introduction to mimo ofdm - university of the ryukyuswada/system13/100-sysarch2013... · 2013. 12....
TRANSCRIPT
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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
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• REVIEW
2013/12/25 Fire Tom Wada, Univ. of the Ryukyus 2
Section 1
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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
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Base Station Receiver
Two path Multi path Channel Example
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Channel Impulse Response = [1, 0.5 , 0, 0]
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Two path Multi path Channel Example
2013/12/25 Fire Tom Wada, Univ. of the Ryukyus 5
)3(
)2(
)1(
)0(
)3(000
0)2(00
00)1(0
000)0(
)3(
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1
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1111
4
1
15.000
015.00
0015.0
5.0001
1
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)3(
)2(
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)3(
)2(
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***
)3(
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)0(
963
642
321
963
642
321
X
X
X
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!
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Additive Noise
2013/12/25 6
)3(
)2(
)1(
)0(
)3(
)2(
)1(
)0(
)3(000
0)2(00
00)1(0
000)0(
)3(
)2(
)1(
)0(
)3(
)2(
)1(
)0(
1
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1
1111
)3(
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1
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1111
4
1
15.000
015.00
0015.0
5.0001
1
1
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1111
)3(
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)3(
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***
)3(
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)0(
963
642
321
963
642
321
963
642
321
N
N
N
N
X
X
X
X
H
H
H
H
Y
Y
Y
Y
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
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How to recover sending signal from receiver signal.- EQUALIZE -
2013/12/25 Fire Tom Wada, Univ. of the Ryukyus 7
)3(
)2(
)1(
)0(
)3(
1000
0)2(
100
00)1(
10
000)0(
1
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)0(
Y
Y
Y
Y
H
H
H
H
X
X
X
X
Then
X
X
X
X
H
H
H
H
X
X
X
X
IFFTChannelFFT
Y
Y
Y
Y
NoiseIgnore
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Summary of Matrix model of OFDM
2013/12/25 Fire Tom Wada, Univ. of the Ryukyus 8
Transmission Antenna
Reception Antenna
Channel
)3(
)2(
)1(
)0(
X
X
X
X
IFFT
Channel
Cyclic
FFT
)3(
1000
0)2(
100
00)1(
10
000)0(
1
H
H
H
H
)3(
)2(
)1(
)0(
X
X
X
X
)3(
)2(
)1(
)0(
)3(
1000
0)2(
100
00)1(
10
000)0(
1
)3(
)2(
)1(
)0(
X
X
X
X
IFFTChannel
CyclicFFT
H
H
H
H
X
X
X
X
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Important Mathematics
2013/12/25 Fire Tom Wada, Univ. of the Ryukyus 9
Cyclic Matrix can be diagonalized by FFT and IFFT.
XH is Hermitian of X, that is, complex conjugate and transpose.
)),1(),0((
)3(000
0)2(00
00)1(0
000)0(
)),1(),0((
)3(000
0)2(00
00)1(0
000)0(
HHdiag
H
H
H
H
UMatrix
CyclicU
UIFFT
H
FFT
Since
HHdiag
H
H
H
H
IFFTMatrix
CyclicFFT
H
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Unitary Matrix Unitary Matrix U can satisfy following property.
2013/12/25 Fire Tom Wada, Univ. of the Ryukyus 10
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), … ).
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• MIMO OFDM Channel Modeling
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Section 2
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SISO Channel OFDM makes Multi-path channel simple complex h(k) for freq=k.
2013/12/25 Fire Tom Wada, Univ. of the Ryukyus 12
)()()(
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0)2(00
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kXkhkY
X
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h
h
h
h
X
X
X
X
IFFTChannelFFT
Y
Y
Y
Y
Transmission Antenna
Reception Antenna
SISO ChannelIFFT FFT
)(kX )(kY
bjakh )(
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MIMO Channel- Nr X Nt SISO Channels for Freq=k -
2013/12/25 Fire Tom Wada, Univ. of the Ryukyus 13
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
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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.
2013/12/25 Fire Tom Wada, Univ. of the Ryukyus 14
NtNtNrNr
K
H
0000
0000
0000
2
1
UVH
.,1HH
K andofvalueeigenis HHHH
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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
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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
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There are K(=rank(H)) independent channel
2013/12/25 Fire Tom Wada, Univ. of the Ryukyus 19
0'
0'
''
''
'
'
'
000
000
00
00
00
'
'
'
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
1
2
1
2
1
2
1
2
1
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
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SVD-MIMO system
2013/12/25 Fire Tom Wada, Univ. of the Ryukyus 20
)('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
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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
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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.
2013/12/25 Fire Tom Wada, Univ. of the Ryukyus 22