iterative compensated mmse channel estimation in lte systems
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Iterative Compensated MMSE Channel Estimation
in LTE Systems
Yang LIUSequans Communications
Email: [email protected]
Serdar SEZGINERSequans Communications
Email: [email protected]
AbstractIn this paper, an iterative channel estimation algo-rithm is considered in LTE systems. In order to take advantageof null sub-carriers (guard band) in LTE systems and make thealgorithm simpler, an iterative compensated MMSE (IC-MMSE)channel estimation algorithm is proposed in frequency domain.Together with a simple linear interpolation in time domain,channel estimates are obtained in an iterative way with lowercomplexity. Theoretical analysis shows that the compensationprocess does not affect the accuracy of iterative MMSE channelestimation. Simulation results confirm that the proposed channel
estimation has very robust performance and approaches the bestachievable performance.
I. INTRODUCTION
In recent years, with the appearance of turbo principle [1],
iterative receivers are becoming more and more popular and
promising because of their excellent performances. Different
mechanisms have been proposed and studied, for example,
iterative detection, iterative multi-input multi-output (MIMO)
equalization, etc. However, these iterative mechanisms are
seriously affected by channel estimator. For example, in [2], it
has been shown that an iterative MIMO equalizer is sensitive
to channel estimation, and the traditional non-iterative channelestimators cannot provide sufficiently accurate channel esti-
mates. This necessitates more accurate channel estimates to
improve system performances.
Recently, iterative channel estimation is being considered
to improve the accuracy of channel estimation, which uses
the soft information of data to improve channel estimation
performance. This type of channel estimation algorithms is
particularly helpful for systems which have fewer and/or
lower powered pilot symbols. For example, in Long Term
Evolution (LTE) systems, at most 2 orthogonal frequency-
division multiplexing (OFDM) symbols carry pilots in a given
resource block (RB = minimum allocation unit over 7 OFDM
symbols with normal cyclic prefix and 12 sub-carriers) and thisdecreases to 1 OFDM symbol for MIMO transmission [3].
With this sparse pilot arrangement, the iterative channel es-
timation can be a good candidate to improve channel esti-
mation performance. Moreover, for future standards, one of
the key features will be power efficiency and, in this manner,
decreasing the power of pilots is one of the possible ways to
improve the power consumption efficiency. In such systems,
the channel estimation algorithms used in current systems will
have less accuracy and more robust algorithms will be needed.
Some iterative channel estimators have already been pro-
posed for OFDM systems by using the soft information from
decoder. Among these iterative algorithms, the iterative mini-
mum mean square error (MMSE) channel estimator provides
excellent performances which approach the performance with
perfect channel state information (CSI). The iterative MMSE
is based on the traditional MMSE channel estimator defined
in [4] and improves the accuracy of channel estimation by
using the soft information from channel decoder. However,the complexity of the iterative MMSE channel estimator is
high due to the matrix inversion which has to be performed at
each iteration. Furthermore, in LTE systems, the distributed
resource allocation is used to vary RB positions in differ-
ent OFDM symbols. With the iterative MMSE algorithm,
the allocated RB positions have to be pinpointed and the
matrix to be inverted is different from one OFDM symbol
to another one. This process adds more complexities to the
iterative MMSE algorithm. Therefore, a novel iterative MMSE
algorithm which is not sensitive to RB positions is desirable.
In order to solve these problems, we propose a so-called
iterative compensated MMSE (IC-MMSE) channel estimation
algorithm. This method takes advantage of the null sub-carriers to reduce the complexity and to improve the accuracy
of the MMSE channel estimation. In the sequel, we refer to
LTE systems as an example. However, it is worth mentioning
that the idea can be generalized to any OFDM(A)-based
communication system.
The rest of the paper is organized as follows. In Section II,
the interpolation based channel estimation method in LTE
systems is briefly explained. Next, in Section III, the proposed
IC-MMSE algorithm is described in both single-input multi-
output (SIMO) and MIMO transmission systems. Then, the
impact of the compensation process and the complexity of the
proposed algorithm are analyzed in Section IV and Section V,
respectively. Simulation results are shown in Section VI.Finally, conclusions are drawn in Section VII.
I I . INTERPOLATION METHOD IN LTE
In a practical system, in order to have simple channel
estimation, usually we implement 2x1D interpolation method.
According to different strategies, we can perform interpola-
tions first in frequency domain or in time domain. In this paper,
we will focus on the approach where frequency interpolation
is applied first, as depicted in Fig. 1. For time domain
IEEE ICC 2012 - Wireless Communications Symposium
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interpolation, we consider the simple linear interpolation;
for frequency domain channel estimation, an MMSE based
iterative algorithm will be discussed in the following.
Frequency domain interpolation
Time domain interpolation
Channel estimates on RS
Fig. 1. Conventional interpolation based channel estimation in an LTE sub-frame.
III. ITERATIVE COMPENSATED MMSE CHANNEL
ESTIMATION
A. Traditional Iterative MMSE
By considering the soft information from the decoder, the
traditional iterative MMSE channel estimation at the (i + 1)th
iteration h(i+1)MMSE can be formulated as
h(i+1)MMSE = L
LR(i)NNL +
2C1gg
1LX(i)y, (1)
where ()
stands for transpose-conjugate and ()
stands for
complex conjugate. Here, y represents the received signal
vector, L is a matrix consisting of the first L (L representing
the delay spread of channel) columns of the FFT matrix,
X(i) represents a diagonal matrix of soft symbols containing
a posteriori probabilities (APPs) of the data Xkk at the ith
iteration, the matrix R(i)NN is defined asR(i)
NN =CQ
AP Pi(X = C)CC. (2)
Here, Q represents the set of all possible codeword matrices of
X and the matrix C stands for one realization from Q. Cgg is
the auto-covariance matrix of channel impulse response g and
2 denotes the noise variance. From (1), the complexity of the
traditional iterative MMSE channel estimator is high due to the
matrix inversion of size L L which has to be performed at
each iteration of the estimation process, because the value of
the matrix
R
(i)NN changes at each iteration. Furthermore, in
LTE systems, the distributed resource allocation is used to vary
RB positions in different OFDM symbols. With the traditional
iterative MMSE algorithm, the allocated RB positions have to
be pinpointed and the matrix to be inverted is different from
one OFDM symbol to another one. This process adds more
complexities to the traditional iterative MMSE algorithm.
B. Iterative Compensated MMSE (IC-MMSE)
In order to solve the problems of the traditional iterative
MMSE, we propose an iterative compensated MMSE channel
estimation algorithm.
1) Compensation Process: One key feature is the com-
pensation process which is applied on the values of the
estimates which correspond to the null sub-carriers, as shown
in Fig. 2. By using some kind of initial channel estimation
algorithms, it is possible to obtain channel estimates over
all sub-carriers based on pilot symbols. For example, the
simple least square (LS) channel estimation can be a candidate.
This initial channel estimate is noted as h(0), which can be
separated into two parts
h(0) =
h(0)TN , h
(0)TDP
T, (3)
where the vector h(0)DP represents the channel estimates on
modulated sub-carriers, including data and pilot symbols and
vector h(0)N represents the channel estimates on the null sub-
carriers. Since the null sub-carriers normally exist at both sides
of bandwidth, the channel estimates h(0)N can be expressed as
h(0)N =
h(0)TN,1 , h
(0)TN,2
T, as shown in Fig. 2.
Compensate
Compensate
h(i)DP
h(i+1)IC-MMSEh
(i)IC-MMSE
h(i)N,0
h(i)
N,1 EP
h(i)
N,1
X(i) DP
y
EPh(i)N,0
Fig. 2. Compensation process of IC-MMSE.
In the traditional iterative MMSE channel estimation, soft
information is produced by channel decoder during current
iteration and used in the next iteration to build soft symbolsX(i)DP. However, no soft information is available on the nullsub-carriers, since no symbol is transmitted on this part. In
order to simplify the calculations made during the iterations,
we propose to copy the channel estimates on the null sub-
carriers from the current iteration h(i)N to the next (i + 1)th
iteration at the same positions as in the (i)th iteration. It isalso assumed that, for the copied part, the channel estimates
are obtained from pilot symbols and the transmitted power on
this part is equal to the power of pilot symbols, denoted as
EP. With this assumption, it can be seen as pilot symbols are
transmitted on the null part where, actually, no symbol exists.
Compared with the actual pilot symbols, these pilot symbols
are considered as virtual pilot symbols. This copy process is
called compensation, as shown in Fig. 2. The compensation
process can be described as:
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1 In the initial channel estimation, use pilot symbols to
obtain channel estimates over all sub-carriers in the
vector h(0);
2 Split current vector h(i) (i = 0, 1, 2, ) into two sub-
vectors
h(i)N , h
(i)DP
, where the sub-vector h
(i)DP relates to
data and pilot symbols and the sub-vector h(i)N relates to
null sub-carriers;
3 Copy the channel estimates on null sub-carrier parth(i)N to the next iteration by considering the power of
virtual pilot symbols EP;4 Implement iterative MMSE channel estimation by con-
sidering soft symbols (X(i)), actual pilot symbols(XP), and virtual pilot symbols, as shown in Sec-
tion III-B2;
5 Make i = i + 1 and repeat step 2), 3), 4) and 5).
2) IC-MMSE: With the compensation process described in
section III-B, the IC-MMSE channel estimation at the (i+1)thiteration h
(i+1)IC-MMSE can be written as
h(i+1)
IC-MMSE =L L EP, R(i)NDPNDPL + 2C1gg
1
L
EPh
(i)N ,
X(i)DP y , (4)where EP = EPINNDP and INNDP stands for an identitymatrix of size N NDP, N and NDP being the number of
all sub-carriers and the number of modulated sub-carriers
respectively, the matrix
EP, R(i)NDPNDP represents a diagonalmatrix defined as
EP,
R(i)NDPNDP
=
EP, 1 0
R
(i)NDPNDP
0 EP, 2
, (5)
andEPh
(i)N ,
X(i)DP y is an N1 vector, where EP, 1 and EP, 2correspond to h
(i)N,1 and h
(i)N,2, the diagonal matrix
X(i) includesall soft symbols and pilot symbols and the vector y represents
the received symbols. Compared with the traditional iterative
MMSE channel estimation in (1), due to the compensation
process, (4) replaces the matrix R(i)NN with
EP, R(i)NDPNDP
and replaces the vector X(i)y with EPh(i)N , X(i)DP y. The di-agonal entries of the matrix
EP, R(i)NDPNDP and the elements
of the vector
EPh
(i)N ,
X(i)DP y
are arranged corresponding to
the correct pattern of pilot and data arrangement.Then, considering that the average transmit powers over all
sub-carriers are the same, (4) can be approximated as
h(i+1)IC-MMSE L
L
L +2
EavC1gg
1L
constanth(i)N ,
R(i)1NDPNDP
X(i)DP y= Lg
(i+1)IC-MMSE, (6)
where Eav stands for the average power of transmitted symbols,
g(i+1)IC-MMSE stands for the time domain channel estimate in the
(i + 1)th iteration, which will be used in section IV. From(6), it can be seen that the first part, which includes a matrix
inverse, is always constant and we do not need to perform
matrix inversion at each iteration. This is the simplification
which allows the reduction of the calculation complexity.
Furthermore, with distributed resource allocation, since the
positions of the allocated non-null part are varying from
one OFDM symbol to another one, the traditional iterative
MMSE channel estimation algorithm in (1) has to choose
different FFT matrices L for each OFDM symbol, resulting
in more calculations. With the proposed IC-MMSE, thanks
to the compensation process, we do not need to choose the
corresponding partial FFT matrices and to re-calculate the
matrix inversion anymore. Thus, the proposed IC-MMSE has
a much lower complexity than the traditional iterative MMSE
channel estimation.
3) IC-MMSE in MIMO: The IC-MMSE algorithm is pro-
posed in SIMO transmission, and it can also be used in MIMO
transmission. With transmit diversity (taking two transmitantenna system as an example), the received symbols on the
rth receive antenna are
yr =
1t=0
Xthrt + nr (0 r 1) , (7)
where Xt stands for the transmitted symbols on the tth
transmit antenna, and hrt represents the channels between
the tth transmit antenna and the rth receive antenna. In the
(i + 1)th iteration, to estimate the channel vector hrt, thereceived symbol vector yrt is approximated as
y(i)rt = yr X(i)1th(i)r(1t) (0 r 1, 0 t 1) , (8)
where X(i)1t and h(i)r(1t) stand for the soft symbols and chan-nel estimates respectively which contain soft information from
the ith iteration. Then, substituting (8) into (6), we obtain IC-
MMSE channel estimates in transmit diversity transmission.
IV. IMPACT OF COMPENSATION
Define an L L matrix Q as
Q =
L
L +2
EavC1gg
1(9)
and an LN matrix A as A = QL. Then, the IC-MMSEchannel estimate in the time domain in (6) can be expressed
as
g(i+1)IC-MMSE A
h(i)N ,
R(i)1NDPNDP
X(i)DP y . (10)According to the arrangement of null sub-carriers, the matrix
A can also be divided into two parts
A =
ANLNN , ADPLNDP
LN
, (11)
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where AN = QLN
, ADP = QLDP
, and NN + NDP = N.Then, (10) can be written as
g(i+1)IC-MMSE ANh
(i)N + ADP
R(i)1NDPNDP
X(i)DP y. (12)With Appendix A, (12) can be expressed as
g(i+1)IC-MMSE N g
(i)IC-MMSE + DP g
(i)DP , (13)
where 0 < N < 1, 0 < DP < 1, and N + DP 1.With the assumption of high SNR, we have perfect APPs of
transmitted symbols. Therefore, the channel estimate g(i)DP will
not change with different i and we denote g(i)DP = gDP. Then,
in the (i + t)th (t > 1) iteration, the time domain channelestimate can be written as
g(i+t)IC-MMSE
t
N g(i)IC-MMSE + DP gDP
t1n=0
nN . (14)
When t , we have
g(i+t)IC-MMSE 0 + DP gDP
1
1 N gDP. (15)
From (15), we see that, through sufficient iterations, IC-MMSE
algorithm can have the same performance as the traditional
iterative MMSE algorithm using only the modulated part.
However, the proposed IC-MMSE algorithm has much lower
complexity, which will be discussed in the next section.
V. COMPLEXITY
In order to assess the complexity of the IC-MMSE, we
check the number of complex multiplications needed in the
traditional iterative MMSE and the IC-MMSE. In (1), for
each iteration, the number of complex multiplications is
N2DPL + NDPL2 +O L
3
. For IC-MMSE in (6), the constantpart can be done offline and will keep the same value for alliterations and all OFDM symbols. Therefore, at each iteration,
the IC-MMSE only needs N2 complex multiplications. Fur-
thermore, if we consider i iterations, the difference between
the traditional iterative MMSE and the IC-MMSE becomes
i
N2DPL + NDPL2 +O
L3 N2
.
VI . SIMULATION RESULTS
To assess performances of the proposed iterative channel es-
timation algorithms, simulations are conducted over extended
vehicular A (EVA) channel model with 5Hz Doppler frequencyand extended typical urban (ETU) channel model with 70HzDoppler frequency which are defined in [5]. The main parame-
ters are summarized in Table I. We simulate 10MHz bandwidthas defined in [5]. Two strategies of resource allocation are
simulated: full allocation which allocates all RBs to one single
user and distributed allocation which allocates only 2 RBs to
one single user and varies positions of allocated RBs from
one OFDM symbol to another one. For the first iteration, the
simple LS channel estimation is used to get channel estimates
over all sub-carriers. With this LS initial channel estimation,
the proposed algorithm is named as LS+IC-MMSE. In order
to obtain channel estimates on the whole sub-frame, linear
TABLE ISIMULATION PARAMETERS.
Parameter Value
FFT size 1024
Number of modulated sub-carriers 600
Allocated RB 50/2
Cyclic prefix 80
Transmission mode SIMO (1 2) / MIMO (2 2)
Modulation scheme 16-QAM
Channel coding rate 1/2
Channel coding type Duo-binary turbo code
Channel typeEVA5/ETU70 (modified Jakes
Doppler spectrum [6])
time interpolation is implemented after IC-MMSE frequency
domain channel estimation.
In Fig. 3, the packet error rate (PER) performance is
shown over ETU70 channel model with 50RB full allocation.
We see that, in SIMO transmission system, even though
the performance of the first iteration with LS is bad, after
5 iterations, the PER performance of the proposed LS+IC-
MMSE algorithm approaches that of perfect channel state
information (PerCSI). Even with 2RB distributed resource
allocation, after 6 iterations, we obtain good performance
which is close to that of PerCSI, as shown in Fig. 4.
10-3
10-2
10-1
100
4.0 6.0 8.0 10.0
PER
Es/N0 (dB)
1x2 50RB 16QAM 1/2 ETU70 PerCSILS+IC-MMSE iter 1LS+IC-MMSE iter 3LS+IC-MMSE iter 5
Fig. 3. Packet error rate with 50RB full allocation over SIMO transmission.
In Fig. 5, the PER performance is shown over MIMO
transmission. Here, the 50RB full allocation strategy is simu-
lated. From this figure, we see that, over MIMO transmission,
the proposed LS+IC-MMSE algorithm also have good perfor-
mances which approaches the performance with PerCSI, even
though the received symbols for a pair of transmit and receive
antennas have to be estimated based on soft information.
VII. CONCLUSION
In this paper, an MMSE based iterative channel estimation
algorithm is proposed in LTE systems. In order to reduce com-
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10-3
10-2
10-1
100
2.0 4.0 6.0 8.0 10.0
PER
Es/N0 (dB)
1x2 2RB 16QAM 1/2 ETU70 PerCSILS+IC-MMSE iter 1LS+IC-MMSE iter 3LS+IC-MMSE iter 6LS+IC-MMSE iter 10
Fig. 4. Packet error rate with 2RB distributed allocation over SIMOtransmission.
10-3
10-2
10-1
100
2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
PER
SNR (dB)
2x2 50RB 16QAM 1/2 EVA5 PerCSILS+IC-MMSE iter 1LS+IC-MMSE iter 3LS+IC-MMSE iter 6
Fig. 5. Packet error rate with 50RB full allocation over MIMO transmission.
plexity and take advantage of null sub-carriers, a compen-
sation process is proposed to simplify the traditional iterative
MMSE channel estimator. After this iterative compensated
MMSE channel estimation in frequency domain, a simple
linear interpolation in time domain is performed to obtain
channel estimates over all OFDM symbols. Simulation results
show that the proposed IC-MMSE channel estimation algo-
rithm has good performances which approach the performance
with perfect channel state information in both SIMO and
MIMO transmission modes.
ACKNOWLEDGMENT
The research leading to these results has received funding
from the European Commissions seventh framework program
FP7-ICT-2009 under grant agreement n 247223 also referred
to as ARTIST4G.
APPENDIX A
DERIVATION OF (13)
In (12), the channel estimate h(i)N is obtained by
h(i)N = LN g
(i)IC-MMSE. (A-1)
Also, we define g(i)DP asR(i)1NDPNDP
X(i)DP y = LDP g(i)DP. (A-2)Then, we get
g(i+1)IC-MMSE ANLN g
(i)IC-MMSE + ADPLDP g
(i)DP. (A-3)
Together with the definitions of AN, we have
ANLN =
L
L +22
EavC1gg
1LN
LN
=
NILL +
22
EavC1gg
1LN
LN . (A-4)
In order to simplify analysis, we assume rectangular channel
profile, which means all taps have the same power1
L. Then,
(A-4) becomes
ANLN =1
N +2L2
Eav
LN
LN . (A-5)
In (A-5), even though LN
LN is not diagonal, we can still
make a diagonal approximation, because its diagonal entries
are more larger than non-diagonal entries. Then, (A-5) can be
further simplified and the scalar value is defined as N:
ANLN NN
N +2L2
Eav
ILL NILL. (A-6)
The same process can be made for ADPLDP and we get
ADPLDP NDP
N +2L2
Eav
ILL DPILL. (A-7)
Since NN + NDP = N, it is evident to have N < 1 andDP < 1. Also, compared with the value of N, the value of2L2
Eavis so small that we can have
N + DP 1. (A-8)
Finally, substituting (A-6) and (A-7) into (A-3), we get (13).
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