1. ADAPTIVE TRANSCEIVER DESIGN AND PERFORMANCE
ANALYSIS FOR OFDM SYSTEMS
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF ELECTRICAL
ENGINEERING
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Wonchae Kim
June 2009

2. c Copyright by Wonchae Kim 2009
All Rights Reserved
ii

3. I certify that I have read this dissertation and that, in my opinion, it is fully
adequate in scope and quality as a dissertation for the degree of Doctor of
Philosophy.
(Donald C. Cox) Principal Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully
adequate in scope and quality as a dissertation for the degree of Doctor of
Philosophy.
(John M. Cioffi)
I certify that I have read this dissertation and that, in my opinion, it is fully
adequate in scope and quality as a dissertation for the degree of Doctor of
Philosophy.
(Ravi Narasimhan)
Approved for the University Committee on Graduate Studies.
iii

4. Abstract
With the enormous demand for wireless access to the Internet for packet data and voice
applications, Wireless Local Area Networks (WLANs) and Wireless Metropolitan Area
Networks (WMANs) are becoming ubiquitous. As is the case in all wireless systems, appli-
cations carried over these networks are subject to impairments such as path-loss, shadowing
and fading in the wireless channel. These impairments lead to transmission errors and con-
sequently, packet loss, which degrades the Quality of Service (QoS) perceived by a user. In
this study, we focus on coded orthogonal frequency division multiplexing (OFDM)-based
WLANs and WMANs. Adaptive transceivers can provide considerable improvements in
the performance of OFDM systems ; however, the design of adaptive OFDM transceivers
can be very complex and challenging due to estimation errors and limited knowledge of
channel information.
The fading characteristics of the indoor wireless channel are very different from the
ones we know from mobile environment. In indoor wireless systems, the transmitter and
receiver are stationary and people are moving around, while in mobile systems the user
is often moving through an environment. As a result, we propose a new model for time
varying indoor channel in order to fit the Doppler spectrum measurements
In the second part of the dissertation, time and frequency synchronization problems
in an OFDM inner receiver will be presented. In the burst packet mode OFDM systems,
synchronization needs to be done very fast to avoid the reduction of the system capacity
and also must be very accurate to minimize interferences. We analyzed effect of estimation
iv

5. error on the system performance and proposed adaptive synchronization methods based on
windowing and Kalman filtering to mitigate estimation errors with reasonable complexity.
For several different channel environments, numerical results show that the proposed meth-
ods can significantly decrease synchronization errors without the need for prior knowledge
of channel conditions.
In the third part of the dissertation, we propose an enhanced DFT-based minimum mean
square error (MMSE) channel estimator using the Kalman smoother. In practical OFDM
systems with virtual carriers (VCs), conventional DFT-based approaches are not directly
applicable as they induce a spectral leakage owing to VCs, which results in an error floor
for the mean square error (MSE) performance. We applied Kalman smoothing to minimize
the leakage effect and time domain MMSE weighting is also used to suppress the channel
noise.
Finally, using Request to Send (RTS) and Clear to Send (CTS) mechanism, we in-
troduce a method to improve throughput performance by adaptively changing constellation
size and power distribution across the sub-carriers without sacrificing throughput due to ex-
plicit feedback. Based on theoretical analysis, part of this complex maximization problem
approximately reduced to a Lagrange equation and the objective function can be solved
by a simple iterative algorithm. Simulation results, using the proposed channel model,
show that this algorithm combined with the proposed estimation methods is a promising
approach to solving throughput optimization problems within practical impairments.
v

6. Acknowledgements
I would like to first thank my adviser, Professor Donald C. Cox. He has been a great mentor,
and I was very fortunate to have him as my principal adviser. His expertise in wireless
communication has been truly valuable in this research, and I have learned everything
from introductory communication theory through standard communication systems and
estimation theory from him. This dissertation would have not been possible without him.
My other members of the reading committee, Professor John M. Cioffi and Professor
Ravi Narasimhan, were very helpful and I would like to thank them for their time. Pro-
fessor Cioffi really introduced me into the field of multi-carrier modulation and I learned a
great deal on mathematical analysis from him, which I used extensively throughout this dis-
sertation. Professor Narasimhan gave me a lot of insights about wireless channel through
his papers, which helped me in finding a good topic for my research. I would like thank
Professor Cioffi for his valuable input as an expert in multi-carrier systems and Professor
Ravi for his comments from his background in wireless LAN system design.
Taking lectures from world famous scholars in Stanford was certainly a privilege for
me. I have taken invaluable classes from a number of professors in Stanford, and these lec-
tures not only prepared me in doing my research, but also increased my general knowledge
in this field.
I thank the members of the wireless communications research group for their helpful
discussions: Mehdi Soltan, Hichan Moon, Ali Faghfuri, Vahideh HosseiniKhah, Hyunok
Lee and Tom McGiffen. I also thank colleagues in different research groups including
vi

7. Eunchul Yoon, Jiwoong Choi and Seongho Moon.
I was really fortunate to have great friends at Stanford. They include, but are not lim-
ited to, Youngjae Kim, Changhwan Sung, Kwangmoo Koh, Wooyul Lee, Woongjun Jang,
Jeunghun Noh and Hochul Shin. They have been great friends, who gave me the courage
to move forward and finish my study. I am also grateful for what I have received from
Samsung Lee Kun Hee Scholarship Foundation, who took part in funding my study at
Stanford.
And finally, I would like to thank my family, my wife Juyoung Ha, my brother Wony-
oung Kim, and my parents Hongryul Kim and Jungsub Lee, for their unconditional love
and encouragement, which led to my Ph.D. degree at Stanford. This doctoral dissertation
is dedicated to my parents.
vii

8. Contents
Abstract iv
Acknowledgements vi
1 Introduction 1
1.1 Why OFDM? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Research Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Indoor Wireless Channel 8
2.1 Types of small scale fading . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 ETSI Channel models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Modeling the Time Varying Channel . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Mobile Radio Channel . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.2 Indoor Wireless Channel . . . . . . . . . . . . . . . . . . . . . . . 16
3 Adaptive Timing Synchronization for OFDM Systems 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Frequency and Timing Synchronization . . . . . . . . . . . . . . . . . . . 28
3.3.1 Coarse Frequency Offset Estimation . . . . . . . . . . . . . . . . . 28
viii

9. 3.3.2 Adaptive Timing Synchronization method . . . . . . . . . . . . . . 29
3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.1 Simulation Environment . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.2 Random Channel Generation . . . . . . . . . . . . . . . . . . . . . 34
3.4.3 Performance Results . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 Residual Frequency Offset and Phase Tracking 42
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 The Effect of Residual Frequency Offset . . . . . . . . . . . . . . . . . . . 45
4.3 The Proposed Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.1 State-Space Modeling . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.2 The Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.3 Complexity Consideration . . . . . . . . . . . . . . . . . . . . . . 54
4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4.1 Simulation Environment . . . . . . . . . . . . . . . . . . . . . . . 55
4.4.2 Performance result . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5 Enhanced DFT-Based MMSE Channel Estimation 61
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Kalman Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3 MMSE Filtering in the time domain . . . . . . . . . . . . . . . . . . . . . 69
5.4 Complexity issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.5.1 Simulation Environment . . . . . . . . . . . . . . . . . . . . . . . 71
5.5.2 Performance result . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
ix

10. 6 Throughput Enhancement for IEEE 802.11a Wireless LANs 76
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.3 Throughput Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3.1 Throughput Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3.2 The Minimum PER Problem . . . . . . . . . . . . . . . . . . . . . 84
6.3.3 Throughput Enhancement Method . . . . . . . . . . . . . . . . . . 86
6.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.4.1 Simulation Environment . . . . . . . . . . . . . . . . . . . . . . . 87
6.4.2 Performance of the Proposed Inner Receiver . . . . . . . . . . . . 88
6.4.3 Performance of Throughput Optimization . . . . . . . . . . . . . . 89
6.4.4 Benefits of Throughput Optimization . . . . . . . . . . . . . . . . 92
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7 Conclusion 97
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Bibliography 102
x

11. List of Tables
2.1 ETSI channel models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.1 IEEE 802.11a PHY parameters . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
xi

12. List of Figures
1.1 Block diagram of an OFDM transceiver . . . . . . . . . . . . . . . . . . . 2
1.2 OFDM as a broadband communication system . . . . . . . . . . . . . . . . 4
2.1 Two types of small scale fading . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Power delay profile for channel A and B . . . . . . . . . . . . . . . . . . . 11
2.3 Power delay profile for channel C and E . . . . . . . . . . . . . . . . . . . 12
2.4 Illustration of Doppler shift . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Geometry of a single ray . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 Doppler power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.7 Autocorrelation function . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8 Comparison between Doppler spectrum measurement and proposed Doppler
spectrum model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1 802.11a - Frame and slot structure . . . . . . . . . . . . . . . . . . . . . . 24
3.2 802.11a - Subcasrrier allocation . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Timing synchronization diagram . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Exemplary plot of D(n) at SNR=10dB . . . . . . . . . . . . . . . . . . . . 31
3.5 QW output when W = 16, 8, 4 and 3 . . . . . . . . . . . . . . . . . . . . . 32
3.6 A time variation technique for simulation . . . . . . . . . . . . . . . . . . 35
3.7 Average packet error rate for 4QAM in Channel B . . . . . . . . . . . . . . 36
3.8 Average packet error rate for 64QAM in Channel B . . . . . . . . . . . . . 37
xii

13. 3.9 Average packet error rate for 4QAM in Channel C . . . . . . . . . . . . . . 37
3.10 Average packet error rate for 64QAM in Channel C . . . . . . . . . . . . . 38
3.11 Histogram of timing estimates for Channel B, SNR=10dB: Proposed . . . . 39
3.12 Histogram of timing estimates for Channel B, SNR=10dB: Conventional . . 40
3.13 Histogram of timing estimates for Channel C, SNR=10dB: Proposed . . . . 40
3.14 Histogram of timing estimates for Channel C, SNR=10dB: Conventional . . 41
4.1 SNR degradation due to RFO . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 64-QAM signal constellation without RFO . . . . . . . . . . . . . . . . . . 48
4.3 64-QAM signal constellation with RFO . . . . . . . . . . . . . . . . . . . 48
4.4 Comparison of phase error for = 0.01: ML . . . . . . . . . . . . . . . . . 50
4.5 Comparison of phase error : Kalman . . . . . . . . . . . . . . . . . . . . . 55
4.6 Average packet error rate for 4-QAM without channel estimation error ( =
0.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.7 Average packet error rate for 64-QAM without channel estimation error
( = 0.05) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.8 Average packet error rate for 4-QAM ( = 0.1) . . . . . . . . . . . . . . . 59
4.9 Average packet error rate for 64-QAM ( = 0.05) . . . . . . . . . . . . . . 59
4.10 Error variance of RFO estimation ( = 0.1) . . . . . . . . . . . . . . . . . 60
5.1 Block diagram of DFT-based method . . . . . . . . . . . . . . . . . . . . . 64
5.2 Existence of virtual subcarriers . . . . . . . . . . . . . . . . . . . . . . . . 65
5.3 Kalman filtering and smoothing output, SNR=10dB . . . . . . . . . . . . . 68
5.4 MSE comparison in Channel A . . . . . . . . . . . . . . . . . . . . . . . . 73
5.5 SER comparison in Channel A . . . . . . . . . . . . . . . . . . . . . . . . 73
5.6 MSE comparison in Channel B . . . . . . . . . . . . . . . . . . . . . . . . 74
5.7 SER comparison in Channel B . . . . . . . . . . . . . . . . . . . . . . . . 75
xiii

14. 6.1 Block diagram of the simulated BIC-OFDM system . . . . . . . . . . . . . 80
6.2 Timing of successful frame transmission . . . . . . . . . . . . . . . . . . . 83
6.3 Timing of frame transmission failure . . . . . . . . . . . . . . . . . . . . . 83
6.4 Flow chart for iterative procedure for finding power distribution for mini-
mizing packet error rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.5 An exemplary plot of constellation size variation with respect to time . . . . 88
6.6 Throughput comparison for improved inner receiver: Channel A & B . . . . 90
6.7 Throughput comparison for conventional system : Channel A & B . . . . . 91
6.8 Throughput comparison for adaptive transceiver in Channel A & B . . . . . 93
6.9 Throughput comparison for adaptive transceiver in Channel A & B . . . . . 94
6.10 Throughput comparison for only adaptive transceiver in Channel A & B . . 95
xiv

15. Chapter 1
Introduction
Wireless communication has gained a momentum in the last decade of the 20th century
with the success of 2nd Generations (2G) of digital cellular mobile services. Worldwide
successes of GSM, IS-95, PDC, and IS-54/137 are of the few examples demonstrating the
advancement of wireless communications and applications. These systems have initiated
an innovative way of life for the new information and communication technology era. The
total number of cellular subscribers was more than 3 billion in 2007 and now is expected
to exceed approximately 4 billion in 2009. In addition, many of these new subscribers
have started using a number of different forms of data services as well as voice services.
Increasing user demands have drawn the industry to search for better solutions to support
data rates in the range of tens of Mbps. This motivated researchers towards finding a better
solution for handling the nature of wireless channels and limited resources such as power
and bandwidth.
The idea of using multi-carrier transmission for high data rate communications has sur-
faced recently in order to overcome the hostile environments of wireless channels. OFDM
is a special form of multi-carrier transmission where all the subcarriers are orthogonal to
each other. OFDM promises a higher user data rate and greater resilience to severe signal
fading effects of the wireless channel at a reasonable level of implementation complexity.
1

16. CHAPTER 1. INTRODUCTION 2
Bit & Power
Allocation
OFDM
Modulator
Channel
Inner
Receiver
Outer
Receiver
Encoder
Interleaver
Timing Synchronization Frequency Offset Estimation Channel Estimation
Figure 1.1: Block diagram of an OFDM transceiver
OFDM has developed into a popular scheme for wideband digital communication, whether
wireless or over copper wires, and has been used in applications such as digital television,
audio broadcasting, wireless networking, and broadband internet access. In addition, wire-
less communication has utilized OFDM as the primary physical layer technology in high
data rate Wireless LAN/MAN standards. For example, IEEE 802.11a has the capability to
operate in a range of a few tens of meters in typical office space environment whereas IEEE
802.16a uses Wideband OFDM (W-OFDM), a patented technology of Wi-LAN, to serve
up to 1 km radius of high data rate fixed wireless connectivity. Furthermore, OFDM may
become the prime technology for 4G. Pure OFDM or hybrid OFDM will be most likely the
choice for physical layer technology in future generations of telecommunications systems.
1.1 Why OFDM?
A simplified OFDM transceiver system is described in Fig. 1.1. In a digital domain, binary
input data are collected and FEC coded with schemes such as convolutional codes. The
coded bit stream is interleaved to obtain diversity. Afterwards, a group of channel coded
bits are gathered together (1 for BPSK, 2 for QPSK, 4 for QPSK, etc.) and mapped to
corresponding constellation points. At this point, the IFFT operation is performed on the
parallel complex data and a cyclic prefix is inserted in every block of data according to the

17. CHAPTER 1. INTRODUCTION 3
system specification. Now, the data is OFDM modulated and ready to be transmitted. After
the transmission of an OFDM signal through a wireless channel, an inner receiver performs
carrier frequency synchronization and symbol timing synchronization. After these steps,
an FFT operation is performed and a channel estimate is obtained. At this point, the com-
plex received data are demapped according to the transmission constellation diagram using
inner receiver estimates. Finally, FEC decoding and deinterleaving are used to recover
the originally transmitted bit stream in the outer receiver. In this thesis, we are going to
present solutions for questions about how to improve inner receiver performance and how
to efficiently allocate bit and power across subcarriers.
OFDM can offer several advantages over single carrier communication systems[1].
First of all, it can efficiently handle frequency selective channels. At high data rates, chan-
nel distortion to data transmission is very significant, and it is difficult to fully recover the
transmitted data with a simple receiver. A very complex receiver structure is needed which
makes use of computationally extensive equalization and channel estimation. OFDM can
drastically simplify the equalization problem by turning the frequency selective channel
into a flat channel. A simple one-tap equalizer is needed to estimate the channel and recover
the data. In addition, in a relatively slow time-varying channel, OFDM can significantly
improve capacity by adapting data rate across subcarriers. This is very useful for multi-
media communications. Furthermore, OFDM is robust against narrowband interference
because such interference affects only a small percentage of the subcarriers. Lastly, OFDM
makes single-frequency networks possible, which is especially attractive for broadcasting
applications.
1.2 Research Challenges
The radio channel has a crucial impact on the transmission of information through it. Multi-
path propagation will occur during a significant part of the time and this causes a frequency

18. CHAPTER 1. INTRODUCTION 4
Sub-carrier
magnitude
Carrier
Channel
Figure 1.2: OFDM as a broadband communication system
and time selective behavior of the channel response. As the phenomena are random, chan-
nel models for the linear time-variant radio channels are required to estimate the perfor-
mance of radio links and radio networks.
Also, if there are some estimation errors in carrier frequency or symbol timing, it will
induce significant errors in communication. The success of wireless OFDM system de-
pends strongly on synchronization. The higher the data rates are, the stricter the synchro-
nization requirements become. In order to build systems to support higher and higher data
rates, there is a need for algorithms and system designs that can facilitate robust estimation
of the synchronization parameters with minimum computational complexity.
Channel estimation is another primary requirement of an OFDM transceiver that per-
forms coherent reception. The capacity of a system is largely dependent on the channel
estimation scheme used in the system. The more accurate the channel estimate is, the bet-
ter the quality of service. OFDM offers a built-in very simple frequency domain channel
estimation scheme. Despite the fact that the scheme is simple enough, it does not perform
accurately under very low SNR conditions.
In 802.11a, the link adaptation algorithm is intentionally left open. Although many
previous studies have been focused on this particular topic, many of them are not directly
applicable to real systems. In addition, the actual optimization benefit that can be realized

19. CHAPTER 1. INTRODUCTION 5
after taking into account complexity always remains a question.
This dissertation explores the applicability of statistical estimation and optimization
techniques to the above mentioned problems in OFDM systems. Using 802.11a as an ex-
ample, we analyze the effect of various estimation errors and propose novel methods to mit-
igate synchronization and channel estimation error with reasonable complexity. Moreover,
we introduce a simple method to improve throughput performance by adaptively chang-
ing constellation size and power distribution across the sub-carriers without sacrificing
throughput due to explicit feedback. By employing the proposed scheme, we examine
the value of optimization with practical impairments.
1.3 Outline of the Thesis
Chapter 1 is a brief introduction and motivation. Chapter 2 considers an indoor wireless
channel model. An indoor wireless channel is always very unpredictable with harsh and
challenging propagation conditions. Measurement results show that an indoor wireless
channel is very different from a mobile channel in many ways. We particularly focused on
a delay spread model in this study and propose a new model for Doppler power spectrum
for an indoor channel. These models in Chapter 2 will be the basis for our discussion on
how we can improve the current systems in later chapters.
Adaptive timing synchronization for frequency selective channels is studied in Chapter
3. In burst packet mode OFDM systems, timing synchronization need to be done within a
single training symbol time to avoid reduction of the system capacity. Due to this stringent
requirement on synchronization time, standards incorporate preambles suitable for corre-
lation to estimate symbol timing. However, in time-dispersive multi-path channels, the
conventional timing synchronization methods might synchronize to a path in the middle
of the overall channel impulse response (CIR). Consequently, the receiver may not capture
some of the multi-path components. This results in an inter-symbol interference (ISI) and

20. CHAPTER 1. INTRODUCTION 6
an inter-carrier interference (ICI). In this chapter, we present a novel timing synchroniza-
tion method for OFDM systems to detect the most significant channel taps by adaptively
changing observation window length. The method does not require any extra channel infor-
mation such as signal to noise ratio (SNR) or average power delay profile, while allowing
detection of the first arrived path position. Additionally estimating maximum delay spread
and total channel power can be used to increase system capacity in other applications.
Chapter 4 moves on to a residual frequency offset and phase tracking problem. In
OFDM systems, carrier frequency offset (CFO) due to mismatch of the local oscillators
causes ICI, which may result in significant performance degradation. Although, several
frequency synchronization schemes were reported in the past, there can remain frequency
offset and that can still generate ICI and induce phase distortion of the OFDM symbols.
In this chapter, we propose a method to compensate both residual frequency offset (RFO)
and RFO induced phase error (PE) for OFDM systems by using the Kalman filter. In our
proposed method, the linear state-space model for RFO and PE is derived using estimated
SNR. After building a state-space model, the Kalman filter is applied to track and estimate
RFO and PE simultaneously. The proposed method allows unknown parameters to evolve
in time due to frequency drift of the local oscillator. The method is an optimal linear
estimator assuming signal and noise are jointly Gaussian. Furthermore, the computation
cost of the proposed method is much lower than that of the LS phase fitting method due to
the small dimension of the state-space model.
Chapter 5 also considers another estimation problem in an OFDM inner receiver. In
practical OFDM systems with virtual carriers (VCs), conventional DFT-based approaches
are not directly applicable for channel estimation as they induce a spectral leakage owing
to the VCs. This results in an error floor for the mean square error (MSE) performance. To
circumvent this problem, we propose an enhanced DFT-based minimum mean square error
(MMSE) channel estimator using the Kalman smoother. Our approach is based on building
a robust state-space model for a channel frequency response (CFR). Kalman filtering and

21. CHAPTER 1. INTRODUCTION 7
smoothing is then applied to minimize the leakage effect. Time domain MMSE weighting
is also used to suppress the channel noise. This proposed method does not require extra
knowledge about the channel statistics and can be implemented with small complexity
while achieving similar performance to the optimal MMSE estimation.
As we mentioned above, OFDM in combination with bit-interleaved coded modula-
tion is an efficient and robust high-speed transmission technique used in the IEEE 802.11a
standard. In Chapter 6, using the request to send (RTS)/ clear to send (CTS) mechanism,
we present a throughput enhancement method by deriving a simplified expression for the
throughput in the 802.11a system. The IEEE 802.11 MAC specifies for the contention-
based distributed coordination function (DCF) access method to exchange short control
frames - RTS/CTS prior to data transmission. RTS/CTS handshaking is essentially a
medium reservation scheme, and this mechanism is one of the effective ways to alleviate the
hidden node problem under DCF. Assuming channel reciprocity, we incorporate this mech-
anism for getting channel information at the transmitter without sacrificing throughput due
to explicit feedback. After acquiring channel knowledge, a simple iterative algorithm is
used to select constellation sizes and power distribution across the sub-carriers to enhance
the throughput.
As a conclusion, we review the results we have obtained and present some ideas for
future research in Chapter 7 and conclude this thesis.

22. Chapter 2
Indoor Wireless Channel
Due to the nature of wireless communications, wireless channels have very different char-
acteristics from wire-line channels. The mechanisms which govern radio propagation are
complex and diverse, and they can generally be attributed to three basic propagation mech-
anism as follows: reflection, diffraction and scattering. One of the most important charac-
teristics of a multi-path channel is the time varying nature of the channel which is called
small-scale variation. This time variation occurs because of the movement of the transmit-
ter or the receiver or the location of the obstacles.
In this chapter, we describe small scale fading characteristics of wireless channels
which are suitable for describing indoor wireless communication. We then give a brief
overview of European Telecommunications Standards Institute (ETSI) channel models and
propose a new autocorrelation model for temporal variation of an indoor wireless channel.
It is important to understand the different characteristics and properties of indoor wire-
less channels because the measurements of indoor channels show distinct differences from
mobile channel measurements.
8

23. CHAPTER 2. INDOOR WIRELESS CHANNEL 9
2.1 Types of small scale fading
The types of fading experienced by a signal propagating through a mobile radio channel
depends on the relation between the signal parameters, such as bandwidth and symbol
period [2][3]. Fig. 2.1 summarizes the types of fading experienced by a signal passing
through mobile radio channels with different characteristics. Based on delay spread, wire-
less channels can be divided into two categories: flat fading and frequency selective fading.
Furthermore, based on Doppler spread, channels can be divided into two other categories:
fast fading and slow fading. Therefore, the time dispersion and frequency dispersion in
a mobile channel lead to four possible distinct effects, which depend on the nature of the
transmitted signals, the channels, and velocities. While multipath delay spread leads to time
dispersion or frequency selective fading, Doppler spread leads to frequency dispersion or
time selective fading. Multipath dispersion can be described using similar mathematical
models for mobile channels with different parameters. However, there are some differ-
ences between the indoor and the mobile channel. First of all, while spatial variation of a
user is more important for a mobile channel, an indoor channel is neither stationary in time
nor in space. This temporal variation comes from motion of people and equipment around
low height portable antennas.
2.2 ETSI Channel models
In this study, power delay profiles for office environment are generated by ETSI models.
The ETSI channel models define five power delay profiles for the small-scale variations of
wireless channels in an office environment and open space[4]. The channel models describe
the delay spread of the channels. The Doppler and angular spreads, large-scale fading and
path-loss are not addressed in the ETSI channel models. In Table.2.1, we outline the five
channels and types of environment represented by these channels.

24. CHAPTER 2. INDOOR WIRELESS CHANNEL 10
Figure 2.1: Two types of small scale fading
Table 2.1: ETSI channel models
Channel RMS delay spread Environment LOS/NLOS
A 50ns Typical office NLOS
B 100ns large open space and office NLOS
C 150ns large open space NLOS
D 140ns large open space LOS
E 250ns large open space NLOS

25. CHAPTER 2. INDOOR WIRELESS CHANNEL 11
0 0.1 0.2 0.3 0.4
0
5
10
15
20
25
30
Delay spread (µ s)
Power(−dB)
(a) ETSI Channel A
0 0.2 0.4 0.6 0.8
0
5
10
15
20
25
Delay spread (µ s)
Power(−dB)
(b) ETSI Channel B
Figure 2.2: Power delay profile for channel A and B

26. CHAPTER 2. INDOOR WIRELESS CHANNEL 12
0 0.5 1 1.5
0
5
10
15
20
25
Delay spread (µ s)
Power(−dB)
(a) ETSI Channel C
0 0.5 1 1.5 2
0
5
10
15
20
25
Delay spread (µ s)
Power(−dB)
(b) ETSI Channel E
Figure 2.3: Power delay profile for channel C and E

27. CHAPTER 2. INDOOR WIRELESS CHANNEL 13
Since Channel C and D have the same power delay profile, the power delay profile for
only Channel A, B, C and E are shown Fig.2.2 and Fig.2.3. From the power delay profile
of channel A in Fig.2.2, we can observe that the maximum delay spread is about 0.4 µ
secs and the power delay profile consists of two clusters of exponentially decaying paths.
Another point worth noticing is the first arrived path for the profile is not the strongest one
except in Channel A. This effect on timing synchronization will be discussed in the next
chapter. We also observe that the maximum delay spread increases from Channels B to
C to E. This increase in frequency selectivity not only increases diversity gains but also
implies an increase in intersymbol interference (ISI). ISI occurs when delayed copies of a
transmitted symbol overlap the next transmitted symbol and usually degrades the perfor-
mance of wireless systems. In addition, while the power delay profiles for Channels C and
D are the same, there is a line of path (LOS) with power of about 10dB higher than the
sum of average power of all paths in the power delay profile for Channel D. Consequently,
the frequency selectivity of the two channels are the same but Channel D is more stable
due to the non-faded path. Therefore, a system in Channel D would perform better than in
Channel C and, hence, Channel D will not be used in this study.
2.3 Modeling the Time Varying Channel
The fading characteristics of indoor wireless channels are very different from the previously
reported mobile cases. However, in indoor wireless systems, the transmitter and receiver
are stationary and people are moving in between, whereas in outdoor mobile systems, the
user is moving through an environment. Although, this sort of time variation has been
observed in the literatures, for example [5] and [6], it is not thoroughly analyzed yet. A
stochastic time variation model was proposed for fixed wireless communication [7]. How-
ever, numerical methods are needed to implement this model and an inclusion of numerical
components will cause additional delay in practical simulations. In this section, we extend

28. CHAPTER 2. INDOOR WIRELESS CHANNEL 14
the method of [7] and derive a closed form stochastic channel model for an indoor wireless
communication simulation.
2.3.1 Mobile Radio Channel
The complex baseband representation of a wireless channel impulse response can be de-
scribed as,
h(t, τ) =
n
αn(t)e−jφn(t)
δ(τ − τn(t)) (2.1)
where τn(t) is the delay of the nth path and αn(t) is its real amplitude. Due to the motion
of the user, αn(t)e−jφn(t)
represents a wide-sense stationary narrowband complex Gaussian
process, which is independent for different path. If the user moves at speed v in the direc-
tion of θ as shown in Fig. 2.4, The phase change of a ray due to the moving receiver can be
easily obtained as
φ(t + ∆t) − φ(t) = 2π
fcv
c
∆t cos θ (2.2)
Therefore, assuming the power of each incident wave is uniformly distributed, the corre-
sponding autocorrelation function and Doppler power spectrum for nth tap are [3],
R(∆t) = E[exp(φ(t + ∆t) − φ(t))]
=
1
2π
2π
0
exp(j2π
fcv
c
∆t cos θ)dθ
= J0 2π
fcv
c
∆t (2.3)
where fc is the carrier frequency. Fourier transforming above equation, we can derive
power spectrum as,

29. CHAPTER 2. INDOOR WIRELESS CHANNEL 15
Receiver
Incident Plane Wave
Figure 2.4: Illustration of Doppler shift
S(f) =
1
π f2
d − f2
(2.4)
where fd denotes Doppler frequency and c is the speed of light. This model is called the
Jake’s model [45] and widely accepted for cellular environments where spatial variation
is more important than temporal variation. However, it deviates from measured Doppler
spectra in indoor wireless channel environments.

30. CHAPTER 2. INDOOR WIRELESS CHANNEL 16
Transmitter Receiver
Reflector
Figure 2.5: Geometry of a single ray
2.3.2 Indoor Wireless Channel
Fig. 2.5 shows the case when the transmitter and receiver are stationary and reflectors are
moving in the direction of θ at speed v. The phase change of a ray due to a moving reflector
can be easily obtained as [7],
φ(t + ∆t) − φ(t) = 4π
fcv
c
∆t cos θ cos ψ (2.5)
Assuming all reflectors are moving in a similar manner and the power of each incident
wave is uniformly distributed, the autocorrelation function and Doppler power spectrum

31. CHAPTER 2. INDOOR WIRELESS CHANNEL 17
can be computed as
R(∆t) = E[exp(φ(t + ∆t) − φ(t))]
=
1
(2π)2
2π
0
2π
0
exp(j4π
fcv
c
∆t cos θ cos ψ)dθdψ
= J2
0 2π
fcv
c
∆t (2.6)
Fourier transforming the above equation, the power spectral density is
S(f) =
fd
−fd
1
πfd 1 − x2
f2
d
·
1
πfd 1 − (f−x)2
f2
d
=
1
π2fd
K 1 − (
f
2fd
)2 (2.7)
However, in reality, some of the received power is from static objects and also reflectors
usually do not move at the same speed. Therefore, we assume that the factor p of the
received power is static and comes from fixed reflectors while the factor (1 − p) of the
received power is time varying and comes from moving reflectors. Based on the above
reasoning, the autocorrelation function of this channel can be represented as sum of the
power from static reflectors and the power from moving reflectors.
R(∆t) = p + (1 − p)E J2
0 2π
fcv
c
∆t (2.8)
Moreover, if we assume velocities of moving reflectors are exponentially distributed with
a parameter a, we can derive a closed form expression for the autocorrelation function as,
R(∆t) = p + (1 − p)
∞
0
1
a
exp −
1
a
v J2
0 2π
fcv
c
∆t dv (2.9)
= p + (1 − p)
2
aπγ
K
4πfc∆t
cγ
(2.10)

32. CHAPTER 2. INDOOR WIRELESS CHANNEL 18
where a is mean velocity of the moving reflectors, K is the complete elliptic integral and
γ = 1
a2 + 4(2πfc
c
∆t)2. Once we have an autocorrelation function, we can generate a ran-
dom process of the channel by spectrum filtering or spectrum sampling [3] and implement
a multipath fading simulator.
The Doppler power spectra and the autocorrelation functions for different environments
are shown in Fig.2.6 and Fig.2.7. The dotted line corresponds to the Jake’s model when a
receiver is moving at 4km/h and the dashed line, referred to as the worst case, represents the
case when p is zero and all reflectors are moving at 4km/h. Finally, the solid line represents
the proposed model when p is zero and a is 4km/h. Note that the proposed model gives
rise to more peaky Doppler spectrum and has wider spread of the power spectrum than the
Jake’s model in the frequency domain. Also, the autocorrelation function of the proposed
model shows less oscillatory behavior than that of the Jake’s model in the time domain.
Fig.2.8 shows a comparison between an indoor channel measurement result in [8] and
the proposed model. p and a are set to be 0.97 and 8km/h respectively and the power
spectral density of the proposed model is normalized to have the same received power
as the measurement. We can see that the proposed model matches the indoor channel
measurement well. In addition, it can be seen that the Jake’s model can not be fitted to this
measurement regardless of fd. Consequently, the proposed model can be used for more
accurate simulations of indoor wireless channels than the Jake’s model. In addition, since
the proposed model has a closed form expression, it has a lower computational complexity
than the model in [7]. This time variation model will be used throughout this dissertation.

33. CHAPTER 2. INDOOR WIRELESS CHANNEL 19
−2 −1 0 1 2
−30
−25
−20
−15
−10
−5
Normalized frequency
Magnitude(dB)
Jakes model
Worst case
Exponentional model with p=0
Figure 2.6: Doppler power spectrum
0 0.05 0.1 0.15 0.2
−0.5
0
0.5
1
Jakes model
Worst case
Exponentional model with p=0
Figure 2.7: Autocorrelation function

34. CHAPTER 2. INDOOR WIRELESS CHANNEL 20
−10 −5 0 5 10
−100
−90
−80
−70
−60
−50
−40
−30
Frequency (Hz)
Doppler Power Spectrum
Figure 2.8: Comparison between Doppler spectrum measurement and proposed Doppler
spectrum model

35. Chapter 3
Adaptive Timing Synchronization for
OFDM Systems
In burst packet mode OFDM systems, timing synchronization needs to be done within a
single training symbol time to avoid reduction of the data throughput. Due to this stringent
requirement on synchronization time, standards incorporate preambles suitable for using
correlation to estimate symbol timing. However, in time-dispersive multi-path channels,
conventional timing synchronization methods may synchronizes to a path in the middle of
the overall channel impulse response (CIR). Consequently, the receiver may not capture
some of the multi-path components. This results in an inter-symbol interference (ISI) and
an inter-carrier interference (ICI). In this chapter, we propose a simple adaptive timing
synchronization method to locate the first arriving path based on the use of one training
symbol in the preamble. Our computer simulation results show that the proposed method
can significantly improve error rate performance. The performance gain becomes higher as
delay spread increases.
21

36. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 22
3.1 Introduction
In OFDM, the modem can invert dispersive broadband channels into parallel narrow band
sub-channels, thus significantly simplifying equalization at the receiver. However, this
inherent immunity of OFDM to time-dispersive multi-path channels comes at the price of
increased sensitivity to synchronization error. Imperfect synchronization causes ISI and
ICI which can result in significant performance degradation [1] [11].
Several approaches have been proposed on the basis of using training symbols or using
the repetition property of cyclic prefixes [12] [13]. In burst packet mode OFDM systems,
the method using a preamble is preferred for fast time and frequency synchronization due
to the stringent requirement to minimize synchronization time. In [12] and [13], an auto-
correlation based timing metric is calculated. This calculation correlates the received sam-
ples and their delayed copies. These algorithms, based on the auto-correlation, inevitably
result in an ambiguity in timing due to a plateau region and to enhanced sensitivity to burst
noise. This ambiguity must be resolved after the auto-correlation process. One solution
to this problem is to use a cross-correlation method, which correlates the received samples
with known training samples. The cross-correlation peak of the received samples is used
for symbol timing. This method has very good performance in an AWGN environment
but has significant drawbacks since it is sensitive to frequency offset and the power delay
profile of the channel. In [23], short training symbols (STS) are used for timing estimation
via a combination of an auto-correlation and a cross-correlation. However, as mentioned
above, frequency offset in the local oscillator can disturb the cross correlation peaks sig-
nificantly, which will significantly affect the accuracy of the timing estimate. Furthermore,
a few sample errors in the coarse timing estimate may cause significant timing errors in
the resulting fine timing estimate. Therefore, in order to use a cross-correlation method to
estimate timing, frequency offset must be kept small, such as within 50Khz for 802.11a
and HiperLAN/2 environments [18].

37. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 23
In [19], after coarse frequency offset is compensated, fine timing estimation is done
using the periodic property of long training symbols (LTS). This algorithm, however, makes
no effort to estimate the position of the first arriving multi-path component, leading to an
inefficient utilization of the guard interval in multi-path channels. This also causes ISI
and ICI in the demodulation process. One intuitive solution to this problem is to shift
a few samples in the appropriate direction from the acquired correlation peak position.
However, since neither average nor instantaneous power delay profiles (PDP) of the channel
are available, it is not obvious how many samples should be shifted. In [16], they used a
double auto-correlation method to estimate the timing and the energy of the CIR to find the
first arriving path of the signal which may not be the strongest. However, this method has
a weakness that the timing estimate can be compensated only after channel estimation and
it is also not straightforward to decide the optimal window size, which is dependent on the
delay spread of the channel.
In this chapter, we present an adaptive timing synchronization method for OFDM sys-
tems using burst packet mode. In our proposed method, before symbol timing estimation,
frequency offset is corrected by a typical maximum likelihood (ML) method. Hence, the
cross-correlation based timing estimation accuracy is not affected by frequency offset, and
the cross-correlation output is used to detect the most significant channel taps by adaptively
changing an observation window. The proposed method in this chapter does not require
any extra channel information such as signal to noise ratio (SNR) or PDP, while allowing
detection of the first arriving path position and additionally estimating the maximum de-
lay spread and total channel power which can be used to increase system throughput in
other applications. We evaluate the performance of our method with the 802.11a standard
[9] in four different indoor PDP scenarios [4]. Simulation results show that our proposed
method significantly outperforms conventional peak selection methods and is robust to var-
ious channel environments with practical impairments.

38. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 24
10 2 3 4 65 7 8 9 GI2 T1 T2
P1 P2 Header Data Data ……. Data
Details of the preamble field
10 short symbols (0.8*10 = 8 s) 2 long symbols (1.6+2*3.2 = 8 s)
Signal detection, AGC, Coarse timing
recovery, Freq. acquisition
Fine timing recovery, Freq. offset
estimation, Channel estimation
8 s 8 s 4 s
4 Pilot sub-carriers for phase tracking
Figure 3.1: 802.11a - Frame and slot structure
3.2 System Model
Fig.3.1 shows an example of OFDM frame and slot structure In the WLAN standard
adopted by the IEEE 802.11a. Each data packet consists of preamble and a payload. The
preamble consist of 10 short training symbols (STS) of length of 16 samples (8µs) and long
training symbols (LTS) of length of 64 samples (8µs) which are all utilized for synchro-
nization and channel estimation. The data carrying part consists of a variable number of
symbols and the length of each data symbol is 64 samples. Note that a short symbol serves
as a cyclic prefix for a subsequent short symbol. For LTS, GI2 is the cyclic prefix for T1
and it contains 32 of the last samples (1.6µs) of T1. In the frequency domain, a data symbol
contains data subcarriers and some known pilot subcarriers that are usually used for phase
tracking. Fig.3.2 shows an example of the subcarrier allocation for the IEEE 802.11a sys-
tem. Out of the 64 possible subcarriers, only 52 subcarriers are used. Of the 52 subcarriers
used, 48 subcarriers are dedicated to data transmission and 4 are pilot subcarriers.
More generally, let’s consider an OFDM system with FFT length N where a total of

39. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 25
−40 −30 −20 −10 0 10 20 30 40
frequency index
Data subcarriers
Guard band
Pilot subcarriers
Figure 3.2: 802.11a - Subcasrrier allocation
Nu subcarriers are used for transmission. The transmitted signal s(n) is generated by an
IFFT of data symbols Ak and a guard interval of length Tg = Ng · Ts is placed in front of
the useful portion Tu = N · Ts of the signal to prevent ISI. Ts denotes the sampling time
period. Then
s(n) =
1
N
Nu/2+1
k=−Nu/2
Ak · exp
j2πnk
N
(3.1)
for −Ng ≤ n ≤ N − 1
The baseband impulse response of the channel is assumed to be in the form of
h(n) =
L−1
l=0
h(l)δ(n − l) (3.2)
where L is the maximum delay spread of the channel and h(l) represents the complex gain
of the lth multi-path component. Assume time invariance over one OFDM symbol. After

40. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 26
transmission over this multi-path channel, the samples at the receiver are
r(n) =
L−1
l=0
s(n − l − nt) · h(l) · exp (
j2π n
N
+ θ) + N(n) (3.3)
where N(n) is complex white Gaussian noise at time n, nt = δt/Ts is the timing offset, θ is
an unknown phase and = NTs ·δf is the normalized carrier frequency offset. If the guard
interval is correctly removed, the signal is then demodulated by FFT resulting in output at
the subcarrier k of
Yk = Hk · Ak + Nk (3.4)
for −Nu/2 ≤ k ≤ Nu/2 + 1 (3.5)
As long as the start position of the FFT window is in the ”Region A” in Fig.3.3, no
ISI or ICI occurs. Changing the start position will only induce phase rotation across the
subcarriers and this rotation can not be distinguished from actual channel phase response
so performance degradation does not occur. However, if the FFT start position is in the
”Region B”, it will cause ISI and ICI [15]. This effect is minimized when the energy of the
channel inside the guard interval of Ng in Fig.3.3 is the maximum.
In the presence of timing estimation error, the post FFT signal can be derived as
Yk = Hk · Ak · α(nt) + Nk + Nnt,k (3.6)
for −Nu/2 ≤ k ≤ Nu/2 + 1
Attenuation, α(nt), can usually be neglected for large N, so the main disturbance comes
from additional noise, Nnt,k. It was shown that this noise can be approximated by Gaussian

41. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 27
NNg
B A B
FFT Window
L
FFT Start Position
Channel Response
Figure 3.3: Timing synchronization diagram
noise with power [14],
σ2
nt
=
i
|hi|2
(2 · g(nt) − g(nt)2
) (3.7)
where g is a linear function depending on relative timing offset. Furthermore, a timing
offset will have another effect on the performance. Since some portion of the effective
channel is shifted, this portion can not contribute to the channel estimate. The resulting
channel estimation error is given by [24],
σ2
c = E[|Hk − H∆,k|2
] =
Nsu
N i∗
|hi∗ |2
(3.8)
SNR ≈
f2
N ( ) · SNR
(1 − f2
N ( )) · SNR · (N/Nsu) + 1
(3.9)
fN ( ) =
sin(π )
N sin(π /N)

42. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 28
3.3 Frequency and Timing Synchronization
The proposed method can be broken into two steps. In order to use a cross-correlation
based timing synchronization method, frequency offset is compensated first using STS.
After successful frequency offset compensation, the FFT start position is found by our
proposed timing synchronization method.
3.3.1 Coarse Frequency Offset Estimation
STS are periodic after Ns samples. Then ML estimate of frequency offset can be obtained
by auto-correlation of the received signal.
A(n) =
Wa−1
m=0
r(n + m)r∗
(n + m + Ns) (3.10)
ˆ =
−N
2πNs
· tan−1
(A(n)) (3.11)
where Ns = 16 for 802.11a [9], Ns = 64 for 802.16a [10] and Wa is the averaging length
which is dependent on the automatic gain control (AGC). During the AGC stabilization
time, the received signal will be corrupted by large gain fluctuations that cause the auto-
correlation output to be unstable. For most AGC systems, this process will last for the
first 48-80 samples of the STS [20]. Therefore, Wa is set to be less than 4 STS periods
for 802.11a systems. Using the method in [21], acquisition range and Cramer-Rao lower
bounds (CRLB) can be obtained as,
| | ≤
N
2Ns
(3.12)
var(ˆ) ≥ (
N
2π · ks · Ns
)2
·
1
Ns · SNR
(3.13)

43. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 29
where ks is the number of STS such that Wa = ks · Ns and SNR is defined as E[|r(n) −
n(n)|2
]/E[|n(n)|2
]. After symbol timing is acquired as described in the next section, the
LTS are used to further reduce the frequency offset estimation error. For this case, Ns is set
to be the length of the LTS, NL, and ks is set to be 1.
3.3.2 Adaptive Timing Synchronization method
After the packet detection algorithm signals the start of the packet, the symbol timing
algorithm refines the timing estimation to a sample period precision. This is conventionally
done by using the cross-correlation between the received signal r(n) and a known reference
tn with length NL. The reference, tn, can be made by concatenating last NL/2 samples of a
LTS with the first NL/2 samples of a LTS. The value of n that corresponds to the maximum
absolute value of the cross-correlation in (3.14) is the symbol timing estimate.
ˆTf = arg max
n
(|
NL−1
m=0
r(n + m)t∗
m|2
) (3.14)
tn = [LNL/2:NL
L0:NL/2−1] (3.15)
where NL = 64 for 802.11a, NL = 128 for 802.16a. If the first arriving path is the strongest
path, this conventional method can detect the boundary between the last STS and the first
LTS, which is n = 161. Since the guard interval for the LTS is 32 samples, the exact FFT
start position, n = 193, can be found. But the conventional method that is mentioned above
fails to find true FFT start position if the first arriving path is not the strongest path. In such
cases , cross-correlation may take the highest correlation value associated with the path that
arrives later than the first path and this may result in severe ISI and ICI.
In order to avoid this problem, we utilize the fact that the cross-correlation output ap-
proximately coincides with scaled instantaneous channel power. Suppose C(n) is defined
as a cross-correlation output. The conditional expectation of C(n), given h(n), is obtained

44. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 30
as the following equation when a multi-path component exists at index n:
E[C(n)|h(n)] = E[|
NL−1
m=0
r(n + m)t∗
m|2
|h(n)] (3.16)
= |h(n)|2
(
Nu
N
)2
+ σ2
n
Nu
N
+ σ2
I (3.17)
However, if a multi-path component does not exist at index n, the above equation be-
comes
E[C(n)|h(n)] = σ2
n
Nu
N
+ σ2
I (3.18)
where the number of used subcarriers, Nsu, is 52 for 802.11a and σI is additional
noise due to the imperfect cross-correlation property of the pseudo random sequence in
the preamble.
Let us define D(n) and QW (n) as
D(n) = C(n) − σ2
n
Nu
N
− σ2
I (3.19)
QW (n) =
W−1
m=0
D(n + m) (3.20)
where W is a summation window length. The QW (n) represents a normalized summation
of W consecutive samples of C(n). Also, the conditional expectation of QW (n) is zero
if a multi-path component does not exist within summation interval, W. Fig.3.4 shows an
exemple plot of D(n) when the maximum channel length is five sample periods.
As long as the window length, W, is greater than the maximum delay spread of the
channel, the maximum of QW (n) does not change except for some fluctuation due to noise.
Meanwhile, if the window length becomes less than the maximum delay spread of the

45. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 31
W
Figure 3.4: Exemplary plot of D(n) at SNR=10dB

46. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 32
Max delay spread = 4
Figure 3.5: QW output when W = 16, 8, 4 and 3

47. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 33
channel, the maximum of QW (n) will be significantly decreased. Therefore, the position of
the first arriving path can be estimated by detecting a significant decrease of the maximum
of QW (n). However, since the noise fluctuation of the maximum of QW (n) could lead to
an incorrect timing estimate, ˜QW , which is defined as the average of the samples whose
magnitudes are greater than 90% of the maximum of QW (n), is used for timing estimation
instead of the maximum of QW (n). Suppose the window size W∗
and the 90% maximum
start decreasing more than ξ%. Then it can be seen that arg maxn(QW∗+1(n)) corresponds
to the first arriving path position since it indicates the starting time of the window which
contains the maximum power of the CIR. For example, Fig.3.5 represents QW (n) output
when {W = 16, 8, 4, 3} according to the D(n) in Fig.3.4. Since QW (n) starts decreasing
when W∗
= 3, the resulting timing estimate can be obtained from arg maxn(QW∗+1(n)) =
193. If we used the conventional peak-detection method, the resulting timing estimate
would be n = 196. The above method can be executed by a binary search algorithm with
high efficiency. It also can provide an estimate of the instantaneous maximum delay spread,
W∗
+ 1 and an estimate of the instantaneous total channel power.
The proposed method is summarized below:
1. Compensate frequency offset by (3.11).
2. Calculate C(n) and D(n).
3. Execute binary search algorithm to find W∗
using initial W = Ng.
4. Declare timing estimate as arg maxn(QW∗+1(n)).

48. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 34
Table 3.1: Simulation Parameters
Sampling time, Ts 50ns
FFT length, N 64
Useful subcarriers, Nu 52
Number of data subcarriers 48
Guard interval length, Ng 16
Auto-correlation window, Wa 32
Threshold, ξ 95
Subcarrier spacing 312.5 KHz
Initial frequency offset, δf 469 KHz
Modulation 4 QAM, 64QAM
Packet length 540 Bytes
Number of packets 10000
Channel coding rate 1/2
3.4 Simulation Results
3.4.1 Simulation Environment
I simulated a transmitter and a receiver according to the parameters established by the
802.11a standard [9]. The simulation parameters are listed in Table 3.1.For the channel
model, only the small-scale fading is considered. Both the distance dependent path loss
and the shadowing are assumed to be constant over the simulation and incorporated into
the SNR. Two different PDPs, Channel B and C in Fig 2.2 and Fig 2.3 [4], are generated
and the time variation model in Chap 2 is employed. Details of random channel generations
are described in the next section. Two long training symbols in the preamble are used for
least-square channel estimation and four pilot subcarriers are used for residual frequency
offset compensation by the ML method [1].
3.4.2 Random Channel Generation
An average power of each tap in an channel impulse response is set according to a given
PDP. In order to generate time evolutions for each tap, an average velocity, a in (2.9), is

49. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 35
Tim
e
Delay
Packet
duration
Figure 3.6: A time variation technique for simulation
set to 4km/h. After obtaining a Doppler power spectrum, S(f), as described in Chap 2, a
spectrum sampling method [3] is used to independently generate time domain samples for
each tap. Consequently, the baseband representation of a channel impulse response at the
kth tap is,
hk(t) = N
n=1 S(fn) · e−j(2πfnt+φn)
where S(·) is a Doppler power spectrum in Chap 2 and φn are random phases on [0, 2π].
Since the channel variation between adjacent two OFDM symbols are small, only two time
samples are generated for a tap inside a packet. The two time samples are chosen to be the
beginning and the ending time of a packet. A time variation inside these two time samples
is obtained by a linear interpolation as shown in Fig.3.6.

50. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 36
6 8 10 12 14 16 18
10
−2
10
−1
10
0
Average SNR (dB)
Averagepacketerrorrate
Proposed
Conv
Ideal
Figure 3.7: Average packet error rate for 4QAM in Channel B
3.4.3 Performance Results
Fig.3.7 - Fig.3.10 show average packet error rate for the proposed method in two differ-
ent delay spread environments. The ordinate represents average packet error rate and the
abscissa represents average SNR. ”Ideal” is the case when ideal timing estimation is avail-
able and ”Conv” is the case when the peak location of the cross-correlation is declared
as the FFT start position. As you can see from the figures, the proposed method method
significantly outperforms the conventional method in all scenarios. This result is expected
from the PDP of the channel since the conventional method tends to be synchronized to a
path in the middle of the overall CIR. The probability that the first arriving path becomes
the strongest path is low for these channels. With incorrect timing estimate, the conven-
tional method experiences ISI and ICI and these interference become larger as the delay
spread increases finally leading to an error floor and this effect is more critical as modu-
lation complexity increases. The performance improvement in the proposed method over

51. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 37
18 20 22 24 26 28 30
10
−2
10
−1
10
0
Average SNR (dB)
Averagepacketerrorrate
Proposed
Conv
Ideal
Figure 3.8: Average packet error rate for 64QAM in Channel B
6 8 10 12 14 16 18
10
−2
10
−1
10
0
Average SNR (dB)
Averagepacketerrorrate
Proposed
Conv
Ideal
Figure 3.9: Average packet error rate for 4QAM in Channel C

52. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 38
18 20 22 24 26 28 30
10
−2
10
−1
10
0
Average SNR (dB)
Averagepacketerrorrate
Proposed
Conv
Ideal
Figure 3.10: Average packet error rate for 64QAM in Channel C
conventional method is a result of capturing all components in the received signal while ISI
and ICI do not exist. Note that the gap between ”Ideal” and ”Proposed” is almost all from
channel estimation error and frequency offset estimation error, which means the proposed
method does not experience noticeable performance degradation from timing estimation
error.
To demonstrate the detailed performances of the proposed method as opposed to the
conventional method, histograms of timing estimate are shown in Fig.3.11 and Fig.3.12.
For these figures, average SNR is set to be 10dB and 10,000 packets are transmitted to
obtain the result for Channel B. Fig.3.11 shows the timing estimate of the proposed method
and Fig.3.12 shows the timing estimate of the conventional method for Channel B when
the true FFT start position is 193 sample. Note that the timing estimate distribution of the
conventional method tends to be shifted to the right side from sample 193. In contrast,
the timing estimate of the proposed method tends to be shifted to the left side from actual

53. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 39
185 190 195 200
0
1000
2000
3000
4000
5000
Sample index
Proposed
Figure 3.11: Histogram of timing estimates for Channel B, SNR=10dB: Proposed
timing at sample 193. It can be seen that the proposed method achieves correct timing
more often than the conventional one. Also, even when the proposed method misses correct
timing, it makes an error in the direction of ”Region A” of Fig.3.3 where timing error may
not affect system performances as long as the channel delay spread is short enough. The
performance gain is larger in Channel C as shown in Fig.3.13 and Fig.3.14. Since the RMS
delay spread of the Channel C is lager than that of the Channel B as shown in the previous
chapter, the conventional synchronization method fails to find the true timing boundary
more often. The probability of finding the correct timing boundary for the proposed method
is around 48% and the probability for conventional method is only 6.4%.

54. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 40
185 190 195 200
0
1000
2000
3000
4000
5000
Sample index
Conventional
Figure 3.12: Histogram of timing estimates for Channel B, SNR=10dB: Conventional
185 190 195 200
0
1000
2000
3000
4000
5000
Sample index
Proposed
Figure 3.13: Histogram of timing estimates for Channel C, SNR=10dB: Proposed

55. CHAPTER 3. ADAPTIVE TIMING SYNCHRONIZATION FOR OFDM SYSTEMS 41
185 190 195 200
0
1000
2000
3000
4000
5000
Sample index
Proposed
Figure 3.14: Histogram of timing estimates for Channel C, SNR=10dB: Conventional
3.5 Conclusions
Here, we propose an adaptive timing estimation method for OFDM systems. By changing
an observation window length, the method can locate the first arriving path, which may not
be the strongest path. The correct timing can effectively avoid ISI and ICI. This method
does not require any prior knowledge, such as SNR or PDP, and our simulation results
show that it is robust to various channel environments. Furthermore, our proposed method
additionally provides an estimate of instantaneous total received power and maximum delay
spread which can be used in other applications to increase system throughput. Although
the simulation is done using parameters for the 802.11a standard, our method can be used
to perform timing synchronization for different burst packet mode OFDM systems.

56. Chapter 4
Residual Frequency Offset and Phase
Tracking
In orthogonal frequency division multiplexing (OFDM) systems, carrier frequency offset
(CFO) due to mismatch of the local oscillators can cause an inter-carrier interference (ICI),
which may result in significant performance degradation. Although several frequency syn-
chronization schemes were reported by previous studies, frequency offset still remains and
generates ICI as well as induces phase distortion of the OFDM symbols. In this chapter, we
propose a method to compensate both residual frequency offset (RFO) and RFO induced
phase error (PE) by using the Kalman filter. Our approach is based on building a simple
robust state-space model and the Kalman filter is then applied to estimate and track the
RFO and PE. Our simulation results show that the proposed method significantly reduces
the performance degradation due to RFO and almost achieves ideal packet error rate (PER)
performance with lower complexity.
42

57. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 43
4.1 Introduction
OFDM is a powerful modulation technique for high data rate transmission over frequency-
selective channels. However OFDM as a multi-carrier system has a different structure than
a single-carrier system. OFDM can tolerate relatively larger timing errors than a single-
carrier system due to a longer symbol period and a cyclic prefix. On the other hand, the
frequency synchronization requirement for OFDM is tighter than a single-carrier system
because the data are transmitted in parallel narrow sub-bands. If there exist a CFO, then
the number of cycles in the FFT interval is no longer an integer, with the result that ICI
occurs after the FFT [1]. Several approaches have been developed to estimate CFO [21]-
[23]. Unfortunately, it is difficult to completely compensate CFO, and CFO remains as a
residual frequency offset (RFO). This RFO can cause ICI and can induce phase error (PE)
in the OFDM symbols after the FFT. In order to decrease RFO effects, a tracking stage is
required in the OFDM receiver because even a very small RFO can cause a phase to rotate
continuously in every OFDM symbol.
In [25], a decision-feedback loop is used to compensate RFO by estimating the phase
differences between two consecutive OFDM symbols. Although this can actually remove
ICI from RFO, the performance is only guaranteed in relatively high signal to noise ra-
tio (SNR) regions due to the decision-feedback structure. Recently, a RFO compensation
scheme using an approximate SAGE algorithm is proposed [26]. It can compensate the per-
formance degradation due to RFO even in low SNR regions. In this scheme, an expectation
step is used to remove ICI and a maximization step is used to estimate RFO. However,
it is based on an iterative process and requires several maximization calculations, which
may not be possible in practical systems due to the inherent complexity and the processing
delay. In [27], assuming ICI from RFO is negligible, phase error (PE) is simply estimated
by averaging instantaneous phase estimates from pilot sub-carriers in each OFDM sym-
bol. Although the influence of AWGN in the instantaneous phase estimates can be reduced

58. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 44
by the averaging process, these estimates may be biased due to channel estimation errors
and, thus, averaging can lead to accumulation of PE. In contrast, an extended Kalman filter
was used to track only RFO in [28]. Although this method can track RFO by a recursive
procedure, this state-space modeling could require considerable computation because of
correlation matrix estimation. Furthermore, the solution for effects of PE was not clearly
addressed. Another solution for this problem is to use least-square (LS) phase fitting [29].
This method does not accumulate PE from channel estimation error and also can estimate
both RFO and PE. However, no claims about optimality can be made and the computation
cost increases as O(n3
), where n is the number of samples used for the line fitting.
In this chapter, we propose a method to compensate both RFO and RFO induced PE
for OFDM systems using a Kalman filter. In our proposed method, the linear state-space
model for RFO and PE is derived using estimated signal to noise ratio (SNR). After building
a state-space model, a Kalman filter is applied to track and estimate RFO and PE simul-
taneously. The proposed method allows unknown parameters to evolve in time to track a
frequency drift of the local oscillator. The method is an optimal linear estimator assuming
signal and noise are jointly Gaussian. Also, the computation cost of the proposed method
is much lower than that of the LS phase fitting method [29] due to the small dimension of
the state-space model. We evaluate the performance of our method with parameters from
the 802.11a standard [9] in typical office environment. Our simulation results show that the
proposed method significantly compensates the performance degradation due to RFO and
almost achieves an ideal performance in terms of packet error rate in the range of signal to
noise ratios we have tested.

59. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 45
4.2 The Effect of Residual Frequency Offset
We consider an OFDM system having an FFT length N. Then the output of the mth OFDM
symbol is given by
sm(n) =
1
N
N−1
k=0
Am(k) · exp
j2πnk
N
(4.1)
for −Ng ≤ n ≤ N − 1
where Am(k) is a data symbol for the kth subcarrier, Ng = Tg/Ts is the guard interval
length in samples. Ts denotes the sampling time period and Tg denotes the guard interval
period. The baseband impulse response of the channel is assumed to be in the form of
h(n) =
L−1
l=0
h(l)δ(n − l) (4.2)
where L is the maximum delay spread of the channel and h(l) represent the complex gain
of the lth multi-path component. Given a normalized RFO, = NTs · δf, and unknown
phase, θ, the received mth OFDM symbol with ideal timing estimation can be expressed by
[24]
rm(n) = (h(n) ∗ sm(n)) · cm( , n) + nm(n) (4.3)
cm( , n) e
j2π n
N · e(j2π m(1+α)+jθ)
(4.4)
where ∗ is the convolution operator, α = Ng/N and n(n) is complex white Gaussian noise
at index n. After correctly removing the guard interval, the signal is demodulated by an

60. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 46
FFT and the resulting output at the subcarrier k is
Ym(k) = (H(k)Am(k)) ⊗
1
N
Cm( , k) + Nm(k) (4.5)
=
1
N
Cm( , 0)H(k)Am(k) + Im(k) + Nm(k)
Cm( , k) =
sin(π( − k))
sin(π( − k)/N)
ejπ( −k)(1−1/N)
·e(j2π m(1+α)+jθ)
(4.6)
Im(k) =
1
N
N−1
u=1
Cm( , u)H(k − u)Am(k − u) (4.7)
where ⊗ is the circular convolution operator. Cm( , k) is the FFT of cm( , n) and Im(k) is
the FFT of ICI.
Without loss of generality, we can assume that the total average channel power is nor-
malized to a constant, L−1
l=0 E[|h(l)|2
] = 1. Then the approximate SNR in the time domain
can be derived using a method similar to [24]
SNR ≈
f2
N ( ) · SNR
(1 − f2
N ( )) · SNR · (N/Nu) + 1
(4.8)
fN ( ) =
sin(π )
N sin(π /N)
(4.9)
SNR
E[|r(n) − n(n)|2
]
E[|n(n)|2]
(4.10)
where Nu is the number of subcarriers used. Fig.4.1 shows comparison between SNR
and SNR with respect to . We can see from Fig.4.1 that should be kept less than 0.01
to avoid SNR degradation due to RFO for SNR < 25dB. However, even a very small
RFO still can be a problem since it causes phase to rotate continuously for every OFDM
symbol in the packet. As the symbol index m increases, the e(j2π m(1+α)+jθ)
term in (4.6)
accumulates PE and finally results in a demodulation error. This effect is more serious

61. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 47
5 10 15 20 25 30
5
10
15
20
25
30
Ideal SNR(dB)
ActualSNR(dB)
ε = 0
ε = 0.1
ε = 0.05
ε = 0.01
Figure 4.1: SNR degradation due to RFO
when the constellation becomes more complex. For example, if the FFT length, N, and
guard interval length, Ng, are 64 and 16 respectively, then a RFO ( = 0.01) rotates the
constellation by 0.0785 radians per one OFDM symbol from (4.6). Therefore, even without
noise, it takes only two OFDM symbols to make a demodulation error for 64-QAM since
0.1342 radians is the minimum PE to cross a decision boundary for 64-QAM modulation.
Therefore, it takes 11 symbols for 4-QAM modulation for PE to make a demodulation
error. While Fig.4.2 shows an ideal 64-QAM signal constellation when = 0 at SNR =
30dB, Fig.4.3 demonstrates the resulting rotation of a 64-QAM signal constellation when
= 0.01 at SNR = 30dB.

62. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 48
−1.5 −1 −0.5 0 0.5 1 1.5
−1.5
−1
−0.5
0
0.5
1
1.5
In−phase
Quadrature
Figure 4.2: 64-QAM signal constellation without RFO
−1.5 −1 −0.5 0 0.5 1 1.5
−1.5
−1
−0.5
0
0.5
1
1.5
In−phase
Quadrature
Figure 4.3: 64-QAM signal constellation with RFO

63. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 49
4.3 The Proposed Method
4.3.1 State-Space Modeling
After CFO estimation and channel estimation are completed using the preamble, the pilot
tones in the OFDM symbols can be used to track the RFO. Suppose there are Np pilot
tones in each OFDM symbol. The mth OFDM symbol output for the kn pilot tone, after
removing the pilot symbol, is
Pm(kn) = Cm( , 0)H(kn)/N + Im(kn) + Nm(kn) (4.11)
for kn ∈ A = {k1 . . . kNp }
Also the estimated channel frequency response from the preamble can be written as
H(k) = Cp( , 0)H(k)/N + Ip(k) + Wp(k) (4.12)
where p denotes the location where PE is zero and Wp(kn) is channel estimation noise.
Moreover, the PE due to RFO for the mth OFDM symbol can be modeled from (4.6) as
φm = φ0 + m · κ (4.13)
κ = 2π (1 + α) (4.14)
for 1 ≤ m ≤ M

64. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 50
0 5 10 15 20 25
0
0.5
1
1.5
2
2.5
OFDM symbol index
Comparison of Phase Error
Phaseerror(rad)
Without Noise
ML
Figure 4.4: Comparison of phase error for = 0.01: ML
where M is the total number of OFDM symbols in one packet. Then the maximum likeli-
hood (ML) estimate of the PE, φm, can be derived as
tan(φm) =
Im[PmH∗
]
Re[PmH∗]
(4.15)
Pm = [Pm(k1)Pm(k2) . . . Pm(kNp )]
H = [H(k1)H(k2) . . . H(kNp )]
Fig. 4.4 shows an example plot of actual PE without noise and the corresponding ML
estimates with respect to the OFDM symbol index when = 0.01. Assuming |φm −φm|
1, the tangent can be approximated by its argument and the ICI can also be approximated as
a zero mean Gaussian random variable for sufficiently large N by the central limit theorem

65. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 51
[21]. The estimation error can be written then as
φm − φm ≈ Dm/Em
where Dm and Em are defined as
Dm
kn∈A
Im[(NfN
( )H∗
(kn) + (Nm(kn) + Im(kn))e−jφm
)
·(f∗
N ( )H∗
(kn) + (Wp(kn) + Ip(kn))∗
)]
Em
kn∈A
Re[(NfN
( )H∗
(kn) + (Nm(kn) + Im(kn))e−jφm
)
·(f∗
N ( )H∗
(kn) + (Wp(kn) + Ip(kn))∗
)]
At high SNR, the above equation can be further approximated by,
φm − φm ≈ Dm/Em
where Dm and Em are defined as
Dm
kn∈A
Im[(Nm(kn) + Im(kn))f∗
N ( )H∗
(kn)e−jφm
+fN ( )H(kn)(Wp(kn) + Ip(kn))∗
]
Em
kn∈A
|fN ( )H(kn)|2
from which we can deduce that the estimate is conditionally unbiased
E[φm − φm| , H(kn)] = 0 (4.16)

66. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 52
Moreover, the conditional variance of the estimate is
σ2
ML = Var(φm − φm| )
≈
2(1 − f2
N ( )) + σ2
N + σ2
W
2Np · f2
N ( )
(4.17)
where σ2
N = E[|N(k)|2
] and σ2
W = E[|Wp(k)|2
]. Note that σ2
ML is a function of . It
is not obvious how to determine σ2
ML since the statistical distribution of is usually un-
known. Therefore, in order to design the robust estimator, σ2
ML should be set according to
the expected worst case value of max in the acquisition range. In addition, the unknown
constants, σ2
N and σ2
W , can be estimated in advance using the auto-correlation output of
the CFO estimation during the preamble. Suppose |J(n)| is the absolute magnitude of the
auto-correlation output,
|J(n)| = |
Wa−1
i=0
r(n + i)r∗
(n + i + Ns)|
≈
Wa−1
i=0
|q(n + i)|2
(4.18)
where Ns is the repeating period, Wa is the averaging length and q(n) = h(n) ∗ s(n). Due
to the repeating property of the preamble, the noise variances can be estimated as follows:
E[|q(n)|2
] ≈ |J(n)|/Wa
E[|r(n)|2
] ≈ (
Wa−1
i=0
|r(n + i)|2
)/Wa
σ2
N = N · (E[|r(n)|2
] − E[|q(n)|2
]) (4.19)
σ2
W = β · σ2
N (4.20)
where β is a known constant which depends on the channel estimation method. Therefore
combining (4.13), (4.17), (4.19) and (4.20), we obtain the following state-space model for

67. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 53
xm = [φm, κ]T
.
xm = Fm−1xm−1 (4.21)
ym = Gmxm + qm (4.22)
where Fm−1 =


1 1
0 1

 , Gm = 1 0 and
E[|q|2
] = σ2
ML =
2(1 − f2
N ( max)) + σ2
N (1 + β)
2Np · f2
N ( max)
4.3.2 The Kalman Filter
From the above state-space model, the consecutive vector xm|m−1 and xm|m, with error
covariance matrix P, are recursively estimated given the measurement history and current
measurement ym through the Kalman filter. Based on this basic state-space representation
for RFO, the conventional Kalman equations are calculated as follows :
• Prediction step is
xm|m−1 = Fm−1xm−1|m−1
Pm|m−1 = Fm−1Pm−1|m−1FT
m−1
• Update step is
Km = Pm|m−1GT
m[GmPm|m−1GT
m + σ2
ML]−1
xm|m = xm|m−1 + Km[ym − Gmxm|m−1]
Pm|m = Pm|m−1 − KmGmPm|m−1

68. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 54
Thus far, we have formulated the Kalman equations for recursively estimating the state vec-
tor xm. All that needs to be done to complete the recursion is to determine how the recursion
should be initialized. Since the CFO estimate is unbiased, E[κ] = 2π(1 + α)E[ ] = 0 and
E[φ0] = E[−pκ] = 0 from (4.13). Also E[κ2
] can be approximately obtained from the
error variance of the CFO estimation [21]. Therefore, the initial estimate and initial value
for the error covariance matrix can be determined as
P0|0 = E[x0xT
0 ] = σ2
κ


p2
−p
−p 1

 (4.23)
x0|0 = E[x0] = E


−pκ
κ

 =


0
0

 (4.24)
To demonstrate the features of the Kalman filtering as opposed to ML method, Fig.
4.5 shows an example plot of phase errors with the different RFO tracking methods when
= 0.01. Note that the proposed method can significantly reduce RFO induced phase error.
It has better performance because it utilizes not only instantaneous phase measurement but
also history of the phase estimation.
4.3.3 Complexity Consideration
In order to measure the computational complexity of different estimation methods, we use
the number of floating-point operations (flops). The LS phase fitting generally requires
O(M3
) flops to obtain parameters where M is the number of samples used for the line
fitting. In contrast, the Kalman filter in the proposed method requires fewer than 100 flops
for each step and the complexity increases linearly with respect to M. Therefore, if more
than ten ML estimates are used for the LS phase fitting, the complexity of the proposed
method is lower than that of the LS phase fitting while achieving better performance. The
Kalman filter is usually of high complexity. But for our particular application, that uses

69. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 55
0 5 10 15 20 25
0
0.5
1
1.5
2
2.5
OFDM symbol index
Phaseerror(rad)
Comparison of Phase Error
Without Noise
Proposed
ML
Figure 4.5: Comparison of phase error : Kalman
a simple state-space model for PE, it can be verified that the Kalman implementation for
refining ML estimates obtained from pilot tones, does not significantly increase computa-
tional complexity. Indeed, due to basic formulation of the PE, the Kalman filter equations
can be largely simplified and complex matrix calculation is avoided. Another interesting
property to note about the Kalman filter is that Km and Pm|m do not depend on the data
xm. Therefore, it is possible for both of these terms to be computed off-line prior to any
filtering. This fact is not used for calculating complexity in this section.
4.4 Simulation Results
4.4.1 Simulation Environment
We simulated a transmitter and receiver according to the parameters established by the
802.11a standard [9]. Details of simulation parameters are listed in Table 5.1. For the

70. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 56
Table 4.1: Simulation Parameters
Sampling time, Ts 50ns
FFT length, N 64
Number of used subcarriers, Nu 52
Number of pilot subcarriers, Np 4
Guard interval length, Ng 16
Subcarrier spacing 312.5 KHz
Modulation 4-QAM, 64-QAM
Packet length 100 symbols
Number of packets 10000
Channel coding rate 1/2
channel model, only small-scale fading is considered. Both the distance dependent path
loss and the shadowing are assumed to be constant over the simulation and incorporated
into the SNR. The power delay profile for typical office environment, based on the ETSI
model A in Fig.2.2 [4], is generated and time variation model in Chap 2 is employed.
Details of random channel generations are described in Sec 3.4. Moreover, one packet is
composed of 100 OFDM symbols. The proposed timing synchronization method in Chap
3 and LS channel estimation are used and SNR estimation is carried out as stated in (4.19).
In order to clearly demonstrate the performance of the proposed method, RFO, , is set at
constant 0.1(= 31.25KHz) for 4-QAM modulation and 0.05(= 15.62KHz) for 64-QAM
modulation before initial CFO estimation.
4.4.2 Performance result
In Fig.4.6 and Fig.4.7, we present average PER of the proposed method for the different
values of SNR with no channel estimation error. Cases are also shown for the ideal per-
formance when is zero, for the performance with ML method using Np(=4) pilots as
specified in the standard, for the performance without RFO compensation. We see that the
proposed method is superior to the conventional method. In particular, the performance for

71. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 57
64-QAM modulation with higher SNR is greatly improved. The difference in the perfor-
mance with respect to SNR is expected from Fig.4.1. While the performance gain in the
low SNR region is mainly due to decreased PE estimation error, the performance gain at
high SNR mostly comes from ICI reduction by RFO compensation.
Fig.4.8 and Fig.4.9 show average PER performance of the proposed method with chan-
nel estimation error. It can be seen that the gap between the proposed method and ”Ideal”
is slightly increased. This effect occurs because the channel estimation error may make the
ML estimate of PE biased and this can cause the PE output of the Kalman filter to converge
to an incorrect estimate. Also, since it takes a few OFDM symbols for the Kalman filter to
correctly estimate the RFO, ICI is not completely compensated during the initial tracking
period. However, These factors induce almost negligible PER performance degradation
and the performance gain over ML method is even increased since the proposed method is
able to update channel estimation result using the estimated RFO.
Fig.4.10 shows the RFO estimation error variance of the proposed method with channel
estimation error when = 0.1. The variance is calculated using the RFO estimates from
the 10th, 50th and the last OFDM symbol in each packet. It can be seen in Fig.4.10 that
the the estimation performance improves as the OFDM symbol index increases since more
measurement history is used for the estimation. For SNR greater than 15dB, the proposed
method can reduce the error variance below 10−8
for the last OFDM symbol in the packet.
The resulting standard deviation of the RFO estimation error is less than 31.25Hz.
4.5 Conclusions
In this chapter, we propose a residual frequency offset (RFO) compensator that compen-
sates both RFO and RFO induced phase error (PE) by using a Kalman filter. In our pro-
posed method, after the state-space model for RFO and PE is derived, a Kalman filter is
applied to track and estimate RFO and PE simultaneously. This method, when compared

72. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 58
4 6 8 10 12 14 16
10
−2
10
−1
10
0
Average SNR (dB)
AveragePER
Proposed
Ideal
ML
Without Comp
Figure 4.6: Average packet error rate for 4-QAM without channel estimation error ( = 0.1)
16 18 20 22 24 26 28
10
−2
10
−1
10
0
Average SNR (dB)
AveragePER
Proposed
Ideal
ML
Without Comp
Figure 4.7: Average packet error rate for 64-QAM without channel estimation error ( =
0.05)

73. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 59
4 6 8 10 12 14 16
10
−2
10
−1
10
0
Average SNR (dB)
AveragePER
Proposed
Ideal
ML
Without Comp
Figure 4.8: Average packet error rate for 4-QAM ( = 0.1)
16 18 20 22 24 26 28
10
−2
10
−1
10
0
Average SNR (dB)
AveragePER
Proposed
Ideal
ML
Without Comp
Figure 4.9: Average packet error rate for 64-QAM ( = 0.05)

74. CHAPTER 4. RESIDUAL FREQUENCY OFFSET AND PHASE TRACKING 60
0 5 10 15 20 25
10
−9
10
−8
10
−7
10
−6
10
−5
10
−4
Average SNR (dB)
Errorvariance
100
th
symbol
50th
symbol
10th
symbol
Figure 4.10: Error variance of RFO estimation ( = 0.1)
with LS phase fitting, offers improved estimation and tracking behavior for RFO with less
complexity. Numerical results show that the proposed method significantly overcomes the
performance degradation due to RFO and almost achieves ideal PER performance.

75. Chapter 5
Enhanced DFT-Based MMSE Channel
Estimation
In practical OFDM systems, in order to limit the interference to adjacent channels, some
subcarriers are set to zero. These non-existent subcarriers are are often referred to as
virtual carriers (VCs). Due to these VCs, conventional DFT-based approaches are not
directly applicable because they induce spectral leakage, which results in an error floor
for the mean square error (MSE) performance. To circumvent this problem, we propose
an enhanced DFT-based minimum mean square error (MMSE) channel estimator using a
Kalman smoother. Our approach is based on building a robust state-space model for chan-
nel frequency response (CFR) followed by Kalman filtering and smoothing to minimize the
effects of leakage. Time domain MMSE weighting is also used to suppress channel noise.
Our simulation results show that the proposed method almost achieves the performance of
the optimal MMSE estimator while having a limited computational complexity.
61

76. CHAPTER 5. ENHANCED DFT-BASED MMSE CHANNEL ESTIMATION 62
5.1 Introduction
OFDM is an effective technique for overcoming multipath fading and achieving high-bit-
rate transmission over wireless channels. However, without channel estimation, OFDM
systems have to use differential phase shift keying (DPSK), which has 3dB SNR loss com-
pared to coherent demodulation. If coherent OFDM is adopted, channel estimation be-
comes a necessary requirement and pilot tones are usually used for channel estimation. In
general, the realization of pilot-aided channel estimation is based on least square (LS) [31]
or minimum mean square error (MMSE) [30]. The LS approach is simple but has poor per-
formance, whereas the MMSE approach has good performance but is complex and requires
a priori knowledge of channel statistics. As a compromise between LS and MMSE, a dis-
crete Fourier transform (DFT) based channel estimation that utilizes the channel impulse
response has been widely studied for OFDM systems [31] [32].
Fig. 5.1 shows a block diagram of DFT-based channel estimation. This method trans-
forms the channel from the frequency domain into the time domain by an inverse discrete
Fourier transform (IDFT). After that, a time domain windowing is applied to the channel
impulse response assuming the window length is longer than the maximum delay spread of
the channel. Finally, this method transforms the channel impulse response back to the fre-
quency domain by a DFT. Hence, the noise in the taps beyond the maximum delay spread
of the channel is filtered out in the time domain and this improves a performance. However,
this method assumes that all subcarriers of the OFDM signal are used for pilot transmis-
sion. Otherwise, after the IDFT operation, the power may spill over all the taps in the time
domain, and the noise filtering process becomes inapplicable. For example, in the WLAN
802.11a standard, in order to limit the interference to adjacent channels, subcarriers at the
band edge of the shaping filter are left unmodulated and set to zero. These unused subcar-
riers are called virtual carriers (VCs) and hence a direct application of DFT-based channel

77. CHAPTER 5. ENHANCED DFT-BASED MMSE CHANNEL ESTIMATION 63
estimation can not be used. Furthermore, the CFR at VCs can not be estimated by conven-
tional channel estimation approaches. Fig. 5.2 shows a CFR of the OFDM system where
11 subcarriers out of total 64 subcarriers are VCs. Although VCs are located at both edges
of the bandwidth, this figure shows the CFR from 0 to 64 instead of from -32 to 32 since
the DFT is periodic. It can be seen that direct application of a DFT to the CFR leads to
leakage in the time domain. In other words, windowing in the frequency domain can lead
to severe distortion in the time domain and hence the resulting channel estimate becomes
erroneous.
A windowed DFT-based channel estimation was proposed in [33] to reduce an aliasing
error and suppress the noise, but it requires searching for an optimal generalized Hanning
window shape to minimize MSE. Furthermore, implementation complexity is too high to
be used in practical systems for the non-interpolation case. In [34], robust Wiener filter-
ing is applied to eliminate the leakage effect due to absence of pilot symbols in the VCs.
This approach tried to improve BER performance by combining Wiener interpolation and
Wiener filtering for interpolation cases.
In this chapter, I present an enhanced DFT-based channel estimation method with sim-
ple Kalman filtering and smoothing. Moreover, a weighting function is applied to the
effective channel impulse response. The weighting function is chosen so that the MSE
between CFR and its estimate is minimized.
5.2 Kalman Smoothing
Autoregressive (AR) modeling is commonly used to model discrete time random processes.
This is due to the simplicity with which the parameters can be computed and due to their
correlation matching property. AR process of order p can be generated as

78. CHAPTER 5. ENHANCED DFT-BASED MMSE CHANNEL ESTIMATION 64
LS IFFT FFT
Freq domain Time domain Freq domain
H(1)
H(N)
H(2)
h(L)
H’(2)
H’(N)
H’(1)
0
h(1)
Figure 5.1: Block diagram of DFT-based method

79. CHAPTER 5. ENHANCED DFT-BASED MMSE CHANNEL ESTIMATION 65
0 20 40 60
0
0.5
1
1.5
2
index
magnitude
Ideal
LS
Figure 5.2: Existence of virtual subcarriers
xn = −
p
k=1
akxn−k + wn (5.1)
where w(n) is a complex white Gaussian noise process with uncorrelated real and imag-
inary components. Using Yule-Walker equations, the corresponding autocorrelation matrix
can be calculated using the autocorrelation function R as [35],
Rxa = −v (5.2)

80. CHAPTER 5. ENHANCED DFT-BASED MMSE CHANNEL ESTIMATION 66
where
Rx =








R(0) R(−1) · · · R(−p + 1)
R(1) R(0) · · · R(−p + 2)
...
...
...
...
R(p − 1) R(p − 2) · · · R(0)








a = a1 a2 · · · ap
T
v = R(1) R(2) · · · R(p)
T
For a non-interpolation channel estimation case, increasing the order of the AR model
not only increases computational complexity but also degrades the estimation performance
due to singularity of the channel correlation matrix, Rx, [36]. Suppose the LS estimate of
the channel frequency response is ˆHk and the channel dynamics follows an AR(1) model,
then a state space model for the CFR can be written as
Hk = A · Hk−1 + qk−1
ˆHk = Hk + rk
where Hk is the CFR at index k, qk−1 ∼ N(0, Qk−1) and rk ∼ N(0, Rk). Using the
above equations, the prediction and update steps of the Kalman filter are :

81. CHAPTER 5. ENHANCED DFT-BASED MMSE CHANNEL ESTIMATION 67
Prediction steps :
mp = A · mk−1
Pp = A · Pk−1 · AH
+ Qk−1
Update steps :
Kk = Pp · (Pp + Rk)−1
mk = mp + Kk(Hk − mp)
Pk = Pp − KkPp
A can be obtained simply as the ratio of correlations of Hk as A ≈ R(1)/R(0) from the
Yule-Walker method. Also, Qk−1 ≈ R(0) − |R(1)|2
/R(0). The two unknown correlation
values, R(1) and R(0), can be estimated from an LS estimate of the channel frequency
response. In practice, the Kalman filter equations easily cope with missing values or VCs.
When missing values occur, the prediction steps are processed as usual, the update steps
are changed as mk = mp and Pk = Pp. After obtaining the Kalman filter output, then the
Kalman smoother is applied to the Kalman filtered output. The Kalman smoother calculates
recursively the state posterior distributions.
p(Hk| ˆH0:N−1) = N(Hk|ms
k, Ps
k ) (5.3)
The difference between filtering and smoothing is that the smoothed outputs are con-
ditioned to all of the measurement data, while the filtered outputs are conditioned only to
the measurement obtained before and at the time step k. So the smoothed output can be
calculated from the Kalman filter result by recursions as follows:

82. CHAPTER 5. ENHANCED DFT-BASED MMSE CHANNEL ESTIMATION 68
0 20 40 60
0
0.5
1
1.5
2
index
magnitude
Ideal
Filter
Filter+Smooth
Figure 5.3: Kalman filtering and smoothing output, SNR=10dB
P−
k+1 = APkAH
+ Qk
Ck = PkAH
(P−
k+1)−1
ms
k = mk + Ck(ms
k+1 − Amk)
Ps
k = Pk + Ck(Ps
k+1 − P−
k+1)CH
k
starting from the last step N−1, with ms
N−1 = mN−1 and Ps
N−1 = PN−1. Fig.5.3 shows
an example plot of output of Kalman filtering and smoothing at SNR=10dB when missing
values occur between k = 27 and k = 37. It can be seen that smoothing significantly
decreases the estimation error of filtering above.

83. CHAPTER 5. ENHANCED DFT-BASED MMSE CHANNEL ESTIMATION 69
5.3 MMSE Filtering in the time domain
By the IDFT operation, we transform the entire channel estimate obtained from the Kalman
prediction into a time domain channel impulse response
hkal(n) =
1
N
N−1
k=0
ms
k exp(j2πkn/N) (5.4)
Since the prediction steps change white noise into colored Gaussian noise, the resulting
correlation vector of the colored noise in the time domain can be obtained by the scaled
IDFT of Ps
k
rnn(m) = E[nl+mn∗
l ] =
1
N2
N−1
k=0
Ps
k exp(j2πkm/N) (5.5)
The MMSE filtering matrix, M, in the time domain can be derived from the following
equations,
min|h − M · hkal(n)|2
(5.6)
By the orthogonality principle, M can be written as
M = Rhh · (Rhh + Rnn)−1
(5.7)
where
Rnn = t(rnn)
Rhh = E[hhH
]

84. CHAPTER 5. ENHANCED DFT-BASED MMSE CHANNEL ESTIMATION 70
where t denotes an operator that transforms a vector into a Toeplitz matrix. Since the
time domain correlation matrix of the channel impulse response is usually unavailable, Rhh
for a uniform channel power delay profile is used instead for robustness of the estimator.
Assuming a WSSUS (wide sense stationary uncorrelated scattering) channel [37], Rhh is
given by
Rhh =


1
N
R(0) 0
0 0

 (5.8)
R(0) = E[HkH∗
k ] =
N−1
n=0
|h(n)|2
(5.9)
where NGI denotes the length of the cyclic prefix and I represents a NGI by NGI iden-
tity matrix. Since the number of virtual carriers at the left and right side for the guard band
are the same, the resulting Rnn is a real matrix. Therefore, the resulting M matrix is a
real NGI by N matrix. Based on the above analysis, the overall estimation step can be
summarized as
ms
k = S( ˆHLS) (5.10)
Htotal = F · M · G · ms
k (5.11)
where S denotes a Kalman smoother, ˆHLS is an initial LS estimate of the channel
frequency response, G is an N by N IDFT matrix and F is an N by NGI DFT matrix.

85. CHAPTER 5. ENHANCED DFT-BASED MMSE CHANNEL ESTIMATION 71
5.4 Complexity issues
In order to measure the computational complexity of different estimation methods, we use
the number of complex multiplications. The MMSE estimation generally requires O(N2
)
and the window method [33] requires Nu + N + (2N/3)log2N + Nu. In contrast, for the
proposed method, a Kalman smoother requires O(N) and MMSE filtering requires NGI ·N.
So the total required complex multiplication is O(N)+N ·NGI +(2N/3)log2N. Although
the proposed method may appear to increases computational complexity over the window
method, the window method is very sensitive to window size which needs to be decided by
a complex optimization procedure. Therefore, implementation complexity of the window
method is much higher than that of the proposed method. To summarize, the proposed
method requires much less computational complexity than MMSE, while achieving almost
equivalent performance and it can be implemented simpler than the window method even if
it increases computational complexity a little in some sense. Note that although the Kalman
equations are usually of high complexity, for our particular application that uses a simple
state-space model for estimation, it can be verified that the Kalman implementation does
not significantly increase computational complexity and has better performance than the
window method. Furthermore, an interesting and useful property of the Kalman imple-
mentation is that, since the Kalman gain and error covariance matrix do not depend on the
data, it is possible for these terms to be calculated off-line prior to any filtering.
5.5 Simulation Results
5.5.1 Simulation Environment
We simulated a transmitter and receiver according to the parameters established by the
802.11a standard [9]. The simulation parameters are listed in Table 5.1. For the channel
model, only small-scale fading is considered. Both the distance dependent path loss and