Antennas and Propagation. Chapter 6b: Path Models Rayleigh, Rician Fading, MIMO

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1 Antennas and Propagation b: Path Models Rayleigh, Rician Fading, MIMO

2 Introduction From last lecture How do we model H p? Discrete path model (physical, plane waves) Random matrix models (forget H p and model H nc directly) Antennas and Propagation Slide 2

3 Discrete Path Models Assume Finite number of paths from transmit to receive Single polarization (but easily extended) N T transmitters, N R receivers Double-directional response φ R Angle of arrival at receiver φ T Angle of departure at transmitter φ R,l Angle of arrival of lth path φ T, l Angle of departure of lth path β l Complex gain of lth path Antennas and Propagation Slide 3

4 Antennas Field Patterns / Steering vectors Coordinates of transmit/receive antennas To keep this discussion simple assume omnidirectional antennas Antennas and Propagation Slide 4

5 Channel Matrix Computation Channel Matrix (without coupling) using sifting property of Delta function Antennas and Propagation Slide 5

6 Deterministic Path Model Use ray-tracing or measurement to find departure, arrival angles φ R,l, φ T, l path gains β l Given known antenna positions Compute site-specific channel response Antennas and Propagation Slide 6

7 Unknown Paths Problem We do not know the exact paths for the channel Statistical Model Choose appropriate distributions for the unknown parameters Derive the distribution of the quantities of interest Channel Gain, SNR, Diversity Gain, Capacity Or, when not possible Monte Carlo Simulation Antennas and Propagation Slide 7

8 Fading Models Scenario Fixed transmitter: x T =0 y T =0 Receiver at pos: x R =xy R =y Fixed set of paths: β l Assume that we have a local receive origin (x 0, y 0 ) But, this receive origin location is random If origin is uniform over several wavelengths Phase θ l is approximately uniform on [0,2π] Antennas and Propagation Slide 8

9 Fading Models (2) Resulting Model Fixed transmitter: x T =0 y T =0 Relative receiver: x R =x y R =y Fixed set of paths: β l (real gains) Random phases: θ l uniform on [0,2π] (i.i.d) We will use x and y instead of x and y for convenience Antennas and Propagation Slide 9

10 Rayleigh Fading Scenario Large set of paths (L large) Approximately equal gain β l Goal Compute the statistics of H or p(h R,H I ) What part of H is most important for communications link? Concentrate on statistics of H Antennas and Propagation Slide 10

11 Rayleigh Fading (2) Consider Receive antenna fixed at x=0 y=0 (random origin) First Moment (mean) = 0 Antennas and Propagation Slide 11

12 Rayleigh Fading (3) Second moment (variance) Antennas and Propagation Slide 12

13 Rayleigh Fading (4) μ R =0 μ I =0 Consider Limit as L H R and H I are sum of many independent random variables What is distribution? Central limit theorem H R and H I tend to Gaussian Know marginals, but what about joint distribution? Antennas and Propagation Slide 13

14 Rayleigh Fading (5) Check covariance of H R and H I Gaussian + Uncorrelated Independent Known as the Complex Gaussian pdf Denoted Antennas and Propagation Slide 14

15 Rayleigh Fading (6) Envelope Distribution Link quality normally depends on H Next Goal: Find marginal distribution of H Need to do change of variables Antennas and Propagation Slide 15

16 Rayleigh Fading (6) Finally, integrate to find marginal distribution Integrate H (note pdf does not depend on it!) This is the familiar Rayleigh PDF Often written as We will always use total channel variance (the first form) But, it is equivalent (just need to be consistent) Antennas and Propagation Slide 16

17 Rayleigh Channel: Spatial Correlation So far... Channel has joint Gaussian statistics Rayeligh pdf Tells of amount/degree of fading to expect But, what about time aspect? How rapidly does fading occur in time? Or with distance? Gaussian Statistics Consider spatial correlation of channels separated in space Gives temporal correlation for constant velocity Antennas and Propagation Slide 17

18 Spatial Correlation (2) Consider two receive points separated by Δx in x and Δy in y Antennas and Propagation Slide 18

19 Spatial Correlation (3) In rich multipath placing antennas at 0.4λ results in uncorrelated channels (signals) Antennas and Propagation Slide 19

20 Rician Fading Scenario Single non-fading path: Gain β 0, phase θ 0 Large set of other paths (L large) Approximately equal gain on other paths β l Goal Compute the statistics of H or p(h R,H I ) And find pdf of H Antennas and Propagation Slide 20

21 Rician Fading (2) Next, find pdf in terms of H, H Antennas and Propagation Slide 21

22 Rician Fading (3) From Jacobian (change of variables) again Finally, integrate H to get p( H ) alone Antennas and Propagation Slide 22

23 Rician Fading (4) Known as Rician pdf Convention: Instead of expressing in terms of μ and σ 2 Use K-factor K=0 Rayleigh (rich multipath) K= Single path (e.g. pure LOS) Antennas and Propagation Slide 23

24 MIMO Channels Multiple-input Multiple-output (MIMO) Have arrays at both transmit and receive Before we considered equivalent circuit (ignored H nc ) Now we will concentrate on this (ignore coupling) Uses of Multiple Antennas Diversity: Send redundant information on multiple antennas Increase (short-time) robustness of link Spatial Multiplexing: Send different information on antennas Increase (longer term) capacity of link Antennas and Propagation Slide 24

25 MIMO Channel (2) Narrowband MIMO Single frequency Extend to finite bandwidth using Fourier techniques Receive Signal Superposition of signals from N T transmitters Additive noise on mth receiver Signal on mth receiver Channel transfer matrix Input signal on nth transmitter Same as model we saw before, or in matrix notation: Antennas and Propagation Slide 25

26 MIMO Channels (3) Noise Model Can also have colored noise, R η is a general Hermitian matrix Antennas and Propagation Slide 26

27 MIMO Capacity Measuring MIMO Channel Quality Difficult, since we have N R xn T matrix of random quantities Capacity: Most important single figure-of-merit Upper bound on achievable (error-free) transmission rate How do multiple antennas increase transmission rate? Assume: Transmitter and receiver know channel Can perform singular value decomposition (SVD) Matrix of left singular vectors Diagonal matrix of singular values Matrix of right singular vectors Antennas and Propagation Slide 27

28 MIMO Capacity (2) Consider what happens when TX sends Means TX sends N T orthogonal patterns Each pattern modulated by an The receiver then computes Resulting effective channel Antennas and Propagation Slide 28

29 MIMO Capacity (3) What does this represent? We have formed parallel channels Channels do not interfere Idea: Send different information on these parallel channels Yields N S times time capacity of a single channel (for high SNR) Antennas and Propagation Slide 29

30 MIMO Capacity (4) From Information Theory (single complex channel) For our multiple (parallel channels) mutual information is Typically transmit power is constrained. Simple constraint: Capacity is max MI over all possible power allocation schemes Antennas and Propagation Slide 30

31 MIMO Capacity (5) Problem How to solve? Directly with Langrange Multipliers Solution: Referred to as water-filling solution due to nice geometric interpretation Antennas and Propagation Slide 31

32 MIMO Capacity (6) Method Identify water level ν that satisfies equations Behavior ν For increasing transmit power P T Power first allocated to least noisy channel Later to noisier channels Eventually to all channels Assuming M least noisy channels used Algorithm: Assume all channels used If any p i < 0, remove noisiest channel (M M-1) Repeat until find admissible solution Antennas and Propagation Slide 32

33 Next Time Random Matrix Models Cluster Models Antennas and Propagation Slide 33

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