國 立 交 通 大 學
電信工程學系
碩 士 論 文
適用於正交分頻多工行動系統之
都普勒偏移估測與干擾消除技術
Doppler Shift Estimation and Interference
Cancellation for Mobile OFDM Systems
研 究 生:李思漢
Student: Sih-Han Li
指導教授:李大嵩 博士 Advisor:
Dr.
Ta-Sung
Lee
適用於正交分頻多工行動系統之都普勒偏移估測
與干擾消除技術
Doppler Shift Estimation and Interference Cancellation
for Mobile OFDM Systems
研 究 生:李思漢
Student: Sih-Han Li
指導教授:李大嵩 博士 Advisor:
Dr.
Ta-Sung
Lee
國立交通大學
電信工程學系碩士班
碩士論文
A Thesis
Submitted to Department of Communication Engineering
College of Electrical Engineering and Computer Engineering
National Chiao Tung University
in Partial Fulfillment of the Requirements
for the Degree of
Master of Science
in
Communication Engineering
June 2008
Hsinchu, Taiwan, Republic of China
適用於正交分頻多工行動系統之都普勒偏移估測
與干擾消除技術
研究生:李思漢
指導教授:李大嵩 博士
國立交通大學電信工程學系碩士班
摘要
近年來,正交分頻多工(Orthogonal Frequency Division Multiplexing, OFDM)
系統在無線寬頻通訊之應用上日漸普及,許多系統與規格均採納此技術,例如數 位電視、數位廣播、IEEE 802.11 a/g/n、IEEE 802.16 等。此外,移動傳輸將是未 來無線通訊系統的趨勢之ㄧ,但正交分頻多工系統在移動的環境下將面臨都普勒 偏移(Doppler Shift)的問題。都普勒偏移使得接收端的子載波間失去正交性,因 而導致子載波間的相互干擾,並限制了系統的效能。為了解決此問題,在本篇論 文中,吾人提出一個最大都普勒偏移的估測方法,此方法可良好地運作在低信噪 比(Signal-to-Noise ratio, SNR)的環境下。且利用此估測出的都普勒資訊,吾人 可使用一個適應性的干擾消除方法,選擇合適的等化器(Equalizer)來消去子載 波的干擾,而不會浪費運算量。吾人也藉由電腦模擬來驗證所提方法可良好運作 在正交分頻多工系統中。
Doppler Shift Estimation and Interference
Cancellation for Mobile OFDM Systems
Student: Sih-Han Li
Advisor: Dr. Ta-Sung Lee
Department of Communication Engineering
National Chiao Tung University
Abstract
In recent years, orthogonal frequency division multiplexing (OFDM) becomes a
popular technique in wireless broadband communications. There are many systems
adopting OFDM techniques, such as digital video broadcasting (DVB), digital audio
broadcasting (DAB), IEEE 802.11 a/g/n, IEEE 802.16, etc. On the other hand, mobile
transmission is a trend in future wireless communications. However, in mobile OFDM
systems, Doppler shift destroys the orthogonality between subcarriers, and this causes
the intercarrier interference (ICI) and limits the performance of the receiver. To deal
with this problem, we proposed a maximum Doppler shift estimation method for
mobile OFDM systems, and the proposed method can work well at low SNR. Then,
with the help of the Doppler information, an adaptive ICI cancellation scheme can be
employed, which can choose an appropriate equalizer to cancel ICI without needing
unnecessary computation. Finally, computer simulation results for mobile OFDM
Acknowledgement
I would like to express my deepest gratitude to my advisor, Dr. Ta-Sung Lee, for
his enthusiastic guidance and great patience. I learn a lot from his positive attitude in
many areas. Heartfelt thanks are also offered to all members in the Communication
System Design and Signal Processing (CSDSP) Lab for their constant encouragement.
Contents
Chinese Abstract
I
English Abstract
II
Acknowledgement III
Contents IV
List of Figures
VII
List of Tables
X
Acronyms Glossary
XI
Notations XII
Chapter 1 Introduction... 1
Chapter 2 Fundamentals of OFDM and DVB-T Systems ... 4
2.1 Overview of OFDM Systems ... 4
2.1.1 Guard Interval and Cyclic Prefix (CP) ... 7
2.1.2 Continuous Time Model of OFDM ... 8
2.1.3 Discrete-Time Model of OFDM ... 9
2.2.1 System Overview ... 10
2.2.2 Frame Structure and Transmission Band ... 13
2.2.3 Reference Signals ... 14
Chapter 3 Proposed Maximum Doppler Shift Estimation Method
17
3.1 System and Channel Models... 183.1.1 Channel Model... 20
3.2 Proposed Doppler Shift Estimation Method Using Doppler Power Spectrum 23 3.2.1 Chirp Transform Algorithm (CTA) ... 26
3.2.2 Averaging and Smoothing Effect of Maximum Doppler Shift Estimation Method ... 29
3.3 Complexity Analyses of FFT and CTA... 31
3.4 Computer Simulations ... 34
3.5 Summary ... 42
Chapter 4 ICI Cancellation in OFDM Systems over Time-Varying
Channels ... 43
4.1 ICI Analysis ... 43
4.1.1 ICI in OFDM Systems ... 44
4.1.2 Analysis of ICI Power... 46
4.2.1 Low-Complexity MMSE ICI Cancellation... 50
4.3 Computer Simulations ... 52
4.4 Summary ... 58
Chapter 5 Conclusion ... 59
List of Figures
Figure 2.1: MCM scheme for high-speed transmission... 5
Figure 2.2 (a): Illustration of OFDM bandwidth efficiency: traditional MCM technique ... 5
Figure 2.2 (b): Illustration of OFDM bandwidth efficiency: OFDM technique... 6
Figure 2.3: OFDM signal spectra ... 6
Figure 2.4: ISI phenomenon of OFDM systems: (a) without guard interval, (b) zero-padded guard interval... 7
Figure 2.5: Guard interval with cyclic prefix ... 8
Figure 2.6: Continuous time OFDM system model... 9
Figure 2.7: Discrete-time OFDM system model ... 10
Figure 2.8: Functional block diagram of DVB-T ... 11
Figure 2.9: Continual pilot of DVB-T ... 15
Figure 2.10: Scattered pilot of DVB-T ... 16
Figure 3.1: OFDM system model ... 18
Figure 3.2: Autocorrelation function of the channel impulse response ... 21
Figure 3.4: Block diagram of proposed Doppler shift estimation method ... 23
Figure 3.5: Piece-wise linear model in one OFDM symbol period ... 24
Figure 3.6: Frequency samples for chirp transform algorithm ... 26
Figure 3.7: Block diagram of chirp transform algorithm... 28
Figure 3.8: Block diagram of chirp transform algorithm for causal finite-length impulse response ... 29
Figure 3.9: Averaging effect of estimated Doppler power spectrum ... 30
Figure 3.10: Normalized MSE versus SNR. The effect of different accumulation size Q is shown. The observation interval K is 128 symbols and the maximum Doppler shift is 0 Hz to 300 Hz. ... 35
Figure 3.11: Normalized MSE versus accumulation size. The effect of different observation interval K is shown. The maximum Doppler shift is 0 Hz to 300 Hz and SNR is 15 dB. ... 36
Figure 3.12: Normalized MSE versus accumulation size for different observation intervals and SNR. The maximum Doppler shift is 0 Hz to 300 Hz. ... 37
Figure 3.13: Tracking ability of the proposed maximum Doppler shift estimation for SNR = 5dB. (a) K = 64 with Q = 3. (b) K = 64 with Q = 5... 38
Figure 3.14: Tracking ability of the proposed maximum Doppler shift estimation for SNR = 5dB. (a) K = 128 with Q = 3. (b) K = 128 with Q = 5. ... 39
Figure 3.15: Tracking ability of the proposed maximum Doppler shift estimation for SNR = 5dB. (a) K = 256 with Q = 3. (b) K = 128 with Q = 5. ... 40
Figure 3.15: Tracking ability of the proposed maximum Doppler shift estimation for
three paths. (K = 256, Q = 5, and SNR = 5 dB) ... 41
Figure 3.16: Normalized MSE versus accumulation size. Compared Park’s method [10] with proposed method (SNR = 15 dB). ... 42
Figure 4.1 (a): ICI variance versus d (d is the index of super- or sub- diagonal) ... 48
Figure 4.1 (b): Zoomed view of Figure 4.1 (a) ... 49
Figure 4.2: Desired structure of frequency domain channel matrix ... 50
Figure 4.3: BER performance versus SNR. The normalized maximum Doppler shift is 0.01. ... 53
Figure 4.4: BER performance versus SNR. The normalized maximum Doppler shift is 0.05. ... 54
Figure 4.5: BER performance versus SNR. The normalized maximum Doppler shift is 0.1. ... 54
Figure 4.6: BER performance versus SNR. The normalized maximum Doppler shift is 0.15. ... 55
Figure 4.7: BER performance of different equalization type with variation of normalized maximum Doppler shift. ... 57
Figure 4.8: Variation of normalized maximum Doppler shift for Figure 4.7. ... 57
Figure 4.9: Average computational complexity of MMSE with D = 3, MMSE with D = 2, and the adaptive ICI cancellation scheme ... 58
List of Tables
Table 2.1: Parameters for 6 MHz channel in DVB-T ... 12
Table 3.1: Maximum Doppler shift with different velocity for DVB-T systems ... 21
Table 3.2: Complexity analyses of CTA and FFT... 32
Table 3.3: Symbol duration for DVB-T systems ... 33
Table 3.4: Parameters of computer simulations... 34
Table 4.1: Parameters of computer simulations... 52
Acronyms Glossary
ACF autocorrelation function AWGN additive white Gaussian noise BER bit-error-rate
CIR channel impulse response CP cyclic prefix
CSI channel state information CTA chirp transform algorithm DFT discrete Fourier transform DAB digital audio broadcasting
DVB-T digital video broadcasting-terrestrial FFT fast Fourier transform
ICI intercarrier interference
IDFT inverse DFT
IFFT inverse FFT
ISI intersymbol interference LS least square
MCM multi-carrier modulation MMSE minimum mean square error MSE mean square error
OFDM orthogonal frequency division multiplexing SNR signal-to-noise ratio
Notations
D number of subcarrier of ICI mitigation
F DFT matrix
fd maximum Doppler shift
fn normalized maximum Doppler shift
hn,l time domain channel tap of l-th tap at time instance n
Hi channel transfer function at the i-th subcarrier
HFD frequency domain channel matrix
HTD time domain channel matrix
Ii ICI term at the i-th subcarrier
K interval observation
L length of channel
M number of CTA output
ν AWGN
N number of subcarrier
NCP number of cyclic prefix
NP number of pilot tone
Pi pilot tone at the i-th subcarrier
Q accumulation size
τ time delay
Ts sampling period
Tsym symbol duration
w angel frequency
x transmitted time domain signal
X transmitted frequency domain signal
Y received signal after FFT
( )
S ⋅ Doppler power spectrum
( )
hh
φ ⋅ autocorrelation of consecutive CIRs
( )
0
J ⋅ zeroth order Bessel function of the first kind
( )⋅* complex conjugate
( )⋅H complex conjugate transpose ( )⋅−1 inverse ( ) arg ⋅ argument ( ) max ⋅ maximum { } DFT ⋅ DFT { } E ⋅ expectation { } FT ⋅ Fourier transform { } FFT ⋅ FFT { } IDFT ⋅ IDFT { } IFFT ⋅ IFFT
Chapter 1
Introduction
Orthogonal frequency division multiplexing (OFDM) is a popular technique in
modern wireless broadband communication systems. Nowadays, OFDM is utilized in
many standards. In wired transmission systems, OFDM is adopted in asymmetric
digital subscriber loop (ADSL) and very high bit-rate digital subscriber loop (VDSL).
In wireless transmission systems, OFDM is utilized in digital audio
broadcasting-terrestrial (DVB-T) [1], [2], [3], digital audio broadcasting-handheld
(DVB-H) [4], digital audio broadcasting (DAB), IEEE 802.11 a/g/n, and IEEE 802.16,
etc. In OFDM systems, the available bandwidth is divided into several orthogonal
subcarriers for transmission. Then, a cyclic prefix (CP) is inserted before each
transmitted OFDM symbol. If the length of CP is equal to or longer than the delay
spread of the channel, the intersymbol interference (ISI) can be eliminated. Moreover,
subcarriers in OFDM systems are orthogonal to each other over time-invariant channels,
thus the conventional OFDM systems could only utilize a simple one-tap equalizer to
recover the transmitted symbol on each subcarrier [5].
In wireless environments, because of the user’s movement, the channel is time
variant. The user’s mobility introduces a Doppler shift, which destroyed the
orthogonality between subcarriers and degraded the bit-error-rate (BER) performance.
insufficient. Therefore, in order to improve the BER performance, the ICI cancellation
technique plays a very important role in mobile OFDM systems. To mitigate ICI in an
efficient manner, it is necessary to know the value of the maximum Doppler shift. With
the help of the Doppler information, an appropriate equalizer can be chosen to cancel
ICI without needing unnecessary computation.
Many methods have been proposed to estimate the maximum Doppler shift. The
autocorrelation function (ACF) based Doppler shift estimation method has been
proposed in [6]. A maximum-likelihood (ML) approach based estimation is exploited in
[7], [8]. However, these methods are sensitive to signal-to-noise ratio (SNR). In OFDM
systems, the autocorrelation between the repeated parts of the symbol due to CP has
been exploited in [9] to estimate the maximum Doppler shift. When the environment
causes long delay spread, the estimation will degrade greatly. Moreover, this method is
also sensitive to SNR. To deal with the noise problem, a method, which uses the
channel power spectrum, is described in [10].
The main topic of this thesis is about the maximum Doppler shift estimation
method for OFDM systems. Since the Doppler power spectrum is related to the
maximum Doppler shift, a maximum Doppler shift estimation method which utilizes
the Doppler power spectrum is proposed in this thesis. By exploiting the characteristics
of the Doppler power spectrum, the ACF of the estimated channel impulse response
(CIR) at the receiver is translated onto the frequency domain by Fourier transform. Two
kinds of Fourier transform schemes will be introduced; one is fast Fourier transform
(FFT) scheme, and the other is chirp transform algorithm (CTA) scheme. These
schemes provide a tradeoff between computational complexity and memory size.
Furthermore, because of using the autocorrelation of the channel impulse responses, the
Doppler shift, an adaptive ICI cancellation scheme is proposed. The estimated
maximum Doppler shift is utilized as a parameter of the adaptive ICI cancellation
scheme, and the parameter can help in selecting the appropriate equalizer technique. In
order to mitigate the ICI effectively, a conventional linear minimum mean square error
(MMSE) equalizer [11], [12] can be employed. The complexity of the conventional
MMSE equalizer requires O N
( )
3 operations, resulting in impractical implementation for large N. Therefore, a low-complexity MMSE equalizer [5] is utilized in this thesisto reduce the computational complexity. DVB-T is selected as the system platform for
computer simulations.
The remainder of this thesis is organized as follows. Chapter 2 provides the
overview of OFDM systems and DVB-T systems. In Chapter 3, the proposed
maximum Doppler shift estimation method is presented and analyzed. In Chapter 4, ICI
analysis is introduced and an adaptive ICI cancellation scheme is discussed. In Chapter
Chapter 2
Fundamentals of OFDM and DVB-T
Systems
In this chapter, we discuss the technique of orthogonal frequency division
multiplexing (OFDM) and a digital video broadcasting system based on OFDM
technique. First, we state the principle of OFDM transmission and its block diagram.
Then, we will make a description of digital video broadcasting-terrestrial (DVB-T)
standard including the system block diagram, frame structure, and reference signal.
2.1 Overview of OFDM Systems
OFDM derived from multi-carrier modulation (MCM) [13] which is a technique
of transmitting high-speed data by dividing the stream into several parallel low-rate
streams and modulating each of these data streams onto individual subcarriers, as
shown in Figure 2.1. Therefore, by making all subcarriers of MCM narrowband, each
serial
to
parallel
high-speed
data streams
parallel
low-rate
streams
0X
1 NX
−D/A
Figure 2.1: MCM scheme for high-speed transmission
Compared with MCM systems, the subcarriers of OFDM systems are overlapping
and orthogonal to each other. For two subcarriers to be orthogonal within [0, T , they ] must be satisfy the orthogonality constraint:
( ) *( ) 0 1, 0, T n k n k w t w t dt n k ⎧ = ⎪⎪ ⋅ = ⎨⎪ ≠ ⎪⎩
∫
. (2.1)With the property of orthogonality, Figure 2.2 shows OFDM systems have better
bandwidth efficiency than traditional MCM systems. The spectra of OFDM signals are
depicted in Figure 2.3. The overlapping sinc shaped spectra have zero inter-subcarrier
interference at the right frequency sampling points.
Frequency
Subchannel 1 2 3 4
Classical nonoverlapping multicarrier technique 5
Figure 2.2 (b): Illustration of OFDM bandwidth efficiency: OFDM technique
2.1.1 Guard Interval and Cyclic Prefix (CP)
Although the problem of multipath fading may be dealt with efficiently in OFDM
systems, however, as in single carrier systems, intersymbol interference (ISI) is still a
major concern for OFDM systems, as shown in Figure 2.4(a). In order to combat ISI,
Figure 2.4(b) shows a zero-padded guard interval is introduced for each OFDM symbol.
The guard interval is set larger than the multipath delay spread, such that the pervious
symbol cannot interfere with the current symbol. However, intercarrier interference
(ICI) arises because of the zero-padded guard interval.
(m-1)-th OFDM symbol
m-th OFDM symbol
ISI
(a)
(m-1)-th OFDM symbol
m-th OFDM symbol
ISI
(b)
guard
time
Figure 2.4: ISI phenomenon of OFDM systems: (a) without guard interval, (b)
ICI is crosstalk between different subcarriers, which means that the orthogonality
between subcarriers disappeared. To eliminate this obstacle, a cyclic prefix (CP) is
introduced in the guard interval. Figure 2.5 illustrates this technique. CP ensures that
there are always an integer number of cycles within a FFT interval, therefore the
orthogonality between subcarriers can be held.
Figure 2.5: Guard interval with cyclic prefix
2.1.2 Continuous Time Model of OFDM
The general form of continuous time model of an OFDM signal can be written as a
set of modulated carriers transmitted in parallel as follows
( ) 1 ,
(
sym)
0 N m k m k k x t − X w t mT = =∑
⋅ − , (2.2) where ( ) for sym)
otherwise 2 0, 0 k j f t k e t T w t π ⎧ ⎡ ⎪ ∈ ⎪ ⎢ ⎪ ⎣ = ⎨ ⎪⎪⎪⎩ (2.3)is the waveforms of the k-th subcarrier, and fk = f0 +k T/ sym, k = 0, 1, 2, ", N −1. In this model, N is the number of subcarriers, Xk m, is the transmitted data on the
k-th subcarriers in the m-th OFDM symbol, Tsym is symbol period, and f is the k carrier frequency of the k-th subcarrier. Consequently, a transmitted stream of OFDM
( ) ( ) 1 ,
(
)
0 N m k m k s m m k x t +∞ x t +∞ − X w t mT =−∞ =−∞ = =∑
=∑ ∑
⋅ − . (2.4)For simplicity, assuming the channel is ideal, and the demodulation of OFDM is based
on the orthogonality constraint in Equation (2.1), and therefore the demodulator may
derive l ( 1) ( ) *( ) , 1 s s m T k m mT k X x t w t dt T + =
∫
⋅ , (2.5)where lXk m, is the received data on the subcarrier k-th in the m-th symbol. Figure 2.6 shows a simple transmission and reception processes of continuous time OFDM system
model. ( ) 0 w t ( ) 1 w t ( ) 1 N w − t 0,m X 1,m X 1, N m X −
∑
Channel ( ) * 0 w t ( ) * 1 w t ( ) * 1 N w − t l0,m X l1,m X lN 1,m X −Figure 2.6: Continuous time OFDM system model
2.1.3 Discrete-Time Model of OFDM
Compare to the continuous time model, the modulator and demodulator of the
discrete-time OFDM system are replaced by an inverse discrete Fourier transform
(IDFT) and a discrete Fourier transform (DFT), respectively. In real implementation of
an OFDM system, DFT is replaced by an appropriately sized fast Fourier transform
(FFT) to reduce calculating complexity. The discrete-time OFDM system model can be
written as
lm =DFT IDFT
(
(
m)
)
where Xlm = ⎢⎡⎣X0,m X1,m " XN−1,m⎤⎥⎦, and X contains N transmitted data in m the m-th symbol. Figure 2.7 shows a simple transmission and reception processes of
discrete-time OFDM system model.
0,m X 1,m X 1, N m X − l0,m X l1,m X lN 1,m X − Figure 2.7: Discrete-time OFDM system model
2.2 Overview of DVB-T Systems
Digital video broadcasting–terrestrial (DVB-T) standard [1] based on coded
OFDM (COFDM) and it was established by European Telecommunication Standard
Institute (ETSI) in 1997. Furthermore, DVB-T is adopted by most countries around the
world including Taiwan.
2.2.1 System Overview
Figure 2.8 depicts the functional block diagram of DVB-T system [14], [15]. The
source coding of audio and video signals is based on ISO MPEG-2 standard. After the
MPEG-2 transport multiplexer, the video, audio and data stream are translated to
packets. The packet length is 188 bytes and it includes one synchronization word
(sync-word, or SYNC) bytes and 187 data bytes. When the transmitting packet has priority, hierarchical transmission would be used. The splitter is used to split the packet
into high priority and low priority parts. These two parts of data could have individual
channel coding and mapping mode. Every eight packet are multiplied by a pseudo
random binary sequence (PRBS) which derived by the polynomial X15 +X14 +1. To provide an initialization signal, the sync-word of the first packet in a group of eight
packets is translated form 47HEX (SYNC) to B8HEX ( SYNC ). This operation is called multiplex adaptation and randomization for energy dispersal.
Figure 2.8: Functional block diagram of DVB-T
After multiplex adaptation, a Reed-Solomon (RS) shortened code (204, 188, t
= 8) and a convolutional byte-wise interleaving with depth I = 12 shall be applied to
generate error protected packets. The inner coder is designed for a range of punctured
convolutional codes, which allows code rate of 1/2, 2/3, 3/4, 5/6, and 7/8. If
hierarchical transmission is used, each of two parallel inner codes has its own code rate.
Then, the inner interleaver is block based bit-wise interleaving. After channel coding
16-QAM, and non-uniform 64-QAM where the “non-uniform” constellation level is
used by hierarchical transmission. Moreover, the transmission channel bandwidth is 6
MHz, 7 MHz, and 8 MHz, respectively.
The DVB-T system based on OFDM technique with various transmission
parameters. Two transmission modes are defined: a 2k mode and an 8k mode. The 2k
mode is suitable for short distance and high mobility transmission because of its short
symbol duration and wide sucarrier spacing. On the other hand, the 8k mode is suitable
for long distance transmission and deep multipath delay spread. Other parameters such
as code rate of inner coder, mapping mode, and length of guard interval would be
determined properly according to the channel condition. The Table 2.1 lists the
parameters in the 2k and 8k modes for 6 MHz channel bandwidth in DVB-T standard.
Table 2.1: Parameters for 6 MHz channel in DVB-T
8k mode 2k mode
Number of subcarriers 6817 1705
Value of carrier number Kmin 0 0
Value of carrier number Kmax 6816 1704
Carrier spacing 1/T u 837.054 Hz 3,348.214 Hz Duration Tu =Ts T− CP 1194.667 sμ 298.6667 sμ Spacing between carriers Kmin and Kmax 5.71 MHz 5.71 MHz
/ CP u T T 1 1 1, , , 1 4 8 16 32 1 1 1 1 , , , 4 8 16 32
2.2.2 Frame Structure and Transmission Band
The transmitted data is organized in frames. Each frame consists of 68 OFDM
symbols and duration of a symbol is Tsym. Each symbol is constituted by a set of 6817 and 1705 carriers in the 8k and 2k mode, respectively. Therefore, the carrier are
indexed by k ∈
[
Kmin;Kmax]
and determined by Kmin = and 0 Kmax =6816 in 8k mode and Kmax =1705 in 2k mode. The emitted signal at time t is described by the following expression:( ) 67 max ( ) min 2 , , , , 0 0 Re c K j f t m l k m l k m l K s t e π ∞ c w t = = ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = ⎨⎪ ⋅ ⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∑ ∑ ∑
⎭ , (2.7) where ( )(
)
( ) ( ) sym 68 sym sym sym 68 68 ' 2 , , 1 0 s U k j t l T m T T m l k e l m T t l m T w t else π −Δ− ⋅ − ⋅ ⋅ ⎧⎪⎪ ⎪⎪ + ⋅ ⋅ ≤ ≤ + ⋅ + ⋅ = ⎨ ⎪⎪ ⎪⎪⎩and k'= −k
(
Kmax +Kmin)
/2. In Equation (2.7), f denotes the central frequency c of RF signal, m is the transmission frame number, l is the OFDM symbol number,k is the carrier number, cm l k, , the complex symbol for carrier k of the l-th data symbol in frame m , T is the inverse of the carrier spacing, and Δ is the duration U of the guard interval.
The system is supposed to use the conventional analog broadcasting ultra high
frequency (UHF) band with 8 MHz bandwidth. The central frequency f of RF signal c is
470 MHz 4 MHz 8 MHz
c
However, some countries define the transmission bandwidth as 6 MHz or 7 MHz. we
could only change the sampling period to satisfy the regulated bandwidth. The
sampling period T is 7 64 s / μ for 8 MHz channels, 1 8 s / μ for 7 MHz channels, s and 7 48 / μ for 6 MHz channels. s
2.2.3 Reference Signals
The main functions of reference signals are synchronization, channel estimation,
and signaling. DVB-T has three kinds of reference signal: continual pilot, scattered
pilot, and transmission reference signaling (TPS). The continual and scattered pilots are
derived from a PRBS Wk =X11+X2 + . 1
z Continual Pilot
There are 177 continual pilots in the 8k mode and 45 continual pilots in the 2k mode.
The continual pilots are placed into fixed subcarrier in every OFDM symbol as shown
in Figure 2.9. The value of continual pilots on subcarrier k is given by
{
, ,}
4 3 2 1 2(
)
Re cm l k = / × / −Wk , (2.9)
and
{
, ,}
0Im cm l k = . (2.10)
Where m is the transmission frame number, l is the OFDM symbol number, and k
is the carrier number. The continual pilots are always transmitted at the boosted power
level so that for these
{
*}
16 9/E c c× = , (2.11)
Figure 2.9: Continual pilot of DVB-T
z Scattered Pilot
Unlike the continual pilots, the scattered pilots are located at different subcarrier
indices in every symbol. For the l-th OFDM symbol, the positions of the scattered
pilots are
( )
[
]
{
kp =Kmin + ×3 1 mod 4 +12i i| integer, i ≥0, kp ∈ Kmin;Kmax}
. (2.12)The values of scattered pilots with subcarrier k are produced in the same way with p continual pilots in Equation (2.9) and Equation (2.10). Figure 2.10 shows the location
of scattered pilots. There is a repetition for the location of scattered pilots every four
OFDM symbols, so the scattered pilots have a period P = in time domain. In t 4 frequency domain, the location of scattered pilots repeats every twelve subcarriers so
Figure 2.10: Scattered pilot of DVB-T
z Transmission Parameter Signaling (TPS)
The TPS carriers are used for the purpose of signaling parameters related to the
transmission scheme, coding and modulation. There are 68 TPS carriers and 17 TPS
carriers for 8k and 2k mode, respectively. Every TPS carrier in the same symbol
informs the same differentially encoded information bit. The TPS carriers convey
information on the following:
1) mapping mode;
2) hierarchy information;
3) length of guard interval;
4) inner code rates;
5) transmission mode;
6) frame number in a super-frame;
Chapter 3
Proposed Maximum Doppler Shift
Estimation Method
In this chapter, a Doppler shift estimation method in digital mobile communication
systems is described. The proposed method uses the Fourier transform of the
autocorrelation function (ACF) of the channel impulse response (CIR), which is related
with the maximum Doppler shift. By utilizing the characteristic of the Doppler power
spectrum, the ACF of the CIR is translated into the frequency domain. Thus, we will
introduce two schemes of Fourier transform, which are fast Fourier transform (FFT)
and chirp transform algorithm (CTA). Moreover, we also analyze the complexity of
these two schemes, which provide a tradeoff between computational complexity and
memory size. Compared with the autocorrelation based Doppler shift estimation
method [6], the proposed method which works in frequency domain is more robust to
noisy environment. Therefore, the proposed method can work well at low
signal-to-noise ratio (SNR).
Section 3.1 presents the OFDM system model and the channel model. In Section
3.2, the proposed maximum Doppler shift estimation method is described. The
complexity analyses of FFT and CTA are provided in Section 3.3. Finally, the computer
3.1 System and Channel Models
Consider the OFDM system model, which is illustrated in Figure 3.1. A set of
N-coded frequency domain symbols
{ }
Xk is collected to form an OFDM symbol0, 1, , 1 T N X X X − ⎡ ⎤ = ⎣ ⎦
X " . And then, the OFDM symbol is converted into time domain samples
{ }
xn according the N-point IFFT operation{ } IFFT n x = X (3.1) CP 1 2 / 0 1 , 1 N j nk N k k X e N n N N π − = =
∑
− ≤ ≤ − (3.2)where NCP denotes the length of cyclic prefix (CP).
Parallel /
S
eri
al
IFFT Serial / Parall
el
FFT
n
ν
Figure 3.1: OFDM system model
The multipath channel is assumed to be time-variant, and the normalized length of
the channel, L, is equal to or less than NCP. Consequently, during an OFDM symbol
period, the discrete-time channel impulse response is given by [16]
( ) 1 ,
(
)
0 , L n l l l h n l − h δ n τ = =∑
⋅ − (3.3)where h is the channel response of the l-th tap at time instance n, and n l, τ is delay l spread of l-th tap. In this thesis, channel model is the typical wide sense stationary
uncorrelated scattering (WSSUS) model [17], [18], which will be described in detail
later in Section 3.1.1. After transmitting through the channel, the received samples are
collected during an OFDM symbol interval:
( , ) n n n y =h n l ∗x +ν (3.4) 1 , 0 , 0 1 L n l n l n l h x ν n N − − = =
∑
⋅ + ≤ ≤ − (3.5)where ∗ denotes convolution, and ν are samples of additive white Gaussian noise n (AWGN) with variance 2
n
ν
σ . Perfect timing synchronization is assumed in this thesis.
Due to the presence of the guard interval, the received signal is not corrupted by ISI in
the range 0≤ ≤n N − . Therefore, Equation (3.5) can be rewrite as 1
( ) 1 1 2 / , 0 0 1 L N j n l k N n n l k n l k y h X e N π ν − − − = = ⎛ ⎞⎟ ⎜ ⎟ = ⋅⎜⎜ ⎟⎟+ ⎜⎝ ⎠
∑
∑
1 1 2 / 2 / , 0 0 1 , 0 1 N L j nk N j lk N k n l n k l X e h e n N N π π ν − − = = =∑
∑
+ ≤ ≤ − . (3.6)By defining H nk( )=
∑
lL=−01h en l, j2πlk N/ , which is Fourier transform of the CIR at time n. Then, Equation (3.6) can be expressed as( ) 1 2 / 0 1 , 0 1 N j nk N n k k n k y H n X e n N N π ν − = =
∑
⋅ + ≤ ≤ − . (3.7)Comparing Equation (3.7) with the transmitted signal Xk, we can find that the time-varying multipath channel introduces a time-varying complex multiplier, H n , k[ ]
at each subcarrier. Thus, after removing the ISI corrupted guard interval, the k-th
subcarrier output from the FFT can be written as
1 2 / 0 1 N j nk N k n n Y y e N π − − = =
∑
(3.8) , 0 1 k k k k H X I V k N = + + ≤ ≤ − , (3.9)where ( ) 1 0 1 N k k n H H n N − = = ⋅
∑
, (3.10) ( ) ( ) 1 1 2 / 0 0 1 N N j n m k N k m m m n m k I X H n e N π − − − = = ≠ = ⋅∑
∑
, (3.11) and 1 2 / 0 1 N j nk N k n n V e N π ν − = = ⋅∑
. (3.12)In Equation (3.9), Hk is the channel transfer function at the k-th subcarrier, Ik is the ICI
term, and Vk represents the AWGN.
3.1.1 Channel Model
The well-known WSSUS model [17], [18] is assumed in this thesis. Under this
assumption, the ACF of the channel impulse response for l -th tap is given by [19]
( )
{
, (* ),}
0(
2 ,)
hh E hn l hn τ l J fd l
φ τ ≡ ⋅ + = π τ (3.13)
where J ⋅ is the zeroth order Bessel function of the first kind, and 0( ) f is the d l,
maximum Doppler shift. Figure 3.2 shows the theoretical value of the autocorrelation
function φ versus time delay. From this figure, we can know that maximum hh Doppler shift f multiplied by time delay τ is constant. Thus, d l, f and τ have a d l, reciprocal relationship. Moreover, the maximum Doppler shift f is determined by d l, the mobile velocity and the wavelength of carrier wave. Table 3.1 lists the maximum
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1 -0.5 0 0.5 1 Time delay, fdτ A u to correl a ti o n
Figure 3.2: Autocorrelation function of the channel impulse response
Table 3.1: Maximum Doppler shift with different velocity for DVB-T systems
Carrier frequency: 584 MHz ~ 710 MHz
Velocity Maximum Doppler shift
50 km/hr 27.04 Hz ~ 32.87 Hz 100 km/hr 54.07 Hz ~ 65.74 Hz 150 km/hr 81.11 Hz ~ 98.61 Hz 200 km/hr 108.15 Hz ~ 131.48 Hz 300 km/hr 162.22 Hz ~ 197.22 Hz 400 km/hr 216.30 Hz ~ 262.96 Hz
The Fourier transform of Equation (3.13) denotes the Doppler power spectrum. In
the case of Rayleigh fading, the Doppler power spectrum can be expressed as [17], [20]
( ) FT
{
( )}
, 2 , , , 1 , , 1 0, . d l d l hh d l d l f f f f S f f f f π φ τ ⎧⎪⎪ ≤ ⎪⎪ ⎛ ⎞ ⎪ ⎜ ⎟ ⎪⎪ ⋅ −⎜ ⎟ = =⎨⎪ ⎜⎜⎝ ⎟⎟⎟⎠ ⎪⎪ ⎪⎪ > ⎪⎪⎩ (3.14)where Ts is the sampling time. Figure 3.3 depicts the Doppler power spectrum S f . ( ) Moreover, in this figure we can find that the position of peak corresponds to the
maximum Doppler shift. Therefore, we can use this characteristic to estimate maximum
Doppler shift.
( )
S f
,
d l
f
f
,
d l
f
−
3.2 Proposed Doppler Shift Estimation Method
Using Doppler Power Spectrum
In this section, the procedure of proposed maximum Doppler shift estimation
method is introduced. First, we use pilot tones to estimate the CIR. After an observation
interval with K OFDM symbols, the consecutive estimated CIRs are used to compute
the ACF. And then, we translate the ACF into the frequency domain. Therefore, the
Doppler power spectrum of estimated CIRs is obtained. In order to improve the
estimated accuracy of the maximum Doppler shift estimation, we could average some
Doppler power spectrum estimated results. Then, using the estimated Doppler power
spectrum finds the maximum Doppler shift. Figure 3.4 shows a block diagram of
proposed maximum Doppler shift estimation method.
Least Square Estimation Autocorrelation avg l
h
φ
hh(
sT
sym)
kY
FFT or CTA Averaging Effect Find Maximum Doppler Shift Smoothed Doppler Valuel
( )
S k
i
( )
S k
, d lf
avg , d lf
Figure 3.4: Block diagram of proposed Doppler shift estimation method
Here, we utilize frequency domain scattered pilot tones [1], [21] of DVB-T
systems to estimate channel response. Therefore, let L ≤NP be the maximum predicted normalized length of channel. Then, the NP ≥(L+ equally spaced pilots, 1)
p
k
P , are inserted at subcarriers k for p 0≤ ≤p
(
NP −1)
. Form Equation (3.9), a least square (LS) estimation of the channel transfer function at the pilot subcarriers canbe obtained as l
(
)
(
)
P , 0 1 p p p p p p p k k k k k k k I V Y H H p N P P + = = + ≤ ≤ − . (3.15) In Equation (3.15), p k I and p kV denotes ICI term and noise term at kp-th subcaeeier, respectively. Then, through an NP-point IFFT, the estimated CIR
avg l h would be l
(
)
P P avg P P 1 2 / 0 1 , 0 1 p N j pl N k l p h H e l N N π − = =∑
≤ ≤ − . (3.16)In this thesis, we approximate channel time-variations with a piece-wise linear
model during one symbol period Tsym. In Figure 3.5, the solid curve is real or
imaginary part of a channel path, and the dashed line is piece-wise linear model. By the
assumption above, E h
(
lavg −hn l,)
is minimized at n =(
N/2−1)
for the l-th channel tap [22]. We approximate h(N/2 1 ,− )l with the estimate of hlavg. Thus, we will have avg 1 , 2 l N l h⎛ ⎞⎟ h ⎜ − ⎟ ⎜ ⎟ ⎜⎝ ⎠ = (3.17)From Equation (3.13) and Equation (3.17), the estimated ACF of the channel
impulse response of l-th tap can be expressed as
(
)
sym sym * 1 , 1 , 2 2 hh s T E h N l h N s T l φ ⎛ ⎞ ⎛ ⎞ ⎟ ⎟ ⎜ − ⎟ ⎜ − + ⋅ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎝ ⎠ ⎝ ⎠ ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⋅ = ⎨ ⋅ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭(
sym)
2( )
0 2 d l, l J πf s T σΔ δ τ = ⋅ ⋅ + ⋅(
sym)
2( )
hh s T l φ σΔ δ τ = ⋅ + ⋅ , (3.18)where s is the difference in OFDM symbol index, Tsym denotes the symbol period, and
2
σΔ is the combined reduced variance of ICI plus AWGN. Here, the estimated Doppler power spectrum over an observation interval of K symbols can be written as
l ( )K FT
{
hh(
sym)
}
S k = φ s T⋅(
sym)
0 ( ) 1 2 / 0 , 1 K j ks K hh s s T e π k K φ − − = =∑
⋅ ≤ ≤ − . (3.19)If the Doppler power spectrum is approximated over sufficient many observation
intervals, the maximum Doppler shift can be estimated by finding the maximum peak
of the estimated Doppler power spectrum series. However, excessively long length of
observation would lead to an estimation delay and a degradation of the Doppler shift
tracking ability.
From Figure 3.3, we can know that the maximum peak of the Doppler power
spectrum locates within a small frequency domain interval. Thus, in order to increase
the accuracy of maximum Doppler shift estimation, the zero-padding would be used.
The K-sample estimated ACF sequence (Equation 3.18) is extended to a KFFT-sample
sequence by padding with
(
KFFT −K)
zero samples. Therefore, we obtain( )
(
)
(
( ))
{
}
FFT FFT 0 , sym , , 1 sym , 0, 0, , 0 K = ⎡⎣⎢φhh φhh T φhh K− ⋅T ⎤⎥⎦ S " " (3.20)However, we only need a particular small frequency domain interval for Doppler shift
estimation. Thus, we introduce the chirp transform algorithm [23] to provide another
alternative choice for Fourier transform.
3.2.1 Chirp Transform Algorithm (CTA)
Unlike the FFT scheme, the CTA scheme [22] can be used to compute any set of
equally spaced samples of the Fourier transform, thus the CTA scheme is more flexible
than the FFT scheme. From Equation (3.19), we denote that
l ( )
(
)
(
( ))
sym sym
0 , , , 1
K ⎡φhh φhh T φhh K T ⎤
Φ = ⎢⎣ " − ⋅ ⎥⎦ (3.21)
as a K-point sequence and S is its Fourier transform. Consider the evaluation of M K samples of that are equally spaced in angle on the unit circle at frequencies SK
0
k
w =w + Δ , (3.22) k w
where w is the start frequency and the frequency increment w0 Δ can be chosen arbitrarily. Figure 3.6 shows the frequency samples for the CTA scheme.
Im Re 0 ω (M− Δ1) ω Unit circle
The Fourier transform corresponding to the more general set of frequency samples can be written as l ( ) l ( ) ( ) 1 0 , 0,1, , 1 k K jw n M K n S k − n e k M = =
∑
Φ = " − . (3.23) Then, substituting Equation (3.22) into Equation (3.23) can obtainl ( ) 1l ( ) 0 0 K jw n nk M K n S k − n e W = =
∑
Φ , (3.24) where W defined as j w W =e− Δ . (3.25)Here, we use the identity
( )2 2 2 1 2 nk = ⎡⎢⎣n +k − k−n ⎤⎥⎦ (3.26) to express Equation (3.24) as l ( ) ( ) ( ) ( ) 2 2 1 2 2 0 , 0,1, , 1 k n w K M n S k W g n W k M − − − = ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ = − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦
∑
" , (3.27) where g n defined as ( ) ( ) l ( ) 2 0 2 n jw n K g n = Φ n e W . (3.28)However, we need only compute the output of the system in Figure 3.7 over a finite
interval for the evaluation of the Fourier transform samples specified in Equation
( ) 2 2 n
h n
=
W
− ( ) Kn
Φ
g n
( ) 2 0 2 n jw ne
−W
2 2 nW
l
M ( )S
n
Figure 3.7: Block diagram of chirp transform algorithm
Since g n is finite duration, only a finite portion of the sequence ( ) W−n2/2 is used
in obtaining g n( )∗W−n2/2 over the interval 0≤ ≤n (M − . Let us define 1)
( ) ( ) ( ) otherwise 2 2, 1 1 , 0, , n W N n M h n − ⎧⎪⎪ ⎪⎪ − − ≤ ≤ − = ⎨ ⎪⎪ ⎪⎪⎩ (3.29) so we have ( ) ( ) ( ) ( ) 2 2 , 0,1, , 1 n g n ∗W− =g n ∗h n n = " M − . (3.30)
However, in Equation (3.29) h n is noncausal, and for certain real-time ( )
implementation it must be modified to obtain a causal system. Therefore, this
modification is easily accomplished by delaying h n by ( ) (N −1) to obtain a causal impulse response: ( ) ( ) ( ) c otherwise 2 1 2 , 0 2 , 0, . n N W n M N h n − + − ⎧⎪⎪ ⎪⎪ ≤ ≤ + − = ⎨ ⎪⎪ ⎪⎪⎩ (3.31)
Since both the chirp demodulation factor at the output and the output signal are also
delayed by (N −1) samples, the results of the CTA scheme are
lM( ) ( 1 ,) 0,1, ,( 1)
By modifying the system of Figure 3.7, we can obtain a causal system as shown in
Figure 3.8.
FIR chirp filter
( ) c
h n
( ) Kn
Φ
g n
( ) 2 0 2 n jw ne
−W
( 1)2 2 n NW
− + ( )G n
Figure 3.8: Block diagram of chirp transform algorithm for causal finite-length impulse
response
3.2.2 Averaging and Smoothing Effect of Maximum
Doppler Shift Estimation Method
If the observation interval with K symbols is not long enough, the Doppler
information does not have a sufficient statistical characteristic on the estimated Doppler
power spectrum SlM( )k . Therefore, the proposed maximum Doppler shift estimation is overlapped every K/2 samples to deal with this problem, as shown in Figure 3.9. As the estimated Doppler power spectrum is accumulated, the variance of the noise and
the interference power spectrums can be reduced over whole spectrum range and the
q-th Observation Interval
(q-1)-th Observation Interval (q+1)-th Observation Interval
(Autocorrelation) and (FTT or CTA) , d l f , d l f − f Noise
q-th Estimated Doppler Power Spectrum Average
Ideal Doppler Power Spectrum , d l f , d l f − f Noise ( ) 1 0 1Q q Q − = ⋅ ∑
Figure 3.9: Averaging effect of estimated Doppler power spectrum
If the Doppler shift does not change during Q times of observation interval, the
ideal Doppler power spectrum can be approximated by averaging Q times of the
estimated Doppler power spectrum. Thus, we obtain
i ( ) 1l( )( ) ( ) 0 1 , 0,1, , 1 Q q M M q S k S k k M Q − = =
∑
= " − , (3.33)where Sl( )Mq ( )k denotes the q-th estimated Doppler power spectrum. Then, the estimated maximum Doppler shift can be obtained as
i ( ) 1 0 , M2 arg max M , 0 1 d l k w w f S k k M M π − − ⎛⎜ ⎞⎟ = ⋅⎜ ⎟⎟⎟ ≤ ≤ − ⎜⎝ ⎠ . (3.34)
The Doppler shift estimation above described provides an instantaneous estimated of
Doppler shift over Q times of observation interval. The smoothed Doppler spread
values over a long period of time can be calculated as [7], [10]
( )
(
( ))
( ) ( ) ( ) , 1 , , 1 avg avg d l d l d l f n = −β Δf ⋅f n +β Δ ⋅f f n− . (3.35)smoothed estimate can be written as ( ) ( ) , , 1 avg d l d l f f n f n Δ = − − , (3.36)
and the smoothing factor 0<β(Δ < that determines the tracking ability is f) 1 expressed as ( ) max max max 1, , , . f f f f f f f β ⎧ Δ > Δ ⎪⎪ ⎪⎪ Δ = ⎨ Δ⎪ Δ ≤ Δ ⎪⎪Δ ⎪⎩ (3.37) max f
Δ is a maximum frequency difference. Note that Δfmax depends on update period.
3.3 Complexity Analyses of FFT and CTA
In this section, the computational complexity and the memory size requirement of
the FFT and the CTA schemes are compared. Here, we denote the number of arithmetic
multiplications and additions as a measure of computational complexity. First, we
consider the complexity analysis of the CTA scheme. From Figure 3.8, the input part of
CTA needs K complex multiplications. Then, the computation of each result of
convolution requires K complex multiplications and K complex additions, and the
output part of CTA needs M complex multiplications. Therefore, the total computation
of the CTA scheme requires (M + ⋅1) K +M complex multiplications and M K⋅ complex additions. In the CTA scheme, the input part and output part require one complex memory respectively; the convolution part needs (K +M − complex 1) memories. Thus, the total memory size of the CTA scheme requires (K +M + 1) complex memories. Next, we consider FFT scheme. A KFFT-point FFT needs
additions. Moreover, the FFT scheme needs KFFT complex memories for input storage and KFFT complex memories for output storage. The total memory size of the FFT scheme requires 2⋅KFFT complex memories for a general case. Here, Table 3.2 lists the above results.
Table 3.2: Complexity analysis of CTA and FFT
CTA scheme FFT scheme
Complex multiplications: (M + ⋅1) K +M Complex multiplications:
(
KFFT/ 2) (
⋅ log2KFFT)
Computation Complex additions: M K⋅ Complex additions:(
)
FFT log2 FFT K ⋅ K Memory size Complex memories: (K +M + 1) Complex memories: FFT 2 K⋅Then, we will give some examples to experience complexity. First, we know that
the CTA scheme has K-sample input and the M-sample output. The observation
interval K is 64, 128, or 256 symbols. The CTA scheme can estimate the maximum
Doppler shift during 0 Hz to 300 Hz, and the frequency domain resolution of CTA is set
equal to or less than 1 Hz per sample. As the result, the number of CTA output, M, must
be equal to or greater than 300. When both of the CTA and the FFT schemes have the
same frequency domain resolution, the parameter KFFT of FFT scheme should be
satisfy
(
FFT)
sym nextpow2 2 1 log K T ⎛ ⎞⎟ ⎜ ⎟ ⎜ = ⎜ ⎟⎟ ⎜ ⎟ ⎝ ⎠, (3.38)where nextpow2(⋅) is the next higher power of 2. Moreover, the symbol duration, Tsym,
for DVB-T systems is listed in Table 3.3. Therefore, the values of KFFT are between
1024 and 8192.
Table 3.3: Symbol duration for DVB-T systems
2K mode 8K mode 6 MHz channels 308 μ (min.) s 373.33 μ (max.) s 1232 μ (min.) s 1493.33 μ (max.) s 7 MHz channels 264 μ (min.) s 320 μ (max.) s 1056 μ (min.) s 1280 μ (max.) s 8 MHz channels 231 μ (min.) s 280 μ (max.) s 924 μ (min.) s 1120 μ (max.) s
According to the above parameters, the minimum computational complexity of the
CTA scheme is 17,614 complex multiplications and 17,280 complex additions; the
maximum computational complexity is 69,646 complex multiplications and 69,120
complex additions. The minimum computational complexity of the FFT scheme is
5,120 complex multiplications and 10,240 complex additions; the maximum
computational complexity is 53,248 complex multiplications and 106,496 complex
additions. In the CTA scheme, the minimum and maximum number of memory is 335
and 527 complex memories, respectively. In the FFT scheme, the minimum and
maximum number of memory is 2,048 and 16,384 complex memories, respectively.
Therefore, the FFT scheme has less computational complexity than the CTA scheme,
but the FFT scheme needs the larger memory size of the CTA scheme. When the
memory size is the more important consideration factor, the CTA scheme is a better
3.4 Computer Simulations
In this section, simulation results of the proposed maximum Doppler shift
estimation will be presented. Through out the all simulations, timing synchronization is
assumed to be perfect. Simulation parameters are summarized in Table 3.4. Consider a
2048-subcarrier (which is the 2K mode of DVB-T systems) OFDM system with
bandwidth 6 MHz. Transmitted data sequences are modulated in QPSK constellation,
and Jakes model is used as the channel model. The maximum Doppler shift in the
simulation varies form 0 Hz to 300 Hz. As the mobile velocity is 456 km/hour in the
carrier frequency 710 MHz case, the corresponding maximum Doppler shift is 300 Hz.
Thus the maximum Doppler shift that we simulated is limited to 300 Hz. The CTA
scheme is used and the frequency domain resolution is 1 Hz. The maximum frequency
difference Δfmax is 50 Hz.
Table 3.4: Parameters of computer simulations
System DVB-T
Transmission mode 2K
Channel bandwidth 6 MHz
CP length 1/4 OFDM symbol
Modulation QPSK
Carrier frequency 710 MHz
Maximum Doppler shift 0 ~ 300 Hz
Figure 3.10 shows the simulation results for the effect of the accumulation size Q
when the observation interval K equals to 128. It is shown that the large Q size can be
sufficiently estimated low SNR (< 10 dB). If the accumulation size is larger than 5
times, the normalized mean square error (MSE) of proposed method does not reduce
obviously. The normalized MSE denote as
ideal ideal normailzed MSE 2 d d d f f E f ⎧ ⎫ ⎪ ⎪ ⎪ − ⎪ ⎪ ⎪ = ⎨⎪ ⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ . (3.39)
Therefore, the 7 times accumulation size is sufficient for observation 128 symbols. The
error floor is due to the imperfect Doppler power spectrum.
0 5 10 15 20 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 SNR(dB) No rm a liz e d M S E K = 128, Q = 1 K = 128, Q = 3 K = 128, Q = 5 K = 128, Q = 7 K = 128, Q = 9
Figure 3.10: Normalized MSE versus SNR. The effect of different accumulation size Q
is shown. The observation interval K is 128 symbols and the maximum Doppler shift is
Figure 3.11 shows the normalized MSE of estimation error versus accumulation
size for various observation intervals in SNR = 15 dB. When the observation interval
and accumulation size become larger, the estimated Doppler power spectrum becomes
closer to the ideal Doppler power spectrum, thus the estimated value is more accuracy,
as shown in Figure 3.11. However, the long observation intervals and the large
accumulation size will increase the estimation delay and the computational complexity.
Therefore, the various observation intervals and the accumulation size should be
chosen carefully by taking the accuracy, estimation delay, and computational
complexity into consideration. From Figure 3.11, we can see that when the observation
interval equals to 256 or 128 symbols, Q = 5 is sufficient, and when the observation
interval equals to 64 symbols, Q = 7 is sufficient.
1 3 5 7 9 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Accumulation size Q N or m al iz ed M S E K = 64, SNR = 15 dB K = 128, SNR = 15 dB K = 256, SNR = 15 dB
Figure 3.11: Normalized MSE versus accumulation size. The effect of different
observation interval K is shown. The maximum Doppler shift is 0 Hz to 300 Hz and
The simulation results of the normalized MSE for different observation interval
and SNR versus accumulation size are presented in Figure 3.12. As the SNR becomes
larger, the estimated CIRs are more accurate. When estimated CIRs are more accurate,
the estimated Doppler power spectrum would also become more accurate. Because the
accuracy of the estimated Doppler power spectrum is greatly influenced by the
observation interval and the accumulation size, these two parameters are main factors
in the estimation scheme. When the Q is larger than 5, the proposed method has a
stable performance. Figure 3.13 to Figure 3.15 show the tracking ability of the
proposed maximum Doppler shift estimation method. The estimated and the actual
Doppler spread profiles are both listed in these figures. As the result, the proposed
method tracks well for Doppler shift changes, even though the SNR is low than 10 dB.
1 3 5 7 9 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Accumulation size Q No rm a liz e d M S E K = 64, SNR = 0dB K = 64, SNR = 10dB K = 64, SNR = 20dB K = 128, SNR = 0dB K = 128, SNR = 10dB K = 128, SNR = 20dB K = 256, SNR = 0dB K = 256, SNR = 10dB K = 256, SNR = 20dB
Figure 3.12: Normalized MSE versus accumulation size for different observation
(a) 0 10 20 30 40 50 60 70 80 90 0 50 100 150 200 250 300 350 Time (Sec.) M a xi m u m D oppl e r shi ft ( H z) Ideal Estimated SNR = 5dB, K = 64, Q = 5 (b)
Figure 3.13: Tracking ability of the proposed maximum Doppler shift estimation
0 10 20 30 40 50 60 70 80 90 0 50 100 150 200 250 300 350 Time (Sec.) M a xi m u m D oppl er shi ft ( H z) Ideal Estimated SNR = 5dB, K = 128, Q = 3 (a) 0 10 20 30 40 50 60 70 80 90 0 50 100 150 200 250 300 350 Time (Sec.) M a xi m u m D oppl e r shi ft ( H z) Ideal Estimated SNR = 5 dB, K = 128, Q = 5 (b)
Figure 3.14: Tracking ability of the proposed maximum Doppler shift estimation
0 10 20 30 40 50 60 70 80 90 0 50 100 150 200 250 300 350 Time (Sec.) M a xi m u m D oppl e r shi ft ( H z) Ideal Estimated SNR = 5dB, K = 256, Q = 3 (a) 0 10 20 30 40 50 60 70 80 90 0 50 100 150 200 250 300 350 Time (Sec.) M a xi m u m D oppl e r shi ft ( H z) Ideal Estimated SNR = 5dB, K = 256, Q = 5 (b)
Figure 3.15: Tracking ability of the proposed maximum Doppler shift estimation
Figure 3.15 shows three paths with different maximum Doppler shifts. The
maximum Doppler shift of path I varies from 0 to 300 Hz. The maximum Doppler shift
of path II varies from 300 to 60 Hz. The maximum Doppler shift of path III varies from
0 to 145 Hz. As can be seen, the proposed method (with K = 256 and Q = 5) still tracks
well in SNR = 5dB. Figure 3.16 shows the comparison between Park’s method [10] and
the proposed method in SNR = 15 dB. The error flow of Park’s method is owing to the
limit of FFT spectrum resolution. The proposed method has a superior frequency
domain resolution of Doppler power spectrum, but the computational complexity is
larger than the computational complexity in Park’s method.
0 20 40 60 80 100 0 50 100 150 200 250 300 350 Time(Sec.) Ma x imu m D o p p le r s h if t (H z ) SNR = 5dB, K = 256, Q = 5 Path II Path I Path III
Figure 3.15: Tracking ability of the proposed maximum Doppler shift estimation
1 3 5 7 9 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Accumulation size Q No rm a liz e d M S E Park's method, K = 64 Park's method, K = 256 Proposed method, K = 64 Proposed method, K = 256 SNR = 15dB
Figure 3.16: Normalized MSE versus accumulation size. Compared Park’s method [10]
with proposed method (SNR = 15 dB).
3.5 Summary
In this chapter, a maximum Doppler shift estimation method using Doppler power
spectrum is proposed. It is shown that the proposed method works well over wide
Doppler shift at low SNR condition (< 10 dB). This method provides a tradeoff
between computational complexity and estimation accuracy by changing the size of
observation interval and accumulation. Furthermore, two schemes of Fourier transform
(FFT scheme and CTA scheme) are proposed to provide a tradeoff between
computational complexity and the memory size. The FFT scheme has lower
computational complexity than the CTA scheme, but it needs larger memory size than
Chapter 4
ICI Cancellation in OFDM Systems
over Time-Varying Channels
In this chapter, we propose an adaptive ICI cancellation scheme based on [5]. The
proposed method provides a tradeoff between the performance of output signals and
processing time. Through the ICI analysis, the relation between ICI and the maximum
Doppler shift will be introduced. Then, from Chapter 3, we can utilize the estimated
maximum Doppler shift as a parameter of the adaptive ICI cancellation scheme.
First, Section 4.1 presents the ICI analysis. In Section 4.2, the proposed adaptive
ICI cancellation scheme is described. The computer simulations and the summary are
shown in Section 4.3 and Section 4.4, respectively.
4.1 ICI Analysis
Signals transmitted in a mobile environment are often impaired by the delay
spread and Doppler shift. In order to keep a near-constant channel in each subcarrier of
OFDM systems, the length of an OFDM symbol should be increase as the delay spread
increases. Since the delay spread increased, the length of the guard interval should also