Multi attribute augmentation for Pre-DFT Combining in Coded SIMO- OFDM Systems

Transcription

1 Multi attribute augmentation for Pre-DFT Combining in Coded SIMO- OFDM Systems M.Arun kumar, Kantipudi MVV Prasad, Dr.V.Sailaja Dept of Electronics &Communication Engineering. GIET, Rajahmundry. ABSTRACT For coded SIMO-OFDM systems, pre- DFT combining was shown to provide a good trade-off between error-rate performance and processing complexity. Max-sum SNR and max-min SNR are two reasonable ways for obtaining these combining weights. In this paper multi attribute augmentation is Employed to further reveal the suitability and limitation of these two criteria. The results show that neither max-sum SNR nor max-min SNR is universally good.for better error-rate performance, the means for weight calculation should be adapted according to the capability of the error-correcting code used, and multi attribute augmentation can help in the determination. Key words SIMO, MIMO, OFDM, pre-dft combining, convolution code 1. INTRODUCTION Orthogonal frequency division multiplexing (OFDM) combined smart antennas i.e with multiple transmit and receive antennas namely, single-input multiple-output (SIMO) OFDM, has been investigated for use in wireless communication systems. It can provide high spectrum efficiency and high data rate for information transmission. On one hand, OFDM divides the entire channel into many parallel sub channels which increases the symbol duration and therefore reduces the inter-symbol interference (ISI) caused by multipath propagation. Besides, since the subcarriers are orthogonal to each other, OFDM can utilize the spectrum very efficiently. On the other hand, SIMO and MIMO along with combining techniques takes advantage of receive spatial diversity and therefore further enhances the performance. Subcarrier-based maximum ratio combining (MRC) performs the best for coded SIMO-OFDM systems. However, it requires high processing complexity. Pre-discrete Fourier transform (DFT) combining was then developed, in which only one DFT block is necessary at the receiver [1]. It was shown to provide a good trade-off between error-rate performance and processing complexity. In this papermulti attribute augmentation is employed to reveal the suitability and limitations of two previously-proposed criteria for obtaining the pre-dft combining weights, i.e., maximization of the sum of sub carrier signal-to-noise ratio (SNR) values (called max-sum SNR hereafter) [1] and maximization of the minimum subcarrier SNR value (called max-min SNR here-after) [2]. The results show that neither max-sum SNR nor max-min SNR is universally good. Furthermore, for better error-rate performance, the means for weight calculation be adapted according to the capability of the error-correcting code used, and Multi attribute augmentation can help in the determination. Monte Carlo simulations are finally provided to verify the correctness of these sayings. Combining for MIMO-OFDM systems

2 is analyzed considering a Rayleigh channel. Bit error rates for SIMO- OFDM and MIMO-OFDM are compared for rician and Rayleigh channel estimates. Boldface letters with over bar, lowercase letters, and upper-case letters are used to denote vectors, matrices, timedomain signals, and frequency-domain signals, respectively. Besides, ( ), ( ), trace( ), rank( ), and diag{ } are used to represent the matrix trans-pose, matrix Hermitian, matrix trace, matrix rank calculation, and diagonal matrix with its main diagonal being the included vector, respectively. II. PRE-DFT COMBINING IN SIMO-OFDM SYSTEMS A an SIMO-OFDM system is considered with M receive antennas.define a N x 1 signal vector S(k)=[S(KN) S(KN +1) S(KN + N -1 )] T as the Kth OFDM data block to be transmitted, where is the number of subcarriers. This data block is first modulated by the inverse DFT (IDFT). With matrix representation, we can write the output of the IDFT as S(k)=[S(KN) S(KN +1) S(KN + N -1 )] T =F H S(K). where F is an N x N DFT matrix with elements [F ]p,q.= (1/ N )exp (- j2πpq/n) for p,q= 0,1,..N-1 and j= - 1. A cyclic prefix (CP) is inserted afterwards and its length L cp is chosen to be longer than the maximal length of the multipath fading channel(l). A N x 1 Matrix is defined with vector hm =( h m ( 0) h m ( 1) h m ( L-1) 0.0) T. where hm(l) represents the L th channel coefficient for the mth receive antenna,l =0,1..L 1. and m = 0,1 M-1. Collecting the channel vectors from the M different receive antennas an N x M channel matrix h = [ h 0 h 1.. h M-1.], and its frequency response as H = [ H 0, H 1.H M-1 ] = f H ( 1). H m = F h m..in an ordinary OFDM signal reception process after CP removal and DFT modulation, the resultant N x 1 signal vector from the m th receive antenna denoted by R m (K) Can be shown to be R m (K) = diag{ S(k) }H m + N m (k) (2) Where N m (k) is an N x 1 complex Gaussian noise vector with zero mean and equal variance for each element. For the considered scenario the M signal vectors are received and form into N x M received matrix as R (k) = [ R o (k) R 1 (k).r M-1 (k) ] (3). Let W = [ w 0 w 1..w M-1 ] T be an M x 1 weight vector With (1) ( 3) the pre DFT combining operation and the resultant N x 1 signal vector is expressed as Y(K) = R (k) W = diag { S(k) }. H W + N (k) W (4) with With N (k)=[ N o (k) N 1 (k).n M-1 (k]. In [1],W was calculated based on Max- Sum SNR.for that case,the optimum W can be shown to be shown to following augmentation problem. Max W H W H H W,subject to W H W=1 (5). In which W H W H H W indicates the sum of signal power in all the N sub carriers.as an alternative pre-dft based combining based on Max-Min SNR was proposed.[2].define a 1 x M vector γ n as the nth row of the channel matrix H given I in (1).With n=0,1.n-1.the optimization of W can be described as max min I γ n W I 2.,subject to W H W=1 (6) in which I γ n W I 2 indicates the signal power of the nth carrier after Pre-DFT combing..it is understood that Max- sum SNR tends to help the good while the Max- Min SNR helps the bad.. Both the criteria are reasonable for obtaining the Pre-DFT combining weights..nevertheless two questions naturally arise. 1) Is one of the two criteria strictly superior to the other? 2) can the error rate performance can be

3 improved with Pre-DFT combining?. It is observed that these queries are answerable with multi attribute augmentation. III. Fig 1. simo ofdm system Rx. MULTI ATTRIBUTE AUGMENTATION Although Max-Min SNR and Max- Sum SNR are both practical, they conflict with each other i.e. improvement in one leads to deterioration in the other. This motivates for the use of multi attribute augmentation for gaining further insight into the two criteria. Formally a multi attribute augmentation problem can be stated as follows -max g(w), subject to W H W=1 (7) With g 1 (W)=min I γ n W I 2 and g 2 (W)= (W H H H H w )/ N. in which g2(w) is normalized for calculation purposes.a trade-off is expected rather a single solution of either Max-sum SNR or max-min SNR. for this g1(w) and g2(w) are considered as pareto optimizers. This set of pareto Optimizers is called pareto front[3]. How ever there is no systematic way to find the pareto front in (7). For this weighted sum method is employed which gives single attribute augmentation. Mathematically speaking the attribute function in this scenario is changed as Max λ g1 (w) + (1- λ) g2 (W), subject to W H W =1 (8). Where λ Є [ 0,1] is a parameter determining the relative importance between Max-sum SNR and Max-Min SNR. Without the loss of generality the augmentation can be recast as Max λ [min trace( Г n W) + (1- λ)[trace( QW)], subject to trace (w)=1 rank(w)=1,w 0 (9).with Г n = γ H n γ n and Q= H H H. In (9) W is a M x M matrix is to determined and the inequality that W 0 means that W is a symmetric positive semidefinite. By dropping the non-convex rank one constraint the weigted sum augmentation problem can be relaxed to Max λ [min trace( Г n W) + (1- λ)[trace( QW)], subject to trace (w)=1, W 0. Let K 1 and K 2 be two scalars. The relaxation is equivalent to Max λ K 1 + (1 λ) K 2,subject to trace( Г n W) K 1 and trace( QW)] K 2.trace(W)= 1, W 0 (11) which becomes convex.. It s not difficult to see that (11) can be categorized as semidefinite programming problem. The optimum value Wopt is obtained using efficient interior point method and a randomization is used to produce an approximate solution to (7). IV. THE Pre-DFT COMBINING FOR MIMO-OFDM SYSTEMS A MIMO system The combining for MIMO- OFDM systems is carried out with a rayleigh

4 fading channel With 1/4 convolution code. the BER analysis is observed for various code rates and the Comparison between rician and rayleigh fading channel is carried out taking into account The error correcting codes used. comparison of BER for turbo, convolution codes,cpc and LDPC codes. The pre-dft combining is carried out for adaptive beamforming and corresponding Channel vector matrix [H] are estimated using decision directed channel estimation. Fig.2 fig.3 BER performance for Various error detecting codes. Channel matrix for MIMO- OFDM For an (N,M) system, the total number of signal paths is N X M 1 d dmax= NM.The higher my diversity gain, the lower is BEROFDM extends directly to MIMO channels with the IDFT/DFT and CP operations being performed at each of the transmit and receive antennas. MIMO-OFDM decouples the frequencyselective MIMO channel into a set of parallel MIMO channels with the input output relation for the ith (i = 0, 2,,L- 1) tone V SIMULATIONS AND DISCUSSION A comparison of bit- error rate ( BER) performance with different pre-dft combing is made by monte-carlo simulations carried out by using 1 x 2 coded OFDM system..quadrature Phase shift Keying(QPSK) is used for modulation.besides N=64,L cp =16 and L= 2 ( independently generated with rayleigh distribution ) are used. Convolution codes with different error correcting capabilities ( different minimum free distance d free ) are used. At the receiver a Viterbi algorithm with Hard decision and Viterbi algorithm with soft decision is compared with it.. for the case of higher error correcting capability. Max- sum SNR performs slightly better than Max-Min SNR. Note that Max- Sum SNR generally focuses on the good. With the relatively large amount of error protection the

5 low sub carrier SNR values may be compensated. Together with the boosted high SNR sub carriers Maxsum SNR provides better BER performance. On the contrary for the case of lower error correcting capability Max- Min SNR outperforms the Max- Sum SNR especially in the high SNR region.the small amount of error protection makes each sub carrier equally essential. Max-Min usually does a good job in the sub carrier SNR values and thus gives better BER performance. More over it is interesting to note that the weighted sum method which successfully captures the advantages of both Maz-Sum SNR and Max-Min SNR is superior to these two criteria. By varying λ there exists some cases in which a lower BER can be achieved. That is to say Multi attribute augmentation can be employed to some better pre-dft combining weights over the pure Max- Sum SNR and Max- Min SNR. By means of exhaustive simulations it is observed that the effect of Max-Min SNR is substantial than Max-Sum SNR in most circumstances. Thus as a rule of thumb λ should be set close to 1 for better BER performance. Also note that in [6] a similar trade was revealed for augmenting the uncoded BER under various channel conditions. Type of coded ofdm Convolution coded OFDM 16 QAM TCOFDM QPSK TCOFDM VI CONCLUSIONS Gain at 10-2 over Uncoded OFDM Gain at 10-3 over Uncoded OFDM 4.8 db 5.2dB 6.5 db 7.5 db 11.5 db 13 db This paper has discussed and compared the error rate performance for coded SIMO-OFDM systems and MIMO- OFDM systems with different pre-dft combining. The results show that Multi attribute augmentation can be used to determine some better pre-dft combining weights which are generally superior to both Max Sum SNR and Max-Min SNR for achieving a low BER.To this end finding a more exact relation between the weighted sum parameter λ and the BER is surely interesting. References: 1] M. Okada and S. Komaki, Pre-DFT combining space diversity assisted COFDM, IEEE Trans. Veh. Technol., vol. 50, pp , Mar ] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge Univ. 3] W. Stadler, Multicriteria Optimization in Engineering and in the Sci ences. New York: Plenum Press, 1988