Synchronization Issues in OFDM Systems

PRESENTED BY:
DEEPTANU DATTA
ROLL NO. 1811EE05
15 May 2019
A
PRESENTATION
on
Synchronization Issues in OFDM Systems
Guided by :
Dr. Preetam Kumar
Associate Professor
Department of Electrical
Engineering
IIT Patna
Indian Institute of Technology Patna
1

Outlines
 Introduction
 Multicarrier Modulation
 Orthogonal Frequency Division Multiplexing
 Symbol Timing Offset and its effects
 Carrier Frequency Offset and its effects
 Estimation of STO and CFO
 Log-likelihood function
 Maximizing log-likelihood function
 Estimator Structure
 Conclusion
 References
15 May 2019
2

Introduction
 In MCM, entire band B is divided into N sub-bands
each of bandwidth
 Frequency of ith subcarrier is fi =
 The various sub-carriers are superimposed to form a
composite signal
 To extract Xk , we extract Fourier series coefficient of
s(t) using the orthogonal property of sinusoids of
different frequencies.
 MCM transmits N symbols using N subcarriers in a
time period of .
N
B
f 
N
Bi.



i
t
N
B
ij
i eXts
2
)(
B
N
15 May 2019
3

Multicarrier Modulation (Contd…)
 So, symbol rate is = B => same as in single
carrier system.
 Let us consider B = 1024 kHz, number of subcarriers is
N=256 and typically coherence bandwidth is Bc = 250 kHz
 For single carrier system as B >>Bc , so it experiences
frequency selective fading channel, leading to ISI
 Bandwidth per subcarrier is kHz << 250 kHz,
so B<< BS , hence each subcarrier experiences
frequency flat fading channel
 Hence, ISI is completely removed from each subcarrier
BN
N
/
15 May 2019
4
4
256
1024

N
B
Bs

Orthogonal Frequency Division Multiplexing
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5
 Implementing N modulators at the transmitter and
N demodulators at the receivers using large number
of RF chains is a challenging task with very high cost
 This was solved by Weinstein and Ebert using IDFT
operation as
 At the receiver, individual carriers are extracted by
simple DFT operation
 Thus, bank of modulators and demodulators is
replaced by simple DSP chip having FFT and IFFT
functionalities



i
N
iu
j
i
i
B
u
N
B
ij
is eXeXuxuTs
2.2
..)()(

Symbol Timing Offset
15 May 2019
6
 STO occurs when there is a mismatch between the
actual starting point of OFDM symbol and estimated
starting point of OFDM symbol at the receiver
 STO of δ in time domain leads to a phase offset of
in the frequency domain
y(n) = x(n+δ) => Y(k) = X(k).
 Depending on the location of estimated starting
point of OFDM symbol, effect of STO can vary
N
2 k
N
k
j
e
2

Effect of STO
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7
 τmax is the maximum delay spread of the channel
 Estimated starting point of OFDM symbol coincides
with its exact starting point
 Orthogonality among different subcarriers are
preserved, thus no ISI or ICI occurs
lth symbolCP
(l+1)th symbol
τmax
τmax τmaxCASE 1

Effect of STO (Contd…)
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8
 Estimated starting point of OFDM symbol is before
the exact starting point
 lth symbol is not overlapped with previous (l-1)th symbol
 So, ISI do not occur and orthogonality is maintained
 yl(n) = xl(n+δ) => Yl(k) = Xl(k).
lth symbolCP
(l+1)th symbol
τmax
τmax τmaxCASE 2
k
N
j
e
2

15 May 2019
9
 Estimated starting point overlaps with the previous
OFDM symbol
 Orthogonality is destroyed by ISI and ICI also occurs
lth symbolCP
(l+1)th symbol
τmax
τmax τmax
CASE 3

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10
 Estimated starting point of OFDM symbol is after the exact starting
point
 lth symbol is overlapped with the next (l+1)th symbol
 Symbol suffers from both ISI and ICI and distortion here is too severe
to be compensated
lth symbolCP
(l+1)th symbol
τmax
τmax τmax
CASE 4

Effect of CFO
15 May 2019
11
 fc and fc’ are the local oscillator frequencies at Tx and
Rx respectively, so foffset = fc - fc’
 foffset occurs due to mismatch in local oscillator
frequencies and Doppler shift
 Normalized CFO i.e., splitted into
integer and fractional parts
 CFO of Ɛ in frequency domain leads to a phase offset
of in the time domain
Y(k) = X(k-Ɛ) => y(n) = x(n).
c
fv
f c
d
.

fi
offset
f
f
 


N
n2
N
n
j
e
2

Effect of IFO (Ɛi) Effect of FFO (Ɛf)
15 May 2019
 X(k) is cyclic shifted by Ɛi
to give X(k- Ɛi)
 But, orthogonality
among different
subcarriers is not
destroyed so no ICI
 Each sub-carrier
completely takes the
position of other sub-
carrier
12
Comparison of IFO and FFO effects
)(.)()(
1
)(
)(21
0
nzekXkH
N
ny l
N
nk
jN
k
lll
f





Taking the FFT of yi(n), we get Yl(k) as
Yl(k) =
Yl(k) =Xl(k) Hl(k)
+Zl(k)+
•Orthogonality is destroyed and ICI occurs
n
N
jkN
k
l eny




21
0
.)(
N
N
j
f
f
f
e
N
N
)1(
.
)sin(
)sin( 

 


N
N
kmj
l
N
kmm f
fN
N
j
emXmH
N
km
N
km
e
f
1
)(1
,0
1
)()(
)
)(
sin(.
))(sin( 




 




Estimation of STO and CFO
15 May 2019
13
 Let s(k) = IDFT of data symbol xk and б2
s= E{|s(k)|2}
 θ = Symbol Timing Offset, Ɛ = normalized carrier
frequency offset, n(k) = AWGN with variance бn
2
 L = length of cyclic prefix, N = number of subcarriers
 In presence of both STO and CFO, the received
symbol r(k) is
 One OFDM symbol has (N+L) samples.
 Let us consider (2N+L) consecutive samples of r(k)
)().()(
2
knekskr N
jk




Estimation of STO and CFO (Contd…)
15 May 2019
14
Let A = set of data indices that are copied into cyclic prefix
= {θ+N , θ+N+1 , …. , θ+N+L-1}
B = set of indices of the symbols in cyclic prefix
= {θ , θ+1 , ….. , θ+L-1}
r = vector containing (2N+L) observed samples
= [r(1) r(2) ….. r(2N+L)]T
AB
r(k)

Log-likelihood function
15 May 2019
15
 f(r|θ,Ɛ) is the conditional pdf of (2N+L) observed
samples in r when θ and Ɛ are known
 Then, the log-likelihood function is the logarithm of
the function f(r|θ,Ɛ)
 Ψ(θ,Ɛ)= log [f(r|θ,Ɛ)] =
 f(.) denotes pdf of its argument.
 After few calculations, Ψ(θ,Ɛ) becomes
Ψ(θ,Ɛ) = |β(θ)|cos (2πƐ + <β(θ) – ρφ(θ))
where |.| is the modulus of a complex number and < denotes the
argument of a complex number


 
1-L
)
N))r(kf(r(k)).f(
N))r(kf(r(k),
log(

k

Log-likelihood function (Contd…)
15 May 2019
16
 β(θ) is the auto-correlation function of r(k) defined as
 ρ is the magnitude of correlation coefficient between
r(k) and r(k+N) defined as
 φ(θ) is the sum of average energies of all the symbols




1
)(*).()(
L
k
Nkrkr



1}|)({(}|)(({|
))(*).((
22
2
22 






SNR
SNR
NkrEkrE
NkrkrE
ns
s



2
1
2
|)(||)(|
2
1
)( Nkrkr
L
k
 






Maximising Log-likelihood function
15 May 2019
17
Maximization of Ψ(θ,Ɛ) is done in two steps as follows
 Maximise Ψ(θ,Ɛ) wrt. Ɛ first to yield
 Maximise Ψ(θ, ) wrt. θ to get
 Wrt. Ɛ, Ψ(θ,Ɛ) is maximum when
cos (2πƐ + <β(θ)) =1 => 2πƐ + <β(θ)=2πn=>
 Then the value of θ that maximises Ψ(θ, ) is given
by
ˆ
ˆ ˆ
))(ˆ,(max),(maxmax),(max
),(



n


 )(
2
1
)(ˆ 
ˆ
)}(|)({|maxargˆ 



Structure of the estimator
15 May 2019
18
β(.)
φ(.)

Conclusion
15 May 2019
19
 Thus, using this technique, we can estimate the
symbol timing offset θ and the carrier frequency
offset Ɛ successfully

References
15 May 2019
20
[1] SB Weinstein and Paul Ebert, “Data Transmission by
Frequency-Division Multiplexing using the Discrete
Fourier Transform” ,Vol-5, Oct 1971
[2] Cho, Kim, Yang, Kang, “MIMO-OFDM Wireless
Communications with MATLAB”, WILEY Publishers
[3] van de Beek, Sandell, Börjesson, “ML Estimation of
Time and Frequency Offset in OFDM Systems”, IEEE
Transaction on Signal Processing, vol-45, July 1997
[4] Paul H. Moose, “A Technique for Orthogonal Frequency
Division Multiplexing Frequency Offset Correction”,
IEEE Transaction on Communications, vol-42, Oct 1994

References (Contd…)
15 May 2019
21
[5] Minn, H., Zeng, M., and Bhargava, V.K. “On timing
offset estimation for OFDM systems”,IEEE
Transaction on Communication,4(5), 242–244
[6] Willink, T.J. and Witteke, P.H, “Optimization and
performance evaluation of multicarrier
transmission”, IEEE Trans. Communication, 43(2),
p. 426–440, 1997
[7] NPTEL Lecture Series on “Advanced 3G/4G
Wireless Communication Systems” by Prof. Aditya
Kumar Jagannatham, IIT Kanpur

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