EM Based Semi-Blind Beamforming Algorithm
in MIMO-OFDM Systems
H. Zamiri-Jafarian 1, 2 , S. Bokharaiee-Najafee1 and S. Pasupathy2
1 Electrical
2 Department
Engineering Department , Ferdowsi University, Mashhad , Iran
of Electrical and Computer Engineering, University of Toronto, Toronto, Canada
Abstract — A new EM based semi-blind beamforming
algorithm is proposed in this paper for multiple input multiple
output (MIMO) systems in which orthogonal frequency division
multiplexing (OFDM) technique is used. Employing singular
value decomposition (SVD) technique, the semi-blind beamforming algorithm eliminates space interference in an iterative
manner by shaping antenna patterns in transmitter and receiver
sides in frequency domain. A 16-QAM MIMO-OFDM system is
considered for computer simulations in order to evaluate the
performance of the new algorithm under different scenarios. The
results show that the proposed algorithm outperforms subspace
based semi-blind beamforming method.
Index Terms — Beamforming, Expectation Maximization
Algorithm , MIMO , OFDM , Semi-blind , SVD
The SVD of channel matrix can be obtained directly from
the channel matrix estimation, however, because of the
nonlinear operation of taking SVD, if the estimation of the
channel matrix is not precise enough, it may create more
errors. One good solution to this problem is to estimate the
SVD matrices directly from the received signal [4],[5].
The organization of this paper is as follows. After
introduction, the CEM estimator is introduced in section II. In
Section III, a MIMO-OFDM system is modeled based on the
SVD method [4] and then the CEM based semi-blind
beamforming algorithm is presented. Computer simulations are
given in section IV and section V concludes the paper.
I. INTRODUCTION
II. THE CONSTRAINED EM ESTIMATOR
ULTIPLE
input
multiple
output
(MIMO)
communication system using orthogonal frequency
division multiplexing (OFDM) technique can employ both
space diversity and frequency diversity to mitigate the
multipath and fading effects.
Interference from other users/antennas is the true challenge
in the MIMO system. The beamforming technique has already
been introduced to resolve this problem. The purpose of
beamforming is to steer the array beams in such a way that just
the desired signal is received with maximum gain while other
signals are mitigated as much as possible.
Various methods for beamforming have been proposed in
recent years. These methods can be classified into training and
blind groups based on the amount of information that is
available at the receiver. While the first group achieves fast
convergence and better performance, the second one preserves
bandwidth. By coupling the training and blind techniques,
another method called semi-blind is derived that benefits from
the advantages of both methods. The semi-blind method can
be designed to preserve the bandwidth while its performance is
very close to that of the training technique.
Signal subspace partitioning is a popular approach that can
be used for blind and semi blind beamforming [1]-[3];
however this method needs a large amount of data to reach to
near the optimum point. A semi-blind beamforming method
employing the Constrained Expectation Maximization (CEM)
algorithm is proposed in this paper via channel singular value
decomposition (SVD) technique. Computer simulations show
that our proposed beamforming algorithm outperforms the
signal subspace based semi-blind approach without using any
coding.
Let consider θ as a column vector of deterministic
parameters to be estimated from the observed data vector x
where is the sample value of I , incomplete data space. The
maximum likelihood ( ML) estimation of θ is given as
M
θˆ ML  arg max  p (x | θ)  arg max LI (θ )
θ
(1)
θ
where LI (θ)  log  p( x | θ)  . Due to x is incomplete
information , the maximization of LI (θ) is not tractable.
Denoting y as a desired vector from D (the additional
information data space needed to complete I space), we have
p(x | θ)  p(x, y | θ) / p (y | x, θ)
(2)
By using logarithmic form, one can write
L (θ)  L (θ)  L (θ)
I
where
C  ( I, D)
C
(3)
D|I
is defined as complete data space,
LC (θ)  log  p ( x, y | θ  and LD | I (θ)  log  p ( x | y , θ  . The main
idea of the EM algorithm is to increase the likelihood function
LI (θ) in an iterative manner [7]. By taking the conditional

[0]
expectation of both side of (3) over x and θ
(the
estimation of θ in lth iteration), we obtain
LI (θ, θ[l ] )  E{LI (θ) | x ,θ[l ]}
 E{LC (θ) | x , θ[l ]}  E{LD|I (θ) | x , θ[l ] }


 Q(θ, θ[ l ] )  F (θ, θ[ l ] )
where
(4)
[l ]
[l ]
l [ ]
Q(θ θ,
E
) LC θ{ x , θ( ) ,| F (θ, θ} ) 
E{LD| I (θ) | x , θ } and l is the iteration index. It can be shown
[l ]
that by maximizing Q(θ, θ ) with respect to θ causes
[l ]
LI (θ, θ ) to be monotonically increasing [7]. Thus the EM
algorithm can be expressed in two steps
[l ]
E-step:
Q(θ, θ[l ] )  E{log  p(x, y | θ  | x , θ[l ] }
(5)
θ[l 1]  arg max{Q(θ, θ[l ] )}
(6)
M-step:
θ
The E-step and M-step are performed successively until at the




l  th iteration θ[ l 1]  θ[ l ] or θ[ l 1]  θ[ l ]   , where  is a

small value and
Thus,
at
convergence
L (θ ML )  L (θ
I
is an appropriate distance measure [7].
I
[ l  1]
,θ
[ l  1]

θ ML  θ[ l 1]
point,
and
) which is a local or the global
where U
and V
are M  P and N  P unitary
(m )
matrices, respectively. Note that P is the rank of H where
P  min( M , N ) .  ( m )  diag( 1( m ) , (2m ) , , (Pm ) ) is
a diagonal matrix containing the singular values of the mth
subcarrier channel matrix.
We begin with the EM based channel SVD estimation to
derive the beamforming weights in transmitter and receiver as
follows.
For the ease of derivation of formulas and without loss of
generality, we omit the index m in (11). Thus, we can
write [4]
H
(12)
H  UΣV H  WV
 UW2H
1
where
W1  UΣ
(13)
W2  VΣ
(14)
(m )
If u i , v i , w1i and
θ
w  u H  iv
H
2i
(15)
H
i
(16)
x(k )  WV s(k )  n(k )
H
(17)
1
After receiving the k  1 signal, at time k , we have
(8)
denotes constraint. Therefore (8) can be


θˆ [l 1]  arg max Q C (θ, θˆ [l ] )
θ
X(k )  WV S(k )  n(k )
H
III. CONSTRAINED EM BASED SEMI-BLIND BEAMFORMING
A MIMO-OFDM system, which has N transmitting
antennas, M receiving antennas and L subcarriers, is
considered. Defining
as a
S(k)  [s(1) (k),..., s(L) (k)]
transmitted
OFDM
symbol
matrix
where
s( m ) (k)  [s1(m) (k ),..., sN(m) (k )]T is a symbol vector that is
in which
S(k )  [s(0), s(1),..., s(k )]
X(k )  [x(0), x(1),..., x(k )]
n(k )  [n(0), n(1),..., n(k )]
H(m)s(m) (k )  n(m) (k )
for m  1,
,L
(10)
where H
is an M  N channel matrix of the mth
subcarrier and n(m) (k) is the M  1 vector of the complex
additive white Gaussian noise with zero-mean and
Rnl (k )   n2 I M  (k )
autocorrelation
matrix
for
M  (k  1)
P  (k  1)
M  (k  1)
(19)
(20)
(21)
The channel SVD estimation can be computed in two steps as
follows.
A. Iterative Estimation of U Matrix
Starting from (18), we multiply S H (k ) in both sides of
(18).
H
X(k )S H (k )  WV
S(k )S H (k )  n(k )S H (k ) (22)
1

H
Now multiplying both sides of (22) in S( k )S ( k )
have
W1  Y1 ( k )  Z1 ( k )
T
transmitted from the mth subcarrier. Note that (.) represents
the transpose operation. After removing the cyclic prefix, the
M  1 received vector of the mth subcarrier, x(m) (k ) ,
becomes
(18)
1
(9)
The E-step and M-step of the constrained EM (CEM)
algorithm are similar to those of the EM algorithm, just using
[l ]
C
[l ]
Q (θ, θˆ ) instead of Q(θ, θˆ ) in (5) and (6).

H
i
For each subchannel, we can write
QC (θ, θˆ [l ] ) Q(θ, θˆ [l ] )  λ H (Bθ  b)
x(m) (k )
2i
w 1i  Hv i   i u i
(7)
In this case we define
Superscript C
rewritten as
w are defined as the ith column of
U , V , W1 and W2 , respectively, we have
maximum of LI (θ) .
The EM algorithm can be constrained to satisfy the required
limitations on estimated parameters as the following:
θˆ [l 1]  arg max Q(θ, θˆ [l ] ) subject to Bθ  b
(m)

1
V , we
(23)
where
Φs (k )  (S(k )S(k ) H ) 1
Y1 (k )  X(k )S H (k )Φs (k )V
Z1 (k )  n(k )S H (k )Φs (k )V
(24)
(25)
(26)
(m)
, L , while I M is the M  M identity matrix.
(m )
The SVD of the mth subcarrier channel matrix, H , can be
m  1,
given as
H ( m )  U ( m ) ( m ) V (m ) H
(11)
Therefore, each column of W1 can be written as
w 1i  y 1i ( k )  z 1i ( k )
(27)
where y 1i ( k ) and z 1i ( k ) are the ith column of Y1 ( k ) and
Z 1 ( k ) , respectively and y1i (k )  X(k )S H (k )Φs (k )v i . It is
straightforward to show that z 1i ( k ) is white Gaussian noise
vector with R z1i   i I M [5]. Rearranging (27) gives
y (k )  w  z (k )
1i
1i
1i
(28)
Now we have come to the problem of estimating the columns
of W1 that are orthogonal to each other. As can be seen
y 1i ( k ) contains the unknown data. Regarding the orthogonally
constraint w1Hi w1 j  0 for i  j , using (7), (8) and (9) the

CEM algorithm for w1[ li ] estimation is as follows
w  arg max
Q c (w , w
w
[l ]
1i
1i
1i
[ l 1]
1i
)
(29)

 H

 H

w[2li]  [I N  Φs (k )W2[il ] ( W2[il ] Φs (k )W2[il ] ) 1 W2[il ] ]
E{Φ s 1 (k ) | X(k ), w[2li1]}1 E{S(k ) | X( k ), w [2li1]}
X (k )u
H
i  1,, P
[l ]
(39)
i



where W2[il ]  [w[2l1] ,, w[2li]1 ] is the matrix that guarantees the

orthogonality of W2 columns. Considering (16) and after
normalization, we have
1
where
Q (w , w )  E{log p(y (k ) | w ) | X(k ), w }
C
[ l 1 ]
[l ]
1i
1i
1i
1i
1i
λ W w
[l ]H
1i
[ l ]H
1i
(30)
1i
[l ]
and λ [1li]  [ λ11
,, λ1[il]1 ]T is the Lagrange multiplier vector in

 [l ]

iteration l. The columns of W1[il ]  [w11
,, w1[li ] 1 ] which are
obtained in iteration l are mutually orthogonal and also

orthogonal to w 1[ li ] . Substituting the previous estimation of v i
in y 1i ( k ) and expanding (29), we obtain

w  arg min
E (y
w1i
[l ]
1i
[ l 1 ]
1i
( k )  w ) R 1i
(y (k )  w ) | X(k ), w
[ l 1]
1i
1i
1
H
1i
[ l 1]
1i
z
 λ
[l ]H
1i
[ l ]H
W w
1i
1i

(31)
After some simple manipulations, we have

 H 
 H

w1[ li ]  [I M  W1[il ] ( W1[il ] W1[il ] ) 1 W1[il ] ]


 X(k ) E{S H (k )Φs (k ) | X(k ), w1[li1] }v[i l 1]
i  1,..., P
(32)

 H  
v [i l ]  ( w [2li] w [2li] ) 2 w [2li]
1

 H 
 i[ l ]  ( w [2li] w [2li] ) 2
i  1,, P (40)
i  1,...,P (41)
The process of semi-blind beamforming can be divided into
two parts as training and blind. In the training part, we use the
derived formulas in this section without the need for taking
expectation due to the known transmitted signals. In the blind
part, we start from the SVD estimation obtained in the training
part as the initial transmit/receive beamforming matrices, then
assuming a specific interval, we send the estimated transmit
beamforming matrix V to the transmitter side at the beginning
of regular time intervals. Note that this interval should be large
enough not to waste the channel bandwidth while it must be
chosen in such a way that channel has a small variation in this
duration. The received signal is then multiplied in the receive
ˆH.
beamforming weight matrix U
It should be mentioned that for implementing (32) and (39),
the expectation is just taken on points of the constellation with
the most probability according to a soft decision approach and
signal to noise ratio. This leads to a significant reduction in
complexity of the proposed algorithm.
Considering (15), after normalization, we get
IV. SIMULATIONS AND RESULTS
1

 H  
u[i l ]  ( w1[ li ] w1[ li ] ) 2 w1[ li ]
i  1,, P
(33)
B. Iterative Estimation of V Matrix
Multiplying both sides of (18) in U H gives
U H X(k )  W2H S(k )  U H n(k )
(34)
Similar to (23), one can show that
W2  Y2 ( k )  Z 2 ( k )
(35)
Y2 (k )  Φs (k )S(k )X H (k )U
Z2 (k )  Φs (k )S(k )n H (k )U
(36)
where
(37)
Each column of W2 can be written as
w 2i  y 2i (k )  z 2i (k )
(38)
in which z 2 i ( k ) is a zero mean Gaussian noise vector with
autocorrelation matrix  n2 Φ s (k ) and
y 2i (k )  s (k )S(k )
X (k )u i . Rearranging (38) and following similar
For simulations, a MIMO-OFDM system with 64 subcarriers
has been considered. A sequence of independent, identically
distributed 16-QAM signal vector s(k ) with R s  I N is sent
from transmitter array antennas. To evaluate the performance
of the algorithm, the Normalized Mean Square Error (NMSE)
criterion [4] along with BER is employed. The channel has
an exponential delay spread profile with utmost 16 paths [4].
NT and ND are the number of packet symbols for training and
blind parts of the algorithm, respectively. Each packet consists
of N individual OFDM symbols that are the minimum number
required for channel estimation Fig.1 and Fig.2 present the
performance of the MIMO-OFDM system with N=M=2,4 in
terms of SNR. The interval for updating V has been considered
5 OFDM packets with NT=1,2 and ND=30. Closer inspection
reveals that the performance of the system enhances when NT
increases from 1 to 2. Fig.3 and Fig.4 compare the signal
subspace based semi-blind method [1] applied to our system
model with our proposed algorithm. As can be seen, our
scheme achieves a much better performance especially at high
SNRs. Fig. 5 shows BER of the system in terms of number of
iterations. As illustrated, the performance of the system
improves significantly when iteration number increases from 1
to 2 and reaches to a rather steady state in iteration 3 and over.
H
estimation procedure as done for W1 , the estimation of each
column of W2 at lth iteration becomes
V. CONCLUSIONS
A novel semi-blind beamforming algorithm has been
proposed in this paper for MIMO-OFDM systems. The
0
10
5
M=N=2,EM
M=N=2,NT=2
-5
M=N=4,Subspace based
M=N=4,NT=1
-10
M=N=4,EM
M=N=4,NT=2
-15
BER
NMSE (dB)
M=N=2,Subspace based
M=N=2,NT=1
0
-20
-1
10
-25
-30
-35
-40
-2
-45
0
10
5
10
15
20
25
30
0
5
10
SNR (dB)
Fig. 1. NMSE versus SNR for ND =30, M=2,4, NT=1,2 , when the
number of iterations is 5.
15
SNR(dB)
20
25
30
Fig. 4. BER versus SNR, comparing the proposed EM based
beamforming algorithm with signal subspace approach [1], NT=1,
ND=30 , when the number of iterations is 5.
0
0
10
10
M=N=2,NT=1
M=N=2,NT=1
M=N=2,NT=2
M=N=2,NT=2
M=N=4,NT=1
-1
M=N=4,NT=1
10
M=N=4,NT=2
BER
BER
M=N=4,NT=2
-1
10
-2
10
-2
10
0
-3
5
10
15
SNR(dB)
20
25
Fig. 2. BER versus SNR for ND =30 , M=2,4, NT=1,2, when the
number of iterations is 5.
M=N=2,Subspace based
M=N=2,EM
M=N=4,Subspace based
-5
NMSE (dB)
2
3
4
Number of Iterations
5
6
7
Fig. 5. BER versus iteration number for ND=30, SNR=30dB, NT=1,
2 and M=2, 4.
M=N=4,EM
-10
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[1]
-15
-20
[2]
-25
[3]
-30
-35
-40
0
1
beamforming method has been derived based on the
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a blind manner. Simulation results demonstrated that the
proposed method achieves a good performance and
outperforms the subspace based semi-blind algorithm.
5
0
10
30
[4]
5
10
15
20
25
30
SNR (dB)
Fig. 3. NMSE versus SNR, comparing the proposed EM based
beamforming algorithm with signal subspace approach [1], NT=1,
ND=30 , when the number of iterations is 5.
[5]
[6]
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