1. No category

FREQUENCY OFFSET ESTIMATION IN OFDM USING
SAMPLE COVARIANCE
G. Levin
D. Wulich
Communications Laboratory
Department of Electrical and Computer Engineering
Ben-Gurion University of the Negev
Beer-Sheva, ISRAEL
Tel: ++972-7-646 1537; Fax: ++972-7-6472949; E-Mail: [email protected]
ABSTRACT
In this paper we consider the performance of the
Parametric Weighted Least Square (lW?LS) algorithm for
Orthogonal
frequency
oflset
estimation
in
Frequency-Division Multiplexing (OFDM). No training
sequence or redundant information is needed. The
correlation between symbols is achieved due to channel
spreading, which is assumed to be a Linear Time
Invariant (ZTfl jilter with known impulse response. We
show that the frequency estimator based on a PWLS
algorithm may be used in a tracking mode of the OFDM
receivers.
INTRODUCTION
Orthogonal frequency division multiplexing (OFDM)
systems have recently gained increased interest. They are
widely used in HDSL, DAB and for digital transmission at
HF. One of the main problems in the design of an OFDM
receiver is the mismatch of the oscillators in the
transmitter and the receiver. The demodulation of a signal
with an offset in the carrier frequency can cause a high bit
error rate due to degradation in the perilorrnance of the
symbol synchronizer and to Inter Channel Interference
(ICI) [1].
Many ffequency offset compensation methods have been
proposed recently; a training sequence [1,2,3,4] or
redundant information [5] is required in many of them. In
this paper we consider the Parametric Weighted Least
Square (PWLS) algorithm for the frequency offset
estimation [6, 7] based on matching, in the weighted least
square sense, a sequence of sample correlation to the
theoretical values. No training sequence or redundant
information is needed. A correlation between symbols is
achieved due to channel influence. We suppose that the
channel is well described by a Linear Time Invariant (LTI)
system and has an a priori known impulse response.
The problem of the estimation of the random amplitude
complex exponent frequency is very well known in
estimation theory and has many solutions [6,8,9]. Here we
use, as stated, the Parametric Weighted Least Square
(PWLS) algorithm [6].
It is always very interesting to establish the ultimate
statistical performance that can be achieved by a given
estimation method. The Cramer-Rao Bound (CRB) [10,
11] has proved to be a usefhl tool because it provides a
lower bound on the covariance of any unbiased estimate of
the parameters vector in question. We use CRB as a
criterion of successfid estimation.
will
THE OFDM SYSTEM BASEBAND MODEL
Fig. 1 illustrates the discrete-time baseband equivalent
OFDM system. Complex data symbols ~k transformed by
the Inverse Discrete Fourier Transform (IDFT) yield a
string of symbols bn , which represent the baseband
equivalent of an OFDM signal with N-parallel subcarriers.
Symbols bn are serially transmitted over a discrete-time
channel whose impulse response h(z)
is known and
assumed to be of length <L. Any mathematical model
can be exploited for channel description; here we use the
Moving Average (MA) process as a most commonly used
model for the LTI channel. The frequency difference
between the local oscillators in the transmitter and the
receiver is expressed by cq which denotes the normalized
frequency as a fraction of the intercarrier frequency
spacing l/N.
In the following analysis we assume that the channel is a
real-valued linear fiker and the transmitted signal is
disturbed only by the complex additive white Gaussian
noise (AWGN) Wn with power density iV/2.
The purpose of the frequency offset estimator is to
evaluate u on the basis of receiving symbols cin in order
to preset the Carrier Recovery System, which then
compensates the residual fi-equency offset and tracks the
random phase to make coherent demodulation possible. In
particular, as a result of such a compensation and tracking
process the ICI is substantially reduced.
0-7803-4902-4/98/$10.00 (c) 1998 IEEE
%,
h
Channel
Cn
d.
h(r)
32’
Fig. 1 Discrete-Time Baseband Equivalent OFDM System
PROBLEM DEFINITION
We assume that symbols ak are independent and jointly
uniformly distributed. Symbols b = [b.,...2 bN_ 1] being a
linear combination of a = [a.,... ,aN_l ] have for large IV
an approximately Gaussian distribution with independent
real and imaginary parts. Moreover, bn constitutes an
i.i.d. sequence. Random sequence Cn is a response of the
LTI filter to bn and therefore forms a colored Gaussian
sequence with independent real and imaginary parts (due
to the real-valued coefficients of the filter).
As assumed by the suggested model, symbols dn, which
act on the input of the frequency offset estimator, are given
by:
dn = Cnexp{–j2mn}
+ Wn
(1)
where:
L–1
Cn = ~bkh(n – k) =
k=O
L–1
L–1
= ~~oRe{bk }h(n - k) +jk~oh{bk)h(n
- k)
(2)
Rdd (k - n)= E{dnd~ } =
No
= Rcc(k -n)exp{j2m(k
-n)) + ~d(k
-n)
(3)
-L+l<(k-n)<L-l
where:
Rcc(k -n)=
~{CnC;
} =
-n]) = cr~Rhh(k -n)
= G~l_~~>~l)h(l+[k
–L+l<(k–n)<L–l
(4)
cr~=E{bn
2}= NE{ak2}=N&
The unbiased sample autocorrelation of dn equals:
i& (t)= —
1
M–t
M-t
Z dnd;+t
~=~
l<t<L–1
(5)
where &fdenotes the number of data points.
The frequency offset estimation method proposed herein is
based on matching sequence fidd (t) to sequence Rdd (t)
Model (1) will be called here the Moving Average
Exponent (MAEXP) process. Frequency estimation of
such processes is described in [6, 7, 8]. In this paper, we
propose to explore the PWLS approach. Such a method
of estimating the ffequency offset is a direct one and there
is no need for decision feedback, which has a potential of
error propagation.
d.
THE ESTIMATION ALGORITHM
is stationary and zero-mean (E{ak }= O). Its
autocorrelation function is expressed by:
with t=(k-n).
Parameter
a is thus estimated by solving the
following least square minimization problem:
~(~) = [~dd - R&j (@]H~&j
& = min J(a)
a
I&
where &d = “dd (1),..., fidd (L – 1)~
- R& (a)]
(6)
is the vector of
the sample autocorrelation and Rdd (cx) denotes the
vector of autocorrelation lags corresponding to a. W is a
positive definite complex weighted matrix. It is shown in
[6], that CRB can be attained by appropriate choice of this
matrix.
0-7803-4902-4/98/$10.00 (c) 1998 IEEE
There is a wide variety of computationally efficient
methods for solving (6). Here we use the Newton-Gauss
minimization algorithm. This method uses exact first-order
derivatives and approximated Hessian to minimize
criterion (6) as described in [7].
Fig. 2 shows the block diagram of the frequency offset
estimator. The received symbols are stored in a buffer of
length L. Then the sample autocorrelation is computed
according to (5). This covariance is then updated with any
new symbol, which appears at the input.
lim A4S(n,k) =
M+ca
.
L-1
= ~=~L+?$
(t)&jd(t
lim AZl(n,k)
Weighted Matrix derivation:
It is shown in [6, 7] that if
X{
+
L-1
Z &*c(t)Rcc(t
t=–L+l
+ [k – u]) +
(11)
Z&c(k – n)o~ + o~d(k –n)]
where o:- AWGN variance.
It can be seen from (5), (8) and(11) that & is an unbiased
d%d(~).
.
,
~
The exact CRB derivation:
-
Denote:
(7)
- R&j)(&~
Vm= Re{dm }
- R~J*)]-l
E{(6 - a)2} = [DWD~]-1
(12)
‘mi-M = ‘{dm]
then:
(8)
According to the definitions of &d and Rdd (equations
(3, 4, 5)) it follows that the elements of the matrix S are:
S(n, k) =
n’Z=o,l,...,l-l
A new real process Vz 1= 0,1, ....2ikf– 1 is Gaussian
for large N and holds:
E(VmVm+M) = O
Therefore
= E{[&d (n) - Rdd (n)][&d (k) - R&j (k)]* } =
the
nZ=o,l,..., Al-l
Fisher Information
(13)
Matrix
of
v~
(1= 0,1, ....2M- 1)may be derived as follows:
1
(9)
‘(M-n)(M-k)x
F(a)
M-n M-k
x ~~1 ~~1 &jd (/ - i?Z)&
k,n=l..
x
estimator of a with standard deviation cr~ cc 0(—
&)
~=~-1=
= [E{(Ji&
= e-~2zaIk-nl
A4+co
DERIVATION OF THE PROPOSED METHOD
Statistical analysis of the suggested method when dn is
described by an auto-regressive process is carried out in
[6, 7, 8]. Here we just compute an appropriate weighted
matrix W for the MAEXP process and derive the exact
Cramer-Rao Bound forgiven frequency offset estimator.
da
k - n)
k,n=l.. L-l
By substitution (3) in (10) we receive:
STATISTICAL ANALYSES AND cm
~ =
+
.
(/ -
m + k -n)
L–l
In order to check the asymptotic behavior of S, which
actually is a covariance matrix of the sample covariance
estimator, we assume that A4 + m (and of course
L<< M). So the following is valid:
= ~trace{r
-lzr-l~}
da
(14)
where r is the correlation matrix of real process V1and is
defined as:
r(n,Z) = E{ VIVn);
n,l= 0...2A1-1
(15)
The CRB equals therefore:
Cl/B(a)
=F(a)-1
SIMULATION
AND NUMERICAL
(16)
RESULTS
As it has been shown in the previous section, the weighted
matrix W depends on the estimated parameter a . Two
ways to perform the minimization in order to get the best
performance of the estimator are considered.
0-7803-4902-4/98/$10.00 (c) 1998 IEEE
r ------
-------
------
-------
------
! Frequency Offset Estimator
1
1
1
:
d!nl
1
1
I
:
>
Buffer
of length L
---.--
------
. . . . . . ------
Sample Covariance
------
. . . . . . .------,
Sample Covariance
Accumulator
Sample Covariance
Calculator
[1
1
II
I
Update
‘ fl~~(t):
li
I
1
CoVariance
Vector
I
i
1
I
I
I
I
1
~
I
I
1
:
I
I
I
&d(t)
1
I
I
I
i
#
I
:
1
Newton-Gauss
Minimization
;
1
1
1--------------------------------------------
------------------------------
.
Fig. 2 Block diagram of the PWLS based Estimator
CONCLUSIONS
1)
To compute the first derivative and approximated
Hessian of cost fimetion J(cr), when W = W(a) and
apply the Newton-Gauss Algorithm for minimization.
2) To set W=l, identity matrix, and solve the problem. To
compute weighted matrix W for next step by
substitution of the estimated value of & from the
previous one. Repeat the steps until convergence.
Both ways require computing the weighted matrix at least
once for all steps. This task may be very complicated for
real time applications and is extremely difficult to
implement in the systems with a high bit rate. We propose
here to use suboptimal estimation method, when ?7’=1
aJways. The simulations shown later indicate that the
estimation accuracy degrades negligibly regarding CRB,
while the algorithm becomes much faster.
The simulation was performed for the following
conditions: N=32, a = 0.2. The symbols a represent
QPSK modulation format and a second order Butterworth
filter is used to represent the LTI channel.
Fig. 3a and Fig, 3b show the estimated mean and the
variance of the estimation error as a function of the
number of received symbols Cn for SNR=5dB. In Fig. 3b
the CRB is also shown for comparison. As can be seen,
accuracy degradation because of a suboptirnal estimation
is not more than 3dB regarding CRB.
Fig. 4a and Fig. 4b show the estimated mean and the
variance of the estimated error as a fiumtion of SNR after
300 symbols have been transmitted. The variance of
estimation error does not converge to zero and reaches a
steady state for large values of SNR The CRB is also
shown for comparison. The difference is not more than
3dB.
We have presented a PWLS estimator of frequency offset
in OFDM. No training or redundant information is needed,
The fi-equency offset compensation is obtained by direct
estimation and therefore is not affected by error
propagation typical of decision feedback estimators.
A computation was performed for an assumption that the
channel is represented by an LTI filter described well by
an MA process with known coefficients and that the
distortion is caused by the AWGN only. For instance, such
an assumption maybe valid in HDSL.
We presented a suboptimal, due to non-optimal choice of
the weighed matrix W, frequency offset estimator. The
computational complexity of such an estimator is low
while its performance is not more than 3dB worst than
CRB.
Estimated fi-equency offset may be used as frequency
preset for conventional carrier recovery systems working
in the phase tracking mode. Consequently coherent OFDM
may be obtained. If differential encoding is used, such an
approach may prevent phase shift accumulation and
decrease the bit error rate.
ACKNOWLEDGMENT
The authors would like to thanks Dr. J. Frances for his
suggestions and help.
0-7803-4902-4/98/$10.00 (c) 1998 IEEE
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02008
-0.2006
---
-- CRB
--- Error Variance
True Parameter
Mean
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,&
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300
400
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600
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9m
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— ——
$00
200
400
am
500
600
700
800
900
D
1
# Transmitted Symbols
Fig. 3b Estimated Error Variance
# Transmitted Symbols
Fig. 3a Estimated Mean
1o“< ,
1
-- CRB
--- Error Variance
k----
10-’, “-.L
0.19971
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10
20
30
40
60
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——
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Besson and P. Stoics, “Sinusoidal Signals with
Random Amplitude: Least-Squares Estimators and
Their Statistical Analysis”, IEEE Trans. Signal
Processing, vol. 43, pp. 2733-2744, Nov. 1995.
[7] O. Besson and P. Stoics, “Estimation the Parameters
of a Random Amplitude Sinusoid from its Samples
Covariance’s”, ICASSP’96, Atlanta, GA, May 7-10,
1996.
[8] O. Besson, “Improved detection of a random
amplitude sinusoid by constrain least squares
technique”, Signal Processing, vol. 45, no. 3, pp.
347-356, Sept, 1995.
[9] S. P. Bruzzone and M. Kaveh, “Information tradeoffs
in using the sample autocorrelation function in
ARMA parameter estimation”, IEEE Trans.,
Acoust. Speech Signal Processing, vol. 32, no. 4,
pp. 701-714, Aug. 1984.
[10] B. Porat, Digital Processing of Random Signals:
Theory and Methods, Prentice Hall, 1994.
[11] T. Soderstrom and P. Stoics, System Identification,
Prentice Hall, London, 1989.
[12] P. Stoics, T. Soderstrom, and V. Symonyte, “On
estimating the noise power in array processing”,
Signal Processing, vol. 26, pp. 205-220, Feb. 1992.
[6] O.
REFERENCES
M. Luise and R. Reggiannini, “Carrier Frequency
Acquisition and Tracking for OFDM Systems”, IEEE
Trans. Communication, vol. 44, pp. 1590-1598, Nov.
1996.
[2] H. Sari, G. Karam, and I. Jeanclaude, “Channel
equalization and carrier synchronization in OFDM
systems”, in Audio and Video Digital Radio
Broadcasting Systems and Techniques, R. De
Gaudenzi and M. Luise, Eds. Amsterdam Elsevier,
1994.
[3] F. Daflara and O. Adarni, “A new fkequency detector
for orthogonal multicarrier transmission techniques”,
in Proc IEEE VTC ’95, Chicago, IL, July 1995.
[4] M. Luise and R. Reggiannini, “Carrier frequency
recovery in all-digital modems for burst-mode
transmission”, IEEE Trans. Commun., vol. 43, pp.
1169-1178, Feb./Mar./Apr. 1995.
[5] J. Beek, M. Sandell and P. O. Borjesson, “ML
Estimation of Time and Frequency Offset in OFDM
Systems”, IEEE Trans. Signal Processing, vol. 45,
pp. 1800-1805, July 1997.
to
—
SNR [dB]
Fig. 4b Estimator Error Variance
SNR [dB]
Fig. 4a Estimated Mean
[1]
-————.—.
0-7803-4902-4/98/$10.00 (c) 1998 IEEE