Communications and Network, 2013, 5, 93-97
http://dx.doi.org/10.4236/cn.2013.53B2018 Published Online September 2013 (http://www.scirp.org/journal/cn)
Heuristic Channel Estimation Based on Compressive
Sensing in LTE Downlink Channel
Lin Wan1, Min Wang1,2, Lifen g Su1, Jun Wu1
1College of Electronics and Information Engineering, Tongji University, Shanghai, China
2School of Mathematics and Computer Science, Gannan Normal University, Ganzhou, China
Received June, 2013
ABSTRACT
Pilot-assisted channel estimation has been investigated to improve the performance of OFDM based LTE systems. LS
and MMSE method do not perform excellently because they do not consider the inherent sparse feature of wireless
channel. The sparse feature of channel impulse response satisfies the requirement of using compressive sensing (CS)
theory, which has recently gained much attention in signal processing. Result in the application of using compressive
sensing to estimate fading channel. And it achieves a much better performance than that with traditional methods. In
this paper, we propose heuristic channel estimation based on CS in LTE Downlink channel. According to the feature of
recovery algorithm in CS, we design a modified pilot placement method. CS recovery algorithms for channel estimation
don’t consider the statistics character of channel. So we proposed an optimization method which combines the CS and
noise reduction. First we get initial channel statistics obtained by LS. Let the channel statistics as the heuristic informa-
tion input of CS recovery algorithm. Then we perform CS recovery algorithm to estimate channel. Simulation results
show this approach significantly reduces the complexity of channel estimation and get a better mean square error (MSE)
performance.
Keywords: Channel Estimation; Compressed Sensing; Sparse Channel; LTE; Noise Reduction
1. Introduction
OFDM modulation is widely used in LTE systems which
suffer from time and frequency fading channels. How-
ever, the performance of wireless system relies heavily
on the validity of OFDM channel estimation. Generally,
we can achieve better performance through two approaches,
e.g., pilot design and channel estimate methods. Several
pilot-aided channel estimation schemes have been discussed
in [1-4], and measured the performance in terms of bit
error rate (BER) and symbol error rate (SER). Many
channel estimation methods are proposed to improve the
performance of wireless communication system. LS
algorithm [1], as the simplest method, has very low
complexity, but it is extremely sensitive to AWGN noise.
The MMSE algorithm [2] yields much better perfor-
mance than LS estimator. Its high complexity hinders the
implementation practically because matrix inversion is
needed each time [3]. Although the complexity of
MMSE is reduced by deriving singular value decom-
position techniques, it highly depends on the detail
channel statistics [4].
Wireless channels in practice are typically sparse [5],
with very few of channel impulse response is nonzero.
Traditional channel estimation methods mentioned above
don’t take the sparse feature of the channel into account,
this result in much more detection errors. Recently years,
CS theory has gained much attention in mathematics and
signal and communication processing. CS theory de-
clares that signals can still be recovered exactly from
sample signal at a rate much less than Nyquist sampling,
if the signals are sparse [6]. Considering the sparse fea-
ture of impulse respond in wireless fading channel, it is
possible that CS can be applied on wireless channel es-
timation. Zhang et al [7] proposed an optimization of or-
thogonal matching pursuit (OMP) algorithm, according
to the channel characteristics, to increase the precision of
the algorithm and reduce the complexity of the algorithm
with the same number of iterations. But the approach still
need much more compute while it cannot find the useful
value in the index set, more iterations mean more com-
plexity and calculating delay. ALL of the available
methods used to channel estimation based on CS are sen-
sitive to AWGN noise [8], especially at low signal to
noise ratio SNRs, as well as sparse recovery algorithms.
Inspired by this, we proposed a novel optimization
method for OMP, which is that just need one iteration
*This work is financially supported by NSFC General Program under
contract No.61173041.
C
opyright © 2013 SciRes. CN

L. WAN ET AL.
94
with the channel characteristics abstracted from LS esti-
mation. From the LS estimation, we can find the useful
value position and an approximate channel impulse re-
spond, which help us to avoid iteration to find the posi-
tion and errors location caused by noise at low SNRs.
This paper is organized as follows: Section II reviews
compressive sensing theory. In addition, introduces the
LTE downlink system model and the sparsely of the
multi-path blocking fading channel. Section III gives
modified pilot placement strategy, and proposes heuristic
channel estimation algorithm based on CS. Section IV is
the simulation and analysis of proposed channel estima-
tion methods. Section V is conclusions.
2. Background
In this section, we first review compressive sensing the-
ory. Then, we present the LTE framework model for
multi-path block-fading channel.
2.1. Compressive Sensing
Compressive sensing is a revitalized theory in signal
processing for sparse and compressible signals. Mathe-
matically let signal
N
x
Rbe a vector of. Assum-
ing signal
1N
x
is k-sparse under the orthogonal basis

.
Let
M
y
R be a set of linear measurements of
x
by
using a measurement matrix

, which is not related to
the sparse basis.

cs
y
xA

   (1)
where

is the signal x projected on the basis of

. A
is considered as the operator of CS to get observation of

. In order to be able to recover the original signal

from the observation y, CS must satisfy two essential
conditions.
 For accurate reconstruction, the number of observa-
tion M should meet

log /
K
cMN M, where c is
sufficiently small, N is the length of original signal.
 A should satisfy the restricted isometric property
(RIP) [9], which is essential to CS recovery performance.
Signal recovery is the center of CS theory research, it
can be considered as optimization problem. Under the
sparsely of k, it can get

from (1) by solving -
norm minimization problem
0

0
min. .
N
xR xstyA
2


 (2)
This is combination and NP hard problem. Donoho et
al [10] proposed that it can be replaced by an optimiza-
tion -norm problem
1

1
min. .
N
xR xstyA
2


 (3)
In recent years, a collection of sparse recovery algo-
rithms has emerged with CS. Methods for solving CS re-
covery problems can be roughly divided into two classes,
including greedy algorithms and convex optimization
algorithms. Orthogonal matching pursuit (OMP) algo-
rithm is commonly employed to recovery signal. Tropp
et al [11] proposed signal recovery from random meas-
urements by OMP. Reference [12, 13] proposed subspace
pursuit (SP) and compressive sampling matching pursuit
(CoSaMP) to reconstruct signal, respectively.
2.2. System Model
In this paper, we only consider single-antenna case in
LTE downlink system, because it is easy extending from
single-antenna case to multi-antenna case. The system
model is shown in Figure 1.
The complex baseband representation of a multi-path
fading channel impulse response can be described by
1
()()
M
mm
ms
g
tt  


T
(4)
where
s
T is the sampling interval, m

is the delay of
path m, m

is a complex value that characterizes the
attenuation and initial phase of path m. denoted as
the time length of a cyclic extension, and
G
T
0ms G TT

.
Let

T
k
X
x

T
n
 and (k = 0,……, N-1)
denote the frequency domain data at the transmitter and
the frequency domain data at the receiver, respectively.
Let

T
k
Yy
g
g, n and (n = 0,……, N-1), denote
the time domain sampled channel Impulse response and
zero-mean, white, complex Gaussian noise, respectively.
Define the DFT matrix as,

nn

T
000( 1)
(1)0 (1)(1)
N
NN
NNN
NN
N
N
WW
F
WW









 

Figure 1. LTE downlink system model.

L. WAN ET AL. 95
,2
1
nkjnk N
N
We
N
 



/
.
Furthermore, define ()
N
F
DFT gFg, the frequency
domain channel impulse respond. NF is the frequency
domain noise. NT is the time domain noise. Under the
assumption that the interferences are completely elimi-
nated, we can derive
(()))
N
N
YDFTIDFTXg N
T
 (5)
F
X
Fg N (6)
F
X
HN (7)
3. Channel Estimation Based on
Compressive Sensing
In this section, we present our channel estimation scheme
base on compressive sensing. We first give the modifica-
tion of OFDM pilot placement in standard LTE system.
Then, we give the derivation the principle of channel
estimation combined with CS, and propose our optimiza-
tion algorithm.
3.1. Pilot Placement
LTE downlink system applies the comb-type pilot place-
ment. Single-antenna system pilot placement in one re-
source block is shown in Figure 2. The left figure depicts
the pilot placement in standard LTE system. We note that
the pilot placement of standard is uniform distribute in
2D plane, which composed by frequency domain and sub
carriers. The channel responses of non pilot sub-carriers
are estimated by interpolating neighboring pilot in sub
channels. Suppose the block fading channel, the channel
coefficients at the same subcarrier position within on
block are all the same. So we can completely estimate the
channel coefficients only by one-dimensional interpola-
tion.
In order to achieve better performance of channel es-
timation by using CS signal recovery algorithms, we
should modify the position of the pilot placement in 2D
plane. There is a important factors to consider in pilot
placement modification. That is the modification does a
little change to the LTE system. So we must considering
the LTE pilot placement and frame structure. [14] Ex-
press that CS-based channel estimation scheme can achieve
better performance when use random pilot placement. In
standard LTE pilot placement, every block has to provide
four indexes to place pilot in twelve placements. So there
is C4 12 combination. Through 1000 Monte carols simu-
lation of random pilot placement; we find that the modi-
fied pilot placement, shown in the right figure of Figure
2, can achieve the best performance of channel estima-
tion.
0

l
0
R
0
R
0
R
0
R
5

l 0

l
0
R
0
R
0
R
0
R
5

l 0

l
0
R
5l0l
0
R
0
R
5l
Figure 2. Pilot placement.
3.2. Heuristic Channel Estimation Based on CS
Traditional LS estimator minimizes the parameter by

ˆˆ
ˆ
arg minH
LSp pp p
H
YXH YXH (8)
where

H

means the conjugate transpose operation,
p
Y and
p
X
denote as received and transmitted pilot
value, respectively. LS estimator get initial estimation
from pilots and interpolate to all the data position, we can
get all the channel frequency respond. The pilot signal
vector can be expressed as
p
s
YX FgN
F

 (9)
where
s

is an identity matrix with dimensions SN

.
01 000
00 1 00
00 0 10
S
SN
















  

Because the pilot has been known at the receiver, the
impulse response of the channel at the pilot points is de-
scribed as
ˆp
p
Y
H
X
 (10)
s
F
F
gN

 (11)
F
g
N

 (12)
where s
F

 is called sensing matrix in the CS the-
ory. Therefore, we can use the solution of CS methods
for channel estimation. Once we recovery the channel
impulse respond from CS algorithm, we can get the CS
estimation.
ˆ()
cs
H
FFT g (13)
If the channel impulse response is k-sparse, the recov-
ery algorithm needs k iteration at least. So the CS-based
channel estimation algorithm still has very high com-
plexity. Furthermore, the accuracy of estimation maybe
much lower at low SNRs, result in the decrease of per-
formance.

L. WAN ET AL.
96
In this subsection, we proposed a fast solution based
on the combination of OMP method and channel statis-
tics. The channel statistics can be obtained from LS es-
timation. First get initial estimation from pilots and then
interpolate to all the data position, we can get all the
channel frequency response. Evaluating the time domain
estimation, we collect k of the most important taps,
where the importance means the valve is large than oth-
ers. And let k equals the length of a cyclic extension. We
record the position of the collected taps and sort them in
descend by module value. Here we called the collected
set as the delay profile of channel, a useful index that
used for OMP recovery. Here, the useful index is denoted
by Θ. Known the sparsely and the delay profile of chan-
nel, it means that we do not need iterate to find the pos-
sible position. With the Θ, we can increase the precision
and reduce the complexity of the algorithm to one.
Algorithm 1 OMP recovery
Input: CS observation y, measurement matrix mn R


,
Index I=Θ, residual r=y, sparse representation
N
g
R.
Output:
1: while stopping criterion false do
2:r = y − Ω(:,I)[Ω(:,I)]†y;
3:g(I) = [Ω(:,I)]†y;
4: end while
5: return sparse representation g.
4. Simulation and Results
We performed computer simulations to verify the per-
formance of the proposed methods applied to the LTE
downlink system. We simulated a LTE downlink system
with 20 MHz bandwidth with N = 2048 sub-carriers. The
channel model uses Vehicle model. Table 1 shows the
profile of the channel parameters. Table II lists the simu-
lation parameters of LTE systems. Under the same experi-
mental condition, we compare the performance among
LS, MMSE and proposed CS method by using normal-
ized MSE and bit errors ratio (BER), respectively. The
normalized MSE is defined as
2
2
ˆ
() ()
()
EiHi Hi
MSE EiHi


 (14)
Figure 3 depicts the comparison of the normalized
MSE for standard pilot placement and optimized pilot
placement. In this simulation, we perform LS and CS
channel estimation algorithm for two pilot placement
strategy respectively, and SNR range form -5 dB to 30
dB. It shows that two pilot strategies have the same per-
formance at low SNRs. But in LS estimation, comparing
to standard pilot placement, the performance of opti-
mized pilot placement is slight worse at high SNRs. The
reason of this phenomenon is that the uniform distribu-
tion averages the noise at LS interpolating, while the
rand distribution does not have the ability when the noise
is low enough at high SNRs.
Figure 4 shows the normalized MSE performance
among LS, MMSE, CS and CS-optimized channel esti-
mation. At low SNRs range, the CS estimation approxi-
mation effect is significantly better than LS, while much
worse than MMSE. The reason is the noise. For the op-
timized CS estimation, its performance is much better
than CS at lows SNRs, and as well at high SNRs. The
reason is that the CS itself can find the right position as
CS-OPT at high SNRs. However CS needs more com-
plexity and iterations.
Figure 3. MSE Performance comparison between standard
pilot placement and optimized pilot placement.
Figure 4. MSE Performance comparison among CS-OPT,
CS, LS and MMSE channel estimation.

L. WAN ET AL.
97
Figure 5. BER performance comparison among CS-OPT,
CS, LS and MMSE channel estimation.
Figure 5 describes the BER performance of LS, MMSE,
CS and CS-optimized channel estimation. It is seen that
the BER performance of CS is much better than LS. It
uses the sparse of the channel and exactly recovery some
of channel impulse respond at low SNRs, and well-done
impulse at high SNRs. The performance of LS is the
worst because it does nothing about noise reduction and
interpolation, which may cause much more errors.
MMSE become the best one except Ideal along with the
noise reduction by the channel autocorrelation. The op-
timized CS estimation. It is seen that at lows SNRs CS-
OPT is much better than CS method; because of at high
SNRs the CS itself can find the right position the same as
CS-OPT.
5. Conclusions
In this paper, we study the channel estimation method
based on compressive sensing theory. We first present
modified pilot placement strategy to suit CS channel es-
timation. We propose an optimization recovery algorithm
based on OMP by using the channel statistics. Our simu-
lation results demonstrate that the optimization of the
CS-based channel estimation algorithm significantly pro-
motes the performance compared to the traditional esti-
mation and the reduction of complexity for CS recovery
contributes to its implementation.
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