The presentation describes the basic synchronization Issues in OFDM like STO and CFO and their estimation techniques using Maximum Likelihood Detection
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Synchronization Issues in OFDM Systems
1. 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
2. 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
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2
3. 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
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3
4. 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
/
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4
4
256
1024
N
B
Bs
5. 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
..)()(
6. Symbol Timing Offset
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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
7. Effect of STO
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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
8. Effect of STO (Contd…)
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τmax is the maximum delay spread of the channel
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
9. Effect of STO (Contd…)
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τmax is the maximum delay spread of the channel
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
10. Effect of STO (Contd…)
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τmax is the maximum delay spread of the channel
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
11. Effect of CFO
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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
n2
N
n
j
e
2
12. Effect of IFO (Ɛi) Effect of FFO (Ɛf)
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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(
13. Estimation of STO and CFO
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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
14. Estimation of STO and CFO (Contd…)
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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)
15. Log-likelihood function
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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
16. Log-likelihood function (Contd…)
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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
17. Maximising Log-likelihood function
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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
19. Conclusion
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Thus, using this technique, we can estimate the
symbol timing offset θ and the carrier frequency
offset Ɛ successfully
20. References
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[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
21. References (Contd…)
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[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