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Blind Estimation of OFDM Carrier Frequency Offset Using Shift-invariance Properties Aifeng Ren

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Blind Estimation of OFDM Carrier Frequency Offset Using Shift-invariance Properties Aifeng Ren
160
ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.5, NO.1 February 2007
Blind Estimation of OFDM Carrier Frequency
Offset Using Shift-invariance Properties
Aifeng Ren1 and Qinye Yin2 , Non-members
ABSTRACT
We address the problem of carrier frequency offset (CFO) synchronization in Orthogonal Frequency
Division Multiplexing (OFDM) communications systems in the context of frequency-selective fading
channels. The mathematic model of OFDM systems
with cyclic prefix (CP) and virtual subcarrier is derived. And we propose the shift-invariance CFO estimation algorithm based on DOA-MATRIX method.
After the rough estimation, it can acquire the frequency offset and matrix including channel information simultaneously, which are favorable for offset
compensation and demodulation of received signals.
Simulation results illustrate performance of this algorithm.
Keywords:
Carrier frequency offset,
invariance, OFDM, DOA-Matrix
Shift-
1. INTRODUCTION
The basic principle of Orthogonal Frequency Division Multiplexing (OFDM) is to split a high-rate data
stream into a number of lower rate streams that are
transmitted simultaneously over a number of subcarriers. Although the total channel has the frequency
selectivity, for each subchannel to say that is flat
(frequency nonselective) fading. One of the principal advantages of OFDM is its utility for transmission at very nearly optimum performance in unequalized channels and in multipath channels [1]. To account for Inter-Block Interference (IBI), OFDM systems rely on the so called cyclic prefix (CP) inserted
at the transmitter, after IFFT modulation. To eliminate IBI, the length of CP is chosen larger than the
FIR channel memory [2][3]. But one of the principal
disadvantages of OFDM is more sensitivity to carrier
frequency offset (CFO) than single carrier modulations. There are two destructive effects caused by
Manuscript received on August 6, 2006 ; revised on October
11, 2006.
This work was supported in part by the National Natural
Sciences Foundation (No. 60572046 and No. 60502022) of
China.
1 The author is with School of Electronic and Engineering, XiDian University, Xi’an 710071, P. R. China Email:
[email protected]
1,2 The authors are with School of Electronics and Information Engineering, Xi’an Jiaotong University, Xi’an
710049, P.R. China Email:
[email protected],
[email protected]
CFO in OFDM systems. One is the reduction of signal amplitude and the other is the introduction of
intercarrier interference (ICI) from the other carriers. The resulting ICI degrades Bit Error Rate (BER)
performance severely [4]. Thus, accurate carrier offset estimation and compensation is more critical in
OFDM systems.
Recently, the available frequency offset estimation algorithm can be divided into two categories:
the non-blind estimation methods and blind estimation methods. The non-blind estimators are based
on pilots[5][6] or on the cyclic prefix, such as Maximum Likelihood (ML) carrier-frequency offset estimator proposed by Jan-Jaap van de Beek[7], which
makes use of the redundant information introduced
by CP availably. In [8] and [9], the channel impulse response (CIR) is first removed from the likelihood function and then by summing over all the
possible signals, the frequency domain transmitted
signals are removed. The blind estimators contain
MUSIC method[10] proposed by H. Liu and ESPRIT
method[12] by U. Tureli, which are based on subspace
technique and have the super distinguishing performance.
On analyzing the signal structure of CP-based
OFDM modulation, using virtual carrier technique
and shift-invariance properties in DOA-MATRIX
method, we present a new CFO estimation algorithm.
2. SIGNAL MODEL
In OFDM systems, the information stream bn
drawn randomly from a finite-alphabet is parsed into
T
blocks s(k) = [b0 (k), b1 (k), . . . , bp−1 (k)] , where [•]T
denotes transpose. By applying an G-point inverse
discrete Fourier transform (IDFT), P parallel data
bits are modulated onto G(G > P ) subcarriers and
the whole bandwidth is evenly divided for G narrowband subcarriers such that the bandwidth for each
subcarrier is relatively small compared to the coherence bandwidth of the channel. The remaining
G − P subcarriers are not used for data modulation
in order to avoid aliasing at the receiver, which are
often referred to as the virtual carriers (VC) [10].
We model the frequency-selective channel as a finite impulse response (FIR) filter with channel CIR
T
h = [h(0), h(1), . . . , h(L)] , where L is the channel order and [•]T denotes the transpose. The channel has
L independent Rayleigh-fading taps and the power
gain of each tap is randomly generated with the total
Blind Estimation of OFDM Carrier Frequency Offset Using Shift-invariance Properties
PL
power of all taps, l=0 |h(l)|2 , normalized to 1. We
assume that the CIR is time invariant over K(≥ 1)
consecutive OFDM symbol blocks, but could vary
from one set of K blocks to the next. After inserting
the cyclic prefix (CP) with the length Ng (Ng ≥ L) in
OFDM symbols at the transmitter and removing CP
at the receiver, we can make a practical assumption
that there is no synchronization error and no intersymbol interference (ISI) from the previous OFDM
symbol block. At the receiver end, the removal of the
CP makes the received sequence the circular convolution of the transmitted sequence with the channel
impulse response {h(l)}L
l=0 . In the absence of the
carrier frequency offset between the receiver and the
transmitter, i.e., ∆f = 0 , we can describe the IBIfree received data blocks as y(k) in the receiver [3].

T
y(k)=WDH b0 (k), . . . , bp−1 (k), 0,
· · · , 0 =WP D̃H s(k)
|
{z
}
G−P
(1)
where DH = diag [H(0, · · · , H(P − 1)]), and H(n) =
PL
−j 2π
G nl denote the complex channel frel=0 h(l)e
quency response at the n-th subcarrier frequency.
And D̃H = diag [H(0), · · · , H(P − 1)]. Wp comprises the first P columns of the G × G IDFT matrix
W = [w0 w1 · · · wp−1 wp · · · wG−1 ] . Then, the effect
of the frequency-selective channel on the OFDM signal is completely captured by scalar multiplications
of the data symbols by the frequency responses of the
channel at the subcarrier frequencies[11].
In radio communications, because of the local oscillators between the transmitter and receiver inaccuracy and Doppler shift, there is typically a frequency
offset after the subcarrier is removed from the received signal, i.e.∆f 6= 0. The received data block
y(k)becomes [12][13]
161
3. SHIFT-INVARIANCE CFO ESTIMATION
In this section, we estimate the CFO using DOAMATRIX method presented by Q. Yin in 1989
for direction-of-arrival (DOA) estimation. Compared with the well-known ESPRIT method, DOAMATRIX is simpler and more generalized [14]. From
(3), we have two received data matrices with (G−1)×
K dimensions, X and Z, which are the first G−1 rows
and the last rows of Y respectively
X = EG−1 W(G−1)×P D̃H S̃ + Vhead
= AS̃ + Vhead
(4)
Z = EG−1 W(G−1)×P D̃H ΦS̃ + Vtail
= AΦS̃ + Vtail
(5)
where A = EG−1 W(G−1)×P D̃H with dimensions
(G − 1) × P . EG−1 and W(G−1)×P can be expressed as E(1 : G − 1; 1 : G) and W(1 : G −
1; 1 : P ) , respectively. and are the first rows
and the last G − 1 rows of the noise matrix V ,
respectively.Φ is a P × P diagonal matrix including the information
of carrier frequency offset, iand
h
P −1
1
j2π∆f j2π( G
+∆f )
Φ = diag e
,e
, · · · , ej2π( G +∆f ) .
Note that Φ is an unitary matrix that relates the
measurements from subarray X to those from subarray Y. From above deduction we can find (4) and (5)
have the ESPRIT structure, in which Φ and A correspond to the shift operator and channel frequency
attenuation matrix respectively. As the shift operator
and the array manifold matrix can be achieved from
eigenvalue-eigenvector pairs using TLS-ESPRIT [15]
or DOA matrix method [14], we can also estimate Φ
and A similar to [14] or [15] by using shift-invariance
property.
The auto-correlation matrix of subarray X and the
cross-correlation matrix of the two subarrays Z and
X can be respectively written as
H
y(k) = EWp D̃H s(k)ej2π(k−1)(G+Ng )∆f
(2)
where E = diag[1, ej2π∆f , · · · , ej2π(G−1)∆f ] is
CFO matrix. The presence of the CFO makes the
energy loss of the desired signal and introduces intercarrier interference (ICI) resulting from other subcarriers.
Without loss of generality, considering the thermal
noise and stacking K IBI-free received OFDM symbol
blocks, the received data matrix can be expressed as
Y = [y(1), · · · , y(K)] = EWp D̃H S̃ + V
RXX = E[XXH ] = AE[S̃S̃ ]AH + σv2 IG−1
(3)
where S̃ = [s(1), · · · , s(K)]diag[1, ej2π(G+Ng )∆f ,
· · · , ej2π(K−1)(G+Ng )∆f ] . Each entry of the matrix V
with dimensions G × K is the independent identically
distributed (i.i.d.) complex zero-mean Gaussian noise
with variance σv2 .
= ARS̃ S̃ AH + σv2 IG−1
(6)
= RXX0 + σv2 IG−1
and
H
RZX = E[ZXH ] = AΦE[S̃S̃ ]AH + σv2 JG−1
= AΦRS̃S̃ AH + σv2 JG−1
= RZX0 +
(7)
σv2 JG−1
where [•]H denotes conjugate transpose and E[•]
H
the expectation operator. RS̃ S̃ = E[SS̃ ] denotes
the auto-correlation matrix S of and is an P × P
Hermitian matrix.IG−1 is an G − 1identity matrix.
JG−1 is an square G − 1 matrix, where all the first
minor diagonal elements on upper right of the main
diagonal are set to 1 and all others to 0.
162
ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.5, NO.1 February 2007
Obviously, the rank of RXX0 in (6) is equal to
P , the number of bits in one OFDM symbol block.
By doing eigen decomposition on the noisy autocorrelation matrix RXX , we can obtain the eigenvalues λi P
and the corresponding eigenvectors ηi , and
G−1
RXX = i=0 λi ηi ηiH . We sort the eigenvalues as
λ1 ≥ λ2 ≥ · · · ≥ λP > λP +1 = · · · = λG−1 ≈ σv2
. Then, the variance of noise, σv2 , can be estimated
PG−1
1
as σ̂v2 = G−P
i=P +1 λi .When RS̃ S̃ is nonsingu−1
lar and G − 1 > P , RXX0 = RXX − σv2 IG−1 =
PP
ARS̃ S̃ AH = i=1 (λi − σv2 ηi ηiH ). Then, we define an
auxiliary matrix R referred to as the DOA-MATRIX
shift-invariance estimator in [14].
R = RZX0 [RXX0 ]+
(8)
4. SIMULATION RESULTS
Numerical simulations are carried out to demonstrate the performance of the proposed shiftinvariance CFO estimation algorithm. In this simulation, we choose BPSK for modulation schemes, and
we use Nt = 50 Monte Carlo simulations to evaluate
the performance of the estimation of CFO in which
we consider OFDM system with G = 64 subcarriers,
Ng = 10 and coping with the FIR composite channel
of length L, (L < Ng ) for each subcarrier channel.
We use the following multipath model to hold for the
channel impulse response within the block of data[16].
h(t) =
L
X
Al δ(t − τl ), Al = αl e−jθ1
(12)
l=1
where [RXX0 ]+ is the Penrose-Moore pseudoinverse of RXX0 . RXX0 , [RXX0 ]+ and RZX0 can
be calculated by
RXX0 =
P
X
(λi − σv2 )ηi ηiH
i=1
[RXX0 ]+ =
P
X
i=1
1
ηi η H
λi − σv2 i
(9)
RZX0 = RZX0 − σv2 JG−1
It is shown in [14] that if the matrix A in (4) and
(5) is column-rank, RS̃ S̃ is nonsingular and there
are not identical terms in the main diagonal of Φ,
the eigenvalues and corresponding eigenvectors of the
auxiliary matrix R are the diagonal elements of Φ
and columns of the matrix A, respectively. That is
RA = AΦ. After the eigen decomposition on matrix
R , we have the estimated matrix Φ̂ and  . Then
the estimated CFO can be calculated from
tr(Φ̂)
˜
ej2π∆f = PP −1 2π
j Gm
m=0 e
(10)
where tr(•) denotes trace of the matrix, or the sum of
the elements of the principal diagonal of the matrix.
Similar to [10], the carrier frequency offset, 2π∆fˆ ,
calculated from (10) may be at all possible values of
ˆ
[0, 2π] on the unit circle ej2π∆f . Eq.(10) shows that
the proposed algorithm can provide the estimation of
CFOs up to integer times the channel spacing.
In (10), the CFO estimation is in closed-form, and
thus is also better than the MUSIC-like searching
method [10] that minimizes the cost function as (11)
for CFO estimation.
P (e
j2π∆fˆ
) = min
(G−1
X
i=P
)
−1
wH
YYH Êwi
i Ê
(11)
where αl and θl are the amplitude and phase of the
channel path associated with the delay τl . The
complex-valued path amplitudes {Al }are modelled
as zero-mean complex-valued Gaussian random variables and are mutually independent. The L different
delays τl = l/B l = 1,2,· · · ,L and B is the bandwidth
occupied by the real bandpass signal, are mutually
independent and generated according to a truncated
exponential distribution[17], with an average delay
τmean and a maximum delay τmax .
In the following simulations, the carrier frequency
offset ∆f is fixed at 0.2ω except where otherwise
noted, where ω = 2π
G is the subcarrier frequency spacing. We evaluate the performance of the estimator by
means of the Mean-Square Error (MSE) as
Nt
1 X
M SE =
Nt i=1
Ã
∆fˆ − ∆f
1G
!2
(13)
Fig. 1 shows the performance of three CFO estimations with the variety of the signal to noise ratio
(SNR) in the case of L = 5 and G − P = 10.
The bit error rate (BER) performance of an
OFDM system, which is retrieved and compensated
for the CFO by using the proposed algorithm based
on L = 5 and K = 100, is shown in Fig.2. For comparison, we also give the BER performance of the
same system with no CFO compensation and with
CFO compensation by using ML CP-based estimation and ESPRIT-like estimation.
Fig. 3 shows the performance of the proposed estimator at different value K. We assume that the
channel fading is slow enough so that the channel
does not vary rapidly while CFO estimation is performed. The MSE of the proposed CFO estimator
decreases as the OFDM blocks is increased.
Fig. 4 shows the performance of the estimated carrier frequency offset estimation algorithm proposed
over the range of 1/2 the subcarrier spacing at the
SNR of 15dB. In order to comparison, we plot the
actual CFO curve.
Blind Estimation of OFDM Carrier Frequency Offset Using Shift-invariance Properties
163
Fig.5 shows the results for estimating the CFO
with the number of virtual carriers (G − P ) and the
channel length L. In Fig.5, the channel has 5, 10
and 15 independent Rayleigh-fading taps respectively,
and the power gain of each tap is randomly generated
with the total power of all taps normalized to 1. The
results show that the DOA-MATRIX CFO estimator
is robust to frequency-selective fading.
Fig.4: Estimated CFO vs. actual CFO
Fig.1: Comparison of MSE vs. SNR
Fig.5: MSE versus the number of virtual carriers
(G − P )
5. CONCLUSIONS
Fig.2: BER Performance of CFO Estimations
In this letter we have presented a DOA-MATRIX
based blind CFO estimation for OFDM system with
virtual carriers. The method can effectively estimate
CFO by exploiting the inherent shift-invariance structure of received signal in time-domain. Based on
these simulation results, we conclude that the proposed algorithm can estimate CFOs that are both
fractional and integer multiples of the carrier spacing.
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[3]
Fig.3: MSE versus SNR at different value K
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Aifeng Ren
was born in Hebei
Province in China, in Nov. 1974. He
received the B.S. and M.S. degrees on
electronic and engineering from XiDian
University in China in 1997 and 2000 respectively, and now pursuing his Ph.D.
degree in electronics and information engineering in Xi’an Jiaotong University,
P.R. China. His research interests lie
in the areas of wireless communications,
multimedia communication, and signal
processing. Email: [email protected]
Qinye Yin
was born in Jiangsu
Province in China, in 1950. He received
the B.S. and M.S. degrees from Xi’an
Jiaotong University, and received Ph.D.
degree from the University of Maryland, U.S.. He is now a professor and
Ph.D. advisor of the School of Electronics and Information Engineering, Xi’an
Jiaotong University. His research interests lie in the areas of signal processing
and communications, space spectrum estimation, neural net and its applications, and time-frequency
analysis. Email: [email protected]
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