This paper proposes an improved carrier frequency offset (CFO) and sampling frequency offset (SFO) estimation scheme for orthogonal frequency division multiplexing (OFDM) based broadcasting system with cyclic delay diversity (CDD) antenna. By exploiting a periodic nature of channel transfer function, cyclic delay and pilot pattern with a maximum channel power are carefully chosen, which helps to enable a robust estimation of CFO and SFO against the frequency selectivity of the channel. As a performance measure, a closedform expression for the achievable mean square error of the proposed scheme is derived and is verified through simulations using the parameters of the digital radio mondiale standard. The comparison results show that the proposed frequency estimator is shown to benefit from properly selected delay parameter and pilot pattern, with a performance better than the existing estimator.
1. Introduction
O
rthogonal frequency division multiplexing (OFDM) has found many applications in mobile radio communication systems. Because of its excellent characteristics, it enables high data rate transmissions over frequencyselective fading channels. OFDM modulation along with guard interval techniques allows the implementation of single frequency networks in digital broadcasting systems such as the digital audio broadcasting (DAB), digital radio mondiale (DRM), and terrestrial digital video broadcasting (DVBT)
[1]

[3]
. DRM offers better sound quality and more reliable reception compared to analog AM and FM. The DRM consortium proposes to add a new VHF mode to the existing DRM modes
[3]
, with a basic 100KHz bandwidth for the DRM robustness mode E, referred to as “DRM+” in this paper.
DRM+ will be used on FM frequency up to 174MHz
[3]
. The bandwidth used in the DRM+ system, approximately 100KHz, is much smaller than the coherence bandwidth of the channels in typical urban environment that is in the order of 1MHz
[4]
, which results in a severe flat fading condition. Thus, the major challenge in FM band is to cope with severe flat fading conditions. One possible solution to combat flat fading is cyclic delay diversity (CDD)
[5]

[11]
, which has been adopted as one of the multipleinput multipleoutput (MIMO) diversity techniques in a wireless communication system. CDD transforms a multipleinput singleoutput (MISO) channel into an equivalent singleinput singleoutput (SISO) channel with increased frequencyselectivity so that the available spatial diversity is transformed into additional frequency diversity. However, the increased frequency selectivity of channel characteristic can be a problem for post fast Fourier transform (FFT) estimation such as the channel estimation and frequency offset estimation
[12]

[19]
. Although they are proposed for general channel conditions, its accuracy heavily depends on frequencyselective multipath distortions. As reported in
[13]

[19]
, moreover, the postFFT pilotaided carrier frequency offset (CFO) and sampling frequency offset (SFO) estimation scheme is not anymore unbiased in the presence of large SFO over frequencyselective channels. The implementation complexity of synchronization mechanisms is an important topic from a practical point of view and there still remains the potential for a better estimator that is computationally efficient as well as is robust to the fading distortion.
This paper deals with how to determine the amount of cyclic delay and the pilot pattern for an improved CFO and SFO estimation in the OFDMbased DRM+ system with CDD transmit antennas. Cyclic delay and pilot pattern with a maximum channel power are chosen by exploiting a periodic nature of channel transfer function. The mean square error (MSE) of the proposed estimation scheme is numerically derived, and its simple expression is calculated. We show via simulations that such a design improves the robustness of the CFO and SFO estimation scheme with the increased frequency selectivity because of the use of CDD.
The rest of the paper is organized as follows. Section 2 introduces the signal model when the CDD is used in the DRM+ system. In Section 3, we give a brief overview of the conventional CFO and SFO estimation algorithm. A robust CFO and SFO estimation scheme is proposed and its performance is analyzed in Sections 4 and 5, respectively. In Section 6, we present simulation results verifying the MSE analysis and the effectiveness of the proposed scheme, while in section 7 some conclusions are drawn.
2. Signal Models
When the CDD scheme is adopted in OFDM systems, cyclic delayed replicas of the time domain data signal
x_{l}
(
n
) over several transmit antennas are transmitted, thus the
l
th OFDM signal of the
t
th transmit antenna is
where
N
is the number of FFT (IFFT) points, (·)
_{N}
is the modulo
N
operation,
N_{T}
is the number of transmit CDD antennas, and
δ_{t}
is the amount of cyclic delay at the
t
th antenna. As discussed in many literatures
[5]

[11]
, choosing
maximizes delay between transmit antennas, thereby
N
/
N_{T}
is the maximum possible cyclic delay.
Consider a discretetime baseband OFDM system with
N
subcarrier and
N_{g}
guard interval (GI) samples. A frequency offset Δ
_{f}
between the transmit signal and the received signal can be introduced by inaccuracies of the receiver’s local oscillator. After compensating the CFO with estimated CFO
at the acquisition stage, hence, only a residual CFO
and possible SFO Δ
_{s}
will remain during the data section of the frame. A frequency offset will appear as a phase shift
ϕ
(
k
) and is comprised of two parts: residual CFO Δ
_{c}
and SFO Δ
_{s}
. The former is the same for all subcarriers, while the latter contributes linearly with the subcarrier index
k
, i.e.,
ϕ
(
k
) ≈ Δ
_{c}
+
k
Δ
_{s}
[20]
. Under the assumption of a quasistationary channel, the channel is constant during one OFDM symbol interval and intercarrier interference (ICI) due to fading channel variations is negligible. At the receiver end, the GI is removed and the received symbol is demodulated using the FFT operation. Therefore, the FFT output
R_{l}
(
k
) during the
l
th period is given by
[18]

[20]
where
α
(
ϕ
(
k
)) =
sin
(
πϕ
(
k
)) / (
Nsin
(
πϕ
(
k
) /
N
)) is the selfdistortion coefficient of each subcarrier due the frequency offsets Δ
_{c}
and Δ
_{s}
,
X_{l}
(
k
) are the complex numbers placed on the
k
th data subcarrier of the
l
th OFDM symbol,
N_{u}
=
N
+
N_{g}
denotes the OFDM symbol length after GI insertion,
H_{l}
(
k
) is the channel’s frequency response with zeromean and variance
σ
^{2}
_{H}
,
W_{l}
(
k
) is is a zeromean complex Gaussian noise with variance
σ
^{2}
_{W}
during the
l
th symbol period, and
l_{I}
(k) is the ICI term introduced by frequency offset. In the presence of small CFO and SFO,
α
(
ϕ
(
k
)) ≈ 1 and the ICI contribution will be omitted in the following discussion, since its power is very small compared with the additive noise power
[20]
. In (3),
H_{l}
(
k
) denotes the equivalent channel transfer function (CTF), which appears to be a superposition of CTFs from all transmit antennas with the corresponding phase shifts
where ,
H_{l ,t}
(
k
) is the frequency channel response from the
t
th transmit antenna with zeromean and variance
σ
^{2}
_{H}
.
3. Conventional Joint CFO and SFO Estimation Scheme
A postFFT pilotassisted joint CFO and SFO estimation scheme considered in
[17]

[19]
is revisited in this section. Since there is no frequency reference cell (FRC) in the DRM+ system, which is mainly used for frequencyoffset estimation as studied in
[21]
, a gain reference cell (GRC) can be a possible candidate as a pilot. Carrier indices for a timefrequency latticetype GRCs are defined as
[3]
where
p
is integervalued,
N_{s}
is the number of symbol per frame, and
D_{f}
and
D_{t}
are the periodicity of the GRC pattern in the frequency and time directions, respectively. Here, we assume that the number of elements in S , equivalently, the number of pilots in one OFDM symbol, is
N_{p}
. The pilot subcarriers with length
N_{p}
are divided into two sets of S
_{0}
and S
_{1}
. Let S
_{0}
and S
_{1}
denote the set of
N_{p}
/ 2 indices in the left half
k
∈[–
N
/ 2,0) and the right half
k
∈(0,
N
/ 2] of the spectrum, respectively.
By using the consecutive GRCs as pilot symbols, we can effortlessly apply the basic concept in
[14]

[19]
to the DRM+ system. Then, a cumulative correlation on S
_{0}
and S
_{1}
is represented by
wth
where
E_{s}
=
X_{l}
(
k
) 
^{2}
and
ρ
= (
N
+
N_{g}
) /
N
. For simple description, we have assumed that
H_{l}
(
k
) ≈
H_{l+Dt}
(
k
) in (6). As originally suggested in
[17]

[19]
, the estimate of Δ
_{c}
and Δ
_{s}
is respectively computed as
and
where
arg
{·} denotes the argument of a complex number and
k_{iN p}
_{/2+1}
(
g
) stands for the first subcarrier index in
S_{i}
(
i
= 0,1 ), depending on the
g
th GRC pattern. Note that the performance of (8) and (9) heavily depends on the index g of combtype GRC patterns because of the periodic nature of the CTF in the frequency direction. Hence, one can expect that a proper choice of
g
will enhance the performance of (8) and (9). In doing so, we consider the issue of selecting the cyclic delay and pilot pattern with a maximum channel power in the following section.
4. Proposed Joint CFO and SFO Estimation Scheme
In order to improve the performance of the CFO and SFO estimation scheme when the CDD is adopted in the DRM+ system, an efficient frequency estimation scheme by using the periodic nature of overall CTF
H_{l}
(
k
) and multiple uniformly distributed combtype pilots referred to as GRC is suggested in this section. For simplicity, it is assumed that
δ
_{0}
= 0 and cyclic delay between adjacent antennas is equidistant. The power of equivalent CTF in (4) can be expressed as
where
θ
_{t1 ,t2}
is the phase of
H
_{l,t1}
(
k
)
H
^{*}
_{l,t2}
(
k
) By using the fact that the DRM+ system suffers from severe flat fading conditions
[4]
, the channels , {
H_{l ,t}
(
k
)} are assumed to be frequencyflat and one can build the channel
H_{l}
(
k
) that is periodic with a period of
As we can see in (10), the channel magnitude of the received signal heavily depends on the cyclic delay in the second term on the righthand side when the system parameters
N_{p}
,
D_{f}
, and
N
are fixed. If we choose
δ
_{0}
= 0 and
δ
_{1}
=
N
/ 2 as in (2) in the case of twotransmit antenna, one can find that the CTFs in (10) are same at all GRC positions.
In order to improve the accuracy of the frequency estimation scheme, first, one can find a minimum cyclic delay D
_{δ}
=
δ
_{t+1}
–
δ_{t}
as
which meets with the condition
H_{l}
(
k
) ≈
H_{l}
(
k
+
D_{f}
) . Then, we suggest to use an integer multiple of
D_{δ}
as a candidate of cyclic delay between neighboring antennas. Note that an integervalued
m
is a fundamental design parameter that deals with the amount of cyclic delay and the receiver performance, depending on the number of CDD antennas. Since the increase in
m
results in the decrease in the periodicity of the channel in (13), substituting (13) into (11) yields the periodicity of the channel
H_{P,m}
=
D_{f}
/
m
, which still guarantees periodically flatfading condition
H_{l}
(
k
) ≈
H_{l}
(
k
+
D_{f}
) . From the above discussion, the transmitantenna specific cyclic delays between adjacent antennas are set as equidistant delays
where
mD_{δ}
≥
N_{g}
+ 1 should be guaranteed
[8]
.
The amount of cyclic delay is chosen as (13) such that the channel in all pilot positions is periodically flatfading and the number of distinguishable channel coefficient is maximized. For a possible index of GRC positions
a careful selection of
m
in (13) leads to the maximization of the number of distinguishable channel coefficient
H_{l}
(
k
+
gD_{f}
/
D_{t}
) at
k
∈ S . In order to maximize the number of distinguishable channel coefficient denoted by
N_{max}
, the periodicity of the channel
H_{P,m}
=
D_{f}
/
m
should not be reduced to lowest terms, i.e.,
where
Dʹ_{f}
and
mʹ
are obtained by reduction of a fraction to the lowest terms. If
H_{P,m}
=
D_{f}
/
m
is reduced to lowest terms such that
H_{P,m}
=
Dʹ_{f}
/
mʹ
one can see that an integer value
Dʹ_{f}
is a multiple of
H_{P,m}
and
H_{l}
(
k
) ≈
H_{l}
(
k
+
Dʹ_{f}
) , which says that the channel
H_{l}
(
k
) is periodically flat every
Dʹ_{f}
subcarriers. If
H_{P,m}
is not reduced to lowest terms,
H_{l}
(
k
) ≈
H_{l}
(
k
+
D_{f}
) . Since
D_{f}
˃
Dʹ_{f}
, one can expect that the number of distinguishable channel coefficient
N_{max}
becomes smaller.
When the cyclic delay is appropriately chosen in accordance with (14) and (15), one GRC pattern which has the maximum channel power is eventually selected. By sensing all possible channel powers of the GRCs, the estimated GRC position with maximum channel power can be obtained. Since pilot symbols are known at the receiver, the estimated GRC position can be practically estimated by
Based on the selected GRC position according to (16), the proposed estimation scheme can be simply constructed by
and
which are in a form identical to (8) and (9) with g replaced by
respectively. In DRM mode E, four GRC patterns are defined as in (5) and the number of available pilots differs from GRC patterns. In order to provide a framework for a fair comparison, we assume that
N_{p}
is evennumbered. A close observation of (5) obviously indicates that there are two possible pilot configurations: (C1)
N_{p}
= 12 when (
s
)
_{Dt}
= 0,3 (C2)
N_{p}
=14 when (
s
)
_{Dt}
=1,2 .
5. MSE Analysis
For simplicity, we consider the CDD scheme employing
N_{T}
= 2 transmit antennas. Since most of OFDM systems use the powerful channel coding scheme and there is a certain amount of channel correlation between transmit antennas in practical deployments, the transmit antennas with
N_{t}
˃ 2 leads only to a small performance improvement
[5]

[7]
.
When the amount of cyclic delay is chosen according to (14) and one GRC pattern
which has the maximum channel power is selected as in (16), the maximum of the channel can be obtained by
which corresponds to when
cos
(2
πk
(
δ
_{t2}
–
δ
_{t1}
) /
N
+
θ
_{t1 ,t2}
) = 1 in (10). Based on assuming that {
H_{l ,t}
(
k
)} are frequencyflat, thereby we have from (19)
where
and
From (20), one can remark that at high SNR’s
where
x^{Q}
denotes the imaginary part of
x
. Then, the error of the SFO estimate takes the expression
which depends on the estimated GRC
. Since GRCs are uniformly distributed as defined in (5), one can easily see that
for all
Based on
E
{
W_{l}^{Q}
(
k
)}= 0 , (26) is simplified into
which leads to
Based on an approximation at high SNR, the product of two noise terms is negligible. After some straightforward calculations, the noise variance can be obtained by
Since
H
_{l ,0}
(
k
) and
H
_{l ,1}
(
k
) are statistically independent complex Gaussiandistributed random variables, x =
H
_{l ,0}
(
k
)  +
H
_{l ,1}
(
k
)  is a sum of two independent Rayleigh random variables, whose probability density function takes the expression
[22]
Hence we have from (30)
Substituting (31) into (29) yields the MSE expression
where
is the average SNR. The final MSE expression can be obtained by averaging (32) over
since
N_{p}
and
is different according to the estimated
. Similarly, the error of CFO estimate can be expressed as
Based on the fact that
it is found that the second term on the right hand side of (33) is zero. Similar to (32), the MSE of (33) is directly given out without providing the detailed derivation as
6. Simulation Results
In order to verify the usefulness of the proposed estimation scheme and the accuracy of MSE analysis, simulations are performed in the DRM robustness mode E, considering
N
= 213,
N_{g}
= 24 ,
D_{f}
=16
D_{t}
= 4 ,
BW
= 96KHz, and center frequency is 90MHz
[3]
.
Table 1
shows the channel models used in our simulations
[3]
. For antenna configuration, the antennas are placed such that their CTFs can be considered as uncorrelated. The CFO Δ
_{c}
is set to of the subcarrier spacing 2% and Δ
_{s}
= 20ppm. For a fair comparison, we assume that there are two possible pilot configurations: (C1) =12
N_{p}
when (
s
)
_{Dt}
= 0,3 (C2)
N_{p}
=14 when (
s
)
_{Dt}
=1,2 .
DRM channel profiles
MSE of the CFO estimators versus m in CM1 when N_{T} = 2.
MSE of the SFO estimators versus m in CM1 when N_{T} = 2
MSE of the CFO estimators versus m in CM1 when SNR = 25dB, N_{T} = 3 and 4 are used.
Figs. 1
and
2
depict the MSE of the CFO and SFO estimation schemes with respect to
m
defined in (14), respectively, when =
N_{T}
= 2. In order to provide a framework for a fair comparison, the mobile speed is not taken into account in
Table 1
, i.e.,
v
= 0 km/h. As shown in these figures, we can observe that the performance of the proposed CFO and SFO estimation scheme depends on the amount of cyclic delay, while the performance of the conventional one is insensitive to
m
. Since one can see from (16) that
Dʹ_{f}
= 2 for
m
= 8 ,
Dʹ_{f}
= 4 for
m
= 4 ,
Dʹ_{f}
= 8 for
m
= 2 and 6 ,
Dʹ_{f}
= 16 for other
m
’s, the number of distinguishable channel coefficient {
H_{l}
(
k
+ 4
g
)} at subcarrier
k
∈ S (
g
= 0,1,2,3 ) is
N_{max}
=1 for
m
= 4 and 8,
N_{max}
= 2 for
m
= 2 and 6, and
N_{max}
= 4 for others. Therefore, the performance when
m
= 3 , 5, and 7 is better than the other values of
m
’s, which says that there are
m
’s that give the minimum MSE regardless of SNRs. Interestingly, the performance of the proposed scheme when
m
= 4 is nearly the same as that of the conventional scheme. This mainly stems from the fact that the channel
H_{l}
(
k
)when
m
= 4 is periodically flat every
Dʹ_{f}
= 4 subcarriers in accordance with (15), which enables the channel power for all
g
’s in (16) to be identical. As expected, the CFO and SFO estimation scheme endowed with cyclic delay (14) and (15) is capable of robustly estimating the frequency offset in the presence of increased frequency selectivity of the channel caused by the CDD.
Fig. 3
shows the MSE of the CFO estimation scheme with respect to
m
when
N_{T}
= 3 and
N_{T}
= 4 are used. These results are qualitatively similar to those reported in
Figs. 1
and
2
, indicating that
m
’s that give the minimum MSE are also independent of
N_{T}
.
Figs. 4
and
5
present the MSE of the CFO and SFO estimation schemes, respectively, when =
N_{T}
= 2. As discussed in
Figs. 1
and
2
, three representative cases
m
= 2,3,4 are considered in the proposed scheme for the purpose of justifying the choice of
m
for varying SNR. In the case of the conventional scheme, the MSE curves only when
m
= 2 are plotted because its performance is independent of
m
, as confirmed in
Figs. 1
and
2
. Also, we assumed that
v
= 0 km/h. When the channel is frequencyflat as in CM1, the use of the criterion (12) and a careful selection of
m
= 3 that satisfies (15) dramatically improve the performance of (8) and (9), which provides a benefit to the DRM+ receiver with CDD. More importantly, we observe that the analytic curves (32) and (34) are very close to the simulated curves when
m
= 3 , or equivalently
N_{max}
= 4 . This is due to the fact that the maximum channel power in (19) is assumed to be obtained when the number of distinguishable channel coefficient is as large as possible in deriving (32) and (34). When the frequency selectivity is increased as in CM2, on the other hand, the performance gap between the conventional and proposed schemes becomes further smaller, and the MSE performance in the frequencyselective channel is slightly worse than that in CM1. The reason for this is that the frequencyflat assumption used in (20) is not valid in CM2. As confirmed in
Figs. 1
and
2
, the results in
Figs. 4
and
5
justify why we choose
m
= 3 in the proposed approach, which is used in the following example.
MSE of the CFO estimators when N_{T} = 2: (a) CM1 (b) CM2
MSE of the SFO estimators when N_{T} = 2: (a) CM1 (b) CM2.
MSE of the CFO estimators in CM1∼CM3: (a) N_{T} = 3 (b) N_{T} = 4.
Fig. 6
shows the MSE of the CFO estimation schemes when
m
= 3 , taking into consideration the mobile speed defined in
Table 1
. The presence of
v
causes a severe performance degradation for all frequency estimators at high SNR. This primarily arises from the impact of the time selectivity of the channel, which is one of the main disadvantages for frequency estimation in OFDM. Nevertheless, the proposed scheme still outperforms the conventional scheme even for the fast fading.
7. Conclusion
In this paper, the problem of frequency detection for OFDMbased FM systems employing with CDD transmit antennas has been considered. The robust CFO and SFO estimation scheme was derived by properly selecting the amount of cyclic delay and a pilot pattern. To verify the effectiveness of the frequency estimation scheme, the MSE performance of the proposed scheme was theoretically analyzed. It was verified by performance analysis and computer simulations that the proposed CFO and SFO estimation scheme endowed with appropriately chosen cyclic delay and pilot pattern is robust to the frequency selectivity of the channel, when compared to the conventional scheme.
BIO
WonJae Shin received the B.S. and M.S. degrees in School of Computer Engineering from Sejong University, Seoul, Korea, in 2007 and 2009, respectively. He is currently working toward the Ph.D. degree in the School of Computer Engineering from Sejong University, Seoul, Korea. His research interests are in the areas of wireless communications systems design, multicarrier transceivers, and system architecture for realizing nextgeneration communications systems.
YoungHwan You received the B.S., M.S., and Ph.D. degrees in electronic engineering from Yonsei University, Seoul, Korea, in 1993, 1995, and 1999, respectively. From 1999 to 2002 he had been a senior researcher at the wireless PAN technology project office, Korea Electronics Technology Institute (KETI), Korea. Since 2002 he has been a professor of the Department of Computer Engineering, Sejong University, Seoul, Korea. His research interests are in the areas of wireless communications systems design, spread spectrum transceivers, and system architecture for realizing advanced digital wireless communications systems, especially, for wireless OFDM.
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