Research Journal of Applied Sciences, Engineering and Technology 4(8): 897-901, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: October 12, 2011
Accepted: November 25, 2011
Published: April 15, 2012
Compressive Channel Estimation for OFDM Cooperation Networks
1,2
Aihua Zhang, 3Guan Gui and 1Shouyi Yang
1
School of Information Engineering, Zhengzhou University, Zhengzhou, 450001, China
2
School of Electronic and Information Engineering, Zhongyuan University of Technology,
ZhengZhou, 450007, China
3
Department of Electrical and Communication Engineering, Graduate School of Engineering,
Tohoku University, Sendai, 980-8579, Japan
Abstract: In this study, we study compressive channel estimation for Orthogonal Frequency Division
Multiplexing (OFDM) modulated Amplify-and-Forward (AF) relay networks. Based upon the mathematical
channel model of P2P and sparseness measure function, by using Monte-Carlo runs, we show that the OFDM
cooperation convoluted channels also exhibit sparsity. In this study, we propose two compressive channel
estimation methods to exploit the inherent sparse structure in multipath fading channels of OFDM cooperation
networks with Amplify-and-Forward (AF) relays, Simulation results demonstrate that the proposed compressive
channel estimation methods provide significant improvement in mean square error (MSE) performance
compared with the conventional channel estimation methods.
Key words: Amplify-and-Forward (AF), cooperation networks, Compressive Sensing (CS), multipath fading
channel, Orthogonal Frequency Division Multiplexing (OFDM)
received signals and then forward them to the destination
(Laneman et al., 2004). Compared with the DF scheme,
the AF scheme is more effective since the cooperative
terminals do not need to decode their received signals.
Therefore, we focus our attention on the AF relay scheme
in this study.
Many relay studies focused on flat-fading channels,
where single-carrier systems are of interest (Seyfi et al.,
2010; Gao et al., 2008). However, the use of relays in
frequency-selective broadband channels is important as
well. It is well known that the popular solution against
frequency-selective fading is OFDM. So, cooperative
OFDM relay networks are increasingly important.
Recently, several OFDM channel estimation methods for
modulating cooperation communication have been
proposed (Gao et al., 2011; Zhang et al., 2009; Chan
et al., 2007) and these OFDM channel estimation methods
using AF protocol are based on the assumption that the
wireless channels between the terminals and the relay
nodes have rich multipath (Cotter and Rao et al., 2002).
However, recent channel measurements have demonstrate
that the wireless channels tend to exhibit a sparse or
cluster-sparse structure (Adachi et al., 2009) in delayspread domain. In order to take advantage of OFDM
cooperation channel sparsity, two novel compressive
channel estimation schemes are proposed in this study.
Simulation results validate the effectiveness of the
proposed methods.
INTRODUCTION
Cooperative relaying has been researched intensively
for wireless networks in recent years due to its capability
of enhancing the transmission capacity and providing the
spatial diversity for single-antenna wireless transceivers
by employing the relay nodes as virtual antennas
(Laneman and Wornell, 2003; Gao et al., 2008). Multipleinput multiple-output (MIMO) system is a well-known
technique which can boost the capacity and diversity of
wireless communications (Foschini et al., 1999).
However, it is difficult to equip multiple antennas at
mobile terminals because of cost and size limits. In order
to reckon with this problem, relay-based cooperation
communication networks have been proposed recently. It
is expected that relay protocols will be a very promising
method for future high data-rate radio communication
systems.
In cooperative communication system, terminals can
be usually divided into three parts: source, relay, and
destination, source and destination can be the base station
and mobile station, respectively, or vice versa, and the
relay receives signal from source and retransmits to
destination. Generally, there are two kinds of protocols in
cooperation networks: the Amplify-and-Forward (AF)
scheme and Decode-and-Forward (DF) scheme. In the AF
relaying, the relay amplifies and retransmits its received
noisy signal without decoding it. In the DF relaying, the
relay terminals demodulate and modulate again the
Corresponding Author: Aihua Zhang, School of Information Engineering, Zhengzhou University, Zhengzhou, 450001, China
897
Res. J. Appl. Sci. Eng. Technol., 4(8): 897-901, 2012
Fig.1: A typical sparse multipath cooperation network adopting OFDM modulation
SYSTEM MODEL
Transmitted signal at the source: Suppose that each
OFDM block contains N information symbols and denote
the frequency domain training vector from S as
~
x [~
x0 , ~
x1 ,..., ~
x N 1 ] . The corresponding time-domain signal
vector can be obtained from the normalized inverse
discrete Fourier transformation (IDFT) as:
Relay transmission model: Consider a three-node
OFDM cooperation network with two terminals S, D and
one relay node R is shown in Fig. 1. We assume that all
the terminals are equipped with only one antenna and
work in the half-duplex mode, therefore, they cannot
receive and transmit simultaneously.
In a relay networks, data transmission is usually
divided into two phases. The source broadcasts its own
information and the relay forwards its received signal to
the destination.
Due to the limited transmit power and the multipath
fading channel, we assume that there is no direct path
between S and D as shown in Fig. 1. Therefore,
frequency-selective multipath channels will generate
multiple delayed and attenuated copies of the transmitted
waveform. Source and relay are assumed to have average
power constraints in PS and PR respectively. Assume that
the channel impulse response between S and R is h1(t) ,
which is constant within one transmission period and
represented by
h1 
L1  1
h
l0
1, l
(t ) (   1, l )
x FH~
x  [~
x0 , ~
x1 ,..., ~
x N 1 ]
(3)
where F is the discrete Fourier transformation (DFT)
matrix with the (m,n) -th entity given by
F  1 / N exp(  j 2mn / N ),(m, n  0,1, ..., N ). In order to
avoid the Inter-block Interference (IBI), S inserts the
cyclic prefix (CP) of length Lp in the front of each OFDM
block before the transmission, and Lp should satisfy that
Lp  max(L1-1,L2-1). After performing DFT and inserting
CP, the transmitted signal vector can be written as:

S  x N  Lp , ,... x N 1 , x0, ..., x N 1

(4)
The received signals at the relay node and the
destination: In the AF relay transmission, the received
signal at the relay is amplified by a relay factor " . Over
a doubly selective channel between the source S and the
relay R, after removing CP, the received signal at the
relay node can be represented as (Adachi et al., 2009)
(1)
where h1,l is the complex-valued path and it satisfies
1
E[ l  0 h1, l ]  1, and J1,l denotes the symbol-spaced
L 1
y1 = H1x+n1
2
where H1 is an N×N circulated channel matrix with
[hT1 0 1×(N-L)]Tas its first column, n1 is the complex additive
Gaussian white noise (AWGN) with zero mean and
covariance matrix E[n1nH1] = F2nIN.
The received signal at the destination D, which is
from relay R, can be written as:
time delay of the lth path, and L1 is the length of the
channel between S and R.
Due to the same channel property as Eq. (1), channel
vector h2 between R and D is given by the follows:
h2 
L2 1
h
l0
2, l
(t ) (   2 , l )
(5)
(2)
y2 = "H2y1+n2 = "H2H1x+n
(6)
where n = "H1n1+n2 is the composite noise with zero
mean and covariance matrix E{nnH} = "2F2n|H2|2+lN.
The amplified positive coefficient " is given by:
where h2,l and J2,l denote the complex-valued path and the
lth path symbol-spaced time delay , respectively. L2 is the
length of the channel between R and D:
898
Res. J. Appl. Sci. Eng. Technol., 4(8): 897-901, 2012


PR
l0
said to satisfy RIP of order S, and accurate channel
estimator with high probability can be obtained by using
CS methods. Although it is quite difficult to judge
whether a given matrix satisfies this condition, it has been
demonstrate that many matrices satisfy the RIC with high
property and few measurements. In particular, it has been
shown that with exponentially high probability, random
Gaussian, Bernoulli, and partial Fourier matrices satisfy
the RIC with number of measurements nearly linear in the
sparsity level.
(7)
 12, l P1   l  0  22, l
L1  1
L2  1
According to the matrix theory, matrices H1 and H2
can be de-composited as H1 = FH 71F and H2 = FH72F
(Gray, 2006), respectively. F is the discrete Fourier
transformation (DFT) matrix. Thus, system model (6) can
be rewritten as:
y2 = FH "7271Fx + n
(8)
Sparse channel estimation: Because the channel impulse
responses h is sparse, its convolution from h1 and h2 have
been verified to be sparse or approximate sparse (Gui
et al., 2011). Numerous practical algorithms exist for the
cooperation convoluted channel h. Generally, these
algorithms can be classified into two kinds. The first kind
is greedy algorithms such as Orthogonal Matching Pursuit
(Omp) and Compressive Sampling Matching Pursuit
(CoSaMP), which selects each dominant coefficient in
channel by iteration. The second kind is convex relation
methods such as Lasso. In this study, we also consider the
LS channel estimator for comparison. The LS estimator is
written as the follows.
We define the convolution channel vector h = " (h1* h2)
with the length of (L1 + L2 -1). After left-multiplying by F,
Eq. (8) can be rewritten as:
~
y  XWh  n  Xh  n
(9)
where X = diag (Fx) denotes the training signal matrix,
~
X = diag (Fx)W, W is a matrix taking the first (L1+L2-1)
columns of N F and n   2 Fn1  Fn2 is a realization of
a complex Gaussian random vector.
COMPRESSIVE CHANNEL ESTIMATION
 X~ Ty
h  
0
Overview of compressive sensing: Compressed Sensing
(CS) theory has recently gained a fast-growing interest in
applied mathematics and signal processing communities.
And it has been applied in various areas, such as imaging,
radar, speech recognition, data acquisition, etc. In
communications, an immediate application of CS is in
wireless sparse multipath channel estimation (Gui et al.,
2011; Bajwa et al., 2008).
In this study, we consider the linear model as (9).
According to the CS, if an unknown signal vector satisfies
sparse or approximate sparse, then a designed
measurement matrix X can accurately capture most of its
dominant information. Hence, these kinds of unknown
signal can be reconstructed from observation signal y
However; the sparsest solution is always a Nondeterministic Polynomial-time hard (NP-hard) problem. In
order to solve this problem, several methods have been
proposed. Donoho (Donoho, 2006) presented a necessary
and sufficient condition (RIP) on the sensing matrix.
Candès and Tao (Cand, 2008; Candes and Tao, 2005)
introduced the Restricted Isometry Constants (RIC) of the
matrix. Suppose that X be a n×p complex-valued
measurement matrix, which has unit R2-norm columns.
The X satisfies the RIP of order S with parameter
*s,(0,1), which can satisfy the following inequality:
2
~ 2
(1   S ) h 2  Xh  (1  s) h
2
2
T  Suup h
others
(11)
where suup(h) denotes the nonzero taps supporting the
channel vector h,XT is the submatrix constructed from the
columns of X, and T denotes the selected subcolumns
corresponding to the nonzero index set of the channel
vector h. The MSE of LS estimator h is given by (Bajwa
et al., 2008):

MSE h   n2 Tr
 X~

T
~
XT

1

(12)
By utilizing CS recovery algorithms for compressive
channel estimation, we propose CCS-Lasso, CCSCoSaMP. The two methods for cooperation convoluted
channel estimate are described as follows:
C
CCS-Lasso estimator hLasso : Given the received
signal y, the unitary DFT matrix F and W, training
~
signal matrix X = diag (Fx)W the regularized
parameter    2log N The CCS-Lasso (Gui et al.,
2011) channel estimator h Lasso given by:
1

~ 2
hLasso  arg min y  Xh 2   h 1 
2


(10)
where 5h522denotes the R2-norm which is given by
2
2
h 2   hi . If (10) is satisfied, the training sequence is
C
899
(13)
CCS-CoSaMP estimator hCosaMP: Given y,F and W
~
and training signal matrix X = diag (Fx)W the
Res. J. Appl. Sci. Eng. Technol., 4(8): 897-901, 2012
C
0
10
B (LS)
B (CCS-Lasso)
B (CCS-CoSaMP)
B (LS:known position)
-1
10
-2
Average MSE
C
maximum number of dominant channel coefficients
is assumed as S. The CCS-CoSaMP method (Gui et
al., 2011) can be described as follows:
Set T0 = Ø, r0 = y,T is nonzero coefficient index, and
r is the residual estimation error. Set the initialize
iteration index k = 1.
Select a column index nk of X which is most
correlated with the residual:
10
-3
10
10-4
-5
10
-6
~
nk  rk 1 , X n and Tk  Tk 1  n k
10
(14)
-7
10
0
C
C
Here we use LS method to calculate a channel
~
estimator as TLs = argmin5y- X h5, and select T
maximum dominant taps hLs . Where TLS denotes the
positions of the selected dominant taps in this
substep.
-2
(15)
Find the dominant channel coefficients, and replace
the left taps T/Tk by zero:
10
30
-3
-4
10
-5
10
-6
10
-7
10
(16)
Update the estimation residual:
5
10
15
SNR (dB)
20
25
30
Fig. 3: MSE perfomance as a function of SNR (number of
nonzero taps of h1 and h2 are 4)
~
rk  y  X Tk hk
(17)
where h and h denote channel vector and its estimator,
respectively. M is the number of Monte Carlo runs and
((L1 + L2-1) is the overall length of channel vector h . In
Fig. 2, the number of nonzero taps of hi,(i = 1, 2) is set to
2, the cooperation convoluted channel also has sparsity.
Figure 2 shows that performance of the proposed CCS
estimator methods outperforms LS estimator and it is
close to the ideal LS estimator by using known position of
the channel. In addition, under low SNR (less than 10dB),
the proposed CCS-Lasso estimator achieves even better
performance than the LS estimator (known position). In
Fig. 3, the number of nonzero taps of hi,(i = 1, 2) , is set
to be 4.
From the simulation results in Fig. 2 and 3, it can be
seen that the proposed estimators can exploit the channel
sparseness. And if channels are dense rather than sparse,
all of the proposed estimators will have the same
performance as LS estimator.
Increment the iteration counter k. Repeat (13)-(16)
until stopping criterion holds and then set
hk.
h
CoSaMP
=
SIMULATION RESULTS
In this study, we adopt 10000 independent MonteCarlo runs for average. The length of training sequence is
N = 256. All of the nonzero taps of sparse channel vectors
h1 and h2 are generated by Gaussian distribution and
subject to 5h1522 = 5h2522=1. The length of the two
channels is L1 = L2 = 32, and the positions of nonzero
channel taps are randomly generated. We set transmit
power equal to AF relay power, that is Ps = PR = P. The
Signal to Noise Ratio (SNR) is defined as S as 10 log
P/F2n). When the number of nonzero taps in cooperation
channels hi,( i = 1,2) is changed, the simulation results are
shown in Fig. 2 and 3.
Channel estimators are evaluated by average Mean
Square Error (MSE) which is defined by:
averageMSE ( h) 
25
10
0
hk = [h]S
20
B (LS)
B (CCS-Lasso)
B (CCS-CoSaMP)
B (LS:known position)
10
2
15
SNR (dB)
0
10
Update the dominant taps by Tk  TLs  Tk
Channel estimation,
h
C
10
-1
~
hk  arg min y  X Tk h
C
5
Fig. 2: MSE perfomance as a function of SNR (number of
nonzero taps of h1 and h2 are 2)
Average MSE
C
h  h
CONCLUSION
2
2
M ( L1  L2  1)
,
In this study, we introduce OFDM channel
estimations for the AF cooperative channel. Distinct from
the conventional linear channel estimation methods, we
(18)
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Res. J. Appl. Sci. Eng. Technol., 4(8): 897-901, 2012
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proposed compressive channel estimation methods for the
OFDM cooperation networks under AF protocol.
Sparseness of OFDM cooperation convoluted channel
was demonstrated by a measure function. The proposed
methods have exploited the sparsity in OFDM
cooperation channel. Simulation results have confirmed
the performance superiority of the proposed method to the
conventional linear LS method.
ACKNOWLEDGMENT
This study is funded by the National Natural Science
Foundation of China (No. 61071175).
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