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KEYHOLES AND MIMO CHANNEL MODELLING

COST 273, Bologna meeting. Alain SIBILLE sibille@ensta.fr ENSTA 32 Bd VICTOR, 75739 PARIS cedex 15, FRANCE. KEYHOLES AND MIMO CHANNEL MODELLING. Outline. Keyholes in MIMO channels viewed as the result of diffraction. Outline. Keyholes in MIMO channels viewed as the result of diffraction

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KEYHOLES AND MIMO CHANNEL MODELLING

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  1. COST 273, Bologna meeting Alain SIBILLE sibille@ensta.fr ENSTA 32 Bd VICTOR, 75739 PARIS cedex 15, FRANCE KEYHOLES AND MIMO CHANNEL MODELLING

  2. Outline Keyholes in MIMO channels viewed as the result of diffraction

  3. Outline Keyholes in MIMO channels viewed as the result of diffraction Channel modelling with multipath junctions in a small Size, uncoupled sensors antenna approximation

  4. Outline Keyholes in MIMO channels viewed as the result of diffraction Channel modelling with multipath junctions in a small Size, uncoupled sensors antenna approximation How to include inter-sensors coupling

  5. Outline Keyholes in MIMO channels viewed as the result of diffraction Channel modelling with multipath junctions in a small Size, uncoupled sensors antenna approximation How to include inter-sensors coupling towards a stochastic MIMO channel model

  6. Outline Keyholes in MIMO channels viewed as the result of diffraction Channel modelling with multipath junctions in a small Size, uncoupled sensors antenna approximation How to include inter-sensors coupling towards a stochastic MIMO channel model Conclusion

  7. A1 B1 A2 B2 B3 A3 K 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. Slit transmittance 1D channel Rank(H)=1 (two null coefficients of characteristic polynomial) : uncorrelated (complex) entries

  8. Rx Tx Keyholesin MIMO channels A simple numerical example of keyhole using Kirchhoff diffraction: • Large slit: no diffraction

  9. Rx Tx Keyholesin MIMO channels junction A simple numerical example of keyhole using Kirchhoff diffraction: • Large slit: no diffraction • Narrow slit: diffraction and multipath junction 1  3 Kij computed by Kirchhoff diffraction

  10. Keyholesin MIMO channels H: (normalized) channel transmission matrix nt=3: number of Tx, Rx radiators SNR = 3 dB Space-variant stochastic ensemble 0.8 0.6 cumulated probability 0.4 case A wide slit little correlation: 3 DF SV: -7, +4.8, +7.6 dB 0.2 0 2.5 3 3.5 4 capacity (b/s/Hz)

  11. Keyholesin MIMO channels B: narrow slit, little correlation : 1 DF SV: -47, -28, +9.5 dB case B 0.8 0.6 cumulated probability 0.4 case A wide slit little correlation: 3 DF SV: -7, +4.8, +7.6 dB 0.2 0 2.5 3 3.5 4 capacity (b/s/Hz)

  12. Keyholesin MIMO channels C: narrow slit, strong correlation : 1 DF SV: -111, -41, +9.5 dB B: narrow slit, little correlation : 1 DF SV: -47, -28, +9.5 dB 1 case B case C 0.8 0.6 cumulated probability 0.4 case A wide slit little correlation: 3 DF SV: -7, +4.8, +7.6 dB 0.2 0 2.5 3 3.5 4 capacity (b/s/Hz)

  13. Keyholesin MIMO channels: capacity vs. Slit width 1 2 5 0.8 Slit width in units of l 0.6 cumulated probability 0.4 0.2 0 2 2.5 3 3.5 4 capacity (b/s/Hz)

  14. Keyholesin MIMO channels: capacity vs. Slit width 1 0.25 0.5 2 5 0.8 Slit width in units of l 0.6 cumulated probability 0.4 0.2 0 2 2.5 3 3.5 4 capacity (b/s/Hz)

  15. Keyholesin MIMO channels: capacity vs. Slit width Slit width in units of l

  16. Keyholesin MIMO channels: capacity vs. Slit width Slit width in units of l • When d< ~ l/2 all incoming waves are diffracted into all exiting waves through a 1-dimensional channel

  17. Keyholesin MIMO channels: capacity vs. Slit width Slit width in units of l • When d< ~ l/2 all incoming waves are diffracted into all exiting waves through a 1-dimensional channel • When d>~2l transmission through the slit occurs through multiple modes and evanescent states and results in greater 3 dimensional effective channel

  18. A1 B1 A2 B2 B3 A3 K Keyholes: correlations or no correlations ? Rank(H)=1 (two null coefficients of characteristic polynomial) junction : uncorrelated (complex) entries

  19. A1 B1 A2 B2 B3 A3 K Keyholes: correlations or no correlations ? Rank(H)=1 (two null coefficients of characteristic polynomial) junction : uncorrelated (complex) entries : correlated amplitudes

  20. Keyholesin MIMO channels B: narrow slit, little correlation : 1 DF SV: -47, -28, +9.5 dB C: narrow slit, strong correlation : 1 DF SV: -111, -41, +9.5 dB 1 case B case C 0.8 case D 0.6 narrow slit, strong correlation, one random phase: 2 DF SV: -40, -3.9, +9.4 dB cumulated probability 0.4 case A wide slit little correlation: 3 DF SV: -7, +4.8, +7.6 dB 0.2 fading 0 2.5 3 3.5 4 capacity (b/s/Hz)

  21. DOD DOA Small antenna, uncoupled sensors approximation: { MIMO channel modelling 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)

  22. m n’ m’ n Rx Tx MIMO channel modelling Example: channel correlation matrix (US approximation, spatial averaging) DOD DOA Receiver sensors positions Transmitter radiators positions

  23. MIMO channel modelling : case of coupled sensors Uncoupled sensors Steering matrix ar (nrXN)

  24. MIMO channel modelling : case of coupled sensors Uncoupled sensors Coupled sensors Steering matrix ar (nrXN) Complex gain matrix Gr (nrXN)

  25. Towards a stochastic MIMO channel model specification of Tx and Rx antennas, either through steering matrices (uncoupled sensors) or through complex gain matrices

  26. Towards a stochastic MIMO channel model 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

  27. Towards a stochastic MIMO channel model 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.

  28. Towards a stochastic MIMO channel model 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

  29. MIMO channel model simplification Introduction of artificial junctions to reduce DOA/DOD number: depend on maximum antenna size/angular resolution

  30. MIMO channel model simplification Introduction of artificial junctions to reduce DOA/DOD number: depend on maximum antenna size/angular resolution

  31. MIMO channel model simplification 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.

  32. Double directional channel measurements and junctions ? • look for a differing number of DOAs and DODs • look for several path delays for the same DOA (or DOD) Y X . Y X . .

  33. Conclusion • 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

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