through the two closed-form equations (2-26) and (2-27). Comparing to iterative algorithms, proposed one has a low complexity and practical to real-time communications with no performance loss.3. Sim u lat ion R e sul t sIn this simulation system, SCM channel model is used 10, which is the interim channel model for Beyond-3G systems. It is a six-path and the maximum delay is no more than 10 TC , where TC is the sampling interval. And the base has 4 antennas, and each mobile has 1 antenna. Assuming that the fading is quasi-static, i.e., it remains constant during the transmission of an OFDM symbol and it changes from a symbol to another. The entire transmit bandwidth, 5MHz, is divided into 512 sub-carriers and an additional 16 cyclic prefix is introduced to eliminate the ISIdue to the multi-path delay spread. Assume that every symbol carry 0, 2, 4, 6, or 8 bits, namely,4QAM, 16QAM, 64QAM and 256QAM are available in our adaptive modulator. For a simple normalization, an OFDM symbol embodies 2048 bits, that is, an average 4 bits per sub-carrier. And the total power of an OFDM symbol is 2048.Consider that 4 independent streams are transmitted from the base, i.e., each user are served with one stream for 4 users system, and two streams for 2 users system, and so on. Theperformance of fixed modulation (FM) and adaptive modulation (AM) are compared in each case.010-110-210-3BER10-410AM for 4 users FM for 4 users-5 AM for 2 users10 FM for 2 usersAM for 1 users FM for 1 users-610-710012345678910SNR (dB)Fig2 The performance of fixed modulation (FM) and adaptiveModulation (AM) for different number of users.From Figure2, we can see that adaptive scheme outperforms the fixed one in the entire range of observed average bit SNR, especially in the higher range. It also shows that adaptive modulation contributes little in the range of low SNR, and has a steep performance gain according to the increase of SNR.On the other hand, accompanying with the decrease of users number, the performances of BER are decayed. Multi-user system improves performance 24 dB. It demonstrates that the free degree of spatial domain increases when more users take part in the transmission. And in single user case, it must adopt all the spatial channels to achieve multi-steam multiplex, although some channels are in deep fading condition.4. C o nclu s ion sIn this paper, a multi-user MIMO-OFDM transmit scheme with the constraint of fixed rate and power, is designed which includes three steps: zero-forcing based multi-user interference cancellation scheme in spatial domain, exploiting multi-user diversity by beam-forming, bit and power allocation among the obtained sub-channels.In conclusion, the scheme presented in this paper is optimal with the goal of minimizing the average BER.MIMO-OFDMA-SDMA system has considerable potential to increase the degree of freedom for the rich scattered channels and facilitates adaptive bit and power allocation, which are multi-user diversity effects. And the proposed discrete bit loading algorithm has low complexity but maintains excellent performance.R e f ere nces 1 B. G. Agee. 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