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ML Estimation and Correction of Frequency Offset for MCCDMA Systems over Fading Channels Qingjiang TIAN1 and Khaled Ben Letaief,2 Senior Member, IEEE 1 Box 527, 1285 Electrical Engineering Building School of Electrical and Computer Engineering Purdue University W-Lafayette, IN, 47906-1285 USA Email: tianq@ecn.purdue.edu 2 Center for Wireless Information Technology The Department of Electrical and Electronic Engineering The Hong Kong University of Science and Technology Clear Water Bay, Kowloon HONG KONG Email: eekhaled@ee.ust.hk Abstract - Multi-Carrier Code Division Multiple Access (MC-CDMA) has been proposed as one of the air interface candidates for the third generation wireless communication systems. However, MC-CDMA systems suffer a lot of performance degradation due to carrier frequency offsets. In this paper, we derive the likelihood function for the carrier frequency offset for MC-CDMA systems for the downlink over Rayleigh flat fading channels. We also propose a gradient algorithm to estimate and minimize the carrier frequency offset in a tracking mode. Simulation results show that system performance can be improved significantly even with only a few iterations. systems. In this paper, we concentrate on the estimation and correction of the frequency offset for an MC-CDMA system for the downlink over Rayleigh flat fading channels in a tracking mode. In the remainder of this paper, it is shown that the proposed method can minimize the normalized 7 frequency offset to less than 10 . Hence, significant improvement in the system performance can be achieved The rest of this paper is organized as follows. We derive the Likelihood Function for the frequency offset in Section II. In Section III, we propose a gradient algorithm to correct the frequency offset. Simulation results and analysis are provided in Section IV. Finally, we draw our conclusion in Section V. I. Introduction Since the Multi-Carrier Modulation (MCM) technique was proposed in the sixties, significant research effort has been involved in its implementation and application. Development and research achievements in the field of digital signal processing and communication theory have resulted in communication systems such as MCCDMA and OFDM [1]. MC-CDMA, similar to other Multi-Carrier schemes, suffers a great deal of performance degradation from the frequency offset due to the large number of subcarriers [2][3]. For MC-CDMA systems, one adverse effect caused by frequency offsets is the reduction of the desired signal amplitude because the sinc(.) functions are shifted and are no longer sampled at the peaks. The other negative effect is the generation of ICI due to the loss of the orthogonality between different subcarriers. There has been a lot of work, which investigated the performance sensitivity to the frequency offset for MCCDMA systems [4][5]. But to the author’s best knowledge, there has been no contribution in regard to the estimation and correction of the frequency offset itself for MC-CDMA II. Likelihood Function Before we derive the likelihood function for the frequency offset, we present some results in regard to the performance degradation due to frequency offsets. Assuming BPSK modulation, the number of users is K and the processing gain is N. We also assume that the receiver estimates the fading amplitude and phase perfectly even though the frequency offset causes a phase rotation in each subcarrier. This means that we assume that the receiver is perfectly synchronized with the transmitter in our later discussion. Since we consider the downlink, we assume that all the users suffer a Rayleigh fading channel in the i-th subcarrier as follows hi (t) ρi exp (jθi ) (1) After coherent demodulation and the combination of signals from all subcarriers, the decision statistic for the first user’s data bit is given by 0 q0,i a k [n ]ck [l ]c0 [i ] k 0 i 0 l 0 K 1N 1N 1 1 Tb ( i 1)Tb iTb i cos(2 ( f l f i f ) l i )dt 10 0 1 sin() q0,i ak [i]ck [l ]c0 [i] i i l cos( l i ) k 0 i 0 l 0 N 1 sin() a0 [i ] i q0,i n 0 BER K 1N 1N 1 10 -1 10 -2 sin() N 1 K q0,i i ak [i]ck [i]c0 [i] i 0 k 1 sin() q0,i l a0 [l ]c0 [l ]c0 [i] i l cos( l i ) i 0 l 0,l i N 1 N 1 10 -3 K = 16,M R C 10 -4 K = 16,E G C K = 4,M R C K = 4,E G C K = 1,M R C K = 1,E G C sin() N 1 K 1 N 1 q0,i l ak [l ]ck [l ]c0 [i] i l cos( l i ) i 0 k 1l 0,l i 10 -5 -4 0 -3 5 -3 0 -2 5 -2 0 -1 5 -1 0 -5 0 N o rm a liz e d fre q u e n c y o ffs e t b y b it ra t e (d B ) Fig. 1: BER versus normalized frequency offset Now, we derive the likelihood function for the carrier frequency offset, which is the objective function we try to maximize based on the ML (Maximum Likelihood) estimation principle. It is worthwhile pointing out that the frequency offset in the tracking process is usually only a few percent of the transmitted bit rate (1/T) since the receiver performs the coarse frequency offset correction with training sequences before data transmission. Frequency offset during transmission mainly results from the fine difference of the oscillators in the transmitter and the receiver. It is assumed, therefore, that the normalized frequency offset (normalized by 1/T) is between –0.5 and 0.5 throughout this paper [7]. Based on some mathematical approximation, which is valid for low SNR conditions, the likelihood function for the carrier frequency offset of MC-CDMA systems is found as follows [7]: ' 2 4 N 1 2 { d m Re{qm }}2 N o m1 1 2 No C 2 No (2) Based on the Gaussian approximation of the interference terms, we may derive the system BER for the EGC case is as follows [6]: N EGC 1 (3) P ( e) erf c( ) b , EGC 2 DEGC where N EGC ( / 4)(sin( N ) / N ) 2 DEGC K sin( N 2 ( ) (2 ) N N 2 K N -1 N 1 sin( N ) )2 ( N 2 n 0 i 0 ,i n ( n i N ) N0 Eb As for MRC, Pb ,MRC (e) 1 erfc ( 2 N MRC ) D MRC (4) where N MRC (sin( N ) / N ) DMRC 2 E{d m 1 N m 1 N m }2 Re{qm }2 m 2K sin( N 2 ( ) N N K N -1 N 1 sin( N ) 2 ( )2 N n 0 i 0 ,i n ( n i N ) N 0 Eb Re{q }2 (6) where q m 0 r ( t ) e j( 2 fmtm ) e j( 2 f0t ) dt T * (5) Fig. 1 shows the system BERs of the MC-CDMA system versus different normalized frequency offsets. In this case, the Processing Gain N in the simulation is 64 and SNR=10dB. It shows that an MC-CDMA system is very sensitive to frequency offsets. It is also found that the system BER does not change much when the frequency offset is below some level. This means that if we can mitigate the frequency offset to less than some level, system performance can be improved significantly. 0 ( l d l e T l1 T N l1 N j ( 2 flt k1 ) e j( 2 f0t ) ) e j( 2 fmtm ) e j( 2 f0t ) dt * 0 l d l e j[ 2 ( fl fm f ) tl m ) dt (7) During the calculation and derivation, some multiplicative positive constants that show up in the expression of the likelihood function are dropped since they do not effect the maximizing argument. The definition of each term in the likelihood function and detailed derivation for the likelihood function are shown in the appendix. The data dependence of (6) is removed by averaging the likelihood function over all possible values of pm with form N T (Re{q m }) f 0* (10) dk p m 2 d l 0 t cos[2(f l f m f ) t l m ]dt l 1 N and all the fading parameters. For BPSK modulation, it is known that d l a k [i]ck [ n ] which k 0 K 2 d l l 1 1 {T cos[2(f l f m f )T l m ] 2(f l f m f ) has values of –K, -K+2… K-2, K. After some calculations, we obtain (8) E[' ] 1 N N 1 cos(2) 2 N 0 m1 l1 ( m l) 2 1 [sin( fT l m ) cos(l m )]} 2(f l f m f ) (11) From the above equation, we can derive an In Fig. 2, the normalized mean value of the likelihood function versus normalized frequency offset Δ is plotted. It can be seen that the likelihood function reaches an absolute maximal value at Δ=0 and possess the other local maximum points. 1 0.9 0.8 0.7 Average MLF iterative algorithm for the optimization of f 0* : (12) f 0*,i1 f 0*,i ki where k is a positive constant and εi k 2 1 Re{q N k2 } pk 2 (13) IV. Simulation Results and Analysis In this section, we will first investigate the converging characterisctics of the gradient algorithm with different system configurations. In Fig. 3 (a), we set the processing gain N=16, 0.6 0.5 0.4 0.3 0.2 0.1 -1 i Re{q m } p m the m 1 N number of users K=4, SNR=10dB, and k=0.0001, where k is normalized by the symbol duration T. In Fig. 3 (b), the simulation setup is the same as that in (a) except that k=0.001. In the two figures, the vertical axis shows the value of the square of the normalized frequency offset. -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 Normalized Frequency offset 0.6 0.8 1 0 Fig. 2: Normalized mean value of the likelihood function versus frequency offset Normalized Frequency Offset Square 10 III. ML Estimation and Correction of the Frequency Offset Because of the local convexity (with respect to the normalized frequency offset error) of the likelihood function between [-0.5 0.5], a gradient method can be applied to maximize the likelihood function, so as to minimize the frequency offset. Taking the derivative of (6) with respect to 10 -5 10 -10 ' 2C N (Re{q m }) 2 Re{q m } * f 0 N o m1 f 0* 2C N 2 Re{q m } p m N o m1 (9) where f 0* , we obtain: 10 -15 10 -20 0 50 100 150 200 250 300 350 The Number of iterations 400 450 500 (a) 10 0 Normalized Frequency Offset Square 10 -5 10 -10 10 -15 is N=128, SNR=10dB, initial normalized frequency offset=0.42, and with EGC. It should be noted that to calculate the system BER, the result from Fig. 3 (a) is used. After a fixed number of iterations, the normalized frequency offset is obtained from Fig. 3 (a), then the system BER is calculated with formula (3) and (4). This is reasonable since even with different system configurations, it is still possible to appropriately adjust the positive constant k. After the same number of iterations, the frequency offsets converge to the similar value in all the cases. It shows that after only 50 iterations (bits), the system BER can be improved around 10dB. 10 -20 V. Conclusion 10 -25 0 50 100 150 200 250 300 350 The Number of iterations 400 450 500 (b) Fig. 3: The frequency offset correction results versus number of iterations These results show that with different constants k, the algorithm converges to the same error floor; the normalized frequency offset can be minimized below but at different convergence speeds. Furthermore, after reaching the error floor, the different constants result in different error variances. The reason is that different constants k mean different step sizes of converging to the optimal value and this results in different convergence speeds. If the constant is too large, the algorithm will overadjust the carrier frequency around the correct carrier frequency. More iterations will make the carrier frequency oscillate around the correct carrier frequency. 10 0 10 7 , MC-CDMA system performance degrades significantly even with frequency offset only of a few percent of the data transmission rate. There will be significant system performance improvement even with a fine frequency-offset correction. The likelihood function for the frequency offset of the MC-CDMA system is derived in the case of low SNR conditions. Simulation results demonstrate that the function reaches a global maximal value when the frequency offset is zero. Based on the ML principle and the characteristics of the likelihood function, a gradient algorithm is proposed to minimize the frequency offset. Simulation results show that the developed algorithm showed can minimize the normalized frequency offset below 10-7, and that different step constants result in different convergence speeds and different variances of the final frequency offset error variance. We also provided the system BER improvement versus the number of iterations. It is showing that the system BER can be improved significantly after only a small number of iterations. Appendix In this appendix, we will derive the likelihood function for the frequency offset for a MC-CDMA system over fading channels. Assume BPSK modulation and i.i.d Rayleigh fading channel in each subcarrier, then the received signal is 10 -1 BER 10 -2 r ( t ) n d[k ]c n [k ]e j ( 2 f t ) e j ( 2 f t ) n ( t ) n n 0 K N k 1 n 1 N n d n e j( 2 f t ) e j( 2 f t ) n ( t ) n n 0 n 1 10 -3 10 -4 No iteration 20 iterations 50 iterations 100 interations 120 iterations 150 iterations 0 20 40 60 80 The number of users K 100 120 (A.1) where dn is the sum of the bits from all the users, which takes value of –K, -K+2….K-2, K. The Log likelihood function (LLF) is: Fig. 4: System performance improvement with different number of iterations In Fig. 4, we present the system BER versus the number of iterations. In this case, the system configuration 2 T * Re{r ( t )s ( t )}dt No 0 (A.2) where s(t) is the estimate of the r(t) with the format of s(t) ρ n d[k]c n [k]e j(2π f t ) e j(2π f t) n n * 0 K N k 1 n 1 N (A.3) The reason for that is that the estimation of data information in each subcarrier is independent of each other. The cross-correlation between them is 0. Taking the derivative of the ML function with respect to ρ n d* e j(2π f t ) e j(2π f t) n n n * 0 f 0* , we get n 1 since it is in a tracking mode, we assume that the receiver may estimate the fading amplitude and phase perfectly. Let ' 2C N (Re{q m }) 2 Re{q m } * f 0 N o m 1 f 0* 2C N Re{q m } p m 2 N o m 1 (A.10) f 0* denote the estimation of the carrrier frequency, and define Then, f f 0 f 0* (A.4) where 2 T * Re{r ( t )s ( t )}dt No 0 2 T * Re{r ( t )s ( t )}dt No 0 N 2 T j ( 2 f * Re{1 d m (r ( t ) e m No 0 m t m pm We finally get N T (Re{q m }) f 0* (A.11) p m 2 d l 0 t cos[2(f l f m f ) t l m ]dt l 1 N ) e j ( 2 f t ) )dt} * 0 2 d l l 1 2 N * T j ( 2 f d m Re( 0 r ( t ) e N o m 1 2 N * d m Re{q m } N o m 1 k 2 t k 2 ) e j ( 2 f t ) dt ) * 0 1 {T cos[2(f l f m f )T l m ] 2(f l f m f ) 1 [sin( fT l m ) cos( l m )]} 2(f l f m f ) (A.12) (A.5) where Reference [1] Baoguo Yang, K. B. Letaief, R. S. Cheng, and Z. Cao, “Channel Estimation for OFDM Transmission in Multipath Fading Channels based on Parametric Channel Modeling,” IEEE Trans. Commun., March 2001. [2] R. Prasad and S. Hara, “An Overview of Multi-Carrier CDMA,” IEEE Commun. Mag., Vol.35, No.12, pp. 126-33, Dec. 1997 [3] N. Yee, J. P. Linnartz, G. Fettweis, “Multi-Carrier CDMA in Indoor Wireless Radio networks,” In Proc. of PIMRC’93, Yokohama Japan, pp. 109-113. [4] Y. Lshida, “Recent Study on Candidate Radio Transmission Technology for IMT-2000,” The 1st Annual CDMA Euro. Cong., London, U.K., Oct. 1997. [5] K. W. Kang, J. Ann, and H. S. Lee, “Decision-directed maximum-likelihood estimation of OFDM frame synchronization offset,” Electronics Letters, Vol.30 No.25, pp. 2153-4, Dec. 1994. [6] J. Jang and K. B. Lee, “ Effect of Frequency Offset on MC/CDMA System Performance,” IEEE Commun. Letters, Vol. 3, No. 7, pp.196-198, July 1999. [7] Y. Kim, S. Choi, C. You and D. Hong, “Effect of Carrier Frequency Offset on the Performance of an MC-CDMA System and its Countermeasure Using Pulse Shaping,” In Proc. ICC’99, Vol.1, pp. 167-171, Vancouver, Canada, 1999. [8] L. Tomba and W. A. Krzymien, “ Sensitivity of the MC-CDMA Access Scheme to Carrier Phase Noise and Frequency Offset,” IEEE Trans. Veh. Tech. Vol. 48, No.5, pp. 1657-1665, Sept. 1999. q m 0 r ( t ) e T T N l 1 T j ( 2 f m t m ) e * j ( 2 f 0 t ) dt m t m 0 ( l d l e j( 2 f t ) e j( 2 f t ) ) e j( 2 f l l 0 ) e j( 2 f t ) dt * 0 0 l d l e j[ 2 ( f f l N m f ) t l m ) l 1 dt (A.6) Hence, the likelihood function (A.7) In the case of low Signal-to-Noise Ratio (SNR), we may expand the exponential into a power series, By ignoring higher series terms more than 2, we get 2 (A.8) ' 1 2 4 Since dropping down some positive constants which is independent of ' ' e f 0* doesn’t effect our result, we get the likelihood function as 2 4 1 N 2 { d m Re{qm }}2 N o m1 1 N E{d m }2 Re{qm }2 2 N o m1 C N Re{qk }2 2 N o k 1 2 2 (A.9)