keyholes and mimo channel modelling alain [email protected]@ensta.fr ensta 32 bd...
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![Page 1: KEYHOLES AND MIMO CHANNEL MODELLING Alain SIBILLEsibille@ensta.frsibille@ensta.fr ENSTA 32 Bd VICTOR, 75739 PARIS cedex 15, FRANCE COST 273, Bologna meeting](https://reader035.vdocument.in/reader035/viewer/2022062423/5697bf791a28abf838c82665/html5/thumbnails/1.jpg)
KEYHOLES AND MIMO CHANNEL MODELLING
Alain SIBILLE [email protected]
ENSTA32 Bd VICTOR, 75739 PARIS cedex 15, FRANCE
COST 273, Bologna meetingCOST 273, Bologna meeting
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Outline
Keyholes in MIMO channels viewed as the result of diffraction
![Page 3: KEYHOLES AND MIMO CHANNEL MODELLING Alain SIBILLEsibille@ensta.frsibille@ensta.fr ENSTA 32 Bd VICTOR, 75739 PARIS cedex 15, FRANCE COST 273, Bologna meeting](https://reader035.vdocument.in/reader035/viewer/2022062423/5697bf791a28abf838c82665/html5/thumbnails/3.jpg)
Outline
Keyholes in MIMO channels viewed as the result of diffraction
Channel modelling with multipath junctions in a smallSize, uncoupled sensors antenna approximation
![Page 4: KEYHOLES AND MIMO CHANNEL MODELLING Alain SIBILLEsibille@ensta.frsibille@ensta.fr ENSTA 32 Bd VICTOR, 75739 PARIS cedex 15, FRANCE COST 273, Bologna meeting](https://reader035.vdocument.in/reader035/viewer/2022062423/5697bf791a28abf838c82665/html5/thumbnails/4.jpg)
Outline
Keyholes in MIMO channels viewed as the result of diffraction
Channel modelling with multipath junctions in a smallSize, uncoupled sensors antenna approximation
How to include inter-sensors coupling
![Page 5: KEYHOLES AND MIMO CHANNEL MODELLING Alain SIBILLEsibille@ensta.frsibille@ensta.fr ENSTA 32 Bd VICTOR, 75739 PARIS cedex 15, FRANCE COST 273, Bologna meeting](https://reader035.vdocument.in/reader035/viewer/2022062423/5697bf791a28abf838c82665/html5/thumbnails/5.jpg)
Outline
Keyholes in MIMO channels viewed as the result of diffraction
Channel modelling with multipath junctions in a smallSize, uncoupled sensors antenna approximation
How to include inter-sensors coupling
towards a stochastic MIMO channel model
![Page 6: KEYHOLES AND MIMO CHANNEL MODELLING Alain SIBILLEsibille@ensta.frsibille@ensta.fr ENSTA 32 Bd VICTOR, 75739 PARIS cedex 15, FRANCE COST 273, Bologna meeting](https://reader035.vdocument.in/reader035/viewer/2022062423/5697bf791a28abf838c82665/html5/thumbnails/6.jpg)
Outline
Keyholes in MIMO channels viewed as the result of diffraction
Channel modelling with multipath junctions in a smallSize, uncoupled sensors antenna approximation
How to include inter-sensors coupling
towards a stochastic MIMO channel model
Conclusion
![Page 7: KEYHOLES AND MIMO CHANNEL MODELLING Alain SIBILLEsibille@ensta.frsibille@ensta.fr ENSTA 32 Bd VICTOR, 75739 PARIS cedex 15, FRANCE COST 273, Bologna meeting](https://reader035.vdocument.in/reader035/viewer/2022062423/5697bf791a28abf838c82665/html5/thumbnails/7.jpg)
333131
232221
131211
321
3
2
1
BABABA
BABABA
BABABA
KAAAK
B
B
B
H
0)()()( 23112311 BAEBAEBABAE : uncorrelated (complex) entries
K
A1A2
A3
B1B2
B3
Rank(H)=1 (two null coefficients of characteristic polynomial)
Slit transmittance
Keyholes : The concept of « Keyholes » has been suggested by Chizhik in order to hightlight the imperfect correspondence between rank and correlation. In a keyhole, the channel matrix has uncorrelated entries, but its rank is one. Such keyholes have therefore intrinsically a small capacity, even in a rich scattering environment.
1D channel
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A simple numerical example of keyhole using Kirchhoff diffraction:
Large slit: no diffraction
Rx Tx
Keyholes in MIMO channels
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A simple numerical example of keyhole using Kirchhoff diffraction:
Large slit: no diffraction Narrow slit: diffraction and multipath junction 1 3
Rx Tx
Rx Tx
)exp()exp()exp()exp(..),,(3
1,1tjrij
jiiijjitr rkjrkjljkjklKTRrrH
junction
Keyholes in MIMO channels
Kij computed by Kirchhoff diffraction
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2.5 3 3.5 40
0.2
0.4
0.6
0.8
cum
ulat
ed p
roba
bilit
y
case A
wide slitlittle correlation : 3 DFSV: -7, +4.8, +7.6 dB
HH
n
SNRIC
tnr
.detlog2
Keyholes in MIMO channels
capacity (b/s/Hz)
H: (normalized) channel transmission matrix
nt=3: number of Tx, Rx radiators
SNR = 3 dB
Space-variant stochastic ensemble
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2.5 3 3.5 40
0.2
0.4
0.6
0.8
cum
ulat
ed p
roba
bilit
y
case A
case B
wide slitlittle correlation : 3 DFSV: -7, +4.8, +7.6 dB
B: narrow slit, little correlation : 1 DFSV: -47, -28, +9.5 dB
Keyholes in MIMO channels
capacity (b/s/Hz)
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2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
cum
ulat
ed p
roba
bilit
y
case A
case C case B
wide slitlittle correlation : 3 DFSV: -7, +4.8, +7.6 dB
C: narrow slit, strong correlation : 1 DFSV: -111, -41, +9.5 dB
B: narrow slit, little correlation : 1 DFSV: -47, -28, +9.5 dB
Keyholes in MIMO channels
capacity (b/s/Hz)
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2 2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
capacity (b/s/Hz)
cum
ula
ted
pro
babi
lity
5 2
Slit width in units of
Keyholes in MIMO channels: capacity vs. Slit width
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2 2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
capacity (b/s/Hz)
cum
ula
ted
pro
babi
lity
5 2 0.25 0.5
Slit width in units of
Keyholes in MIMO channels: capacity vs. Slit width
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2 2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
capacity (b/s/Hz)
cum
ula
ted
pro
ba
bili
ty
5 2 1 0.25 0.5
Slit width in units of
Keyholes in MIMO channels: capacity vs. Slit width
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2 2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
capacity (b/s/Hz)
cum
ula
ted
pro
ba
bili
ty
5 2 1 0.25 0.5
Slit width in units of
When d< ~ /2 all incoming waves are diffracted into all exiting waves through a 1-dimensional channel
Keyholes in MIMO channels: capacity vs. Slit width
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2 2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
capacity (b/s/Hz)
cum
ula
ted
pro
ba
bili
ty
5 2 1 0.25 0.5
Slit width in units of
When d< ~ /2 all incoming waves are diffracted into all exiting waves through a 1-dimensional channel
When d>~2 transmission through the slit occurs through multiple modes and evanescent states and results in greater 3 dimensional effective channel
Keyholes in MIMO channels: capacity vs. Slit width
![Page 18: KEYHOLES AND MIMO CHANNEL MODELLING Alain SIBILLEsibille@ensta.frsibille@ensta.fr ENSTA 32 Bd VICTOR, 75739 PARIS cedex 15, FRANCE COST 273, Bologna meeting](https://reader035.vdocument.in/reader035/viewer/2022062423/5697bf791a28abf838c82665/html5/thumbnails/18.jpg)
333131
232221
131211
321
3
2
1
BABABA
BABABA
BABABA
KAAAK
B
B
B
H
0)()()( 23112311 BAEBAEBABAE : uncorrelated (complex) entries
K
A1A2
A3
B1B2
B3
Rank(H)=1 (two null coefficients of characteristic polynomial)junction
Keyholes : correlations or no correlations ?
![Page 19: KEYHOLES AND MIMO CHANNEL MODELLING Alain SIBILLEsibille@ensta.frsibille@ensta.fr ENSTA 32 Bd VICTOR, 75739 PARIS cedex 15, FRANCE COST 273, Bologna meeting](https://reader035.vdocument.in/reader035/viewer/2022062423/5697bf791a28abf838c82665/html5/thumbnails/19.jpg)
333131
232221
131211
321
3
2
1
BABABA
BABABA
BABABA
KAAAK
B
B
B
H
0)()()( 23112311 BAEBAEBABAE
0
.2
1
2
121
21112111
AEAEBEBE
BAEBAEBABAE
: uncorrelated (complex) entries
: correlated amplitudes
K
A1A2
A3
B1B2
B3
Rank(H)=1 (two null coefficients of characteristic polynomial)junction
Keyholes : correlations or no correlations ?
![Page 20: KEYHOLES AND MIMO CHANNEL MODELLING Alain SIBILLEsibille@ensta.frsibille@ensta.fr ENSTA 32 Bd VICTOR, 75739 PARIS cedex 15, FRANCE COST 273, Bologna meeting](https://reader035.vdocument.in/reader035/viewer/2022062423/5697bf791a28abf838c82665/html5/thumbnails/20.jpg)
2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
capacity (b/s/Hz)
cum
ulat
ed p
roba
bilit
y
case A
case D
case C case B
wide slitlittle correlation : 3 DFSV: -7, +4.8, +7.6 dB
C: narrow slit, strong correlation : 1 DFSV: -111, -41, +9.5 dB
B: narrow slit, little correlation : 1 DFSV: -47, -28, +9.5 dB
narrow slit, strong correlation, one random phase: 2 DFSV: -40, -3.9, +9.4 dB
fading
Keyholes in MIMO channels
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)()()()( Ttr aWaH
Small antenna, uncoupled sensors approximation: {ar , at : steering matrices for N DOAs and M DODs (nrXN , MXnt )
W : wave connecting matrix (NXM) : complex attenuations from all DODs to all DOAs
W is in general rectangular in the presence of path junctions (diffraction, refraction …)
Rank(H) Min(nr , nt , N , M)
All MIMO properties determined
by the geometry of sensors and by W (DOD, DOA, complex amplitudes)
)()()()( Ttr aWaH
MIMO channel modelling
DOD
00000
00000
0000
0000
00000
X
X
XX
XX
X
DOA
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Example: channel correlation matrix(US approximation, spatial averaging)
ii
iiii
iinmmn WW,
2
,,
2
, /expR tnntitrmmrri rrkjrrkj
)()()()()()( ** HE trTtr aWaaWaR
n’
nm’
m
DOA DOD
MIMO channel modelling
Receiver sensors positions Transmitter radiators positions
Rx Tx
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ik
Steering matrix ar (nrXN)
)exp(,,ra rij rkjji
Uncoupled sensors
MIMO channel modelling : case of coupled sensors
)()()()( Ttr aWaH
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ik
ik
Steering matrix ar (nrXN) Complex gain matrix Gr (nrXN)
)exp(,,ra rij rkjji
)(,rG,,rG jkiji
Uncoupled sensors Coupled sensors
MIMO channel modelling : case of coupled sensors
)()()()( Ttr aWaH )()(W)()(H T
tr GG
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specification of Tx and Rx antennas, either through steering matrices (uncoupled sensors) or through complex gain matrices
Towards a stochastic MIMO channel model
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specification of Tx and Rx antennas, either through steering matrices (uncoupled sensors) or through complex gain matrices
double directional model of emitted and received waves, specifying the statistical laws of angular distributions on both sides of the radio link
Towards a stochastic MIMO channel model
![Page 27: KEYHOLES AND MIMO CHANNEL MODELLING Alain SIBILLEsibille@ensta.frsibille@ensta.fr ENSTA 32 Bd VICTOR, 75739 PARIS cedex 15, FRANCE COST 273, Bologna meeting](https://reader035.vdocument.in/reader035/viewer/2022062423/5697bf791a28abf838c82665/html5/thumbnails/27.jpg)
specification of Tx and Rx antennas, either through steering matrices (uncoupled sensors) or through complex gain matrices
double directional model of emitted and received waves, specifying the statistical laws of angular distributions on both sides of the radio link
statistical model for the wave connecting matrix , specifying the distribution of complex entries of the matrix, especially the number of non zero entries for the various columns or lines and their relative amplitudes.
Towards a stochastic MIMO channel model
![Page 28: KEYHOLES AND MIMO CHANNEL MODELLING Alain SIBILLEsibille@ensta.frsibille@ensta.fr ENSTA 32 Bd VICTOR, 75739 PARIS cedex 15, FRANCE COST 273, Bologna meeting](https://reader035.vdocument.in/reader035/viewer/2022062423/5697bf791a28abf838c82665/html5/thumbnails/28.jpg)
specification of Tx and Rx antennas, either through steering matrices (uncoupled sensors) or through complex gain matrices
double directional model of emitted and received waves, specifying the statistical laws of angular distributions on both sides of the radio link
statistical model for the wave connecting matrix , specifying the distribution of complex entries of the matrix, especially the number of non zero entries for the various columns or lines and their relative amplitudes.
statistical model for the distribution of delays involved in the non zero entries of )(W
Towards a stochastic MIMO channel model
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Introduction of artificial junctions to reduce DOA/DOD number: depend on maximum antenna size/angular resolution
Tx Rx
MIMO channel model simplification
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Introduction of artificial junctions to reduce DOA/DOD number: depend on maximum antenna size/angular resolution
Tx Rx
Tx Rx
MIMO channel model simplification
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Introduction of artificial junctions to reduce DOA/DOD number: depend on maximum antenna size/angular resolution
Limitation on the dynamic range of wave amplitudes: substitution of numerous small amplitude waves by one or a few Rayleigh distributed waves of random DOA/DOD.
MIMO channel model simplification
Tx Rx
Tx Rx
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X
Y
Y
X
.
.
.
• look for a differing number of DOAs and DODs
• look for several path delays for the same DOA (or DOD)
Double directional channel measurements and junctions ?
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Analysis of keyholes through Kirchhoff diffraction: continuous variation of channel matrix effective rank with slit width
Small antenna approximation yields a MIMO channel description entirely based on DOA, DOD and antennas geometry
Junctions in multipath structure is responsible for the rectangular or non diagonal character of the « wave connecting matrix »
Coupling between sensors readily incorporated
Stochastic channel model. Simplifications as a function of precision requirements
May feed simpler MIMO channel models with environment dependent channel correlation matrices
Conclusion