Embed Size (px)
Citation preview
Fire Tom WadaProfessor, Information Engineering, Univ. of the
RyukyusChief Scientist at Magna Design Net, Inc
[email protected]://www.ie.u-ryukyu.ac.jp/~wada/
04/21/23 1Fire Tom Wada, Univ. of the Ryukyus
• Matrix Based OFDM Modeling• Channel Matrix diagonalization by
Unitary Matrix FFT
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 2
04/21/23 3
Base Station
MobileReception
Path 2
Path 3
Direct Path
Building
Transmission
Antenna
Reception AntennaSingle Input and
Single Output(SISO)
Channel
Fire Tom Wada, Univ. of the Ryukyus
04/21/23 4
MMAAPP
SS//PP
IFFTIFFT
PP//SS
Generate Generate Complex symbol Complex symbol dd00~d~d Ｎ－１ Ｎ－１
Bit Bit streamstream
Copyto make Guard
Interval
OFDM symbol (1/fOFDM symbol (1/f00))TTgg
Fire Tom Wada, Univ. of the Ryukyus
04/21/23 5
Base Station
MobileReception
Path 2
Path 3
Direct Path
Building
OFDM symbol (1/fOFDM symbol (1/f00))TTgg
OFDM symbol (1/fOFDM symbol (1/f00))TTgg
Fire Tom Wada, Univ. of the Ryukyus
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 6
SS//PP FFTFFT
EqualizeEqualize
DDEEMMAAPP
Bit Bit StreamStream
OFDM symbol (1/fOFDM symbol (1/f00))TTgg
PP//SS
NoiseNoise
Remove Guard Interval
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 7
)1(
)1(
)0(
1
)1(
)1(
)0(
)1(
)1(
)0(
,;1
)1(
)1(
)0(
)1(
)1(
)0(
)1(*)1()1(0
)1(10
000
)1(*)1(
Nx
x
x
N
NY
Y
Y
Nx
x
x
columnlrowkN
Nx
x
x
FFT
NY
Y
Y
NNN
N
lk
Nj
eHere
2
,
04/21/23 8
)1(
)1(
)0(
1
)1(
)1(
)0(
)1(
)1(
)0(
,;1
)1(
)1(
)0(
)1(
)1(
)0(
)1(*)1()1(0
)1(10
000
)1(*)1(
NY
Y
Y
N
Nx
x
x
NY
Y
Y
columnlrowkN
NY
Y
Y
IFFT
Nx
x
x
NNN
N
lk
Mj
eHere
2
,
Fire Tom Wada, Univ. of the Ryukyus
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 9
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 10
Symbol n
Symbol n-1
GI of n
GI of n-1
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 11
Channel Matrix is Cyclic Matrix by GI.
Base Station Receiv
er
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 12
Channel Impulse Response = [1, 0.5 , 0, 0]
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 13
)3(
)2(
)1(
)0(
)3(000
0)2(00
00)1(0
000)0(
)3(
)2(
)1(
)0(
)3(
)2(
)1(
)0(
1
1
1
1111
4
1
15.000
015.00
0015.0
5.0001
1
1
1
1111
4
1
)3(
)2(
)1(
)0(
)3(
)2(
)1(
)0(
***
)3(
)2(
)1(
)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!
04/21/23 14
)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
1
1
1111
4
1
)3(
)2(
)1(
)0(
1
1
1
1111
4
1
15.000
015.00
0015.0
5.0001
1
1
1
1111
4
1
)3(
)2(
)1(
)0(
)3(
)2(
)1(
)0(
)3(
)2(
)1(
)0(
***
)3(
)2(
)1(
)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
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 15
)3(
)2(
)1(
)0(
)3(
1000
0)2(
100
00)1(
10
000)0(
1
)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(
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
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 16
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
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 17
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
Unitary MatrixUnitary Matrix U can satisfy following property.
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 18
IUUUU HH
)(0
)(1
,
1
ji
ji
then
vectorverticaliswhen
jHi
N
i
ee
eeU
e
U
H
H
H
H
UMatrix
Cyclic H
)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 Channel Modeling
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 19
SISO ChannelOFDM makes Multi-path channel simple complex h(k) for freq=k.
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 20
)()()(
)3(
)2(
)1(
)0(
)3(000
0)2(00
00)1(0
000)0(
)3(
)2(
)1(
)0(
***
)3(
)2(
)1(
)0(
kXkhkY
X
X
X
X
h
h
h
h
X
X
X
X
IFFTChannelFFT
Y
Y
Y
Y
Transmission
Antenna
Reception Antenna
SISO ChannelIFFT FFT
)(kX )(kY
bjakh )(
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 21
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 HNr 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.
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 22
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
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 23
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
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 24
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 25
)(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
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 26
)('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
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 27
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
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
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
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 28
)('1 kX
)('2 kX
)('3 kX
)(' kX Nt
)('1 kY
)('2 kY
)('3 kY
)(' kYNr
H=VΣU
H
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 systemSpace Division Multiplexing by MIMO (K stream)Orthogonal Frequency Division Multplxing (OFDM)
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 29
)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
SummaryThis 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.
04/21/23 Fire Tom Wada, Univ. of the Ryukyus 30
![ETSSMaterial110131a.ppt [互換モード]ie.u-ryukyu.ac.jp/~wada/digsys17/Lab/ETSSMaterial110131a.pdfTitle Microsoft PowerPoint - ETSSMaterial110131a.ppt [互換モード] Author WADA](https://img.dokumen.tips/doc/110x75/600b0d76ca5c001b3d0e2ac2/fffieu-ryukyuacjpwadadigsys17labetssmaterial110131apdf-title.jpg)
