[62]

CHAPTER 3 TIMING OFFSET

ESTIMATION

iming synchronization is one of the major issue of OFDM system. Imperfect

synchronization destroys the orthogonality of sub-carriers and degrades the performance of

OFDM system. Timing synchronization includes symbol timing offset estimation and correction. This

chapter presents the effects of STO on the performance of OFDM system and different methods of timing

offset estimation. One new method of timing offset estimation has been proposed and compared with

other existing methods.

3.1 INTRODUCTION

The synchronization plays an important role in the design of digital communication system with

either single-carrier (SC) or multi-carrier (MC) modulation schemes and its overall performance metric.

In SC system, the timing synchronization often refers to the symbol or frame synchronization. In MC

(particularly in OFDM) system, the timing synchronization means to find the correct starting point of

DFT window at the receiver.

In an OFDM system, the IDFT and DFT are the fundamental functions used for the modulation

and demodulation of OFDM symbol at transmitter and receiver respectively. In order to demodulate an

OFDM symbol correctly at the receiver using N-point DFT, it is very much required to take exact samples

of transmitted OFDM symbol. The correct starting point of DFT window is required to preserve the

orthogonality among the sub-carriers. There is lot of advantages of OFDM system over SC system

however all these advantages can be useful only when the orthogonality among sub-carriers is

maintained. If one DFT window takes sample of two different OFDM symbol then it will generate ICI

and ISI as shown in next section. Hence it can be stated that the DFT window timing synchronization in

OFDM system is corresponds to symbol timing synchronization in single carrier transmission system.

However, it is much more difficult, because there is always an ‘‘eye opening’’ in each single carrier

modulated symbol, whereas there are many ‘‘zero crossings’’ in each OFDM symbol. Therefore, normal

synchronization algorithms such as zero-forcing cannot be adopted [113]. The effects of timing offset in

the performance of OFDM system is shown in next section.

T

[63]

3.2 SYMBOL TIMING OFFSET ANALYSIS

The symbol timing estimator in an OFDM system is used to determine the starting point of FFT

window at the receiver. By applying timing estimation method on received signal 푟(푛), the starting point

of the FFT window is determined as the sample 푟(훿) , where ′훿 is defined as an integer-valued STO.

The effect of STO (‘δ’) is depends on the location of the estimated starting point of OFDM symbol. Four

different cases of timing offset are shown in Figure- 3.2.1. The effect of timing error can be understood

easily in frequency domain (i.e. after FFT). To show the effects of STO, the effect of CFO has been

neglected for present study [139].

Figure-3.2.1: Four different cases of STO

Case-I: First consider the case when there is no timing error (i.e. timing offset δ = 0). This is the case

when the estimated starting point of OFDM symbol coincides with the exact timing, preserving the

orthogonality among sub-carriers, therefore the OFDM symbol can be perfectly recovered without any

type of interference. In this case, the received sampled signal, after removing the cyclic prefix during the

time of one complete OFDM symbol, can be expressed in the form of a vector as –

푟 = {푟(푖, 0), 푟(푖, 1), … , 푟(푖,푁 − 1)} , (3.2.1)

where, 푟 represents the current vector for FFT operation and 푟(푖, 푛) is the 푛 sample of 푖 OFDM

symbol which can be expressed as –

[64]

푟(푖,푛) = ℎ (휏) 푥(푖,푛 − 휏 ) + 푤(푖,푛),

푓표푟 푛 = −퐺,−퐺 + 1, … , 0, 1, … ,푁 − 1 (3.2.2)

where, 푥(푖,푛) is the transmitted OFDM signal, ℎ (푛) is the impulse response of multipath fading channel,

휏 is the path delay of 푙 path, and 푤(i,푛) is a zero-mean, complex value Gaussian noise process with

variance 휎 . The output of FFT can be written as-

푌(푖, 푝) = 푟(푖,푛) 푒 (– / ) ,

푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.3)

After substituting 푟(푖,푛), it becomes –

푌(푖, 푝) = ℎ (휏) 푥(푖,푛 − 휏 ) + 푤(푖,푛) 푒 ( / ) ,

푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.4)

After separating the AWGN term –

푌(푖,푝) = ℎ (휏) 푥(푖,푛 − 휏 ) 푒— 푗 2 휋 푝 푛푁 + 푤(푖,푛) 푒— 푗 2 휋 푝 푛

푁 ,

( , )

푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.5)

Now, put the value of 푥(푖,푛 − 휏 ) from (2.2.2.2) –

푌(푖,푝) = ℎ (휏) 1푁

푋(푖, 푘) 푒푗 2 휋 푘 ( 푛−휏푙)

푁 푒— 푗 2 휋 푝 푛푁 + 푊(푖,푝),

푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.6)

[65]

After interchanging the summation, (3.2.6) can be expressed as -

푌(푖,푝) = (1/푁) 푋(푖,푘) ℎ (휏) 푒−푗 2 휋 푘 휏푙)

푁

( )

1 × 푒푗 2 휋 푛 (푘−푝)

푁

+ 푊(푖,푝), 푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.7)

Substituting 퐻(푘) –

푌(푖,푝) =1푁

푋(푖, 푘) 퐻(푘) 1 × 푒푗 2 휋 푛 (푘−푝)

푁 + 푊(푖,푝),

푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.8)

After rearranging (3.2.8) –

푌(푖,푝) = 푋(푖,푝) 퐻(푝) + 1푁

푋(푖,푘) 퐻(푘) 1 × 푒푗 2 휋 푛 (푘−푝)

푁 ,

+ 푊(푖,푝), 푓표푟 푝 = 0, 1, … ,푁 − 1

(3.2.9)

Finally the output of FFT can be expressed as –

푌(푖,푝) = 퐻(푝) 푋(푖,푝) + 푊(푖,푝), 푓표푟 푝 = 0, 1, … ,푁 − 1, (3.2.10)

With the help of following identity –

푒 ( )

= 푒 ( ) ( ) 푆푖푛 휋 (푘 − 푝)

푆푖푛 휋푘 − 푝푁

= 푁, 푓표푟 푘 = 푝

0, 푓표푟 푘 ≠ 푝

(3.2.11)

[66]

From (3.2.10), it is clear that output will be distortion free when there is no timing and frequency offset.

Now consider the other cases when some timing error exists (i.e. timing offset δ ≠ 0 ). Due to this timing

offset, the samples in FFT window will not start from exact point (i.e. zero indexes). There are three

different possibilities with this case [139].

Case-II: This is the case when the estimated starting point of OFDM symbol is before the exact point,

yet after the end of the (lagged) channel response to the previous OFDM symbol. In this case, the 푖

symbol is not overlapped with the previous (푖 − 1) OFDM symbol, that is, without incurring any ISI by

the previous symbol in this case. In order to see the effects of the STO, consider the received signal in the

frequency domain by taking the FFT of the time domain received samples 푟(푖,푛 + 훿), given as-

푌(푖,푝) = 푟(푖,푛 + 훿) 푒−푗 2 휋 푝 푛 / 푁 , 푓표푟 푝 = 0, 1, … ,푁 − 1

(3.2.12)

After substituting the value of 푟(푖,푛 + 훿) –

푌(푖,푝) = ℎ (휏) 푥(푖,푛 + 훿 − 휏 ) + 푤(푖,푛 + 훿) 푒−푗 2 휋 푝 푛 / 푁 ,

푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.13)

Separating the AWGN term –

푌(푖,푝) = ℎ (휏) 푥(푖,푛 + 훿 − 휏 ) 푒−푗 2 휋 푝 푛 / 푁 + 푤(푖,푛 + 훿) 푒−푗 2 휋 푝 푛 / 푁

( , )

,

푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.14)

Now, put the value of 푥(푖,푛 + 훿 − 휏 ) and interchange the summation –

푌(푖,푝) = ℎ (휏) 1푁

푋(푖,푘) 푒푗 2 휋 푘 ( 푛+훿−휏푙)

푁 푒−푗 2 휋 푝 푛 / 푁 + 푊(푖,푝) ,

푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.15)

[67]

푌(푖, 푝) = 1푁

푋(푖,푘) 푒푗 2 휋 푘 훿 / 푁

⎩⎪⎨

⎪⎧

ℎ (휏) 푒−푗 2 휋 푘 휏푙 / 푁

( ) ⎭⎪⎬

⎪⎫

1 × 푒푗 2 휋 푛 (푘−푝)

푁

+ 푊(푖,푝), 푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.16)

After substituting 퐻(푘) –

푌(푖,푝) =1푁

푋(푖,푘) 푒푗 2 휋 푘 훿 / 푁 퐻(푘) 1 × 푒푗 2 휋 푛 (푘−푝)

푁 + 푊(푖,푝),

푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.17)

After breaking the summation in two terms, the (3.2.17) becomes –

푌(푖,푝) = 푒푗 2 휋 푝 훿 / 푁 푋(푖,푝)퐻(푝)

+ 1푁

푋(푖,푘) 푒푗 2 휋 푘 훿 / 푁

,

퐻(푘) 1 × 푒푗 2 휋 푛 (푘−푝)

푁 + 푊(푖,푝),

푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.18)

Finally, it becomes –

푌(푖,푝) = 푒푗 2 휋 푝 훿 / 푁 푋(푖,푝)퐻(푝) + 푊(푖,푝), 푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.19)

From (3.2.19), it is clear that the orthogonality among sub-carrier can be completely preserved.

However, there exists a phase offset which is proportional to the STO ‘δ’ and sub-carrier index ‘k’, forcing

the signal constellation to be rotated around the origin. This phase offset can be compensated by a single

tap frequency domain equalizer.

Case-III: This is the case when the starting point of the OFDMsymbol is estimated to exist prior to the end

of the (lagged) channel response to the previous OFDMsymbol, and thus, the symbol timing is too early to

avoid the ISI. In this case, the orthogonality among subcarrier components is destroyed by the ISI (from

the previous symbol) and furthermore, ICI occurs.

[68]

Case-IV: This is the case when the starting point of the OFDMsymbol is estimated just after the exact

point. In this case, the samples for current FFT operation interval is consists of a part of the current OFDM

symbol 푥(푖,푛) and a part of next symbol 푥(푖 + 1,푛). Therefore the received sampled signal for current

FFT operation or interval can be expressed in the form of a vector as –

푟 = {푟(푖, 훿), 푟(푖, 훿 + 1), … , 푟(푖,푁 − 1), 푟(푖 + 1,−퐺), 푟(푖 + 1,−퐺 + 1), … , 푟(푖 + 1,−퐺 + 훿 − 1)}

(3.2.20)

Alternatively, the current vector 푟 can be written as –

푟 =푟(푖,푛 + 훿), 푓표푟 0 ≤ 푛 ≤ 푁 − 1 − 훿

푟(푖 + 1,푛 − 퐺 −푁 + 훿), 푓표푟 푁 − 훿 ≤ 푛 ≤ 푁 − 1

(3.2.21)

where, 푟(푖,푛) is the 푛 sample of 푖 OFDM symbol and 푟 represents the current vector for FFT

operation. The output of FFT can be written as –

푌(푖,푝) = 푟(푖,푛 + 훿)푒−푗 2 휋 푝 푛 / 푁 + 푟(푖 + 1,푛 − 퐺 −푁 + 훿)푒−푗 2 휋 푝 푛 / 푁 ,

푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.22)

After substituting the value of 푟(푖,푛 + 훿) and 푟(푖 + 1,푛 − 퐺 − 푁 + 훿) and further simplification –

푌(푖,푝) = ℎ (휏) 푥(푖,푛 + 훿 − 휏 )푒−푗 2 휋 푝 푛 / 푁 + 푤(푖,푛 + 훿) 푒−푗 2 휋 푝 푛 / 푁

( , )

+ ℎ (휏) 푥(푖 + 1,푛 − 퐺 − 푁 + 훿 − 휏 ) 푒−푗 2 휋 푝 푛 / 푁

+ 푤(푖 + 1,푛 − 퐺 − 푁 + 훿) 푒−푗 2 휋 푝 푛 / 푁

( , )

(3.2.23)

[69]

Now, put the value of 푥(푖,푛 + 훿 − 휏 ) and 푥(푖 + 1,푛 − 퐺 − 푁 + 훿 − 휏 ) in (3.2.23) –

푌(푖,푝) = ℎ (휏) 1푁

푋(푖,푘)푒 2 휋 푘 ( ) / 푁 푒−푗 2 휋 푝 푛 / 푁

+ ℎ (휏) 1푁

푋(푖 + 1,푘)푒 2 휋 푘 ( ) / 푁 푒−푗 2 휋 푝 푛 / 푁

+ 푊(푖,푝) + 푊(푖 + 1,푝), 푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.24)

푌(푖,푝) =1푁

푋(푖,푘) 푒푗 2 휋 푘 훿 / 푁

⎩⎪⎨

⎪⎧

ℎ (휏) 푒−푗 2 휋 푘 휏푙 / 푁

( , ) ⎭⎪⎬

⎪⎫

1 × 푒푗 2 휋 푛 (푘−푝)

푁

+

⎝

⎜⎛1푁

푋(푖 + 1, 푘) 푒푗 2 휋 푘 (훿−퐺−푁) / 푁

⎩⎪⎨

⎪⎧

ℎ (휏) 푒−푗 2 휋 푘 휏푙 / 푁

( , ) ⎭⎪⎬

⎪⎫

× 1 × 푒푗 2 휋 푛 (푘−푝)

푁

⎠

⎟⎞

+ 푊(푖,푝) + 푊(푖 + 1,푝),

푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.25)

Further simplification gives –

푌(푖,푝) =1푁

푋(푖,푘) 푒푗 2 휋 푘 훿 / 푁 퐻(푖,푘) 1 × 푒푗 2 휋 푛 (푘−푝)

푁

+ 1푁

푋(푖 + 1,푘) 푒푗 2 휋 푘 (훿−퐺−푁) / 푁 퐻(푖 + 1,푘) 1 × 푒푗 2 휋 푛 (푘−푝)

푁

+ 푊(푖,푝) + 푊(푖 + 1,푝), 푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.26)

[70]

After rearranging (3.2.26) –

푌(푖,푝) = 푁 − 훿푁

푋(푖,푝) 퐻(푖,푝) 푒푗 2 휋 푝 훿 / 푁

+ 1푁

푋(푖,푘) ,

퐻(푖,푘) 푒푗 2 휋 푘 훿 / 푁 1 × 푒푗 2 휋 푛 (푘−푝)

푁

+ 1푁

푋(푖 + 1,푘) 퐻(푖 + 1,푘) 푒푗 2 휋 푘 (훿−퐺−푁) / 푁 1 × 푒푗 2 휋 푛 (푘−푝)

푁

+ 푊(푖,푝) + 푊(푖 + 1,푝), 푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.27)

After combining Gaussian noise terms –

푌(푖,푝) = 푁 − 훿푁

푋(푖,푝) 퐻(푖,푝) 푒푗 2 휋 푝 훿 / 푁

+ 1푁

푋(푖,푘) ,

퐻(푖,푘) 푒푗 2 휋 푘 훿 / 푁 1 × 푒푗 2 휋 푛 (푘−푝)

푁

+ 1푁

푋(푖 + 1,푘) 퐻(푖 + 1,푘) 푒 푗 2 휋 푘 (푛+훿−퐺−푁) / 푁 1 × 푒−푗 2 휋 푝 푛

푁

+ 푊 (푖,푝), 푓표푟 푝 = 0, 1, … ,푁 − 1 (3.2.28)

where, 푊 (푖,푝) = 푊(푖,푝) + 푊(푖 + 1,푝)

By applying the following identity –

푒 ( )

= 푒 ( ) 푆푖푛 (푁 − 훿) 휋 (푘 − 푝)/ 푁

푆푖푛( 휋 (푘 − 푝)/ 푁)

=

푁 − 훿, 푓표푟 푘 = 푝

푁표푛 푍푒푟표, 푓표푟 푘 ≠ 푝 (3.2.29)

[71]

The first term in the left hand side of (3.2.28) indicates that timing offset results in phase offset and

attenuation of 푘 sub-carrier output of 푖 OFDM symbol. The second additive term represents ICI while

the third term stands for ISI. The last term represents the white Gaussian noise component.

3.3 DIFFERENT TIMING OFFSET ESTIMATION METHODS

Several methods of timing offset estimation which are based on training symbol transmission are

available in the literature [1, 2, 6, 17, 19, 20, 35, 36, 41, 42, 66, 71, 79, 81, 92, 107, 125, and 142]. These

methods are called DA methods. The training symbol based methods are generally used with packet

oriented applications such as IEEE 802.11(a) and HIPERLAN/2.

In DA methods, for timing synchronization, some training symbols or preambles are generally

transmitted before the transmission of actual information data symbols. At the receiver, the concept of

auto-correlation is used to detect these training symbols in incoming signal. In order to be detectable or

separately identifiable within the incoming signal, these training symbols consist of some repetitive

patterns or blocks. The timing estimation algorithms generally find the maxima of auto-correlation of

incoming signal to detect the starting of training symbol. The most popular DA method for timing

synchronization was given by Schmidl & Cox [125]. Later some more methods have been proposed in the

literature which performs better that Schmidl & Cox. In order to understand more about these methods,

their training symbol pattern and expression of timing metric is described below.

3.3.1 Schmidl and Cox Method

The form of time domain preamble proposed by Schmidl & Cox [125] is as follows -

푃푟푒푎푚푏푙푒 & = [퐴 / 퐴 / ] (3.3.1.1)

where, 퐴 / represents samples of length N/2 and is generated by method described in [125]. The timing

metric of Schmidl & Cox is given as-

푀 & (푑) =|푃 & (푑)| 푅 & (푑) ,

(3.3.1.2)

where,

[72]

푃 & (푑) = 푟∗(푑 + 푘) . 푟 푑 + 푘 +푁2

,

(3.3.1.3)

where, 푟∗(. ) is the conjugate of 푟(. )

푅 & (푑) = 푟 푑 + 푘 +푁2

(3.3.1.4)

The starting point of symbol is given by the maximum of timing metric 푀 & (푑). The timing metric of

Schmidl & Cox has a large plateau. The timing metric has its peak for entire interval of CP.

3.3.2 Minn’s Method

In order to reduce the plateau appear in the timing metric of Schmidl & Cox, Minn et.al [41 and

42] have proposed a training symbol with more than two identical segments along with flipping the signs

of last two segments to obtain a steeper roll-off trajectory at the correct timing position in the timing

metric. The preamble of Minn et.al has following form -

푃푟푒푎푚푏푙푒 = [퐴 / 퐴 / − 퐴 / − 퐴 / ] (3.3.2.1)

where, 퐴 / represents a PN sequence of length N/4. The timing metric of Minn et.al is given as-

푀 (푑) = | ( )| ( )

(3.3.2.2)

where,

푃 (푑) = 푟∗ 푑 + 푘 +푚 푁

2 . 푟 푑 + 푘 +

푚 푁2

+푁4

/

(3.3.2.3)

푅 (푑) = 푟 푑 + 푘 +푚 푁

2+푁4

/

(3.3.2.4)

[73]

The Minn’s method has its peak at the correct starting point for the OFDM symbol, since correlation of

some samples results in negative values. For this reason, Minn’s method eliminates the peak plateau of the

timing metric, hence resulting in a smaller MSE.

3.3.3 Reverse Auto-correlation Method

In order to have sharp peak at the correct timing point, Park et.al [17] proposed a different timing

metric and modified preamble according to this timing metric. The Park’s method is based on reverse auto-

correlation method and performs better than Schmidl & Cox and Minn’s method. The preamble of Park’s

method has the following form -

푃푟푒푎푚푏푙푒 = [퐴 / 퐵 / 퐴∗ / 퐵∗ / ] (3.3.3.1)

where, 퐴 / represents samples of length N/4 generated by IFFT of a PN sequence, and 퐴∗ / represents

a conjugate of 퐴 / . To get impulse-shaped timing metric,퐵 / is designed to be symmetric with 퐴 / .

푀 (푑) =|푃 (푑)| (푅 (푑))

(3.3.3.2)

where,

푃 (푑) = 푟(푑 − 푘) . 푟(푑 + 푘)/

(3.3.3.3)

푅 (푑) = |푟(푑 + 푘)| /

(3.3.3.4)

The timing metric 푃 (푑) is designed such that there are N/2 different pairs of product between two

adjacent values. It has maximum different pairs of product. Therefore, this timing metric has its peak value

at the correct symbol timing, while the values are almost zero at all other positions.

[74]

3.3.4 Shi and Serpedin Method

In the year 2004, K. Shi and E. Serpedin [66] modified the Minn’s scheme with outcome of a more

advanced timing metric based on the maximum likelihood criterion. Shi and Serpedin adopted the

following preamble structure from Minn’s -

푃푟푒푎푚푏푙푒 & = [퐴 / 퐴 / − 퐴 / 퐴 / ] (3.3.4.1)

where, A / represents a PN sequence of length N/4. In this method, the received time domain sample

collects in to four vectors 푟 , 푓표푟 푖 = 0, 1, 2 and 3 . The timing metric of Shi and Serpedin is given as-

푀 & (푑) = 푃 & (푑) 32

|푟 |

where, (3.3.4.2)

푃 & (푑) = |푅 (푑)| + |푅 (푑)| + |푅 (푑)|

푅 (푑) = 푟 (푑) 푟 (푑) − 푟 (푑) 푟 (푑) − 푟 (푑) 푟 (푑)

푅 (푑) = 푟 (푑) 푟 (푑) − 푟 (푑) 푟 (푑)

푅 (푑) = 푟 (푑) 푟 (푑)

Here, (. ) represents the conjugate transposition.

3.3.5 Restricted Cross-Correlation Method

The method proposed by A. B. Awoseyila et.al [1, 2] is a multi-stage method. In this method, a

simple autocorrelation technique and a restricted cross-correlation technique are combined to achieve

enhanced estimation performance. The preamble used in this method is same as given by Schmidl & Cox

method. The coarse timing estimation is performed in first stage. An auto-correlation similar to Schmidl &

Cox and its integration is taken in this stage. After that fine timing estimation is carried out in second stage.

The restricted cross-correlation is performed in second stage. The coarse timing estimate is given as –

푃 & (푑) = 푟∗(푑 + 푘) . 푟 푑 + 푘 +푁2

,/

(3.3.5.1)

[75]

푀 (푑) =1

퐺 + 1 |푃 & (푑 − 푘)| ,

(3.3.5.2)

푑 = 푎푟푔푚푎푥 {푀 (푑)} , (3.3.5.3)

where, 푀 (푑) is the integration of auto-correlation function 푃 & (푑) over the length of the cyclic prefix in

order to eliminate its uncertainty plateau and achieve a coarse timing metric 푀 (푑), whose peak indicates

the coarse timing estimate 푑 . The restricted cross-correlation stage is summarized as –

푃 (푑) = 푟 (푑 + 푘) 푆∗(푘)

(3.3.5.4)

푀 (푑) = |푃 (푑)| 푀 (푑) (3.3.5.5)

푑 = arg max 푀 (푑) , 푓표푟 푑 ∈ 푑 − , 푑 + (3.3.5.6)

푑 = arg max { | 푃 (푑) > 푇 | }, 푓표푟 푑 ∈ 푑 − 휆,푑 (3.3.5.7)

푇 = −4휋

ln(푃 ) (푚푒푎푛 { | 푃 (푑) | } ),

푓표푟 푑 ∈ 푑 − + 휆 + 1,푑 − 휆 − 1 (3.3.5.8)

where, 푟 is the total-frequency corrected signal, 푃 (푑) is the restricted cross-correlation, 푀 (푑) is

the filtered timing metric and ‘d’ is chosen in all equations to ensure that all relevant timing points that

could be the ideal timing are tracked. The sample ‘d’ is chosen in last equation such as to track the first

arriving path (d ) which may not always be the strongest path (푑 ) , wherein all channel paths are

expected to be received within λ+ 1 samples. The threshold 푇 is used to detect the ideal timing 푑 .

[76]

3.4 PROPOSED TIMING OFFSET ESTIMATION METHOD

The proposed timing offset estimator is based on the localization of chirp signal with the help of

FRFT. The emphasis of proposed estimator is on the use of fractional Fourier transform based correlation

and chirp signal. The application of chirp signal for the synchronization in OFDM system along with its

advantages is very well documented in [107]. The chirp signals have good PAPR property. Due to the

constant envelope of chirp signal, the linearity and power back-off requirements of the power amplifier is

reduced at the transmitter and the automatic gain controller at the receiver. Due to its flat spectrum, it can

be transmitted either in the time or frequency domain.

The FRFT has found applications in the domain of non-stationary signals (chirp signal) analysis,

especially in filtering, radar signal processing and time delay estimation [7, 8, 40, 63, 67, and 100], due to

the fact that the FRFT localizes chirp signal nicely in fractional Fourier domain. This characteristic of

FRFT on chirp signal has been exploited in the proposed estimator. Before explaining the proposed

estimator, a small review of FRFT and analysis of chirp signal is required, as given in next sub-section.

3.4.1 Correlation Theorem of FRFT

The fractional Fourier transform is a generalization of the conventional Fourier transform in time-

frequency plane and found to be numerous applications in the areas of signal processing, image processing,

and electromagnetic wave propagation [7, 67, 68 and 100]. Various properties of the FRFT have already

been derived and established, as reported in [7] and [67]. Recently, a new weighted convolution and

correlation theorems in the FRFT domain are given by Singh et.al as documented in [7] and [8]. The FRFT

of a signal 푥(푡) with angle parameter ′훾′ , represented by X (u) is defined as [67] –

ℑ[푥(푡)] = 푋 (푢) = 푥(푡) 퐾 (푡,푢) 푑푡

(3.4.1.1)

where, K (t, u) is the kernel of FRFT, which is given as –

[77]

퐾 (푡,푢) =

⎩⎪⎪⎪⎨

⎪⎪⎪⎧ 1− 푗 퐶표푡 (훾)

2휋 푒푥푝

푗2

{(푡 + 푢 ) 퐶표푡 (훾) – 2 푡 푢 퐶표푠푒푐(훾)} ,

푖푓 훾 ≠ 푛 휋

훿(푡 − 푢), 푖푓 훾 = 2 푛 휋

훿(푡 + 푢), 푖푓 훾 = (2 푛 + 1) 휋

(3.4.1.2)

The correlation theorem for FRFT domain [8] is given as –

푟 (휏) ↔ 푅 (푢)

(3.4.1.3)

where, 푟 (휏) is defined as weighted auto-correlation of signal 푥(푡) and 푅 (푢) is defined as FRFT of

r (τ), which are defined as –

푟 (휏) = 푥(푡) 푥(푡 + 휏) e ( ) ( ) 푑푡

(3.4.1.4)

푅 (푢) =2 휋

1 − 푗 퐶표푡 (γ) 푒 ( ) 푋 (−푢) 푋 (푢)

(3.4.1.5)

3.4.2 Chirp Signal and its FRFT

A chirp signal has an inherent characteristic of time-varying frequency components. The frequency

of a chirp signal is depending on time according to some function. Its frequency may increases ('up-chirp')

or decreases ('down-chirp') with time. Mathematically, the chirp signal of chirp rate ‘2a’ is defined as -

푥(푡) = 푒 푟푒푐푡 , (3.4.2.1)

where, 푟푒푐푡 =1, |푡| ≤ 휏

0, 표푡ℎ푒푟푤푖푠푒

[78]

where, ′휏′ is the duration of chirp, ′2 푎 휏′ is the bandwidth of chirp , and ′푏′ is the center frequency of

chirp signal. It is known that a finite duration chirp signal can be concentrated maximally in the fractional

Fourier domain with an angle γ (Optimum value of ′γ′ ) which is determined by the chirp rate ′2푎′ of

the signal satisfying the relation [40 and 100] -

퐶표푡 γ = −4 휋 푎 (3.4.2.2)

To illustrate this behavior of chirp signal, a simulation exercise has been performed by considering the

chirp signal with chirp rate of 110 and unity duration. Thereafter, the FRFT of chirp signal at optimum

angle γ = 3.14015 and two nearby angles (γ = 3.13 and γ = 3.15) is determined, as shown in

Figure- 3.4.2.1.

Figure – 3.4.2.1: FRFT of a chirp signal (3.4.2.1) for different angle 후 with a = 55, b=75

It can easily be depicted from the results that at an optimum angle, the FRFT of chirp has a sharp

and pronounced peak in comparison to other nearby angles. This is in conformity of the fact that the FRFT

transforms a chirp signal into a delta function at an optimum angle corresponding to the chirp rate

associated with the chirp signal. In the following sub-sections the timing estimations are described.

[79]

3.4.3 Proposed Algorithm

The proposed timing offset estimation method has exploited the basic property of the chirp signal

which shows that it peaks in the fractional Fourier domain corresponding to an optimum angle ′γ ′

associated with the FRFT kernel. This characteristic of the FRFT on chirp signal has been used for

searching the start of training sequence in the received signal. Therefore, at the receiver side, the FRFT of

auto-correlation of received signal is determined and its peak is observed.

In the proposed estimation method, a constant envelope signal (i.e. chirp signal) is used as a

training sequence and the DFRFT is utilized as a tool to locate the training sequence in the received signal.

The proposed training symbol (preamble) block contains two identical halves, similar to the Schmidl &

Cox method [125]. The first half of preamble will be used for timing offset estimation whereas both

halves will be utilized for frequency offset estimation. The training symbol has the following structure-

푃푟푒푎푚푏푙푒 = 푆 = [퐴 퐴] (3.4.3.1)

where, 퐴 is a chirp signal of length N with chirp rate ‘2a’. For the simulation studies in OFDM system,

the discrete time chirp signal has been used which is defined as –

퐴 = 푥(푛) = 푒 ( / ) ( / ) , 푓표푟 푛 = 0, 1, 2, … ,푁 − 1 (3.4.3.2)

where, ′푏′ is the center frequency of chirp signal. To make the transform technique compatible with the

discrete signal as normally encountered in signal processing applications and also in our present

application (OFDM), the discrete version of the transform technique is needed. Therefore, the sampling

type discrete FRFT (DFRFT) algorithm [40] has been considered for the calculation of the FRFT of the

signal in the proposed scheme.

The correlation theorem for FRFT domain [8], as given in section 3.4.1, is used for timing metric

generation. From (3.4.1.5), it is evident that the FRFT at an angle γ of the time-domain weighted auto-

correlation {푟 (휏)} is equivalent to the multiplication of signal’s FRFT 푋 (푢) , the mirror image of

signal’s FRFT 푋 (−푢) , a chirp function and a scaling factor. The proposed timing estimation method is

based on the auto-correlation of received signal. Therefore, the timing metric is generated by using

(3.4.1.5). The algorithm to generate timing metric is described in the following paragraph.

A sliding window (N samples) method is used to generate timing metric 푀 (푑) (where,

′d′ is a time index). First, the DFRFT of every frame or block of N samples of received signal is

[80]

determined at the optimum angle γ , then this DFRFT is used to calculate the 푅 (푢) as given in

(3.4.1.5). The maximum of |푅 (푢)| is used to give the value of timing metric corresponding to index ′푑′.

This window slides sample by sample as the receiver searches for the training symbol. Therefore, the

timing metric of proposed method is defined as -

푀 (푑) = 푚푎푥 2 휋

1− 푗 퐶표푡 (γ) 푒 ( ) 푋 ,(푢) 푋 ,

(−푢)

(3.4.3.3)

where, 푋 ,(푢) is a N-point DFRFT of 푥(푛 + 푑). The maximum of timing metric 푀 (푑) gives

the starting point of the training symbol. The timing metric of proposed method is shown in Figure-

3.4.3.1, under no noise and no channel imperfection condition. The 512 sub-carrier OFDM system with

64 cyclic prefix has been considered for the generation of this timing metric. The correct timing point is

indexed as 0 in the figure. The plots of timing metrics of [17, 41, 66, and 125] have also been shown in

Figure 3.4.3.1 for the comparison. It is clearly visible from the plots that the timing metric of proposed

method has a sharp peak with no side peaks as appear in Minn et.al and Park et.al timing metric.

Figure – 3.4.3.1: Timing metric of different estimators

[81]

In order to explain the superiority of the proposed timing metric over other included methods as

shown in Figure-3.4.3.1, the DFRFT of an OFDM symbol and of a chirp signal under both the conditions

(no channel distortion and channel distortion with noise) have been shown in Figure- 3.4.3.2. An OFDM

symbol with 512 sub-carriers has been considered for this analysis and a chirp signal defined in (3.4.2.1) is

taken with values of ‘a’ = 435 and ‘b’ =76.

Figure – 3.4.3.2: DFRFT of an OFDM and chirp signal

It is clearly visible from the plots included in Figure- 3.4.3.2, that the maximum value of DFRFT

of chirp signal is much larger than the maximum value of DFRFT of OFDM symbol. This concept is used

to search the preamble in received signal.

[82]

3.5 PERFORMANCE EVALUATION

In this section, the performance of proposed method is presented and compared with the methods

given by Schmidl & Cox [125], Minn et.al [41], Park et.al [17], Shi and Serpedin [66] and Awoseyila

et.al [1]. For this comparison, an OFDM system with 64 sub-carriers and 16 cyclic prefix with QPSK

modulation is considered. The HIPERLAN/2 indoor channel model [23] is used for simulations. A

normalized frequency offset of 0.1 is considered to explore the robustness of proposed method. The mean

and mean square error has been taken as performance evaluation parameter.

Figure- 3.5.1 and Figure- 3.5.2 shows the comparison of mean and MSE of timing offset with all

four estimators in the HIPERLAN/2 indoor channel-A. It is clearly visible from Figure- 3.5.1, that the

value of mean with proposed method is approximately equal to zero at all the signal to noise ratios (SNR).

The values of MSE of timing offset with proposed method are much better as compared to other

estimators, as shown in the Figure- 3.5.2.

Figure-3.5.1: Mean of timing offset estimation of different methods in HIPERLAN/2 indoor Channel-A

[83]

Figure-3.5.2: MSE of timing offset estimation of different methods in HIPERLAN/2 indoor Channel-A

Simulation results for an advance OFDM system like Wi-Max with 256 sub-carriers are also

presented in Figure- 3.5.3 and Figure- 3.5.4. An ISI channel model [55], consisting of L=8 paths with

path delays of m = 0,1, … , L− 1 samples and an exponential power delay profile having average power

of 푒 / has been considered for simulation. It is clearly visible from the plots that the mean and MSE

with proposed algorithm in this channel model is less than all other methods.

The proposed algorithm performs better than existing methods both in Indoor and Outdoor fading

channel model. This confirms the robustness of proposed algorithm against multipath fading.

[84]

Figure-3.5.3: Mean of timing offset estimation of different methods in Wi-Max system

Figure-3.5.4: MSE of timing offset estimation of different methods in Wi-Max system

[85]

The computational complexity of various estimators is compared with proposed algorithm and

results are shown in Table-3.1.

Table- 3.1: Approximate computational complexity of various estimators.

Methods Complex Multiplication Complex Addition

Schmidl & Cox [125] N/2 N/2− 1

Minn et.al [41] N/2 N/2− 1

Park et.al [17] N/2 + 1 N/2

Shi & Serpedin [66] 3N 2⁄ 3N/2− 1

Awoseyila et.al [1] (7N + 8) 4⁄ (7N − 8) 4⁄

Proposed (N log N) + 2N + 1 N log N

The proposed method outperforms all other existing methods at the cost of computational

complexity. The reason for this increase in complexity is due to the fact that other methods generate

timing metric by taking auto-correlation in time domain directly whereas the proposed method first takes

the DFRFT of received signal and then generate timing metric.

3.6 SUMMARY

A new method is proposed for timing offset estimation in an OFDM system. The proposed

method is based on the localization of chirp signal in the FRFT domain. The suggested algorithm for

timing offset estimation in both HIPERLAN/2 indoor channel-A and in strong fading channel (Wi-Max

with 256 sub-carriers) provides lowest MSE in comparison to other available methods. However, to

obtain such improvements in MSE of timing offset estimation, the computational complexity increases as

included in Table-3.1.

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