Carrier Frequency Offset Estimation Using Hidden Pilots for MC-DS-CDMA Systems Licheng Liu*†
Xianhua Dai* †
*School of Information and Science Technology Sun Yat-sen University Guang Zhou,P.R.China Email:
[email protected]
School of Information Engineering Guangdong University of Technology Guang Zhou,P.R.China Email:
[email protected]
proposed in [6], in which the specially designed pilots are inserted to do the CFO estimation before the transmission of data symbol block. One disadvantage of the estimator is that the extra transmission bandwidth is needed, which may reduce data transmission efficiency.
Abstract—This paper develops a new hidden pilots based algorithm for the carrier frequency offset (CFO) estimation of multicarrier DS-CDMA (MC-DS-CDMA) systems. Multiple CFO of different users in the uplink of MC-DS-CDMA systems may violate the orthogonality among subcarriers and induce both inter-carrier interference (ICI) and inter-user interference (IUI). This algorithm utilizes mutually orthogonal hidden pilots for each user and implements fast Fourier transform (FFT) to the received signal matrix after the average operation on its column vectors, so the interference including the IUI caused by users’ data symbols are greatly depressed and CFOs can be estimated with the performance close to the conventional pilot-based CFO estimator without the requirement of extra transmission bandwidth. The simulation results verify the effectiveness of the proposed approach with CFOs in the fine tracking phase and the slowly time-varying flat-fading channels. Keywords- hidden pilots;carrier estimation;OFDM;MC-DS-CDMA
I.
frequency
The superimposed training or hidden pilot technique which adds pilots to data symbols arithmetically has recently attracted wide attention for its bandwidth saving advantage. Although some part of the transmission power has been allocated for the hidden pilots resulting in the reduction of the power for the data symbols, the quality of the data symbol demodulation can be ensured with advanced signal processing methods. In [7], a single CFO estimation based on superimposed training for OFDM system has been presented. But the estimator can get the acceptable result only when the CFO is small and the power ratio of hidden pilots to data symbols is rather high. And it can not be used for the multiple CFOs scenarios directly due to the IUI effects existing in the multiple user OFDM systems.
offset(CFO)
INTRODUCTION
Motivated by the estimator in [7], we propose a novel hidden pilot scheme in this paper for the multiple CFO estimation in the uplink of MC-DS-CDMA systems. In this scheme, mutually orthogonal sequences are designed to compose the pilot blocks for users, which are arithmetically added to the data symbol blocks before they are spread in the transmitter. At the receiver, the received signal matrix within a whole spreading code period is reconstructed with the hidden pilot block of the interested user, and an average operation is applied to the column vectors of the received signal matrix to restrain the interference of all users’ data symbols to the pilot block of the interested user. Then the fast Fourier transform (FFT) operation is exploited to obtain the CFO estimation of the interested user. By appropriately choosing the average power ratio of hidden pilots to data symbols, the high accurate CFO estimation can be achieved and the impacts to the demodulation of data symbols can be reduced to the least as well. The simulation results illustrate that when the average power ratio of hidden pilots to data symbols is 5%, the mean value of the CFO estimation error is less than 0.04% subcarrier spacing even when the Signal to Noise Ratio (SNR) is í5 dB.
Carrier frequency offset (CFO) resulting from the Doppler shift or the mismatch of the carriers between the transmitter and receiver in orthogonal frequency division multiplexing (OFDM) systems, will cause the inter-carrier interference (ICI) and degrade the system performance seriously. It is also a key disadvantage in multicarrier DS-CDMA (MC-DS-CDMA) systems [1], which, like multicarrier CDMA (MC-CDMA) systems, is one promising candidate of multiple user OFDM systems. So far, there have been many proposals for the estimation of single CFO in single user OFDM systems, which can be divided into two main categories: data aided schemes [2] and blind schemes [3]. Mainly due to the inter-user interference (IUI) existing in multiple user OFDM systems, those single CFO estimators can not work properly under the multiple CFO environment. For the multiple CFO estimation in the uplink of multiple user OFDM systems, some CFO estimators for OFDMA or MC-CDMA systems have been proposed recently, belonging to either data aided (pilot) [4] or blind[5].In general, as the data symbol transmission mechanisms are different among these multiple user OFDM systems, those CFO estimators are not applicable for the uplink of MC-DS-CDMA systems directly. Recently, a pilot-based CFO estimation algorithm for the uplink of MC-DS-CDMA systems has been
1-4244-2424-5/08/$20.00 ©2008 IEEE
Throughout the paper, bold and italic letters denote the matrices or vectors. The superscripts [·]T represents the transpose operation and F[ ⋅ ] denotes the FFT operation on [ ⋅ ].
49
ICCS 2008
II.
where
SYSTEM MODEL WITH CFO EFFECTS
Nu
In the uplink of MC-DS-CDMA systems, all users transmit their data on the same subcarriers in the same way. Suppose there are Nc subcarriers and Nu users. A sequence of Nc data symbols of user n, An, is serial-to-parallel converted onto Nc sub-streams. Within each sub-stream, the data symbols are spread by the user-specific spreading code of length Ns chips denoted as {cn,t, t=0,1, …, Nsí1}[8]. Then the Nc data symbols spread by the same chip on different sub-streams do the inverse fast Fourier transform (IFFT) operation to form an OFDM block, totally Ns OFDM blocks within a whole spreading code period. So the tth OFDM block (0tNsí1) of the bth block of Nc data symbols parallel transmitted by user n can be denoted by
Rt ( k ) = ¦ cn ,t H n e n =1
the FFT value of vt ( k ) . Wn ( k ) = F [e
T
= F [e
After a cyclic prefix (CP) is inserted into the OFDM block, the sequence is transmitted serially through a liner channel. At the receiver end, the received composite signal matrix containing Ns OFDM blocks within a whole spreading code period, after discarding the CP sequences, can be written as
Pn =
P
where Gn represents the channel gain factor of user n, and we assume the channels of all users here are flat-fading and slowly time-varying channels; vt ( k ) is the additive white noise
A
pn = [ pn ,1 , pn ,2 , " , pn , N ] (n=1,2,…,Nu) , should be mutually T
P
variable with zero mean value. ε n is the unknown normalized CFO of nth user and assumed to be limited to (í0.5,0.5).The assumption is usually satisfied after the CFO coarse synchronization. In addition, we have also assumed perfect time alignment or synchronization [9].
orthogonal. The superposition of data symbol block An and pilot block Pn is spread and forms into Ns OFDM blocks within a whole spreading code period. Considering the received signal matrix R in (5), we draw out the row vectors corresponding to the subcarriers with non-zero pilot tones to form a newNP×Ns
By performing the fast Fourier transform (FFT) operation on each column of the received signal matrix in (2), we can obtain the following FFT demodulated outputs:
(6)
≥L
(m=1,2,…,NP); N P = «ª N c /( L + 1) »º ( ‘ ª º ’denotes ceiling operation), the number of the non-zero pilot tones, is a positive integer . And the number of zero pilot tones between two neighboring non-zero pilot tones is L(except for that between the “tail” and the “head” non-zero pilot tones, which may consist of zero pilot tones more than L). It is required that NP Nu in order to distinguish users. And the vectors composed of the non-zero pilot tones of different users, i.e.
(4)
Rt = [ Rt (0)," , Rt ( k )," , Rt ( N c − 1)]T
P
L
pilot tones of the pilot block, P + P = 1 , and pn , m = ±1
where
(5)
P
P [ pn ,1 , 0, " , 0 , pn ,2 , 0, " , 0 , " , pn , N , 0, " , 0] (8) N N N
where P P is the normalized average power of the non-zero
(3)
R = [ R0 ,", Rt ,", RNs −1 ] ,
HIDDEN PILOT SCHEME AND CFO ESTIMATION
L
(2)
+ vt (k )
] , n=1,…,Nu. We refer to
In this scheme, we design a pilot block Pn with the length of Nc, which is added arithmetically to the data symbol block An of user n. And the structure of Pn is designed as follows
and An , respectively.
n =1
]
ALGORITHM
j 2 π kt / N c
j 2πε n ( k + tNc ) / Nc
," , e
j 2πε n ( N c −1) / Nc
j 2πε n m / N c
To apply the new estimator, we need to assume the following assumptions: A1) The data symbol to be sent in subcarrier, i.e. an(k), k=0,…,Ncí1, is independent identically distributed (i.i.d) random variable with zero mean value;A2) The data symbols and the hidden pilot tones are mutually independent for all users.
the IFFT matrix with [ F −1 ]k ,t = e and j 2 = −1 . As CFO estimation is limited within only a block of Nc data symbols in this study, the data symbol block index b will be omitted for simplification, and xnb (t , k ) and Anb are denoted by xn (t , k )
Nu
j 2πε n 0 / N c
III.
(1)
normalized average power of P A and an (bN c + k ) = 1 . F −1 is
rt (k ) = ¦ xn (t , k )Gn e
] is the FFT
Wn(0) as a CFO vector associated with ε n and Wn(k) is a cyclicshifted vector of Wn(0) with a shifting length k. “ ⊗ ” represents the periodic convolution.
]T is the bth block of Nc data symbols of user n, with the
rt = [ rt (0)," , rt ( k )," , rt ( N c − 1)]T
j 2π ( ε n + k ) m / N c
n , and Wn (0) = [ wn (0), " , wn ( N c − 1) ] = F [e
A
r = [ r0 ," , rt ," , rN s −1 ] ,
(7)
vector of the complex exponential function with CFO of user
where A = P [an (bNc + 0),", an (bNc + k ),", an (bNc + Nc − 1) b n
[ An ⊗ Wn ( k )] + vt ( k )
where Hn is the channel frequency response of user n; vt ( k ) is
X nb (t ) = [ xnb (t , 0)," , xnb (t , k )," , xnb (t , N c − 1)] = F −1 Anb cn ,t , n = 1,! N u , t = 0,! N s − 1
j 2πε n t
Nu
matrix R which can be decomposed as R = ¦ R n + v , where v n=1
is the additive noise matrix, and R n is the signal component corresponding to user n.
50
To estimate the CFO of user n, firstly we define row vector C n = [1 cn ,0 ,1 cn ,1 ," ,1 cn , N s −1 ] , matrix pn C n , and the Schur
According to the assumptions A1 and A2, the absolute value of Bn , l , t can be treated as a random variable, which varies
product or element-by-element product ( pn C n )ΘR which is
from 0~
Nu
¦ [( p C )ΘR ] + p C v ; l
equal to
n
l =1
n
n
small value towards 0 when NP is large enough. And due to the mutually orthogonal property of Pn , Dn,l,t=0 when ln; and
Then we calculate the
n
P A wl (0) H l and has a high probability to be a
def
average values of the column vectors in matrix ( pn C n )ΘR ,
P P wn (0) H n = Dn when l= n .So we have
Dn,l,t =
i.e. E[( pn C n )ΘR ] ; At last, we use the above results to get the CFO estimation of user n. To do so, we follow the procedures below step by step.
E[( pn C n )ΘR ≈
Considering calculating ( pn C n )ΘR firstly, we need to l
[B
n , l ,0
l
l
l
T
(10)
j 2πε n t
j 2πε n t
, " , Bn , n , N s −1e
, " , Dn e
j 2πε n ⋅( N s −1)
j 2πε n ⋅( N s −1)
º¼
º¼
(16)
l =1
≈
Nu
¦[B
n , l ,0
l =1
+ ª¬ Dn , " , Dn e
, " , Bn , l ,t e j 2πε n t
j 2πε l t
, " , Dn e
, " , Bn ,l , N s −1e
j 2πε n ⋅( N s −1)
j 2πε l ⋅( N s −1)
º¼
º¼ + E[ pn C n v ] (17)
In (17), as E[ pn C n v ] is a row vector associated with the additive noise and each element of it can be treated as a random variable with zero mean value, we can omit it when doing the CFO estimation. Notice that after E[ pn C n v ] is removed from (17), (17) can be denoted as the superposition of Nu+1 complex exponential function sequences with a length of Ns:
Wn (0) ≈ [ wn (0), " , wn (T ), 0, " , 0, wn ( N c − T ), " , wn ( N c − 1) ] (12)
For the convenience of deduction, here we assume that only wn (0) is considered. According to the above assumption for Wn(0), the pilot block structure, and the definition of periodic convolution, (11) can be simplified as Rt ( m) ≈ cl ,t H l e [ P al ( I m ) + P pl,m ]wl (0) From (12) ~ (13), we have
, " , Bn , n , t e
Nu
T
P
(15)
= ¦ E[( pn C n )ΘR l ] + E[ pn C n v ]
As mentioned before that we focus only on fine-frequency tracking phase and the normalized CFO of the nth user ε n ∈ ( −0.5,0.5) , the principal components or major lobe of Wn(0) are concentrated on its two ends (low-frequency components) with index kę(0,1,…,T) and kę(NcíT,…,Ncí1) (Assuming TL is a small positive integer). Thus, Wn(0) can be approximated as[10]
A
º¼
E[( pn C n )ΘR ]
Thus
j 2πε t
j 2πε l t
n , n ,0
+ ª¬ Dn , " , Dn e
where Rtl ( m) = cl ,t H l e l [( Al + Pl ) ⊗ Wl ( I m )] (11) and Im=(mí1)(L+1) (m=1,2,…,NP), is the subcarrier index with non-zero pilot tones.
l
[B
≈
l
Rt = [ Rt (1)," , Rt ( m)," , Rt ( N P )] l
j 2πε l ⋅( N s −1)
n
R = [ R0 ,", Rt ,", RNs −1 ] (9) l
, " , Bn , l , N s −1e
E[( pn C n )ΘR ]
obtain the expression of R (l=1,…,Nu ), which can be deduced from (5)~(7) as : l
, " , Bn , l , t e
] j 2πε l t
and
l
l
l ,l ≠ n
Nu
E[( pn C n )ΘR ] ≈
(13)
¦B l =1
n ,l
(t ) e
j 2πε l t
+ Dn e j 2πε nt
t = 0, ! N s − 1 (18)
E[( pn C n )ΘR ] l
≈
[ B ," , B e + [ D ," , D e
j 2πε l t
n , l ,0
n ,l ,t
n , l ,0
n ,l , t
, " , Bn , l , N s −1e
j 2πε l t
j 2πε l ⋅( N s −1)
, " , Dn ,l , N s −1e
Now, we apply the FFT operation to (18) in order to obtain the CFO estimation of user n.
º¼
j 2πε l ⋅( N s −1)
[
º¼ (14)
where E[ ⋅ ] denotes average operation on each column of [ ⋅ ],and Bn , l , t = Dn , l , t =
PP § NP
PA § NP
]
ª Nu º ¦ Bn ,l (t )e j 2πεlt » + F ª¬ Dn e j 2πεnt º¼ ¬ l =1 ¼
F E[( pn C n )ΘR ] = F « Nu
NP
· ¨ ¦ pn , m al ( I m ) ¸ wl (0) H l cl ,t / cn ,t , © m =1 ¹
=
j 2πε t j 2πε t F ª¬ Dn e n º¼ ≈ F ª¬ Dn e n º¼ (19) ¼º +
l =1
Part B Part B
¦ F ¬ª B
n ,l
(t ) e
j 2πε l t
Part A
NP
· ¨ ¦ pn , m pl , m ¸ wl (0) H l cl ,t / cn ,t . © m =1 ¹
By treating Part A in (19) as the interference of data symbols of all users and omitting it (the reason will be
51
discussed later at the end of this section), we focus on Part B in (19) to do the CFO estimation of user n.
case with a very high probability. However, the analytical conclusion on appropriately choosing the power ratio still needs further studying. Our simulation results in the next section show that when the average power ratio of P P to P A is 5%, the mean value of the CFO estimation error is still less than 0.04% subcarrier spacing even when the Signal to Noise Ratio (SNR) is í5 dB.
According to the duality of time and frequency, the Part B can be further expressed as F [ Dn e
j 2πε n t
] = F [ Dn e
= Dn ⋅ F [e
j 2𠫪ε n N s »º t / N s
j 2π « ªε n N s »º t / N s
⋅e
j 2π {ε n N s − «ªε n N s »º }t / N s
] ⊗ F [e
]
j 2 π {ε n N s − « ªε n N s »º }t / N s
]
Another claim to be worthy of raising is that our proposed CFO estimation algorithm can work effectively not only when the channels are flat fading but also when the channels are frequency selective fading. The demonstration in theory may be complicated, but it has been demonstrated by simulations. Due to the limitation to paper pages, we’ll not discuss it deeply here in this paper.
t = 0, ! N s − 1 (20) In (20), the index of the maximum of the magnitude frequency response of F [e
j 2π « ªε n N s »º t / N s
] is {ª«ε n N s º» + 1} (if
ε n > 0 ) or { N s + ª«ε n N s »º + 1} (if ε n ≤ 0 ), while in other indexes the corresponding values are zero; The magnitude j 2π {ε
To sum up, the hidden pilots based multiple CFO estimation algorithm includes two main procedures:
N − ªε N º}t / N
n s « n s» s ] is similar to frequency response of F [e Wn(0), i.e. in most indexes the corresponding values are very small except for the maximum value in index 1 and subj 2πε t maximum values in its neighboring indexes. As F [ Dn e n ]
is
the
with F [e
period
convolution
j 2π {ε n N s − «ªε n N s »º }t / N s
of
F [e
j 2𠪫ε n N s º» t / N s
Procedure 1: At the transmitter, Nu pilot blocks Pn (n=1,2,…,Nu) are designed and added to the data symbol blocks of Nu users before the step of spreading in the MC-DSCDMA system, respectively. The pilot blocks are constructed according to the low-pass property of the Wn(0) vector as well as the property of orthogonal sequences.
]
] , the index of the maximum of the
magnitude frequency response of F [ Dn e
j 2πε nt
Procedure 2: At the receiver, vectors pn and Cn are constructed firstly, then element-by-element multiplied with matrix R , which consists of the row vectors corresponding to subcarriers with non-zero pilots tones of received signal matrix R. The average operations are taken to each column of the matrix ( pn C n )ΘR , i.e. E[( pn C n )ΘR ] . And the CFO estimation of user n is finally got by a search of the index of maximum of the magnitude frequency response of F [ E[( pn C n )ΘR ]] .
] is the one in j 2π ªε N º t / N
which the magnitude frequency response of F [e « n s » s ] reaches its maximum value. That is to say, the index of the maximum of the magnitude frequency response of j 2πε t F [ Dn e n ] is {«ªε n N s »º + 1} (if ε n > 0 ) or { N s + «ªε n N s »º + 1} (if ε n ≤ 0 ). estimated by
So the CFO of user n can be
( Indexn − 1) / N s , if Indexn < N s / 2
εˆn = ®
¯( Indexn − 1 − N s ) / N s , if Indexn ≥ N s / 2
(21)
IV.
where “Indexn” denotes the index of the maximum of the magnitude frequency response of F [ E[( pn C n ) ΘR ]] .
In the simulation, a MC-DS-CDMA system with user number Nu=5 is used as the experiment model. Each user transmits the data stream using the QPSK modulation constellation with Hardama sequence {cn,t | cn,t=±1} as spreading code sequence having the length of Ns=64. The nonzero pilot tone sequences used here for different users are also Hardama sequences, which meet the requirement of being mutually orthogonal. And the average power ratio of hidden pilot block to data symbol block is 5%, i.e. P P = 0.047
Now we briefly discuss the effects of Part A in (19) to the CFO estimation of user n. As we mentioned before that Bn , l (t ) ,i.e. Bn , l , t , can be treated as a random variable with small absolute value in the case of large NP , the magnitude
frequency response of the FFT of Bn , l (t ) ( t = 0, ! N s − 1) is
fluctuant and has many local peak values, which leads to the fact that the magnitude frequency response of the FFT of j 2πε t Bn , l (t )e l is also fluctuant and has many local peak values according to the similar analysis for the Part B . Such result is also suitable for the Part A since it is the sum of F ª¬ Bn , l (t )e
j 2πε l t
SIMULATION RESULTS
and P A = 0.953 . All users’ CFOs and channel pulse response gain factors are listed in Table 1. Fig.1 shows the mean value of CFO estimation error of the 1st user, i.e. E[| ε1 − εˆ1 |] under the different SNR, using the proposed hidden pilot algorithm. The mean value is got by an averaging operation over 400 independent simulations. And the T is equal to 1 and 3 respectively, L=T. It is obvious that the
º¼ ( l = 1, ! , N u ) . So the CFO estimation of
user n will be seriously interfered only if at least one peak value of that of the Part A is beyond that of the Part B. On the premise that the power ratio of hidden pilots to data symbols is as small as possible, by appropriately choosing the average power ratio of P P to P A , we can avoid the occurrence of such
User CFO İn Gn/G1
52
Table 1: Five users CFOs and gain factors 1st 2nd 3rd 4th 0.46 0.38 -0.22 -0.42 1.0 0.85 0.75 0.65
5th -0.32 0.55
new estimator is robust against the additive noise. Though SNR decreases from 20 dB to ˉ 5 dB, the mean value of CFO estimation error still keeps at the low value of 0.04% subcarrier spacing. This can be attributed to the fact that the additive noise is uniform-distributed and has little impact to the index of the maximum of the magnitude frequency response of F [ E[( pn C n )ΘR ]] if it is not very strong. We can also observe that the estimation is better with larger number of NP ( N P = ª« N c /( L + 1)º») .It can be explained with the evidence
Mean value of CFO estimation error
10
that large NP helps to get small values of Bn , l (t ) and lower the peak values of the magnitude frequency response of the FFT of the Part A, so as to decrease the interference to that of the Part B. And it is also shown in Fig.1 that when NP is large enough, for instance, NP256, the estimation errors are almost the same for different NP. So in practice, we may pay more attention to choose the appropriate value of NP via Nc and T. Generally, T=1 or 2 is enough.
10
-2
-3
-4
-10
Mean value of CFO estimation error
10
CONCLUSION
-5
5 SNR,dB
10
15
20
− εˆ1 |] vs. SNR: Hidden pilot estimator.
0
10
10
10
10
-1
-2
-3
-4
-5
-10
Fig.2. [3]
0
Pilot-based Hidden pilot-based
10
-5
0
5 SNR, dB
10
15
20
E[| ε1 − εˆ1 |] vs. SNR. Subcarrier num Nc=512 and T=1.
B.Chen, “Maximum likelihood estimation of OFDM carrier frequency offset,” IEEE Signal Processing Letters, Vol.9,pp.123-126,2002. [4] X.H.Dai, “Carrier frequency offset estimation and correction for OFDMA uplink,” IET Commun.,Vol.2 ,pp.273-281,2007. [5] Y.Ma, R.Tafazolli, “Estimation of carrier frequency Offset for multicarrier CDMA uplink,” IEEE Transactions on signal processing,Vol.55,pp.2617-2627,2007. [6] L.C. Liu, X.H. Dai, “A novel pilot-based carrier frequency offset estimation in MC-DS-CDMA uplink,” IET CCWMSN 07, pp.10391042, Dec. 2007. [7] M.Ahmadi, A.S.Mehr, “Superimposed training aided carrier frequency offset estimation in OFDM systems,”IEEE EIT 2007, pp.296-299,2007. [8] I.Barbancho, “Code shift for intercarrier interference cancellation in MC-DS-CDMA,” Signal Processing, vol.84,pp.2449-2453,2004. [9] H.Steendam, H.Bruneel and M.Moeneclaey, “A comparison between uplink and downlink MC-DS-CDMA sensitivity to static timing and clock frequency offsets,” IEEE Trans. on signal processing, Vol.53,pp.3869-3878,Oct. 2005. [10] X.H.Dai, “Carrier frequency offset estimation for OFDM/SDMA systems using consecutive pilots,” IET Proc-Commun.,vol. 152(5),pp.624-632,2005.
ACKNOWLEDGMENT This work is supported by National Science Foundation of China (NSFC), Grant 60772132, Science & Technology Project of Guangdong Province, Grant 2007-B010200055, Industrial & Academic Research Project, Grant 2007A090302116, and Natural Science Foundation Key Project of Guangdong Province, Grant 8151027501000102. REFERENCES
[2]
10
Fig.1. E[| ε1
The problem of multiple CFO estimation for the uplink MC-DS-CDMA systems was investigated in this work. Utilizing hidden pilots, this paper has proposed a new multiple CFO estimator for MC-DS-CDMA uplink in slow time-varying flat-fading channels. It has been established that the CFO is identifiable by applying mutually orthogonal hidden pilots and the FFT operation to the column average values of the selected received signal matrix. Compared with pilot aided scheme, both theoretical and simulation results have shown the proposed CFO estimator has an excellent performance without losing transmission efficiency.
[1]
Nc=512,T=1 Nc=512,T=3 Nc=1024,T=1 Nc=1024,T=3
10
Fig.2 compares the pilot-based algorithm in [6] with the new hidden pilot algorithm. It is shown that the pilot-based algorithm outperforms the proposed hidden pilot-based algorithm only after SNR is beyond 4 dB. But when SNR reduces from 4 dB toí5 dB, the CFO estimation error of the former rises from 0.03% to 5% subcarrier spacing dramatically, while that of the latter keeps at 0.03% subcarrier spacing level. Thus it demonstrates that in the aspect of the stability of estimation error, the new hidden pilot algorithm outperforms the pilot-based algorithm in [6].
V.
-1
H.Steendam,M.Moeneclaey, “ The effect of carrier frequency offsets on downlink and uplink MC-DS-CDMA,” Selected Areas in Communications, IEEE, Vol.19, pp.1528-1536,2001. P.H.Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. on Commun.,Vol.42,pp.2908-2914,1994.
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