Research Journal of Applied Sciences, Engineering and Technology 4(10): 1250-1259, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: November 25, 2011
Accepted: December 27, 2011
Published: May 15, 2012
New Restructure of Transmitted Sequences for CP-based LS Channel
Estimation Method in OFDM System
Wang-Xing Zhao, Zhang-Xin Chen and Qun Wan
Department of EE, University of Electronic Science and Technology of China, Chengdu, China
Abstract: This study proceeded to investigate a study and a signal processing on channel estimation problem
of OFDM system in wireless communication area. We gave an optimization stretching into total transmitted
sequences restructure which aimed to improve Cyclic-prefix Least Square (CPLS) channel estimation method
proposed in this paper. By contrast to conventional Training Sequences (TS) methods especially frequency TS,
like sub-carriers TS, which directly occupy sub-carrier data sequences, CPLS method can greater the usage of
cyclic-prefix thus saves system resources. In detail, we first gave a deduction and Mean Square Error (MSE)
compare of both CPLS and Equally Spaced Training Sequences (ESTS) method based on LS principle. Then
according to the deduction, we mainly concerned the restructure of total transmitted sequences using
optimization tools, the Lagrange resolving, before which an constrain model was established, effects of our
restructure were that it largely lower down the channel estimation MSE by using CPLS, while system BER was
also improved. In the last part simulations given showed the correctness of our restructure theory.
Key words: Channel estimation, constrain model, Cyclic-prefix Least Square (Cpls) method, Equally Spaced
Training Sequences (ESTS) method, lagrange resolving, Orthogonal Frequency Division
Multiplexing (OFDM), transmitted sequences restructure
INTRODUCTION
Orthogonal Frequency Division Multiplexing
(OFDM) is considered to be a key technical in the next
4G or B3G wireless communication systems. And before
this, it has already been used in European Digital
Broadcast Audios (DBA) system etc. Superiority of this
new technical is its high usage rate of frequency, due to
its function that transforms a frequency band with a
certain width into orthogonal sub-carriers; sub-carriers
carry data symbols and are modulated into OFDM
symbols.
As a necessary part of OFDM systems, channel
estimation can reduce detection Bits Error Rate (BER) in
receiver thus gains a large significance of investigation.
By now, means or methods of estimation are already
abundant. Among them, the most prevalent way is the
method that inserts comb pilots and interpolates so as to
imitate the fast time-varied channel. This method uses
assistant information in frequency and performs well
particularly in time-varied channel. In the methods that
based on training sequences, in different principles
discussed in Wang et al. (2008), like Least-Square (LS),
Linear Minimal Mean Square Error (LMMSE), or
Maximal Likelihood (ML), computer in both receiver and
transmitted training sequences, it can make out result of
estimation. Nevertheless, drawbacks of this method are its
occupying of valid data sequences. Other sorts of method
basically cope with channel as a FIR filter and explicit
adaptive signal processing ways to estimate it, that is, do
iterations to make closest to real channel value, as
mentioned in article Sun et al. (2005), LMS adaptive
algorithm, and in paper Way and Yumin (2004), the
Kalman filtering method. In the field of blind channel
estimation, conventional method is to transform the
received signal into high order to generate a new
correlation matrix and decompose it into sub-spaces then
recover the channel information see Kathryn and Ying
(2005). Due to matrix theory, paper Feng (2009) abstracts
OFDM system into matrix perturbation model and
resolves the model so as to reach the purpose of
estimation.
Cyclic-prefix only plays a role as to inhibit the
symbols and sub-carriers interferences and is seldom used
to estimate channel. However, paper Toshiaki and
Andreas (2011) deduces the channel time presentation
through convex optimum modeling. Paper Christopher
and Steger (2004) specifically excavates the relations
between cyclic-prefix and channel in time domain. Paper
Borching (2008) concludes how the completeness of
cyclic-prefix influents channels estimation and data
recovery in receiver. However, unlike the methods based
on cyclic-prefix above, the paper reuse cyclic-prefix in a
simple way, that is, transform both received cyclic-prefix
and transmitted cyclic-prefix in frequency sequences and
directly do LS estimation, we call it Cyclic-prefix LS
(CPLS), as shown in Fig. 1.
Corresponding Author: Wang-Xing Zhao, Department of EE, University of Electronic Science and Technology of China,
Chengdu, China
1250
Res. J. Appl. Sci. Eng. Technol., 4(10): 1250-1259, 2012
LS channel estimator and input condition: In OFDM
system, according to paper ¤LKER (2005), combine (2)
channel estimator based on frequency LS principle can be
presented as:
Y
H ls 
X
Fig. 1: A simplified CPLS channel estimation model
Method of CPLS is easy to accomplish and for
OFDM system, it only need to transform CP into
frequency domain and do LS estimation. Thus this method
can greatly increase the usage effectiveness of cyclicprefix and save extra cost. However, compare to the
conventional TS or the most optimized Equally Spaced
TS (ESTS), MSE performance of CPLS is not always
better. Paper Wang-xing and Fang (2011) gives a proof
that in LS frame, whatever transmitted sequences are, it
can always find a certain number of training sequences
whose performances are better than CPLS. Actually in
this paper we compare deeply the MSE of same point
equally spaced TS and CPLS method under same
transmitted sequences.
Under this background, this paper proposes a
transmitted sequence restructure method that make CPLS
and ESTS as close as possible or in MSE even better,
moreover, in constrain model constraints are made aiming
to better CPLS under LS frame. We repeatedly use
Lagrange resolving mathematical tool to resolve our
models, that is, use the tool for two times. In the first, a
subtle way is introduced to resolve the result as we
wanted, while in the second resolving process we
proposed a resolving algorithm by which we escape large
amount of computing complexity and obtain final result.
At last, simulations given verify our theories respectively.
Moreover, before restructure, we deduce the MSE of LS
frame and deduce the characteristic representation of
ESTS as follows. These are roots for our restructure
theory.
In (3), Y is received frequency sequences, X means
transmitted frequency sequences, the action is to do a
division in the corresponding sides, then according to the
definition of MSE:

H

N
N
 

E   H   H  H   H 
 

X
X
 

 N  H  N    N  H  N 
 E          
  X   X    X   X 

N
2
X
2

(4)
1
SNR
Equation (4) means that when MSE of noise vector is
stationed, MSE of LS channel estimation only depends on
the signal power. Besides, (4) shows MSE of LS estimator
only relates to transmitted symbols. Actually, selections
of transmitted sequences can be arbitrary in LS estimator,
if and only if they are linearly combined from all
transmitted sequences.
Consider attach CP frequency sequences to
transmitted sequences, we have:

1


1
X p  Fp * * 1
N 
1


1
e
j
e
2 *( N  p  1)
N
j
2 *( N  p  2 )
N
...
2 * N
j
e N
...
...
...
...
2 *( N  p  1)( N  1)
N



2 *( N  p  2 )( N  1) 
j

N
e

...


2 *( N  1)
j

N
e

e
j
(5)
* X N  Fp  N * X N
General channel transmitted model of OFDM system:
(1)
In formulation (1), y indicates receiver time domain
symbols, s, the transmitted time domain symbols; h
indicates channel, n, Gauss white noise, H presents
convolution calculation. Particularly, when transmitted
symbols are continuous, according to cyclic convolution
property, in frequency domain, (1) can be presented as:
Y = HS + N

M SE ls E ( H ls  H ) H ( H ls  H ) 
LS ESTIMATOR
y=sqh+n
(3)
(2)
In Eq. (5), N presents number of transmitted symbols
or number of sub-carriers, p is the length of cyclic prefix,
Xp equals to left side of (5) and presents the cyclic prefix
frequency sequences, XN, total transmitted sequences, F
p×N is a linear transformation matrix. Therefore, Eq.(5)
indicates that CP frequency sequences can be linearly
combined by transmitted sequences, it satisfies input
conditions of LS estimator.
Consistency condition of LS estimator: From the LS
estimators using conventional training sequences in
1251
Res. J. Appl. Sci. Eng. Technol., 4(10): 1250-1259, 2012
frequency domain we find that it must satify an
consistency condition, that is, training sequences for LS
division of both receiver and transimtted part must be
conresponding to each other, otherwise, it may not
comform to the purpose of what we want. So here we
have to analysis the consitence condition satification of
CPLS.
From the property of eccess of OFDM system, what
we need to prove is the receiver cyclic-prefix also equals
to the last part of time domain OFDM symbols of
receiver. Now we prove based on transimtted model in
time channel model.
We set receive time domain OFDM symbol with
cyclic-prefix as:
yN+p = (y(0), y(1), … , y(N+p-1))
(6)
then cyclic-prefix of receiver can be presented as:
yCP = (y(0), y(1), … , y(p-1))
(7)
Meanwhile the last P point of time symbols y N+p is:
yp = (y(N-1) , y (N) , … , y(N+p-1))
(8)
Then, Y CP = Fp* yCP meanwhile:
F p × N* YN = Fp *yp
(9)
So we only need to compare to prove yCP = yP
.Actually in OFDM time domain transmitted model it is
not hard to prove. First consider cyclic-prefix of the
transmitted side xg OFDM time domain symbols without
cyclic-prefix is x, then:
 x ( N  p  n) , 0  n  p  1
xg (n)  
 x ( n  p) , p  n  N  p  1
y N + p (m) = y N+P (m+N-1), m = 0,…, p-1, p > l
So yCP = yp which means receiver side also satisfies
the cyclic-prefix added trait. Therefore this proof of
consistence conditions of CPLS provides that CPLS can
be totally feasible to compare with the conventional
training sequences like ESTS.
CHARACTERISTICS SEQUENCES
As mentioned above, here define two characteristic
sequences, all of them can do LS estimation; they are
Equally Spaced Training Sequences (Ests) and Cyclic
Prefix Frequency Sequences (CPFS), as shown in Fig. 2
where transmitted sequences named in this paper are
actually sub-carriers symbols in OFDM system. First, we
deduce their expression and make a compare of both
sequences.
Set length of IDFT M which equals to number of subcarriers, as P<M, thus, we can always selected P training
sequences from M transmitted sequences in equal space.
Each adjacent training sequences space in length, Q =
[M/P], [•] presents Gauss ceiling.
To easily deduce, we set M = P*Q, in reality we just
need to adjust the length of P so that it is large than the
length of paths number of multi-paths channel.
Let Xi, i = 1, 2, …, M presents total transmitted
sequences, then, ESTS can be presented as:
X m+(i-1)*Q, 1# m # Q, I = 1, 2,…, P
(13)
M presents originate position of the sequences P
means the block number, Q is space. Generally, for
arbitrary m, use DFT:
X

 m

X
 m Q



 X m( P 1)Q 
(10)
m*0
W M

( mQ )*0
W M

...
  M ( p 1)*Q *0
W M
Therefore time domain transmitted model can be
presented as:
l 1
y g (n)   h(i ) * x g (n  i )  w(n)
(12)
(11)
m*( M 1)
WM

  x1 
 x 2 
...


... ...
 ... 


[ m( P 1)*Q ]*( M 1)   x M 
... W M

...
(14)
( mQ )*( M 1)
WM
i0
Length of yg is N+p+l-1, in general, take the
sequences from l to N+p+l-1, that is , y N + p (n) = yg (n+l1) n = 0,…, N+p-1.Therefore according to (10) and (11)
where cyclic-prefixes are added in transmitted side, if
p > l there must be:
WMi*k  e
 j *2  *i *k
M
Equation (14) means xQ = F xM. xM. is time domain
transmitted OFDM symbols. Consider separate xM into
pieces, that is:
1252
Res. J. Appl. Sci. Eng. Technol., 4(10): 1250-1259, 2012
Fig. 2: Two characteristic sequences to be compared


 

f
  0 
x
 1   f1 

x M   x2  
   ... 
 ...   f 
 x   Q2 
 M f
 Q 1 
 WMm*0
0
0
0 


m*1
0
0
0
W


M
F0  FP 
 FPW
0
0
0 
...


0
0 WMm*( P 1) 
 0
(15)
Meanwhile, it is not hard to find out that property as:
In Eq. (15), f Q-1 = x CP, means the cyclic prefix. f 0,…,
f Q-2 are non-CP part. Therefore x M is divided into Q
pieces, the last one is cyclic prefix.
Do the same separations to F, that is, block F into Q
blocks:
F = [F0 F1 … FQ-2 FQ-1]
W Mm*0
...
 ( m Q)*0
W M
...
F0  
...
...
W [ M  ( p 1)*Q]*0 ...
 M
W M( m Q)*( P  2 )
...
[ m ( P  1)*Q*( P  2 )]
WM
m* P
 WM

0
F1  F0 
0

0
0
0
m* P
WM
0
0
...
0
0



0
  F0
0

m* P 
WM 
0
In the same way, F i F i-1N , i = 1,…, Q , thus, combine Eq.
(16) and (19), we subtly get the relation that:


( m Q )*( P  1)

WM

...

[ m ( P  1)*Q ]*( P  1) 
WM

W Mm*( P 1)
F = F p W [I p, N , …, N Q-1]
(17)
X Q  FPWf 0  FP W f 1  ...  FPW Q 1 f Q 1
 FP W[ I ,  ,...,  Q 1 ] X M
WM[ m (i 1)*Q ]*( j 1)
W
(22)
while according to the definition of cyclic prefix:
Unit expression of elements in F0 is, WM [m+(i-1)*Q]*(j-1) , = 1,
2, …, P
m*( j 1)
M
(21)
Therefore, combine Eq. (14) and (15) ultimately we get
deduction of presentation for ESTS as following:
XCP = Fp f Q-1 = FP [0, 0,…, I] XM
 WMm*( j 1)WM( j 1)*(11)*Q
(20)
(16)
In Eq. (16), F0corresponds to multiply f0 in detail:
W Mm*( P  2 )
(19)
(18)
( j 1)*( i 1)
P
W
Set Fp be P points DFT matrix, combine (18) we can tear
F0 as:
(23)
Then in Appendix A we give a proof of MSE of LS
estimation for XQ and XCP in detail, the proof shows that
MSE of CPLS is not always larger than ESTS method
except that under our following restructure we force the
MSE of former smaller than the latter. Simulation will
show the correctness. While in Appendix B we give a
optimization concerned proof that under same number of
TS selected, ESTS can make MSE of LS smallest,
therefore we need not find other TS from transmitted
sequences. These can be the origins and roots for our
restructure in the next part of article.
1253
Res. J. Appl. Sci. Eng. Technol., 4(10): 1250-1259, 2012
Therefore combine Eq. (4) and (5), we can establish
constrains as following:
NEW RESTRUCTURE OF TRANSMITTED
SEQUENCES
Restructure method analysis: Deduction in appendix
shows that under same condition MSE of ESTS and CPLS
are not certain while in the selection of direct TS (training
sequences) ESTS is the optimized. Therefore, here we
consider restructure a formation of new transmitted
sequences using convex optimization. We need to set up
a principle function, as well as constrains, all of them
composed an optimized model.
First, our target function, we need to restructure
transmitted sequences that make representation of CPFS
is as similar as possible to ESTS or in MSE the former
even better than the latter. So our participation goes like
this: a first target function make both method as close as
possible in formation, and a second target function we aim
to make CPLS surpass the other in MSE performance.
Then, for constrains, according to LS estimator
property, especially the MSE of LS we deduced before,
we aim to make CPLS of new transmitted sequences not
larger than ESTS in MSE. Meanwhile, remain structure of
the new restructured sequence the same as before so that
BER can be controlled.
Combine the first target function and constrains we
can compose an initial optimizing models. Then Lagrange
resolving method tool is introduced to resolve our
constrain model respectively.
Restrictions model: Let new transmitted sequences be x,
for the description of method, combine Eq. (5) and (22),
new x must satisfy:

 Fp1 F p  N X

AX
x op  arg min 

1

X

  W F N X
(26)
" maps time domain transmitted
sequences into sequences with cyclic prefix, * is a
constant assured by input sequences.
 FN1 X /  FN1 X N
Presents the ratio of time domain power of new
transmitted and original sequences which is also the ratio
of variances of noise because when signal-to-noise ratio
is stationed the value of latter is directly related to the
former. W  FN1 X
N
Is the original power of original
ESTS.
Therefore combine Eq. (24) (25) and (26), a
constrained model is set up as following respectively:
min AX
X
st . X  X N  
and
MX   NX
(27)
Lagrange resolving: To make out the result of (27), here
we first restructure Lagrange function where the method
Lagrange resolving is a common method to solve convex
problem:
2
 ( X  X N
  ( MX   2 NX
2
2
 2)
2
 2 )
(28)
Here 8, : are Lagrange factor, , is one increment
element, introduced by the second inequality formulation.
Now according to the essence of Lagrange resolving
extremism, grade the (28) by X and differential (28) by 8
and :, we have:
 A H AX  X  X N  M H MX 

  2 N H NX  0
.....( a )

2
2
......(b)
 X  XN 

.....(c)
 MX 2   2 NX 2   2
(25)
õ is a small real positive number.
Further, in order to make MSE of CPLS smaller than
that of ESTS method, according to deduction of MSE for
LS, it is determined by two factors: time domain power of
input sequences of LS and variance of noise.
FN1 X N
N   FN1 ,
(24)
This goal function makes both sequences closer to
each other. Our goal is to restructure the transmitted
sequences under which the two sequences ESTS and
CPFS mentioned will be close toN each other.
Second, keep the total structure as similar as original:
FN 1 X
* W FN1 X N   NX
f ( X ,  ,  )  AX
5 5Means the second norm of vector, N [N 0 , … , NQ-1].
5X!XN5#0
Fp 1 F p  N X  MX 
(29)
Next we will utilize a subtle way to resolve these
relations in (29), the deduction of result X op is as
following:
First, (29) (a) is meant to:
1254
Res. J. Appl. Sci. Eng. Technol., 4(10): 1250-1259, 2012
X
N
 X
A H A  M
M  
H

2
NH N
X  BX
However, this condition is somewhat off to our
optimizing goals, our goal is to let CPLS surpass ESTS
method under LS. So here we have to introduce a more
direct and strong target function as (39):
(30)
Then substitute (30) into (29) (b):
X H B H BX   2
(31)
X  op
Fold (29) (c):
X H (M H M   2 N H N) X  2
(32)
arg min
x




(39)
Then, a repeated Lagrange resolving model can be:
Therefore compare (31) and (32), we discover that:
g ( x )  A X  b

BH B  2 (M H M   2 N H N)

 F 1 F
p N X
 p
 A `X  b

W
FN1 X N



2
2
 

   X H X N  X H X  X H zX 

 
(40)
(33)
We grade both sizes by x,
Important analysis: N= " F N is (N+P)×N matrix with
rank N, so MHM - *2 NHN must be a full rank matrix with
rank N, thus, we certainly find a matrix z which satisfy
MHM - *2 NHN z2 N×N, here z is full rank matrix also with
rank N.
Meanwhile, according to (30), BHB is a symmetric
matrix, thus, (33) is:
-1
 
B 2   z
 

(35)
A H A   M H M   2 N H N
with XH

in the left side and X in right side, according to (32):
Till now, multiply
X H A H AX   2   X H

zX

X H A H AX   2
X H XN  X H X
C
C
(36)
(41)
2X
H
A
H
A X  2 X
H
A H b
(42)
X H XN
set g0 = 0.0001; X(0)op = ones (N, 1), F(0) = 1;
While g (X(k)op) > g0 do


( k  1)  ( k )
X op
   I   ( k ) z  A H A 



1
  (k )


X N  A H b
 2

(37)
H
2 X ( k ) A  H A X 
 ( k 1) 
Therefore combine (36) and (37) we get the relation
which is rather important:

X H X N  X H X  X H zX

1
Nevertheless, it seems we have no ways to solve the
final result. But actually, thanks to computer simulations
tool, here we need not to get the mathematical
presentation and avoid the complex calculations that are
required, what we need is to design an approximate
algorithm to make out the drawing near result as follows,
which exists equal to what we want:
Meanwhile according to (29) (a), multiply XH in the left
side, we resolute:




  I   z  A `H A  



Here F is a Lagrange constraint factor:
(34)

z

op


 X N  A `H b

2
2
Because B is also a full matrix, thus respectively, (34) is
meant to:
B
X
C
2 X ( k ) H A H b
X (k ) H X N
k = k + 1, If
(k )
g ( X op
) 0
stop
(43)
(38)
Definitely (38) have multitude resolutions of x. And
the mission of (38) is done to make the CPLS as close as
possible to ESTS under same transmitted sequences.
Though algorithm (43), we get our final resolution, X  op ,
 out . Then in next part, we will give simulations to verify
the algorithms (43) to see whether it is conform to our
aims or not.
1255
Res. J. Appl. Sci. Eng. Technol., 4(10): 1250-1259, 2012
Table 1: Simulation conditions
1
Front modulation
2
Number of sub-carriers
3
Insert pilots or not
4
Length of cyclic prefix
5
Length of equal space
6
Channel type
7
Number of channel paths
8
Signal distribution
9
Maximum SNR
10
Space of SNR
11
Channel interpolations
CPLS after restructure
Equally spaced TS after restructure
CPLS before restructure
Equally spaced TS before restructure
100
-1
BER
10
-2
10
final result. Whatever, only four methods are compared in
our simulations, they are: CPLS and ESTS method before
and after restructuring.
-3
10
5
0
10
15
20
25
SNR
30
35
40
Fig. 3: BER compare of methods before or after restructure for
basic simulation
CPLS after restructure
Equally spaced TS after restructure
CPLS before restructure
Equally spaced TS before restructure
101
0
10
MSE
-1
10
-2
10
-3
10
-4
10
0
5
10
15
20
25
SNR
30
35
40
Fig. 4: MSE compare of methods before or after restructure for
basic simulation
CPLS after restructure
Equally spaced TS after restructure
CPLS before restructure
Equally spaced TS before restructure
101
0
MSE
-1
10
-2
10
-3
10
-4
10
5
10
15
20
25
SNR
30
VARIED CONDITIONS AND THE
SIMULATION RESULTS
Basic stationed simulation conditions: We find that the
total restructure resolution contains some parameters that
need to be assured, but unfortunately we have seldom
found the principles. Here we just give some analysis in
that not only under stationed conditions but also under
varied conditions our total performance compare should
be the same.
Actually, from algorithms (43), definitely 0, g should
be certain first, they are what we called varied parameters.
we know that they are both small positive number,
and the ratio 0/g directly determine the effects and
strengths of matrix z in final resolution. So here we set
them under two conditions:
C
C
0 = 0.01, g = 0.01
0 = 0.0001, g = 0.1
Besides, from the final restructure result while other
stationed results are stationed, no other parameters should
be certain and it seems that only the final simulation
should be done. Nevertheless, there is one noticing which
should be highlighted.
That is, for the BER compare, we only need to
consider the BER performance of restructure sequences
without reduction in receiver although in transmitted side
transmitted sequences have been restructured.
10
0
16-QAM
256
no
P = 16
Q = 16
Rayleigh
H = 16
random
40db
Space of SNR
DFT-based
35
40
Fig. 5: MSE compare of methods before or after restructure for
extension simulation
Simulations: To verify the theory of our restructure
method, we give simulations. First set simulation
conditions. Table 1, we unify in the article some
conditions are stationed therefore need not be discussed
while other conditions are varied which may influent the
Basic simulation: In this simulation we set 0 = 0.01, g =
0.01 as the varied conditions. Judge the MSE and BER of
both CPLS of new transmitted sequences and ESTS of old
ones based on our stationed conditions as followings.
From Fig. 3, definitely, BER of CPLS of new
transmitted sequences is better than that of ESTS (Equally
spaced TS) method of original transmitted sequences.
This means after restructure if in receiver we deal with
some reduction inversely BER of OFDM system can be
more optimized, however here we do not discuss the
reduction algorithms.
1256
Res. J. Appl. Sci. Eng. Technol., 4(10): 1250-1259, 2012
on it, we established a serials of constrain model; used
Lagrange resolving method to get the resolution.
Simulation shows that for CPLS in new transmitted
sequences performance is much better than the
conventional LS methods under original transmitted
sequences in both MSE and BER. Furthermore this
method can increase utilization of cyclic prefix and save
and balance OFDM resources although it requires some
complexity for restructuring process. Therefore this new
method our paper discussed must be a good choice to
fulfill the requirement of channel estimation.
Fig. 6: A new simplified OFDM scheme after transmitted
sequences restructured
Furthermore, from Fig. 4, MSE of CPLS after new
transmitted sequences restructured is indeed smaller than
that of ESTS method, this means CPLS can also estimate
channel more accurately.
Figure 4 also proved that under different transmitted
sequences (before or after restructure), CPLS is not
always larger than ESTS method except after restructure
by which we force it to be smaller, though it approach our
purpose kindly. This verified conclusion from the
deduction of Appendix A.
Extension simulation: Unlike the conditions of basic
simulation analysis, here we set 0 = 0.01, g = 0.01 while
keep other conditions stationed in Table 1. Then we do
our simulation.
Gladly we find that CPLS after restructure is the
optimized among the totals although compare to the
former, superiority is not that obvious.
An system model: Conclude Fig. 3 and 4, 5, and combine
Fig. 6, the structure of new OFDM system, we gladly
discover its value for system model. Figure 6 where '
means our restructure process and Rd., the reduction part.
It can be expected that if Rd. process does not make
impact to total system BER, the performance is mainly
determined by CPLS as discussed above.
Therefore, it can be predicted that total system BER
will be lower down.
Appendix A:
From deduction of equally spaced training sequences presentation,
according to MSE of LS estimator, when transmitted sequences are
certain, MSE of LS only determined by input sequences power.
According to Parseval theory, power in time domain relates to frequency
domain as:
 xT
T
2

1
P
 XF
2
F
So we only need compare the power of XQ XCP. Consider following
transformation:
|XQ |2- |XCP| 2 =(XQ – X CP)H (XQ + X CP)-(HHQ XCP - XHCP XQ)
Calculate each part. Because f --and W are diagnosis matrix, therefore
f
H
  H   1 , WH =W-1, besides, f has a period of Q below:
NQ-1
= WMmP(Q-i) = WMmP(-i) = N Q-1
Therefore, using property above:
(XQ!XCP)H (XQ+XCP)
H
 XM
[ I ,  ,..., Q 1  W 1 ]H W H FPH FPW
[ I ,  ,...,  Q 1  W  1] X M
H
 XM
[ I ,  ,..., Q 1  W 1 ]H P[ I ,  ,..., Q 1  W 1] X M
H
 PX M
[ I ,   1,...,  1 Q  W ]T [ I ,  ,..., Q 1  W 1]  m
H
 PX M
AX M
and
H
X QH X CP  X M
[ I ,  ,...,  Q 1 ] H
W H FPH FP [0,0,..., I ] X M
H
 XM
[ I ,  1 ,...,  1Q ]T W 1 P
H
[0,0,..., I ]x M  Px M
Bx M
Therefore,
CONCLUSION
This study proposed a new restructuring method that
improves the performance of cyclic prefix based channel
estimation. We first introduced the cyclic prefix LS
method and then deduce the equally spaced training
sequences presentation to get the relation with it, then
gave a proof of comparing MSE of both methods. Based
H
H H
X CP
X Q  ( X QH X CP ) H  PX M
B XM
Thus,
|XQ |2!|XCP |2 = PXHM (A!B+BH)xM = PxHM CxM
In detail, calculate,
1257
Res. J. Appl. Sci. Eng. Technol., 4(10): 1250-1259, 2012
I
 1


1

I

...
 ...
 1Q
 W  2Q  W

Q 1
QL×L. Pp×L is a matrix whose rank must be smaller than p, thus, QL×L =
PH p×L Pp×L has properties as:
QL×L = Q HL×L , diag (QL×L) = p Rank (QL×L)# Rank (Pp×L ) # p
1


W

Q 2
1 1 

 W

...


1
1
W  W

...
...
...
...
which means it was a hermit matrix, dialogize elements are all p and the
rank is smaller than p.Consider do Eigen value decomposition to QL×L as
follows:
 1 0 

0  
QL  L  q L  L 
  s

0 0 
 0 0 ...
W 1 


1 1
 0 0 ...  W 
B

...
 ... ... ...


1Q 1 
W 
 0 0 ... 
 C  A  B  B
H
 I
 1


 ...
 1Q

 q L  L  q LH L , s  p

I
...
 2 Q
...  Q 1 

...  Q 2 

...
... 

0 
...
Among, qL×L is unitary matrix and because Pp×L is complex matrix,
thus Eigen values of QL×L must be complexes. Meanwhile, the summary
of Eigen value equal to all summary of diagnosis elements. Define the
size of one matrix in the spectrum norm; we get the restriction relations:
 imax ( QL  L  max i )
2

s

st .
i  L * p


i 1
Then, we see that under same x M , the result only determined by the
matrix C, whether it is positive or negative. According the traits of
matrix C, we reform it as:
 I
 1

C
 ...
  1Q


I
...
 2 Q
...  Q 1   I
 
...  Q 2   0

...
...   ...
0   0
...
 ...  Q 1 
0 ...
... ...
0 ...
Appendix B:
We select number p training sequences from number L total transmitted
sequences composing assemble {k1,k2,..., kp}, then:
X p  ( X k 1 , X k 1 ,..., X kp ) 

1


1 1
*
L 



1
2 *k 1*1
j
L
e

2 *k 2*1
j
L
e



2 *kp*1
j
L
e

1
L

Definitely, if QL×L has only one Eigen value, all power will decline
to the Eigen value which make the spectrum norm the biggest. Then rank
of QL×L is one, according to the definition, it means rank of Pp×L must
be one.
Return to find the presentation of Pp×L , its rank is one, we must let:

0 
D
... 
 I 
We find that C and D is equal in their Eigen value, because D has
negative Eigen values -1, therefore it is not positive, therefore C is also
not positive.
Above proved that the result is not always positive, therefore we
get the conclusion that under LS frame MSE of CPLS is not always
smaller nor larger than that of ESTS method although the latter is the
optimized.
T
0

0 H
q
 L L

0
e
2 * k 1
*j
L
e
2 *k 2
*j
L
 ...  e
2 *kp
*j
L
j  1,..., L  1
ki 0 {0, 1, ..., L!1}
]ki = k1 + L / p!1 * (i!1) , i = 1,..., p
This means we have to select the training sequences equally-spaced.
Then, we subtly prove our theory.
ACKNOWLEDGEMENT
* Pp  L * x L
We gratefully thank lecturer Zhang-xin Chen of
Department of EE, University of Electronic Science and
Technology of China, for his Fundamental Research
Funds for the Central Universities (No. ZYGX2011J016)
2 *k 1*( L  1)
j
L
e



2 *k 2*( L  1) 
j
L

e



2 *kp*( L  1) 
j

L
e

REFERENCES
Here |Xp |2 can be represent as:
H
|Xp |2 = XHp = 1/L * XHL PH p×L Pp×L xL = 1/L * x L QL×L xL
From above, whatever x L is, the result |Xp |2 is only determined by
Borching, S., 2008. Blind Channel Estimation using
Redundant Recoding: New Algorithms, Analysis and
Theory [D]. Institute of Technology Pasadena,
California, pp: 1-80.
Christopher, B.S, 2004. Wireless Downlink Schemes in a
class of Frequency Selective Channels with uncertain
Channel State Information [D]. Rice University,
Houston, pp: 1-80.
1258
Res. J. Appl. Sci. Eng. Technol., 4(10): 1250-1259, 2012
Feng, W., 2009. Signal-Perturbation-Free semi-Blind
Channel Estimation for MIMO-OFDM Systems [D].
Concordia University, Montreal, pp: 1-60.
¤LKER, G., 2005. Channel estimation for OFDM systems
[D]. Middle East Technical University, pp: 1-78.
Kathryn, K. and L. Ying, 2005. Channel Estimation of
Frequency Selective Channels for MIMO-OFDM
[D]. University of Calgary, Alberta, pp: 1-62.
Sun, G.C., C.Z. Zhou and Sun-Bei, 2005. LMS adaptive
channel estimation algorithm and simulation. J.
Comp. Appli., 25(6): 1468-1470.
Toshiaki, K.A. and F. Andreas, 2011. Molisch, Chun-jie
Duan, Zhi-feng Tao, Philip Orlik. Capacity, MSE and
secrecy analysis of linear block precoding for
distributed antenna systems in multi-user frequen
selective fading channels, J. IEEE Trans. Commun.,
59(3): 888-900.
Wang, L.G., H.S. Liu and L.X. Zhu, 2008. Design
methods of channel estimation device based on
OFDM system. J. Shipboard Electronic Counter
Measure, 31(4): 101-104.
Wang-Xing, Z. and F. Fang, 2011. A diverse proof of
OFDM channel estimation comparing CP-sequences
and training-sequences methods under LS frame [C].
Proc. of Information Theory and Artificial
Intelligence Computing, IEEE Press, 2: 380-384.
Way, H.H. and L. Yumin, 2004. Low-complexity kalman
channel estimator structures for OFDM systems with
and without virtual carriers. C. Proc. of Communi.
Soci., IEEE Press, pp: 2447-2451.
1259