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CHAPTER 2
CARRIER FREQUENCY OFFSET ESTIMATION
IN OFDM SYSTEMS
2.1 INTRODUCTION
Orthogonal Frequency Division Multiplexing (OFDM) is
multicarrier modulation scheme for combating channel impairments such as
severe multipath fading and impulsive noise. However, the principal
disadvantage of OFDM is that it is highly susceptible to carrier frequency
offset (CFO). Carrier frequency offset occurs due to frequency discrepancies
between transmitter and receiver and Doppler shift of the mobile channel. The
impact of CFO are the loss of orthogonality among subcarriers, inter
subcarrier interference, accumulation of phase error over successive symbols.
These effects can degrade system performance to a significant extent. In order
to rectify the aforementioned issues, a signal processing algorithm is to be
developed to estimate frequency offset and correct it.
2.2 LITERATURE REVIEW
The signal processing algorithms for CFO estimation in OFDM
systems are grouped either as blind or data aided. In blind estimation
algorithms, the periodic structure of Cyclic Prefix (CP) is used to estimate CFO
(Beek et al 1997). In literature, many blind algorithms have been reported
(Bölcskei 2001, Huang and Latif 2006, Huang and Latif 2006a, Lee and Kim
2006). Traditionally, blind estimation algorithms are bandwidth efficient and
do not require additional overhead (Zeng 2008).
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In data aided CFO estimation algorithms, a known preamble, or
pilot symbol is inserted in front of each data packet such that it can easily be
employed by the receiver to achieve synchronization, thereby making it
suitable for applications involving packet based transmission. The major
drawback of the data aided estimation algorithm is the overhead associated
with the pilots or training in the OFDM symbols. In data aided algorithms,
CFO estimation is carried out in two stages namely a coarse estimation or
acquisition stage and fine estimation or tracking stage (Gao et al 2008). This
chapter addresses the CFO estimation in packet based transmission and
proposes a novel data aided algorithm for fine estimation.
2.2.1 Data Aided CFO Estimation
The data aided method uses correlation between repetitive slots to
achieve the CFO estimation.
2.2.1.1 Coarse CFO estimation methods
A Maximum Likelihood (ML) frequency offset estimator based on
the use of two consecutive and identical symbols was presented by Moose
(Moose 1994). The maximum frequency offset that can be handled is 12
f ,
where f is the subcarrier spacing. When the training symbols are shortened
by a factor of two, acquisition range of CFO is doubled. However, when
symbols are shorter, there are fewer samples over which average has to be
performed. The training symbols need to be kept longer than the guard
interval so that channel impulse response does not cause distortion while
estimating the frequency offset. To overcome this drawback, a null symbol
based method was proposed by Nogami and Nagashima(1995). In this
method, the CFO is estimated in the frequency domain after applying a
Hanning window and taking the Fast Fourier Transform. However, this
26
approach requires an extra overhead for null symbol and increase in
computational complexity.
Schmidl’s method (Schmidl and Cox 1997) of CFO estimation
investigates the usage of two training symbols. The first has two identical
halves and is used to estimate a frequency offset less than the subcarrier
spacing while the second symbol contains a pseudo noise sequence used to
increase the range of estimation. The drawback of this method is that it
consumes more overhead due to the usage of two training symbols. Improved
frequency offset estimation was proposed by Morelli and Mengali based on
the Best Linear Unbiased Estimation (BLUE) principle at the cost of
increased complexity (Morelli and Mengali 1999). In continuation, Minn et al
has developed three methods based on the BLUE principle. The frequency
offset estimation and MSE performance of the first method is almost same as
in the method by Morelli (Morelli and Mengali 1999). But the other two
methods show better performance, especially at low SNR values (Minn et al
2002). This work also analyzes the effects of number of identical parts
contained in the training symbol on the frequency offset estimation
performance. This gives an insight on how the training symbols should be
designed in order to achieve a better MSE performance with the same amount
of training overhead.
A burst format for OFDM transmission was initially proposed for
frequency synchronization with a large estimation range and good accuracy
(Lambrette et al 1997). In the sequel, Averaged Decision-Directed Channel
Estimation (ADDCE) technique for burst data was proposed to track
time-variation of a wireless channel as well as to reduce noise effects at sub-channels (Song et al 2000).
Bang et al (2001) proposed a coarse CFO estimation algorithm that
is robust for any symbol timing offset that falls within an allowed range. The
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proposed algorithm uses the concept of the coherence phase bandwidth for
reducing the effect of a symbol timing offset.
Liu et al (2004) proposed a CFO estimation algorithm using a
multi-stage synchronization in time and frequency domain for OFDM. This
algorithm uses all pilots including continual pilots and scattered pilots to
estimate the CFO in frequency domain, and it can acquire more accurate
estimation results than the conventional algorithm which uses continual pilots.
Lottici et al (2005) presented an algorithm considering the selectivity of the
channel, leading to the use of a weighted window instead of a rectangular one.
Shi et al (2005) proposed a new Decision Directed (DD)
post-FFT CFO synchronization scheme without relying on pilots. It is shown
that the proposed CFO estimator is approximately unbiased in both AWGN as
well as frequency selective channels.
Lin (2006) developed an effective technique for frequency
acquisition based on ML detection for mobile OFDM. The proposed
technique employs a frequency acquisition stage and a tracking stage. By
exploiting the differential coherent detection of a single synchronization
sequence, where a Pseudo-Noise (PN) sequence is used as a synchronization
sequence. Data aided frequency acquisition with frequency directional PN
matched filters reduce probability of false alarm and probability of miss on a
channel whose coherence bandwidth is sufficiently wide.
Laourine et al (2007) proposed a new data aided CFO scheme for
OFDM communications suitable for frequency selective channels. It is based
on the transmission of a specially designed synchronization symbol that
generates a particular signal structure between the received observation
samples at the receiver (Laourine et al 2007). The proposed work offers a
wide acquisition range with reduced computational complexity. Sevillano et
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al (2007) proposed ML based carrier frequency estimator for OFDM systems
with the preambles formed by Short Training Symbols (STS). The FFT
processor in the OFDM receiver is used for efficient implementation of the
proposed algorithm. Ghogho et al (2009) proposed a method to design
optimal preamble using the CRB as a metric. This involves optimizing the
number of repetitive slots and the power loading. They provided closed-form
expressions illustrating the impact of multipath diversity on estimation
performance.
2.2.2 Data Aided Fine Frequency Estimation
After the acquisition stage of CFO estimation, residue CFO is
present either due to the insufficient accuracy during the coarse estimation, or
the time varying nature of the surrounding environment. The residue CFO, if
not compensated, may still lead to performance degradation. Hence, many
existing standards reserve a limited number of scattered pilot symbols in each
OFDM blocks to improve the system robustness in different aspects. For
example, in IEEE 802.11a WLAN standards , four pilots are placed at the
subcarriers with indices {7, 21, 43, 57} for the purpose of combating the
residue CFO and the phase noise.
Classen and Meyr’s (1994) proposed a method for fine CFO
estimation assumed that the Channel Impulse Response (CIR) remains
constant for two consecutive OFDM blocks over a slow fading channel. For a
small CFO (much less than one subcarrier spacing) and a low SNR, the Inter
Carrier Interference (ICI) induced by the CFO can be ignored as opposed to
the large additive noise. Therefore, the CFO can be estimated by comparing
the received symbols on the pilot carriers from the two consecutive OFDM
blocks.
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Gao et al (2008) proposed a novel CFO tracking algorithm using
the scattered pilot carriers embedded in each OFDM block. Identifiability of
this algorithm was studied for the noise free case, and a constellation rotation
strategy was proposed to eliminate a major type of CFO ambiguity for widely
used constellations. To improve the performance of the CFO estimation and
enhance the robustness to the CFO ambiguity, the virtual carriers existing in
practical OFDM standards were used. Later, merits of both the algorithms
were combined by exploiting both scattered pilots and virtual carriers.
In summary, there exist a large variety of algorithms for course
CFO estimation and a few algorithms for fine CFO estimation. Each of the
algorithm attempts to reduce the mean square error and increase the range of
CFO estimation with reduced computational complexity. However there
exists tradeoff between the MSE and computational complexity.
2.3 ISSUES AND PROBLEM FORMULATION
Though much work has been done on coarse CFO estimation in
OFDM, there exist very few algorithms for fine CFO estimation. Further
these algorithms are computationally intensive due to the ML search
operation. Hence, they can not be directly applied to develop low cost
solutions to Wireless standards such as WLAN and WiMAX standards (IEEE
2004). Further, the algorithms in the literature assume the channel is
uncorrelated, static and timing synchronization is perfect. However, the real
time channels are correlated due to the mobility of the communication
transceivers and the practical timing estimation algorithms results in a finite
residual timing error. Hence, CFO estimation algorithms need to be developed
for correlated fading channels, considering the timing errors.
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2.4 SYSTEM MODEL
This section introduces the system model that is considered for
CFO estimation. A detailed derivation on the effect of much smaller CFO in
terms of ICI is presented. This analysis motivates the development of a novel
algorithm for the CFO estimation.
The information bit stream is multiplexed into N symbol streams,
each with symbol period T, modulating a set of N sub-carriers that are spaced
by 1 NT . Using the Inverse Discrete Fourier Transform (IDFT), the OFDM
transmitted symbol is given by
1
0
1 ( )exp 2 0,1,2... 1N
m
mx n X m j n n NN N
(2.1)
where ( )X m are the discrete baseband symbols on each sub-carrier, that are
derived from a modulation alphabet of size M . A cyclic prefix is added
before transmission to combat Inter Symbol Interference (ISI). At the
receiver, the cyclic prefix is removed followed by Discrete Fourier Transform
(DFT) processing. After DFT processing the received signal is expressed as
1
0
exp 2 0,1,..., 1N
n
nY k y n j k k NN
(2.2)
Due to the frequency offset, the received baseband signal is given by
exp 2 ny n x n j u nN
(2.3)
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where D LOf ff
is the normalized frequency offset with respect to
sub-carrier spacing f , D LOf f is the total frequency offset due to both the
doppler frequency, Df , and the local oscillator mismatch between the
transmitter and the receiver, LOf . u n represents the complex Gaussian
noise with zero mean and variance 2 . Substituting Equation (2.3) into
Equation (2.2),
1
0exp 2 exp 2
N
n
n nY k x n j j k U kN N
(2.4)
where U k is the DFT of the noise u n . Substituting Equation (2.1),
Equation (2.4) can now be written as,
1 1
0 0
1 1
0 0
1 exp 2 exp 2
1 exp 2
N N
n m
N N
m n
n nY k X m j m j k U kN N N
nX m j m k U kN N
(2.5)
Using the identity,
1
0
11
NNn
n
aaa
(2.6)
The term 1
0
exp 2N
n
nj m kN
can be expressed as,
1
0
1 exp 21 1exp 211 exp 2
N
n
j m knj m kN N N j m k
N
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Using the trigonometric identity sin2
jx jxe exj
,
1
0
sin1 1 1exp 2 exp 11sin
N
n
m knj m k j m kN N N N m k
N
(2.7)
Substituting Equation (2.7) into Equation (2.5), yields
1
0
sin1 1exp 11sin
N
m
m kY k X m j m k U k
N N m kN
(2.8)
Thus, the received signal in Equation (2.8) can then be decomposed as,
1
0,
0N
m m k
Y k X k X m m k U k (2.9)
where m k are the ICI coefficients between the thm and thk sub-carriers
which is given by,
sin 1exp 1sin
m km k j m k
NN m kN
(2.10)
The first term in Equation (2.10) denotes the desired signal, while
the second term represents the ICI which appears as a result of the frequency
offset and the third term is the noise. Ignoring the noise term and focusing
only on the ICI as a source of impairment, the carrier-to-interference ratio is
given by
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2
21
0,
0
N
m m k
E X kCI
E X m m k
(2.11)
where .E represent the expectation operator.If the transmitted data
symbols are assumed to have zero mean and are statistically independent,
Equation (2.11) becomes
2 2
1 12 2
0, 1
0 0N N
m m k m
CI m k m
(2.12)
A plot of the carrier to interference ratio versus frequency offset for
FFT size of 1024N is shown in Figure 2.1 In the WiMAX system (IEEE
2004), the subcarrier spacing 10.94f kHz, and D LOf f =200Hz, thus
0.02 (i.e. 2% of the sub-carrier spacing). It can be seen from Figure 2.1
that at 200Hz frequency offset the carrier to interference ratio due to the ICI is
about 29dB. The impact of ICI on each sub-carrier is negligible even for the
200Hz frequency offset. Hence, the effect of ICI can be eliminated while
deriving CFO estimation algorithms for applications such as WiMAX and
LTE.
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Figure 2.1 C/I as a function of frequency offset
2.5 PROPOSED ALGORITHM FOR FREQUENCY OFFSET
ESTIMATION
In this section, a novel algorithm for CFO estimation is proposed.
At first the Goa algorithm (Gao 2008) for fine CFO estimation is explained.
Then the proposed algorithm is discussed. The Gao method uses a ML
approach for the fine CFO estimation.It can be described as
2'
0
ˆ arg min , , , 1 , 1N
k
X k l Y k l X k l Y k l (2.12a)
where1
'
0, 1 , 1 exp 2 exp 2
N
n
nY k l y n l j j kN
(2.12b)
The Gao algorithm searches for a CFO which compensates the additional
CFO in a adjacent symbol as in Equation (2.12b) and minimizes the
deferential metric defined in Equation (2.12a). The ML operation in (2.12a)
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is computational intensive and hence new method based on a novel signal
model which results due to very less CFO is proposed as below.
When the CFO is very less, it has been shown in section 2.1 that the ICI
can be ignored. By ignoring the ICI term, Equation (2.9) is written as,
0Y k X k U k (2.13)
Further, for a small frequency offset, 1, (0) can be written as,
sin 1 10 exp 1 expsin
Nj jN NN
N
(2.14)
For a small frequency offset 0 1.Considering the channel
frequency response at the kth subcarrier of lth OFDM symbol, the received
symbol after DFT is represented as the product of channel frequency
response, transmitted symbol and the frequency offset accumulated up to lth
symbol. It is given by,
, exp 2 , , ,Y k l j l H k l X k l U k l (2.15)
where ,H k l is the channel frequency response at thk subcarrier of thl OFDM
symbol. The FFT output at thk subcarrier of thl symbol is the product of
transmitted pilot, channel frequency response at respective subcarriers and a
phase term due to frequency offset at the thl symbol. This motivates the idea to
propose the estimation of CFO by cross correlating the same carriers in
adjacent symbols. Since the same subcarriers at adjacent symbols carry
different pilots an intermediary symbol is proposed as
* *, , , , 1 , 1Z k l Y k l X k l Y k l X k l (2.16)
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Assuming perfect timing synchronization, and the fact that the
channel is almost same across two symbol durations, and assuming that the
modulation symbols are known pilots, Equation (2.16) can be written as,
2, exp 2 , ,Z k l j H k l V k l (2.17)
where the noise term ,V k l is due to the cross-product in Equation (2.16).
The effect of noise is reduced by averaging over L pilot symbols and N
number of subcarriers. It is given by
1 1
0 0
1 ,N L
k lZ Z k l
NL (2.18)
Equation (2.18) represent the averaging over continual pilots. A
similar averaging can also be applied for scattered pilots which are defined
for estimating smaller CFO. It is noted that if there are N number of
scattered pilots, the averaging can also be done for N number of scattered
pilots in L symbols. Hence this method can also be applied for tracking or
fine CFO estimation. By applying the ML principle (Kay 1993) the estimate
of the normalized CFO is proposed as
1ˆ arg2
Z (2.19)
This estimate can be used to compensate the CFO at the DFT
output, as
ˆ, exp 2 ,Y k l j l Y k l (2.20)
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2.6 PERFORMANCE ANALYSIS OF THE PROPOSED
ESTIMATOR
In this section, the MSE performance of the proposed CFO
estimation algorithm is analyzed. The DFT output at thk subcarrier of thl and
( 1)thl OFDM symbols are given by
( , ) exp 2 ( , ) ( , ) ( , )Y k l j l H k l X k l U k l (2.21)
( , 1) exp 2 ( , 1) ( , 1) ( , 1)Y k l j l H k l X k l U k l (2.22)
It can be assumed that , , 1H k l H k l and
2 2, , 1 1X k l X k l . Substituting Equation (2.21) and Equation (2.22),
equation (2.16) can be written as,
2 *
* * * *
( , ) exp 2 ( , ) exp 2 ( , ) ( , 1) ( , 1)
exp 2 1 ( , ) ( , ) ( , ) ( , ) ( , ) ( , 1) ( , 1)
Z k l j H k l j l H k l U k l X k l
j l H k l U k l X k l U k l X k l U k l X k l
(2.23)
At high SNR, the last term can be ignored. Then, Equation (2.23)
can be written as,
2 *
* *
( , ) exp 2 [ ( , ) exp 2 1 ( , ) ( , 1) ( , 1)
exp 2 ( , ) ( , ) ( , ) ]
Z k l j H k l j l H k l U k l X k l
j l H k l U k l X k l
(2.24)
Define
*( , ) : exp 2 ( , ) ( , ) ( , )v k l j l H k l X k l U k l (2.25)
38
Substituting Equation (2.25) in Equation (2.24), ( , )Z k l can be
simplified as
2 *( , ) exp 2 [ ( , ) ( , 1) ( , ) ]Z k l j H k l v k l v k l (2.26)
Substituting Equation (2.26) in Equation (2.18), Z can be written as
1 12 *
0 0
exp 2[ ( , ) ( , 1) ( , ) ]
N L
k l
jZ H k l v k l v k l
NL (2.27)
Substituting Equation (2.27) in Equation (2.19), the normalized
frequency offset estimate is given by
1 12 *
0 0
1 1ˆ arg [ ( , ) ( , 1) ( , ) ]2
N L
k l
H k l v k l v k lNL
(2.28)
Assuming that 1 2( , ) ( , ) ( , )v k l v k l jv k l the estimation error is
written as,
1ˆ 12
P QjA A
(2.29)
where C represent angle of the complex number C,
1 12
0 0
1 ( , )N L
k lA H k l
N L (2.30)
1 1
1 10 0
( ( , 1) ( , ))N L
k lP v k l v k l (2.31)
1 1
2 20 0
( ( , 1) ( , ))N L
k lQ v k l v k l (2.32)
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Since the term 1( , )v k l is the product of signal and noise, at high
SNR its value is less and hence both P and Q are much smaller, and
1 P QjA A
can be approximated as QA
(Tretter 1985, Rosnes and Vahlin
2006); substituting it in Equation (2.29), the estimation error is given by
1ˆ2
QA
(2.33)
Further, if N and L are large, A can be approximated as
1 12 2 2
0 0
1 ( , ) [| ( , ) | ]N L
hk l
A H k l E H k lN L
(2.34)
Then, MSE of the normalized frequency offset estimation is
given by
22
2 4
1 [ ]ˆ[( ) ]4 h
E QE (2.35)
Using Equation (2.32), 2[ ]E Q is written as
21 12
2 20 0
, 1 ,N L
k lE Q E v k l v k l
Since2 2 2 2
2 21 2( , ) 0, ; ( , ) 0, ; ( , ) 0,
2 2u h u h
u hv k l N v k l N v k l N ;
1( , )v k l and 2 ( , )v k l are uncorrelated, 2[ ]E Q is computed to be
2 22
2[( )] u hE QNL
(2.36)
40
Substituting Equation (2.36) in Equation (2.35), the MSE of the
proposed frequency offset estimator is given by,
22
2 2
1ˆ[( ) ]( 2 )
u
h
EN L
(2.37)
2.7 EFFECT OF TIMING ERROR ON THE FREQUENCY
OFFSET ESTIMATE
The analysis of MSE performance assumes that the received
OFDM symbol is free from timing errors. But in practice, most of the timing
synchronization algorithms for OFDM results in residual timing errors. In this
section, the robustness of the proposed algorithm in the presence of residual
timing error is analyzed.
Consider the case of timing error of m sampling periods. Let cpN
be the length of cyclic prefix and the allowable range of the timing error
be cp cpN m N . It is shown by Mastofi and Cox (Mastofi and Cox 2006)
that for cpm N , the SIR is in the order of 20dB to 30 dB and the
corresponding ICI and ISI can be neglected. Hence, Equation (2.15) is
rewritten as
22, , , ,
mkjj l NN mY k l e e H k l X k l U k l
N (2.38)
For ( 1)thl symbol, Equation (2.38) can be rewritten as
22 1, 1 , 1 , 1 , 1
mkjj l NN mY k l e e H k l X k l U k lN
(2.39)
41
Substituting Equations (2.38) and (2.39) in Equation (2.16), it is
written as,
2( 2 ) ' ' *( , ) e [ ( , ) ( , 1) ( , ) ]jZ k l H k l v k l v k l (2.40)
where
2( 2 ) *'( , ) : e ( , ) ( , ) ( , )
mkjj l Nv k l e H k l X k l U k l (2.41)
Since the statistics of ( , )v k l and '( , )v k l are similar, the MSE of
the CFO estimate ˆ given by Equation (2.37) is applicable for the cases with
residual timing offset. This proves the robustness of the proposed algorithm
for CFO estimation.
2.8 CRAMER RAO LOWER BOUND (CRLB) FOR THE CFO
ESTIMATE
The FFT output at thk subcarrier in the thl OFDM symbol is
rewritten as
2, ( , ) ( , ) ( , ) 0 1,0 1j lY k l e H k l X k l U k l k N l L (2.42)
Let X be a LN LN diagonal matrix with elements
0,1 0,2 ( 1),, ,..., L NX X X , A be a LN LN diagonal matrix with elements
,exp 2 ,......exp ( 1) 2j L jN N N1 1 1 ; h be a 1LN channel frequency
response vector with elements 0,1 0,2 ( 1),[ , ,..., ]L NH H H , u be a 1LN with
noise vector elements 0,1 0,2 ( 1),[ , ,..., ]L NU U U .Then the 1LN vector
representation of Equation (2.42) is given as
y XAh u X (2.43)
42
where 1,1 1,2 ,[ , ,..., ]TL N ;
,2, ,
,
l nj ll n l n
l n
nH e
X (2.44)
The CRLB of the CFO estimation is given by (Key 1993)
1CRLB J
where the Jacobian J =2
2
ln /PE
y
The observation vector ‘ y ’ conditioned on the normalized
frequency offset ‘ ’ is Gaussian distributed with zero mean and covariance
matrix yyR . Its likelihood function can be expressed as
1122
1 exp2 det
LNP RH
yy
yy
y y yR
(2.45)
By neglecting the terms independent of ‘ ’, the log likelihood
function can be written as
1ln HP y yyy R y (2.46)
where
( )( )H HE EyyR yy XAh u XAh u 0( )H HLNNhA XR X I A (2.47)
Substituting Equation (2.47) in Equation (2.46), log likelihood
function is written as
43
ln /P y 10( )H H H H
LNNhAX XR X I XA (2.48)
where 22 2 101 12 ( 1)( , ..., )L Ndiag X X X . For simplicity, log likelihood function
can be written as
ln H HP y ABA (2.49)
where 1 10 0( ) ( )H H
LNN Nh hB X XR X I X R . Substituting the
definitions of &A B , the log likelihood function can be written as
ln /P y = '
' '' ' '
1 1 1 1* ' ' 2
l,n ( ),00 1 0
2Re (( 1) , ( 1) )L L N N
j ll l n
nl l l n
B l l N n l N m e
='
'
1' 2
0
2Re ( )L
j l
l
g l e (2.50)
where ' '' '
' * ' 'l,n ( ),
11 1
( ) (( 1) ,( 1) )L N N
l l nnl l n
g l l l N n l N nB
The first derivative of Equation (2.50) is
'
'
1' 2
0
ln4 Im ( )
Lj l
l
Pg l e
y (2.51)
The second derivative of Equation (2.50) is calculated as
2 12 2
20
ln /8 Re ( )
Lj l
l
Pg l e
y (2.52)
The statistical average of Equation (2.52) is expressed as
44
'
'
'
' '' ' '
21
2 ' 22
0
1 1 1 12 * ' ' 2
l,n ( ),00 1 0
ln8 Re ( )
8 Re [ ] (( 1) ,( 1) )
Lj l
l
L L N Nj l
l l nnl l l n
PE E g l e
E B l l N n l N n e
y
(2.53)The expectation of l,n is calculated to be (Hoeher 1997)
'
' '* ' ' 2
l,n max 0( ),[ ] 2 2 j l
b Dl l nE E sinc f n n J f T l e
(2.54)
where bE is the average bit energy, max is the maximum delay spread. Using
Equation (2.53) and Equation (2.54), the Jacobian operator is written as
' ' '
2
2
1 1 1 12 ' '
00 1 0
ln
8 Re ( , , ) ( , , )L L N N
nl l l n
PJ E
E l n n F l n n
y
(2.55)
where '( , , )E l n n'' ' 2
max 02 2 j lb dE sinc F n n J f T l e
'' ' ' 2( , , ) (( 1) ,( 1) ) j lF l n n B l l N n l N n e
The CRLB for CFO estimate in frequency selective correlated fading channel
is calculated as 1CRLB J . From Equation (2.55) it is noted that the CRLB
depends on the maximum delay spread, Doppler spread, subcarrier spacing,
signal to noise ratio and the correlation matrix of the wireless channel.
45
2.9 RESULTS AND DISCUSSION
The MSE performance of the proposed CFO estimation algorithm
is evaluated using Monte Carlo simulation in MATLAB. The simulation
parameters are shown in Table 2.1(Hoeher 1997).
Table 2.1 Simulation parameters for CFO estimation
S.No Parameters Values 1 Number of subcarriers 1282 Modulation QPSK 3 Normalized offset 0.05=5%4 Number of OFDM symbols 25 Maximum delay spread max 1µsec
6 Maximum Doppler spread df 320Hz
7 OFDM symbol duration T 260 µsec 8 Subcarrier spacing f 10kHz
9 Number of Monte Carlo runs 10000
Figure 2.2 shows the MSE performance of the proposed CFO estimator in a
static channel. It shows the simulated MSE, theoretical MSE and the CRLB
for the proposed CFO estimation method. The SNR required to attain a MSE
of 10-5 is 12.8 dB. The simulated MSE is in close agreement with theoretical
MSE at high SNR.
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0 5 10 15 20 25 3010
-7
10-6
10-5
10-4
10-3
SNR(dB)
MSE vs SNR over static channel
simulationtheoreticalCRLB
Figure 2.2 Performance analysis of proposed CFO estimate in static channel
Figure 2.3 shows the MSE performance of the proposed CFO
estimator in a frequency selective fading channel. It shows the simulated
MSE, theoretical MSE and the CRLB for the proposed CFO estimation
method. The SNR required to attain a MSE of 10-5 is 13 dB. The simulated
MSE is in close agreement with theoretical MSE at high SNR. There exists
consistent 0.4 dB difference with that of the CRLB. This is due to the high
SNR assumption made in deriving the MSE equation.
Figure 2.4 shows the performance comparison with recent data
aided fine frequency offset estimation method by Gao et al(2008). The
method by Gao et al(2008) is a ML method and is computationally intensive
due to search. The accuracy of the algorithm depends on the search interval.
Here, it is chosen as 0.01 to maintain marginal computational complexity.
Since the proposed method is a closed form solution, it requires less
computation. With the given search interval, the proposed method gives
0.3dB gain over method by Gao et al(2008) at a MSE of 10-4 . Moreover at
47
5 dB of SNR the proposed method gives an MSE of 10-4.164.However at very
high SNR the method by Gao et al (2008) outperforms the proposed method
since assumption of negligible ICI is no longer valid.
Figure 2.5 shows the performance comparison with Gao method
with normalized CFO of 0.3. With the given search interval of 0.01 for CFO
estimation, the proposed method gives 0.26dB gain over method by Gao et al
(2008) at a MSE of 10-4. The performance of the proposed algorithm is
similar to the performance with normalized CFO of 0.05. However the gain
over Goa method is 0.04dB lesser than that of the performance at a CFO of
0.05. Moreover at 5 dB of SNR the proposed method gives an MSE of10-4.126
. This reduction in the performance at higher normalized CFO is due to the
fact that the proposed method is derived with low CFO assumption. At higher
values of CFO the validity of the approximation reduces causing minor
performance degradation.
0 5 10 1510
-6
10-5
10-4
10-3
SNR(dB)
MSE vs SNR over correlated frequency selective channel
simulationtheoreticalCRLB
Figure 2.3 Performance analysis of proposed CFO estimate in frequency selective fading channel
48
0 5 10 1510
-6
10-5
10-4
10-3
SNR(dB)
MSE vs SNR over fading channel
Proposed-SimulationGaoProposed-Theoretical
Figure 2.4 Performance comparison of proposed algorithm with Gao method
0 5 10 1510-6
10-5
10-4
10-3
SNR(dB)
MSE vs SNR over fading channel
Proposed-SimulationGaoProposed-Theoretical
Figure 2.5 Performance comparison of proposed algorithm with Gao
method with normalized CFO of 0.3
49
Figure 2.6 shows the effect of timing offset on the MSE performance of
proposed algorithms. The timing offset considered is 0,1,2 and 5 samples. It is
observed that there exists 0.1dB loss at the MSE of 10-6 when the timing error
is 5. This loss is negligible and thus the proposed algorithm is robust against
timing error.
0 5 10 15 20 25 3010
-7
10-6
10-5
10-4
10-3
SNR(dB)
timing error=0timing error=1timing error=2timing error=5
Figure 2.6 Effect of timing error on the performance of CFO estimate
Figure 2.7 shows the range of frequency offset that can be
estimated at 10 dB of SNR. When the normalized frequency offset is -0.5 to
0.5 the estimated frequency offset is equal to the true frequency offset with an
error of 0.9*10-5 .When the range is extended there exists larger error
between the true and the estimated CFO. Hence, the CFO estimation range is
found to be -0.5 to 0.5 of the subcarrier spacing. The breakdown region of the
proposed algorithm for the normalized CFO is greater than +0.5 and less than
-0.5.
50
-1.5 -1 -0.5 0 0.5 1 1.5-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
true frequency offset
Figure 2.7 Estimation range of the proposed estimator
2.10 SUMMARY
In this chapter, low complex data aided fine CFO estimation
algorithm with improved accuracy is developed. The proposed algorithm is
based on continual pilots and can also be used for CFO estimation based on
scattered pilots. It is assumed that the CFO is very less when compared to
subcarrier spacing and the channel response is constant for two consecutive
OFDM symbols. The impact of the very less CFO on OFDM in terms of
carrier to interference ratio is characterized. A frequency domain CFO
estimation algorithm with improved performance is proposed. An analytical
expression for MSE performance of the algorithm is obtained. Cramer-Rao
Lower Bound (CRLB) on the CFO estimation in frequency selective fading
channel is also derived and it is shown that simulation results are matching
with the analytical expressions. For a specific search interval of 0.0l, the
proposed method gives 0.3dB gain over method by Gao et al (2008) at a
MSE of 10-4 .
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