1 OFDM Synchronization Speaker:. Wireless Access Tech. Lab. CCU Wireless Access Tech. Lab. 2 Outline...

1

OFDM Synchronization

Speaker: 官雯彗

2

Wireless Access Tech. Lab.

CCU Wireless Access Tech. Lab.

Outline

OFDM System DescriptionSynchronization

What is Synchronization? Symbol Synchronization

Non-data-aided Method (Maximum likelihood, BPSK-OFDM case)

Frequency Synchronization Non-data-aided Method (Via oversampling) Data-aided Method (Special training-symbol-block)

Joint Symbol and Frequency Synchronization Non-data-aided Method (Exploiting second-order cyclostationarity)

3



OFDM System Block

S/P IFFTP/S

channel

P/S FFTS/P

iX

ix

'ix

iY

4



OFDM System Description

An OFDM symbol structure

CP

Symbol interval

Cyclic prefix

interval

xi,0,xi,1,……………………………,xi,N-1

copy

)1(,1,0, ,,, Niiii xxx x

5



Signal Model

signal. domain time is . --

signal. domain frequency is . --

iii

N

k

nkN

j

kini

iii

N

n

nkN

j

niki

NN

NNN

NN

NN

NN

NNNNN

NNNNN

nkN

πj

knN

eXN

x

exN

X

WWWW

WWWW

WWWW

,N-,,, keN

W

xXWx

XWxX

W

*1

0

2

,,

1

0

2

,,

)1()1(2)1(1)1(0)1(

)1(1211101

)1(0201000

2

1

1

12101

FFT

IFFT

6








7



What is Synchronization?

Symbol synchronization Symbol Synchronization refers to the task of finding the pr

ecise moment of when individual OFDM symbols start and end.

Frequency synchronization Let the transmitted signal be sn, then the complex baseban

d model of the passband signal yn is

Then the received complex baseband signal rn is

offset. frequency is f and ly,respective

frequency carrier receiver and rtransmitte the is f and f where

es

eesr

rxtx

fj2n

f-j2fj2nn

rxtx

s

ss

nT

nTnT

snTtxfj2nn esy

8



Received Signal

Transmitted signal

Received signal

)1(,1,0,)1(,)(, ,,,,,, NiiiNiCPNi xxxxx Symbol interval

Cyclic prefix interval

term. AWGN is

symbol, OFDM th in sample th represents

offset, frequency is

phase, initial is where

ki

ki

ki

nN

jj

kiki

w

ikx

wexx

,

,

,

2

,',

9



CP ith OFDM symbol CP

:0 and 0, 0, sampling, Corrert

)1(,

2,

1,

0,

)1(,

2,

1,

0,

)1()1(2)1(1)1(0)1(

)1(2221202

)1(1211101

)1(0201000

'

,

Ni

i

i

i

i

Ni

i

i

i

NNN

NN

NN

NN

NNNNN

NNNNN

NNNNN

ki

X

X

X

X

x

x

x

x

WWWW

WWWW

WWWW

WWWW

w

XWx

10




iaN

Ni

iaN

aNi

i

aNi

aNNN

NN

NNN

aNNN

aNNN

aNNN

NN

aNN

aNN

aNNN

NN

aNN

aNN

NaNN

aNN

aNN

aNi

i

aNi

NNN

NN

NN

NN

NNNNN

NNNNN

NNNNN

ki

X

X

x

x

x

WWWWW

WWWWW

WWWWW

W

W

W

x

x

x

WWWW

WWWW

WWWW

WWWW

aw

XPP

Wx

)(

)1(,

0,)(

)1(,

0,

)(,

)1()1(0)1()1()1()1()1()()1(

)1(101)1(1)1(1)(1

)1(000)1(0)1(0)(0

)1()(

1)(

0)(

)1(,

0,

)(,

)1()1(2)1(1)1(0)1(

)1(2221202

)1(1211101

)1(0201000

'

,

00

00

00

:number integer an is 0, 0, 0, sampling, Error

11




)(,1

)(,1

)1(,

,

)1()1(2)1(1)1(0)1(

)1(2221202

)1(1211101

)1(0201000

'

,

aCPNi

CPNi

Ni

ai

NNN

NN

NN

NN

NNNNN

NNNNN

NNNNN

ki

x

x

x

x

WWWW

WWWW

WWWW

WWWW

aw

Wx

:number integer an is 0, 0, 0, sampling, Error

12




)1(2

22

2

)1()1(2)1(1)1(0)1(

)1(2221202

)1(1211101

)1(0201000

)1(2

22

2

)1()1(2)1(1)1(0)1(

)1(2221202

)1(1211101

)1(0201000

,

]1[

]2[

]1[

]0[

]1[

]2[

]1[

]0[

'

NN

j

Nj

Nj

NNN

NN

NN

NN

NNNNN

NNNNN

NNNNN

j

NN

jj

Njj

Njj

j

NNN

NN

NN

NN

NNNNN

NNNNN

NNNNN

ki

eNx

ex

ex

x

WWWW

WWWW

WWWW

WWWW

e

eNx

ex

ex

ex

WWWW

WWWW

WWWW

WWWW

w

:0 sampling, Corrert

Wx

13



Synchronization task

Data-aided method

Non-data-aided method

14








15



Non-data-aided Method (Maximum likelihood)

term. AWGN isand

ly,respective offset frequency carrier and shift timing are and

signal, dtransmitte is

s,subcarrier of number the is

prefix, cyclic of length the is

,

where

signal received the consider To

k w

ks

N

CP

NCPkCPCPks

CPkCPNksk s

kweksk r

t

kN

j

t

)(

)(

1,)(

10,)()(

)()()()(

2

16




otherwise ,

,

,

0

0

)()(, 22

22

* Nme

m

mkrkrEk js

ns

Symbol iSymbol i-1 Symbol i+1

'

1 2N+CP

Observation interval

17




and,

variables, the of function desity yprobabilit the denotes where

is function likelihood-log The

NkrNkrkrekr

Nkrkrf

kr

krf

f

krfNkrfkrf

Nkrkrf

krfNkrkrf

rf

ns

ns

j

ns

ns

kk

kk

)1)((

)1)((

)()()(Re2)(exp

)(),(

)(

)(

)(exp

)(

)(

)()()(

)(),(log

)()(),(log

),|(log),(

2222

222

2*22

22

22

2

'

18




t.coefficien ncorrelatio the of magnitude the is

and

,

,

number, complex a of argument the deonteswhere

,

Thus,

1)()(

)()(

)()(2

1)(

)()()(

)())(2cos()(),(

22

2

22

*

122

1*

SNR

SNR

NkrEkrE

NkrkrE

Nkrkrm

Nkrkrm

ns

s

Lm

mk

Lm

mk

19




is timing symbol estimated the and

integer, an is

is offset frequency estimated the Thue,

,

,,,

Because

),(

nn

ML

ML

ML

)()(maxargˆ

,)(2

1ˆ

)())(2cos()()(

)),(ˆ(max)(maxmax)(max

20








21



Non-data-aided Method (BPSK-OFDM case)

Signal model in BPSK-OFDM system

value. real is used, is mappingBPSK Because

interval. symbol th in subcarrier th of data the presents

symbol, OFDM th of sample th presents

size, window IFFT the denotes where

,

is signal dtransmitte The

ni

ni

i

N

n

nkN

j

nii

x

inx

ikks

N

NkexN

ks

,

,

1

0

2

,

)(

101

)(

22




1

0,

1

0

2

,

1

0

22

,

1

0

)(2

,

1

0,

1

0

2

,

1

0

22

,

1

0

)(2

,

1

0,

1

0

2

,

1

0,

1

0

2

,

)2

sin(11

11)(

)2

cos(11

11)(

)2

sin(11

)(

)2

cos(11

)(

N

nni

N

n

nkN

j

ni

N

n

njnk

Nj

ni

N

n

kNnN

j

nii

N

nni

N

n

nkN

j

ni

N

n

njnk

Nj

ni

N

n

kNnN

j

nii

N

nni

N

n

nkN

j

nii

N

nni

N

n

nkN

j

nii

nkN

xN

exN

imag

eexN

imagexN

imagkNsimag

nkN

xN

exN

real

eexN

realexN

realkNsreal

nkN

xN

exN

imagksimag

nkN

xN

exN

realksreal

)(ksreal i

)(ksimag i

23




. or ,

as written be can symbol

OFDM-BPSK in sticcharacteri symmetry the Thus,

20)()( * N

k kNsks ii

24




ly.respective of phase the and magnitude the denote and where

or ,

as expressed be can , or when Because

)(

)2(

20)()(

)0(

)(2

0)()(

,,

2,

,,

0,

*

2,

,,

0,

ksA

eANs

Nk eAkNs eAks

eAs

ksN

k kNsks

ikiki

jNii

jkii

jkii

jii

iii

Ni

kiki

i

25




term. AWGN is and

symbol, OFDM th of sample th presents

offset, frequency carrier is

phase, initial is

shift, timing is where

is signal received The

)(

)(

)()(

)()()(2

kw

ikks

ekskv

kwkvkr

i

i

kN

jj

ii

iii

CP

)()0()( iii wvr

26



,

character the has it , and

samples opposite two the for point central the as taken isWhen

:iprelationsh symmetry of group Second

,

character the has it , and

samples opposite two the for point central the as taken isWhen

:iprelationsh symmetry of group First

Nk eAkNvkv

kNvkv

N v

CP k eAkvkv

kvkv

v

jjkiii

ii

i

jkiii

ii

i

121)()(

)()(

)2(

1)()(

)()(

)0(

222,

22,


CP

2

N

CP

2

N

27




There are three kinds of useful property can be employed as follows:

The first group of symmetry relationship.

The second group of symmetry relationship.

The cyclic prefix copied from the tail of OFDM symbol.

28



Non-data-aided Method (BPSK-OFDM case)- Using angle

is function cost The

is of mean The

where

12

1,

12

1,

12,

1,

12,2,1,

12

1

)12()12(

)1()1(

,,,

N

mkmkk

N

mmkk

iiNk

iik

Nkkkk

ECf

f

CN

E

NNkrNkrangleC

NkrkrangleC

CCC

CC

C

C

C

C

.121)()( 222, Nk eAkNvk v jjkiii where

ip,relationsh symmetry of group second In

case. free-noise for

zero isfunction cost the and , of phase

initial twice the is of mean the

,is samples opposite of point central the When

fNr

CCC

N r

i

N

i

C

C

)2(

,,,

)2(

12,2,1,

CP

2

N

CP

29



Non-data-aided Method (BPSK-OFDM case) - Using angle

CP

mkmkk

CP

mmkk

iimk

N

mkmkk

N

mmkk

iimk

CP

mkmkk

CP

mmkk

iimk

EDf

DCP

E

mNkrmkrangleD

ECf

CN

E

mNkrmkrangleC

EBf

BCP

E

mkrmkrangleB

1,

1,

*,

12

1,

12

1,

,

1,

1,

,

1

)()(

12

1

)()(

1

)()(

DD

D

CC

C

BB

B

:property useful Third

:property useful Second

:property useful First

kkkk fffF DCB

CP

2

N

CP

2

N

CP

2

N

30



Non-data-aided Method (BPSK-OFDM case) - Using correlation

12

1

)()()(N

mii mNkrmkrkC

.121)()( 222, Nk eAkNvk v jjkiii where

ip,relationsh symmetry of group second In

region. nobservatio in value

maximum the has , , is

samples opposite of point central the When

)()2( CkNri

CP

2

N

CP

31



Non-data-aided Method (BPSK-OFDM case) - Using correlation

CP

mii

N

mii

CP

mii

mNkrmkrkD

mNkrmkrkC

mkrmkrkB

1

*

12

1

1

)()()(

)()()(

)()()(

:property useful Third

:property useful Second

:property useful First

)()()( kDkCkBFk

CP

2

N

CP

2

N

CP

2

N

kF

maxargˆ

32



Performance Analysis

0 1 2 3 4 5 6 7 8 9 1010

-6

10-5

10-4

10-3

10-2

10-1

100

SNR(dB)

Lost

sym

bol t

imin

g ra

te

AWGN channel

Using angle

Using correlationML

0 1 2 3 4 5 6 7 8 9 1010

-3

10-2

10-1

100

101

102

103

SNR(dB)

Mea

n-sq

uare

d er

ror

AWGN channel

Using angle

Using correlationML

33



0 50 100 1500

1

2

3

4

5

6

7

8x 10

4

Time index, n

Sta

tistic

s

Maximum likelihood algorithm

20 40 60 80 100 120 1400

1

2

3

4

5

6

7

8

9

10x 10

4

Time index, n

Sta

tistic

s

Using angle (Using two groups of symmetry relationship)

CP CPCP

34



Solution

Real part

Image part

: A pair of sample which has symmetry characteristic in noise free

case

35



Solution

Real part

Image part

: A pair of sample which has symmetry characteristic in noise free

case

36








37



Non-data-aided Method (Via oversampling)

interval). (DFT/IDFT duration symbol OFDM the is

time), guard the (excluding

symbol OFDM current the of time starting the is

subcarrier the modulate to used is where

symbol OFDM baseband complex time continuous the

,subcarrier with modulation OFDM based DFT the Using

T

t

ed

edN

ts

N

s

Tkjk

N

k

ttT

kj

k

s

,

1)(

2

1

0

)(2

38




1

0

)2

1(

2

2

1

0

2

1

1)(

22

1)(

1,,0

)(

N

k

nN

kj

k

s

N

k

N

nkj

k

ss

s

edN

ns

N

TT

edN

ns

Nn

nTttts

NTT

model signal the get we case, this In .

model symbol OFDM IDFT usual the is This 0.

. on depending nsobservatio time-discrete of sets

two have we , and shift time initial the is where

at are for instances sampling the that Assume

interval. symbol the as define We

TTNTT sss )1( 2 0

)( )( 21 nsns

39




N

Nj

Nj

TN

eediag

dd )1(

10

1

,,

E

dW

WEd

Wd

and ;

vector symbol the is matrix; IDFT the is where

s

s

have we form, matrix in Written

2

1

40




offset.

frequency the is and subcarrier to ingcorrespond

response frequency channel the is where

then, is signal received baseband

time continuous the offset, frequency carrier and noise,

additive channel, dispersive time of presence the In

k

kH

tzedkHN

txN

k

ttT

kj

k

s

)(

)()(1

)(1

0

))(2

(

41




. as defined We other.

each to eduncorrelat be to assumed usually are and noise

Gaussian complex additive are and where

2T

0.

. on depending nsobservatio time-discrete of sets

two have we , at time sampling the Under

s

s

N

k

nTN

kj

k

N

k

nTN

kj

k

ss

T

nznz

nzedkHN

nx

N

T

nzedkHN

nx

nTtt

s

s

)()(

)()(1

)(

2

)()(1

)(

21

2

1

0

)2

1)(

2(

2

1

1

0

)2

(

1

42




Where

. constant scalar a by differs offset frequency

the to due shift phase the while s,frequencie subcarrier in difference

the to due and between shift phase the reflects matrix The

. element diagonal with matrix diagonal a being with where

(2)

(1)

:follows as

form matrix compact more a in written be can model signal above The

),,,1(

)(~

~

~

)1(

2

21

22

2

11

Njj

j

j

eediag

e

kH

e

P

xxE

HHdd

zdPWEx

zdPWx

43




Perfect recovery of carrier frequency offset in the absence of noise

equal. are

and resulting the such find to intuitive is it Thus

(5)

noise of absence the in that verify to rwardstraightfo is It

(4)

(3)

Define . offset frequency the of estimation blind to want

we and unknown, is that notice we (2), and (1) From

21

21

2HHH

2

1HH

1

yy

dyy

xPWEy

xPWy

d

~

~

2

je

44




Frequency offset estimation in the present of noise

section. next the in provided is principles

likelihood maximum using criterion onminimizati Above

and between distance the minimize to

y,intuitivel propose, we However, . of value any for

true be will (5) that unlikely is it noise, of presence In

)y(y)y(y

yy

21H

21

21

min

.

45








46



Data-aided Method (Special training-symbol-block)

At the transmitter, the training-symbol-block contains two equal-length training symbols in time domain, and the second training symbol is the inverse repeat of the first one.

symbols. training of length DFT the denotes where

:block-symbol-training The

N

,s,s,N,sN,s,,ss )0()1()1()1()1()0( S

47



At receiver, without considering of the channel attenuation and additive AWGN, the relationship between corresponding samples in a received training-symbol-block is :

symbols. training of spacing subcarrier

a to normalized offset frequency carrier the

denotes and sample th the denotes where

kkr

NkekrkNr NkNj

)(

]1,0[)()12( /)212(2


48



A timing metric is defined:

samples. of window a in

sample first the to ingcorrespond index time a is where

N

d

kr

edk

kr

dkrkdNr

dM

Nd

dk

NkdNj

Nd

dk

N

k

2

)(

)(d)r-r(k2

)(

)()12(2

)(

122

1-N

0k

/)2122(2*

122

1

0

*


49




50



lobe. main the of center the in stands which vicinity

small a in only satisfied is assumed), is

,generality of loss (without period one Within

. of tindependen constant a is of nexpectatio The

appears.

lobe main a period, each within and , of period with of

function a is of nexpectatio the that 1, Fig. in Shown

)0()0(

4,4

)0(

2/

)0(

dMM

NN

dM

N

M


51




52



buffered. being block-symbol-training integral

one guarantees of length The buffered. are sequence

training the of samples receiver, the At d.transmitte

are blocks-symbol-training identical 2 r,transmitte the At

N

N

3

3


53



interval. lookup the denotes

and where

)

(

:values dpresuppose

:used is lookup of method a n,acquisitio of start

the at unknown fully is offset frequency carrier the Because

))12(2(04

ˆ,)1(ˆ,,ˆ

,0ˆ,ˆ,,)1(ˆ,ˆ

)12(

11

01)1(

NNNk

kk

k-kε

k

kk

kk


54



n.acquisitio of output the also is block-symbol

-training integral that of start estimated The algorithm.

nacquisitio by estimated offst frequency carrier the

denotes then , each for if ; block

-symbol-training integral an of appearance the implies

which as drepresente is )( maximum

the sequence, training dcompensate )( th For

0ˆ

10)(

d

qkpk

NddM

kjkj

qpq

j


55



After acquisition, if the remaining carrier frequency offset exceeds the tracking range, tracking algorithm will not work correctly, and this is called Missing Lock.

For usable SNR, if N is large enough, the probability of Missing Lock is negligible.


56



After acquisition, the remaining frequency offset needs to be further corrected.

. function

density yprobabilit the of logarithm the is , offset

frequency carrier the for function likelihood-log the condition,

SNR certain under block-symbol-training received a For

1

0

1

0

)12(),(log

)12(),(log

)|(log)(

)|(

)(

N

k

N

k

kNrkrf

kNrkrf

f

f


57



where

ondistributi Guassian valued-complex D The

2

2

22

2

22

*

22222

222

2*/)212(22

1

)12()(

)12()(

1

1

)12()12()(Re2)(exp

)12(),(

2

n

s

ns

s

ns

ns

NkNj

SNR

SNR

SNR

kNrEkrE

kNrkrE

kNrkNrkrekr

kNrkrf


58



spacing. subcarrier is range tracking the ,

satisfied, should each for

correctly, work algorithm tracking proposed the make to order In

)))12(2/((

1,0)12()(

212)12()(2

212)12()()12()(ˆ

*

1

0

2*

1

0

**

NNi.e.

NkkNrkrangle

kNkNrkr

kNkNrkranglekNrkrN

N

k

N

k


59



is bound lower Rao-Cramer The

SNRN

N

N

NeVar

)14(4

3

)12(2|

22


60








61



Non-data-aided Method(Exploiting second order cyclostationarity)

order. channel is

channel, of response impulse the is where

by given is channel

fading selective-frequency a through dtransmitte

signal OFDM equivalent baseband time-discrete The

L

nh

mnxmhn yL

m

][

][][][1

0

62



symbols. ninformatio complex the are

carrier,-sub of number the is

filter, shaping pulse rtransmitte the is where

is signal dtransmitte The

lk

N

k l

knN

j

lk

c

N

ng

elPngcnx

nx

,

1

0

2

,

][

][][

][


63



phase. initial the is

offset, frequency the is

symbol, a of time arrival unknown valued-integer the is where

receiver the At

n

nwennyny nj ][][][ˆ )2(


64



. and with eduncorrelat is 3.

. variance with variable random

gaussian tindependen mean-zero is each , For 2.

. variance with ion,constellat

complex alphabet-finite a from drawn values with sequence

d(i.i.d.)distribute yidenticall tlyindependen mean-zero a is 1.

:paper this in hold are sassumption following The

][][][

][102

][

2

,

nwnhnx

nhLn

c

nh

c

lk


65



filter.

shaping pulse by caused signal OFDM receiver the of

onaritycyclostati on based is algorithm proposed The

and estimate to is ationsynchroniz of goal The . n


66



][][

];[

][][][

][ˆ][ˆ];[

ˆ

*

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where


67



1

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where

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68



. of transform Fourier denotes where

theorem Parseval's Using

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69



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defining Upon

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as expressed is ncorrelatio cyclic Then

. Let


70



n.correlatio cyclic of phase the

affect not does channel of response impluse the and n,correlatio

cyclic of phase the as appear offset frequency and offset Timing

as derived be can offset frequency the , offset timing Given

for

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71



].;[];[ˆlim

];[ˆ][ˆ

];[ˆ

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i.e. sense, square mean in consistent and unbiased

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set data from obtain We samples. data finite from it estimate

should we quantity, cyclic ensemble to access have not can we Since


72



.estimation reliable more makes Large

estimator. the of complexity

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for

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ypotentiall be can ncorrelatio cyclic of onimperfecti of effect The

Q

Q

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Q

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73



Reference

1) J.J. van de Beek, M. Sandell and P. O. Börjesson, “ML estimation of time and frequency offset in OFDM systems,” IEEE Trans. Signal Processing, vol. 45, no. 7, pp. 1800-1805, July 1997.

2) B. Chen, H. Wang, “Blind OFDM carrier frequency offset estimation via oversampling”, 2001 IEEE, vol. 2, pp. 1465 – 1469, Nov. 2001.

3) Z. Zhang, M. Zhao, H. Zhou, Y. Liu, J. Gao, “Frequency offset estimation with fast acquisition in OFDM system”, IEEE Communications Letters, vol. 8, no. 3, pp. 171 – 173, Mar. 2004.

4) B. Park, H. Cheon, E. Ko, C. Kang, D. Hong, “A blind OFDM synchronization algorithm based on cyclic correlation”, IEEE Signal Processing Letters, vol. 11, no. 2, pp. 83 – 85, Feb. 2004.

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