Joint Time-Frequency Synchronization and Channel ... - CiteSeerX
2014 IEEE 25th International Symposium on Personal, Indoor and Mobile Radio Communications
Joint Time-Frequency Synchronization and Channel Estimation for FBMC Yonghong Zeng and Meng Wah Chia Institute for Infocomm Research, A∗ STAR, 1 Fusionopolis Way, Singapore 138632 Emails: {yhzeng, mwchia}@i2r.a-star.edu.sg
Abstract—Preamble with repeated time domain blocks is a very common design for joint time and frequency synchronization in various communication systems and standards. While it is easy to design two or more repeated time domain blocks in orthogonal frequency division multiplexing (OFDM) and some other systems, it is not straightforward to design filter bank multicarrier (FBMC) signal that has the repetition property in time domain, as FBMC modulation spreads one block of frequency domain signal into multiple blocks in time domain. In this paper, a new design is proposed for constructing repeated time domain blocks in FBMC. The preamble based on this design can be used for joint time-frequency synchronization and channel estimation as well. Extensive simulations are shown to verify the performance of the design. I. I NTRODUCTION In recent years, filter bank multicarrier (FBMC) has recaptured widespread interests for its possible applications in areas like cognitive radio, dynamic spectrum access and heterogenous network [1–3]. Like orthogonal frequency division multiplexing (OFDM), FBMC is also a multicarrier scheme. However, FBMC uses a much well localized prototype filter than OFDM. The localization of the FBMC signal in both time and frequency makes it more robust to synchronization errors and having better out-off-band emission control. Time-frequency synchronization and channel estimation are critical in most communication systems. Although there are many researches on blind synchronization and channel estimation, preamble-based method is the most widely used approach in practical systems and standards. Although time synchronization, frequency synchronization and channel estimation are different problems, they are cross-related and dependent on each other. Thus the preamble design and estimation algorithm should consider the three problems as a whole. There have been researches on the synchronization and channel estimation for FBMC, for example [4–8]. However, some of them only consider channel estimation problem [4, 7, 8]. [5] proposes a preamble-based method for joint estimations. The method does not need special preamble, but it requires high complexity twodimensional search in general, unless the frequency offset is very small. [6] proposes a pilot-based method that aims for fine synchronization and equalization after the initial synchronization and channel estimation. It can be used for channel tracking, but it is not intended for initial synchronization and channel estimation. In systems like OFDM, a common preamble design is a repeated multiple blocks in time domain [9–12]. Based on
978-1-4799-4912-0/14/$31.00 ©2014 IEEE
the repetition property, an approximate maximum likelihood estimation for the three parameters have been found in a simple form [9–12]. In systems like OFDM, it is easy to generate two or more repeated time domain blocks. However, it is more complicated in FBMC as one frequency domain block spreads into multiple time domain blocks. Thus repeated blocks in frequency domain do not generate repeated blocks in time domain, as the signals from different blocks interfere with each other in time domain. It is not straightforward to design the FBMC signal that has the repetition property in time. As far as we know, there is still no existing good design in literature yet. Hence, in this paper, we propose a new design to construct repeated blocks in time domain for general FBMC systems. Based on the new design, joint time-synchronization is done with very low complexity like that in OFDM systems, and the channel is then estimated by the same preamble. The rest of the paper is organized as follows. We briefly review the theory and discrete implementation of FBMC in Section II. The new preamble design and estimation methods are discussed in Section III. Simulation results are shown in Section IV. Finally, we give conclusions in Section V. II. A
BRIEF REVIEW OF
FBMC
AND ITS DISCRETE
IMPLEMENTATION
In this section, we first briefly review the theory of FBMC and then give the discrete implementation procedure of FBMC transmitter that is closely related to the preamble design.
A. A brief review of FBMC Recently FBMC with localized prototype filter has regained popularity due to its potential in applications like cognitive radio, spectrum sharing and heterogenous networks [1–3]. The design of a FBMC system is to select a few parameters including prototype filter p(t), time duration T and subcarrier space F [13–16]. Assume that we are going to transmit a data sequence using N subcarriers. The data sequence is divided into blocks of length N and denoted by: s¯n (k), k = 0, 1, · · · , N − 1; n = 0, 1, · · · , M − 1, where N is the number of subcarriers in the FBMC system. Thus N F is the whole bandwidth used for the system. The data sequence is then modified to sn (k) = s¯n (k)dn (k), where dn (k) are modulation coefficients independent on the data. The modified data sequence is modulated to an analog signal expressed as
413
x(t) =
N −1 M−1 X X k=0 n=0
sn (k)p(t − nT )ej2πkF t .
(1)
The expressions here are very general and can be applied to different systems. In literature, there are some systems like FMT (Filtered Multiple Tones) and OFDM/OQAM (Offset Quadrature Amplitude Modulation). They are treated as special cases here using the expressions above. For general FBMC or FMT, s¯n (k) can be any type of modulation (complex or real) and dn (k) = 1. For OFDM/OQAM, s¯n (k) must be real √ and dn (k) = j n+k , where j = −1. At the receiver, let x ¯(t) be the received signal. The signal is matched with the prototype filter to obtain: Z ∞ zn (k) = cn (k) x ¯(t)p(t − nT )e−j2πkF t dt, (2) −∞
where cn (k) is a constant to normalize the output. For OFDM/OQAM, only real part of the matched signal is used for equalization, and the image part is discarded. That is, for the special case of OFDM/OQAM, we compute Z ∞ −(n+k) −j2πkF t zn (k) = R j x ¯(t)p(t − nT )e dt . (3)
We divide the output into blocks and denote yl (m) sˆn (m)
B. Discrete fast implementation for the transmitter In practical implementation, only samples of the analog signal (1) is computed and a digital to analog converter (DAC) is used to convert the digital signal into analog signal. Fast algorithms for computing the discrete signal have been discussed in literature, for example [2, 16]. Since the discrete implementation directly determines the preamble signal property, we give its procedure here. The sampled signal of (1) is: x(mTs ) = =
N −1 M−1 X X
k=0 n=0 N −1 M−1 X X
sn (k)p(mTs − nT )e
sn (k)p0
k=0 n=0
j2πkm N
√ j2πkm m √ −n λ e N . λN
(4)
(5) (6)
k=0
m = 0, 1, · · · , N − 1. Then we have yl (m) =
M−1 X
sˆn (m)p0
n=0
Defining gk (m) = p0
lN + m − nλN √ λN
k m √ +√ λ λN
.
,
(7)
(8)
we obatin yl (m) =
M2 X
sˆn (m)gl−λn (m).
(9)
n=M1
−∞
Note that the prototype filter is chosen such that the signals at different subcarriers are orthogonal. Thus usually a single tap equalization is used to recover s¯n (k) from zn (k). A good FBMC system needs to have a few properties like high bandwidth efficiency, robust to double selective channel, robust to synchronization errors, low complexity etc.. Technically the basic parameters in a FBMC are the prototype filter p(t), the time duration T and subcarrier space F . For fixed bandwidth efficiency, we can choose different T or F for different channel conditions with given timefrequency product: T F = λ. For cognitive radio applications, we may need to change T or F at different scenarios. For this purpose, we can design a normalized prototype filter p0 (t)√for normalized time duration and frequency space: T = F = λ. Using p0 (t), we can construct the√prototype filter for any T and F with T F = λ: p(t) = p0 ( Tλ t) = p0 ( √Fλ t). Thus we only need to design one prototype filter with given λ. We know that OFDM/OQAM has the highest bandwidth efficiency. However, OFDM/OQAM is less robust to channel estimation and synchronization errors than FMT. Furthermore, using OFDM/OQAM for multiple input multiple output (MIMO) also faces some problems [1, 17].
= x((lN + m)Ts ), N −1 X j2πkm = sn (k)e N ,
III. J OINT
TIME - FREQUENCY SYNCHRONIZATION AND CHANNEL ESTIMATION
Let r(n) be the received signal. Then r(n)
=
ej2πǫn
L X l=0
h(l)x(n − l − τ0 ) + η(n),
(10)
where ǫ is the normalized carrier frequency offsets (CFO), h(l) is the channel (including filters), τ0 is the time delay, and η(n) is the noise. In order to recover the signal, we need to estimate the time delay τ0 , the CFO ǫ and the channel h(l). From the equation, we know that the three parameters are cross-related and dependent on each other. Thus we cannot simply estimate a single parameter by assuming that the others are known.
A. Preamble with repeated time domain blocks Using specially designed preamble for the estimations is the most popular approach in existing systems and standards. A very common preamble design is a repeated multiple blocks in time domain [9–12]. Based on the repetition property, an approximate maximum likelihood estimation for the three parameters have been found in a simple form [9–12]. Thus it is natural to consider constructing similar preambles for FBMC system. In systems like OFDM, it is easy to design two or more repeated time domain blocks for time-frequency synchronization and channel estimation. However, it is more complicated in FBMC as one frequency domain block spreads into multiple time domain blocks and different frequency domain signals interfere with each other in time. It is not straightforward to design the FBMC signal that has the repetition property. To overcome this difficulty, we have two options. (1) We do not use FBMC modulation for the preamble signal that is used for time and frequency synchronization. If so, the synchronization preamble cannot be used for channel estimation, and we need at least another dedicated preamble in FBMC modulation for channel estimation. Furthermore, it is necessary to insert a gap between the time-frequency synchronization preamble and
414
the channel estimation preamble, as the FBMC modulation spreads the signal into multiple blocks, which interferes the time-frequency synchronization preamble if there is no gap. The additional preamble and the gap is a loss of data rate. (2) We use FBMC modulation for the preamble of time and frequency synchronization. Then we need to specially design the preamble such that repetition property is there and the same preamble can be used for channel estimation as well. Obviously this approach can reduce the preamble length and thus increase data rate. Hence, we choose the second approach to maximize data rate. Here we consider using two frequency domain blocks as the preamble. The design can be directly extended to using multiple blocks. Let s0 (k) and s1 (k), k = 0, 1, · · · , N − 1, be the frequency domain preamble. The time domain signal generated by s0 (k) and s1 (k) are denoted by u0 (m) and u1 (m), respectively. We divide the time domain signal into blocks of length N . As shown in the last section, the generated signals can be expressed as lN + m √ u0 (lN + m) = sˆ0 (m)p0 , (11) λN lN + m − λN √ , (12) u1 (lN + m) = sˆ1 (m)p0 λN m = 0, 1, · · · , N − 1, where sˆ0 (m) and sˆ1 (m) are the inverse discrete Fourier transform (IDFT) of s0 (k) and s1 (k), respectively. From the equations we have lN + m − (λ − 1)N √ u1 ((l + 1)N + m) = sˆ1 (m)p0 . (13) λN
Hence, the lower bound and upper bound of l are respectively √ √ L1 = ⌈−∆ λ − 1⌉ and L2 = ⌊∆ λ⌋. (19) However, we see in the last section that the time domain transmitted signal x(n) is a superposition of signals generated by multiple blocks. That is, x(n) = u0 (n) + u1 (n) + w(n), where w(n) is the interference by other blocks following s0 (k) and s1 (k). Hence, the transmitted signal only has the periodical property approximately (with interference): x((l + 1)N + m + βN ) ≈ x(lN + m),
m = 0, 1, · · · , (1 − β)N − 1; l = L1 , · · · , L2 .
B. Joint time-frequency synchronization The periodical property (20) allows us using similar derivation in [9–12] to find an approximate maximum likelihood estimation for the three parameters. The estimation of τ0 is obtained as τˆ0 = arg max A(τ ), τ
A(τ )
m = 0, 1, · · · , (1 − β)N − 1. To satisfy (15), we can choose the frequency domain blocks such that s1 (k) = s0 (k)e−j2πkβ , k = 0, 1, · · · , N − 1.
(17)
r∗ (lN + m + τ )
m=0
L2 X
(1−β)N −1
l=L1 L2 (1−β)N X X −1 n=0
X
n=0
|r(lN + m + τ )|2
|r((l + 1)N + m + βN + τ )|2 .
(23)
We then compensate the time delay and get the CFO estimation as ǫˆ =
∠(
L2 X
(1−β)N −1
l=L1
X
r∗ (lN + m + τˆ0 )
m=0
·r((l + 1)N + m + βN + τˆ0 ))/(2πλN ). (24)
we have (16)
X
l=L1
V (τ ) =
sˆ0 (m) = sˆ1 (m + βN ), m = 0, 1, · · · , (1 − β)N − 1, (15) u1 ((l + 1)N + m + βN ) = u0 (lN + m),
(1−β)N −1
L2 X
and
l=L1
To simplify notations, we denote β = λ − 1. From the equations above, if the two frequency domain blocks satisfy
= |
·r((l + 1)N + m + βN + τ )|/V (τ ), (22)
+ (14)
(21)
where
Thus u1 ((l + 1)N + m + (λ − 1)N ) lN + m √ = sˆ1 (m + (λ − 1)N )p0 . λN
(20)
Although u0 (lN + m) 6= 0 for L1 ≤ l ≤ L2 , some of them may have low power. It is not good to use the low power signals for the synchronization. Thus it may be better to only choose part of the blocks such that the chosen signals have relatively high power, which is related to the property of the prototype filter.
C. Channel estimation
This can be easily proved by using the property of DFT. Thus the specially designed preamble (17) has the periodical property (16). Note that the periodical property holds for all l. However, the prototype filter p0 (t) is supported in a limited time interval t ∈ (−∆, ∆). Thus u0 (lN + m) have limited length with index l being constrained by √ √ (18) −∆ λ − 1 < l < ∆ λ.
After obtaining the estimations of time delay and CFO, we compensate the received signal with the estimated parameters to obtain
415
rˆ(n) = e−j2πˆǫ(n+ˆτ0 ) r(n + τˆ0 ) L X = ej2π(ǫ−ˆǫ)(n+ˆτ0 ) h(l)x(n − l + τˆ0 − τ0 ). l=0
(25)
1 0.9 1
10
0.8 0.7
0
10
PSD
0.5
0
p (t)
0.6
0.4 0.3
−1
10
−2
10
0.2 −3
0.1
10
0 −4
−5
−4
−3
−2
−1
0
1
2
3
4
10
5
−4
t
Fig. 1.
Fig. 2.
This compensated signal together with the preambles are used for the channel estimation. For general FBMC (FMT), we do not need to impose constraints on s0 (k) for channel estimation. However, it is easy to show that using constant amplitude signal (|s0 (k)| =constant) is the best. Let Ri (k) be the signal after the FBMC receiving process [2, 16] for rˆ(n), i = 0, 1, · · · ; n = 0, 1, · · · , N − 1. The frequency domain channel (DFT of h(l)) is estimated as = (s∗0 (k)R0 (k) + s∗1 (k)R1 (k)) /(|s0 (k)|2 + |s1 (k)|2 ),
−2
−1
0
1
2
3
4
Frequency (MHz)
Normalized prototype filter (λ = 1.25)
H(k)
−3
(26)
k = 0, 1, · · · , N − 1.
For OFDM/OQAM, the channel estimation is more complicated. Methods can be found in [4, 7, 8]. Based on the estimated channel, the transmitted signal can be recovered by using a one-tap equalization or more complicated method [1, 3]. IV. S IMULATIONS In the following we show some simulation results for the performance of the new design. We consider general FBMC (FMT) systems. The prototype filter is designed based on the IOTA (Isotropic Orthogonal Transform Algorithm) [13–15]. √As an example, the normalized prototype filter for T = F = 1.25 is shown in Figure 1. The system parameters are chosen to mimic the DVBT standard with reduced number of subcarriers. The carrier frequency is 1GHz and the bandwidth of the channel is 8MHz. The sampling rate is 64/7 MHz. The time-frequency product is chosen as λ = 1.25, the number of subcarriers is N = 256. Among the 256 subcarriers, there are 43 null subcarriers for controling the out-of-band emission. The data sequence is divided into data frames. Each frame contains 100 blocks that corresponds to sampled data in 3.5 milli seconds (ms). In a data frame, there are two preamble blocks. In each data block, there are 5 pilot symbols. The power spectrum density (PSD) of the transmitted FBMC signal is shown in Figure 2. From the figure, we see that, with the 43 null subcarriers and fast decay
PSD of the transmitted FBMC signal
prototype filter, the out-of-band emission of the transmitted signal is well controlled. The two preamble blocks are used for synchronization and channel estimation, while the pilots are used for phase and amplitude correction. First, a joint time-frequency synchronization is done by using the proposed method at the beginning of a frame. Then a one-tap maximum ratio combining (MRC) method (26) is used to estimate the frequency domain channel by using the two preambles. As the frequency synchronization may not be absolutely accurate, it is necessary to do phase correction for data blocks. We use the inserted pilots to get a phase estimation and then compensate the signal with the estimated phase term. Finally we use the one-tap equalization to recover the signal based on the estimated channel. We assume that the maximum CFO is 10kHz, which is equivalent to the oscillator accuracy of 10 ppm (parts per million) at 1GHz carrier frequency. The actual CFO at each Monte-Carlo test is randomly generated in [−10, 10)kHz with even distribution. The maximum timing error is 50 samples. The actual timing error at each Monte-Carlo test is randomly generated in [−50, 50) with even distribution. We consider a Rayleigh fading frequency selective channel h(l) with 65 taps and exponential power delay profile: E(|h(l)|2 ) = e−dl , l = 0, 1, · · · , 64. For comparison, we also simulate the case when the signal is perfectly synchronized without time offset and frequency error, while the channel is estimated by using the preambles. At this case, it is unnecessary to do phase estimation and compensation. First, time-invariant channel is considered. The symbol error rates (SER) for 4QAM modulation are shown in Figure 3 and Figure 4, respectively for d = 0.4 and d = 1.2. From the figures, we see that the residual time offset and frequency error only slightly degrade the SER performance, which means that the accuracy of time and frequency offset estimation is good enough in practical applications. Secondly, a time-variant channel with Doppler frequency 9.26Hz (corresponding to moving speed 10km/h at carrier frequency 1GHz) is used. The Matlab function “ricianchan”,
416
FBMC−Actual Sync FBMC−Perfect Sync
FBMC−Actual Sync FBMC−Perfect Sync
Symbol error rate (SER)
Symbol error rate (SER)
−1
10
−2
10
−1
10
−2
10 −3
10
0
5
10
15
20
25
30
0
SNR
Fig. 3.
Fig. 5.
FBMC−Actual Sync FBMC−Perfect Sync −1
Symbol error rate (SER)
10
−2
10
−3
10
5
10
15
20
25
30
SNR
Fig. 4.
10
15
20
25
30
SNR
SER vs SNR (4QAM, d = 0.4)
0
5
SER vs SNR (4QAM, d = 1.2)
which is based on Jake’s model, is used to generate the channel. The profile of the channel is the same as described above with d = 1.2. The result is shown in Figure 5. Here there is almost no difference between the SER performances of preamble-based synchronization and ideal synchronization. In summary, the design and estimation method can meet the requirements in systems with practical channel conditions. V. C ONCLUSIONS We have proposed a new design for constructing repeated time domain blocks in FBMC. The preambles based on this design can be used for joint time-frequency synchronization and channel estimation as well. Simulations have shown that the design and estimation method can meet the performance requirements in practical applications. ACKNOWLEDGEMENT Thank Dr. The Hanh Pham for providing the prototype filter. R EFERENCES [1] B. Farhang-Boroujeny, “OFDM versus filter bank multicarrier,” Signal Processing Magazine, pp. 92–112, May 2011.
SER vs SNR (4QAM, d = 1.2, Doppler frequency 9.26Hz)
[2] E. Gutierrez, J. A. Lopez-Salcedo, and G. Seco-Granados, “Unified framework for flexible multi-carrier communication system,” in 8th International Workshop on Multi-Carrier Systems & Solutions (MC-SS), (Spain), May 2011. [3] PHYDYAS, “PHYDYAS-physical layer for dynamic spectrum access and cognitive radio,” in http://www.ict-phydyas.org/, 2013. [4] C. Lele, J.-P. Javaudin, R. Legouable, A. Skrzypczak, and P. Siohan, “Channel estimation methods for preamble-based OFDM/OQAM modulations,” European Transactionson Telecommunications, vol. 19, pp. 741–750, 2008. [5] T. Fusco, A. Petrella, and M. Tanda, “Joint symbol timing and CFO estimation for OFDM/OQAM systems in multipath channels,” EURASIP Journal on Advances in Signal Processing, no. Article ID 897607, 2010. [6] T. H. Stitz, T. Ihalainen, A. Viholainen, and M. Renfors, “Pilot-based synchronization and equalization in filter bank multicarrier communications,” EURASIP Journal on Advances in Signal Processing, no. Article ID 741429, 2010. [7] D. Li, J. J. Chen, K. Y. Wu, L. Y. Cai, and Y. J. Han, “Single-symbol preamble design and channele stimation for filter bank multi-carrier modulations,” in IEEE PIMRC, (Sydney), Sept. 2012. [8] E. Kofidis, D. Katselis, and A. Rontogiannis, “Preamble-based channel estimation in OFDM/OQAM systems: a review,” Signal Processing, vol. 93, pp. 2038–2054, 2013. [9] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., vol. 45, no. 12, pp. 1613– 1621, 1997. [10] M. Morelli and U. Mengali, “An improved frequency offset estimator for OFDM applications,” IEEE Communications Letters, vol. 3, no. 3, pp. 75–77, 1999. [11] H. Minn, M. Zeng, and V. K. Bhargava, “On timing offset for OFDM systems,” IEEE Communication Letters, vol. 4, no. 7, pp. 242–244, 2000. [12] Y. H. Zeng, W. Leon, Y.-C. Liang, and A. R. Leyman, “A new method for frequency offset and chanenl estimation in OFDM,” in IEEE Intern. Conference on Communications, 2006. [13] B. L. Floch, M. Alard, and C. Berrou, “Coded orthogonal frequency division multiplex,” Proceedings of IEEE, vol. 83, no. 6, pp. 982–996, 1995. [14] R. Haas and J.-C. Belfiore, “A time-frequency well-localized pulse for multiple carrier transmission,” Wireless Personal Communications, vol. 5, pp. 1–18, 1997. [15] P. Siohan and C. Roche, “Cosine-modulated filterbanks based on extended gaussian functions,” IEEE Trans. Signal Processing, vol. 48, no. 11, pp. 3052–3061, 2000. [16] Y. H. Zeng, Y.-C. Liang, M. W. Chia, and E. C. Y. Peh, “Unified structure and parallel algorithms for FBMC transmitter and receiver,” in IEEE PIMRC, (London), Sept 2013. [17] M. Payaro and A. Pascual-Iserte, “Performance comparison between OFDM and OFDM/OQAM in MIMO systems under channel uncertainty,” in IEEE International Conf. Communications, (South Africa), May 2010.
IEEE
417
Joint Time-Frequency Synchronization and Channel Estimation for FBMC Yonghong Zeng and Meng Wah Chia Institute for Infocomm Research, A∗ STAR, 1 Fusionopolis Way, Singapore 138632 Emails: {yhzeng, mwchia}@i2r.a-star.edu.sg
Abstract—Preamble with repeated time domain blocks is a very common design for joint time and frequency synchronization in various communication systems and standards. While it is easy to design two or more repeated time domain blocks in orthogonal frequency division multiplexing (OFDM) and some other systems, it is not straightforward to design filter bank multicarrier (FBMC) signal that has the repetition property in time domain, as FBMC modulation spreads one block of frequency domain signal into multiple blocks in time domain. In this paper, a new design is proposed for constructing repeated time domain blocks in FBMC. The preamble based on this design can be used for joint time-frequency synchronization and channel estimation as well. Extensive simulations are shown to verify the performance of the design. I. I NTRODUCTION In recent years, filter bank multicarrier (FBMC) has recaptured widespread interests for its possible applications in areas like cognitive radio, dynamic spectrum access and heterogenous network [1–3]. Like orthogonal frequency division multiplexing (OFDM), FBMC is also a multicarrier scheme. However, FBMC uses a much well localized prototype filter than OFDM. The localization of the FBMC signal in both time and frequency makes it more robust to synchronization errors and having better out-off-band emission control. Time-frequency synchronization and channel estimation are critical in most communication systems. Although there are many researches on blind synchronization and channel estimation, preamble-based method is the most widely used approach in practical systems and standards. Although time synchronization, frequency synchronization and channel estimation are different problems, they are cross-related and dependent on each other. Thus the preamble design and estimation algorithm should consider the three problems as a whole. There have been researches on the synchronization and channel estimation for FBMC, for example [4–8]. However, some of them only consider channel estimation problem [4, 7, 8]. [5] proposes a preamble-based method for joint estimations. The method does not need special preamble, but it requires high complexity twodimensional search in general, unless the frequency offset is very small. [6] proposes a pilot-based method that aims for fine synchronization and equalization after the initial synchronization and channel estimation. It can be used for channel tracking, but it is not intended for initial synchronization and channel estimation. In systems like OFDM, a common preamble design is a repeated multiple blocks in time domain [9–12]. Based on
978-1-4799-4912-0/14/$31.00 ©2014 IEEE
the repetition property, an approximate maximum likelihood estimation for the three parameters have been found in a simple form [9–12]. In systems like OFDM, it is easy to generate two or more repeated time domain blocks. However, it is more complicated in FBMC as one frequency domain block spreads into multiple time domain blocks. Thus repeated blocks in frequency domain do not generate repeated blocks in time domain, as the signals from different blocks interfere with each other in time domain. It is not straightforward to design the FBMC signal that has the repetition property in time. As far as we know, there is still no existing good design in literature yet. Hence, in this paper, we propose a new design to construct repeated blocks in time domain for general FBMC systems. Based on the new design, joint time-synchronization is done with very low complexity like that in OFDM systems, and the channel is then estimated by the same preamble. The rest of the paper is organized as follows. We briefly review the theory and discrete implementation of FBMC in Section II. The new preamble design and estimation methods are discussed in Section III. Simulation results are shown in Section IV. Finally, we give conclusions in Section V. II. A
BRIEF REVIEW OF
FBMC
AND ITS DISCRETE
IMPLEMENTATION
In this section, we first briefly review the theory of FBMC and then give the discrete implementation procedure of FBMC transmitter that is closely related to the preamble design.
A. A brief review of FBMC Recently FBMC with localized prototype filter has regained popularity due to its potential in applications like cognitive radio, spectrum sharing and heterogenous networks [1–3]. The design of a FBMC system is to select a few parameters including prototype filter p(t), time duration T and subcarrier space F [13–16]. Assume that we are going to transmit a data sequence using N subcarriers. The data sequence is divided into blocks of length N and denoted by: s¯n (k), k = 0, 1, · · · , N − 1; n = 0, 1, · · · , M − 1, where N is the number of subcarriers in the FBMC system. Thus N F is the whole bandwidth used for the system. The data sequence is then modified to sn (k) = s¯n (k)dn (k), where dn (k) are modulation coefficients independent on the data. The modified data sequence is modulated to an analog signal expressed as
413
x(t) =
N −1 M−1 X X k=0 n=0
sn (k)p(t − nT )ej2πkF t .
(1)
The expressions here are very general and can be applied to different systems. In literature, there are some systems like FMT (Filtered Multiple Tones) and OFDM/OQAM (Offset Quadrature Amplitude Modulation). They are treated as special cases here using the expressions above. For general FBMC or FMT, s¯n (k) can be any type of modulation (complex or real) and dn (k) = 1. For OFDM/OQAM, s¯n (k) must be real √ and dn (k) = j n+k , where j = −1. At the receiver, let x ¯(t) be the received signal. The signal is matched with the prototype filter to obtain: Z ∞ zn (k) = cn (k) x ¯(t)p(t − nT )e−j2πkF t dt, (2) −∞
where cn (k) is a constant to normalize the output. For OFDM/OQAM, only real part of the matched signal is used for equalization, and the image part is discarded. That is, for the special case of OFDM/OQAM, we compute Z ∞ −(n+k) −j2πkF t zn (k) = R j x ¯(t)p(t − nT )e dt . (3)
We divide the output into blocks and denote yl (m) sˆn (m)
B. Discrete fast implementation for the transmitter In practical implementation, only samples of the analog signal (1) is computed and a digital to analog converter (DAC) is used to convert the digital signal into analog signal. Fast algorithms for computing the discrete signal have been discussed in literature, for example [2, 16]. Since the discrete implementation directly determines the preamble signal property, we give its procedure here. The sampled signal of (1) is: x(mTs ) = =
N −1 M−1 X X
k=0 n=0 N −1 M−1 X X
sn (k)p(mTs − nT )e
sn (k)p0
k=0 n=0
j2πkm N
√ j2πkm m √ −n λ e N . λN
(4)
(5) (6)
k=0
m = 0, 1, · · · , N − 1. Then we have yl (m) =
M−1 X
sˆn (m)p0
n=0
Defining gk (m) = p0
lN + m − nλN √ λN
k m √ +√ λ λN
.
,
(7)
(8)
we obatin yl (m) =
M2 X
sˆn (m)gl−λn (m).
(9)
n=M1
−∞
Note that the prototype filter is chosen such that the signals at different subcarriers are orthogonal. Thus usually a single tap equalization is used to recover s¯n (k) from zn (k). A good FBMC system needs to have a few properties like high bandwidth efficiency, robust to double selective channel, robust to synchronization errors, low complexity etc.. Technically the basic parameters in a FBMC are the prototype filter p(t), the time duration T and subcarrier space F . For fixed bandwidth efficiency, we can choose different T or F for different channel conditions with given timefrequency product: T F = λ. For cognitive radio applications, we may need to change T or F at different scenarios. For this purpose, we can design a normalized prototype filter p0 (t)√for normalized time duration and frequency space: T = F = λ. Using p0 (t), we can construct the√prototype filter for any T and F with T F = λ: p(t) = p0 ( Tλ t) = p0 ( √Fλ t). Thus we only need to design one prototype filter with given λ. We know that OFDM/OQAM has the highest bandwidth efficiency. However, OFDM/OQAM is less robust to channel estimation and synchronization errors than FMT. Furthermore, using OFDM/OQAM for multiple input multiple output (MIMO) also faces some problems [1, 17].
= x((lN + m)Ts ), N −1 X j2πkm = sn (k)e N ,
III. J OINT
TIME - FREQUENCY SYNCHRONIZATION AND CHANNEL ESTIMATION
Let r(n) be the received signal. Then r(n)
=
ej2πǫn
L X l=0
h(l)x(n − l − τ0 ) + η(n),
(10)
where ǫ is the normalized carrier frequency offsets (CFO), h(l) is the channel (including filters), τ0 is the time delay, and η(n) is the noise. In order to recover the signal, we need to estimate the time delay τ0 , the CFO ǫ and the channel h(l). From the equation, we know that the three parameters are cross-related and dependent on each other. Thus we cannot simply estimate a single parameter by assuming that the others are known.
A. Preamble with repeated time domain blocks Using specially designed preamble for the estimations is the most popular approach in existing systems and standards. A very common preamble design is a repeated multiple blocks in time domain [9–12]. Based on the repetition property, an approximate maximum likelihood estimation for the three parameters have been found in a simple form [9–12]. Thus it is natural to consider constructing similar preambles for FBMC system. In systems like OFDM, it is easy to design two or more repeated time domain blocks for time-frequency synchronization and channel estimation. However, it is more complicated in FBMC as one frequency domain block spreads into multiple time domain blocks and different frequency domain signals interfere with each other in time. It is not straightforward to design the FBMC signal that has the repetition property. To overcome this difficulty, we have two options. (1) We do not use FBMC modulation for the preamble signal that is used for time and frequency synchronization. If so, the synchronization preamble cannot be used for channel estimation, and we need at least another dedicated preamble in FBMC modulation for channel estimation. Furthermore, it is necessary to insert a gap between the time-frequency synchronization preamble and
414
the channel estimation preamble, as the FBMC modulation spreads the signal into multiple blocks, which interferes the time-frequency synchronization preamble if there is no gap. The additional preamble and the gap is a loss of data rate. (2) We use FBMC modulation for the preamble of time and frequency synchronization. Then we need to specially design the preamble such that repetition property is there and the same preamble can be used for channel estimation as well. Obviously this approach can reduce the preamble length and thus increase data rate. Hence, we choose the second approach to maximize data rate. Here we consider using two frequency domain blocks as the preamble. The design can be directly extended to using multiple blocks. Let s0 (k) and s1 (k), k = 0, 1, · · · , N − 1, be the frequency domain preamble. The time domain signal generated by s0 (k) and s1 (k) are denoted by u0 (m) and u1 (m), respectively. We divide the time domain signal into blocks of length N . As shown in the last section, the generated signals can be expressed as lN + m √ u0 (lN + m) = sˆ0 (m)p0 , (11) λN lN + m − λN √ , (12) u1 (lN + m) = sˆ1 (m)p0 λN m = 0, 1, · · · , N − 1, where sˆ0 (m) and sˆ1 (m) are the inverse discrete Fourier transform (IDFT) of s0 (k) and s1 (k), respectively. From the equations we have lN + m − (λ − 1)N √ u1 ((l + 1)N + m) = sˆ1 (m)p0 . (13) λN
Hence, the lower bound and upper bound of l are respectively √ √ L1 = ⌈−∆ λ − 1⌉ and L2 = ⌊∆ λ⌋. (19) However, we see in the last section that the time domain transmitted signal x(n) is a superposition of signals generated by multiple blocks. That is, x(n) = u0 (n) + u1 (n) + w(n), where w(n) is the interference by other blocks following s0 (k) and s1 (k). Hence, the transmitted signal only has the periodical property approximately (with interference): x((l + 1)N + m + βN ) ≈ x(lN + m),
m = 0, 1, · · · , (1 − β)N − 1; l = L1 , · · · , L2 .
B. Joint time-frequency synchronization The periodical property (20) allows us using similar derivation in [9–12] to find an approximate maximum likelihood estimation for the three parameters. The estimation of τ0 is obtained as τˆ0 = arg max A(τ ), τ
A(τ )
m = 0, 1, · · · , (1 − β)N − 1. To satisfy (15), we can choose the frequency domain blocks such that s1 (k) = s0 (k)e−j2πkβ , k = 0, 1, · · · , N − 1.
(17)
r∗ (lN + m + τ )
m=0
L2 X
(1−β)N −1
l=L1 L2 (1−β)N X X −1 n=0
X
n=0
|r(lN + m + τ )|2
|r((l + 1)N + m + βN + τ )|2 .
(23)
We then compensate the time delay and get the CFO estimation as ǫˆ =
∠(
L2 X
(1−β)N −1
l=L1
X
r∗ (lN + m + τˆ0 )
m=0
·r((l + 1)N + m + βN + τˆ0 ))/(2πλN ). (24)
we have (16)
X
l=L1
V (τ ) =
sˆ0 (m) = sˆ1 (m + βN ), m = 0, 1, · · · , (1 − β)N − 1, (15) u1 ((l + 1)N + m + βN ) = u0 (lN + m),
(1−β)N −1
L2 X
and
l=L1
To simplify notations, we denote β = λ − 1. From the equations above, if the two frequency domain blocks satisfy
= |
·r((l + 1)N + m + βN + τ )|/V (τ ), (22)
+ (14)
(21)
where
Thus u1 ((l + 1)N + m + (λ − 1)N ) lN + m √ = sˆ1 (m + (λ − 1)N )p0 . λN
(20)
Although u0 (lN + m) 6= 0 for L1 ≤ l ≤ L2 , some of them may have low power. It is not good to use the low power signals for the synchronization. Thus it may be better to only choose part of the blocks such that the chosen signals have relatively high power, which is related to the property of the prototype filter.
C. Channel estimation
This can be easily proved by using the property of DFT. Thus the specially designed preamble (17) has the periodical property (16). Note that the periodical property holds for all l. However, the prototype filter p0 (t) is supported in a limited time interval t ∈ (−∆, ∆). Thus u0 (lN + m) have limited length with index l being constrained by √ √ (18) −∆ λ − 1 < l < ∆ λ.
After obtaining the estimations of time delay and CFO, we compensate the received signal with the estimated parameters to obtain
415
rˆ(n) = e−j2πˆǫ(n+ˆτ0 ) r(n + τˆ0 ) L X = ej2π(ǫ−ˆǫ)(n+ˆτ0 ) h(l)x(n − l + τˆ0 − τ0 ). l=0
(25)
1 0.9 1
10
0.8 0.7
0
10
PSD
0.5
0
p (t)
0.6
0.4 0.3
−1
10
−2
10
0.2 −3
0.1
10
0 −4
−5
−4
−3
−2
−1
0
1
2
3
4
10
5
−4
t
Fig. 1.
Fig. 2.
This compensated signal together with the preambles are used for the channel estimation. For general FBMC (FMT), we do not need to impose constraints on s0 (k) for channel estimation. However, it is easy to show that using constant amplitude signal (|s0 (k)| =constant) is the best. Let Ri (k) be the signal after the FBMC receiving process [2, 16] for rˆ(n), i = 0, 1, · · · ; n = 0, 1, · · · , N − 1. The frequency domain channel (DFT of h(l)) is estimated as = (s∗0 (k)R0 (k) + s∗1 (k)R1 (k)) /(|s0 (k)|2 + |s1 (k)|2 ),
−2
−1
0
1
2
3
4
Frequency (MHz)
Normalized prototype filter (λ = 1.25)
H(k)
−3
(26)
k = 0, 1, · · · , N − 1.
For OFDM/OQAM, the channel estimation is more complicated. Methods can be found in [4, 7, 8]. Based on the estimated channel, the transmitted signal can be recovered by using a one-tap equalization or more complicated method [1, 3]. IV. S IMULATIONS In the following we show some simulation results for the performance of the new design. We consider general FBMC (FMT) systems. The prototype filter is designed based on the IOTA (Isotropic Orthogonal Transform Algorithm) [13–15]. √As an example, the normalized prototype filter for T = F = 1.25 is shown in Figure 1. The system parameters are chosen to mimic the DVBT standard with reduced number of subcarriers. The carrier frequency is 1GHz and the bandwidth of the channel is 8MHz. The sampling rate is 64/7 MHz. The time-frequency product is chosen as λ = 1.25, the number of subcarriers is N = 256. Among the 256 subcarriers, there are 43 null subcarriers for controling the out-of-band emission. The data sequence is divided into data frames. Each frame contains 100 blocks that corresponds to sampled data in 3.5 milli seconds (ms). In a data frame, there are two preamble blocks. In each data block, there are 5 pilot symbols. The power spectrum density (PSD) of the transmitted FBMC signal is shown in Figure 2. From the figure, we see that, with the 43 null subcarriers and fast decay
PSD of the transmitted FBMC signal
prototype filter, the out-of-band emission of the transmitted signal is well controlled. The two preamble blocks are used for synchronization and channel estimation, while the pilots are used for phase and amplitude correction. First, a joint time-frequency synchronization is done by using the proposed method at the beginning of a frame. Then a one-tap maximum ratio combining (MRC) method (26) is used to estimate the frequency domain channel by using the two preambles. As the frequency synchronization may not be absolutely accurate, it is necessary to do phase correction for data blocks. We use the inserted pilots to get a phase estimation and then compensate the signal with the estimated phase term. Finally we use the one-tap equalization to recover the signal based on the estimated channel. We assume that the maximum CFO is 10kHz, which is equivalent to the oscillator accuracy of 10 ppm (parts per million) at 1GHz carrier frequency. The actual CFO at each Monte-Carlo test is randomly generated in [−10, 10)kHz with even distribution. The maximum timing error is 50 samples. The actual timing error at each Monte-Carlo test is randomly generated in [−50, 50) with even distribution. We consider a Rayleigh fading frequency selective channel h(l) with 65 taps and exponential power delay profile: E(|h(l)|2 ) = e−dl , l = 0, 1, · · · , 64. For comparison, we also simulate the case when the signal is perfectly synchronized without time offset and frequency error, while the channel is estimated by using the preambles. At this case, it is unnecessary to do phase estimation and compensation. First, time-invariant channel is considered. The symbol error rates (SER) for 4QAM modulation are shown in Figure 3 and Figure 4, respectively for d = 0.4 and d = 1.2. From the figures, we see that the residual time offset and frequency error only slightly degrade the SER performance, which means that the accuracy of time and frequency offset estimation is good enough in practical applications. Secondly, a time-variant channel with Doppler frequency 9.26Hz (corresponding to moving speed 10km/h at carrier frequency 1GHz) is used. The Matlab function “ricianchan”,
416
FBMC−Actual Sync FBMC−Perfect Sync
FBMC−Actual Sync FBMC−Perfect Sync
Symbol error rate (SER)
Symbol error rate (SER)
−1
10
−2
10
−1
10
−2
10 −3
10
0
5
10
15
20
25
30
0
SNR
Fig. 3.
Fig. 5.
FBMC−Actual Sync FBMC−Perfect Sync −1
Symbol error rate (SER)
10
−2
10
−3
10
5
10
15
20
25
30
SNR
Fig. 4.
10
15
20
25
30
SNR
SER vs SNR (4QAM, d = 0.4)
0
5
SER vs SNR (4QAM, d = 1.2)
which is based on Jake’s model, is used to generate the channel. The profile of the channel is the same as described above with d = 1.2. The result is shown in Figure 5. Here there is almost no difference between the SER performances of preamble-based synchronization and ideal synchronization. In summary, the design and estimation method can meet the requirements in systems with practical channel conditions. V. C ONCLUSIONS We have proposed a new design for constructing repeated time domain blocks in FBMC. The preambles based on this design can be used for joint time-frequency synchronization and channel estimation as well. Simulations have shown that the design and estimation method can meet the performance requirements in practical applications. ACKNOWLEDGEMENT Thank Dr. The Hanh Pham for providing the prototype filter. R EFERENCES [1] B. Farhang-Boroujeny, “OFDM versus filter bank multicarrier,” Signal Processing Magazine, pp. 92–112, May 2011.
SER vs SNR (4QAM, d = 1.2, Doppler frequency 9.26Hz)
[2] E. Gutierrez, J. A. Lopez-Salcedo, and G. Seco-Granados, “Unified framework for flexible multi-carrier communication system,” in 8th International Workshop on Multi-Carrier Systems & Solutions (MC-SS), (Spain), May 2011. [3] PHYDYAS, “PHYDYAS-physical layer for dynamic spectrum access and cognitive radio,” in http://www.ict-phydyas.org/, 2013. [4] C. Lele, J.-P. Javaudin, R. Legouable, A. Skrzypczak, and P. Siohan, “Channel estimation methods for preamble-based OFDM/OQAM modulations,” European Transactionson Telecommunications, vol. 19, pp. 741–750, 2008. [5] T. Fusco, A. Petrella, and M. Tanda, “Joint symbol timing and CFO estimation for OFDM/OQAM systems in multipath channels,” EURASIP Journal on Advances in Signal Processing, no. Article ID 897607, 2010. [6] T. H. Stitz, T. Ihalainen, A. Viholainen, and M. Renfors, “Pilot-based synchronization and equalization in filter bank multicarrier communications,” EURASIP Journal on Advances in Signal Processing, no. Article ID 741429, 2010. [7] D. Li, J. J. Chen, K. Y. Wu, L. Y. Cai, and Y. J. Han, “Single-symbol preamble design and channele stimation for filter bank multi-carrier modulations,” in IEEE PIMRC, (Sydney), Sept. 2012. [8] E. Kofidis, D. Katselis, and A. Rontogiannis, “Preamble-based channel estimation in OFDM/OQAM systems: a review,” Signal Processing, vol. 93, pp. 2038–2054, 2013. [9] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., vol. 45, no. 12, pp. 1613– 1621, 1997. [10] M. Morelli and U. Mengali, “An improved frequency offset estimator for OFDM applications,” IEEE Communications Letters, vol. 3, no. 3, pp. 75–77, 1999. [11] H. Minn, M. Zeng, and V. K. Bhargava, “On timing offset for OFDM systems,” IEEE Communication Letters, vol. 4, no. 7, pp. 242–244, 2000. [12] Y. H. Zeng, W. Leon, Y.-C. Liang, and A. R. Leyman, “A new method for frequency offset and chanenl estimation in OFDM,” in IEEE Intern. Conference on Communications, 2006. [13] B. L. Floch, M. Alard, and C. Berrou, “Coded orthogonal frequency division multiplex,” Proceedings of IEEE, vol. 83, no. 6, pp. 982–996, 1995. [14] R. Haas and J.-C. Belfiore, “A time-frequency well-localized pulse for multiple carrier transmission,” Wireless Personal Communications, vol. 5, pp. 1–18, 1997. [15] P. Siohan and C. Roche, “Cosine-modulated filterbanks based on extended gaussian functions,” IEEE Trans. Signal Processing, vol. 48, no. 11, pp. 3052–3061, 2000. [16] Y. H. Zeng, Y.-C. Liang, M. W. Chia, and E. C. Y. Peh, “Unified structure and parallel algorithms for FBMC transmitter and receiver,” in IEEE PIMRC, (London), Sept 2013. [17] M. Payaro and A. Pascual-Iserte, “Performance comparison between OFDM and OFDM/OQAM in MIMO systems under channel uncertainty,” in IEEE International Conf. Communications, (South Africa), May 2010.
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