### 中 華 大 學 碩 士 論 文

### 題目：應用 FFT 為 IEEE 802.11a 無線區域網路的內 插器做取樣時間偏移之預估

### Use of the FFT in the Sampling Time Offset Estimation for the Interpolator of IEEE 802.11a WLAN

### 系 所 別：電機工程學系碩士班 學號姓名：M09101028 蔣忠易 指導教授：鍾英漢 博士

### 中華民國 九十四 年 十二 月

**摘 要 **

近年來，正交分頻多工技術已經被廣泛地運用在無線傳輸上。眾 所周知，精確的時間及頻率同步對正交分頻多工接受器的正常運作相 當重要。於此論文中我們將提出一個可估測符元時間偏移和取樣時間 偏移的方法。此方法利用 IEEE 802.11a 無線區域網路標準中前置短封 包資料的相關比較來估測取樣時間偏移，並將其估測的值交給內插器 來還原所傳送的原始資料。我們的估測方法是屬於前饋式，因此適用 於要求迅速同步的突發模式傳輸。此方法的特點是利用正交分頻多工 接收端內的快速傅立葉轉換處理器，在其正常工作外的閒置期間來從 事估測取樣時間偏移的任務。這大大簡化內插器硬體實現的複雜度。

我們用模擬來評估此新策略在可加性高斯白色雜訊通道及多路徑通 道時的位元錯誤率和取樣時間偏移誤差等重要效能參數。模擬的結果 顯示，此新方法優於其他研究人員所曾提出的方法。此外，我們也研 究了利用曲線適合函數來更進一步加強所提新方法的效能的技巧。針 對採用線性、二次、及三次等曲線適合函數所能提供好處，我們用實 例加以比較和討論。

**Abstract **

In recent years, Orthogonal Frequency Division Multiplexing (OFDM) techniques have frequently been used for wireless transmissions. It is well known that accurate timing and frequency synchronization is important for an OFDM receiver to operate properly. In this paper we propose a scheme to estimate the symbol time offset and the sampling time offset. This scheme employs correlations on the short preamble, which is specified in the IEEE 802.11a WLAN standard, to estimate the sampling time offset. The estimated sampling time offset is used in the interpolator to recover the originally transmitted data. Since this scheme uses feed forward estimating approach, it is suitable for burst-mode transmissions that necessitate rapid synchronization. An important feature of this scheme is that the FFT processor in the OFDM receiver has been utilized to carry out sampling time offset estimation during its idle period. This significantly simplifies the hardware implementation complexity of the interpolator. Important performance measures such as the bit error rate and sampling time offset jitter have been simulated for both AWGN channels and a multipath channel. Numerical results show that the new scheme outperforms a referenced scheme proposed by other researchers. In addition, to further enhance the performance of this new scheme, we also study the case in which curve-fitting functions are applied. Three curve-fitting functions, linear, quadratic, and cubic functions are considered. Performance improvements resulted from applications of these functions are also illustrated via numerical examples.

**Contents **

摘要………....i

Abstract………..ii

Contents……….iii

Figure Contents……….vi

Table Contents………...ix

**Chapter 1 Introduction**

……….……….1
**Chapter 2 OFDM**

……….………5
**2.1 OFDM History………5 **

**2.2 OFDM Principle………...…..6 **

2.2.1 Multicarrier Modulation………..6

2.2.2 Orthogonal Frequency Division Multiplexing………8

2.2.3 Guard Interval and Cyclic Prefix………...10

**2.3 Baseband Model of OFDM system……….…12 **

2.3.1 Coding / Decoding……….13

2.3.2 Interleaving / Deinterleaving……….13

2.3.3 Mapping / Demapping………...13

2.3.4 Pilot………14

2.3.5 IFFT / FFT……….14

2.3.6 Channel………..14

**Chapter 3 IEEE 802.11a STD**

……….………19
** 3.1 Introduction……….…….………..19 **

** 3.2 Protocol Architecture……….………20 **

**3.3 PLCP Sublayer………..………..20 **

3.3.1 PLCP Frame Format……….…….…..…20

** 3.3.1.1 PLCP Preamble**……….……..21

** 3.3.1.2 Signal**……….………….23

** 3.3.1.3 Data**………23

3.3.1.3.1 Service, Tail, and Pad………24

3.3.1.3.2 Data Scrambler………25

3.3.1.3.3 Convolutional Encoder……….…….25

3.3.1.3.4 Data Interleaving……….……….27

3.3.1.3.5 Mapping……….………28

3.3.1.3.6 Pilot………..…….30

**3.4 Operating Frequcecies……….………..…30 **

**Chapter 4 Interpolation**

……….……...33
** 4.1 Polynomial-Based Interpolation Filter……….33 **

** 4.2 Filter Structure………..…..…34 **

** 4.3 Another Interpolation Filter………..36 **

4.3.1 Lower Degree Interpolator………..36

4.3.2 Efficient Implementation of Interpolator……….36

**Chapter 5 STO Estimation**

………..39
** 5.1 Feedback Estimation………..…39 **

** 5.2 Feedforward Estimation………40 **

5.2.1 Maximum likelihood Estimation……….…42

5.2.2 Fourier Transform Estimation……….42

** 5.3 Simulation and Some Results………48 **

** 5.4 Use of Curve Fitting Functions……….55 **

**Chapter 6 Conclusion**

………60
**Reference**

………...61
**Figure Contents **

Figure 1.1 A feedback timing recovery diagram for an

OFDM receiver………..……3

Figure 1.2 The input and output of the interpolator………3

Figure 2.1 Multicarrier modulation……….7

Figure 2.2 Single OFDM subcarrier………...…….7

Figure 2.3 Orthogonal multicarrier……….………….7

Figure 2.4 Bandwidth saving of OFDM………..9

Figure 2.5 OFDM demodulation……….…………9

Figure 2.6 Cyclic prefix of OFDM symbol………..…….10

Figure 2.7 Inserting GI to avoid ISI………...………11

Figure 2.8 Multipath with empty GI causes ICI………..……..11

Figure 2.9 Inserting CP to avoid ICI……….12

Figure 2.10 Baseband architecture of OFDM system………...…12

Figure 2.11 The AWGN channel model………14

Figure 2.12 The multipath transmittance……….……….15

Figure 2.13 The multipath channel impulse response…………...……16

**Figure 2.14 The fading channel model………..…………16 **

**Figure 3.1 PPDU frame format……….……..22 **

**Figure 3.2 PLCP preamble………...……...……22 **

**Figure 3.3 SIGNAL field bits………..24 **

**Figure 3.4 Data scrambler………..……….………25 **

**Figure 3.5 Convolutional encoder……….……..26 **

**Figure 3.6 Puncturing (R=2/3)………..……..26 **

**Figure 3.7 Puncturing (R=3/4)………27 **

**Figure 3.8 BPSK & QPSK constellation………...……..28 **

**Figure 3.9 16 QAM constellation………29 **

**Figure 3.10 64 QAM constellation……….….……29 **

**Figure 3.11 Pilot subcarriers………30 **

**Figure 3.12 The 802.11a frequency channel……….….…….32 **

**Figure 4.1 Farrow structure of cubic interpolation filter……….…..…35 **

**Figure 4.2 The simpler structure of 2-degree interpolation filter….….38 **
**Figure 4.3 Farrow structure of Interpolator I………....….38 **

**Figure 5.1 Block diagram of the ML estimation………..…….42 **

**Figure 5.2 The stricture of STO estimation in [9]……….………..43 **

Figure 5.3 Structure of using the FFT for the sampling time
**offset estimation………...…44 **

**Figure 5.4 The result of using Dengwei’s estimation and FFT estimation…45 **
**Figure 5.5 Preamble of the IEEE 802.11a WLAN………46 **

**Figure 5.6 The butterfly structure of 64-points FFT………...………47 **

Figure 5.7 Structure of using the FFT for the sampling time
**offset estimation………...47 **

Figure 5.8 BER of FT and ML methods for QPSK in AWGN
**channels with N=96………...49 **

Figure 5.9 BER of FT and ML methods for QPSK in AWGN
**channels with N=32……….….…50 **

Figure 5.10 BER of FT and ML for QPSK and 16-QAM in
**AWGN channels………...50 **
Figure 5.11 BER of FT and ML methods for 16-QAM in

**Fading channels………...….51 **
Figure 5.12 BER of FT and ML methods for QPSK in

**Fading channels………..…..51 **
**Figure 5.13 Jitter of sampling time offset estimation………..…..52 **
Figure 5.14 The BER comparison of the 2-degree interpolator

**with 16-QAM………...53 **
Figure 5.15 The BER comparison of the 2-degree interpolator

**with QPSK………....54 **
Figure 5.16 The BER comparison of the symmetrical interpolator

**with 16-QAM………..….54 **
Figure 5.17 The BER comparison of the symmetrical interpolator

**with QPSK……….…………...55 **
Figure 5.18 The actual estimation value of FT method and

**ideal estimation value………...…57 **
Figure 5.19 Estimation error of FT estimation with curve-fitting

**function amendment……….…57 **
Figure 5.20 Estimation error of the simple estimation with

curve-fitting function………..…..58 Figure 5.21 The BER of simplified estimation with cubic

curve-fitting function………..…..59

**Table Contents **

Table 2.1 DAB specifications………...…….17

Table 3.1 Rate-dependent parameters………...…21

Table 3.2 Timing-related parameters………..……..21

Table 3.3 RATE bits………....….23

Table 3.4 Block interleaving……….………27

Table 3.5 Mapping normalization factor………...…..28

Table 3.6 The 802.11a operating bands……….………..….….31

Table 4.1 Filter coefficients b*l**(i) for cubic interpolator……….35 *

Table 4.2 The coefficients (b*l**(i)) of 2-degree interpolation filter.…….36 *

Table 4.3 Degree and length of two interpolator……….………..37

Table 4.4 The filter coefficient of interpolator I………..……..37

Table 4.5 The filter coefficient of interpolator II……….……..37

**Acknowledgement **

I would like to express my gratitude to my advisor, Dr. In-Hang Chung, for his invaluable suggestions and great patience. His knowledge and experience have benefited me tremendously.

Besides, I appreciate Professor Chi-Kuang Hwang, Tung-Chou Chen, and Instructor Ming-Ching Yen for providing me immediate support when I need. I also appreciate my classmates and the members of the Communication Laboratory, Yu-Yi Chao, Chun-Hao Huang, Tao-Lun Chin, Shih-Ming Fang, Chiu-Hua Li, Ying-Tang Shih, Yi-Fang Wang, Yung-Pin Wang, Yu-Cheng Wang, Yuan-Lien Hsieh, and Ming-Yen Hsu. They gave me opportunities to share their far-reaching knowledge with me during the three years.

Finally, I would like to give my special appreciation to my family and my girl friend, Miss Yu-Hsin Wei, for their endless love and encouragement.

**Chapter 1 **

**Introduction **

Orthogonal Frequency Division Multiplexing (OFDM) techniques possess the capability of combating frequency selective fading or time dispersion in wireless channels [1]. Consequently, OFDM has been adopted as the transmission techniques for various standardized networks or systems, such as the IEEE 802.11a Wireless Local Area Networks (WLAN) [2]. It is well known, however, that an OFDM receive is sensitive to frequency or timing offsets [3] [4]. Inaccurate synchronization will cause Inter-Channel Interference (ICI) or Inter-Symbol Interference (ISI). This will degrade the performance or even worse can result in malfunction of an OFDM system.

This thesis concentrates on timing synchronization issues for an OFDM receiver in which the received signal is sampled with period Ts. Usually, the timing offsets can be divided into the integer part and the fractional part, in which the former is called symbol time offset and the latter is called sampling time offset. Here we consider an 802.11a WLAN receiver that employs an interpolator to recover the originally transmitted data in the presence of sampling time offset (STO).

In a fully digital OFDM receiver, the received signal is sampled by a free running oscillator with fixed clock period. Due to mismatches of the oscillators used

in the transmitter and the receiver, the transmitted data and the sampled ones are not always synchronous. For proper operation of the receiver, an interpolator which can recover the originally transmitted data based on the sampled signals and the estimated CFOs can be used. A schematic block diagram of an OFDM receiver that adopts an interpolator is depicted in Fig. 1.1 [5]. In this figure, the input x(t) is the signal transmitted in the four pilot channels. The interpolator uses the sampled input signal,

) mT (

x _{s} and the estimated STO , )(0 ,1 , to calculate the interpolants y(kT_{i}).
After removal of the GI, the interpolated time domain signals are transformed into
frequency domain by the FFT processor to obtain the phase rotation resulted from the
time offset. This estimated time offset can be detected by the Timing Error Detector
(TED) [6] and is divided into the integer part and the fractional part. The former,
which is usually called the symbol time offset, is used by a Window Controller to
adjust the FFT window. On the other hand, the fractional part of the estimated time
offset is passed through a Loop Filter to filter out the high frequency noise. The
resultant offset is used by a Numerical Control Oscillator (NCO) to calculate the
sampling time offset (STO) . The performance of the interpolator depends highly
on the accuracy of the STO estimation [7]. The relationship between the input and
output of the interpolator is illustrated in Fig. 1.2. The black rectangle points mean the
sampled data at incorrect points, and the white circle points mean the recovered data
after the interpolator. Accurate sampling timing synchronization is important for an
OFDM receiver to operate properly.

Since the architecture for the estimation of STO appeared in Fig. 1.1 requires input data that are produced after the FFT processor, this configuration is termed feedback. Feedback estimation has the drawback of long acquisition time. For burst-mode transmissions, the receiver needs to achieve synchronization rapidly. Thus,

a feedforward estimation approach is desirable. A feedforward estimating method that employs an interpolator to produce a log-likelihood function to calculate the STO is presented in [8]. Another feedforward method that uses correlation of the preamble in frequency domain (after FFT) can be found in [9].

Figure 1.1 A feedback timing recovery diagram for an OFDM receiver.

Figure 1.2 The input and output of the interpolator

In this thesis, we propose a new feed-forward scheme to estimate the STO of an 802.11a. system using the correlations of the short preamble sequence. The

estimated STO is provided to the interpolator for necessary calculations. A significant contribution of this scheme is that the parameters required for the STO estimation are carried out by the FFT processor during its idle period. This considerably reduces the implementation complexity of the receiver. Important performance measures such as the bit error rate (BER) and the STO estimation jitter have been simulated for both AWGN channels and a multipath channel. Effects of variation in some system parameters on the performance are investigated. Numerical results show that the new scheme outperforms a referenced scheme proposed by other researchers. In addition, to further enhance the performance of this new scheme, we also study the case in which curve-fitting functions are applied. By using curve-fitting functions, the computation complexity required by the FFT processor can be considerably reduced.

We have considered linear, quadratic, and cubic three curve-fitting functions.

Performance improvements resulted from applications of these functions are also demonstrated via numerical examples.

OFDM principles and specifications of IEEE 802.11a WLAN standard are described in Chapter 2 and Chapter 3, respectively. Operation principles of the interpolator and another simple structures of interpolator will be presented in Chapter 4. Simulated numerical examples showing bit error rate and the jitter of STO estimation are presented and discussed in Chapter 5. Three curve-fitting functions, linear, quadratic, and cubic that are adopted to enhance the performance of the scheme are also portrayed in this chapter. Therein the benefit of applying these algorithms is also presented via numerical examples. Finally, Chapter 6 concludes this thesis.

**Chapter 2 **

**OFDM **

**2.1 OFDM History **

In 1958, OFDM was first used for military high frequency (HF) communication systems, such as the Kineplex system which is a multicarrier modulation for data transmission [10]. In 1966, Chang proposed the original OFDM principles and successfully achieved a patent in 1970 [11]. Later on, Saltzberg analyzed the OFDM performance and observed that the crosstalk was the severe problem in the system [12]. In 1971, Weinstein and Ebert proposed a modified OFDM system in which the discrete Fourier Transform (DFT) was applied to generate the orthogonal subcarriers waveforms [13]. In 1980, Peled and Ruiz introduced cyclic prefix (CP) also known as guard interval (GI) for the OFDM system. Hirosaki introduced an equalization algorithm to suppress both Inter Symbol Interference (ISI) and Inter Carrier Interference (ICI). Hirosaki also applied QAM modulation, pilot tone, and trellis coding techniques to the high speed OFDM system [14]. In 1985, Cimini introduced a pilot-based method to reduce the multipath interference and used it for mobile communication [15]. In 1987, Al1ard and Lasalle used OFDM for digital broadcasting [16]. In 1989, Kalet suggested a subcarrier-selective allocating scheme

which suffers less channel distortion [17]. In 1995, the ETSI (European Telecommunication Standard Institute) DAB (Digital Audio Broadcasting) standard was proposed which was the first OFDM-based standard [18]. In 1997, the ETSI DVB-T (Digital Video Broadcasting-Terrestrial) standard was proposed [19]. In 1998, Magic WAND demonstrated OFDM modems for wireless LAN and the access points took place in the 5 GHz frequency range at a nominal transmission speed of 20 Mbps and a range of up to 50 meters [20]. In 1999, the IEEE 802.11a and HIPERLAN/2 standard for WLAN were proposed. In 2000, V-OFDM (Vector OFDM) for fixed wireless access was proposed. In 2001, OFDM are considered for new IEEE 802.11 and 802.16 standards.

**2.2 OFDM Principle **

### 2.2.1 Multicarrier Modulation

Multicarrier Modulation (MCM) is the principle of transmitting data by dividing the data stream into several parallel bit streams, each of which has a much lower bit rate and a lower bandwidth than the original data stream. These streams are modulated by several carriers, and each stream represents a carrier independently.

These modulated carriers are summed for transmission, described in Fig. 2.1. In the receiver, the subchannels are separated and demodulated into independent and parallel streams to obtain the transmitted data. There are two reasons for choosing multicarrier modulation in the system. Firstly, a single carrier signal has the enhancement of interference that is caused by linear equalizer. Secondly, the long symbol time used in multicarrier modulation produces a much greater immunity to impulse noise fast fading [21].

Figure 2.1 Multicarrier modulation

### Figure 2.2 Single OFDM subcarrier

### Figure 2.3 Orthogonal multicarrier

Mathematically, each subcarrier (Fig. 2.2) can be described as a complex wave,

S_{c}(t)=A_{c}(t) exp( j [ω_{c}t + Φ_{c}(t)] ). (2.1)

Where Ac(t) and Φc(t) are the amplitude and phase of this subcarrier. A transmitted signal of OFDM consists of many subcarriers shown in Fig. 2.3. Thus the complex signal can be represented as

S_{s}(t)=

*N*

1 ^{1}

0
*N*

*n*

A_{n}(t) exp( j [ω_{n}t + Φ_{n}(t)] ), (2.2)

where ωn = ω0+ nΔω. If we consider the waveforms of each component over one
symbol period, the variables, Ac(t) and Φc(t), would be fixed values. ( An(t) = An and
Φ_{n}(t) =Φ_{n }) If the sampling frequency is 1/T, then the signal is represented by

S_{s}(kT)=

*N* 1

^{1}

0
*N*

*n*

A_{n} exp( j [(ω0+nΔω)kT +Φ_{n}]). (2.3)

### 2.2.2 Orthogonal Frequency Division Multiplexing

Frequency Division Multiplexing (FDM) is a technology that multiple signals are transmitted simultaneously over a single transmission path. Each signal is traveled within its own unique frequency range. In Orthogonal Frequency Division Multiplexing (OFDM) transmissions, data are distributed over a large number of carriers that are spaced apart at precise frequencies. The waveform of subcarriers must have overlapping transmitting spectra, and all waveforms are orthogonal and don’t obstruct each other. From Fig. 2.4, we can find that the transmissions of OFDM have lower bandwidths than FDM.

The demodulation of OFDM consists of a bank of N matched filters described in Fig. 2.5. Thus the OFDM modulation will become very complex for a

large number of subcarriers. So the baseband modulators and demodulators can be replaced with IDFT and DFT processors to make the OFDM structure simple [1].

### Figure 2.4 Bandwidth saving of OFDM

### Figure 2.5 OFDM demodulation

### 2.2.3 Guard Interval and Cyclic Prefix

The transmitted signals arrive to the receiver in various paths of different length in a multipath channel. The effect of the multipath transmission cases ISI, and it also forces the subcarries to be orthogonal no longer and results ICI [22].

To avoid ISI and ICI, OFDM symbols are separated by guard intervals (GI).

The guard interval here is also called as cyclic prefix (CP). A cyclic prefix is a copy of the last part of the OFDM symbol illustrated in Fig. 2.6, and it is added in the front of the transmitted symbol and removed in the receiver before the demodulation.

### Figure 2.6 Cyclic prefix of OFDM symbol

The cyclic prefix should be at least as long as the channel impulse response described in Fig. 2.7. If the cyclic prefix is shorter than channel impulse response, the convolution would not be a circular one and ISI would be caused. However, if the sample number of CP is large, the data transmission rate will decrease with a factor of R (R = N / (N+M), where N is the length of the OFDM symbol and M is the length of CP). Thus it is important to choose the minimum length of CP to maximize the system efficiency.

### Figure 2.7 Inserting GI to avoid ISI

If an empty guard interval is used for preventing ISI in an OFDM symbol, the ICI problem will rise. For example, there is a two-paths channel, and the delayed signal from subcarrier2 interferes with the signal from subcarrier1 illustrated in Fig.

2.8. In order to eliminate ICI, a cyclic prefix is introduced as a guard interval to keep each subcarrier orthogonal illustrated in Fig. 2.9 [23].

### Figure 2.8 Multipath with empty GI causes ICI

### Figure 2.9 Inserting CP to avoid ICI

**2.3 Baseband Model of OFDM System **

Typical baseband architectures of OFDM system include transmitters and receivers, and the two parts are symmetrical. Block diagram of a typical OFDM system is illustrated in Fig. 2.10, and each block function is described in the following sections.

### Figure 2.10 Baseband architecture of OFDM system

### 2.3.1 Coding / Decoding

In the coding processor, a forward error correction (FEC) code is applied to decrease the bit error rate (BER). There are two types of FEC code in coding methods, block coding and convolutional coding. Block coding represents that the coding processor sends blocks of bytes at a time and adds extra bytes to the end of the block.

In the receiver, the extra bytes are removed directly for block decoding.

Convolutional coding represents that the data sequence multiply a pseudorandom sequence in coding processor, hence the data stream is added extra bits for data protection. Viterbi algorithm is used to track the transmitted data in convolutional decoding processor in the receiver [24].

### 2.3.2 Interleaving / Deinterleaving

Interleaving processor is applied to distribute burst errors, and the interleaving process makes sure that adjacent bits are distributed over the OFDM symbol [25].

### 2.3.3 Mapping / Demapping

A major disadvantage of OFDM is the high peak-to-average power rate (PAPR) of the transmitted signal in the transmitter, and adopting mapping processor can reduce the PAPR. A data block is mapped onto a point among associated points in the constellation that involves generating a large set of data vectors. This technique is very simple and does not require any side information from the transmitter to the receiver.

### 2.3.4 Pilot

Pilot symbols are inserted into the OFDM symbols, and they are used to correct frequency and time offset in the receiver. Carrier frequency and time offsets are estimated by calculating the different phases of pilot, and the estimated offsets is adopted to track the transmitted signal in the synchronizer.

### 2.3.5 IFFT / FFT

The OFDM subcarriers must be orthogonal in the frequency spectrum. The Inverse Fast Fourier Transform (IFFT) is a fast and efficient implementation of the Inverse Discrete Fourier Transform (IDFT) to generate the orthogonal carriers for OFDM transmissions.

### 2.3.6 Channel

The simplest channel model for a wireless transmission is additive white Gaussian noise (AWGN) channel described in Fig. 2.11. The characteristics of the white gaussian noise are statistically independent in any two noise samples and constant power spectral density.

### Figure 2.11 The AWGN channel model

In wireless communications, the transmitted signals arrive to the receiver

from various directions over a multiplicity of paths illustrated in Fig. 2.12. One such
type of reduction is called the multipath fading, and the phenomenon is known as
Rayleigh fading. In IEEE 802.11a standard, the model of the channel impulse
response is an exponential decay Rayleigh fading described in Fig 2.13 [26]. The
black illustrates the magnitudes of σ*k*2

and the gray illustrates the magnitudes of h* _{ k}*,
which is the impulse response of the channel with random uniform distribution in

*phase and Rayleigh distribution in magnitude. The k*max=10×TRMS/TS

*, k*max is the maximum number of paths, T

_{RMS}is the root mean square delay spread, and T

_{S}is the

*sampling time. The kth path in mathematics is shown as:*

h* k* = N(0,
2
1σ*k*2

) + j * N(0,
2
1σ*k*2

)

σ_{k}^{2} =σ_{0}^{2 }* exp(-kT_{S}/T_{RMS})

σ02

=1－exp(-TS/TRMS) where N(0,

2
1σ*k*2

) is a zero mean Gaussian random variable with variance
2
1σ*k*2

, which can be produced by generating a N(0,1) random variable and multiplying by

σ*k */√2 .[27]

### Figure 2.12 The multipath transmittance

### Figure 2.13 The multipath channel impulse response

As Fig 2.14 shows, the received signal after multipath fading channel can be expressed by the convolution of the signal with the impulse response and adding the Gaussian noise.

### Figure 2.14 The fading channel model

**2.4 OFDM** **For Wireless **

The most important wireless applications of OFDM are Digital Audio

Broadcasting (DAB), Digital Video Broadcasting-Terrestrial (DVB-T), Wireless Local Area Networks (WLAN).

### 2.4.1 DAB

DAB systems use ISO/MPEG1 audio coding for source coding and OFDM for modulation. MPEG1 is a highly efficient source coding technique. The audio frequency range is divided into 32 subcarriers. There are four transmission modes in DAB format (see Table 2.1), each of which supports a bandwidth up to 3GHz. [28]

Mode 1 Mode 2 Mode 3 Mode 4

Freq. Range 375MHz 1.5GHz 3GHz 1.5GHz Suitable

Communication Applications

Single freq.

Terrestrial Networks

Terrestrial / Satellite Communication

Satellite Communication

Terrestrial Communication

### Table 2.1 DAB specifications

### 2.4.2 DVB-T

DVB-T is a technique using OFDM as the basic modulation format in terrestrial networks and MPEG2 for source coding. There are two modes, the so-called 2k and 8k modes, using 1705 and 6817 carriers respectively. The common modulation for the carrier is QPSK, 16QAM, or 64-QAM. It is applied in digital terrestrial television, portable reception, and mobile reception. [29]

### 2.4.3 WLAN

802.11b WLAN has a problem that the speed is lower than wired LAN. In order to apply in high-speed network, IEEE 802.11a and IEEE 802.11g are provided latter.

IEEE 802.11a provides a data transfer rate of 54 Mbps and it works in a 5 GHz frequency band (The detail is introduced in next chapter). IEEE 802.11g can provide high data rate (up to 54 Mbps) at a lower frequency band, 2.4 GHz.

**Chapter 3 **

**IEEE 802.11a STD **

**3.1 Introduction **

OFDM technology is employed to transmit signal across 52 separate subcarriers to provide transmission of data at a rate of 6, 9, 12, 18, 24, 36, 48, or 54 Mbps in the IEEE 802.11a WLAN standard [2]. The operating frequencies fall into three 100MHz unlicensed national information structure (U-NII) bands: 5.12-5.25, 5.25-5.35, and 5.725-5.825 GHz. The 52 transmitted subcarriers are modulated by using binary phase shift keying (BPSK), quadrature phase shift keying (QPSK), 16-quadrature amplitude modulation (16-QAM), or 64-QAM. Four of the 52 subcarriers are pilot subcarriers, and they are used to detect the frequency or time shifts of the signal during transmission in the receiver. The other 48 subcarriers are used for sending the signal in a parallel mode. One channel bandwidth is 20MHz divided into 64 subcarrier frequency slots (52 subcarriers, 11 guard band subcarriers set to zero, and DC subcarrier), and every separated frequency slot is 0.3125 MHz.

Convolutional coding with a coding rate of 1/2, 2/3, or 3/4 is used for FEC coding[30].

**3.2 Protocol Architecture **

OFDM physical layer (PHY) is uesed to transfer media access control (MAC) protocol data units (MPDUs) into PHY convergence protocol data units (PPDUs) directed by the MAC layer. The OFDM PHY of the 802.11a standard is divided into two elements, the physical layer convergence protocol (PLCP) and the physical medium dependent (PMD) sublayers. PLCP defines a method of mapping the PHY sublayer service data units (PSDUs) into a framing format suitable for sending and receiving data and management information between two or more stations using the associated PMD system. PMD defines the characteristics and method of transmitting and receiving data through a wireless medium between two or more stations each using the same station system.

**3.3 PLCP Sublayer **

### 3.3.1 PLCP Frame Format

Fig.3.1 shows the frame format for 802.11a frame including the PLCP preamble, the SIGNAL, and the DATA. The PLCP preamble consists of 12 symbols, ten short symbols and two long symbols, and they are employed for synchronization in receiver. The SIGNAL consists of 24 bits defining data rate and length of the PSDU.

(The BPSK, QPSK, and QAM are used to combine the data rate as Table 3.1.) The first 4 bits (R1-R4) are used to encode the rate. The next bit is a reserved bit, 12 bits for length, one bit for parity, and 6 bits for tail. The Data contains 16 bits for SERVICE, PSDU, Tail bits and Pad bits. Table 3.2 is the list of timing parameters associated with the OFDM PLCP.

Data Rate

(Mbits/s) Modulation

Coding Rate (R)

Coded bits per subcarrier

(N_{BPSC})

Coded bits per OFDM

symbol(N_{CBPS})

Data bits per OFDM

symbol(N_{CBPS})
6 BPSK 1/2 1 48 24
9 BPSK 3/4 1 48 36
12 QPSK 1/2 2 96 48
18 QPSK 3/4 2 96 72
24 16-QAM 1/2 4 192 96
36 16-QAM 3/4 4 192 144
48 64-QAM 2/3 6 288 192
54 64-QAM 3/4 6 288 216

### Table 3.1 Rate-dependent parameters

Parameter Value NSD：Number of data subcarriers 48

NSP：Number of pilot subcarriers 4

N_{ST}：Number of subcarriers, total 52 (N_{SD }+ N_{SP})

Δ_{F}：Subcarrier frequency spacing 0.3125 MHz (20MHz / 64)
TFFT：IFFT / FFT period 3.2μs (1 /ΔF)

TPREAMBLE：PLCP preamble duration 1.6μs (TSHORT + TLONG)
T_{SIGNAL}：Duration of the SIGNAL-BPSK

OFDM symbol

4.0μs (T_{GI }+ T_{FFT})

TGI：GI duration 0.8μs (TFFT / 4)
TGI2：Training symbol GI duration 1.6μs (TFFT / 2)
TSYM：Symbol interval 4μs (TGI + TFFT)
T_{SHORT}：Short training sequence duration 8μs (10×T_{FFT }/ 4)
T_{LONG}：Long training sequence duration 8μs (T_{GI2}+ 2×T_{FFT})

### Table 3.2 Timing-related parameters

### 3.3.1.1 PLCP Preamble

The preamble consists of 12 symbols illustrated in Fig. 3.2. Ten of the symbols are short preamble for establishing automatic gain control (AGC), diversity selection, timing recovery, and the coarse frequency estimation. The other two

symbols are long preamble for channel estimation and fine frequency estimation.

### Figure 3.1 PPDU frame format

### Figure 3.2 PLCP preamble

The short preamble consists of ten repeated short training symbols. The short training symbols are constituted by 12 nonzero subcarriers modulated by the elements of the sequence S, given by

S-26.26= (13/6)×{0,0,1+j,0,0,0,-1-j,0,0,0,1+j,0,0,0,-1-j,0,0,0,-1-j,0,0,0,1+j,0,0,0, 0,0,0,0,-1-j,0,0,0,-1-j,0,0,0,1+j,0,0,0,1+j,0,0,0,1+j,0,0,0,1+j,0,0}.

The multiplication by “ (13/6) ” is in order to normalize the average power.

The long preamble consists of two repeated long training symbols. The long training symbols are constituted by 53 subcarriers (including DC) modulated by the elements of the sequence L, given by

L-26.26= {1,1,-1,-1,1,1,-1,1,-1,1,1,1,1,1,1,-1,-1,1,1,-1,1,-1,1,1,1,1,0, 1,-1-1,1,1,-1,1,-1,1,-1,-1,-1,-1,-1,1,1,-1,-1,1,-1,1,-1,1,1,1,1}.

### 2.3.1.2 Signal

SIGNAL field consists of 24 bits shown in Fig. 3.3. The first 4 bits (R1-R4) are used to encode the transmitting rate (Table 3.3). Bit 4 shall be “Reserved” for future use. The LENGTH has the length of 12 bits to indicate the number of octets in the PSDU. Bit 17 shall be a positive parity bit for bit 0 to bit 16. The last 6 bits are signal tail that shall be set to zeros

**Rate ( Mbits / s ) ** **R1-R4 **

6 1101

9 1111

12 0101

18 0111

24 1001

36 1011

48 0001

54 0011

### Table 3.3 RATE bits

### 3.3.1.3 Data

The DATA field contains the SERVICE field, PSDU, TAIL bits, and PAD

bits.

### Figure 3.3 SIGNAL field bits

3.3.1.3.1 Service, Tail, and Pad

The SERVICE field has the length of 16 bits. The bits 0-6 are used to synchronize the descrambler in the receiver and bits 7-15 shall be “Reserved” for future use.

The TAIL is a 6-bits field. All of the bits shall be set to zeros to ensure that the convolutional encoder is back to zero state.

Pad bits shall be inserted after the convolutional encoder and puncturer to
ensure that the encoded data stream is a multiple of the number of coded bits in an
OFDM symbol (N_{CBPS}). The number of OFDM symbols (N_{SYM}), the number of data
bits per OFDM symbol (NDBPS), the number of bits in the DATA field (NDATA) and
the number of pad bits (N_{PAD}), are computed from the length of the PSDU (LENGTH)
as follows:

NSYM = Ceiling [(16 + 8 LENGTH + 6)] / NDBPS] NDATA = NSYM NDBPS

NPAD = NDATA –(16 + 8 LENGTH + 6)

The function Ceiling ( ) is a function that returns the smallest integer value greater than or equal to its argument value. The appended bits (pad bits) are set to zeros and are subsequently scrambled with the rest of the bits in the MAC frame payload.

3.3.1.3.2 Data Scrambler

The DATA shall be scrambled with a length-127 scrambler. The generator
polynomial of the scrambler is S(x) = x^{7 }+ x^{4} +1, which is operated with all ‘1’ for
initial state to generate a 127 bit sequence. The sequence is: 00001110 11110010
11001001 00000010 00100110 00101110 10110110 00001100 11010100 11100111
10110100 00101010 11111010 01010001 10111000 11111111. The data scrambler
adopts the 127 bits sequence generator to scramble all bits in the data field to
randomize the bit patterns avoiding long streams of 1s and 0s.The scrambler is
described in Fig. 3.4.

### Figure 3.4 Data scrambler

3.3.1.3.3 Convolutional Encoder

Data are coded with a convolutional encoder of coding rate R = 1/2, 2/3, or
3/4. Fig. 3.5 illustrates the convolutional encoder of R = 1/2 using the
*industry-standard generator polynomials, g*0 = 1338* and g*1 = 1718. The bit denoted as

“A” should be the first bit generated by the encoder, followed by the bit denoted as

“B”. Another coding rates are derived from the basic rate of 1/2 using a technique called “puncturing”. Puncturing is “stealing” some of the encoded bits in the transmitter (thus reducing the number of transmitted bits and increasing the coding

rate) and inserting a dummy “zero” into the decoder bits in place of the stolen bits in the receiver (Fig. 3.6 & Fig. 3.7). Decoding the received signal is adopting the Viterbi algorithm to track the transmitted data.

### Figure 3.5 Convolutional encoder

### Figure 3.6 Puncturing (R=2/3)

### Figure 3.7 Puncturing (R=3/4)

3.3.1.3.4 Data Interleaving

In IEEE 802.11a OFDM, the main purpose of the interleaving is used to avoid burst errors due to channel fading in the frequency domain. The data bits are interleaved by a block interleaver (the data bits is written in columns and read in rows), and the block size is correspondent to the number of bits in one OFDM symbol (NCBPS). Table 3.4 is an example of 6×8 block interleaver.

0 6 12 18 24 30 36 42 1 7 13 19 25 31 37 43 2 8 14 20 26 32 38 44 3 9 15 21 27 33 39 45 4 10 16 22 28 34 40 46 5 11 17 23 29 35 41 47

### Table 3.4 Block interleaving

3.3.1.3.5 Mapping

OFDM modulation is employing BPSK, QPSK, 16-QAM, or 64-QAM. The
binary data sequence is divided into group symbols of 1, 2, 4,or 6 bits, depending on
the chosen data rate, and then the group symbols are converted into complex number
points of the BPSK, QPSK, 16-QAM, or 64-QAM constellations. The group symbols
of the constellation points are coded by Gray code illustrated in Fig. 3.8-3.10. The
converted complex values are the form of ”I+jQ” multiplied by a normalization factor
K_{MOD} listed in Table 3.5.

Modulation KMOD

BPSK 1 QPSK 1/√2 16 QAM 1/√10 64 QAM 1/√42

### Table 3.5 Mapping normalization factor

### Figure 3.8 BPSK & QPSK constellation

### Figure 3.9 16 QAM constellation

3.3.1.3.6 Pilot

In the IEEE 802.11a standard, pilot symbols are put in subcarrier –21, -7, 7,
*and 21 (Fig. 3.11). The pilot subcarriers for the nth OFDM symbol are produced by *
the sequence P*n *after Fourier transformation, given by

P* _{n –26,26}* ={ 0,0,0,0,0,p

*,0,0,0,0,0,0,0,0,0,0,0,0,0,p*

_{n}*,0,0,0,0,0,0,0, 0,0,0,0,0,0,p*

_{n}*n*,0,0,0,0,0,0,0,0,0,0,0,0,0,-p

*n*0,0,0,0,0 }

The sequence p* _{n }*can be generated by the scrambler with the “all ones” initial state,
and replacing “1” with “-1” and “0” with “1”, given by

p0-126= {1,1,1,1, -1,-1,-1,1, -1,-1,-1,-1, 1,1,-1,1, -1,-1,1,1, -1,1,1,-1, 1,1,1,1, 1,1,-1,1, 1,1,-1,1, 1,-1,-1,1, 1,1,-1,1, -1,-1,-1,1, -1,1,-1,-1, 1,-1,-1,1, 1,1,1,1, -1,-1,1,1, -1,-1,1,-1, 1,-1,1,1, -1,-1,-1,1, 1,-1,-1,-1, -1,1,-1,-1, 1,-1,1,1, 1,1,-1,1,-1,1,-1,1, -1,-1,-1,-1, -1,1,-1,1, 1,-1,1,-1, 1,1,1,-1, -1,1,-1,-1, -1,1,1,1, -1,-1,-1,-1, -1-1-1 }

### Figure 3.11 Pilot subcarriers

**3.4 Operating Frequencies **

The frequency band of 5 GHz U-NII is specified in the IEEE 802.11a standard. In the United States, the FCC (Federal Communication Commission) is the

agency responsible for the allocation of the 5 GHz U-NII bands. The 5 GHz U-NII frequency band is segmented into three 100 MHz bands. The lower band range is 5.15-5.25 GHz, the middle band range is 5.25-5.35 GHz, and the upper band range is 5.725-5.825 GHz. The lower and middle bands provide 8 carriers in a total bandwidth of 200 MHz and the upper band provides 4 carriers in a 100MHz bandwidth. The frequency channel center frequencies are spaced by a 20 MHz frequency band. The two outermost carrier centers of the lower and middle bands are at a distance of 30 MHz from the band edges and the upper bands are 20 MHz from the band edges (Fig.

3.12). Transmitting power levels are specified as: 40 mW (lower band), 200 mW (middle band), and 800 mW (upper band) (Table 3.6).

Band (GHz) Channel Number Center Freq. (MHz) Max Output Power (mW) 36 5180

40 5200 44 5220 Lower

Band (5.12-5.25)

48 5240

40

(2.5mW/MHz)

52 5260 56 5280 60 5300 Middle

Band (5.25-5.35)

64 5320

200

(12.5mW/MHz)

149 5745 153 5765 157 5785 Upper

Band (5.725-5.825)

161 5805

800

(50mW/MHz)

### Table 3.6 The 802.11a operating bands

### Figure 3.12 The 802.11a frequency channel

**Chapter 4 ** ** **

** **

**Interpolation **

**4.1 Polynomial-Based Interpolation Filter **

The interpolator, which is used to recover the originally transmitted data in the presence of STO can be regarded as a finite impulse response (FIR) filter [31]. For the purpose of simplifying the filter coefficients, a polynomial-based filter can be

employed [32]-[35]. The continuous-time output of the filter is shown as )

( ) ( )

( _{I}_{s}

*m*

*s* *h* *t* *mT*

*T*
*m*
*x*
*t*

*y* , (4.1)

where {x*(mT**s*)} *is a sequence of signal samples taken at intervals T**s* , and *h*_{I}*(t*) is
the finite impulse response of a time-continuous analog interpolation filter. Sampling
*y(t)* *at time instants t= kT**i ,* the sampled data would become as

) (

) ( )

( _{I}_{i}_{s}

*m*

*s*

*i* *x* *mT* *h* *kT* *mT*

*T*
*k*

*y* . (4.2)

Refer to Fig. 1.1 and 1.2, the interpolated data after the interpolation filter can be expressed as

] ) (

[ )

(*kT*_{i}*y* *m* *T*_{s}

*y*

] ) [(

] ) [(

2

1

*s*
*I*

*s*
*I*

*I*
*i*

*T*
*i*
*h*
*T*
*i*
*m*

*x* . (4.3)

*In (4.3), m is the largest positive integer which is less than or equal to * T_{i}/T_{s}, is

the estimated STO, I_{1} and I_{2} are fixed numbers (I_{1}=-M/2 and I_{2}=M/2-1, M is the length
of the filter impulse response). Let I = I2－I1＋1. Then, I is the number of branches
used for the interpolation. Moreover, N=I－1 is the degree of the interpolator. If N =1
and 3, the filters are called “linear interpolation filter” and “cubic interpolation filter”,
respectively.

**4.2 Filter Structure **

According to equation (4.3), we find that the continuous-time impulse
response )*h*_{I}*(t* is a function of variable, .In other word, the filter coefficients of the
interpolation filter are changeful because of different value of . In order to simplify the structure,
the interpolation filter coefficient might be evaluated by a suitable approach devised
by Farrow [36]. The interpolation filter impulse response would become as

] ) ( [ )

( _{I}_{s}

*I* *t* *h* *i* *T*

*h* ^{l}

*N*

*l*

*l* *i*

*b*

0

)

( , (4.4)

*where N is the degree of interpolation filter, and b** _{l}*(i) are filter coefficients for

*h*

_{I}*(t*). Note that all the filter coefficients are fixed. In practical implementation, the number of multipliers can be reduced if the number of calculations required for the determination of filter coefficients can be decreased. In this case, the interpolated data can be written as

2

1

) (

) ( ) (

*I*

*I*
*i*

*k*

*i* *x* *m* *i*

*T*
*k*
*y*
*k*

*y* ^{l}

*N*

*l*

*l* *i*

*b*

0

) (

*N*

*l*
*l*
0

2

1

) ( ) (

*I*

*I*
*i*

*l* *i* *x* *m* *i*

*b* . (4.5)

Obviously, the recovered data depends highly on . Consequently, effective interpolation necessitates accurate STO estimation. An example of the cubic

interpolation filter with the Farrow structure is shown in Fig. 4.1, and the filter coefficients are listed in Table 4.1. In this figure, the degree of filter is N=3 and the length of the filter impulse response is I=4. Thus, the number of filter branch is 4 and each filter branch has four taps. We will apply this cubic interpolation filter for our simulation.

Figure 4.1 Farrow structure of cubic interpolation filter

* i * * l=0 * *l=1 * *l=2 * *l=3 * -2 0 -1/6 0 1/6 -1 0 1 1/2 -1/2 0 1 -1/2 -1 1/2 1 0 -1/3 1/2 -1/6

*l*

*(i)*for cubic interpolator

**4.3 Another Interpolation Filter **

### 4.3.1 Lower Degree Interpolator

*In the previous section, the cubic interpolation filter (N=3) was discussed, *
and a good 2-degree interpolation filter is introduced in this section. In [37], the
authors provide a fixed coefficient of the 2-degree filter, and the filter coefficients are
shown in Table 4.2. Simpler structure of this 2-degree interpolation filter is shown in
Fig. 4.2 [38].

*i* *l=0* *l=0* *l=0*

### -2 0 -0.6741 0.6741 -1 0 1.4542 -0.4542 0 1 -0.5458 -0.4542 1 0 -0.6741 0.6741

*l*

*(i))*of 2-degree interpolation filter

### 4.3.2 Efficient Implementation of Interpolator

There is a new method to make Farrow structure more efficient by symmetrizing the fixed coefficients. Table 4.3 describes two different length and degree of interpolation filters called Interpolator I and Interpolator II. The coefficients of each interpolation filter are shown in Table 4.4 and Table 4.5. [39]

Degree Length Interpolator I 2 4 Interpolator II 3 8

* i=0 * * i=1 *

### b

0*(i) 0.58071 * -0.09914 b

1*(i) * 0.5 0

### b

2*(i) -0.008071 * 0.09914

*i=0 * *i=1 * *i=2 * *i=3 *

### b

0*(i) 0.61442 -0.15277 0.04856 -0.01132 * b

1*(i) * -0.60188 0.04089 -0.00402 -0.00076 b

_{2}

*(i) -0.11442 * 0.15277 -0.04856 0.01132 b

3*(i) 0.10188 -0.04089 0.00402 0.00076 *

The Farrow structure of Interpolator I is shown in Fig. 4.3. Obviously, we
can find that the coefficients of the structure are fixed and symmetrical. The structure
consists three FIR filter branches multiplying by (2 －1)^{n} and then summing them.

Figure 4.2 The simpler structure of 2-degree interpolation filter

Figure 4.3 Farrow structure of Interpolator I

**Chapter 5 **

**STO Estimation **

From Fig. 1.1 we can see that STO is the only parameter which will influence the calculation of the interpolator. For effective operation of the interpolator, the STO must be estimated accurately. In the specification of IEEE 802.11a WLAN standard, four pilots and short preamble are used for time offset estimation. Using the four pilots for estimation is so-called feedback estimation, and applying short preamble to estimate is called feedforward estimation.

**5.1 Feedback Estimation **

In feedback timing recovery loop, the STOis estimated by calculating the difference of
the phase rotation between two of the four-pilots in the TED, shown in Fig. 1.1. The
**estimated STO of the (n-2)th symbol using the pilots of (n-1)th and nth symbols is given by **

### μ

(n-2)=average (2

2 *p*1 *p*

*N* (

1 2

,*p* *p*
*pilot*
*p*

( ˆ , 1

*p*

*X**n* -

, 2

ˆ*n**p*

*X* ) - (

, 1

ˆ 1
*p*

*X**n* -

, 2

ˆ 1
*p*

*X**n* ) ) )

** **

**(5.1)**

*where p*1

*, p*2

*= -21, -7, 7, and 21, N is the length of OFDM symbol , and*

, 1

ˆ*n**p*

*X*
*means the phase of p*_{1}* pilot in the nth symbol. With those four pilot subcarries, there *

exists a total of six terms in the summation. [40] [41]

**5.2 Feedforward Estimation **

Feedback estimation has the drawback of long acquisition time. In the burst-mode transmissions, the receiver needs to achieve synchronization rapidly, and the feedforward estimation can accomplish it. So we adopt feedforward method to get the only changeable parameter ( ) of interpolator. In this section, we introduce two feedforward estimating methods and modify one of them to suit the OFDM system.

### 5.2.1 Maximum Likelihood Estimation

Synchronization algorithms of estimating the STO are based on the maximum likelihood estimation (MLE) theory, and these algorithms are used to find the STO which maximizes the log-likelihood function (Λ(τ)). [8]

Λ(τ) =

*N*

*n 1*

a^{*}*(n) m(n,τ) (5.2) *

where a^{*}*(n) are the conjugate of the deterministic known preamble sequence, m(n,τ)*
are the received short preamble sequence, andτ(0≦τ＜T_{s}*)is the timing error. N is *
the number of used samples and Ts is the sampling interval.

In this system, the received signals are sampled at the double sampling rate and a cubic interpolation filter is used for an interpolator. Because of the double sampling, we can use two different polynomial approximations for the log-likelihood functions as

Λ_{1}(2τ) =

*N*

*n 1*

a^{*}*(n) m(2n-1,2τ) , for 0≦τ＜0.5 *

Λ_{2}(2τ-1) =

*N*

*n 1*

a^{*}*(n) m(2n,2τ-1), for 0.5≦τ＜1 (5.3) *

The range of 2τand 2τ-1 are in the interval [0,1), and therefore, 2τand 2τ-1 can be
*replaced by μ. The values of m(n,μ) are produced by the cubic interpolator in Farrow *
structure using the received short preamble as input.

m(n,μ) =

3

0
*i*

μ* ^{i}* f

*i*

*(n) (5.4)*

where f* i**(n) are the branch filter outputs of cubic interpolation filter. Consequently, the *
log-likelihood function become:

Λ1(μ) =μ^{3 }X ^{3 1 }+μ^{2 }X ^{2 1 }+μX ^{1 1 }+ X ^{0 1 }

Λ_{2}(μ) =μ^{3 }X 3 2 +μ^{2 }X 2 2 +μX 1 2 + X 0 2 (5.5)
where

X*i 1 *=

*N*

*n 1*

a^{*}*(n) f** i**(2n-1) for i = 0,….,3 *

X* i 2 *=

*N*

*n 1*

a^{*}*(n) f** i**(2n) for i = 0,….,3 (5.6) *

The block diagram of maximum likelihood estimation is shown in Fig 5.1 that achieves the following:

1. When the short preamble is received,

A . computes the outputs of the interpolator branches f* i**(n) *
B. divides f_{ i}*(n) into two parts (odd and even). *

C. computes X* i 1 *and X* i 2*

2. obtain the timing offset , μ

A .Λ1(μ) orΛ2(μ) which has the larger maximum value that is the log-likelihood function we select.

B. find the timing offset μ which maximizes the log-likelihood function

3. using μ to track in the interpolator

Figure 5.1 Block diagram of the ML estimation

### 5.2.2 Fourier Transform Estimation

In [9], authors proposed a feedforward approach to estimate the STO using correlations of the known preamble sequences preceding a data packet is proposed.

Let *a be the deterministic known preamble sequence, and let N be the length of the *_{m}
sequence used in the correlation. Furthermore, let x(t) be the received sequence and
let )r_{xx}(*i denote the correlation. Then, the correlation can be described as *

)
(
r_{xx} *i =*

1

0
*N*

*m*

a^{*}_{m }*x( iT*_{s} + 2mT_{s}), . (5.7)

in which ^{*} denotes complex conjugate. Assume that each symbol consists of two
samples. That is, the symbol period is 2T . After some derivations, we find that the _{s}
STO, , can be represented as

)]

R(e [ 2 arg j

, (5.8)

where R(*e** ^{j}* ) is the Fourier transform of rxx

*(i), shown as:*

R(*e** ^{j}* ) =

*i*

r_{xx}*(i) * *e*^{j}* ^{i}*. (5.9)

Based on (5), enumeration of R(*e** ^{j}* ) requires infinite number of terms. To make the
Fourier transform implementation realistic, the authors truncate the r

_{xx}(

*i sequence*) before taking Fourier transform. Specifically, only four samples of r

_{xx}(

*i sequence*) are used in the original work, as shown in (5.10).

RT( e ^{j}^{π}^{/2}) = [rxx(0) - rxx(2)] + j[rxx(-1) - rxx(1)]. (5.10)

Thus the estimating equation of STO would become as (5.11), and the estimation structure is described as Fig. 5.2.

μ= 2

arg( RT( e ^{j}^{π}^{/2}) ) (5.11)

Figure 5.2 The stricture of STOestimation in [9]

Obviously, the structure of the Fourier transform becomes much simpler.

This, however, calls for an extra hardware circuit for the receiver. Recall that in an OFDM receiver, there is an FFT processor used for demodulation. For the 802.11a

OFDM system, operation of the FFT processor is not necessary during the STS period.

This motivates us to use the FFT processor for the computation of the Fourier
transform during its idle period. To probe further, we take correlation of the STSs to
get the r_{xx}(*i sequences and then use the existent FFT processor to enumerate the *)
Fourier transform, depicted in Fig. 5.3. In this way, the performances shown in Fig 5.4
are very close, but the additional hardware requirement described in [39] can be
eliminated.

Figure 5.3 Structure of using the FFT for the sampling time offset estimation

It is necessary to address the feasibility of the new scheme. Specifically, we
need to justify that the FFT processor in the standardized 802.11a receiver can achieve
our mission. According to the IEEE 802.11a specifications, the received signal should
be detected within the STS t to _{1} t (see Fig. 5.5). This implies that the symbol _{7}
time offset can be estimated during the interval t to _{1} t and the FFT window must _{7}
be adjusted accordingly, except the possible STO. For our scheme, the symbol time
can be detected by observing the peaks of the r_{xx}(*i sequence. This information is *)
used to control the FFT window start position to within an STO interval. Thus,
misalignment of the FFT window start position will be only STO, if ant, after the t _{7}
STS.

Figure 5.4 The result of using Dengwei’s estimation and FFT estimation

Another issue that should be considered regarding to the feasibility of our
scheme is that whether there is sufficient time for the FFT processor to carry out the
task without obstructing its duty work. Recall that the FFT processor is always
functioning during the LTS interval for channel estimation or residual frequency
offset correction. This means that the FFT processor can be employed to do the
computation only before the start of the LTS. From the above description, we can see
that the STS from t_{8} to t_{10} and the guard interval G12 prior to the LTS are
available. Thus, there are 4μs (=2.4μs+1.6μs) in total available for our need. This
is greater than the FFT window size 3.2μs specified in the standard. Thus, there is
enough duration for the FFT processor to accomplish the STO calculation. In the
following, we present detailed operation of STO estimation using the FFT processor.

Figure 5.5 Preamble of the IEEE 802.11a WLAN

Equation (5.10) can be obtained by dividing the four truncated samples,
rxx(-1)~ rxx(2), into odd part (multiplied by “j”) and even part. Then, a 4-points FFT
can be employed to do the transformation. An alternative is to use a 32-points FFT,
which is a half of the 64-points FFT specified in the IEEE 802.11a, to perform the
computation. We take correlations of the STSs to provide correlation sequences based
on (3). Then, choose 32 terms of the correlation sequence, rxx(-16) ~ rxx(15), and
divide these 32 terms into an odd part and an even part. The odd part consists of the
number of the FFT input from 0 to 15 (after being multiplied by j), and the even part
is comprised of the input from 16 to 31. All the remainder inputs of the FFT are set to
zeroes. Based on this input arrangement, we can find that the 64-points FFT is
equivalent to two 32-points FFT, and we choose the upper 32-points FFT for our
purpose. Since the number of the adopted correlation sequence terms is eight times of
the number of terms dictated by (5.10), we can use eight terms of the FFT outputs for
estimation. If the output pin number (0), (8), (16), (24), (32), (40), (48), and (56) of
the FFT (these terms are the upper eight output pins of a 64-points FFT with butterfly
structure depicted in Fig. 5.6 [15]) are selected, only four inputs, rxx(-16), rxx(-15),
r_{xx}(0), and r_{xx}(1) are needed. It is noted that these four inputs represent two peak
positions of estimation equation (5.7) which we need. Alternatively, if we only select

the first seven terms of the above eight outputs, the value of (5.7) will depend heavily on the four “peak” input terms and little on the other 28 input terms. Gathering the minor contribution resulted from these 28 inputs can reduce the effect of channel noise. This can improve the STO estimation accuracy. In this work, an angle is computed based on the average of the selected seven output terms. Multiplying the angle by 4/π yields the vale of as illustrated in Fig. 5.7

Figure 5.6 The butterfly structure of 64-points FFT

Figure 5.7 Structure of using the FFT for the sampling time offset estimation

**5.3 Simulation and Some Results **

We consider burst-mode transmissions of the OFDM-based IEEE 802.11a WLAN for our simulations. The number of burst block is 250, and each burst block contains 25 OFDM symbols. Performance of the system that uses the FFT processor to calculate the STO will be compared to that which can be achieved by using a Maximum-Likelihood (ML) estimation method. In the sequel, the notation “FT” and

“ML” will be used to denote a system that employs the FFT calculation and the ML algorithm, respectively.

For AWGN channels and QPSK modulation, Fig. 5.8 and 5.9 display the bit error rate (BER) of a system that can be achieved by using FT and ML method, as a function of signal to noise ratio (SNR). We have simulated for two values of N, which is the number of STS samples used in the calculation, N=96 and 32. From these figures we can see that for any value of N, the FT method outperforms the ML method. In addition, we can also find that larger N will result in better performance.

This is the direct consequence that large number of samples will average out the variance of AWGN noise. Using N=96 and in AWGN channels, we compare the BER performance of FT and ML with QPSK and 16-QAM in Fig. 5.10. It can be observed that the FT method can perform better than the ML method regardless of the modulation modes. Similar characteristics can be found for the BER performance under a Rayleigh multipath fading channel [26] shown in Fig. 5.11 and 5.12. But we can see from Fig. 5.11 and 5.12 that the BER performance will level off when the SNR is larger than some specific values. This is because that when SNR is high (AWGN noise is low), the interference resulted from the multipath effect will dominate the performance. Thus, further increase of SNR will not significantly