Bài giảng về hệ thống Má

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

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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

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TSKS14 Multiple Antenna Communications

Problems with point-to-point MIMO
• Multiplexing gain: 𝑆 = rank 𝑮
• Line-of-sight: 𝑆 ≈ 1
• Non-line-of-sight: 𝑆 = min 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

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TSKS14 Multiple Antenna Communications

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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

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Orthogonal multiple access
• Two users want to communicate with base station
𝛽

• Power per user: 𝑃
• Bandwidth: 𝐵
• Noise power spectral density: 𝑁A
User 1
• Divide bandwidth: 𝛼𝐵 to user 1, (1 − 𝛼)𝐵 to user 2
𝑃𝛽
𝑅* = 𝛼𝐵 log J 1 +
𝛼𝐵𝑁A
𝑃𝛽
𝑅J = 1 − 𝛼 𝐵 log J 1 +
1 − 𝛼 𝐵𝑁A

𝛽

User 2


TSKS14 Multiple Antenna Communications

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Orthogonal multiple access: Rate region
• Rates depend on 𝛼:
𝑃𝛽
𝛼𝐵𝑁A
𝑃𝛽
1+
1 − 𝛼 𝐵𝑁A

𝑅* = 𝛼𝐵 log J 1 +

𝑅J = 1 − 𝛼 𝐵 log J

𝛼=0

1
𝛼=
2

What is the preferred
operating point?

𝑃𝛽
= 10 dB
𝐵𝑁A

𝛼=1

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Non-orthogonal multiple access

From user 1
𝑥* ~𝐶𝑁(0, 𝑃)

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Noise
(power 𝐵𝑁A )

𝑦 = 𝑥* + 𝑥J + 𝑛

• Let both users transmit simultaneously:
Received signal

From user 2
𝑥J ~𝐶𝑁(0, 𝑃)

• Strategy:
1. Decode signal from user 1, treat interference as noise
𝑃
𝑅* = log J 1 +
𝑃 + 𝐵𝑁A
2. Subtract 𝑥*: 𝑦 − 𝑥* = 𝑥J + 𝑛. Decode signal from user 2:
𝑃
𝑅J = log J 1 +
𝐵𝑁A
Called: Successive interference cancelation

We can change the user order


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Non-orthogonal multiple access: Rate region
Four operating points 𝑅*, 𝑅J :
1.

W
log J 1 + XY
Z

2.

0, log J 1 + XY

3.
4.

2

3

,0
W

Z

W
W
log J 1 + W[XY , log J 1 + XY
Z

Z
W
W
log J 1 + XY , log J 1 + W[XY
Z
Z

4

Time sharing:
We can achieve all points in between
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TSKS14 Multiple Antenna Communications

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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

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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

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What is the difference from point-to-point MIMO?
• Difference 1: Users do 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 ≤ 1
• Difference 4: The channel matrix 𝑮 is modeled differently
• Each column can be modeled as a SIMO channel

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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


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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

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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

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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

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Uplink Capacity Region with 𝐾 = 2
• Region contains all (𝑅*, 𝑅J) satisfying:
𝑅* ≤ log J 1 + 𝜌cd 𝒈* J
𝑅J ≤ log J 1 + 𝜌cd 𝒈J J
𝑅* + 𝑅J ≤ log J det 𝑰1 + 𝜌cd 𝑮𝑮&
𝑅J

2

3

Limited by sum capacity
4

𝑮 = 𝒈* 𝒈J

1

𝑅*

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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 5

TSKS14 Multiple Antenna
Communications