Time Synch Metrics Slides

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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




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




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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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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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




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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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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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




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




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.

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