7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 1 7. MIMO: Spatial Multiplexing and Channel Modeling

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 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.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 3 Capacity of AWGN Channel Capacity of AWGN channel If average transmit power constraint is watts and noise psd is watts/Hz,

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 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).

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 5 Spatial Parallel Channel Capacity is achieved by waterfilling over the eigenmodes of H. (Analogy to frequency-selective channels.)

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 6 Rank and Condition Number At high SNR, equal power allocation is optimal: where k is the number of nonzero i 2 's, i.e. the rank of H. The closer the condition number: to 1, the higher the capacity.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 7 Example 1: SIMO, Line-of-sight h is along the receive spatial signature in the direction  := cos  : n r –fold power gain.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 8 Example 2: MISO, Line-of-Sight h is along the transmit spatial signature in the direction  := cos  : n t – fold power gain.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 9 Example 3: MIMO, Line-of-Sight Rank 1, only one degree of freedom. No spatial multiplexing gain. n r n t – fold power gain

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 10 Beamforming Patterns The receive beamforming pattern associated with e r (  0 ): Beamforming pattern gives the antenna gain in different directions

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 11 Line-of-Sight: Power Gain Energy is focused along a narrow beam. Power gain but no degree-of-freedom gain.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 12 Example 4: MIMO, Tx Antennas Apart h i is the receive spatial signature from Tx antenna i along direction  i = cos  ri : Two degrees of freedom if h 1 and h 2 are different.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 13 Example 5: Two-Path MIMO A scattering environment provides multiple degrees of freedom even when the antennas are close together.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 14 Example 5: Two-Path MIMO A scattering environment provides multiple degrees of freedom even when the antennas are close together.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 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.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 16 Back to Example 4 h i is the receive spatial signature from Tx antenna i along direction  i = cos  ri : Condition number depends on

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 17 Beamforming Patterns The receive beamforming pattern associated with e r (  0 ): L r 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.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 18 Angular Resolution Antenna array of length L r provides angular resolution of 1/ L r : paths that arrive at angles closer is not very distinguishable.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 19 Varying Antenna Separation Decreasing antenna separation beyond /2 has no impact on angular resolvability. Assume /2 separation from now on (so n=2L).

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 20 Channel H is well conditioned if i.e. the signals from the two Tx antennas can be resolved. Back to Example 4

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 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 h l is an aggregation of several physical paths and can be modeled as Gaussian. We can follow the same approach for MIMO channels.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 22 MIMO Modeling in Angular Domain The outgoing paths are grouped into resolvable bins of angular width 1/ L t The incoming paths are grouped into resolvable bins of angular width 1/ L r. The ( k,l ) th entry of H a 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 H a.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 23 Spatial-Angular Domain Transformation What is the relationship between angular H a and spatial H? 2L t £ 2L t transmit angular basis matrix (orthonormal): 2L r £ 2L r receive angular basis matrix (orthonormal): Input,output in angular domain: so

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 24 Angular Basis The angular transformation decomposes the received (transmit) signals into components arriving (leaving) in different directions.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 25 Examples

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 26 More Examples

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 27 I.I.D. Rayleigh Model Scatterers at all angles from Tx and Rx. H a i.i.d. Rayleigh $ H i.i.d. Rayleigh

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 28 Correlated Fading When scattering only comes from certain angles, H a 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.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 29 Clustered Model How many degrees of freedom are there in this channel?

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 30 Dependency on Antenna Size

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 31 Clustered Model For L t,L r 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) device environment

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 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.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 33 Dependency on Carrier Frequency Measurements by Poon and Ho 2003.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 34 Diversity and Dof

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 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.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 36 Diversity-Multiplexing Tradeoff Richly scattered environment: L t  t = n t, L r  r = n r

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 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 n 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?

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 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.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 39 Infrastructure Cooperation Base-stations cooperate to form a macro-array with large angular spread at each mobile. MIMO BS

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 40 User Cooperation Users relay information for each other and act as virtual scatterers to increase the effective angular spread.

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 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 n 2/3. (Aeron & Saligrama 06) n source nodes n destination nodes

7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 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.