Iterative qr decompostion channel estimation for

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 02 Issue: 08 | Aug-2013, Available @ http://www.ijret.org 56
ITERATIVE QR DECOMPOSTION CHANNEL ESTIMATION FOR
MIMO-OFDM SYSTEMS
R. Prakash Kumar1
, I Raghu2
, M. Vinod Kumar Reddy3
1, 2, 3
Assistant Professor, ECE Department, CVR College of Engineering, AP, India,
prakash.rachmagdu@gmail.com, raghu.indraganti303@gmail.com, vinodkumarreddy488@gmail.com
Abstract
Channel estimation algorithms have a key role in signal detection in MIMO-OFDM systems. In this system, the number of channel
components which need to be estimated is much more than conventional SISO wireless systems. Consequently, the computational
process of channel estimation is highly intensive. In addition, the high performance channel estimation algorithms mostly suffer from
high computational complexity. In the other words, the system undergoes intensive computations if high performance efficiency is
desired. However, there is an alternative solution to achieve both high performance efficiency and relatively low level of
computational complexity. In this solution, high efficient channel estimation is firstly designed, and then it is simplified using
alternative mathematical expressions. In this paper, Iterative channel estimation based on QR decomposition for MIMO-OFDM
systems is proposed. From simulation results, the iterative QRD channel estimation algorithm can provide better mean-square-error
and bit error rate performance than conventional methods.
Index Terms: MIMO, OFDM, QRD,Least squre Channel estimation
-----------------------------------------------------------------------***-----------------------------------------------------------------------
1. INTRODUCTION
Orthogonal frequency division multiplexing (OFDM)
technology has been used widely in many wireless
communication systems, such as digital audio broadcasting
(DAB), digital video broadcasting (DVB), wireless local area
network (WLAN), asymmetric digital subscriber loop
(ADSL), and future 4G systems. OFDM systems can provide
higher bandwidth efficiency and achieve higher data
throughput. In order to enhance the data rate, multi-antenna
technique is applied to existing systems. Multiple-input
multiple-output (MIMO) communication refers to wireless
communication systems using an array of antennas (i.e.
multiple antennas) at either the transmitter or the receiver.
Multiplexing would cause interference, but MIMO systems
use smart selection and/or combining techniques at the
receiving end to transmit more information and to improve
signal quality.
Therefore, MIMO OFDM systems are regarded as attractive
systems for high speed transmission. Hence, the integration of
these two technologies has the potential to meet the ever
growing demands of future communication systems [1]. If
space-time coding is used at the transmitter, the channel
knowledge is required at the receiver to decode the transmitted
symbols. Therefore, accurate channel estimation plays a key
role in data detection especially in MIMO-OFDM system
where the number of channel coefficients is M×N time more
than SISO system. (M and N are the number of transmitted
and received antenna respectively). Technically, there are four
types of channel estimation [2]; training-based, blind, semi-
blind and data–aided channel estimation. In wireless
communications, signals are always distorted by channel. The
wireless channel is time or location variant the channel state
information to compensate the channel distortion. Pilot signals
can be spaced separated in the transmitted symbols. In the
receiver, the channel impulse response can be estimated at the
positions of pilot signals. The other channel information at the
data signals can be obtained by interpolating the estimated
channel impulse response. However, error caused by channel
interpolation cannot be avoided. A good channel estimation
method can provide higher reliable data detection.
In this paper, we proposed iterative channel estimation based
on QR Decomposition for MIMO OFDM systems. The aim of
this paper is to investigate the effectiveness of QRD (QR
decomposition) to reduce the computational complexity of
channel estimation algorithms in MIMO-OFDM system, and
design high performance channel estimation for this system by
using iterative technique.
2. SYSTEM MODEL
Figure.1 shows the basic model of MIMO-OFDM system with
M and Nr number of antenna at the transmitter and receiver
respectively. In this model, MIMO transmission is assumed to
be OSTBC (Orthogonal Space-Time Block Coded). Therefore
the block of user information after mapping in MPSK
modulator is coded by the MIMO-STBC encoder with the
matrix dimension of P×M. Where P is the number of time

__________________________________________________________________________________________
interval needed to transmit this matrix by M number of
transmit antenna.
S/P C= Serial to Parallel Converter
P/S C= Parallel to Serial Converter
FFT= Fast Fourier Transform
IFFT= Inverse Fast Fourier Transform
Figure 1.Block diagram of an MIMO-OFDM transceiver
If a column of encoded matrix which enter to the OFDM block
is (X1, X2,…,Xm)T in frequency domain then the output of
OFDM module will be (x1, x2,…,xm)T in time domain. Each
element of encoded matrix Xk before OFDM module has a
length of N= 64 symbols while after OFDM module change to
xk in time domain with the length of 80 symbols. i.e received
signal after distortion by frequency selective channel and
AWG noise at antenna j from antenna i can be represented by
“Equation (1)”.
)1(,....2,1,)()()()(
1
0
Minwlnxnhny
L
l
jiji
l
ji
=+−⋅= ∑
−
=
Where is )(nh ji
l is lth channel coefficient between received
antenna j and transmitted antenna i at time n. WJ(n) is AWGN
with zero mean and variance one. The “Equation (1)” in vector
form can be rewritten by “Equation (2), (3), (4), (5)”.
Received signal vector at time interval t by antenna j=1,
2,…,Nr can be represented as
[ ]
[ ]
[ ] 

















=
+
+
+
TN
NN
NN
T
NN
T
NN
r
CP
rr
CP
CP
yyy
yyy
yyy
y
......,
.
.
......,,,
.....,,
10
22
1
2
0
11
,1
1
0
(10)
[ ]
[ ]
[ ] 























=
+
+
=
+
=
∑
∑
∑
i
NN
iiiNr
i
NN
ii
M
i
i
i
NN
ii
M
i
i
CP
CP
CP
xxxhtoeplitz
xxxhtoeplitz
xxxhtoeplitz
y
......,(
.
.
......,,,)(
.....,,).(
10
10
1
2
,10
1
1
Toeplitz function is a channel matrix function which can be
defined as toeplitz (h) =


























−
−
−
−
)0()1()1(0000
00
0)1(0
00)1(0
000)1()1(
0000)0()1(
00000)0()1(
000000)0(
ijijij
ij
ij
ijij
ijij
ijij
ij
hhlh
lh
lh
hlh
hh
hh
h
L
OOMOMM
MOOMM
LOOM
LOM
LM
L
L
After removing cyclic prefix and FFT transformation by
OFDM demodulator, received signal in frequency domain can
be represented as
∑=
+=
M
i
j
k
i
K
ji
K
j
k WSHY
1
.
(12)
Wkj is AWGN in frequency domain and it can be calculated
using
∑
−
=
−
=
1
0
/2
.
1 N
n
Nknjjj
k ew
N
W π
(13)
Receive signal in vector form can be represented as
jji
K
M
i
ij
WHSY += ∑=1 (14)
=S.HJ +WJ
In more detail each of variable in Equation (14) can be written
as in Equations (15),(16),(17) and (18).
[ ]Tj
N
jjj
nyynYY )(,....,),( ,11=
(15)
(11)

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[ ] [ ]( )TJM
N
JMJMJ
N
JJji
K HHHHHHH ),(),(
2
),(
1
),1(),1(
2
),1(
1 ,,,,,,,, LLL= (16)














=
)(00)1(00
00
00)(0)(0
00)(00)(
1
1
1
1
2
1
1
1
nSS
nSnS
nSnS
S
M
N
M
M
LLL
MOMMMOM
MLL
LLL
(17)
[ ]Tj
N
jjj
nWnWnWW )(),(),( 21 L= (18)
Received signal by antenna [1, 2 …Nr] in frequency domain
after removing cyclic prefix and FFT transformation can be
written as
Y = S × H +W (19)
In more detail each of variables in Equation (19) can be
written as in Equations (17), (20), (21) and (22)














=
)()()(
)()()(
)()()(
21
21
2
21
1
nynyny
nynyny
nynynY
Y
r
r
r
N
NNN
N
NN
N
NN
L
MMMM
L
L
(20)






























=
),()2,()1,(
),(
2
)2,(
2
)1,(
2
),(
1
)2,(
1
)1,(
1
),1()2,1()1,1(
),1(
2
)2,1(
2
)1,1(
2
),1(
1
)2,1(
1
)1,1(
1
r
r
r
r
r
r
NM
N
M
N
M
N
NMMM
NMMM
N
NNN
N
N
HHH
HHH
HHH
HHH
HHH
HHH
H
L
MMMM
L
L
MMMM
L
MMMM
L
L
(21)














=
)()()(
)()()(
)()()(
21
21
2
21
1
nWnWnW
nWnWnW
nWnWnW
W
r
r
r
N
NNN
N
NN
N
NN
L
MMMM
L
L
(22)
3. LS CHANNEL ESTIMATION FOR MIMO-
OFDM
From Equation (12), it can be seen that for estimation of
channel component between receive antenna j and transmit
antenna i=1, 2,…M, the number of subcarriers which has to be
estimated is M×N. where N is the number of subcarriers. In
the other words for every receive antenna j=1, 2,…Nr
vector
ji
kH
in Equation(16) has to be estimated.
If one OFDM training block with N subcarriers transmitted
from every of transmitted antenna, then from the model for
every receive antenna there will be N equation with N×M
unknown, hence these equations are under determined and
cannot be solved. For solving this problem there are two
solutions, first solution is transmitting M OFDM blocks which
in practical case is not applicable. Second solution is reducing
the unknown elements by looking at an alternate
representation of the received signal, called the transform-
domain estimator that was first proposed by van de Beek in [4]
for OFDM systems and well explained in [2] for MIMO-
OFDM system. Base on this method CFR (Channel Frequency
Response) can be expressed in terms of the CIR (Channel
Impulse Response) through the Fourier transformation. Hence,
the received signal model in “Equation (14)” can be expressed
in terms of the CIR. The benefit of this representation is that
usually the length of the CIR is much less than the number of
subcarriers of the system. CIR representation can be achieved
using following transformation
),(),( ijij
hFH ⋅= (23)
Where h(j,i) is the (L×1) channel impulse vector and F is
Fourier transform in vector form, and it can be represented as
LN
N
LNj
N
Nj
N
Nj
N
Lj
N
Lj
N
j
ee
e
e
ee
F
×
−−−−−
−−
−−
−−−


















=
)1)(1(2)1)(1(2
)2)(1(2
)1)(2(2
)1)(1(2)1)(1(2
1
1
1
111
ππ
π
π
ππ
L
ML
MMM
L
L
(24)
To extend the matrix Fourier transform to operate on multiple
channels following matrix in Equation (25) can be defined as
)(
00
00
00
FDIMM
F
F
F
×














=Γ
L
MOLM
L
L
(25)
By using this definition, transformation of CFR to CIR in
Equation (14) can be done as
jjjjj
K
jj
WhXWHSY +Γ=+= ..
jJ
WhA +. (26)
By applying LS algorithm on Equation (26), channel
component can be estimated using Equation (27). By this
transformation one OFDM block is enough to estimate the
channel. The only condition is N≥M×L
YAAAh HHj
..).(
~ 1−
= (27)

__________________________________________________________________________________________
4. QR DECOMPOSITION
QR decomposition is just an alternative for calculating matrix
inversion. There are different methods for QR decomposition.
Here Householder algorithm is used. The steps of QRD
algorithm to solve LS problem can be presented as follow [5]:
Algorithm
1. Making the LS error function for Equation (27) as
represented in Equation (28).
hAY
~
−=ε And if hAY
~
0 =⇒=ε (28)
2. Decompose W into Hermitian matrix Q and upper
triangular matrix R using Householder algorithm as Equation
(29).
h
R
QhAY
NM
MM
~
.
0
~
×
× 





⋅==
(29)
3. Multiply Hermitian of Q to both side of Equation (29). The
result can be represented as in Equation (30).
YQh
R H
MM
NM
⋅=⋅





×
×
~
0
(30)
4. Finally, solve the channel using back substitution
5. COMPLEXITY COMPARISON BETWEEN LS
AND QRD ALGORITHM
The advantage of using QR decomposition is to reduce the
computational complexity of the LS channel estimation. In
this research, the computational complexity in terms of
number of mathematical operations has been measured. The
derivations are based on an Mt-by-Mr MIMO-OFDM system
with N subcarriers and a channel length of L. The known
matrix A has dimensions (N×L.Mt). For simplicity in notation
L.Mt is denoted by M. For a consistent comparison, the
complex operations are converted to real operation
equivalents.
Table 2 shows the real equivalent operations for the various
complex operations. In addition, each type of real operations
has different levels of complexity when implemented in the
hardware. For example multiplications, additions, and
subtractions can be set to 1 FLOPs (Floating Point
Operations), divisions to 6 FLOPs, and square roots to 10
FLOPs (table-1). It should be emphasized that counting of the
number operations is only an estimate of the computational
complexity of the algorithms. A more exact measure would be
to implement the algorithm in hardware and count the number
of instructions and processing time required. However, in
computer simulations, FLOP counts can give a good
indication of the relative complexity of different algorithm.
Table-1: Number of flops in every real operation
Operation # No of Flops
Multiplication, addition
and subtraction
1
Division 6
Square root 10
Table-2: Number of real operations in every complex
operation
Complex
Operation
No of Real Operations
Multipli
cation
Division Subtraction
& Addition
Multiplication 4 2 0
Division 6 3 2
Subtraction
/Addition
0 0 2
Complex
magnitude
2 0 1
6. ITERATIVE QRD CHANNEL ESTIMATION
In order to complete data-aided channel estimation, pilot
signals can be spaced separated in the transmitted symbols. In
the receiver, the channel impulse response can be estimated at
the positions of pilot signals by several algorithms, such as
least square method. The other channel information at the data
signals can be obtained by interpolating the estimated channel
impulse response [6].
In this paper, we propose a iterative LS-QRD channel
estimation algorithm for MIMO OFDM system. At first step,
an LS-QRD channel estimate is obtained by using (31)
jj
QRD hH
~ˆ Γ=
(31)
h
~
can be obtained from
YQh
R H
MM
NM
⋅=⋅





×
×
~
0
Where
10,10,
00
00 /2
−≤≤−≤≤=












=Γ −
LnNmeF
F
F
FFF
Nmnj π
L
MOMM
L
L
And then the channel estimate is set as initial
0,ˆ )(
=kH
kj
QRD
.Secondly, the receiver uses the estimated
channel to help the detection/decision of data signals. The
detection data can be obtained by zero forcing method
jk
QRD
i
ZF YHX
Φ
= )(ˆˆ
(32)

__________________________________________________________________________________________
Where
Φ)(ˆ k
QRDH
is pseudo inverse of
j
QRDHˆ














=
)(,
,
)(,2
,
)(,1
,
)(2,
,
)(2,2
,
)(2,1
,
)(1
,
)(1,2
,
)(1,1
,
)(
ˆˆˆ
ˆˆˆ
ˆˆˆ
ˆ
,
kNN
diagQRD
kN
diagQRD
kN
diagQRD
kNr
diagQRD
k
diagQRD
k
diagQRD
kN
diagQRD
k
diagQRD
k
diagQRD
k
QRD
rtrr
t
HHH
HHH
HHH
H
L
MMMM
L
L
And then the channel estimation treats the detected signals as
known data to perform a next stage channel estimation
iteratively and the index k adds 1. Go to the first step and
repeat the process till the mean-square-error of channel
estimate is converged or the expected iterations reach. By
utilizing the iterative channel estimation and signal detection
process we can reduce the estimation error caused by channel
interpolation between pilots. The accuracy of the channel
estimation can be improved by increasing the number of
iteration process.
7. SIMULATION RESULTS
The system specification for this simulation can be
summarized in Table 3. For this simulation the channel has
L=16 paths where the amplitude of each path varies
independently according to the Rayleigh distribution with an
exponential power delay profile [10], and can be represented
as in Equation (33). The results can be classified into two
parts; Performance comparison and complexity comparison
results. These are presented in the next sections.
A. Iterative QRD Algorithm
The bit error rate (BER) and MSE Performance of iterative
QRD channel estimation method are shown in figure 2 and
3.Iteration number r=0 means the conventional QRD channel
estimation method . After about 2 iterations ,the BER and
MSE performance of iterative channel estimation are much
closer to that of ideal one .
Figure-2: BER Performance of iterative channel estimation
with different iteration numbers
Table-3: Simulation parameters for MIMO-OFDM
System MIMO(STBC)-OFDM
#Rx Antenna 2
#Tx Antenna 2
Channel Frequency selective
,Rayleigh fading
Noise AWGN
#Sub carrier 64
#Cyclic Prefixes 16
Cahnnel length 16
Trms (RMS delay spread) 25 ns
Ts-Sampling Frequency 1/80 MHz
1,......2,1)2/1,0()2/1,0(
22
−=+= LljNNh li σσ (33)
Where
RMS
s
RMS
s
T
LT
T
T
l ee
−−
×−= )1(2
σ and for
normalization
∑
−
=
=
1
0
2
1
L
l
lσ
and L approximated by L= s
rms
T
T10
Figure-3: MSE Performance of iterative channel estimation
with different iteration numbers.
The use of iterative QRD method improves the performance in
terms of lower channel estimation error. From the results it
can be concluded that the iterative QRD channel estimation
algorithm have high performance efficiency in terms of BER
and MSE. However, in the next section the benefit of QRD,
which is the significant reduction of the complexity of the
system, is portrayed.
B. Complexity Comparison between LS and QRD Algorithm
Using the system parameters for the MIMO-OFDM system
specified in Table 3, the number of operations for a 2 transmit
antenna system with a channel length of 6 and 16 was

__________________________________________________________________________________________
calculated for the two algorithms. In this section, the
complexity comparison in terms of FLOPs count is performed
for two algorithms. Figures 4 and 5 shows the complexity
comparison of both algorithms
Figure-4: Complexity comparisons between LS and QRD
with channel length=6
The results in Figure 5 is more highlighted which the number
of channel length increase to 16. Increasing the channel length
increases the number of unknown parameters, thereby will
increase the complexity of the channel estimation. It shows
that the LS increases exponentially as the channel length
increases and has much higher complexity than the QRD for
long channel lengths
with channel length=16
with TX=2
Figure 6 shows the simulation result using 2 transmit antenna
while the number of channel component vary from 1 to 16.
The previous conclusion for computational complexity can be
made here. In Figure 7, the number of transmit antenna is
increased to 8 while the channel is changed from 1 to 16. As
expected, when the number of antennas increases, both
estimation techniques increase in complexity because the size
of the unknown matrix A increases. The general trend of the
QRD method is that it increases almost linearly with the
number of transmits antennas of the system. The LS method
increases exponentially at a considerably higher rate than the
QRD methods. Therefore, the QRD is especially preferable for
higher number of transmit antennas since it does not explode
in complexity as the LS solution. Finally the Numerical
example for computational complexity comparison between
two channel estimation algorithms is provided in Table 4.
Table-4: Number of complex operations and Flops in two
algorithms with L=16 and Tx =4 (A: # of complex ultiply, B:
# of add/sub, C: # of complex division, D: # of square root, E:
# of complex magnitude, F: # of flops).
Algorith
m
A B C D E F
LS 79052
8
782277
2
617
6
0 0 1437356
8
QRD 18722
0
184047
1
192 214
4
12
8
3376126
The results prove that QRD method is lower in complexity
than LS method. The results in Table 4 show that the total
number of operation for the LS method is much higher than
the QRD method. For this simulation scenario using QRD
achieves a complexity reduction by approximately 77%. This
verifies that the QRD has significantly lower complexity than
that of direct LS estimation via the pseudo inverse, hence a
better option for channel estimation.

__________________________________________________________________________________________
with TX=8
CONCLUSIONS
The simulation results proved that Iterative QRD channel
estimation algorithm has good performance efficiency it can
provide better mean square error and bit error rate
performance than conventional methods. However the
computational complexity of the QRD channel estimation is
much lower than LS algorithm. In addition, computational
complexity for QRD channel estimation is approximately
linearly proportional with number of transmit antenna and
channel length, whereas for LS algorithm is exponentially
proportional with the number of transmit antenna and channel
length. As finding indicate; using QRD channel estimation,
computational complexity of the system for above particular
scenario which mentioned in table-4 can dramatically decrease
by 77 %.
Finally it can be concluded that Iterative QR decomposition
can be an ultimate solution for high performance efficiency
and reduction computational complexity.
REFERENCES
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Ingram, and T. G. Pratt, “Broadband MIMO-OFDM wireless
communications,” Proceedings of IEEE, vol. 92, no. 2, pp.
271-294, Feb. 2004.
[2] T. Y. Al-Naffouri, “Adaptive algorithms for wireless
channel estimation,” Department of Electrical Engineering,
Stanford University, Dec. 2004, PHD thesis.
[3] Kathryn Kar Ying Lo “Channel Estimation of Frequency
Selective Channels for MIMO-OFDM” University og Calgary.
Master thesis 2005
[4]. J.-J. van de Beek, O. Edfors, M. Sandell, S.K. Wilson, and
P.O. Borjesson, “On channel estimation in OFDM system,” in
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