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

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. : uncorrelated (complex) entries K A1A1 A2A2 A3A3 B1B1 B2B2 B3B3 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

8. A simple numerical example of keyhole using Kirchhoff diffraction:  Large slit: no diffraction Rx Tx Keyholes in MIMO channels

9. A simple numerical example of keyhole using Kirchhoff diffraction:  Large slit: no diffraction  Narrow slit: diffraction and multipath junction 1  3 Rx Tx junction Keyholes in MIMO channels K ij computed by Kirchhoff diffraction

10. 2.533.54 0 0.2 0.4 0.6 0.8 cumulated probability case A wide slit little correlation : 3 DF SV: -7, +4.8, +7.6 dB Keyholes in MIMO channels capacity (b/s/Hz) H: (normalized) channel transmission matrix n t =3: number of Tx, Rx radiators SNR = 3 dB Space-variant stochastic ensemble

11. 2.533.54 0 0.2 0.4 0.6 0.8 cumulated probability case A case B wide slit little correlation : 3 DF SV: -7, +4.8, +7.6 dB B: narrow slit, little correlation : 1 DF SV: -47, -28, +9.5 dB Keyholes in MIMO channels capacity (b/s/Hz)

12. 2.533.54 0 0.2 0.4 0.6 0.8 1 cumulated probability case A case C case B wide slit little correlation : 3 DF SV: -7, +4.8, +7.6 dB 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 Keyholes in MIMO channels capacity (b/s/Hz)

13. 22.533.54 0 0.2 0.4 0.6 0.8 1 capacity (b/s/Hz) cumulated probability 52 Slit width in units of Keyholes in MIMO channels: capacity vs. Slit width

14. 22.533.54 0 0.2 0.4 0.6 0.8 1 capacity (b/s/Hz) cumulated probability 52 0.25 0.5 Slit width in units of Keyholes in MIMO channels: capacity vs. Slit width

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

16. 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

17. 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

18. : uncorrelated (complex) entries K A1A1 A2A2 A3A3 B1B1 B2B2 B3B3 Rank(H)=1 (two null coefficients of characteristic polynomial) junction Keyholes : correlations or no correlations ?

19. : uncorrelated (complex) entries : correlated amplitudes K A1A1 A2A2 A3A3 B1B1 B2B2 B3B3 Rank(H)=1 (two null coefficients of characteristic polynomial) junction Keyholes : correlations or no correlations ?

20. 2.533.54 0 0.2 0.4 0.6 0.8 1 capacity (b/s/Hz) cumulated probability case A case D case C case B wide slit little correlation : 3 DF SV: -7, +4.8, +7.6 dB 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 narrow slit, strong correlation, one random phase: 2 DF SV: -40, -3.9, +9.4 dB fading Keyholes in MIMO channels

21. Small antenna, uncoupled sensors approximation: { a r, a t : steering matrices for N DOAs and M DODs (n r X N, M X n t ) W : wave connecting matrix (N X M) : complex attenuations from all DODs to all DOAs W is in general rectangular in the presence of path junctions (diffraction, refraction …) Rank(H)  Min(n r, n t, N, M)  All MIMO properties determined by the geometry of sensors and by W (DOD, DOA, complex amplitudes) MIMO channel modelling DOD DOA

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

23. Steering matrix a r (n r X N) Uncoupled sensors MIMO channel modelling : case of coupled sensors

24. Steering matrix a r (n r X N)Complex gain matrix G r (n r X N) Uncoupled sensorsCoupled sensors MIMO channel modelling : case of coupled sensors

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

26.  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

27.  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

28.  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 Towards a stochastic MIMO channel model

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

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

31.  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

32. 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 ?

33.  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