Communications and Network, 2013, 5, 398-402
http://dx.doi.org/10.4236/cn.2013.53B2073 Published Online September 2013 (http://www.sci r p.org/journal/cn)
Copyright © 2013 SciRes. CN
Compari sons of Short-P refix Based Channel Estimation
in Single-Carrier Communication Syst em s
Huilei Li, Shiwen Fan, Lisha Gong, Guobing Cheng, Shaoqian Li
National Key Laboratory of Science and Technology on Communications,
University of Electronic Science and Technology of China, Chengdu, China
Email: leehuilei@163.com
Received August, 2013
ABSTRACT
In this paper, we compare the performance between channel estimation based on compressed sensing (CS) and
time-domain least square (LS) for single-carrier (SC) communication system. Unlike the conventional channel estima-
tion t ech nique s suc h as fr eque nc y do main LS whic h is used in the cond itio n tha t the l engt h of p ilo t seq uence i s equa l to
data sequence, the estimation scheme based on CS requires smaller length of pilot sequence. In this paper, the corres-
ponding system structure is presented. Zadoff-Chu sequence is used to generate the pilot sequence, which is shown to
perform better i n forming measurement matri x of CS than pseudo random seque nce. S imu lation results demonstrate that
channel estimation based on CS achieves a better bit error rate (BER) performance than time domain LS with a smaller
pilot sequence and thus raising data rate of the SC communication system.
Keywords: Compre ssed Sensing; Time Domain Least Square; Single-carrier Communication Syst em; Zadoff-Chu Se-
quence
1. Introduction
Recently the single-carrier with frequency domain equa-
lization (SC-FDE) communication system has drawn a
great attention as an alternative to the orthogonal fre-
quency division multiplexing (OFDM) communication
system, especially in the uplink communications, be-
cause it alleviates carrier synchronization problem and
the peak-to-average power ratio (PAPR) of transmitted
sequences is much lower than that of OFDM[1].
SC-FDE system requires precise channel state infor-
mation (CSI) for the equalization. To estimate the CSI,
the pilot blocks of SC-FDE are inserted between several
data blocks in time domain. With the assumption that
there are only L number of channel taps where L is
smaller than the length of cyclic prefix (CP), the tradi-
tional time domain LS estimation can be improved to be
used in the condition that the length of pilot sequence is
smaller than data sequence but still larger than cyclic
pr efix.
In thi s pa per , we compare channel estimation based o n
compressed sensing (CS) and time-domain least square
(LS) for the SC communication system. The length of
pilot sequence of the compressed method can be smaller
than c yclic prefix and can thus raising the data rate of the
system. By choosing a good pilot sequence and cyclic
prefix to form the C S measurement matr ix, we can obtain
the sparse channel impulse response (CIR) vector.
The rest of the paper is organized as follows. We start
in Section 2 by describin g the system model. In Section 3,
we review traditional frequency domain LS channel es-
timation and time domain LS channel estimation which
suits the condition that the length of pilot sequence is
smaller than data sequence and larger than cyclic prefix.
In Section 4, we introduce the idea of compressed chan-
nel est i mat i o n re q uir i ng t he l e ngt h o f p ilo t se q ue nc e e ve n
smaller than c yclic prefix. Si mulation result s are given in
Section 5 and the paper is concluded in Section 6.
2. System Model
Figure 1 shows the bloc k diagr am of the transmitter and
receiver of the considered SC communication system.
Modulation
Information bits
(a) Transmitter
Insert PilotAdd CP
Remove
CP DFT Frequency
Domain
Equalization IDFT Demodulation
(b) Receiver
Figure 1. The considered SC commu nication syste m.
H. L. LI ET AL.
Copyright © 2013 SciRes. CN
399
We consider the single-carrier b lock transmission over
the multipath Rayleigh fading channel with additive
white Gaussian noise. At the transmitter, infor mation bits
are mapped into data symbols depending on the modula-
tion type. Each length-N data symbols appended with a
length-
cp
N
cyclic prefix forms a data block. The cyclic
prefix is assumed to be larger than the known channel
impulse response length L to eliminate inter-block inter-
ference (IBI). This is achieved by discarding the first
cp
N
received symbols corresponding to the cyclic prefix
[2]. The pilot block is formed the same way as the data
block while the pilot sequence of each pilot block is of
length
p
N
. After inserti ng pilo t a nd ad ding c yclic p refix,
the transmit block is generated. Figure 2 shows the
transmit block format o f the system.
As depicted in Figure 2, the tr ansmit block consists o f
pilot blocks and data blocks. Pilot sequence
{ }
(0), (1),..., (1)
p
pppN
added by a length-
cp
N
CP forms the pilot block and
{ }
(0), (1),..., (1),1,...,
ii i
x xxNiK−=
, each with the same
length of CP as the pilot block, forms K data blocks. At
the receiver, after removing CP, channel estimation, fre-
quency domain equalization which is the frequency do-
main analog of what is done by a conventional time do-
main equalizer[3], we obtain the estimated d ata symbols.
By demodulation, these symbols are changed into the
estimated in formation bits.
3. Channel Estimation of the Pilot Sequence
Based on Least-square Channel
Estimation
In the case that the length of pilot sequence is larger than
cyclic prefix, i.e.
, the cyclic prefix of pilot
block are copies of the last
cp
N
symbols of the pilot
sequence.
At receiver, after removing the c yclic prefix, the input-
output relationship of the pilot sequence can be ex-
presse d in matri x for m as foll o ws:
,
ppp p
yhx n= +
(1)
p
y
,
p
x
, and
p
n
are length-
p
N
sequence of re-
ceived, input, and noise symbols, respectively. Note that
the
pp
NN×
channel matrix
p
h
is circulant with first
column equal to the channel impulse response (CIR) ap-
pended by (
p
NL
) zeros, therefore it can be diagona-
lized by the orthonormal discrete Fourier trans-
form(DFT)
p
Cyclic
Prefix of
Data Block
( 1)
k
xN
(0)
k
x
Cyclic
Prefix of
Data Block
1
( 1)xN
1
(0)x
Cyclic
Prefix of
Pilot Block
( 1)
p
pN
(0)p
N
N
cp
N
N
cp
N
cp
N
Data Block KData Block 1Pilot Block
Figure 2. Block for mat of the considered sy s tem.
matrices where the diagonal elements are the DFTs of the
channel impulse response. That is,
0
1
**
1
00
00
00 0
00 p
pp
N
H
H
hFHF FF
H



== 



(2)
where (.)* denotes complex-co njugate transpo se; F is the
DFT matrix whose element is givenby
2 (1)(1)/
1
(, )p
jik N
p
FFi ke
N
π
− −−
=
, 1, p
ik N≤≤.
3.1. Traditional Frequency Domain LS
Channel Estimation
In the case that the length of pilot sequence is equal to
that of data sequence, i.e.
p
NN=
, after removing cyc-
lic prefix, the received time-domain pilot sequence
p
y
is transformed to the frequency domain by applying DFT
to (1)
,
p
ppppppp
YFyFh xFnHXN== +=+
(3)
p
Y
, p
X
and
p
N are length-
p
N
frequency-domain
response of received, input, and noise symbols, respec-
tively.
Therefore, the traditional LS channel estimation of
p
H
at the pilot sequenceis
_
,
p
p LS
p
Y
HX
=
(4)
3.2. Time Domain Least-square Channel
Estimation
Usin g the ti me do main LS c hannel esti matio n, the le ngt h
of pilot sequence can be reduced to be smaller than data
sequence, i.e.
p
NN<
. Therefore, the equation (1)-(3)
are still satisfied.
With t he a s su mp tion tha t t he c han nel i mp ul se r e sp ons e
length is smaller than
cp
N
[4], the input-output rela-
tionship of the system in frequency domain can be ex-
pressed as
() ,
p
p pp
YdiagXFhN= +
(5)
where
()
p
diag X
stands for diagonal matrix with the
column vector
p
X
on its diagonal;
p
F
is an
×
p cp
NN
matrix which retains only the first
cp
N
columns of F;
and h is a length-
cp
N
column vector which retains the
first
cp
N
elements of
p
h
. Then the improved LS chan-
nel estimatio n is
1
_
() ,
HH
p LSp
hQQ QY
=
(6)
H. L. LI ET AL.
Copyright © 2013 SciRes. CN
400
where
( ),
pp
Qdiag XF=
(7)
Then, by doing
N
-points DFT of
_p LS
h
, we obtain
the estimated channel frequenc y response
_p LS
H
of the
pilot sequence. That is,
__,
p LSp LS
N
H Fh= (8)
where
2 (1)(1)/
( ,),1,1
jik N
N cp
F ikeiNkN
π
− −−
=≤≤≤≤
.
4. Channel Estimation of the Pilot Sequence
Based on Compressed Sensing
When t he length ofpilo t sequence is not smaller tha n that
of cyclic prefix, i.e.
p cp
NN
, the cyclic prefix of the
pilot sequence is the copy of the last
cp
N
symbols of
the pilot sequence, and then (1) is equal to
,
pp p
yTh n= +
(9)
where
01 1
10 2
12
,
pp cp
p cp
ppp cp
N NN
NN
p
N NNN
xx x
xx x
T
xx x
− −+
−+
−− −




=




 
 
(10)
is a
p cp
NN×
Toeplitz matrix formed by the pilot se-
quence.
Now we consider the case that the length of the pilot
sequence is smaller than cyclic prefix, i.e.
.
Assuming the cyclic prefix of the pilot block in the form
of vector is
01 1
[ , ,...,]
cp
T
N
ccc
, then the Toeplitz matrix
p
T
in (9) can be modified to be
011 1
10
1
1201 1
cpcp p
cp
ppcpcp p
N NN
pN
N NNNN
xc cc
xx
Tc
xx xcc
− −+
− −−−+



=




 
 
(11)
which is a
p cp
NN×
Toeplitz matrix for med by both the
pilot sequence and its cyclic prefix.
It is known that CS can be effectively utilized in linear,
time-invariant system identification problems provided
the impulse response of the system is (approximately or
exactly) sparse. An immediate application is in wireless
multipath channel estimation[5]. As the length-
cp
N
CIR
h of the sparse wireless channel has only few of its taps
nonzero, then using (9) we can change the matter of
channel estimation into the matter of recovery of h based
on compr essed sensing.
,
pp
y Th
(12)
Therefore,
p
T
is considered as the CS measurement
matrix. The channel estimation based on CS introduced
in this p ape r uses t he or thogon al matching pursuit (OMP)
algorithm to solve the problem of compressed sensing.
4.1. Selection of Pilot Sequence
Advances in CS show that the sparse
h
can be recon-
structed from the measurement
p
y
with high probabili-
ty when measurement matrix
p
T
satisfies RIP[6].
However , there is no known method to t est in polynomial
time whether a given matrix satisfies RIP . An alter native
approach we adopt here is to minimize the coherence of
p
T
[7]. The coherence of a
p cp
NN×
measurement
matrix
p
T
is the maximum absolute cross-correlation
between the normalized columns of
p
T
:
1, 22
|, |
max ,
pcp
lk
lk
TlkNlk
ϕϕ
µϕϕ
≤≤
=
(13)
The smaller the coherence p
T
µ
is, the better the
measurement matrix preserves the information of the
sparse vector h in the produced samples
p
y
[7].
p
T
is formed by elements of the pilot block, so the
selection of pilot seq uence an d its cyclic p refix is impo r-
tant. In this paper, we choose Zadoff-Chu sequence
which has constant envelope and uniform spectrum[8] to
form the pilot sequence and its cyclic prefix. Compared
with pseudo random sequence, measurement matrix
formed by Chu sequence has smaller
p
T
µ
.
A length-
p
N
Chu sequence
{ }
1
0
p
N
kk
P
=
used as the
pilot sequence is re p r esented as
2
/,
( 1)/,
,
pp
pp
jrkNfor even N
kjrkkNfor oddN
e
Pe
π
π
+
=
(14)
where r is relatively prime to
p
N
. T he lengt h-
cp
N
cyc-
lic prefix of the pilot sequence is generated the same way
as the pilot sequence using a length-
cp
N
Chu se-
quence
{ }
1
0
cp
N
kk
c
=
.
Simulation results show that measurement matrix
formed by Chu sequence performs better in the com-
pressed channel estimation than pseudo random se-
quence.
4.2. Proposed Compressed Channel
Estimation
In this paper, the compressed channel estimation is based
on OMP algor ithm. O MP is a n iterati ve greed y algor ithm
that selects at each step the column, which is most corre-
lated with the current residuals[9]. With OMP, faster
H. L. LI ET AL.
Copyright © 2013 SciRes. CN
401
convergence to a sparse solution is obtained by eliminat-
ing the re-selec tion problem[10].
For (9), with the knowledge of the received time
domain pilots
p
y
and the Toeplitz matrix p
T formed
by pilot block, we use OMP algorithm to estimate the
positions and values of the taps of the sparse vector h and
then transform it into frequency domain the same way as
(8).
5. Simulation Results
In this paper, we do simulations to compare the perfor-
mance of traditio nal LS channel estimatio n, ti me d omain
LS channel estimation and the compressed channel esti-
mation. Suppose the coherent time of the sparse sin-
gle-carrier channel lasts about 1 0 d ata b locks, the number
of nonzero taps of the channel is
L=6
, the length of
each data sequence is
N= 256
and the length of CP for
both the pilot sequence and data sequence is 64
cp
N=.
The transmission block contains 12 data blocks and 3
pilot blocks inserted between each 6 data blocks. Infor-
mation about the Rayleigh channel of the system is de-
scribed in Table 1 with additive white Guassian noise.
We assume that the time delays of the channel are
integral times of the s ystem sampling pe r iod
9.7657 07
s
T es=− for conve nie nce.
At the receiver, we apply first order linear interpola-
tion to gain the channel of data sequence in frequency
domain and Minimum mean square error (MMSE) equa-
lization as the equalization algorithm. Let
p
Nm=
and
consider the following groups of simulations:
1) We apply traditonal LS channel estimation on the
pilot block for
256m=
;
2) We apply time domain LS channel estimation on
the pilot blo c k for
128,64m=
;
3) We apply the compressed channel estimation on
the pilot block for
128,64,32m=
;
The BER performance results of the methodsabove are
given by Figure 3. “FLS” means traditional LS channel
estimation, “TLS” means time domain LS channel esti-
mation, and “CS” means the estimation method based on
compressed sensing. “H_ideal” means ideal channel es-
timation.
Table 1. Parameters of th e multipath Rayleigh channel .
Path Table Column Head
Delay Ti me Average Power(dB)
1 0 0
2
2
s
T
-3
3
8
s
T
-6
4
16
s
T
-9
Path Table Column Head
Delay Ti me Average Power(dB)
5
32 s
T
-12
6
48 s
T
-15
0246810 12 14 16 18 20
SNR gain, dB
BER
SC System
m=32 CS
m=64 CS
m=128 CS
m=128 TLS
Ideal
m=64 TLS
m=256 FLS
m=256 TLS
10
-7
10
-6
10
-5
10
-4
10
-2
10
-3
10
-1
Figure 3. BER performances of t he methods above.
Figure 3 shows that b y using ti me do main LS me thod
or the compressed method, the BER performance of the
system can be improved compared with traditional LS
method even reducing the length of the pilot sequence.
For time domain LS method, the larger the len gth of pilot
sequence is, the better its BER performance is. For the
compressed method, when the lengthof pilot sequence is
larger than or equal to its cyclic pr efix, it performs better
than time domain LS method. As the length of pilot se-
quence reduces to
32m=
which is smaller than its cyc-
lic prefix, the BER perfor mance o f thecompre ssed channel
estimation is almost the same with time domain LS
channel estimation with
256m=
.
6. Conclusions
In this paper, a comparison among channel estimation
based on traditional LS method, time domain LS and
compressed sensing in SC communication system has
been done for the sparse channel. We showed that among
these three estimation methods, both time domain LS and
compressed method perform better than traditional LS
method. Moreover, the compressed method based on
OMP algorithm reduces the length of pilot sequence re-
quired for channel estimation compared with time do-
main LS method without performance degradation. Si-
mulation results showed that t he compressed method has
significant performance gains over time domain LS esti-
mation method under the same condition of the length of
pilot sequence.
7. Acknowledgements
H. L. LI ET AL.
Copyright © 2013 SciRes. CN
402
This work is supported in part by the National Science
Foundation of China under Grant number 61101101,
National Grand Special Science and Technology Project
of China under Grant No. 2010ZX03006-002-02, Pro-
gra m for N ew Cent ury E xcel lent Ta lents in U nive rsit y o f
China ((NCET110058), the Foundation Project of Na-
tional Key Laboratory of Science and Technology on
Communications under Grant 9140C020404120C0201,
and Key Laboratory of Universal Wireless Communica-
tions, Beijing university of Posts and Telecommunica-
tions, Ministry of Education, P.R.China (No. KFKT-
2012102).
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