A new digital signal processing implementation of OFDM timing recovery

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A

New Digital Signal Processing

Implementation of

OFDM

Timing

Recovery

Yi-Ching Liao, Kwang-Cheng Chen

Institute of Communications Engineering, College of Electrical Engineering

National Taiwan University, Taipei, Taiwan, 10617,

R.O.C.

FAX:

+886-2-23683834

E-mail:

ycliao@santos

.

ee

.

ntu.

edu.

t w

, chenkcQcc

.

ee

.

ntu.

edu.

t w Abstract-In this paper, we present a novel and pure digi-

tal timing recovery scheme in orthogonal frequency drutsron mul- trplezrng (OFDM) system without additional pilots. This op- timum estimator is derived based on the maximum likeli- hood (ML) criterion with t h e decision directed approach. Sub-optimum schemes based on correlation extraction and SCCL algorithm are also developed. Simulations show that the estimation variance of t h e optimum scheme approaches zero for signal-to-noise ratio greater than 6 d B with num- ber of subcarriers equals t o 256. And the SCCL-based sub- optimum scheme is simpler and efficient t o apply in closed loop operation.

I . INTRODUCTION

In recent years, with the growing demand for wireless Internet access and wireless multimedia services, consid- erable attention has been focused on the design of broadband wireless communication system to support high rate

transmission up to tens or hundreds of Mbps with high

carrier frequency at the level of GHz. In such condition,

the channel is generally frequency selective and dispersive which causes serious performance degradation and impair- ment (inter-symbol interference, for instance) to reliable

transmission of a broadband signal. Adaptive equalization

techniques can be used to correct this distortion. However, for data rates up to tens of Mbps, adaptive equalization introduces practical difficulties for compact size and low power consumption.

OFDM which was proposed by Chang [lo] is a princi-

ple for transmitting information via a synthesis of mul-

tiple band-limited signals through a linear band-limited

channel without inter-symbol interference (ISI) and inter- channel interference (ICI). Enhancements such as using digital Fourier transform (DFT) to perform baseband mod-

ulation [ll] and demodulation and introducing cyclic prefix

to maintain the orthogonality [12] make OFDM more at- tractive.

The main advantage of OFDM is its capability of multi- fold increasing the symbol duration. With the increase of the number of subcarriers, the frequency selectivity of the channel may be reduced so that each subcarrier experi- ences flat fading. Consequently, the complexity of channel

This paper was supported by the Ministry of Education under the project contract 89-E-FA06-2-4.

equalization can be greatly reduced. OFDM has been sug- gested for a variety of applications such as asymmetric dig- ital subscriber loop (ADSL) and very high bit rate digital

subscriber loop (HDSL) [9] in telephone network, digital

audio broadcasting (DAB) [8] and digital video broadcast-

ing (DVB) (21 in broadcasting system and mobile communication systems [7]. And the most important application for future wireless multimedia is high speed data transmission over fading channels.

In OFDM, orthogonality is obtained by having the sub- carrier spacing as the baud rate, and having the F F T de- modulation process accumulate over exactly one OFDM symbol. However, the orthogonality can only occur if the demodulator clock is synchronized t o modulator and no

frequency offset is present. If a timing offset occurs, it will

cause a phase rotation which has been analyzed in [12].

Therefore, it is extremely important to estimate the exact

timing offset to force the receiver t o start its FFT processor

at the right time and mitigate the effect of phase rotation. Another critical problem of OFDM transceiver is that the synchronization scheme introduces significant complexity and is difficult to implement. It is extremely important to find out it synchronization scheme which is simple, effective and compatible with DSP realization of OFDM. Hence, we start from the ML-sense optimum approach for timing synchronization in ideal OFDM system, and then derive sub-optimum architectures from the optimum scheme

based on correlation extraction and SCCL algorithm [6]

which is simple and proved an effective tracking scheme

for symbol timing. In Section 11, the OFDM transmission

model is described. The synchronization methods both the optimum approach and the sub-optimum approaches are

presented iii Section 111. Performance simulation and con-

clusion are contained in Section

Iv

and V respectively.

11. OFDM SYSTEM MODEL

The OFDM transmitted baseband complex signal can be written as +m N-1 s ( t ) =

-

C

x l , k e J z n f k t g ( t

-

1 ~ ) (1) I = - 0 0 k = O N 0-7803-5718-3/00/$10.00 02000 IEEE. 1517 VTC2000

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where j k =

$,

T is the OFDM symbol duration, & , k is the

complex symbol modulating the k t h carrier with frequency f k , at the lth time slot of duration

T ,

N is the number of

subcarriers and g ( t ) the rectangular pulse given by

The block diagram of an OFDM transmission system is

illustrated in Fig. 1.

The transmitted data symbols are modulated on N sub-

carriers via an inverse FFT processor. After parallel to

serial conversion, digital to analog conversion and transmission pulse filtering, the complex baseband transmitted

signal s ( t ) is obtained. The channel response c ( 7 , t ) is as-

sumed to affect s ( t ) by a delay T E [O,T) and a complex,

AWGN, n ( t ) , i.e., c ( ~ , t ) = b(t - 7). At the receiving end,

the inverse signal processing is performed on the received

signal r ( t ) . As shown in Fig. 1, after analog to digital con-

version and serial to parallel conversion, N baseband time

domain samples are applied to a F F T processor to get the received data symbols { R ~ , o ,

..,

R L , N - I } .

111. SYNCHRONIZATION METHODS

A Optzmum Approach Based on M L Crzterzon

When a pnor information about the estimated param-

eter is not available, the maximum likelihood estimation (MLE) [4] is often applied. Different from conventional ML timing estimator for OFDM, we develope a new version of the optimum symbol timing offseet estimator based on ML criterion with decision directed approach. The proposed optimum timing estimation based on MLE is to test all possible candidates simultaneously in present OFDM symbol and choose the paremeter with the largest likelihood function value as the estimation for next symbol timing.

Under our assumption, the received complex signal is

function corresponds to the lth symbol.

where x , k is given by

The detailed derivation is shown in appendix. And the decision directed optimum symbol timing estimator and the

nth branch of it are shown in Fig. 2 and Fig. 3 respectively, The optimum symbol timing estimator can be im-

plemented by dividing the observation interval (O,T] into N

sub-intervals and each interval corresponds to a timing can-

didate T,, n E {0,1,

..,

N - 1). After all the log-likelihood

functions corresponding to each candidate are calculated, we choose the one with the largest log-likelihood function

as the estimated timing.

1 I

Fig. 2. Block diagram of the optimum synchronizer.

We can convert proposed optimum synchronization scheme to an equivalent digital counterpart. It proceeds by the following approximation.

N - 1 x , k ( 7 ) = r[ZN

+

n

+

T I ~ - J ~ ~ % (6) n=O N - 1 ( r ( t

+

T)I2dt = ( r [ l N

+

n

+

-r]Iz (7) n=O

where n ( t ) is the complex additive white Gaussian noise.

Based on ML criterion [5], we can derive the log-likelihood

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I I I

Fig. 3. Block diagram of the nth branch in the optimum synchronizer.

Where r [ l N

+

n] denotes the nth digital sample in the l t h

symbol period from the analog to digital converter at the receiver.

From this approximation,

X , ~ ( T )

is exactly the kth spec-

tral element of the discrete Fourier transform of { r [ l N

+

71, . . , r [ ( l + l) N

+

7

-

l]}. Thus, we can use F F T to calcu-

late

X , ~ ( T )

and a moving sum unit to calculate the power

in ( 7 ) . The pure DSP realization of the proposed opti-

mum synchronization is illustrated in Fig. 4. The proposed

structure can be operated as an open loop system.

: I L I I : I

...

~ ... !??:I!."!":

Fig. 4. Block diagram of the digital realization of the n f h branch in the optimum synchronizer.

B. Sub-optimum Approach Based on Correlation Extrac-

As shown in Fig. 4, the log-likelihood function in each

branch of the optimum synchronizer is exactly composed of two individual parts: one is the correlation calculating part,

the other is the power calculating part. The relationship

of the two constituting part is tion

where ALc(n) denotes the output of the correlation calculating part and ALp(n) is the output of the power calculating part. Consequently, it motivates us to investigate the waveforms of the three log-likelihood functions in equation (8) to see if we can extract any information for a simplified implementation of the optimum synchronizer. A simula-

0-7803-5718-3/00/$10.00 02000 IEEE. 1519

tion result of the waveforms mentioned above with the as-

sumption of 64 subcarriers and QPSK modulation without

noise is shown in Fig. 5.

*,2t

...

i

...

;

...

j

... i ...

_2;"

-

1

... ... ... ... ...

:

i

... i ... 2: ... i ... 4 ... 0 I W 200 300 U X I so0 I- n- (OFDM wri

Fig. 5. The waveforms of the three log-likelihood functions with N=64, QPSK modulation without noise, timing offset= 1/16

symbol period.

After comparing the waveforms of the three Iog-

likelihood functions, we can find that Aic(n)l like AL(n),

generates periodic peak with period equals t o the OFDM symbol duration, although it contains the perturbation of

signal power which equals t o ALp(n). Therefore, a sub-

optimum scheme which maximizes ALc(n) t o find the es-

timated timing offset can be realized with some perfor-

mance/complexity trade off. The simplified implementa-

tion can be achieved by retaining the correlation calculat,-

ing part and omit the power calculating part in Fig. 4.

C. Sub-optimum Approach Based on SCCL Algorithm a sub-optimum tracking algorithm, SCCL.

In order t o simplify the above implementation, we adopt

The Algorithm of SCCL:

ft

separates the whole observation interval t o many sub-

observation intervals.

In each sub-observation interval, the SCCL inspects three

possible linked candidates and choose the one with the maximum likelihood function as the winner.

In next sub-interval, the winner and two of its linked neighboring candidates are inspected and the one with the maximum likelihood function is chosen as the new winner.

The process proceeds.

Please note that the likelihood function in

SCCL

is ex-

actly the log-likelihood function in the above optimum scheme. What we need t o do is t o change the the param-

eter from T t o the timing offset candidate i l which is the

estimation of the l t h sub-observation interval results form

the previous symbol. 2 N - l

-

C

Ir(lN

+

n

+

n)12

A L ( t

+

j ) = S[Ri,k(il + j ) X , k ( + ~

+

j ) ] (9) k=O N - 1 n=O VTC2000

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Where i l E {O,l,

...,

N

-

1) is the timing estimation

of SCCL at the l t h sub-observation interval. And J E

{ - l , O , l } . Also note that x , k ( + l

+

j ) just denotes the correlation of the advanced version of the received signal

corresponding to the (+l

+

j ) t h timing candidate with the

kth subcarrier at the l f h symbol period at the receiving

end. Assume the sub-observation interval is T . The block

diagram of the proposed SCCL scheme is shown in Fig.

6. Moreover, because the SCCL-based synchronizer only

needs to calculate tree likelihood function values in a sym-

bol duration, the implementation in Fig. 6 can be greatly

simplified by adopting single branch and reusing the func- tional blocks of the demodulator.

= a r q A 1 u

{ZJI = - l . i l . l }

uduunce

rl+l = 71 + I

Fig. 6. The block diagram of the proposed SCCL scheme.

IV. PERFORMANCE SIMULATION

A . The Optimum Synchronizer in open loop operation

An OFDM system consisting of 64 subcarriers and the

modulation scheme of QPSK is considered here. White,

complex Gaussian noise is added and the normalized mean squared error as a function of the signal to noise ratio is simulated. The signal to noise ratio is defined as

where E { l s ( k ) I 2 } and E{[n(k)12} denote the average sym-

bol power and average noise power respectively. Each value of SNR is simulated for 10,000 symbols. The simulation re-

sults are shown in Fig. 7, where we use the sub-optimum

scheme which maximizes h;,(n) as a reference system.

The tendency of the curves of both the optimum scheme and the sub-optimum scheme basically coincides with the

adopted decision directed approach. For SNR

<

5 dB,

the normalized mean squared error of the estimated timing

maintains a level of about

lo-'

of (l/symbol period)'. This

tallies with the fact that the feed-backed decision in such SNR condition has a significant error within. While for

SNR

>

7 dB, the normalized mean squared error of the

estimated timing decreases sharp, and for SNR

>

9 dB, the

simulated normalized mean squared error of the estimated timing equals to zero. This shows that when the error in the feed-backed decision less than a specific level, we can find a dramatically elevation of the performance of the optimum synchronizer. Compared to the optimum scheme, the sub- optimum implementation has little performance difference

USE n SNR m h A W N 1 00 - I I

;

10'' 4 I \

\

I

t

1

Fig. 7. Normalized mean squared error of the timing estimation vs SNR in AWGN.

for low SNR environment, while for high SNR, it is about

2 dB in average inferior to the optimum synchronizer.

Another simulation about the comparison of normalized mean squared error of the estimated timing versus SNR

and the number of subcarriers is illustrated in Fig. 8. The

simulation result shows that the proposed synchronizer can

get superior performance with a larger number of subcar-

riers. The performance gap between systems with N=64, N=128, and N=256 is about 1.5 dB/double.

0-7803-571 8-3/00/$10.00 02OOO IEEE. 1520

I i \ i

= 10''

Fig. 8. Normalized mean squared error of the timing estimation vs

SNR and No. of subcarriers in AWGN.

B. SCCL Synchronizer in closed loop operation

The mean acquisition time of the proposed SCCL-based

synchronizer operates in a closed loop environment with

initial residual timing offsets of 17, 20 and 31 samples is

shown in Fig. 9. Where we assume SNR=O dB, 2048 sub-

carriers and QPSK modulation in AWGN channel. Com-

pared to [3], the proposed SCCL-based synchronizer has better performance with 17 samples of residual timing offset and is simpler with the single branch implementation.

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V. CONCLUSION

In this paper, we have proposed a novel symbol tim-

ing recovery scheme for OFDM system with a pure digital realization. The proposed optimum scheme is derived based on maximum likelihood criterion with decision feed- back and requires no further pilots to assist the estimation. Two sub-optimum schemes, one based on correlation extraction, while the other based on the SCCL algorithm are also developed. Simulation results show that the proposed optimum scheme has excellent performance in estimation

variance for SNR

>

6 dB in AWGN and the correlation ex-

traction based sub-optimum approach has simplified struc- ture at the price around 2 dB away from optimal value. The SCCL-based sub-optimum approach with simpler implementation is also shown to be effective in closed loop operation.

APPENDIX ’

Based on ML criterion, The log-likelihood function cor-

responds to the l t h symbol can be derived as follows.

1 1 A ~ ( T ) = exp{--

/

lr(t)

-

s ( t - r)l”t) NO To NO To = exp{--

/

Ir(t)\’cit

+

&%[Lo

T ( t ) S * ( t -

M I 1

(-4.1)

where NO is the one-sided power spectrum density of noise,

To

is the observation interval and we assume To E [lT

+

T , (1

+

M ) T

+ T I ,

with M a positive integer. And the log

likelihood function is

we denote the first term in (A.2) by

A i c ( ~ )

and it can be

further derived.

g ( t

-

T

-

1T)dt) M-1 N-1

Based on the decision directed approach, which is to as-

sume X L , ~ has been estimated in the absence of demodu-

lation errors, i.e., X 1 , k =

R ! , ~ ( T ) ,

where R l , k ( ~ ) is the de- modulated symbol from F F T processor corresponding to

the timing delay T . Therefore the log-likelihood function

becomes Fig 111

PI

[31 141 [51 161 (‘4.3) 191 where l ’ i , k ( ~ ) is given by (1+1)T & ( T ) =

lT

r(t

+

T ) e - J i = f k t d t (‘4.4) [lo1 1111

Please note, in the above derivations, we neglect all the multiplicative and constant terms since they have no influ- ence on the estimation of the parameter. And if we choose

M to be 1, the log-likelihood function becomes

[I21

Ir(t

+

7-))2dt

k=O

(44.5)

9. Mean acquisition time of SCCL with residual estimation error

= 17, 20, 31 samples and SNR = 0 d B in AWGN.

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