Iterative Carrier Frequency Offset Estimation for OFDMA Uplink based on Null Subcarriers Di Niu and Xianhua Dai Department of Electronics and Communication Engineering, Sun Yat-sen University, Guangzhou, 510275 P. R. China E-mail:
[email protected],
[email protected]
Abstract—In the uplink of an orthogonal frequency division multiple access (OFDMA) system, because of multiple access interference (MAI), carrier frequency offset (CFO) estimation is much more difficult than that in single-user systems. This paper proposes a null subcarrier-based iterative CFO estimation method for the uplink of OFDMA system adopting sub-band based carrier assignment scheme. Rather than applying independent estimations for users separated by bandpass filters as in [5], we first obtain an inaccurate initial CFO estimate for each user. Then, we run an iterative scheme which could update the estimates and reduce errors iteration by iteration. The performance of the proposed method is evaluated by simulations. Keywords-OFDM systems; multiple access; frequency offset estimation; null subcarriers.
I. INTRODUCTION Frequency synchronization is important for orthogonal frequency division multiple access (OFDMA) systems. Currently, many methods [1], [2], [3] are available for singleuser systems or OFDMA downlink. But the problem is much more complicated in OFDMA uplink. This is because in the uplink of OFDMA, due to the effect of Doppler shift, each user has a different carrier frequency offset (CFO), which introduces multiple access interference (MAI) [4]. Even in the absence of noise, MAI alone makes CFO estimation algorithms devised for single-user systems unsuitable for multiuser systems. As far as the broadcast channel of OFDMA is concerned, the null-subcarrier-based algorithm [5] by S. Barbarossa, M. Pompili and G. B. Giannakis exhibits good performance. Their basic idea is that the energy falling across null subcarriers will reach its minimum when the CFO is compensated completely at the receiver. When this method is extended to OFDMA uplink, a sub-band based carrier assignment scheme is assumed in [5], in which signals from different users occupy non-overlapping frequency bands in a similar fashion as traditional FDMA. Guard intervals are put in the edge of each user’s sub-band so that signals of different users can be separated by filter banks and an independent CFO estimation algorithm can be applied to each user. However, the filtering operation cannot be perfect because of the MAI, and thus a serious performance loss is inevitable
with respect to the downlink case or single user system. The need for long guard intervals also costs the transmission rate. In this paper, we propose an iterative CFO estimation method for OFDMA uplink which solves the problem caused by MAI. Just like in [5], we also assume a sub-band based carrier assignment scheme and exploit the presence of null subcarriers for CFO estimation. However, we never isolate users with filter banks. Instead, we first allow inaccuracy in the initial CFO estimate for each user. Then, we run an iterative scheme which could suppress MAI among users and reduce estimation errors iteration by iteration. By using this iterative scheme, we improve the performance of the nullsubcarrier-based CFO estimation algorithm in multiuser orthogonal frequency division multiplexing (OFDM) systems greatly. The rest of the paper is organized as follows. The signal model of OFDMA uplink is given in Section II. Next, in Section III, we review the null-subcarrier-based CFO estimation proposed in [5], followed by the discussion about its limitation in OFDMA uplink. In Section IV, we propose our iterative estimation scheme which solves the problem caused by MAI and, finally, in Section V, we report simulation results to analyze the performance of our method.
II. SIGNAL MODEL FOR THE UPLINK OF OFDMA We suppose the OFDMA system in concern has M user terminals (UT) and one base station (BS). Each transmission block consists of N+L samples, where the first L samples constitute the cyclic prefix. Among the N subcarriers, some are null or virtual subcarriers, while others carry data symbols for the users. Supposing the duration of one OFDM block is T, the frequency spacing f between adjacent subcarriers is 1/T. For analysis simplicity, we assign the same number of subcarriers, J 1 subcarriers to each user. We adopt the subband based carrier assignment scheme, i.e., if we denote with um(p;l) the lth symbol transmitted by the mth user within the pth OFDM block, the frequency index associated to it is mJa+l+ip, where Ja ＝J+J0 and J0 is the length of the guard interval composed of J0 null subcarriers put between adjacent sub-bands, i p p, ( p 0,1,..., N 1) is a frequency hopping
This work was supported by National Science Foundation of China (NSFC) under grant number 60272068.
index. As has been demonstrated by [5], frequency hopping is instrumental to resisting deep fading. We use quasi-synchronous systems for discussion. In such systems, the UTs attempt to synchronize with a pilot signal sent by the BS, before initiating the communication link. Consequently, the relative time offsets among users are limited to a few chips (fraction of a symbol), and can, thus, be incorporated as part of the unknown channel impulse response. In such cases, the time offset is compensated as part of the equalization performed at the receiver. This approach entails a small efficiency loss, because it requires guard intervals longer than the channel delay spread (to account for the relative delays among users). However, it simplifies the analysis by allowing us to fully concentrate on the frequency offset estimation. Now we will give the expression of the sequence received at the BS from all the M user. We use similar expressions as [5] does. Please refer to [5] for details. The pth block sent by the mth UT has entries J 1
xm ( p; n ) u m ( p ; l ) e
j
2 ( l mJ a i p ) n N
, n = -L,…,N-1.
(1)
l 0
Denoting with cm (t ) the continuous-time baseband equivalent channel between the mth UT and the BS, the transfer function of the mth channel is L
Cm (l ) cm (r )e
2 j rl N
, l 0,1,..., N 1 .
We denote the CFO between the BS and the mth UT with f m and the symbol transmission rate with 1/ Ts N / T . Then we introduce the dimensionless CFO of user m, vm f mTs , m [0, M 1] . The qth block received at the BS from the mth UT can be expressed as J 1
j
2 ( l mJ a iq ) i N
l 0
where
vm (q; i ) ,
i = －L,…,N-1 (3) the symbols
um ( q; l ) um (q; l )Cm (l mJ a iq ) are
filtered by the channel, vm (q; i ) is additive noise. Superimposing all the ym (q; i ) from the M users and removing the cyclic prefix, we arrive at the following baseband discretetime equivalent signal received at the BS M 1
J 1
m 0
l 0
y (q; i ) e j 2 vm ( q ( N L ) i ) um (q; l )e
j
2 ( l mJ a iq ) i N
or 1, and use Fig.1 to explain how the method in [5] works and to show its limitation in multi-user OFDM systems. Estimator m is used to generate vm . First, we assume that during transmissions only user m carries data symbols, while the other user merely carries null symbols. So we will find the estimator in Fig.1 becomes exactly the estimator for singleuser systems in [5]. We first compensate each inputting block j 2 v ( n )( q ( N L ) i ) by multiplying it by e generated by numerically controlled oscillator (NCO), where v(n) is the guess of the mth user’s CFO in the nth search. Then we do the FFT transform on the resulting sequence and obtain data symbols in frequency domain Y (q; k ) . We let v(n) explore the range [-1/2N,1/2N] with certain step as n increases and calculate the average energy falling in the virtual subcarriers beside the mth user’s sub-band, denoted by J m , until we find a v(i ) which minimizes J m . Then we take v(i ) as the value of
vm . In the nth search,
J m (n)
(2)
r 0
ym (q; i ) e j 2 vm ( q ( N L ) i ) um (q; l )e
first. And in the end of Section IV, we will extend our method to systems with more than two users. We denote the estimate of the mth user’s CFO by vm , m=0
1 Nb
mJ a iq
N b 1
| Y ( q; k ) |2
is the average taken over Nb consecutive blocks. As spectra assignment hops from block to block, the concerned null subcarriers occupy different frequency indexes in different blocks, and thus the effect of deep fading could be mitigated [5]. In case of no noise, perfect synchronization leads to a null energy in the band occupied by null carriers. And CFO estimation is performed based on this conclusion. But if both users carry data symbols, the system has two real users now. Because each user has a different CFO, even if user m’s CFO is compensated completely, i.e., we multiply j 2 vm ( q ( N L ) i ) y(q;i) by e so that no energy of user m falls into its neighboring null subcarriers, we still can not observe a y(q;i) Y ( q; k ) S/P
e
FFT
j 2 v ( n )( q ( N L ) i )
NCO
Yes
vm v( n)
v(q; i ) ,
i = 0,…,N-1 (4) In OFDMA uplink, we should estimate each vm ( m [0, M 1] ) which varies among different users.
III. NULL-SUBCARRIER-BASED CFO ESTIMATION We start with reviewing the null-subcarrier-based CFO estimation method in [5], followed by the discussion about its limitation in OFDMA uplink due to the MAI. Then, we will propose an iterative scheme in the next section to tackle this problem. For simplicity, we use a two-user system for analysis
(5)
q 0 k ( m -1) J a J 1 iq
n=n+1
No
Jm (n)reaches min?
Compute
J m ( n)
Fig.1. Block diagram of Estimator m
null energy in user m’s neighboring null subcarriers, because interference from the other user still falls into this band. In other words, perfect synchronization of user m does not necessarily coincide with a null energy in user m’s neighboring null subcarriers. Therefore, we cannot get an enough accurate vm using Estimator m alone. If we separate users with bandpass filters and applying independent estimations to them as [5] does, the errors still remain high.
Because, although guard intervals are put between neighboring users’ sub-bands, interferences from all users still fall into the guard intervals and make the filtering imperfect.
IV. ITERATIVE CFO ESTIMATION FOR OFDMA UPLINK
e
NCO
y(q; i )
e
FFT
j 2 vr ( q ( N L ) i )
Y ( q; k )
modulus of the signals
S/P
y(q;i)
beside user r’s sub-band, even though a very little residue of it still remains since the guess cannot be totally perfect. After setting the CFOs back to their original values in the third step, we now obtain a sequence y(q; i ) which is very close to the signal of user m. And the problem becomes the CFO estimation for single-user systems. Because y(q; i ) is so close to the signal of user m that we will obtain a very accurate vm .
Suppressor
j 2 vr ( q ( N L ) i ) P/S
Y ( q; k ) IFFT
Fig.2. Block diagram of Eliminator r
contribution from user r is zero, and thus user r’s energy is only concentrated on its own sub-band. Therefore, to eliminate user r, we only need to eliminate user r’s data symbols in its own sub-band. However, in this sub-band there is also interference contribution from the other user, namely user m. But when the two users’ sub-bands are separated far enough (the threshold of this separation is analyzed in the simulations in Section V), within the sub-band of user r, the interference amplitude from user m( m r ) will remain low with respect to the amplitude of user r’s data symbol and will vary slightly among the subcarriers. So the interference from user m in user r’s sub-band is approximately the same as the interference from user m in the null subcarriers beside user r’s sub-band. Specifically, we use a linear approximation in (6) to substitute the sum signal within user r’s sub-band with the guess of interference from user m in this band. By doing this we suppress the signal of user r to the interference level
0
0
10
20 30 frequency index
40
50
4 2 0
0
10
20 30 frequency index
40
50
Fig. b. Frequency-domain symbols after the first step in Eliminator r if
vr vr . modulus of the signals
6 4 2 0
0
10
20 30 frequency index
40
50
Fig. c. Frequency-domain symbols after the second step in Eliminator r. Here the sum signal is Y ( q; k ) . 6
modulus of the signals
j 2 v ( q ( N L ) i )
r resulting sequence by e to set the CFOs back to their original values. Because vr is the estimate of vr , vr is close to vr . So, after the first step, user r will have a very small frequency offset with most of its energy concentrated on its own subband, which is the spectra pertaining to its own data symbols. Assuming that vr vr , after the first step, the interference
2
6
e j 2 vr ( q ( N L ) i ) and then obtain frequency-domain data symbols Y ( q; k ) . Second, in the qth block, we set Y ( q; k ) Y ( q; rJ a 1) ( k ( rJ a 1)) Y ( q; rJ a J ) Y ( q; rJ a 1) , J 1 k rJ a , rJ a 1,..., rJ a J 1 otherwise. (6) Y (q; k ) , At last, we do IFFT transform on Y ( q; k ) , and multiply the
4
The sum signal 6 The signal of user m O The signal of user r X Data symbols and null symbols in case of perfect synchronization Fig. a. Frequency-domain symbols input into Eliminator r modulus of the signals
In order to suppress MAI, noticing y(q;i) is the sum signal of the signals of user 1 and user 2, we try to eliminate user r’s signal with Eliminator r (r=0 or 1 and r m ), shown in Fig.2, while estimating user m’s CFO. There are three steps in Eliminator r. First, we compensate y(q;i) by multiplying it by
6
4 2 0
0
10
20 30 frequency index
40
50
Fig. d. Frequency-domain symbols when user m’s CFO is compensated completely in Y ( q; k ) .
Nonetheless, in practice, vr is also a parameter that we need to estimate and we cannot know its value at first, i.e.,
vr vr . Therefore, after the first step, user r’s energy will still spread on null subcarriers and thus equation (6) becomes less effective to suppress user r’s signal. But in a system that is not varying the average symbol power across different users, there is always more energy on user r’s sub-band than on the guard intervals, where the only contributions of energy are the interferences from user data symbols and noise. Therefore, by using (6) to suppress the signals within user r’s sub-band to the interference level beside it, we still reduce a part of user r’s energy. This process is shown in Fig. a, b and c. Thus, when we input y(q; i ) into Estimator m and compensate it by j 2 v ( n )( q ( N L ) i )
multiplying it with e in the first step of estimation, we will have less interference from user r falling on the null subcarriers beside user m’s sub-band, which is the band we use to calculate J m . This is shown in Fig. d. Therefore, we still get a more accurate vm than we estimate
vm without suppressing user r at all. We have three important remarks here. Remark 1: Even when vr vr , Eliminator r still helps to yield a more accurate
vm than we do the estimation without suppressing user r at all. Remark 2: the closer the value of vr gets to vr , the lower the amount of user r’s energy will leak into the null subcarriers after the first step in Eliminator r and thus the more effectively equation (6) will suppress the signal of user r. Remark 3: The more effectively Eliminator r suppresses the signal of user r, the more accurately we can estimate vm . y(q;i) Eliminator 1
Estimator 0
y(q;i) Eliminator 0
Estimator 1
v0
v1
Fig.3. Block diagram of the proposed OFDMA receiver
Based on these three remarks, it is possible for us to design an iterative method to improve the accuracy of estimation iteration by iteration. First, we input y(q;i) directly into Estimator m and take the output as the initial value of vm , though it is not accurate. The initial value of vr is obtained using the same method. Then, we run the iterative scheme in Fig.3 to update the estimates. To obtain a new vm , we simply let y(q;i) go through Eliminator r to suppress user r and input the resulting sequence into Estimator m so that the new value of vm is more accurate than the initial one, according to Remark 1. Next, we feed back this more accurate vm into Eliminator m so that it can suppress user m more effectively, based on Remark 2. According to Remark 3, this helps to yield a more accurate vr , which is also fed back into Eliminator r that in turn helps to obtain an even more accurate
vm . We run such iterations until both vm and vr become stable. Because the linear approximation in (6) does not give a totally perfect guess of the interference from user m in user r’s sub-band, even if vr vr , Eliminator r cannot eliminate the signal of user r completely from y(q;i). Therefore, in case of no noise, instead of converging to vm , vm actually converges to some value very near vm . Despite of this approximation, simulation results show that this method achieves very good performance on fading multipath channels with white Gaussian noise, provided enough separation between neighboring users’ sub-bands exists. This is because, when the estimates become more accurate as the iterations go, the error of the guess becomes neglectable with respect to the noise level and can be incorporated as a part of the noise. Moreover, the errors of the guess are mitigated as we use Nb consecutive OFDM blocks instead of one block for the estimation. In addition, simulation results show that the threshold of the length of the guard interval is actually very low. In systems with more than two users, in order to update vm , we use Eliminator m to suppress user m one after another, where m explores the range [0, M 1] and m m , until all users except for user m have been suppressed. By doing this, we will get a more accurate vm through Estimator m . With the new vm , we can eliminate the signal of user m more effectively, which is instrumental to estimating other users’ CFOs. In simulation, we have found that after two iteration cycles, the estimates will change little. Therefore, the computational complexity is fairly low.
V. SIMULATION RESULTS AND CONCLUSIONS To evaluate the performance our method, we first compare our method with the method in [5] which uses filter banks to separate users and apply independent estimation to each user. We have simulated an OFDMA uplink channel with the following parameters: M=3, N=96, Ja=32, J=24, J0=8, SNR=10dB, channel order L=15. The system employs 16QAM. The CFOs of the three users are generated as independent random variables, distributed uniformly in (1/2T,1/2T). If we denote the normalized CFO of user m with f m f m / f , apparently f m is distributed uniformly in (0.5,0.5). And we define the normalized estimation error of user m as em fm f m / f , where fm is the estimate of f m . We have simulated 50 independent transmissions, in each of which we calculate estimation errors em (m=0,1,2) of both our method and the method in [5]. Then we give the estimation error of one specific user, which is randomly chosen, in every transmission in Fig. 4. We can see our method has improved the performance greatly. Then we give the relationship between SNR and the average estimation error. The system parameters are the same as those in the previous one. For each SNR, we have done 40
independent experiments and calculated the average estimation error which equals to E{1/ M M 1 em } . The result m0 is shown in Fig. 5.
Estimation Error
6
x 10
(a) The New Method
-3
4
REFERENCES
2 0 0
10
20
30
40
50
40
50
Experiment index (b) The Filter Bank Method
Estimation Error
0.8 0.6 0.4 0.2 0 0
10
20
30
Experiment index
Fig.4. The normalized estimation errors of one specific user in 50 independent experiments x 10
Average Estimation Error
1.5
-3
Estimation Error-SNR Relation
1.4 1.3 1.2 1.1 1 0.9
5
10
15 20 SNR(dB)
25
30
Fig.5. The relationship of the estimation error and the SNR Estimation Error-Guard Interval Relation 0.025 Estimation Error
performance and the transmission efficiency, we would prefer to set the length of the guard interval between 6 and 8. In conclusion, the CFO estimation algorithm proposed in this paper successfully circumvents the difficulty caused by MAI and exhibits reliable performance in the uplink of OFDMA systems. Interesting further developments include a more careful choice of initial estimate generation algorithm and the design of a more effective suppressor in the eliminator.
0.02
0.015 0.01
0.005 0
2
4 6 8 10 The Length of the Guard Interval
12
Fig.6. The relationship of the estimation error and the guard interval
Lastly, we show the relationship between the estimation error and the length of the guard interval put between neighboring users’ sub-bands. We set SNR=10dB, N=96, M=3, Ja=32. Other system parameters are the same as those in the previous simulation. We calculate the average estimation error for each J0 which explores the range {2, 4, 6, 8, 10, 12}. The result is shown in Fig. 6. Considering both the system
[1] F.Classen and H. Meyr,“Frequency synchronization algorithms for OFDM systems suitable for communication over frequency-selective fading channel,” in Proc. 44th Vehicular Technology Conf., 1994, pp. 1655-1659. [2] H.Roh, K.Cheun, and J.Park,“An MMSE fine carrier frequency synchronization algorithm for OFDM systems,”IEEE Trans. Consumer Electron., vol.43, pp.761-766, Aug.1997 [3] T.M.Schmidl and D.C.Cox,“Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., vol.45, pp.16131621, Dec. 1997. [4] A.M.Tonello, N.Laurenti, and S.Pupolin,“Analysis of the uplink of an asynchronous multi-user DMT OFDMA system impaired by time offsets, frequency offsets, and multi-path fading.” in VTC Conference Record, October 2000 Fall, vol.3, pp. 1094-1099. [5] S.Barbarossa, M.Pompili and G.B.Giannakis, “Channel-independent Synchronization of Orthogonal Frequency Division Multiple Access Systems,” IEEE Journal on Selected Areas in Communications, vol.20, no.2, February 2002.
