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Optimum Symbol TimingEstimation with VariousPerformance Measures for
OFDM systems
Jungwon Lee, Dimitris Toumpakaris, and Hui-Ling Lou
May 18, 2005
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Outline
I. Introduction
II. System Model
III. Symbol Timing Synchronization
IV. Various Symbol Timing Estimators
V. Simulation Results
VI. Conclusion
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Introduction OFDM
Orthogonal Frequency Division Multiplexing (OFDM)
Widely used in many wireless systems such as WLAN, DAB, DVB, etc.
Combats inter-symbol interference using computationally efficient fastFourier transform (FFT) and cyclic prefix.
Symbol timing synchronization is necessary.
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Introduction
Symbol Timing Synchronization Symbol timing offset
Difference between the current FFT window position and the correctposition.
Introduces phase change, inter-carrier interference (ICI), and/or inter-symbol interference (ISI).
Maximum likelihood (ML) estimator of symbol timing offset
Maximizes the probability that the estimate is correct. Does not care about the error magnitude. Weighs all non-zero timing
errors equally.
Well-known but may not be the best choice.
This paper derives the optimal symbol timing estimators withvarious performance measures.
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Outline
I. Introduction
II. System Model
III. Symbol Timing Synchronization
IV. Various Symbol Timing Estimators
V. Simulation Results
VI. Conclusion
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System Model (1)
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System Model (2)
Received signal with unknown delay
: received signal
: transmit signal
: noise
: uniformly distributed in .
Objective
Estimation of with the observation of M received samples .
][][][ nznxny +=
][ny
][nx
][nz
]1,0[ + gNN
][ny
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Outline
I. Introduction
II. System Model
III. Symbol Timing Synchronization
IV. Various Symbol Timing Estimators
V. Simulation Results
VI. Conclusion
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Effects of Symbol Timing Offset (1)
Three cases
gN 0
12/1 ++ NNN gg
2/1 N
OFDM Symbol 0
cyclic prefix
OFDM Symbol 0
OFDM Symbol 0
OFDM Symbol 1
OFDM Symbol 1
OFDM Symbol 1
OFDM Symbol 2
OFDM Symbol 2
OFDM Symbol 2
FFT window
FFT window
FFT window
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Effects of Symbol Timing Offset (2)
Case 1 Inside the region of the cyclic prefix samples and thefirst useful sample
Linear phase change in the frequency domain
gN 0
OFDM Symbol 0
cyclic prefix
OFDM Symbol 1 OFDM Symbol 2
FFT window
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Effects of Symbol Timing Offset (3)
Case 2 & 3 Outside of the region consisting of the cyclicprefix samples and the first useful sample
Linear phase change in the frequency domain
Inter-symbol interference (ISI) and inter-carrier interference (ICI)
ISI and ICI increase as the error magnitude increases.
12/1 ++ NNN gg
2/1 N
OFDM Symbol 0
OFDM Symbol 0
OFDM Symbol 1
OFDM Symbol 1 OFDM Symbol 2
OFDM Symbol 2
FFT window
FFT window
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Goals of Symbol Timing Synchronization
A linear phase may not be a problem in some cases.
The FFT window beginning can be placed anywhere in the regionconsisting of cyclic prefix samples and the first useful sample.
ICI and ISI gradually increases as the symbol timing errorincreases.
It may not be a good idea to treat all the symbol timing errors equally
regardless of their magnitudes.
In some cases, it may be preferable to have frequent small errors ratherthan infrequent large errors.
The goal of symbol timing synchronization can be differentdepending on OFDM systems considered.
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Outline
I. Introduction
II. System Model
III. Symbol Timing Synchronization
IV. Various Symbol Timing Estimators
V. Simulation Results
VI. Conclusion
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Maximum A Posteriori (MAP)
Estimator MAP estimator maximizes , the probability that the
estimate is correct.
MAP estimator
: observation vector of Msamples : conditional probability mass function of given y
MAP estimator becomes the maximum likelihood (ML)
estimator for uniformly distributed
.
ML estimator was derived by Van de Beek et al.
{ })|(maxarg y
p=
y)|( yp
)( =P
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Minimum Failure Probability (MFP)
Estimator MFP estimator maximizes , the probability
that the FFT window beginning lies in the region of cyclicprefix samples and the first useful sample.
MFP estimator
MFP estimator can be derived using the expression for ,
which is available in the literature for AWGN and fast fadingRayleigh channel.
]))[(( gN NP t
=
+
=
g
t
N
k
Nkp
)|))(((maxarg | yY
)|( yf
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FFT window
cyclic prefix
Minimum Mean Square Error (MMSE)
Estimator MMSE estimator minimizes
MMSE estimator can be easily
derived using .
2}]))((,))[min{(()(tt NtN
Nc =
)]([ cE
)|(
yf
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Minimum Modified Mean Square
Error (MMMSE) Estimator MMMSE estimator minimizes
MMSE estimator can also be
easily derived using .
)]([ cE
)|(
yf
2}]0},))((,))([max{min{()(tt NtN
Nc =
FFT window
cyclic prefix
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General Bayesian Estimator (1)
General Bayesian estimator minimizes .
For any cost function, a Bayesian estimator can be found fortiming synchronization.
All the previous estimators can be explained in the context of
a general Bayesian estimator.
)]([ cE
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General Bayesian Estimator (2)
Cost function for ML estimator Cost function for MFP Estimator
=otherwise,1
))((,0)(
gN Nc t
=
=otherwise,1
0))((,0)( t
Nc
FFT window
cyclic prefix
FFT window
cyclic prefix
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General Bayesian Estimator (3)
Other goals of synchronization of interest
Maximizing signal-to-noise ratio (SNR)
Minimizing average bit error rate (BER)
The estimators for maximum SNR and minimum average BERcan be found as a special case of a general Bayesianestimator.
No analytic expressions for the cost functions are available.
The cost functions can be found by simulation.
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Outline
I. Introduction
II. System Model
III. Symbol Timing Synchronization
IV. Various Symbol Timing Estimators
V. Simulation Results
VI. Conclusion
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Simulation Results Failure Probability
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Simulation Results Mean Square
Error
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Simulation Results Modified Mean
Square Error
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Outline
I. Introduction
II. System Model
III. Symbol Timing Synchronization
IV. Various Symbol Timing Estimators
V. Simulation Results
VI. Conclusion
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Conclusion
Developed optimum symbol timing estimators for variousperformance measures.
Minimum failure probability estimator
Minimum mean square error estimator
Modified minimum mean square error estimator
Derived a general Bayesian estimator and described all the
above estimators and ML estimator in the general context ofBayesian estimation theory.
Showed that different estimators should be chosen depending
on the goal of the symbol timing synchronization.