### Joint Weighted Least Squares Estimation of

### Frequency and Timing Offset

### for

### OFDM Systems over Fading Channels

Pei-Yun Tsai, Hsin-Yu

### Kang and

### Tzi-Dar Chiueh

...,.. ,. i , . and Graduate Institute of Electronics Engineering,i Department

### of

Electrical### Engineering

! , -

### .

. . National Taiwan University, Taipei, Taiwan, 10617.

Absfracr-This paper presents an algorithm for joint ferences are then averaged among all pilot subcarriers or
estimation of carrier frequency Offset and timing fre- over m w a l **OFDM **svmhok to =<timate the carrier fre-

## ~~~.

~~~~~~ ~ - . ..~~~~~~~~~~~~~. . near the**Cram&-Rao bound in**the variance of estimation

**errors: Simulations of several estimation algorithms in a **

estimation. Also the complex data of the pilot subcarriers, instead of their phases, are averaged. However, these multipath fading channel show'that this algorithm provides

the most accurate estimation in both the carrier frequency offset and the timing frequency offset.

. . ...

2 . **I. INTRODUCTION **

,'The OFDM modulation technique offers an attrac-
tive solution to high-rate data access for its robust-
ness against frequency-selective multipath fading and its
simple equalization scheme. Moreover, it is also very
efficient in spectrum utilization since the spectra of
adjacentwbcarriers overlap. OFDM has therefore been
adopted in several standards such as DVB-T, VDSL,
and'IEEE'802.1 la. However, it is also well known that
OFDM systems are veiy sensitive to synchronization
ernjrs,. which can cause * .inter-carrier interfejerence *(ICI)
and degrade system performance.

### ,,

All the OFDM standards mentioned above have ded- icated pilot subcarriers to facilitate**the**synchronization t&ks in the receivers. Various pilot-aided carrier fre- quency offset estimation and timing frequency offset estimation algorithms have been proposed [I], 121,

### 131,

141, 151,*[71. Most of them utilized the detected phase of the received frequency-domain complex data in the pilot subcarriers. The phase shift in the received complex data due to carrier frequency offset is identical*

**[ 6 ] ,**### in all subcarriers of an

OFDM symbol**if IC1 is**neglected. Classen[l], Kapoor[2] and Moose(3J have taken advan- tage of this fact to build their algorithms. The phases of the received pilot-subcarrier data are extracted and the

**phase **difference between two consecutive symbols in a
pilot subcarrier is computed in [I] and

**[Z]. **

The phase dif-
### -

algorithms may produce biased estimation when there exists timing frequency offset, which occurs frequently in most communication systems.

Timing frequency offset, unlike carrier frequency off-
set, causes phase shifts that are proportional to the
subcarrier index **as **well **as **the offset itself. A very
popular class of schemes estimate the timing offset by
computing a slope from the plot of measured pilot sub-
carrier phases versus pilot subcarrier index [41, 151, [61,
[7]. In 141, the slope is obtained by averaging over phase
differences between pairs of adjacent pilot subcarriers.
On the other hand, both the phase and magnitude of
the pilot subcarriers in a single OFDM symbol are used
in the slope calculation in [51. In **161 and 171 linear **
regression is adopted in the estimation of **the slope. All **

of these four algorithms examine only the phases of the
pilot subcarriers in one symbol, which are influenced by
not only the timing offset but also frequency selective
fading. Therefore, the estimated timing frequency offset
can **be **severely degraded. One way to get around this
problem is to first take the phase difference of the same
pilot subcarrier in two adjacent

**OFDM **

symbols so that
the frequency selective fading, being same in the two
symbols, will he canceled.
Sliskovic[S] proposed to jointly estimate the carrier frequency offset and the timing frequency offset. In addition, a weight is assigned to each pilot subcarrier dur- ing averaging since the subcarriers suffer from different levels of fading and thus may have different

**SNR. **

The
timing frequency offset is estimated by first computing
the phase difference between a pair of adjacent pilot
**SNR.**

suhcarriers, computing the. difference of that amount in two OFDM symbols, and then weighted averaging. The carrier frequency offset can he estimated by first calculating the pilot-suhcarrier phase difference between two OFDM symbols, removing the quantity contributed by the previously estimated timing frequency offset, and finally weighted averaging. Thus

### in

this algorithm the accuracy of carrier frequency offset estimation depends on not only the noise contained in the measurements hut also the accuracy of timing frequency offset estimation. In this paper, a weighted least squares algorithm, also based on the pilot-aided scheme, is proposed. Both AWGN and multipath fading channels are considered and the optimal weight assignment that achieves the Cramtr- Rao bound is presented.The rest of the paper is organized as follows. The OFDM signals with both carrier and timing synchro- nization errors are analyzed in Section 11. The proposed weighted least squares algorithm is presented in Sec- tion 111. In Section IV, the simulation results that demon- strate the potential of the proposed method comparing to others are shown. Finally, a conclusion is given in Section V.

**11. OFDM SIGNALS W I T H SYNCHRONIZATION **
**ERRORS **

An OFDM baseband signal is given by modulating N
complex data * ( A h + ) *using the

**inverse discrete F o u r i e r***(IDFI') on N subcarriers*

**t r a n s f o r m****as **

shown in Fig. I .
The suhcamiers spacing is 1/T, which is the inverse of
the symbol duration. The n-th time-domain sample of
the i-th symbol can be expressed as
**N I 2 *** Ak,i&2n"k/N *n =

_{-N .. }

_{-N .. }

_{0 : }### .

,,%,-I*d , . = -*

*''*

**k = - N / 2 + 1****N **

**N**

**(1)**Note that in order to combat

**inter-symbol interference**(1%). a cyclic prefix of

**Ng **

samples is inserted at the
beginning of a symbol.
**Ng**

Assume that the received signal is influenced by a mul- tipath fading channel with a channel impulse response

*h ( t ) *=

**Ch,(t). **

**Ch,(t).**

*6 ( t*-

*r,(t)).*

**(2)**

The amplitude and delay of the r-th path

**are **

denoted by
* h , ( t ) *and

*~ ~ ( t ) . *

Consequently after convolving with the
channel impulse response, the received signal takes the
form of
*Z ( t )*=

**Eh$) **

' **Eh$)**

*-*

**d ( t**

**r r ( t ) )**### +

*" ( t ) ,*

**(3)**

**d-,,,**I

**Fig. I.**

**subcarrien.**

**System model of a **OF'DM **communication system with ****N **

**In **addition to multipath channel fading, oscillator
mismatch and Doppler effect inflict the received signal
with carrier frequency offset and timing frequency offset.'
If some carrier frequency offset A f and timing frequency
offset *6.T, *exist, where *T, is *the sampling interval in the
transmitter, the m-th received sample of the 1-th symbol
(counting from the end of the cyclic prefix) is given by

**Zm,l **= *z ( t ) , ***P A f f It=lN,(l+6)T,+N,(l+6)T.+m(l+6)T. ** I

m = 0,1,.

### .. ,

*N *

- **1,**

**(4) **

where

**NT **

is equal to N **NT**

### +

**N o . **

The receiver then drops
the cyclic prefix and passes the **N o .**

### N

samples to an**N - **

point **N -**

*(DFT) block. After DFT, the complex data on the*

**discrete F o u r i e r****rmnsfom****k-th **

subcarrier consists of three
components: signal term **k-th**

*St,!,*interference term

*and*

**I k , l**noise term * V k J . * . ,

**z k , l ****sk,L ****&,t **

### +

**vh.1.**

**( 5 )**If * I f k *represents the channel complex response

### on

subcamier**k, **

then the signal term can he expressed **k,**

**as**

**sk,l **

= **sk,l**

**H , .****e j W i ( l N r + N . ) b t t**

**( 6 ) **

**&k ****(1 **f * 6 ) *'

*-*

**(&P)***(7)*

**k**### .

**e I n + h k ( l - + )**### .

**S ( n d k k ) ,**where

Note that **f **= Af

### .

T is the normalized frequencythe distonion of the kth-subcarrier data in phase and
magnitude, respectively. Moreover, the synchronization
error destroys the orthogonality between subcarriers and
the IC1 term, * Z k , I 9 *is induced and it is given by.

* z k , l *=

**A ~ , ~ ,****H,,**### .

**e j 2 n k ( l N r + i v g ) O p p**' * p = - N / Z + l , p # k * (9)

### .

**, p d p d - k )**

### .

**S ( T $ p k )****Note **

that in the case with small 6 and

**E ,****S(T$kk)****is close to 1 and s(?iQpk)**is near zero, and the IC1 term can he ignored. Consider the case when only the

*carrier frequency offset exists, Sk.1 will he rotated by (in*addition to

**H k )**(10) 1

**2 s ; ( l N r **

**2 s ; ( l N r**

### +

**N s )**### +

m(1 - -)**N ' **

Note that this phase is independent of * k and is identical *
in every subcarrier.

**On **

the contrary, in the case when
only some timing frequency offset 6 exists, *will be rotated by (in addition to*

**S k , l**

**H k )*** 6k * 1

### Za-(/Nr+

*- -)*

**N g ) + r r 6 k ( l****(11)**

N N ' .

which is proportional to the subcarrier index

**k **

as well
**k**

*as the symbol index 1.*

111.

**JOINT **

**ESTIMATION ALGORITHMS**In the AWGN channel,

*=*

**H k**

**1 for all le and****&,I**is distorted in phase and amplitude caused only by
synchronization emors. This effect is clearly seen in
Fig. 2, which illustrates the phases of suhcarrier data in
two consecutive OFDM symbols when they are distorted
by noise, carrier frequency offset, and timing frequency
offset. The carrier frequency offset is *0.05 subcarrier *

spacing

**and **

the timing frequency offset is **100**p,p.m. The received data contain IC1 and noise, therefore the extracted phases deviate from the two ideal straight lines. From the extracted subcarrier data phases, the

**least-****squares **

**(LS) **

method can estimate the two straight lines,
from which the carrier frequency offset and the timing
frequency offset can be derived. In the multipath fading
channel, the complex channel response * H k *distorts the
received data in magnitude and in phase. Moreover,

**signals on the deeply-faded subcarriers have low**

**S N R , **

**S N R ,**

while those on the subcarriers with little fading have high

**SNR. **

Obviously, the data phases of the subcarriers with
**higher SNR are much more reliable than those with lower**

**S N R **

**S N R**

### in

the estimation process.We believe that the concept of weighting the data in each subcarrier is necessary because data of deeply- faded subcarriers should he assigned smaller weights to

**Pilot Phase ****(~=0.05 ****and S = l w m ) **

**sub-carrier **index

**Fig. 2. Phases of rubcarrier data in two OFDM symbols with 6 **= **0.05. **
* 6 *=

**100ppm. and Gaussian noise.**

minimize their adverse effect on **estimation accuracy. In **
addition, estimation based **on **the phase difference across
two OFDM symbols has the advantage of removing
the common channel phase. Finally, linear regression
provides better estimation since it can find simultane-
ously the best slope and intercept in terms of least.
squared ermr. In light of all the'above considerations,
we propose a weighted least squares **(WLS) **algorithm
for joint estimation of the carrier frequency offset and
timing frequency offset.

~ e t **y **= [yo **y1 **

### . . .

yJ-*represents J pilot Sukar- rier phase differences between two consecutive OFDM*

**, I T****symbols and y,**=

**$(Z=i,iZ;i,,-l), **

where **$(Z=i,iZ;i,,-l),**

### $0

is the phase of its argument and**zj**is the j-th pilot subcarrier index. Also let b =

*[m,d]'*be an estimation of the slope m and the intercept d of the optimal straight line in the

y, **versus pilot suhcrrier index plot, then y = **

**Xb **

### +

**n, **

where

The phase noise component

**n **

=

**[eo****el**### .

### .

### .

**e**

**~***rep-*

**-**

**~**

**]***resents the errors in phase contributed by IC1 and AWGN*

**~**and * e j * =

**yj**-

**$(Sz,,tS;2,1-,).**The Gauss-Markov

solution for minimum-variance least-squares estimation, which achieves the Cram&-Rao bound [9], is given by

**G **

= **G**

**(XTWX)-lX*Wy,' **

**(13)**

For simplicity, assume that the residual synchroniza-
tion error is so small that IC1 can he neglected. The off-
diagonal terms of **E{nnT} **are contributed by IC1 and
can be ignored, hence the matrix inverse operation can he
avoided and the j-th diagonal component of W, denoted

**as wj, **is given by

**(14) **

W . - **o - 2 **

3 - * e j : *
where

**02; **

is the variance of ej.
**02;**

Assume that the channel response is almost static in the duration of two symbols, then y j is approximated by.

Y j =

**$(ZZj,I **

' **$(ZZj,I**

**z:+l) **

### =

*$(s,j,l *

' s;j,l-l ### +

**SZjJ**

### 'v;,,l-l

**+ ~ ~ j , l ' ~ ; , , ~ ~ l + ~ ~ , , f ' ~ ~ j , l - l ~**=

**$(EsIHzj/**

**3 '****'S2(.4.,,.,)**

### +

*SZj,l*'

*V&] *

### ,+

V,,J '### s:&l

### +

*'*

**Vz;J****V,:,f-l)**

**z e j 2 9 r 3 + .**. = . (15)

The first term in the right-hand side of the above equation
is the desired output component with a phase related to
the two synchronization errors * L *and

**6 **

in *and*

**&,,zj**### a

power equal to the squared received signal power. In high-SNR cases; the noise component in y j depends mainly on the two signal-noise product terms, whose power scales with**E,NoIH,,12. **

Thus, assuming high
**E,NoIH,,12.**

**SNR, wj can he approximated by [IO]**In the AWGN channel, wj is proportional to the inverse of !he Guassian

### noise

power in the j-th pilot subcarrier. On the other h'and, in the multipath channelw j scales with * 'IHSjIz *and'inversely with the Guassian
noise power., In this case, the carrier frequency offset

**L****as: **well **as the timing frequency offset **

**6 **

can ### be

derived**from'(l3) **andwe **have **

and

**IV. **COMPARISON OF SIMULATED PERFORMANCE
Assume that the carrier frequency offset and the timing
frequency offset both exist, which is the typical d i e in
communication systems. The **root-mean-square **

**(RMS) **

errors in the estimated offsets using different algorithms
over a multipath fading channel are shown in Fig. 3
and Fig. 4. A carrier frequency error **t **= 0.05 and

### a

timing frequency error**6 **

= 20 p.p.m. are introduced in
the simulation. As can he seen from the two figures,
the proposed **WLS**algorithm achieves, in most cases, the minimum RMS error in both carrier frequency offset estimation and timing frequency offset estimation. Note that even though a small-IC1 assumption 'is made in deriving the WLS joint estimation algorithm, the residual sydchronization error is so small in the tracking mode that the

**IC1 **

can indeed he neglected.
V. CONCLUSION

In this paper, we proposed a joint estimation al- gorithm that can estimates both the carrier frequency offset and the timing frequency offset. With a weighted least squares technique, the proposed algorithm indeed generates offset estimates with minimum

**R M S **

errors.
Therefore, we believe that this algorithm'can greatly en-
hance the performance of OFDM-based communication
receivers by reducing residual synchronization error and
thus suppressing **ICI.**

**Carrier frequency onset estimation **
,... .

### Ih

**16**18 22 24

**26**

**J8**

**SNR (dB) **
Fig. 3.

algorithms

Timing **frequency oilset estimalion **

* Fig. 4 . *
algorithms.

### .

**I .**Simiilated timing frequency estimation **ermn by different **

R E F E R E N C E S

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**for**OFDM systems suitable

**for communication over frequency se-**kclive fading

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**of 19981EEE 48lh Vuhiwlor Trrh,!alogy Coq'cruncr. ****vol. ****3. ****1998. **

**pp. 2077-2080. **

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