Spatial Multiplexing and Channel Modeling docx

1 7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 7. MIMO I: Spatial Multiplexing and Channel Modeling 2 7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 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 7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath Capacity of AWGN Channel Capacity of AWGN channel If average transmit power constraint is watts and noise psd is watts/Hz, 4 7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 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 7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath Spatial Parallel Channel Capacity is achieved by waterfilling over the eigenmodes of H. (Analogy to frequency-selective channels.) 6 7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath Rank and Condition Number At high SNR, equal power allocation is optimal: where k is the number of nonzero λ i 2 's, i.e. the rank of H. The closer the condition number: to 1, the higher the capacity. 7 7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath Example 1: SIMO, Line-of-sight h is along the receive spatial signature in the direction Ω:= cos φ: n r –fold power gain. 8 7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath Example 2: MISO, Line-of-Sight h is along the transmit spatial signature in the direction Ω := cos φ: n t – fold power gain. 9 7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath Example 3: MIMO, Line-of-Sight Rank 1, only one degree of freedom. No spatial multiplexing gain. n r n t – fold power gain 10 7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath Beamforming Patterns The receive beamforming pattern associated with e r (Ω 0 ): Beamforming pattern gives the antenna gain in different directions [...]... I: Spatial Multiplexing and Channel Modeling Clustered Model For Lt,Lr large, number of d.o.f.: where Fundamentals of Wireless Communication, Tse&Viswanath 29 7: MIMO I: Spatial Multiplexing and Channel Modeling Dependency on Carrier Frequency Measurements by Poon and Ho 2003 Fundamentals of Wireless Communication, Tse&Viswanath 30 7: MIMO I: Spatial Multiplexing and Channel Modeling Diversity and. .. I: Spatial Multiplexing and Channel Modeling Angular Basis • The angular transformation decomposes the received (transmit) signals into components arriving (leaving) in different directions Fundamentals of Wireless Communication, Tse&Viswanath 24 7: MIMO I: Spatial Multiplexing and Channel Modeling Examples Fundamentals of Wireless Communication, Tse&Viswanath 25 7: MIMO I: Spatial Multiplexing and Channel. .. Communication, Tse&Viswanath 19 7: MIMO I: Spatial Multiplexing and Channel Modeling Back to Example 4 Channel H is well conditioned if i.e the signals from the two Tx antennas can be resolved Fundamentals of Wireless Communication, Tse&Viswanath 20 7: MIMO I: Spatial Multiplexing and Channel Modeling MIMO Channel Modeling • • • • Recall how we modeled multipath channels in Chapter 2 Start with a deterministic... Tse&Viswanath 25 7: MIMO I: Spatial Multiplexing and Channel Modeling More Examples Fundamentals of Wireless Communication, Tse&Viswanath 26 7: MIMO I: Spatial Multiplexing and Channel Modeling Clustered Model How many degrees of freedom are there in this channel? Fundamentals of Wireless Communication, Tse&Viswanath 27 7: MIMO I: Spatial Multiplexing and Channel Modeling Dependency on Antenna Size Fundamentals... Tse&Viswanath 31 7: MIMO I: Spatial Multiplexing and Channel Modeling I.I.D Rayleigh Model Scatterers at all angles from Tx and Rx Ha i.i.d Rayleigh $ H i.i.d Rayleigh Fundamentals of Wireless Communication, Tse&Viswanath 32 7: MIMO I: Spatial Multiplexing and Channel Modeling Correlated Fading • When scattering only comes from certain angles, Ha has zero entries • Corresponding spatial H has correlated... Fundamentals of Wireless Communication, Tse&Viswanath 15 7: MIMO I: Spatial Multiplexing and Channel Modeling Back to Example 4 hi is the receive spatial signature from Tx antenna i along direction Ωi = cos φri: Condition number depends on Fundamentals of Wireless Communication, Tse&Viswanath 16 7: MIMO I: Spatial Multiplexing and Channel Modeling Beamforming Patterns The receive beamforming pattern associated...7: MIMO I: Spatial Multiplexing and Channel Modeling Line-of-Sight: Power Gain Energy is focused along a narrow beam Power gain but no degree-of-freedom gain Fundamentals of Wireless Communication, Tse&Viswanath 11 7: MIMO I: Spatial Multiplexing and Channel Modeling Example 4: MIMO, Tx Antennas Apart hi is the receive spatial signature from Tx antenna i along direction... degrees of freedom if h1 and h2 are different Fundamentals of Wireless Communication, Tse&Viswanath 12 7: MIMO I: Spatial Multiplexing and Channel Modeling Example 5: Two-Path MIMO A scattering environment provides multiple degrees of freedom even when the antennas are close together Fundamentals of Wireless Communication, Tse&Viswanath 13 7: MIMO I: Spatial Multiplexing and Channel Modeling Example 5:... Fundamentals of Wireless Communication, Tse&Viswanath 17 7: MIMO I: Spatial Multiplexing and Channel Modeling Angular Resolution Antenna array of length Lr provides angular resolution of 1/Lr: paths that arrive at angles closer is not very distinguishable Fundamentals of Wireless Communication, Tse&Viswanath 18 7: MIMO I: Spatial Multiplexing and Channel Modeling Varying Antenna Separation Decreasing antenna separation... provides a physical explanation of correlation Fundamentals of Wireless Communication, Tse&Viswanath 33 7: MIMO I: Spatial Multiplexing and Channel Modeling Analogy with Time-Frequency Channel Modeling Time-Frequency Spatial- Angular Domains Time Frequency Angular Spatial Resources signal duration T bandwidth W angular spreads Ωt, Ωr antenna array lengths Lt,Lr Resolution of multipaths into delay bins of 1/W . Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath 7. MIMO I: Spatial Multiplexing and Channel Modeling 2 7: MIMO I: Spatial Multiplexing and. watts and noise psd is watts/Hz, 4 7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals of Wireless Communication, Tse&Viswanath MIMO Capacity via SVD Narrowband MIMO channel: . Tse&Viswanath Spatial Parallel Channel Capacity is achieved by waterfilling over the eigenmodes of H. (Analogy to frequency-selective channels.) 6 7: MIMO I: Spatial Multiplexing and Channel Modeling Fundamentals