Advanced topics in OFDM systems - Sites personnels de ... / OFDM systems 37 Bibliography (2) ¾Synchronization algorithms suited to OFDM [BEE-97] JJ Van De Beek, M sandell, PO Borjesson

OFDM SystemsMarie-Laure Boucheret

IRIT/ENSEEIHTEmail : [email protected]

Synchronization / OFDM systems 2

Contents

Recall : OFDM systems• multipath mobile channel• Principles of OFDM systems• OFDM systems and filter banks• OFDM systems with guard interval• Advantages/drawbacks of OFDM systems

Synchronization aspects in OFDM systems• Specificity of OFDM system w.r.t synchronization• Impact of synchronization errors (frequency, sampling time) on

OFDM systems• Synchronization algorithms


( )1

mc

Tf

=∆

Recall on multipath mobile channels (1)

Coherence bandwidth : (∆f)c

–Two carriers separated by (∆f)c are affected by « more orless » the same attenuation.

W : occupied bandwidthW<< (∆f)c => non frequency selective channelsW>> (∆f)c => frequency selective channels

Nota : (∆f)c is not related to the relative mobility emitter/receiver(ex: cables)


Recall on multipath mobile channels (2)

Coherence time (∆t)c

Two signal samples separated by less than (∆t)c are affected by « more or less « the same attenuation.

Bd: doppler bandwidth

( )1

dc

Bt

=∆


Principles of OFDM systems (1)

Frequency selective channels⇒ Use of multiple carriersThe « elementary channel » (one carrier) is now non frequency selective.

Spectral efficiency⇒ Use of overlapping orthogonal carriers

Diversity⇒ Use of ECC

COFDM



Expression of OFDM signal (complex envelop)

Carrier #i :

h(t): rectangle of width T (NRZ)fi=i/T

Frequency multiplex

( )( ) ( )exp 2i ik ik

x t d h t kT j f tπ= −∑

( )1

0( ) ( )exp 2

N

ik ii k

x t d h t kT j f tπ−

=

= −∑∑





Modulator / demodulator for carrier # l (ideal case)

h(t)

fl

x(t)( )ikd h t kT

∞

−∞

−∑

h*(-t) decisionˆ

kd

kT-fl

x(t)+n(t)


OFDM systems and filter banks (1)

OFDM modulator/demodulator can be seen as a synthesis/analysis filter bank (no guard time, no coding)

h(t)

h(t)

h(t)

f1

fN-1

Channel receiver

emitter

fi=i/T



Receiver for carrier n°l

Efficiently implemented via FFT-1 (emitter) and FFT (receiver)

h*(-t) decisionˆ

kd

kT-fl

{hi(n)} : polyphase implementation of h(n

h0(n)

h1(n)

hN-

1(n)

IFFT

Channel 0

Channel 1

Channel N-1emitter



OFDM receiver

h0(t)

h1(t)

hN-1(t)

FFT

Channel 0

Channel N-1

Channel 1



Application : classical OFDM

Implementation with polyphase+FFT filter bankst

h(t)

T0

Fe=N/T

h(n)=1 for n=0,…,N-1

hi(n)=1 for n=0

hi(n)=0 elsewhere

IFFT

Channel 0

Channel 1

Channel N-1

FFT

Channel 0

Channel N-1

Channel 1


OFDM system with guard interval (1)

Guard interval is used to removed residual intersymbol interference (ISI)

Guard interval is inserted by copying the [kT, kT+∆T[ part of original OFDM symbol => no discontinuity in the signal!

Resulting OFDM symbol period is T+∆T (∆T : guard interval)





The FFT output is (symbol # i, carrier #j):

Xi,j=Hjsi,j (without noise)

=> flat fading channel at sub-carrier level

Cyclic prefix is used in order to:– Avoid equalization– Increase robustness against sampling time error


Advantages/drawbacks of OFDM systems

Advantages:– Emitter and receiver are efficiently implemented with FFT/IFFT– No equalization is required– Spectral efficiency– Diversity

Drawbacks– Sensitivity to synchronization errors– Sensitivity to non linearities (Amplifiers)– Mainly used in broadcasting applications


Receiver Architecture (1)

Differential demodulation (ex: DAB)

Diff.

encoderIFFT CP channel

CP-1 Frequency and timing correction FFT Diff.demo decoder

In non-coherent communication, differential encoding/decoding avoids the use of channel estimation.


Receiver Architecture (2)

Coherent demodulation (ex: DVB-T)

IFFT CP channel

CP-1 Frequency and timing correction FFT

Channel estimation/

compensationdecoder


Specificity of OFDM system w.r.t synchronization issue

•OFDM systems are much more sensitive to synchronization errors than single carrier systems.

•Synchronization algorithms suited to single carrier systems are inefficient for OFDM.


Impact of a synchronization error (1)

System model (Gaussian channel)–Carrier : n° l–Frequency offset : ∆f–Timing error : τ

h*(-t)

fl+∆f kT+τ

ˆkd

-∆f



Timing error τ

−τ<∆-L : phase rotation (compensated by channel estimation/correction=

−τ>∆-L : nth symbol, carrier n° i

SNR loss

ICI/ISI

2 ( / ), , , , ( , )j n N

i n i n i n i nNY e X H n n i n

Nπ τ

ττ−

= + +



Frequency error : ∆f

Ym,l=p(∆f)exp[2jπ(m+1/2)∆fT]dml+ICI

with ( ) ( )( ) ( )exp 2 ( )( 1/ 2) sin , ( ) sinc c

n lICI j k l m n l fT p f fTπ π π

≠= − + − + ∆ ∆ = ∆∑

( ){ }( ), ,

122

, , , ,0 0

For (G: guard time)

sin

1 1 24

n i k

b bn n k n i k

i n

G

n l f TI

n l f T

E ETEB erfc I IN N

τ

π

π

−

≠

<

− + ∆ × = − + ∆ ×

= × +

∑



BER degradation due to a frequency error(gaussian channel)



BER degradation due to a frequency error(gaussian channel) : single and MC case

1: single carrier

2: OFDM, N=100

3: OFDM, N=256

3: OFDM, N=512

4: OFDM, N=1024



Impact of phase noise10 11 4 (OFDM)

ln10 60

10 11 4 (SC)ln10 60

s

O

s

O

ENR N

DE

R N

βπ

βπ

≈

β : 3 dB BW (SSB) in Wiener model

1: single carrier

2: OFDM, N=100

3: OFDM, N=256

3: OFDM, N=512

4: OFDM, N=1024


Timing/frequency estimators (1)

Estimators using pilot symbolsMooseSchmidl et Cox

Estimators not using pilot symbolsVan de Beek

These estimators are suited to frequency selective channelsGuard time is necessary for other reasonEach elementary channel (FFT output) is modelled by a differentcomplex multiplicative coefficient.


Moose estimator (1)

Principle : Emission of 2 identical OFDM symbols

Timing has to be corrected first

Hypothesis : the channel impulse response is constant over some OFDM symbols


Moose estimator (2)

First OFDM received symbol : [r0 r1…rN-1]

Second OFDM received symbol : [rN rN+1…r2N-1]

CIR constant over 1 OFDM symbol => rn+N=rnexp(2jπ∆fNTe)= rnexp(2jπε)

with ε=1/T (inter carrier spacing)

FFT output (first symbol) :

FFT output (second symbol):

y(k+N)=y(k)exp(2jπε) k∈{0,1,…,N-1} =>The signal and ICI are affected exactly in the same way by the frequency offset.

1

0( ) exp 2

N

nn

nky k r jN

π−

=

=

∑

1

0( ) exp 2

N

n Nn

nky k N r jN

π−

+=

+ =

∑


Moose estimator (3)

MLE estimator:

N-1*

k=0

1ˆ Arg y(k+N)y ( ) 2

1 1 11 2 2

k

f fT T T

επ

ε

=

< ⇒ ∆ < ⇒ − < ∆ <

∑

Frequency unbiguity has to be removed.


Schmidl et Cox estimator (1)

Estimation of both timing and frequency errors

Principle:

2 dedicated pilot symbols•First symbol : null odd carriers

•Second symbol : 2 interleaved PN sequences (odd/even carriers)

Estimation•First symbol is used for timing and frequency estimation (2/T ambiguity)

•Second symbol is used to remove ambiguity on frequency estimation



First symbol : null odd carriers

1

0/ 2 1

20

exp 2

= exp 2/ 2

N

n kkN

kk

nky x jN

nkx jN

π

π

−

=−

=

=

∑

∑

yn+N/2=yn => OFDM symbols with 2 identical halves



( )

n

2 / 2 12

/ 220

Received OFDM symbol: r ,0 1Timing metric:

( )( ) ( )

( )

N

d m Nm

n N

P dM d R d r

R d

−

+ +=

≤ ≤ −

= = ∑

( )*

Z-N/2

/2 1

0

N −

∑ P(d)rn+d

{ }ˆ arg max( ( ))d

d=

{ }1 ˆˆ ( )

2 1 1/ 2 1

angle P d

f fT T T

επ

ε

=

< ⇒ ∆ < ⇒ − < ∆ <

d MTiming estimate:

Frequency estimate:


Van de Beek estimator

τ

Moving sum

L samples

Moving sum

L samples

|•|2

|•|2

Argmax

ZN

-1/2π

*

γ(.)

r(k)

Φ(.)

∆f

{ }

( )ˆ arg max ( ) ( )

1ˆ ˆ2

ML

ML MLf

τ γ θ θ

γ θπ

= −Φ

∆ = − ≺


Yang estimator (timing) (1)

Idea : exploit the fact that a timing error introduces a phase error at the FFT output which depends on the carrier number.

ADC

FFT

FFTWindowcontroller

Phaserotation

Corr.

Corr.

()2

()2

pilotes

filtre1/(z-1)

CoarseSymbolEstim.

(1)

(2)

DLL


Yang estimator (timing) (2)

( ) ( )( )( )

( )( )( )

2 2sin sin

,sin / sin /

SM M M M

π ε ξ π ε ξε ξ

π ε ξ π ε ξ

+ −= − + −


Bibliography (1)

Impact of synchronization errors on performances[BOU-01] S Bougeard “Modélisation du bruit de phase des oscillateurs

hyperfréquence et optimisation des systèmes de communications numériques”, Thèse de Doctorat, INSA Rennes, décembre 2001.

[MOE-97] M Moeneclaey “The effect of synchronisation errors on the performance of orthogonal frequency-division multiplex (OFDM) systems”, COST 254 conference, Toulouse, July 1997

[POL-94] T pollet, P Spruyt, M Moeneclaey “The BER performance of OFDM systems using non-synchronised sampling”

[POL-95] T Pollet, M Van Bladel, M Moeneclaey « BER sensitivity of OFDM systems to carrier frequency offset and Wiener phase noise », IEEE on COM , fev/mars/avril 1995

[SPE-97] M Speth, F Classen, H Meyr “Frame synchronization of OFDM systems in frequency selective fading channels”, VTC97

[YAN-00] B Yang, KB Letaief, RS Cheng, Z Cao “Timing recovery for OFDM transmission”, IEEE JSAC, Novembre 2000


Bibliography (2)

Synchronization algorithms suited to OFDM[BEE-97] JJ Van De Beek, M sandell, PO Borjesson « ML estimation of time and

frequency offset in OFDM systems », IEEE on signal processing, juillet 1997

[DAF-93] F Daffara, A Chouly “ML frequency detectors for orthogonal multicarrier systems”

[MOR-99] M Morelli, U Mengali “An improved frequency offset estimator for OFDM applications”, IEEE com letters, mars 1999

[MOO-94] PH Moose “A technique for orthogonal frequency division multiplexing frequency offset correction”,IEEE on COM, octobre 1994

[SCH-97]T Schmidl, D cox “robust frequency and timing synchronization for OFDM”, IEEE on COM, décembre 1997

[YAN-00] B Yang, KB Letaief, RS Cheng, Z Cao “Timing recovery for OFDM transmission”, IEEE JSAC, Novembre 2000

Documents

Advanced topics in OFDM systems - Sites personnels de ... / OFDM systems 37 Bibliography (2) ¾Synchronization algorithms suited to OFDM [BEE-97] JJ Van De Beek, M sandell, PO Borjesson