pulse making the requirements for timing recovery easier. Although OFDM is a ..... 2274-2278. [2] Y. S. Lim, J.H. Lee, âAn Efficient Carrier Frequency Offset.
SIMPLE COARSE FREQUENCY OFFSET ESTIMATION SCHEMES FOR OFDM BURST TRANSMISSION Zbigniew Długaszewski, Krzysztof Wesołowski Institute of Electronics and Telecommunications, Poznań University of Technology, ul. Piotrowo 3A, 60-965, Poznań, Poland, {zdlugasz, wesolows}@et.put.poznan.pl
Abstract - Simple methods of the estimation of that part of the frequency offset, which is a multiple of the subcarrier spacing in the OFDM (Orthogonal Frequency Division Multiplexing) systems have been proposed. Described methods are aimed at burst transmission systems and take use of only one preamble symbol also required for the timing offset estimation. The methods are based on the frequency domain search for the unused subcarriers in the received signal spectrum. The proposed algorithms are immune to a constant phase shift (e.g. caused by timing errors). The estimation range depends on the algorithm but in general can be made as large as desired (but less than a half of the signal bandwidth). Simulation results show that the proposed algorithms estimate an integer part of the frequency offset in the AWGN channel even in the presence of some amount of the InterCarrier Interference (ICI). Satisfactory performance has been obtained also for OFDM transmission on multipath channels.
and divided into two parts: the integer and the fractional ones. Several methods of the frequency offset estimation in OFDM systems can be found in the literature [1-8]. Some of these methods are able to estimate the fractional or integer part of the frequency offset and some others estimate both of them. Generally, the algorithms operating in the frequency domain are able to estimate integer part of the frequency offset only and require the fractional part to be removed prior to the FFT.
Keywords - OFDM, frequency offset estimation, synchronization.
Most of the frequency offset estimation algorithms presented in the literature are based on the Maximum Likelihood (ML) criterion. Many algorithms are performed by the search of the maximum of the correlation between received samples, pilot tones or some functions of the received pilot tones. One has to stress that these algorithms correlate complex values and they are much more computationally demanding than the methods proposed below. The methods described in this paper rely on the search for the unused subcarriers in the received signal spectrum.
I. INTRODUCTION
II. RECEIVER ARCHITECTURE
OFDM is an efficient method of fast data transmission over dispersive channels. Due to the division of the data stream into many substreams transmitted at a low data rate on different subcarriers and thanks to the application of a guard period preceding the data pulse, intersymbol interference can be avoided. In most of the OFDM systems the guard period is filled with the samples from the end of the data pulse making the requirements for timing recovery easier.
The receiver architecture and the distinction between time and frequency domain processing have been shown in Figure 1.
Although OFDM is a spectrally efficient method of digital modulation, it has some serious drawbacks, such as sensitivity to the frequency offset. The frequency offset is unavoidable in burst transmission. Its main sources are the Doppler shift and differences between frequencies of the oscillators at the transmitter and receiver. Such offset destroys the orthogonality between OFDM subcarriers and introduces ICI at the output of the OFDM demodulator. When the frequency offset is larger than the subcarrier spacing ∆f, a circular shift of the output samples at the FFT output is additionally observed. The total frequency offset can be normalized to the subcarrier spacing
0-7803-7589-0/02/$17.00 ©2002 IEEE
Input samples
S/P
FFT
coarse freq. estimation
fine freq. estimation offset < ±∆f/2
time domain
frequency domain
Fig. 1. Frequency offset estimation – receiver architecture The fine frequency offset ML estimation algorithm operates in the time domain. Such an algorithm is able to estimate a fractional part of the frequency offset (smaller than a half of the subcarrier spacing ∆f). After this estimation, the fractional part of the frequency offset is removed from the input signal and in this manner the FFT output is almost free of the ICI. The algorithms described in this paper are placed after the FFT and are denoted as coarse frequency estima-
PIMRC 2002
It will be shown, that the performance of these algorithms drops when the frequency offset walks away from the integer multiple of ∆f. This is why another algorithm estimating and correcting a fractional part of the normalized frequency offset should be applied at the beginning of the synchronization process. Simulation results presented in this paper have been obtained when no cyclic prefix is present in front of the preamble symbol, as it is often the case for the OFDM symbol preambles. The impact of the cyclic prefix on the algorithm performance has been analyzed in the last section. III. PREAMBLE SYMBOL All algorithms described in this paper use one OFDM symbol, which is the first part of the synchronization preamble placed at the beginning of the signal burst. This symbol consists of two identical short preamble symbols, which are used by the timing offset estimation algorithm. Two consecutive short preamble symbols create one full OFDM symbol, in which only even subcarriers are used – see figure below.
IV. SYNCHRONIZATION ALGORITHMS AND DISCUSSION ON THEIR PERFORMANCE The general scheme of the integer frequency offset estimation has been shown below. We assume that the fractional part of the frequency offset has been already removed in front of the FFT. After OFDM demodulation we calculate the power received at each subcarrier and the metrics M(n), where n ranges within the interval of subcarrier indices determining the possible position of the DC subcarrier in the received OFDM signal. Minimum of this metric indicates the estimated frequency offset or, equivalently, the location of the null in the middle of the received signal spectrum. One can notice that all proposed algorithms are immune to a constant phase shift (e.g. caused by timing errors) since they apply the received subcarrier power only.
Input data
FFT
Metrics Calculation
|x|2
One can notice in Figure 2 that three consecutive subcarriers in the middle of the spectrum are switched off, since only even subcarriers are transmitted. Thus, we define our metric as a sum of powers of three consecutive subcarriers: +1
∑x
i = −1
-6
-4
-2
0
2
4
6
8
Fig. 2. Preamble symbol – frequency domain representation Generally, some subcarriers have to be switched off at the edges of the signal spectrum. This is caused by the requirements imposed by signal filtering and is aimed at decreasing interference introduced into neighboring channels. To obtain two identical short preamble symbols the DC subcarrier has to be switched off as well. The first algorithm presented in the next section searches for the DC subcarrier in the received signal spectrum. Similar idea and architecture was presented in [1]. However, in [1] a bank of filters designed for the Doppler shift that is expected in the channel has been reported. This causes that the correct estimate can be obtained after a relatively long time period. Additionally, oversampling was applied in the algorithm presented in [1] and a channel without selective fading was assumed in the simulations. Simulation results obtained for the WIND-FLEX channel models [9] clearly show that the last assumption does not hold in the WIND-FLEX modem SOHO environment.
offset
Fig. 3. Coarse frequency offset estimation
M (n) =
-8
minimum
tion. These algorithms allow for estimation of the integer part of the frequency offset. Since in many practical systems the FFT introduces significant delay, the estimates can not be used in the same frame.
2
(1)
n +i
where i denotes the subcarrier index. However, it can occur that due to the multipath propagation, the presence of additive noise and ICI caused by the uncompensated fractional part of the frequency offset, an incorrect frequency offset estimate can be determined. To improve the accuracy of our scheme we can switch off some additional subcarriers and broaden our spectral null around the DC. Let us define the parameter ex as a number of subcarriers, which are switched off on a single side of the DC subcarrier. The preamble symbol for ex equal 2 (Figure 4) and the corresponding metric have been shown below. 1+ 2*ex
∑
M ex (n ) =
x n +i
2
(2)
i = −1− 2*ex
-8
-6
-4
-2
0
2
4
6
8
Fig. 4. Preamble symbol – frequency domain representation for ex = 2 Selected simulation results have been presented in the next section. As we will see, thanks to increasing the width of the spectral null around DC subcarrier we improve the algorithm performance. There is almost no improvement beyond
ex equal 2 for high SNR values. However, for low SNR value such an improvement is still visible. This phenomenon is caused by averaging of the larger amount of noise while increasing the summing window width. The results obtained for the AWGN channel show 100% correctness of the estimates for high and medium SNR, even in the presence of small ICI. However, some incorrect estimates occur even for high SNR for the multipath channels and systems without ICI. Another important conclusion, which can be drawn from these observations is the following. It is clear that selective fading can distort the received signal and it can prevent correct frequency offset estimation. The impact of the selective fading on the performance of the algorithm depends on the SNR value. The lower SNR (higher level of noise) the shallower fades can distort the estimate. In practical systems we can not exclude too many subcarriers. That is why we propose the first modification of our algorithm. To improve the algorithm performance we have to increase the summing range to average larger amount of noise. Since we know that odd subcarriers are not transmitted we can add the noise power received at these subcarriers. Our new metric can be written as follows:
M R ,ex ( n ) =
2*ex
∑x
i = 2*ex ; i even
2 n +i
+
1+ 2* R
∑
xn+ j
2
,
(3)
j = −1− 2* R ; j odd
where R is the number of the summed odd subcarriers on one side of the DC subcarrier. Simulations have been performed for different channel models and different levels of ICI. The results partially confirm our presumptions. The performance improvement is negligible for small R and high SNR values (e.g. 16.5 dB) and increases for higher values of R. For the low SNR (e.g. 6.5 dB) and ex=2 the probability of the correct estimate increases by 0.5%, 1% or even by 2% depending on the value of parameter R and the SNR. This impact can be observed also for higher SNRs and for small uncompensated fractional part of the frequency offset. One can compare the results obtained for different values of the uncompensated fractional part of the frequency offset. This offset causes ICI and attenuates subcarrier amplitudes on the FFT output. In the systems with many subcarriers ICI can be treated as an AWGN and that is why the results obtained at the presence of low fractional part of the frequency offset do not differ much among themselves. However for higher SNR high values of an uncompensated fractional part of the frequency offset can cause the performance degradation in comparison with the previous algorithm. This is caused by the increase of the ICI (when the ICI distortions become significant in the comparison with the thermal noise). Since now we have calculated our metrics using the power of unused subcarriers only. Why shouldn’t we apply the
metrics using subcarriers with transmitted tones? Different subcarriers suffer from different fading on the multipath channels. The performed analysis shows that taking into account the transmitted subcarriers as well decreases the probability of correct estimates for small values of R. One could also imagine the use of all received subcarriers: the used and unused ones. Such a procedure would rely on the correlation between the received symbol power spectrum and some mask with +1 for unused subcarriers and –1 for the used subcarriers. Although we calculate the correlation, in fact we have to sum and subtract all consecutive subcarrier powers. In this case the metric can be written as follows:
M F ,ex ( n) =
N / 2 −1
∑
i = − N / 2; i unused
x n +i
2
N / 2 −1
−
∑
j = − N / 2; j used
xn+ j
2
,
(4)
where N denotes the FFT size. The maximum expected frequency offset in a real system is limited and known. This allows for significant simplification of the calculation procedure. One can notice that in fact all masks (and thus correlations) for even and separately for odd offsets n differ between themselves at a few positions only (two masks differ exactly on 2*ex+1 positions). The common element MF represents the sum of powers of all odd subcarriers minus the sum of all even subcarriers. It is described by the expression:
MF =
N / 2 −1
∑
xi
2
N / 2 −1
−
i = − N / 2; i odd
∑
xj
2
,
(5)
j = − N / 2; j even
so, the applied metric can be written as follows: 2*ex 2 M 2 * x n +i ; + F i = −2*ex ; i even M F ,ex ( n ) = 2*ex 2 − M + 2 * x n +i ; F i = −2*ex ; i even
∑
∑
n even (6)
n odd
One can expect that the results obtained with this method are more reliable in comparison to the previous ones. Simulation results confirm that. They prove that for the low SNR (6.5 dB) and ex equal 2 the probability of correct estimates increases by 1.5% in comparison with the first proposed method. One can notice that the method is more reliable also for high SNR, however, some residual errors caused by deep selective fades remain. The third method has another important feature. We have noticed that since even and odd metrics use common element MF with the opposite sign, using the metric MF,ex we are only able to make an error, which is an even multiple of the subcarrier spacing.
V. SIMULATION RESULTS Most of the presented simulation results have been obtained for the NLOS (Non Line-of-Sight) channel model. In the LOS (Line-of-Sight) channel model and in the AWGN channel model the performance of the studied algorithms is higher than in the NLOS channel model. For the AWGN channel 100% of estimates are correct for high SNR values because many erroneous estimates are caused by selective fading. Similarly for the LOS channel the better performance is caused by a lower number of selective fades influencing the received signal power spectrum. Both the LOS and the NLOS channel models have been created within the WIND-FLEX project for indoor radio channels in the 17 GHz band and have been described in [9]. Because the coarse frequency offset estimation using the proposed algorithms is a single-shot process we measure the number of successful attempts of the synchronization algorithm. Therefore, the simulation results are shown in the form of tables. All algorithms have been simulated for 100 000 different channel realizations All tables presented below show the percentage of correct estimates for different SNR values and different uncompensated fractional parts of the frequency offset f. A fractional part of the frequency offset is expressed as a percentage of the subcarrier spacing. Table 1 and Table 2 show the results obtained for the first algorithm presented above, which was determined by the metric Mex. Table 1. The basic algorithm, ex=2, AWGN channel SNR [dB] 1.5 dB 6.5 dB 11.5 dB 16.5 dB
f = 10% 98.4% 100.0% 100.0% 100.0%
f = 5% 98.9% 100.0% 100.0% 100.0%
f = 0% 99.0% 100.0% 100.0% 100.0%
Table 2. The basic algorithm, ex=2, NLOS channel model SNR [dB] 1.5 6.5 11.5 16.5
f = 10% 82.2% 93.8% 97.5% 98.5%
f = 5% 83.5% 94.7% 98.2% 99.2%
f = 0% 84.0% 95.0% 98.4% 99.4%
Simulation results obtained for the first modification of the algorithm, in which the metric MR,ex was applied, have been summarized in Table 3, Table 4 and Table 5. In all cases the parameter ex was equal 2. Results obtained for ex=4 have been shown in Table 6. Table 7 and Table 8 show the results obtained using the metric MF,ex. One can compare the algorithm performance for different value of ex.
Table 3. The first modification, ex=2, R=25, NLOS channel model SNR 1.5 dB 6.5 dB 11.5 dB 16.5 dB
f = 10% 85.9% 95.5% 98.5% 99.3%
f = 5% 86.6% 95.9% 98.7% 99.5%
f = 0% 86.6% 96.0% 98.8% 99.5%
Table 4. The first modification, ex=2, NLOS channel model SNR 6.5 dB; R=5 6.5 dB; R=10 6.5 dB; R=20 16.5 dB; R=5 16.5 dB; R=10 16.5 dB; R=20
f = 10% 93.7% 93.8% 94.6% 97.7% 97.7% 98.1%
f = 5% 95.3% 95.4% 95.8% 99.1% 99.1% 99.2%
f = 0% 95.9% 95.9% 96.1% 99.5% 99.5% 99.5%
Table 5. The first modification, ex=2, LOS channel model SNR [dB] 1.5 dB; R = 10 6.5 dB; R = 10 11.5 dB; R = 10 16.5 dB; R = 10 1.5 dB; R = 20 6.5 dB; R = 20 11.5 dB; R = 20 16.5 dB; R = 20
f = 10 % 86.0% 95.2% 97.8% 98.5% 86.8% 95.5% 98.0% 98.7%
f = 5% 87.5% 96.3% 98.7% 99.4% 88.0% 96.5% 98.8% 99.4%
f = 0% 88.0% 96.6% 99.0% 99.6% 88.5% 96.7% 99.0% 99.6%
Table 6. The first modification, ex=4, R=20, LOS channel model SNR [dB] 1.5 dB 6.5 dB 11.5 dB 16.5 dB
f = 10 % 87.6% 95.5% 97.8% 98.5%
f = 5% 89.0% 96.7% 98.8% 99.4%
f = 0% 89.5% 97.1% 99.1% 99.7%
Table 7. The last modification, ex=2, LOS and AWGN channels SNR [dB] 1.5 dB; AWGN 6.5 dB; AWGN 11.5 dB; AWGN 16.5 dB; AWGN 1.5 dB; LOS 6.5 dB; LOS 11.5 dB; LOS 16.5 dB; LOS
f = 10 % 99.4% 100.0% 100.0% 100.0% 89.4% 96.9% 99.0% 99.5%
f = 5% 99.6% 100.0% 100.0% 100.0% 89.9% 97.1% 99.1% 99.7%
f = 0% 99.6% 100.0% 100.0% 100.0% 90.2% 97.2% 99.2% 99.7%
Table 8. The last modification, different ex values, NLOS channel model SNR 1.5 dB, ex=1 6.5 dB, ex=1 11.5 dB, ex=1 16.5 dB, ex=1
f = 10% 85.5% 95.3% 98.4% 99.3%
f = 5% 86.1% 95.7% 98.6% 99.5%
f = 0% 86.3% 95.8% 98.6% 99.5%
SNR 1.5 dB, ex=2 6.5 dB, ex=2 11.5 dB, ex=2 16.5 dB, ex=2 1.5 dB, ex=3 6.5 dB, ex=3 11.5 dB, ex=3 16.5 dB, ex=3
f = 10% 88.2% 96.2% 98.7% 99.4% 88.8% 96.4% 98.6% 99.4%
f = 5% 88.7% 96.5% 98.9% 99.5% 89.4% 96.7% 98.9% 99.5%
f = 0% 89.0% 96.7% 99.0% 99.5% 89.7% 96.8% 99.0% 99.5%
VI. THE USE OF CYCLIC PREFIX Cyclic prefix will not be used in the first preamble symbol in the WIND-FLEX modem frame. This is caused by the requirements of the timing offset estimation algorithm. However, for the completeness of our investigations we have also analyzed the influence of the cyclic prefix on the performance of the proposed algorithms. Generally, one can expect that the presence of the cyclic prefix would increase the accuracy of the frequency offset estimates since it improves the received OFDM symbol spectrum. The obtained simulation results confirm that presumption. For high SNR value (16.5 dB), almost no fractional frequency offset and the NLOS channel model the accuracy was increased by about 0.2%. For low SNR (6.5 dB) the improvement is greater than 0.5%. In all simulated cases the parameter ex was equal 2.
Because of lack of space we did not present the results obtained with the second modification or for higher values of an uncompensated fractional part of the frequency offset. The second proposed algorithm modification allows to obtain very high probabililty of correct estimates for selected values of paramater R only. Higer values of the uncompensated fractional part of the frequency offset can cause signficant performance degradation for some of the presented algorithms. All these methods can be categorized as blind since they are based on the analysis of the received preamble spectrum only. The only condition that has to be fulfilled is the shape of the spectrum of the first preamble symbol, which is determined by the selection of subcarriers, which carry data and those, which are left unused. ACKNOWLEDGMENTS This work has been done within the WIND-FLEX project [10] (Wireless Indoor Flexible High Bitrate Modem Architectures IST-1999-10025). The first author was awarded by The Polish Science Foundation with the Annual Stipend for Young Scientists in year 2002. REFERENCES [1] Z. Sayeed, “Fast, Accurate and Simple Carrier Acquisition
VII. CONCLUSIONS The proposed schemes for the estimation of the integer part of the frequency offset in the OFDM systems require very low hardware complexity (two real multiplications for the subcarrier power calculation and only additions for metric calculation). These methods are especially suitable for the burst transmission since they require only one preamble symbol, the same which is used by the timing offset estimation algorithm. Simulation results show that in the presence of small uncompensated fractional part of the frequency offset for medium SNR (11.5 dB) the proposed methods offer about 99% of correct estimates while for 16.5 dB over 99.5% of correct estimates is obtained for the NLOS channel model characterized by deep selective fading. 100 percent of correct estimates have been obtained in the AWGN channel for medium and high SNR values. All simulation results show that the increase of the width of the spectral null allows to reduce the influence of noise and fading on the estimate. Increasing of the number of summed components in the calculation of appropriate metric allows to obtain similar effect. This can be observed in case of the first modification. However, one has to stress that the first proposed modification implies the increase of the obtained performance for low SNR or small fractional part of the frequeny offset only.
Algorithm for OFDM Systems”, Proc. of Globecom 99, pp. 2274-2278 [2] Y. S. Lim, J.H. Lee, “An Efficient Carrier Frequency Offset Estimation Scheme for an OFDM System”, Proc. of VTC’2000-Fall, Boston, September 2000 [3] K. Bang, N. Cho, J. Cho, H. Jun, K. Kim, H. Park, D. Hong, “A Coarse Frequency Offset Estimation in an OFDM System Using the Concept of the Coherence Phase Bandwidth”, IEEE Trans. Commun., Vol. 49, No. 8, pp. 1320-1324, August 2001 [4] H. Nogami, T. Nagashima, “ A Frequency and Timing Period Acquisition Technique for OFDM Systems”, Proc. of PIMRC’95, pp. 1010-1015 [5] T. M. Schmidl, D. C. Cox, “Robust Frequency and Timing Synchronization for OFDM”, IEEE Trans. Commun., Vol. 45, No. 12, pp. 1613-1621, December 1997 [6] P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction”, IEEE Trans. Commun., Vol. 42, pp. 2908-2914, October 1994 [7] J. J. Van Beek, M. Sandell, P. O. Borjesson, “ML estimation of time and frequency offset in OFDM system”, IEEE Trans. Signal Processing, Vol. 45, pp. 1800-1805, July 1997 [8] M. Morelli, U. Mengali, “Frequency Ambiguity Resolution in OFDM System”, IEEE Commun. Letters, Vol. 3, pp. 75-77, March 1999 [9] M. Lobeira, A. Armada, R. Torres, J. L. Garcia, “Parameter estimation and indoor channel modeling at 17 GHz for OFDMbased broadband WLAN”, Proc. of IST Mobile Communications Summit, Galway, October 2000 [10] I. Saarinen, at al., “Main approaches for the design of wireless indoor flexible high bit rate WIND-FLEX modem architecture”, Proc. of IST Mobile Communications Summit, Galway, October 2000
