Limitation of Maasive MIMO in Antenna Correlation

1. Limitation of Massive MIMO System under Antenna
Correlation
Varun Kumar
National Institute of Technology, Rourkela (India)
1

2. Outlines:
 Introduction
 Problem Definition
 System Model
 Simulation Results
 Conclusion
 References
2

3. Challenges in Massive MIMO:
3
 Architecture Aspect
 Aspect of CSI Acquisition
 Antenna Correlation 
 Detection Algorithm
 Downlink Precoding Algorithm
 Channel Modelling
 Resource Allocation

4. Diversity vs Antenna Correlation :
4
Spatial Correlation– Due to proximity of antenna as a signal source
Mutual Coupling– Due to the proximity of the antennas as electrical sources
Massive MIMO:
 Large number of antenna integration across BS in limited physical space constraints.
 Is
𝜆
2
physical spacing provides the perfectly uncorrelated channel for large antenna system?
 Slight frequency/phase offset produces correlation effect.
 Is very large number of antenna provide as much gain as we want?

5. System Model
5
Uplink Receive Signal Vector
𝑌 = 𝑝 𝑢 𝐺𝑥 + 𝑛
𝑝 𝑢 =Transmitting power by per user terminal
𝐺 = 𝑀 × 𝐾 uplink channel matrix in correlated scenario
𝑥 = 𝐾 × 1 transmit signal vector
𝑛 = 𝑀 × 1 CSCG random noise vector CSCG Circularly Symmetric Complex Gaussian
𝑈1
𝑈2
𝑈 𝐾
𝑈3
𝑈 𝐾−1
𝑀
M= Total number of antenna across BS (Very Large)
K=Total number of localized users

6. Continued--
6
G = 𝜁𝑟
1
2
𝐺𝜁𝑡
1
2
where
𝜁𝑟 = 𝑀 × 𝑀 Receive Correlation matrix
𝐺 = 𝑀 × 𝐾 IID channel matrix
𝜁𝑡 = 𝐾 × 𝐾 Receive Correlation matrix
Kronecker Model:[1]
For point to point MIMO under limited number of 𝑇𝑋/𝑅 𝑋 antenna.
Overall Correlation matrix of size 𝑀𝐾 × 𝑀𝐾
𝑅 = 𝜁𝑟 ⊗ 𝜁𝑡
Very large dimensionality (Massive MIMO)

7. Antenna Correlation :
7
If X and Y are two RV then the correlation 𝜌 can be expressed as
𝜌 =
𝔼 𝑋𝑌 − 𝜇 𝑋 𝜇 𝑌
𝜎 𝑋 𝜎 𝑌
If 𝑋 = 𝑣𝑖 and 𝑌 = 𝑣𝑗 keeping 𝜇 𝑣 𝑖
= 0 & 𝜇 𝑣 𝑗
= 0
𝜌𝑖𝑘,𝑗𝑘 =
𝔼 ℎ𝑖,𝑘ℎ𝑗,𝑘
∗
𝔼 ℎ𝑖,𝑘ℎ𝑖,𝑘
∗
𝔼(ℎ𝑗,𝑘ℎ𝑗,𝑘
∗
)
=
𝔼 𝑣𝑖 𝑣𝑗
∗
𝔼 𝑣𝑖 𝑣𝑖
∗
𝔼(𝑣𝑗 𝑣𝑗
∗
)
In general scattering environment [10]
𝑣𝑖, 𝑣𝑗 = 𝑓(𝜙, 𝜓)
where 𝜙 = Polar angle & 𝜓 = Azimuth angle
𝜌 =
𝜙 𝜓
𝑣𝑖 𝜙, 𝜓 𝑣𝑗 𝜙, 𝜓 ∗
𝑃 𝜙, 𝜓 sin 𝜙 𝑑𝜙𝑑𝜓
Here 𝑃(𝜙, 𝜓) be the joint PDF for two RV 𝜙 and 𝜓
M
Uplink Scenario
𝑣𝑗
𝑣𝑖
𝑈1
𝑈2
𝑈 𝑘
𝑈 𝐾
𝑈 𝐾−1BS
𝑑𝑖,𝑗
ℎ𝑗,𝑘
ℎ𝑖,𝑘

8. Correlation vs Relative Antenna Spacing:
8
Another simplified expression for the correlation between two
antenna (Across BS)
M
Uplink Scenario
𝑣𝑗
𝑣𝑖
𝑈1
𝑈2
𝑈 𝑘
𝑈 𝐾
𝑈 𝐾−1BS
𝑑𝑖,𝑗
𝜌𝑖,𝑗 = 𝐽0(𝑘𝑑𝑖,𝑗)
𝑑𝑖,𝑗 = 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐴𝑛𝑡𝑒𝑛𝑛𝑎 𝑆𝑝𝑎𝑐𝑖𝑛𝑔
𝑘 =
2𝜋
𝜆
& 𝜆 =
𝑐
𝑓𝑐
, 𝑓𝑐 = 𝐶𝑎𝑟𝑟𝑖𝑒𝑟 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
𝑅 𝑋
𝑇𝑋

9. Correlation vs Antenna Spacing
9
𝜌𝑖,𝑗 = 𝐽0(𝑘𝑑𝑖,𝑗)
𝑑𝑖,𝑗 = 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐴𝑛𝑡𝑒𝑛𝑛𝑎 𝑆𝑝𝑎𝑐𝑖𝑛𝑔
𝑘 =
2𝜋
𝜆
& 𝜆 =
𝑐
𝑓𝑐
, 𝑓𝑐 = 𝐶𝑎𝑟𝑟𝑖𝑒𝑟 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
𝑑0 ≠
𝜆
2

10. Correlation vs Antenna Spacing
10
We have M number of antenna across the BS
From above correlation expression
1. Let 2 𝑛𝑑
, 3 𝑟𝑑
, … … … 𝑀 𝑡ℎ
antennas are placed at a distance 𝑑0, 𝑑1, … … 𝑑 𝑀−2 then 1 𝑠𝑡
antenna will remain
uncorrelated to the all antennas terminal.
2. Here (𝑑2−𝑑1) = (𝑑3−𝑑2) = ⋯ = (𝑑 𝑀−2−𝑑 𝑀−3) =
𝜆
2
3. But 𝑑1 − 𝑑0 ≠
𝜆
2
4. Ex----𝑑0 = 0.3823 𝜆, 𝑑1 = 0.8845 𝜆, 𝑑2 = 1.3849 𝜆 and so on.
Another challenge
 In case of massive MIMO we can’t place the large number of antenna in linear fashion to maintain
maximum separation.
 Large number of antenna placement is done through compact placement geometry.

11. Receive Correlation Matrix:
11
Since 𝐺 = 𝜁𝑟
1
2
𝐺𝜁𝑡
1
2
(Channel matrix under correlation environment)
𝐴 =
𝐴11 𝐴12 𝐴13
𝐴21 𝐴22 𝐴23
⋮
𝐴 𝑀1
⋮
𝐴 𝑀2
⋮
𝐴 𝑀2
…
⋯
⋱
⋯
𝐴1𝑀
𝐴2𝑀
⋮
𝐴 𝑀𝑀 𝑀×𝑀
= 𝜁𝑟
Let 𝜆1, 𝜆2, … … , 𝜆 𝑀 be the eigen value of matrix A and let Q be the another 𝑀 × 𝑀 diagonal matrix
such that
𝑄 =
± 𝜆1 0 0
0 ± 𝜆2 0
⋮
0
⋮
0
⋮
0
⋯ 0
⋯ 0
⋮
⋯
⋮
± 𝜆 𝑀
𝑀×𝑀
Hence 𝜁𝑟
1
2
= 𝐴𝑄𝐴−1
Note : In case massive MIMO all user terminal (UT) may be considered to widely separated.
Hence 𝜁𝑡 = 𝐼 𝐾 (Identity matrix)
In such scenario
𝐺 = 𝜁𝑟
1
2
𝐺

12. 12
Case Study:

13. Channel Gain Scaling due to Antenna Correlation:
13
𝑔 𝑚,𝑘 = 𝑔 𝑚 𝑚,𝑘 + 𝑔 𝑚,𝑘 (Channel coefficient between the 𝑘 𝑡ℎ
user (transmitting the signal) and 𝑚 𝑡ℎ
antenna of the BS (receiving the signal))
Here 𝑔 𝑚 𝑚,𝑘 = 𝜌 𝑚 𝑚,𝑘 𝑔 𝑚,𝑘 ∀ (1 < 𝑚 ≤ 𝑀, 1 < 𝑘 ≤ 𝐾)
where 𝜌 𝑚 𝑚,𝑘 is the correlation coefficient due to 𝑚 𝑡ℎ
base station to itself for the 𝑘 𝑡ℎ
user.
The channel error coefficient for the 𝑘 𝑡ℎ
user observed across the 𝑚 𝑡ℎ
antenna terminal can be expressed as
𝑔 𝑚,𝑘 =
𝑗=1,𝑗≠𝑚
𝑀
𝜌 𝑚 𝑗,𝑘 𝑔 𝑚,𝑘
𝜌 𝑚 𝑗,𝑘 be the correlation experienced across 𝑚 𝑡ℎ
antenna due to 𝑗 𝑡ℎ
antenna element of the BS.

14. Continued--
14
In this practice the modified and scaled version of channel coefficient can be expressed as
𝑔,
𝑚,𝑘
= 𝔼
𝜌 𝑚 𝑚
𝑔 𝑚,𝑘
1 + | 𝑔 𝑚,𝑘|
o We can observe that the channel is Rayleigh but not identically distributed (IND).
o 𝑔,
𝑚,𝑘
be the IID channel to channel correlation error ratio between 𝑘 𝑡ℎ
user to 𝑚 𝑡ℎ
BS antenna.
New matrix formulation
Let 𝐺′
= [𝑔1, 𝑔2, … … … 𝑔 𝐾] be the modified 𝑀 × 𝐾 channel matrix.
where 𝑔 𝑘 = 𝑔1,𝑘, 𝑔2,𝑘, … … 𝑔 𝑀,𝑘
𝑇
∀ 𝑘 = 1,2, … 𝐾

15. Achievable Rate and Power Efficiency Formulation:
15
Achievable rate [9]
𝐶 𝑘 = log2 1 +
𝑝 𝑢
𝔼 𝐺′ 𝐻 𝐺′ −1
𝑘𝑘
Using ZF detection technique.
Power Efficiency
𝜂 𝐸𝐸 = 𝑘=1
𝐾
𝐶 𝑘 𝐵
𝑃
B=Occupied Bandwidth, P= Total transmit power by all user

16. Capacity vs SNR (Single user)
16

17. EE vs SNR (Single user)
17

18. Impact of User Enhancement over Capacity under Different Correlation Scenario
18

19. Impact of User Enhancement over EE under Different Correlation Scenario
19

20. Conclusion:
20
1. We have analysed the impact of antenna placement geometry on achievable rate and power efficiency. The
observed performance metrics under correlated environment highly deviates from the uncorrelated IID channel.
2. Large number of antenna integration across BS in fixed physical space is also the major problem in massive
MIMO system.
3. In spite of better diversity due to large number of antenna across BS but vicinity of antenna element causes
antenna correlation and mutual coupling.
4. Our result also justify that
𝜆
2
physical spacing is not the only solution for getting uncorrelated IID channel.
5. We can only achieve larger diversity or IID response when antennas are widely spread across BS but under
limited physical space constraint it is not feasible.

21. References
21
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