TSKS14 Multiple Antenna Communications Lecture 5, 2020 Emil Björnson Outline of this lecture • Practical issues with point to point MIMO • Introduction to multi user MIMO • Uplink and downlink • Ortho.
TSKS14 Multiple Antenna Communications Lecture 5, 2020 Emil Björnson TSKS14 Multiple Antenna Communications Outline of this lecture • Practical issues with point-to-point MIMO • Introduction to multi-user MIMO • Uplink and downlink • Orthogonal access versus spatial multiplexing • Capacity region • Operating points • Uplink capacity region 2020-04-17 TSKS14 Multiple Antenna Communications Recall: Point-to-Point MIMO Capacity • Compute SVD of channel matrix: + 𝑮 = 𝑼𝜮𝑽& = ' 𝑠( 𝒖( 𝒗& ( ()* • 𝜮 “diagonal” with 𝑠*, … 𝑠+ • 𝑼 = 𝒖* … 𝒖1 • 𝑽 = 𝒗* … 𝒗2 Decompose the channel into 𝑆 parallel channels 2020-04-17 TSKS14 Multiple Antenna Communications Problems with point-to-point MIMO • Multiplexing gain: 𝑆 = rank 𝑮 • Line-of-sight: 𝑆 ≈ • Non-line-of-sight: 𝑆 = 𝑀, 𝐾 Mainly beamforming gain: • High SNR: Likely to be in line-of-sight (LOS) • Low SNR: Likely to be non-line-of-sight (NLOS) Not scalable: User devices are small, cannot fit many antennas 2020-04-17 TSKS14 Multiple Antenna Communications 2020-04-17 Multiuser MIMO Communication • Uplink • From users to base station • Multipoint-to-point MIMO Multiantenna base station • D0wnlink • From base station to users • Point-to-multipoint MIMO TSKS14 Multiple Antenna Communications 2020-04-17 Orthogonal multiple access • Two users want to communicate with base station 𝛽 • Power per user: 𝑃 • Bandwidth: 𝐵 • Noise power spectral density: 𝑁A User • Divide bandwidth: 𝛼𝐵 to user 1, (1 − 𝛼)𝐵 to user 𝑃𝛽 𝑅* = 𝛼𝐵 log J + 𝛼𝐵𝑁A 𝑃𝛽 𝑅J = − 𝛼 𝐵 log J + − 𝛼 𝐵𝑁A 𝛽 User TSKS14 Multiple Antenna Communications 2020-04-17 Orthogonal multiple access: Rate region • Rates depend on 𝛼: 𝑃𝛽 𝛼𝐵𝑁A 𝑃𝛽 1+ − 𝛼 𝐵𝑁A 𝑅* = 𝛼𝐵 log J + 𝑅J = − 𝛼 𝐵 log J 𝛼=0 𝛼= What is the preferred operating point? 𝑃𝛽 = 10 dB 𝐵𝑁A 𝛼=1 TSKS14 Multiple Antenna Communications 2020-04-17 Non-orthogonal multiple access From user 𝑥* ~𝐶𝑁(0, 𝑃) Noise (power 𝐵𝑁A ) 𝑦 = 𝑥* + 𝑥J + 𝑛 • Let both users transmit simultaneously: Received signal From user 𝑥J ~𝐶𝑁(0, 𝑃) • Strategy: Decode signal from user 1, treat interference as noise 𝑃 𝑅* = log J + 𝑃 + 𝐵𝑁A Subtract 𝑥*: 𝑦 − 𝑥* = 𝑥J + 𝑛 Decode signal from user 2: 𝑃 𝑅J = log J + 𝐵𝑁A Called: Successive interference cancelation We can change the user order TSKS14 Multiple Antenna Communications 2020-04-17 Non-orthogonal multiple access: Rate region Four operating points 𝑅*, 𝑅J : W log J + XY Z 0, log J + XY ,0 W Z W W log J + W[XY , log J + XY Z Z W W log J + XY , log J + W[XY Z Z Time sharing: We can achieve all points in between TSKS14 Multiple Antenna Communications 2020-04-17 Uplink in Multiuser MIMO 𝑥* 𝑦* 𝑦J Transmitters 𝑥] 𝑦1 𝑥2 • Notation: • 𝐾 single-antenna users, 𝑀 base station antennas ^ • Channel response 𝑔] from user 𝑖 to antenna 𝑗 • Data signals 𝑥*, … , 𝑥2 , received signals 𝑦*, … , 𝑦1 Receiver 10 TSKS14 Multiple Antenna Communications 2020-04-17 Uplink Multiuser MIMO: System model • Received signal: where 𝑦* 𝒚= ⋮ 𝑦1 𝒚 = 𝜌cd 𝑮𝒙 + 𝒘 𝑔** 𝑮= ⋮ 𝑔*1 * ⋯ 𝑔2 ⋱ ⋮ ⋯ 𝑔21 𝑥* 𝒙= ⋮ 𝑥2 • Parameters are normalized: SNR is 𝜌cd • Each users signal is power-limited as 𝔼 𝑥( • Normalized noise: 𝒘 ∼ 𝐶𝑁(𝟎, 𝑰1 ) J ≤1 𝑤* 𝒘= ⋮ 𝑤1 11 TSKS14 Multiple Antenna Communications 2020-04-17 What is the difference from point-to-point MIMO? • Difference 1: Users not cooperate • 𝑥*, … , 𝑥2 are independent signals • Difference 2: Each user cares about its own performance • 𝐾 user capacities instead of one capacity • Difference 3: Each user has its own power budget • Power constraint: 𝐸 𝑥( J ≤ • Difference 4: The channel matrix 𝑮 is modeled differently • Each column can be modeled as a SIMO channel 12 TSKS14 Multiple Antenna Communications 2020-04-17 13 Motivating example Multi-user MIMO Orthogonal access Non-orthogonal access 𝑅J 𝑅J 𝑅J 𝑅* Capacity or rate region? Capacity region is the largest possible rate region 𝑅* 𝑅* Two benefits: Beamforming gain for every user Smaller interference loss TSKS14 Multiple Antenna Communications 2020-04-17 Shape of capacity region • One can pick two points and use them fractions of the time • Similar to time-sharing • Hence: Line between any two points are in the region • Conclusion: Region must be a convex set Possible shape Impossible shape 14 TSKS14 Multiple Antenna Communications 2020-04-17 15 Points in the capacity region • Combinations (𝑅*, 𝑅J) of rates that can be simultaneously achieved 𝑅J Max sum rate (Largest 𝑅* + 𝑅J ) Largest 𝑅J Capacity region Max-min rate (Largest 𝑅* = 𝑅J ) Many choices! Only rule: Always pick something on the outer boundary (Pareto boundary)! Max sum rate Largest 𝑅* 𝑅* Fairness scale: Max-min fairness TSKS14 Multiple Antenna Communications 2020-04-17 Sum Capacity of Uplink Multiuser MIMO • Recall: Received signal: 𝒚 = 𝜌cd 𝑮𝒙 + 𝒘 • Assume a deterministic 𝑮 • Let all users transmit with full power: 𝒙~𝐶𝑁(𝟎, 𝑰2 ) Like a point-to-point MIMO channel But with a “suboptimal” signal covariance matrix 𝑸 = 𝑰1 • Sum rate: 𝑅* + ⋯ + 𝑅2 = log J det 𝑰1 + 𝜌cd 𝑮𝑮& This is the sum capacity! Achieved by successive interference cancellation Decoding order determines who gets which share 16 TSKS14 Multiple Antenna Communications 2020-04-17 Uplink Capacity Region with 𝐾 = • Region contains all (𝑅*, 𝑅J) satisfying: 𝑅* ≤ log J + 𝜌cd 𝒈* J 𝑅J ≤ log J + 𝜌cd 𝒈J J 𝑅* + 𝑅J ≤ log J det 𝑰1 + 𝜌cd 𝑮𝑮& 𝑅J Limited by sum capacity 𝑮 = 𝒈* 𝒈J 𝑅* 17 TSKS14 Multiple Antenna Communications 2020-04-17 18 Summary • Point-to-point MIMO channels • Large multiplexing gains are hard to achieve in practice • Multi-user MIMO channels • Similar system model • Key differences: Independent users, different power, different performance • Capacity and rate regions • Orthogonal and non-orthogonal access End of Lecture TSKS14 Multiple Antenna Communications