24

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).

25

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

27

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

28

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.

29

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.

30

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)

31

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

32

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

33

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.

34

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)

35

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)

36

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)

37

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)

39

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.

46

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 .