1. 1 On the MIMO Channel Capacity Predicted by Kronecker and Müller Models Müge KARAMAN ÇOLAKOĞLU Prof. Dr. Mehmet ŞAFAK COST 289 4th Workshop, Gothenburg, Sweden April 11-12, 2007

2. 2 Outline MIMO Channel Models Kronecker model Müller model Results Conclusion

3. 3 Kronecker Model-1 Assumptions: Flat fading channel Only doubly-scattered rays are considered LOS multipath component ignored Single scattered signals ignored Source of fading  Local scatterers Number of scatterers  Typically >10 Fading correlations are separated Tx have no CSI, Rx have CSI

4. 4 Kronecker Model-2

5. 5 Kronecker Model-3 Sensitivity against the model parameters (wavelength=0.15 m)

6. 6 Müller Model-1 Assumptions: Frequency-selective fading channel Scatterers can be distinguished in time and space (Different locations and delays) Only singly-scattered rays are considered No LOS, no multiple scattering Tx and Rx at the foci of concentric (equi-delay) ellipses Asymptotic in the number of scatterers, and of the transmit- and receive antennas

7. 7 Müller Model-2 Delay and space coordinates for the Müller model Propagation coefficient between th Tx and th Rx antenna Received signal at time :

8. 8 Müller Model-3 Singular values of the random channel matrix H show fewer fluctuations, become deterministic as its size goes infinity Approximates finite size matrices Singular value distributions can be calculated analytically Only the surviving physical parameters show significant influence on the singular value distribution and characterise the MIMO channel In the asymptotic limit, singular values of H for flat fading and frequency-selective fading channels are the same

9. 9 Müller Model-4 Surviving physical parameters that dominate the value of the channel capacity: System load: Total richness: Attenuation distribution (assumed): (for all )

10. 10 Results-1 Parameters used for the Kronecker model (m) d t (m) d r (m) R t0 (m) R r0 (m) D t (m) D r (m) R (m) 0.15 50 50000

11. 11 Results-2 Effects of the number of Tx and Rx antennas for

12. 12 Results-3 Effect of the number of scatterers for

13. 13 Results-4 Effect of the number of Tx antennas

14. 14 Results-5 Effect of the number of Rx antennas

15. 15 Results-6 Effect of the number of scatterers

16. 16 Conclusion-1 Kronecker model: Valid for flat-fading channels May be more appropriate for urban channels May lead to pessimistic capacity predictions in suburban areas Some measurement results show that model fails under certain circumstances

17. 17 Conclusion-2 Müller model: Valid for frequency-selective fading channels May describe suburban channels more accurately Simple and characterizes the channel by the number of Tx antennas the number of Rx antennas the chanel richness (usually ignored in other models) Capacity predictions by the Müller model may be higher compared with the Kronecker model

18. 18 Thank You...

19. 19 Channel Capacity-1 Finding average capacity-Method 1: Replace mean value of by deterministic correlation matrix  Find eigenvalues and the capacity Issue: How to model the correlation matrix ? (correlation between antenna elements, angular spread of signals, scattering richness)

20. 20 Channel Capacity-2 Finding average capacity-Method 2: Elements of are zero mean Gaussian random variables  is central Wishart matrix. Joint pdf of the ordered eigenvalues of a complex Wishart matrix is known. Determine the capacity by using joint pdf of the ordered eigenvalues. Issue: Hard to determine the marginal pdf’s analytically.

21. 21 Kronecker Model-3 Isolates the fading correlations Simplify the simulation and the analysis  Underestimates the channel capacity (high corelation )  Should be used at low correlation chanels Assumes double scattering from local scatterers  More suited for urban channels

22. 22 Kronecker Model-4 Sensitivity against the model parameters (wavelength=0.15 m)

23. 23 Müller Model-4 It is assumed that Eigenvalue distribution of the space-time channel matrix does not changes if the delay times of particular paths vary. No need to distinguish between the distributions of path attenuations conditioned on different delays. A uniform power delay profile is assumed. The paths that have same delay are assumed attenuated at the same rate.