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CHAPTER 5
SYMBOL TIMING ESTIMATION IN MULTIBAND
OFDM BASED UWB SYSTEM
5.1 INTRODUCTION
Multi-band orthogonal frequency division multiplexing
(MB-OFDM) based transmission for ultra-wideband (UWB) is commercially
successful for high speed Wireless Personal Area Networks (WPAN). It
offers enhanced coexistence with traditional and protected radio services by
dynamical turn on/off of subcarriers. The distinct feature of MB-OFDM is the
use of Zero Padding (ZP) instead of Cyclic Prefix (CP). In a CP based
OFDM, there exists a correlation between CP and OFDM signal. This
manifests in the form of ripples in the Power Spectral Density (PSD) of the
transmitted signal and hence results in additional power back-off. It has been
reported that the use of ZP has reduced the ripples in PSD to zero (Batra et al
2003). In MB-OFDM, the availability of varying channel responses across
different sub bands provide diversity gain. However, MB-OFDM system is
highly sensitive to timing synchronization errors causing Inter Carrier
Interference (ICI) as in OFDM systems. Symbol timing estimation in MB-
OFDM based UWB is difficult due to the presence of strongest path at
delayed instant. Hence signal processing algorithm to estimate and correct the
timing has to be developed.
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5.2 LITERTURE REVIEW
Symbol timing estimation in MB-OFDM based UWB system
corresponds to locating the start of the OFDM symbol in a band. There are
two stages namely, a coarse timing estimation and a fine timing estimation.
Coarse estimation finds the approximate start of the OFDM symbol and the
fine estimation refines it to obtain an accurate estimation. Most of the data
aided algorithms for symbol timing estimation are based on the preamble
structure defined in IEEE 802.15.3a. The UWB channel is characterized with
dense multipath and the first arriving path need not be the strongest path. This
complicates the timing estimation for conventional correlation based
algorithms as they lock into the significant path.
Wee et al (2005) proposed an algorithm based on distinguishing the
first significant path to address the critical issue of symbol timing
synchronization in UWB MB-OFDM systems. It pinpoints the frame
synchronization sequence and the start of its fast Fourier transform (FFT)
window by accumulating multipath energies and then discerning for first
significant multipath component through threshold comparison between
consecutive accumulated energy samples.
Berger et al (2006) addressed synchronization issues in the
presence of frequency selective channels. They rely on the algorithm, for
coarse timing to capture a data block that contains circular convolution
between the channel and the OFDM data symbols. They then apply the
maximum-likelihood (ML) principle for joint channel and delay estimation.
The mathematical analysis of fractional timing errors in MB OFDM based
UWB system in terms of signal to interference ratio (SIR) is derived by Zhou
et al (2006).
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Jacobs et al (2007) presented a preamble-based low complexity
synchronization method. It consists of sync detection, coarse timing
estimation, fine timing estimation. The distinctive features of MB-OFDM
systems and the interplay between the timing and carrier frequency hopping at
the receiver are judiciously incorporated in the synchronization method. The
Probability Density Functions (PDFs) of the delays of the UWB channel paths
are derived which are used in optimizing the synchronization method.
Sen et al (2008) studied wideband channel delay characteristics and
delay parameters are found considerably different over frequency
bands 3.1-4.6 GHz. Based on this observation, an adaptive timing
synchronization scheme (ATS) which estimates and maintains the timing
delays of each band separately is proposed. Sen et al (2008a) modified his
ATS algorithm by optimal selection of the threshold for timing estimation to
reduce the mean-squared-error (MSE) and increase the synchronization
probability of the estimator. Ye et al (2008) investigated the low-complexity
synchronization design based on auto-correlation-function. The key
component is a parallel signal detector structure in which multiple auto-
correlation units are instantiated and their outputs are shared by other
functional units in the synchronizer, including time-frequency pattern
detection, symbol timing, carrier frequency offset estimation and correction
and frame synchronization. Berger et al (2008) developed a fine
synchronization algorithm for multiband OFDM transmission in the presence
of frequency selective channels. This algorithm is based on Maximum a
Posteriori (MAP) joint timing and channel estimation that incorporates
channel statistical information, leading to considerable performance
enhancement relative to existing maximum likelihood (ML) approaches.
Sen et al (2008) proposed a low-complexity correlation based
symbol timing synchronization. The correlation based scheme attempts to
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locate the start of the FFT window during frame synchronization sequence
(FS) of the received preamble by firstly correlating the received samples with
known training samples and then identifying the first significant multi-path
component by comparing the correlated samples of two consecutive OFDM
symbols against a predefined threshold. Performance of the proposed
algorithm is measured through MSE of timing estimation and probability of
synchronization.
Zhang and Lau (2008) proposed a three-stage cross correlation-
based timing synchronization algorithm for multiband-OFDM UWB systems.
By utilizing the time-domain base sequence of the packet/frame
synchronization preamble, the algorithm first performs a coarse estimation on
the reference symbol boundary, followed by a fine estimation on locating the
exact boundary of the FFT window for the symbol, and a final check if the
exact boundary is within an inter-symbol interference-free zone.
Xiao et al (2011) proposed a low hardware consumption. Most of
the existing algorithms for symbol timing synchronization cost a huge number
of operations on cross-correlation or energy computing, so that, a great
amount of multipliers and logic resources would be consumed if these
algorithms are implemented on FPGA. The scheme proposed locates the
strongest multi-path by sign cross-correlation and then finds the start of FFT
window.
In summary, the methods available in literature for timing
synchronization in MB-OFDM systems essentially has two steps namely
coarse synchronization and fine synchronization. The performance of the
algorithms are measured in terms of mean square error, probability of correct
detection and computational complexity. There exists tradeoff between mean
square error and computational complexity.
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5.3 ISSUES IN TIMING SYNCHRONIZATION OF MB-OFDM
SYSTEM AND PROBLEM FORMULATION
It is observed that the performance of timing estimation algorithms
for ZP MB-OFDM essentially depend on the manual thresholding. This
results in performance degradation. The extension of timing estimation
algorithm for OFDM to ZP multiband OFDM is not straight forward due to
presence of colored noise which results when overlap add operation is
performed for ZP OFDM.
The residual timing errors left uncorrected by timing
synchronization algorithms are integer errors. The sensitivity of MB-OFDM
system to the residual integer error need to be analyzed.
The algorithms that are developed for timing estimation in
MB-OFDM based UWB systems perform reliably at high SNR. However, the
UWB systems are operated at near 0dB SNR due to the FCC regulations.
Hence, it is required to develop algorithm which performs reliably at low
SNR.
The UWB devices have to coexist with the existing licensed
narrowband communication devices. The performance of existing timing
estimation algorithms deteriorate due to the presence of narrowband
interference. Moreover the band hopping for multiple UWB devices creates
MAI. Hence robust timing estimation algorithm against the narrowband
interference and MAI has to be developed.
The presence of CFO affects the performance of timing estimation
algorithms. Hence robust timing estimation algorithm in the presence of CFO
has to be developed.
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Among the aforementioned issues, this chapter addresses the fine
timing estimation in colored noise and the characterization of residual timing
error. A maximum likelihood framework for the fine timing estimation in
colored noise is proposed. It utilizes the delay embedded in the estimated
Channel Impulse Response (CIR) in time domain. An exact mathematical
analysis for the effect of residual timing error in terms of signal to
interference ratio (SIR) in MB-OFDM is derived. The performance of the
proposed method is analyzed in terms of probability of correct timing
detection. It is observed that the proposed technique has better probability of
correct timing detection performance compared to the existing algorithm, in
all the UWB channel models proposed by IEEE 802.15.3a working group for
UWB systems.
5.4 SYSTEM MODEL FOR MULTIBAND OFDM
MB-OFDM system transmits data across multiple sub-bands and
provides frequency diversity. OFDM symbol in each band consists of N data
subcarriers, G number of zero guard samples with pre sufG G G , where
sufG and preG are the number of zeros appended as suffix and prefix of the
OFDM symbol to overcome the multipath inter symbol interference (ISI) and
to allow sufficient time for the switching of oscillators between multiple
bands respectively.
The low-pass-equivalent time-domain received vector in the thi
band of UWB system is given by,
' ' ' ;i i ii B1 i Ny x h w (5.1)
where 'i
x is the ( ) 1N G vector consisting of N transmitted training
sequence appended with G zeroes, ih is the 1C channel impulse response
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and represents convolution. Discarding the last C number of zeros in the
linear convolution, iy is a ( ) 1N G vector and 'i
w is the ( ) 1N G noise
vector with variance N+GI . BN is the number of sub bands.
5.5 SYMBOL TIMING ESTIMATION IN MB-OFDM SYSTEMS
In this section a brief outline of Li method (Li et al 2008)of symbol
timing estimation in MB-OFDM system is discussed followed by the
presentation of the proposed method.
5.5.1 Li method of timing estimation for MB-OFDM system
Li method suggests a three step procedure for the timing estimation,
namely a sync detection, coarse timing estimation and a fine timing
estimation. In sync detection the arrival of a preamble is detected. In coarse
timing estimation an initial estimate of the start of the OFDM symbol is
calculated and the errors in coarse timing estimation are corrected in fine
estimation. Let ( )qr k is the thk received sample at q-th band. The auto
correlation of the received samples is given in Equation (4.13a).
The sync detection is done in the first two preambles of part a by
finding the sample index in which the magnitude of the correlation metric is
above a predetermined threshold SDM . The correlation metric is defined as
{ , (1)} { }
*( ) ( (1))q pl
q qk k d t
k r k r k d (5.1a)
Let SDk be the sample index of the sync detection. Given the SDk
and k , the course timing estimation algorithm also uses the part a of the
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preamble pattern and calculates a rough timing estimate by using the
Equation(5.1b)
arg maxSD
CTK k k M
k k (5.5b)
The fine timing estimate is calculated by using the part b of
preamble. The fine timing estimate FTk is given by
1 2
3
1*( ) ( (1))arg max
CT a CT a
FT q qqK D W k K D W
k r k r k d (5.1c)
where , aD are the time adjustment factor, length of part a preamble
respectively and 1W and 2W are the window lengths.
As Li method is based on correlation metric, it results in error in
fine timing calculation when there exist delayed strongest multipath in
channel under consideration. Hence, fine timing based on energy of the
estimated channel impulse response is proposed.
5.5.2 Proposed maximum likelihood estimation for fine timing
estimation in MB-OFDM system
The initial operation in a MB-OFDM receiver is to determine the
start of the OFDM symbol in a specific band. This is performed in two steps,
namely coarse synchronization and fine synchronization. The coarse
synchronization is performed using a correlator to yield an initial timing
estimate (Li et al 2008). It provides an approximate estimate of the start of the
OFDM symbol in a band.
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Consider 1sufN G received signal vector chosen from the
start of the OFDM symbol specified by coarse synchronization. This vector is
considered to be having a timing error of ‘ d ’samples. A simple model with
the timing error is developed by adding the last sufG number of samples with
the first sufG number of samples of the received vector. The resulting
1N vector in ith band with the timing error of ‘ d ’is represented as
;1di i i Bi Ny XJ Th w (5.2)
where X is a N N circulant matrix of training samples,
1 ( 1)
( 1) ( 1) 1
1N
N N
0J
I 0 is the N N circular shift matrix and
( )
C
N C C
IT
0 is
the N C tail zero insertion matrix. diJ Th represents the circular shift of ih
with delay d and diXJ Th is the circular convolution between training
samples and delayed impulse response. iw is the colored noise in ith band
with a covariance matrix iwwR . The noise is colored due to overlap and adds
operation. The circular cross correlation of the received vector iy with the
training sequence is given by,
di i i
H H HX y X XJ Th X w (5.3)
The training sequence is approximately flat in spectrum
(Batra 2004) and HX X can be approximated as NNI . Hence Equation (5.3)
can be represented as
di N i iNH HX y I J Th X w (5.4)
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The term iHX w in equation (5.4) is the colored noise with
covariance matrix iww
HXR X . Though equation (5.4) appears to be traditional
cross correlation, it gives knowledge about the delay embedded in the channel
impulse response. The estimate of channel impulse response with a timing
error of ‘ d ’ is given by,
1ˆ 1d di i i i i N
NH
Bh X y J Th w (5.5)
Where iw is colored noise with covariance 21 ,1i i
ww ww i NN
HBR XR X .
Assuming that the channel coefficients of each band are uncorrelated and d
as deterministic unknown, the estimate of the channel impulse response
vector ˆ dih for 1... Bi N is Gaussian distributed with mean zero and covariance
matrix ihhR . The likelihood function for the estimation of d is given by
1
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1ˆ ˆ ˆ ˆ( ,..., ; ) expdet
B B
B
N N Hd d d i dN i hh ii
ii hh
dh h h R hR (5.6)
Where
21ˆ ˆ , 1
Hi d d i ihh i i hh wwE i N
Nd H -d H
BR h h J TR T J XR X (5.7)
Here, ihhR is the correlation of the thi band channel. The log
likelihood function of Equation (5.6) is
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1
111
1ˆ ˆ ˆ ˆln ( ,..., ; ) lndet
B B
B
N N Hd d d i dN i hh ii
ii hh
dh h h R hR
(5.8)
Since UWB channel taps are uncorrelated (Molisch 2003), ihhR is
independent of d . Hence considering the second term in equation (5.8), the
maximum likelihood (ML) estimate of delay d is given by,
1
1
ˆ ˆ ˆarg minBN Hd i d
i hh id i
d h R h (5.9)
The estimation of ‘ d ’ is computationally complex due to1i
hhR .
The close observation of ihhR reveals that it is diagonally dominant. For a
diagonally dominant auto correlation matrix, the matrix inversion can be
approximated by inverting its diagonal elements (Molisch 2003). Considering
diagonal elements of ihhR as 0 1 ( 1), ....i i i Nr r r and 1 0 1 ( 1)
ˆ , ....Td
i i i Nh h hh , the
optimal estimate of ‘ d ’ for the MB-OFDM based UWB system is given by,
21
1
ˆ arg minBN d C
in
d i n d in
hd
r (5.10)
Where C is the length of the impulse response.
5.6 SIR ANALYSIS IN THE PRESENCE OF RESIDUAL
TIMING ERROR
In spite of coarse and fine synchronization, there exists a residual
timing error ‘m’ in the received vector due to noise. This causes ICI and ISI in
the OFDM symbol in a band. In this section, a closed form expression for the
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resultant SIR due to residual timing error is derived. In this analysis, the
additive noise is not considered to characterize the ICI and ISI. Figure 5.1
shows the received ZP-OFDM symbol without timing error. In a ZP-OFDM
receiver, the samples contained in the sufG region are added with the first
sufG samples of the ZP-OFDM symbol (Zhou et al 2007). It results in the
circular convolution of transmitted sequence with the CIR and is denoted
as nz , 0 1n N . Figure 5.2 shows the received ZP-OFDM symbol with
the timing error of ‘m’. The terms 0 1 2 1, , ,....... mz z z z are missed due to timing
error. The symbol with timing error consists of 1 2 1, , ,.......m m m Nz z z z and the
samples 1 1, ,.......N m N m Ns s s which are elements in sufG region.
Figure 5.1 ZP-OFDM symbols without timing error
N
N
ZN-1
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Figure 5.2 ZP-OFDM symbol with timing error
Let nv be the thn sample of the ZP-OFDM symbol with the residual
timing error. It can be represented as,
( )Ln n m n nv z s (5.11)
where,
1 0 10 1
0 0 1( ) 1
n
n curr
n N mN m n N
n N ms
z n N m N m n N
where ( )currz n represents the thn sample in the zero guard region of the
current OFDM symbol. Let
1 2 1 0 1 1 0 1 1
0 1 1 1 2 1 0 1 1
, ,......, ; [ , ,......, ] ; [ , ,....., ]
[ , ,......, ] ; , ,......, ; [ , ,....., ]
T T TN C N
TT TN N N
v v v h h h x x x
s s s z z z
v h x
s z
N
N
N
ZN-1
ZN-1
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Let V,H,X,S, ,Z be the N point FFT of v,h,x,s, , z respectively.
The thk element of is given by,
1
0
1 exp 2 /1 1
1 exp 2 /exp 2 /0
N m
k ni
j mk Nk N
j k Nj nk NN m k
(5.12)
The thk element of Z is,
exp 2 /k k kZ H X j mk N (5.13)
Using Equations (5.12) and (5.13), the thk element of V is given
by,
( * )k k k kV Z S
01 exp 2 /k kH X j mk NN
1
( ) ( )1
1 exp 2 ( ) /N
N
n k n N k n kn
H X j m k n N SN
(5.14)
In Equation (5.14), the first term corresponds to signal component,
second and third term correspond to ICI. Let
1
( ) ( )1
1 exp 2 ( ) /N N
N
k n k n k nn
I H X j m k n NN
(5.15)
where iH and iX ’s are assumed to be jointly and individually uncorrelated.
The total interference power is represented as,
2 2 2 *2Rek k k k k kI S I S I S (5.16)
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The first term in right hand side of Equation (5.16) is determined
to be
1 12 2 2* 2 2 2 2
2 21 1
1 1N N
k k k n h x h x nn n
I I IN N
(5.17)
Using Equation (5.12),
1 12 2 2 2
01 0
( ) ( )N N
n nn n
N N m N m m N m (5.18)
Substituting Equation (5.18) in Equation (5.17), the expression for 2
kI is written as
2 2 2 2[ ] ( ) /k h xI m N m N (5.19)
The second term in right hand side of Equation (5.16) is determined
to be
''
' '
2 21 1 1 122 2
0 01 1k
m C m Cx x
k hki ik i k i
S hN N
(5.20)
The cross correlation between kI and kS is calculated as,
''
2 1 1* 2
0 1
( ( ))k
m Cx
k k hi k i
I S N mN
(5.21)
Substituting Equations (5.19) to (5.21), Equation (5.16) is
determined to be
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''
2 1 12 2 2
22 0 1
2
( 2 )( )[ ]
k
m Cx
h x hi k ih
k k
N mN mI S
N (5.22)
Where 2ih is the power of thi channel tap.
Hence, the signal to interference ratio (SIR) is
''
2
1 12
20 1
( )SIR( 2 )( )
k
m C
hi k ih
N mN mN m m
(5.23)
5.7 RESULTS AND DISCUSSION
In this section, the performance of the proposed algorithm is
analyzed using Monte Carlo simulation. The simulation parameters are as
given in Table 5.1.
Table 5.1 Simulation parameters for symbol timing estimation
Sl.No Parameters Value
1. No. of Subcarriers, L 128
2. Length of suffix, sufG 32
3. Length of prefix, preG 5
4. Training sequence TFC1(IEEE 802.15.3a)
5. Channel model CM1,CM2, CM3 and CM4 (Molisch 2003)
6. Number of UWB realizations 100
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The performance of the proposed algorithm is compared with the Li
method (Li et al 2008) where in the fine timing is estimated by finding the
timing instant at which maximum of timing metric occurs. The timing metric
is a sum of magnitude of autocorrelation of the received sequences in three
consecutive bands. The additional parameters required to simulate the method
by Li et al (2008) are correlation window length and the correction constant
term .The value of depends on the type of channel. For CM1 model, it is
chosen to be 4.The correlation window length is chosen to be 128.
Figure 5.3 shows the probability of correct timing detection of the
proposed algorithm in CM1 channel by varying the number of bands
considered for detecting the fine timing. It is observed that the probability of
detection increases exponentially at low SNR region and reaches a constant at
high SNR region. The SNR required to attain a detection probability of 0.9 is
12.2dB when the number of bands is one. However, it is attained for a SNR
of 9.4dB and 8.6dB itself when the number of bands is two and three
respectively. The exponential gain in the SNR as the numbers of bands are
increased is similar to the gain obtained in detection of data transmitted
through fading channels with diversity. Hence, the improvement in the
probability of correct timing detection is due to the inherent frequency
diversity in MB-OFDM. Since the channel impulse response in each band is
independent, there is a possibility that the first path in any one of the bands
may not be in deep fade. It is noted that the proposed algorithm with three
band provides a SNR gain of 10dB for a detection probability of 0.6 when
compared to method by Li et al( 2008).This performance improvement is due
to the usage of delay embedded in CIR.
Figures 5.4 - 5.6 show the detection performance of the proposed
algorithm in CM2, CM3 and CM4 channel models respectively. For CM2, CM3
and CM4 channel models, the value of is chosen to be 8, 9 and 14 respectively.
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Figure 5.3 Detection performance of proposed algorithm in CM1 channel model
Figure 5.4 Detection performance of the proposed algorithm in CM2 channel model
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Figure 5.5 Detection performance of the proposed algorithm in CM3 channel model
Figure 5.6 Detection performance of the proposed algorithm in CM4 channel model
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Figure 5.7 shows the effect of residual timing error in
ZP MB-OFDM based UWB systems in terms of SIR in CM1 channel model.
It is observed that, when the timing error is more than 5 samples, the SIR
drastically reduces to less than 20dB indicating that a timing error of more
than 5 samples should not be left uncorrected. The simulation result has a
0.5dB difference in SIR, when compared to the theoretical analysis. This is
due to the consideration of channel impulse response as uncorrelated
Gaussian for analytical tractability.
Figure 5.7 Effect of timing error in terms of SIR
5.8 SUMMARY
In this chapter, a preamble based novel fine symbol timing
estimation technique for MB-OFDM based UWB systems is proposed. An
analytical expression for the effect of residual timing error in terms of SIR is
derived. It is observed that the proposed algorithm provides significant
improvement in probability of correct timing detection when compared to
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existing method. The proposed algorithm with single band hopping provides
a SNR gain of 10dB for a detection probability of 0.6 when compared to
method by Li et al( 2008).It is also observed that the timing performance
increases with the number of bands, due to the inherent frequency diversity.