Fire Tom Wada
Professor, Information Engineering, Univ. of the Ryukyus
Chief Scientist at Magna Design Net, Inc
http://www.ie.u-ryukyu.ac.jp/~wada/
2013/12/25 1Fire Tom Wada, Univ. of the Ryukyus
Introduction to MIMO OFDM
• REVIEW
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Section 1
SISO Channel
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Base Stat ion
Mobile
Recept ion
Path 2
Path 3
Direct Path
Building
Transmission Antenna
Reception Antenna
Single Input andSingle Output
(SISO)Channel
Fire Tom Wada, Univ. of the Ryukyus
Base Station Receiver
Two path Multi path Channel Example
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Channel Impulse Response = [1, 0.5 , 0, 0]
Two path Multi path Channel Example
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15.000
015.00
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963
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X
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X
H
H
H
H
Y
Y
Y
Y
X
X
X
X
Y
Y
Y
Y
X
X
X
X
IFFTChannelFFT
Y
Y
Y
Y
If time domain channel matrix is cyclic, Frequency Domain Channel Matrix is diagonal!
Additive Noise
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H
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H
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Y
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noise
noise
noise
noise
X
X
X
X
Y
Y
Y
Y
noise
noise
noise
noise
X
X
X
X
IFFTChannelFFT
Y
Y
Y
Y
Fire Tom Wada, Univ. of the Ryukyus
How to recover sending signal from receiver signal.- EQUALIZE -
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)3(
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H
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H
X
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Then
X
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H
H
H
H
X
X
X
X
IFFTChannelFFT
Y
Y
Y
Y
NoiseIgnore
Summary of Matrix model of OFDM
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Transmission Antenna
Reception Antenna
Channel
)3(
)2(
)1(
)0(
X
X
X
X
IFFT
Channel
Cyclic
FFT
)3(
1000
0)2(
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10
000)0(
1
H
H
H
H
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IFFTChannel
CyclicFFT
H
H
H
H
X
X
X
X
Important Mathematics
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Cyclic Matrix can be diagonalized by FFT and IFFT.
XH is Hermitian of X, that is, complex conjugate and transpose.
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HHdiag
H
H
H
H
UMatrix
CyclicU
UIFFT
H
FFT
Since
HHdiag
H
H
H
H
IFFTMatrix
CyclicFFT
H
Unitary Matrix Unitary Matrix U can satisfy following property.
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IUUUU HH
)(0
)(1
,
1
ji
ji
then
vectorverticaliswhen
j
H
i
N
i
ee
eeU
e
U
H
H
H
H
UMatrix
CyclicH
)3(000
0)2(00
00)1(0
000)0(
Eigen value of Channel Cyclic Matrix is Channel Transfer Function as (H(0), H(1), H(2), … ).
• MIMO OFDM Channel Modeling
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Section 2
SISO Channel OFDM makes Multi-path channel simple complex h(k) for freq=k.
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kXkhkY
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h
h
h
h
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IFFTChannelFFT
Y
Y
Y
Y
Transmission Antenna
Reception Antenna
SISO ChannelIFFT FFT
)(kX )(kY
bjakh )(
MIMO Channel- Nr X Nt SISO Channels for Freq=k -
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NtNtNrNtNrNr
Nt
Nt
Nr X
X
X
X
X
X
hhh
hhh
hhh
Y
Y
Y
2
1
2
1
21
22221
11211
2
1
0
00
0
0
H
)(1 kX)(11 kh
)(2 kX
)(3 kX
)(kX Nt
)(1 kY
)(2 kY
)(3 kY
)(kYNr
)(21 kh
)(32 kh
)(khNrNt
H
Singular value decomposition of Nr x Nt Matrix H Nr x Nt matrix H can be decomposed as below using
Nr x Nr Unitary matrix V and Nt x Nt Unitary matrix U.
Σ is Nr x Nt diagonal matrix.
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NtNtNrNr
K
H
0000
0000
0000
2
1
UVH
.,1HH
K andofvalueeigenis HHHH
SVD Example by Matlab(1) H =
1 2 3
2 4 5
>> [U,S,V] = svd(H)
U =
-0.4863 -0.8738
-0.8738 0.4863
S =
7.6756 0 0
0 0.2913 0
V =
-0.2910 0.3396 -0.8944
-0.5821 0.6791 0.4472
-0.7593 -0.6508 -0.0000
H =
1 2
2 4
3 5
>> [U,S,V] = svd(H)
U =
-0.2910 -0.3396 -0.8944
-0.5821 -0.6791 0.4472
-0.7593 0.6508 -0.0000
S =
7.6756 0
0 0.2913
0 0
V =
-0.4863 0.8738
-0.8738 -0.4863
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SVD Example by Matlab(2) H =
1.0000 + 1.0000i 2.0000 + 1.0000i
1.0000 - 3.0000i 3.0000 - 1.0000i
>> [U,S,V] = svd(H)
U =
-0.4616 - 0.0659i -0.4907 + 0.7361i
-0.3956 + 0.7913i -0.2863 - 0.3680i
S =
5.0000 0
0 1.4142
V =
-0.6594 0.7518
-0.5934 + 0.4616i -0.5205 + 0.4048i
>> U*S*V'
ans =
1.0000 + 1.0000i 2.0000 + 1.0000i
1.0000 - 3.0000i 3.0000 - 1.0000i
>> U*S*V'
ans =
1.0000 + 1.0000i 2.0000 + 1.0000i
1.0000 - 3.0000i 3.0000 - 1.0000i
>> U'*U
ans =
1.0000 0.0000 - 0.0000i
0.0000 + 0.0000i 1.0000
>> U*U'
ans =
1.0000 0.0000 - 0.0000i
0.0000 + 0.0000i 1.0000
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MIMO communication
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)(1 kX
)(2 kX
)(3 kX
)(kX Nt
)(1 kY
)(2 kY
)(3 kY
)(kYNr
NtNr X
X
X
Y
Y
Y
2
1
2
1
H
HMIMO
Channel
Introduce pre-processing and post-processing
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)('1 kX
)('2 kX
)('3 kX
)(' kX Nt
)('1 kY
)('2 kY
)('3 kY
)(' kYNr
'
'
'
'
'
'
2
1
2
1
Nt
H
Nr X
X
X
Y
Y
Y
HUV
HMIMO
Channel
NtxNt
UNrxNr
HV
There are K(=rank(H)) independent channel
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0'
0'
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'
000
000
00
00
00
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1
111
2
11
2
1
Nr
K
KKK
Nt
K
Nr
Y
Y
XY
XY
X
X
X
Y
Y
Y
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2
1
2
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NtNr
Nt
HH
Nr
Nt
H
Nr
X
X
X
Y
Y
Y
X
X
X
Y
Y
Y
X
X
X
Y
Y
Y
UUVV
HUV
SVD-MIMO system
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)('1 kX
)('2 kX
)('3 kX
)(' kX Nt
)('1 kY
)('2 kY
)('3 kY
)(' kYNr
H=VΣUH
MIMOChannel
NtxNt
UNrxNr
HV
1
K
)('1 kX
)('2 kX
)('3 kX
)(' kX Nt
)('1 kY
)('2 kY
)('3 kY
)(' kYNr
Put them altogetherMIMO-OFDM system Space Division Multiplexing by MIMO (K stream)
Orthogonal Frequency Division Multplxing (OFDM)
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)1:0('1 NX
)1:0('2 NX
)1:0('1 NY
)1:0('2 NY
IFFT FFT
IFFT
IFFT
IFFT
FFT
FFT
FFT
NtxK
NtxK
NtxK
NtxK
NtxK
NtxK
NtxK
KxNr
U HV
Summary This presentation shows matrix based modeling for both
OFDM and MIMO and there are many similarity in mathematics.
1. OFDM realizes many parallel communication channels in frequency domain.
2. OFDM converts multi-path channel to simple one tap channel such as h(k)=a+bj for Frequency=k.
3. Then OFDM-based MIMO system can focus on simple channel matrix.
4. By singular value decomposition (SVD), MIMO channel matrix H can be decomposed to V*Σ*UH.
5. Non-zero elements of Σ (rank of H) indicates parallel communication channel in space.
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