Proc. of the IEEE Global Telecommunications Conference GLOBECOM ’99, Rio de Janeiro, Brazil, pp. 857-862, Dec. 1999.
FREQUENCY–DOMAIN FRAME SYNCHRONIZATION FOR OPTIMUM FREQUENCY–DIFFERENTIAL DEMODULATION OF OFDM Stefan H. M¨ uller–Weinfurtner Laboratorium f¨ ur Nachrichtentechnik, Universit¨ at Erlangen–N¨ urnberg Cauerstraße 7, D–91058 Erlangen, Germany e-mail:
[email protected], WWW: http://www.lnt.de/˜dcg Abstract — A frame synchronization for Orthogonal Frequency–Division Multiplexing (OFDM) is addressed which exploits the correlation of adjacent subcarriers. A lower bound for the variance of the estimated frame start position is provided. The estimate is found to be biased according to the center of power delay gravity of the channel impulse response but this enables an optimum differential demodulation in frequency direction with respect to a minimum mean–squared error (MMSE) in the differentiated decision variables. 1.
Introduction
OFDM will be applied in high–rate mobile data transmission [9] such as wireless Asynchronous Transfer Mode (ATM) [7]. ATM is a packet–oriented transmission scheme and the non–continuous traffic in wireless ATM scenarios requires reliable single–shot synchronization with low burst overhead. In bursty OFDM transmission with differential modulation schemes in time direction, the phase reference symbol is a major source of inefficiency [3] so that differentially encoded modulation across the subcarriers, i.e., in frequency direction, becomes an interesting alternative. For frame- and carrier frequency synchronization, preamble schemes with periodic signal repetitions [5, 10, 11] are often used. With such repeated structures, a coarse frame synchronization can be achieved by finding the extremum of an appropriate time–domain metric [6, 1]. The demodulation window for the discrete Fourier transform (DFT) in the receiver is positioned according to this initial frame synchronization estimate. Unfortunately, it is too inaccurate such that interfering samples from adjacent OFDM symbols are included in the window which leads to degraded demodulation performance. Concludingly, it is highly desirable to improve the initial estimate by a second frame synchronization stage, like it is proposed in this paper. This work was supported by Ericsson Eurolab Deutschland GmbH in N¨ urnberg and is funded in parts by the national research initiative ATMmobil of the German Ministry for Research and Education (BMBF).
The paper is organized as follows: After description of the transmission model and a fundamental channel parameter in Section 2, the frame synchronization improvement algorithm is given in Section 3 together with the derivation of its optimality for differential demodulation in frequency direction. Section 4 provides simulation results and Section 5 is dedicated to conclusions. 2. 2.1.
Transmission Model
Orth. Frequency–Division Multiplexing (OFDM)
The transmission of one single OFDM symbol generated with a D–point inverse DFT (IDFT) is considered. Subcarrier number ν is modulated by the complex–valued signal point Aν . The subcarriers are arranged around zero so that −D/2 ≤ ν < D/2. We assume the signal points for all Du (≤ D) active subcarriers to be taken from the same zero–mean signal set. The D −Du virtual subcarriers [9] are set to zero. The OFDM symbol is equipped with a guard interval [9] with a total of Dg samples. In our case, they are split up into a cyclic prefix and a cyclic postfix of Dpr and Dpo samples, respectively. The reason for the simultaneous use of a pre- and postfix lies in the estimated window position which will be systematically delayed when the proposed estimator is used. This systematic misplacement necessitates the cyclic postfix. The discrete–time complex baseband transmit samples 1 sρ = √ D
D/2−1
X
2π
Aν e+j D νρ , −Dpr ≤ ρ < D+Dpo (1)
ν=−D/2
are generated by a D–point IDFT plus cyclic repetition of the respective samples to form the pre- and postfix. 2 The average signal power is E{|sρ | } = σs2 = Es /T , where Es is the average energy per channel symbol. A dispersive channel is assumed which is modelled with the overall discrete–time channel impulse response (CIR) hρ , ρ = 0, . . . , De − 1 with excess length De . The sample sequence sρ is convolved with the CIR to yield the noiseless receive signal r˜ρ . Perfect frequency synchronization is assumed in the receiver. At the receiver input, samples nρ of additive white Gaussian noise are added to
obtain the received sample rρ = r˜ρ + nρ . The CIR is energy–normalized such that D e −1 X
2
|hρ | = 1
(2)
not the autocorrelation function of the channel gains. The latter would be obtained by averaging CH [ν0 ] over all channel realizations. With (5), (6) is rewritten as
ρ=0 def
2
and, thus, E{|˜ rρ | } = σs2 . The noise power is σn2 = 2 E{|nρ | } = N0 /T , where N0 is the one–sided power spectral density of the white noise. The receiver signal–to– noise power ratio (SNR) is Es /N0 = σs2 /σn2 . A coarse frame synchronization unit in the receiver locates the beginning of the useful part of the OFDM symbol at discrete time ξt . We refer to ξt as normalized time offset (NTO) which corresponds to a left–shift of the signal by ξt samples. Clearly, ξt = 0 represents the “perfect” frame synchronization in the sense of natural positioning of the demodulation window. It will be found later that this time position is not optimum when we consider the overall error performance with conventional differential demodulation in frequency direction. The OFDM receiver yields the noisy subcarrier amplitudes (−D/2 ≤ ν < D/2) via DFT processing as Yν Y˜ν
= =
D−1 2π 1 X √ rρ+ξt e−j D νρ = Y˜ν + Nν D ρ=0
Aν Hν e
+j 2π D ξt ν
,
D e −1 X
2π
(5)
ρ=0
and Dpr and Dpo are assumed to be sufficiently large such that even with ξt 6= 0 the linear convolution of noiseless OFDM signal and CIR still is equal to their circular convolution [9]. From (4) follows that ξt 6= 0 causes a linear phase ramp across the subcarrier axis ν. 2.2. Channel Correlation Parameter The cyclic correlation sum of the channel gains1 1 CH [ν0 ] = D def
D/2−1
X
Hν∗ H(ν+ν0 ) mod D
(6)
ν=−D/2
is a characteristic parameter for a specific CIR. So far, the CIR is no stochastic process and, hence, CH [ν0 ] is 1 Here,
ρ1 =0 ρ2 =0
=
D e −1 X
2π
h∗ρ1 hρ2 e−j D ν0 ρ2 2π
|hρ | e−j D ν0 ρ . 2
D/2−1 1 X j 2π ν(ρ1 −ρ2 ) e D D ν=−D/2 | {z } = δ[ρ1 −ρ2 ]
(7)
ρ=0
Clearly, δ[k] represents the unit impulse function, which is obtained due to the sum orthogonality of the exponential functions. Hence, CH [ν0 ] represents a channel characteristic, which depends purely on the delay power profile of the current CIR. P De −1 2 |hρ | = 1 follows and, According to (2), CH [0] = ρ=0 thus, it is obvious that |CH [ν0 ]| ≤ 1 ∀ν0 . The argument of CH [ν0 ] is non–positive. For ν0 De ¿ D/4, (7) is approximated2 by CH [ν0 ] ≈
D e −1 X ρ=0
(4)
hρ e−j D νρ
D e −1 D e −1 X X
2
|hρ |
¶ µ 2π 2π ξc , (8) 1 − j ν0 ρ = 1 − jν0 D D
(3)
where Nν is the resulting noise (sample) in subcarrier ν and Y˜ν is the noiseless received subcarrier amplitude. Furthermore, the channel gain at subcarrier ν is Hν =
CH [ν0 ] =
the modulo operator x mod D shall reduce x to the basis interval [−D/2, D/2 − 1].
where the center of power delay gravity (CPDG) parameter is D e −1 X 2 ρ |hρ | . (9) ξc = ρ=0
It follows from this result that arg (CH [ν0 ]) ≈ −ν0 2π D ξc is proportional to the CPDG of the specific CIR hρ . 3.
Frequency–Domain Frame Synchronization
The following frame synchronization algorithm in frequency domain requires some initial coarse frame start position from a robust time–domain metric [6], a Null symbol or a simple maximum power detector. Then, a first DFT is performed according to this rough estimate and an NTO estimate is obtained from correlating subcarrier amplitudes. This estimate is used to readjust the DFT window for final data demodulation. Optionally, an iterative improvement procedure can be implemented. 3.1. The Frame Synchronization Algorithm We intend to exploit the linear phase property in (4) for frequency–domain frame synchronization like it is already investigated in [13]. There, the authors apply linear regression methods to find axis crossings and 2 We
use e+jx ≈ 1 + jx, for x ¿ 1.
slope of the linear phase component across the subcarrier multiplex. From these quantities, they derive estimates for carrier phase and time offset. Clearly, a proper phase unwrapping procedure needs to be implemented for |ξt | > 1. Other publications which exploit this phase property include [12, 4]. Here, the subcarrier transition from subcarrier amplitude Aν to Aν+1 is interpreted to be differentially encoded in frequency direction with some complex–valued information amplitude Iν so that Aν+1 = Aν Iν ,
−D/2 ≤ ν ≤ D/2 − 2.
(10)
Some of the differential subcarrier transitions in (10) are assumed to be estimated in the receiver as Ibν . In the case that they are known (Ibν = Iν ) we call them differential pilots. Equivalently, the respective absolute subcarrier values Aν could be known (e.g., pilot tones) so that Iν can be calculated from two pilot tones. The index set ¯ i = 0, . . . , D − 1 D ¯ cg } (11) I = {ν = − + Dcs i + j ¯ 2 j = 0, . . . , Dcig − 1 expresses that Dcg correlation groups with Dcig (correlations in group) directly adjacent differential information symbols Ibν are used for synchronization. The correlation blocks have to be spaced Dcs > Dcig +1 subcarriers apart such that the single blocks are actually separated. The principle arrangement of differential pilots within a subcarrier multiplex is depicted in Fig. 1. D cig
D cg correlation groups
Iν Aν D cig+1
D cs
subcarrier ν
Figure 1: Differential pilots Iν in an OFDM symbol for the proposed frame synchronization algorithm. Exemplarily, we have Dcig = 2 in the depicted set I. Considering (4), it is obvious that the correlation of adjacent subcarrier amplitudes provides a reasonable value for estimating the relative time offset ξt . For the noiseless received subcarrier amplitudes, one such correlation product reads Y˜ν∗ Y˜ν+1
=
A∗ν Aν+1
| {z }
2π Hν∗ Hν+1 e+j D ξt ,
(12)
=|Aν |2 Iν
and we observe that the linear increasing phase is transformed into a constant phase offset. The dependency on the differential information symbol Iν must be cancelled (modulation removal), e.g., by multiplication with the complex conjugate of an estimate Ibν of Iν . In a non– dispersive channel, where the discrete channel frequency
response is Hν = 1, ∀ν, the perfect estimate of ξt can be obtained from the argument of the expression in (12). We restrict our discussion to two well–known modulation removal techniques and one combination of both: Data Aided (DA): The Iν are known in the receiver such that the synchronization overhead is Dcg Dcig . With a frequency–selective channel, it is advantageous to spread the pilot groups across the frequency axis to combat deep fades. Under the constraint Dcg Dcig = const, a reasonable choice for a high diversity factor is Dcg > Dcig . The DA approach is very robust and works if the ratio ξt /D is considerably smaller than 12 . Decision Directed (DD): The estimator operates on decided differential symbols. With coherent modulation, the differential encoding with Iν in frequency direction can be fictive but, nonetheless, it can be estimated and exploited for frame synchronization. Thus, no dedicated redundancy is needed and DD could be used to estimate the time offset from any information–carrying symbol. DD works satisfactory for large OFDM symbols (D ≥ 128 carriers), moderate signal constellations (e.g., QPSK) and small initial NTOs, only. This is due to 1 when M is the number of phase angles in ξt /D < 2M the signal set. Theoretically, the estimation variance of DD can be better than that of DA, as more differential symbol decisions Ibν can be exploited for estimation without spoiling transmission efficiency. Clearly, the estimate of DD suffers not only from Gaussian noise and interference due to DFT window offsets. Wrong demodulator decisions produce feedback noise so that DD is not as robust as DA. Hybrid Solution: This scheme is a mixture of DA and DD and should be used in conjunction with at least two iteration steps. The first steps only use the known differential pilots for first frame offset estimates. After the according frame adjustments, a final reestimation based on the joint use of pilots and symbol decisions is performed. Due to noise enhancement of non–decision directed estimators, nonlinear modulation removal methods like M th–power law [8, p.357] or [4] are not considered. With the multiplicative modulation removal, the investigated NTO estimator evaluates X def Yν∗ Yν+1 Ibν∗ (13) L = ∀ν∈I
ξbt
=
D arg (L) . 2π
(14)
For the readjustment of the DFT window, the NTO estimate ξbt needs to be rounded to a multiple of the receiver sample spacing.
In [12], an entire training OFDM symbol is used so that Dcg = 1, Dcig = Du − 1. In the case that subcarrier ν = 0 (DC) is not used, we would have Dcg = 2, Dcig = (Du − 2)/2. A parabolic weighting of correlations belonging to one group in (13) would be optimum [2] but, here, we refrain from weighting due the dispersive channel. It requires large correlation block sizes Dcig to make parabolic weighting superior to uniform weighting. To achieve large diversity in dispersive channels we should rather use a large number (e.g., Dcg ≥ 5) of small (e.g., Dcig ≤ 2) widely separated correlation blocks. To analyze the estimator properties, we assume X Hν∗ Hν+1 ≈ Dcg Dcig CH [1] (15) ∀ν∈I
which says that I comprises a representative set of all available subcarriers, so that this reduced correlation sum can be approximated with CH [1], which itself is determined by averaging over all (cyclic) subcarrier transitions as defined in (6). If we assume correct (error– free) modulation removal, the expected correlation reads 2π E {L} ∼ e+j D ξt CH [1] and incorporating this into (14) yields n o (8) D arg (CH [1]) ≈ ξt − ξc , (16) E ξbt ≈ ξt + 2π
the decision variable and arrive at a MMSE frequency– differential demodulation rule. We apply a phase correction coefficient e−jϕ in the differential demodulation process so that we obtain a new noiseless differential decision variable 2π (12) 2 e−jϕ Y˜ν∗ Y˜ν+1 = Iν |Aν | e−j(ϕ− D ξt ) Hν∗ Hν+1 h ³ ´i 2π 2 2 = Iν |Aν | |Hν | + Hν∗ e−j(ϕ− D ξt ) Hν+1 −Hν .
2
|Hν | is real–valued and does not degrade3 the demodulation performance of the desired information Iν . The second component in the rectangular brackets must be interpreted as a “noise–generating” component caused by the frequency selectivity of the channel. This component will actually be the reason for the flattening of the BER performance for the uncompensated demodulation which will be presented in Fig. 4. To optimize the frequency–differential demodulation process, the overall noise power is to be minimized by minimizing the sum of all squared noise–generating factors in one entire OFDM symbol. If all subcarriers are active this is equivalent to minimize the cost function 1 D
D 2π
arg (CH [1]) ≈ −ξc is used. The negative bias where ξc is constant for one specific static CIR. Section 3.3 illuminates that the frame synchronization estimate found by this estimator is MMSE–optimum for differential demodulation in frequency direction. 3.2.
Lower Bound for Frame Synchronization Variance
We want to provide a lower bound for the NTO estimation variance of the estimator in (14). A long calculation def reveals the NTO variance σξ2t = E{(ξbt − E{ξbt })2 } to be lower bounded by 1 DDu 1 1 Du2 1 + 2 . 2 2 4π Dcg Dcig Es /N0 8π Dcg Dcig (Es /N0 )2 (17) This purely theoretical lower limit assumes a perfectly aligned demodulation window without any interference power from preceeding or subsequent OFDM symbols. This is clearly not true for the cases of practical operation. In non–dispersive channels this bound is a very good approximation of the actual variance. σξ2t ≥
3.3.
Optimum Differential Demod. in Frequ. Direction
Now, differential data demodulation in frequency direction is considered from the optimum demodulation point of view. We want to minimize the mean–squared error in
D/2−2
X
¯ ¯2 ¯ −j(ϕ− 2π ¯ D ξt ) H − H ¯e ν+1 ν¯
ν=−D/2 (6)
≈
³ n o´ 2π 2 · 1 − < e−j(ϕ− D ξt ) CH [1] .
(18)
PD−1 PD/2−1 2 2 (We exploited ν=−D/2 |Hν | = D ρ=0 |hρ | = D.) As only Du subcarriers are active, (18) is an approximate cost function. The optimum ϕ maximizes the real part and, hence, a zero–valued argument must be enforced. For MMSE frequency–differential data demodulation, we obtain ϕ=
(8) 2π 2π ξt + arg (CH [1]) ≈ (ξt − ξc ) . D D
(19)
With the optimum choice of ϕ, we obtain 2·(1 − |CH [1]|) as the minimum value which the cost function in Eq. (18) can take. Hence, the channel characteristic |CH [1]| determines an invincible “multipath–induced noise floor” for frequency–differential OFDM data demodulation in dispersive channels. According to (19), the optimum subcarrier phase correction by −ϕ can be accomplished two–fold: 1) If another frame synchronization algorithm enforces the demodulation window to be positioned at ρ = 0, the 3 This only means that the real–valued factor |H |2 does not inν troduce degrading phase errors. It clearly affects the performance in terms of useful receive power, as it represents the amplitude attenuation/amplification introduced by the channel.
1
−→
10
0
10
lower bound first estimate second estimate Dm = 0 (ideal) Dm = 4 Dm = 6 Dm = 8 Dm = 10 Dm = 12
σ ξt
entire phase rotation must be performed on subcarrier amplitudes in frequency domain. 2) When we apply the frequency–domain frame synchronization algorithm from (14) and correct that part which corresponds to an integer number of sample shifts in time domain by shifting the demodulation window, only the remaining (“fractional”) part needs to be corrected in frequency domain. Concludingly, the investigated frame synchronizer inherently introduces an approximately optimum phase rotation which can easily be verified by comparing Eqs. (19) and (16).
−1
10
0
2
4
Simulation Results
8
10
12
14
16
−→
18
20
Figure 2: Stdv of ξbt for multipath channel with initial misplacement Dm . This estimator works with Dcg = 5 and Dcig = 2 (lower bound according to (17)). Dcg Dcig = 10. The initial NTO estimate is fixed as ξt = Dm = 6. It is obvious that it is advantageous to use Dcig = 2, as firstly the bound is lowered and secondly the performance gain is actually achievable for medium SNRs. The variance floor is caused by the same effect as above, and we observe that due to the lower “diversity factor” with Dcg = 5 the “approximation noise” in (15) is more severe. 1
lower bound first estimate second estimate Dcg = 10, Dcig = 1 Dcg = 5, Dcig = 2
10
−→
For simulation, D = 64 with Du = 53 active subcarriers and a guard interval of Dg = 8 is used. The guard interval is partitioned into a prefix of Dpr = 6 and a postfix of Dpo = 2 samples. Depending on the expected CPDG of the CIR, an optimum setting of preand postfix exists. The type of modulation in the subcarriers was 8DPSK in frequency direction and we scattered Dcg Dcig = 10 (known) differential pilots into the very first of six OFDM symbols per burst. A fixed initial misplacement ξt = Dm is used to start the (iterative) frequency–domain improvement of the NTO. A discrete–time exponentially decaying channel power delay profile of 8 samples length is applied which decays with −3 dB per T –spaced tap, i.e., the average power of each complex–valued, randomly generated and uncorrelated Rayleigh fading tap is halved from one tap to the next in positive time direction. This is often used to model an indoor radio communications channel. The taps are scaled jointly after generation such that they fulfil (2). The CIR is static during the entire burst. In Fig. 2, we demonstrate the achievable standard deviation (stdv) for the first and second (iterated) frequency– domain NTO estimate versus channel SNR with respect to the lower bound. The initial NTO estimate is parameter. The stdvs are dramatically decreasing during the iterative process and the lower bound is approached for medium SNRs. The estimator exhibits a sufficient large lock–in range for initial NTOs at least up to Dm = 12. For practical systems with such short OFDM symbols, stdvs σξt ≤ 0.5 can be considered to be sufficient. There is a flattening for large SNRs which is due to the fact that only 10 differential pilots are used, whereas E{ξbt } is calculated using the actual value of CH [1] from (7). Casually speaking, this variance floor represents the “noise” of the approximation in (15). In a non–dispersive channel, no variance floor will be encountered for Dm = 0 and the bound becomes a perfect asymptote. In Fig. 3, we compare stdvs for two alternative distributions of differential pilots with identical overhead
6
10 log10 (Es /N0 ) [dB]
0
10
σ ξt
4.
−1
10
0
2
4
6
8
10
12
10 log10 (Es /N0 ) [dB]
14
16
−→
18
20
Figure 3: Stdv of ξbt for multipath channel with Dm = 6. Dcg = 10, Dcig = 1 is compared vs. Dcg = 5, Dcig = 2 (lower bounds according to (17)). For the results given in Fig. 4, we used a convolutional code with 64 states, rate 1/2 and Gray–mapping onto 8DPSK symbols. The receiver performs soft–decision Viterbi decoding and the error rates prior to decoding (BER) and after decoding (BRER) are plotted versus channel SNR to compare differential encoding across time direction to the frequency direction. For all simulations ξt = 0, i.e., a “perfect” NTO, is used to separate the effects of synchronization from the power efficiency of demodulation and forward error correction. The dif-
ferential encoding in time direction has an advantage over the frequency direction in static dispersive channels, as the demodulation in frequency direction suffers from the invincible degradation according to the non– zero cost function in (18). But clearly, the inefficiency for the transmission of a redundant phase reference symbol in the case of differential encoding in time direction can be high for very short bursts and this loss is not incorporated here; depending on the burst length, differential encoding in frequency direction might become more efficient. We distinguish between two approaches for differential demodulation in frequency direction: Conventional demodulation employs ϕ = 0, wheras MMSE–optimum demodulation applies a phase correction according to (19), and we clearly see the advantage of the MMSE– optimum demodulation rule which minimizes the “noise” in the decision variables. It must be mentioned that the MMSE–demodulation is automatically performed when the frequency–domain NTO estimator is in operation and the BRER curve for MMSE, in Fig. 4, is approached within 0.2 dB with operational synchronization. −1
BER, BRER
−→
10
−2
10
−3
10
−4
10
8DPSK, time 8DPSK, frequ. 8DPSK, frequ., MMSE BER (uncoded) BRER (coded)
−5
10
−6
10
4
5
6
7
8
9
10
11
12
13
14
15
16
10 log10 (Es /N0 ) [dB]
17
18
19
20
21
22
−→
Figure 4: Bit error performance of (un)coded 8DPSK across time vs. across frequency (with conventional and with MMSE demodulation) over multipath. 5.
Summary and Conclusions
In this paper, a frequency–domain frame synchronization algorithm for OFDM is investigated and a lower bound for the NTO estimation variance is provided. The estimate is biased according to the CPDG of the CIR. At first glance, this property appeared undesirable, but it enables an MMSE–optimum differential demodulation in frequency direction. Due to the frame synchronization bias, the cyclic prefix should be reduced in favour of a postfix. The combination of differentially encoded modulation across the subcarriers together with this frame synchro-
nization algorithm leads to a robust and easy implementable burst mode transmission system. Especially, if the bursts are very short or even consist of single OFDM symbols, such a system is very efficient. References [1] A. Czylwik. Low Overhead Pilot–Aided Synchronization for Single Carrier Modulation with Frequency Domain Equalization. In Proceedings of the Global Telecommunications Conference (GLOBECOM’98), pages 2068– 2073, Sydney, Australia, November 1998. [2] S. Kay. A Fast and Accurate Single Frequency Estimator. IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.37,no.12:pp.1987–1990, 1989. [3] A. S. Macedo and E. S. Sousa. Coded OFDM for Broadband Indoor Wireless Systems. In Proceedings of the International Conference on Communications (ICC’97), Montreal, Canada, 1997. [4] B. McNair, L. J. Cimini, Jr., and N. Sollenberger. A Robust Timing and Frequency Offset Estimation Scheme for Orthogonal Frequency Division Multiplexing (OFDM) Systems. In Proceedings of the Vehicular Technology Conference (VTC’99–Spring), pages 690– 694, Houston, Texas, USA, 1999. [5] P. H. Moose. A Technique for Orthogonal Frequency Division Multiplexing Frequency Offset Correction. IEEE Trans. on Commun., vol.42,no.10:pp.2908–2914, 1994. [6] S. M¨ uller–Weinfurtner. On the Optimality of Metrics for Coarse Frame Synchronization in OFDM: A Comparison. In Proceedings of the International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’98), pages 533–537, Boston, Mass., USA, September 1998. [7] K. Pahlavan, A. Zahedi, and P. Krishnamurthy. Wideband Local Access: Wireless LAN and Wireless ATM. IEEE Commun. Magazine, vol.35,no.11:pp.34–40, 1997. [8] J. G. Proakis. Digital Communications. McGraw–Hill, Inc., 1995. [9] H. Sari, G. Karam, and I. Jeanclaude. Transmission Techniques for Digital Terrestrial TV Broadcasting. IEEE Communications Magazine, vol.33,no.2:pp.100– 109, 1995. [10] T. M. Schmidl and D. C. Cox. Robust Frequency and Timing Synchronization for OFDM. IEEE Trans. on Commun., vol.45,no.12:pp.1613–1621, 1997. [11] M. Speth, F. Classen, and H. Meyr. Frame synchronization of OFDM systems in frequency selective fading channels. In Proceedings of the Vehicular Technology Conference (VTC’97), pages 1807–1811, Phoenix, Arizona, USA, 1997. [12] B. Stantchev and G. Fettweis. Burst Synchronization for OFDM–based Cellular Systems with Separate Signaling Channel. In Proceedings of the Vehicular Technology Conference (VTC’98), Ottawa, Canada, May 1998. [13] S. Zaman and K. Yates. Multitone Synchronization for Fading Channels. In Proceedings of the International Conference on Communications (ICC’94), pages 946– 949, New Orleans, USA, 1994.
