7. MIMO: Spatial Multiplexing and Channel Modeling

1. 7. MIMO: Spatial Multiplexing and Channel Modeling

2. Main Story • So far we have only considered single-input multi-output (SIMO) and multi-input single-output (MISO) channels. • They provide diversity and power gains but no degree-of-freedom (d.o.f.) gain. • D.o.f gain is most useful in the high SNR regime. • MIMO channels have a potential to provide d.o.f gain. • We would like to understand how the d.o.f gain depends on the physical environment and come up with statistical models that capture the properties succinctly. • We start with deterministic models and then progress to statistical ones.

3. Capacity of AWGN Channel Capacity of AWGN channel If average transmit power constraint is watts and noise psd is watts/Hz,

4. MIMO Capacity via SVD Narrowband MIMO channel: is by , fixed channel matrix. Singular value decomposition: are complex orthogonal matrices and real diagonal (singular values).

5. Spatial Parallel Channel Capacity is achieved by waterfilling over the eigenmodes of H. (Analogy to frequency-selective channels.)

6. Rank and Condition Number At high SNR, equal power allocation is optimal: where k is the number of nonzero i2 's, i.e. the rank of H. The closer the condition number: to 1, the higher the capacity.

7. Example 1: SIMO, Line-of-sight h is along the receive spatial signature in the direction := cos : nr –fold power gain.

8. Example 2: MISO, Line-of-Sight h is along the transmit spatial signature in the direction  := cos : nt – fold power gain.

9. Example 3: MIMO, Line-of-Sight nr nt – fold power gain Rank 1, only one degree of freedom. No spatial multiplexing gain.

10. Beamforming Patterns The receive beamforming pattern associated with er(0): Beamforming pattern gives the antenna gain in different directions

11. Line-of-Sight: Power Gain Energy is focused along a narrow beam. Power gain but no degree-of-freedom gain.

12. Example 4: MIMO, Tx Antennas Apart hi is the receive spatial signature from Tx antenna i along direction i = cos ri: Two degrees of freedom if h1 and h2 are different.

13. Example 5: Two-Path MIMO A scattering environment provides multiple degrees of freedom even when the antennas are close together.

14. Example 5: Two-Path MIMO A scattering environment provides multiple degrees of freedom even when the antennas are close together.

15. Rank and Conditioning • Question: Does spatial multiplexing gain increase without bound as the number of multipaths increase? • The rank of H increases but looking at the rank by itself is not enough. • The condition number matters. • As the angular separation of the paths decreases, the condition number gets worse.

16. Back to Example 4 hi is the receive spatial signature from Tx antenna i along direction i = cos ri: Condition number depends on

17. Beamforming Patterns The receive beamforming pattern associated with er(0): Lr is the length of the antenna array, normalized to the carrier wavelength. • Beamforming pattern gives the antenna gain in different directions. • But it also tells us about angular resolvability.

18. Angular Resolution Antenna array of length Lr provides angular resolution of 1/Lr: paths that arrive at angles closer is not very distinguishable.

19. Varying Antenna Separation Decreasing antenna separation beyond /2 has no impact on angular resolvability. Assume /2 separation from now on (so n=2L).

20. Back to Example 4 Channel His well conditioned if i.e. the signals from the two Tx antennas can be resolved.

21. MIMO Channel Modeling • Recall how we modeled multipath channels in Chapter 2. • Start with a deterministic continuous-time model. • Sample to get a discrete-time tap delay line model. • The physical paths are grouped into delay bins of width 1/W seconds, one for each tap. • Each tap gain hl is an aggregation of several physical paths and can be modeled as Gaussian. • We can follow the same approach for MIMO channels.

22. MIMO Modeling in Angular Domain The outgoing paths are grouped into resolvable bins of angular width 1/Lt The incoming paths are grouped into resolvable bins of angular width 1/Lr. The (k,l)th entry of Ha is (approximately) the aggregation of paths in Can statistically model each entry as independent and Gaussian. Bins that have no paths will have zero entries in Ha.

23. Spatial-Angular Domain Transformation What is the relationship between angular Ha and spatial H? 2Lt£ 2Lt transmit angular basis matrix (orthonormal): 2Lr£ 2Lr receive angular basis matrix (orthonormal): Input,output in angular domain: so

24. Angular Basis • The angular transformation decomposes the received (transmit) signals into components arriving (leaving) in different directions.

25. Examples

26. More Examples

27. I.I.D. Rayleigh Model Scatterers at all angles from Tx and Rx. Ha i.i.d. Rayleigh $H i.i.d. Rayleigh

28. Correlated Fading • When scattering only comes from certain angles, Ha has zero entries. • Corresponding spatial H has correlated entries. • Same happens when antenna separation is less than /2 (but can be reduced to a lower-dimensional i.i.d. matrix) • Angular domain model provides a physical explanation of correlation.

29. Clustered Model How many degrees of freedom are there in this channel?

30. Dependency on Antenna Size

31. Clustered Model device environment For Lt,Lr large, number of d.o.f.: where t, r are the total angular spreads of the scatterers at the transmitter and the receiver. (Poon,Brodersen,Tse 05)

32. Spatial Channel Resource • Single-antenna: T seconds of transmission over a channel of bandwidth W yields WT degrees of freedom (Nyquist). • MIMO: Antenna array of size L over a channel with angular spread  yields L spatial degrees of freedom per second per Hz.

33. Dependency on Carrier Frequency Measurements by Poon and Ho 2003.

34. Diversity and Dof

35. Diversity and Multiplexing:Old Meets New • MIMO allows spatial multiplexing • But MIMO provides diversity as well. • In a richly scattered environment, there are resolvable angular paths. • This is the maximum amount of diversity available. • Increasing the amount of spatial multiplexing reduces the amount of diversity.

36. Diversity-Multiplexing Tradeoff Richly scattered environment: Ltt = nt , Lrr = nr

37. System Considerations • MIMO makes sense in indoor environments with high SNR and rich scattering. • MIMO-based products have started to appear in the WiFi space. (emerging 802.11n standard) • In wide-area cellular networks, users have wide ranges of SNR’s and angular spreads, so system design becomes more challenging. • How to get spatial degrees of freedom gain even when there is limited angular spread?

38. Space-Division Multiple Access • SDMA exploits the geographical separation of users. • Increase system throughput. • But how to get high per-user peak rate when there is limited angular spread? • Idea: cooperation.

39. Infrastructure Cooperation BS Base-stations cooperate to form a macro-array with large angular spread at each mobile. BS MIMO MIMO BS

40. User Cooperation Users relay information for each other and act as virtual scatterers to increase the effective angular spread.

41. Distributed MIMO • Node cooperation can increase effective angular spread. • Can it also be used to overcome device limitation? • Each single-antenna source node wants to talk to a specific destination node. • Without cooperation, total capacity is bounded irrespective of n. (interference-limited) • With joint processing, capacity grows linearly with n. (MIMO gain) • Interestingly, cooperation can achieve a capacity scaling of at least n2/3. (Aeron & Saligrama 06) n destination nodes n source nodes

42. Conclusions • Modern wireless communication theory exploits fading to increase spectral efficiency. • Real advances require marriage of theory with understanding of system issues. • The new point of view even suggests that fading can be induced by appropriate system design.