Transcript 投影片 1 - 清華大學電機系

Block-by-Block Blind Channel Estimation
Algorithm Using Subcarrier Averaging for
Multi-user OFDM Systems
Presenter: Teng-Han Tsai (蔡騰漢)
Institute of Communications Engineering &
Department of Electrical Engineering
National Tsing Hua University
Hsinchu, Taiwan 30013, R.O.C.
E-mail: [email protected]
1
Outline
1. Introduction
2. Review of MVDR beamformer and MUSIC algorithm
3. MIMO Model for Post-FFT Beamforming Structure
4. Proposed Blind Channel Estimation Algorithm by Subcarrier
Averaging
5. Simulation Results
6. Conclusions
MVDR: Minimum Variance Distortionless Response
MUSIC: Multiple Signal Classification
MIMO: Multiple-input Multiple-output
2
Introduction
3
Wireless Environment
Tx
signals
MS 1
sP [ n]
x1[n]
L1
1,1
user 1

user P
 1,1
ULA (Uniform Linear Array)
BS
 1,1
x2 [n]
1,L1 1,L1  1,L1
s1[n]

 P ,1  P ,1

 P ,1
MS P
LP
Rx
signals
xQ [n]

P ,LP P ,LP
 P ,LP
P : number of users
 p ,l : path gain of the lth path of user p
Noise
Q : number of antennas
 p ,l : Direction of arrivals (DOA) of the lth path of user p ( / 2   p, l   / 2)
 p ,l : time delay of the lth path of user p
L p : number of paths (or DOAs) associated with user p.
L : ( L1  L 2    L P ) total number of paths (or DOAs) of all the users.
4
Assumptions
u p [k ] , p  1, 2,..., P
(A1)
are QPSK ( BPSK ) zero-mean independent identically
2
distributed (i.i.d.) random sequences with E{|u p [k ]| }  1 , and u p [k ] is
statistically independent of uq [k ] for q p .
(A2)
 q , m  p ,l
(A3) 0
for all (q, m)
( p, l ) , Q
 p ,1  p , 2 ...  p ,L p
L , and L is known.
N g , p .
(A4) w[n] is zero-mean white Gaussian and statistically independent of
u p [k ] , p 1, 2, ..., P,
and
E{w[n]wH [n]]}   w2 IQ .
BPSK : Binary phase shift keying
QPSK : Quadriphase shift keying
IQ : Q
Q identity matrix
5
Review of MVDR Beamformer
and MUSIC Algorithm
6
Beamforming (1/3) [2][3]
 Interference suppression
 Signal separation and extraction
 Antenna gain enhancement
 Spectral efficiency increase
u1[n ]
(Desired signal)
1
 P P
u P [n ]
r[n ]
1
#1
Beamformer
v
#2
#Q


y[n ]  1u1 [n ]
(Interfering signal)
7
Beamforming (2/3)
 MIMO Model
path gain
u1[n ]
DOA (Direction of Arrival)
x[n]
1
(source 1)
#1
1
Beamformer
v
y [n ]
#2
 P P
#Q
u P [n ]


Beamformer
Output
(source P )
P
 Assumptions:
x[n]   pa( P )u p [n]  w[n]
p 1
(M1) u p [n ] , p 1, 2, ..., P are wide-sense stationary random processes, and
uq [n ] is statistically independent of u p [n ] for q p.
(M2)
 j i for all j i , and Q
P.
2
(M3) w[n ] is zero-mean white Gaussian with E{w[n ]w H [n ]]}  w
IQ
and statistically independent of u p [n ], p 1, 2, ..., P .
8
Beamforming (3/3)
 MVDR Beamformer:
Criterion:
By Lagrange
multiplier
min E{| y[n] |2 }  min v H R xx v
v
v
subject to
where
v H a(1 )  1
v MV
R xx1a(1 )
 H
a (1 )R xx1a(1 )
Rxx  E{x[n]xH[n]} : correlation matrix of x[n]
1 : (known in advance) DOA of the path of user 1 (Desired source)
Under the assumption (M1) and (M2),
y[n]  vHMV x[n]  1u1[n]
as
 w2  0
which implies the MVDR beamformer can perfectly extract
the desired signal
1u1 [n ] by processing x[n]
MVDR : Minimum Variance Distortionless Response
.
9
DOA Estimation - MUSIC Method (1/2)
 EVD (Eigenvalue Decomposition) of Correlation matrix :
R xx  E{x[n]xH[n]}  ARuu AH   w2IQ
Q
R xx   i ei eiH
i 1
where
Ruu  E{u[n]uH[n]}
R xx  e1
1

eQ   0
0

rank (Ruu )  P
0
0
0  e1H 
 
0  
Q  eQH 
1. e1 ,..., eP , eP 1 ,..., eQ are orthonormal basis.
2.
i   w2 if i  1,..., P
eigenvalues  
2



otherwise
w
 i
10
DOA Estimation - MUSIC Method (2/2)
 Signal subspace is orthogonal to noise subspace :
e Hj a(i )  0
, i  1, 2,..., P , j  P  1, P  2,...,Q
 Construct projection matrix PN :
PN 

Q
ei eiH

i P 1
1 , 2 ,..., P may be found by solving
PN a( )  0 for  .
 Compute MUSIC spectrum :
S MUSIC ( ) 
1
PN a( )
2

1
a H ( ) PN a( )
and search for “infinitely high” spectral peaks.
11
MIMO Model for Post-FFT
Beamforming Structure
12
Post-FFT Beamforming Structure (1/2)
X[k ]
x[n]
channel information at
subcarrier 0 for user p
X[0]
S/P
N-point
FFT
…
GI
Removal
v 2( 0)

P/S
u p [k ]



…
A/D
S/P
u p [0]
v1( 0)

…

N-point
FFT
S/P
…

GI
Removal
N-point
FFT
…
A/D
GI
Removal
…
A/D
beamformer v (p0 )
.. (0)
vQ
N beamformers
13
Post-FFT Beamforming Structure (2/2)
 MIMO model for each subcarrier k:
P
FFT
Lp
x[n]   p ,l a( p ,l )s p [n   p ,l ]  w[n]
p 1 l 1
P Lp
X[k ]   p,l a( p ,l )e
 j 2 p ,l / N
p 1 l 1
u p [k ]  w[k ]
X[k ]  A( k ) u( k ) [k ]  w[k ]
where
A( k )  [a1( k ) , a2( k ) , ..., aP( k ) ] ( Q  P channel matrix)
Lp
Channel response of
user p at subcarrier k
a (pk )    p ,l a( p ,l )e
l 1
j
2k
 p ,l
N
( Q  1 vector )
u( k ) [k ]  [u1[k ], u2[k ] ,... , uP [k ]]T ( P  1 source vector)
w[k ]
( Q  1 white Gaussian noise vector)
14
Proposed Blind Channel Estimation
Algorithm by Subcarrier Averaging
15
MIMO Model (1/4)
 MIMO Model
P Lp
X[k ]    p ,l a ( p ,l )u p [k ]e
p 1 l 1
re-expression
X[k ]  Au[k ]  w[k ]
where
A  [ A1 , A 2 ,, A P ] (Q
j
2k
 p ,l
N
 w[k ]
k  0, 1, ..., N  1
L DOA matrix)
A p  [ p,1a( p ,1 ), p, 2 a( p , 2 ), ..., p,L p a( p ,L p )] ( Q
full column rank
by Assumption
(A2)
Lp )
u[k ]  [ u1[k ],u 2 [k ],, uP [k ] ]T ( L 1 source vector)
u p [k ]  [ u p ,1 [k ], u p , 2 [k ], ... , u p ,L p [k ] ]T ( L p 1 )
u p ,l [k ]  u p [k ]e
 j 2k p ,l / N
(component)
w[k ] ( Q 1 white Gaussian noise vector )
16
MIMO Model (2/4)
 Source Vector u[k ] :
 Under Assumption (A1) and Assumption (A3), check the
components of the L × 1 source vector u[k ] by statistical
averaging :
1. Different users:
E{u p ,i [k ](uq , j [k ])* }  0,
pq
2. Same user but different path:
E {(u p ,i [k ])( u p , j [k ]) * }  e
 j 2k ( p ,i  p , j ) / N
 0,
The components of the source vector u[k ] are
statistically correlated.
ij
MVDR
17
MIMO Model (3/4)
 Define the subcarrier averaging of u[k ] :
1
u[k ] 
N
N 1
u[k ]

k 0
Under Assumption (A1) and Assumption (A3), check the
components of the L × 1 source vector u[k ] by subcarrier
averaging :
P
u p ,i [k ](u q , j [k ])* 

0, ( p , i )  (q , j )
where
P


denotes “convergence in probability” as N   .
Each components of u[k ] can be “de-correlated”
by subcarrier averaging.
MVDR
18
MIMO Model (4/4)
 MVDR and MUSIC methods by subcarrier averaging:
By subcarrier averaging, MVDR and MUSIC methods can be
applied to post-FFT beamforming structure by processing
X[k ]  Au[k ]  w[k ]
where
A  [ A1 , A 2 ,, A P ] (Q L DOA matrix)
A p  [ p,1a( p ,1 ), p, 2 a( p , 2 ), ..., p,L p a( p ,L p )] ( Q
full column rank
by Assumption
(A2)
Lp )
u[k ]  [ u1[k ],u 2 [k ],, uP [k ] ]T ( L 1 source vector)
u p [k ]  [ u p ,1 [k ], u p , 2 [k ], ... , u p ,L p [k ] ]T ( L p 1 )
u p ,l [k ]  u p [k ]e
 j 2k p ,l / N
(component)
w[k ] ( Q 1 white Gaussian noise vector )
19
Algorithm Procedure
X[k ], k  0, 1, ..., N  1
(received signal vector)
MUSIC
DOA
Estimation
ˆ ( k ) , k  0, 1, ..., N  1
A
(estimate of channel matrix)
ˆ (l )
MVDR Beamformer
ˆ (l )
Source
Extraction
Path Gain
Estimation
Time Delay
Estimation
Classification
and Grouping
ˆ (l )
l  1, 2, ..., L
20
Proposed Algorithm –
DOA Estimation
 MUSIC method :
1
S MUSIC ( )  H
a ( ) Pˆ N a( )
( MUSIC spectrum )
Q
ˆ ˆ ˆH
ˆ
where R
XX  X[ k ]X [ k ]   i ei ei : EVD of correlation matrix
H
i 1
ˆ1    ˆL  ˆL 1    ˆQ
( the smallest Q - L eigenvalues )
eˆ 1 ,  , eˆ L , eˆ L 1 ,  , eˆ Q
( noise eigenvectors )
Pˆ N 
Q

i  L 1
eˆ i eˆ iH : projection matrix
All the DOAs can be estimated by finding the L
largest local maxima of SMUSIC(θ) .
EVD : eigenvalue decomposition
21
Proposed Algorithm –
Source Extraction
ˆ , we have MVDR beamformer
By ˆ (l ) and R
XX
v
(l )
ˆ 1 a (ˆ (l ) )
R
 H (lXX
ˆ 1 a (ˆ (l ) )
a (ˆ ) ) R
H
ˆ
R
XX  X[k ]X [k ]
XX
and MVDR beamformer output
(l )
y [k ]  ( v
where
(l ) H
(l )
(l )
) X[k ]   u [k ]e
 j 2k ( l ) / N
 (l ) : path gain associated with DOA ˆ (l )
(l )
and   {1,1 ,, 1,L1 ,,  P ,1 ,,  P ,LP }
u (l ) [k ] : data associated with DOA ˆ (l )
and u (l ) [k ] {u1[k ],, uP [k ]}
 (l ) : time delay associated with DOA ˆ (l )
(l )
and   { 1,1 ,, 1,L1 ,, P ,1 ,, P ,LP }
22
Proposed Algorithm –
Time Delay Estimation (1/3)
 Estimate time delay by processing y (l ) [k ] :
 Data sequence (QPSK signals):
u(l ) [k ] {1, 1, j,  j}
(u (l ) [k ])4  1
 Estimate time delay ˆ (l ) :
ˆ(l )  arg max{J QPSK ( )},
  0, 1,, N g

(l )
(l )
  u [k ]e
8 k
( y [k ] ) exp{ j
}
N
(l )
where
J QPSK ( ) 
 j 2k ( l ) / N
4

(l )
y [k ]
2

2

0,
   (l )
1,
   (l )
23
Proposed Algorithm –
Time Delay Estimation (2/3)
 Estimate time delay by processing y (l ) [k ] :
 Data sequence (BPSK signals):
u(l ) [k ] {1, 1}
(u (l ) [k ])2  1
 Estimate time delay ˆ (l ) :
ˆ(l )  arg max{J BPSK ( )},

  0, 1,, N g
(l )
where
J BPSK ( ) 
(l )
 j 2k ( l ) / N
  u [k ]e
4k
( y (l ) [k ] ) 2 exp{j
}
N
(l )
y [k ]
2

0,
   (l )
1,
   (l )
24
Proposed Algorithm –
Time Delay Estimation (3/3)

Time compensated beamformer output yc(l ) [k ] associated
with ˆ (l ):
yc(l ) [k ] 
where
2k (l )
y [k ] exp{ j
ˆ }   (l ) u (l ) [k ]
N
(l )
 (l )
: path gain associated with DOA ˆ
(l )
and   {1,1 ,, 1,L1 ,,  P ,1 ,,  P ,LP }
(l )
u (l ) [k ] : data associated with DOA ˆ (l )
and u (l ) [k ] {u1[k ],, uP [k ]}
25
Proposed Algorithm –
Classification and Grouping
 Procedures of classification and grouping:
 Define
l , p 
yc(l ) [k ]g *p [k ]
2
(l )
yc [k ]
2
*
g p [k ]
(Step 1)
Select a path yc [k ] and set it to be g1[k ] .
(Step 2)
Calculate
(Step 3)
(Step 4)
(Step 5)
(Step 6)
(1)
l , p for all the paths to be analyzed.
Extract all the paths that havel , p  0.5 and assign
them as a new group.
(i )
From the remaining paths, select another path yc [k ]
and set it to be gi [k ] , where i = 2, …, P.
Go to Step 2 until there is no more path to group.
Finally, there will be P groups where all the paths of a
group belongs to the same user.
26
Proposed Algorithm –
Path Gain Estimation (1/9)
 After Classification and Grouping:
 It is obtained P groups, where in each group there are Lp
sequences with the same data symbol information u p [k ]
multiplied by different coefficient:
 yc(1) [k ]  1,1u1 [k ]


 y L1 [k ]   u [k ]
1, L1 1
 c
Group 1
 yc(1) [k ]   P ,1uP [k ]


 y LP [k ]   u [k ]
P , LP P
 c
Group P
27
Proposed Algorithm –
Path Gain Estimation (2/9)
 Estimate path gain by processing yc(l ) [k ] :
 Data sequence (QPSK signals):
u(l ) [k ] {1, 1, j,  j}
(u (l ) [k ])4  1
 Estimate path gain ˆ (l ) :
  (l )u (l ) [k ]
EQPSK  ( yc( l ) [k ] ) 4  ( ( l ) ) 4
j
j
j
j
(l )
(l )
(l )
,  (l ) e
ˆ (l )   e ,  e ,  e
l ,1
l ,2
l ,3
l ,4
 The 4 solutions have the same magnitude but different
phase angle. Therefore, it needs to choose one of them.
28
Proposed Algorithm –
Path Gain Estimation (3/9)
 Decision of the path gain phase angle:
 For QPSK case:
(Step 1)
From the 4 phase angle solutions {1,1 ,1,2 ,
, 1,3 ,1,4 }
select an angle  1,i , for i = 1, …, 4.
(Step 2)
Select its corresponding path yc [k ] (first path)
and rotate the path by its corresponding phase
Ambiguity phase
angle estimate  1,i .
(1)
yrot ,1[ k ]  yc(1) [ k ]  e
 j 1,i
Im
  (1) u p [ k ]  e j
Im
 1,i
Re
After rotation
3 
 
  0, ,  , 

2
2 
 l ,i
Re
29
Proposed Algorithm –
Path Gain Estimation (4/9)
 Decision of the path gain phase angle:
Im
 For QPSK case:
(Step 3)
(l )
Select another path yc [k ], where
l = 2, …, Lp and rotate it by its 4
possible path gain phase angles
{ l ,1, l ,2 , l ,3 , l ,4} .
 l ,i
Re
yrot ,l ,1[k ]
Im
Im
 l ,1
yrot ,l ,2 [k ]
Im
Im
Re
 l ,3
 l ,2
Re
Re
yrot ,l ,4 [k ]
 l ,4
Re
yrot ,l ,3[k ]
30
Proposed Algorithm –
Path Gain Estimation (5/9)
 Decision of the path gain phase angle:
 For QPSK case:
(Step 4)
(Step 5)
Perform the inner product of the first rotated path yrot ,1[k ]
and the four l-th rotated paths yrot ,l [k ], for l= 2, …, Lp.
. of the four inner products
Calculate the phase angle
yrot ,l [k ]  yrot ,1[k ], which the results will be
H
approximated to {0, π/2, π, 3π/2}.
(Step 6)
Im
Choose the path gain phase angle  1,i whose phase angle of
inner product is closed to 0.
Im
 l ,1
 1,i
Re
Re
They are in phase
yrot ,1,1[k ]
yrot ,l ,1[k ]
31
Proposed Algorithm –
Path Gain Estimation (6/9)
 Decision of the path gain phase angle:
 For QPSK case:
(Step 7)
(Step 8)
Go to the Step 2 until there is no more paths to rotate in
the group.
.
Finally, the path gain phase
angle  1,i for each path of the
group will be obtain.
Note: As the proposed algorithm is a blind method, the estimated path
j
gain has an ambiguity scalar e . This value depends on the choice of the
Phase angle solution in Step 1.
32
Proposed Algorithm –
Path Gain Estimation (7/9)
 Estimate path gain by processing yc(l ) [k ] :
 Data sequence (BPSK signals):
u(l ) [k ] {1, 1}
(u (l ) [k ])2  1
 Estimate path gain ˆ (l ) :
  ( l ) u ( l ) [k ]
EBPSK  ( y c(l ) [ k ] )
2
 ( ( l ) ) 2
ˆ (l )   (l ) or   (l )
 The 2 solutions have the same magnitude but different
phase angle. Therefore, it needs to choose one of them.
33
Proposed Algorithm –
Path Gain Estimation (8/9)
 Decision of the path gain phase angle:
 For BPSK case:
(Step 1)
From the 2 phase angle solutions {1,1 ,1,2 },
select a solution  1,i , for i = 1, 2.
(Step 2)
Select its corresponding path ymc,1[k ] (first path)
and rotate the path by its corresponding phase
Ambiguity phase
angle estimate  1,i .
yrot ,1[k ]  ymc,1[k ]  e
 j1,i
 u p [k ]  e
j p
 p 0,  
(Step 3)
Select another path ymc,l [k ], where l = 2, …, Lp and
rotate it by its 2 possible path gain phase angles
{ l ,1 , l ,2 } .
(Step 4)
Perform the inner product of the first rotated path yrot ,1[k ]
and the four l-th rotated paths yrot ,l [k ], for l= 2, …, Lp.
.
34
Proposed Algorithm –
Path Gain Estimation (9/9)
 Decision of the path gain phase angle:
 For BPSK case:
(Step 5)
Calculate the phase angle of the four inner products
H
yrot ,l [k ]  yrot ,1[k ], which the results will be
approximated to {0, π}.
(Step 6)
Choose the path gain phase angle  1,i whose phase angle of
inner product is closed to 0.
(Step 7)
Go to the Step 2 until there is no more paths to rotate in
the group.
.
Finally, the path gain phase
angle  1,i for each path of the
group will be obtain.
(Step 8)
Note: As the proposed algorithm is a blind method, the estimated path
gain has an ambiguity scalar e
j p
Phase angle solution in Step 1.
. This value depends on the choice of the
35
Proposed Algorithm –
Channel Recovery
ˆ (k ) :
 Estimate of channel matrix A
With the estimated DOA, time delay and path gain of each path,
the channel matrix
ˆ (k )can be obtain:
A
ˆ (k )  [aˆ (k ) , aˆ (k ) ,, aˆ (k ) ]
A
1
2
P
A
where
(k )
P
A(k )  [a1(k ) , a2(k ) ,...,aP(k ) ]
user index
group index
P is P × P unknown
permutation matrix.
( PPH  I P )
Note: The permutation matrix P can be obtained using the information
of the transmitted sources after the data sequence detection.
36
Data Sequence Detection
ˆ and the noise power  n (from
Once the channel information A
MUSIC) are obtained. A MMSE beamformer can be applied:
(k )
(k )
MMSE
V
 (R xx   I )  A
2
n
1
(k )
2
 (A
(k )
A
(k )H
  n2 I ) 1  A ( k )
Then the estimated data sequence for a user p will be:
(k )
H
ˆ
u p [k ]  (VMMSE )  X[k ]  u p [k ]  e
for p  1,..., P
3 
 
 p  0, ,  , 

 p 0,  
2
2 
Ambiguity phase
j p
For QPSK
For BPSK
37
Simulation Results
38
Performance Index
 Definition of Normalized Mean Square Error (NMSE):
1
NMSE 
N
where
N 1

k 0
A
(k )
~ (k )
A
A
2
F
(k ) 2
F
~
ˆ (k ) P H : estimate of channel matrix A (k )
A (k )  A

F
: Frobenius norm
39
Parameters Used
 A two-user (P =2) OFDM system
 Ng= 20





2
w[n ]: i.i.d. zero-mean Gaussian with E{w[n ]w H [n ]}  w
IQ .
Q = 10.
N = 1024.
L = 6 ( L1  3, L2  3) .
DOA randomly generated for all the users   / 2:  / 2 .
 Time delay randomly generated for all the users p,l  N g .
 Path gain randomly generated for all the users  l 1|  p , l |2 = 1 .
Lp
 Input SNR:
2
Lp




E   p,l a( p ,l ) s p [n   p ,l ] 
l 1




SNR 
E{||w[n] ||2 }
40
NMSE of A
41
Symbol Error Ratio
42
Conclusions
 We have presented blind channel estimation algorithm by
subcarrier averaging for the post-FFT beamforming
structure over one OFDM block. This proposed algorithm
basically includes DOA estimation using MUSIC method, source
extraction using MVDR beamformer, time delay estimation and
compensation, classification and grouping, path gain estimation
and channel recovery.
 The proposed channel estimation algorithm only needs one
OFDM block to estimate channel with good performance.
 Some simulation results were provided to support the blind
beamformer designed by the proposed channel estimation
algorithm, and its performance is very closed to the
performance of the MMSE beamforming using perfect
channel.
43
References (1/3)
[1] R. V. Nee and R. Prasad, OFDM for Wireless Multimedia Communications . Boston:
Artech House, 1999.
[2] J. C. Liberti and T. S. Rappaport, Smart Antennas for Wireless Communications: IS95 and Third Generation CDMA Applications. New Jersey: Prentice Hall, 1999.
[3] L. C. Godara, “ Application of antenna arrays to mobile communications, Part II:
Beam-forming and direction-of-arrival considerations,” IEEE Proceeding, vol. 85,
No. 8, pp 1195-1245, Aug. 1997.
[4] Ralph O. Schmidt, “Multiple emitter location and signal parameter,” Proc. IEEE Trans.
Antennas and Propagation, vol. AP-34, No. 3, pp. 3381-3391, Dec. 1999.
[5] Shinsuke Hara, Montree Budsabathon, and Yoshitaka Hara, “A pre-FFT OFDM
adaptive antenna array with eigenvector combining,” Proc. IEEE International
Conference on Communication., vol. 4, pp. 2412-2416, June. 2004.
[6] Ming LEI, Ping ZHANG, and Hiroshi HARADA, and Hiromitsu WAKANA, “LMS
adaptive beamforming based on pre-FFT combining for ultra high-data-eate OFDM
system,” Proc. IEEE 60th Vehicular Technology Conference, vol. 5, Los Angeles,
California, USA, Sept. 26-29, 2004, pp. 3664-3668.
44
References (2/3)
[7] Fred W. Vook and Kevin L. Baum, “Adaptive antennas for OFDM,” Proc. IEEE 48th
Vehicular Technology Conference, vol. 1, Ottawa, Ont., May 18-21, 1998, pp. 606-610.
[8] Chan Kyu Kim, Kwangchun Lee, and Yong Soo Cho, “Adaptive Beamforming Algorithm
for OFDM Systems with Antenna Arrays,” IEEE Trans. Consumer Electronics, vol. 46,
No. 4, pp. 1052-1058, Nov. 2000.
[9] Hidehiro Matsuoka and Hiroki Shoki, “Comparison of pre-FFT and post-FFT
processing adaptive arrays for OFDM systems in the presence of co-channel
interference,” Proc. IEEE 14th International Symposium on Personal, Indoor and
Mobile Radio Communications, vol. 2, Beijing, China, Sept. 7-10, 2003, pp. 1603-1607.
.
[10] Zhongding Lei and Francois P.S. Chin, “Post and pre-FFT beamforming in an OFDM
system,” Proc. IEEE 59th Vehicular Technology Conference, vol. 1, Milan, Italy, May
17-19, 2004, pp. 39-43. .
[11] Matthias Munster and Lajos Hanzo, “Performance of SDMA multi-user detection
techniques for Walsh-Hadamard-Spread OFDM Schemes,” Proc. IEEE 54th
Vehicular Technology Conference, vol. 4, Atlantic City, NJ, USA, Oct. 7-11, 2001, pp.
2219-2323.
45
References (3/3)
[12] Samir Kapoor, Daniel J. Marchok, and Yih-Fang Huang, “Adaptive interference
suppression in multiuser wireless OFDM systems using antenna arrays,” IEEE Trans.
Signal Processing, vol. 47, No. 12, pp. 3381-3391, Dec. 1999.
[13] Shenghao Yang and Yuping Zhao, “Channel estimation method for 802.11a WLAN
with multiple-antenna,” Proc. 5th International Symposium on Multi-Dimensional
Mobile Communications, 29 Aug.-1 Sept, 2004, pp. 297-300.
[14] Bassem R. Mahafza and Atef Z. Elsherbeni, Matlab Simulations for Radar Systems
Design. Boca Raton, FL :CRC Press/Chapman & Hall, 2004.
[15] Kai-Kit Wong, Roger S.-K. Cheng, Khaled Ben Letaief, and Ross D. Murch, “Adaptive
antennas at the mobile and base stations in an OFDM/TDMA system,” IEEE Trans.
Signal Processing, vol. 49, No. 1, pp. 195-206, Jan. 2001.
[16] Shiann-Shiun Jeng, Garret Toshio Okamoto, Guanghan Xu, Hsin-Piao Lin, and
Wolfhard J. Vogel, “Experimental evaluation of smart antenna system performance
for wireless communications,” IEEE Trans. Antennas and Propagation, vol. 46, No. 6,
pp. 749-757, June 1998.
46
Thank you very much
47
Transmitter of OFDM Systems
Ng
User p
N-point
IFFT
…
S/P
…
u p [k ,m]
P/S
1
s p [ n] 
N
s p [n ,m]
GI
Insertion
D/A & Up
Converter
N 1
j 2kn / N
u
[
k
]
e
 p
k 0
p  1, 2, ..., P , n   N g ,  N g  1, ..., N 1 ,
k : frequency-domain sample index u p [k , m ] : data sequence of user p
n : time-domain sample index
N g : length of GI ( N / 2)
N : number of subcarriers
48
Symbol Error Ratio (QPSK)
49
Symbol Error Ratio (BPSK)
50