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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

111

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.