0. How many users are needed for non-trivial performance of random beamforming in highly-directional mm-wave MIMO downlink?
Gilwon Lee
School of Electrical Engineering
KAIST
Oct. 14, 2015
Joint work with Prof. Youngchul Sung and Junyeong Seo
Information Theory Workshop 2015, Jeju island, Korea

1. 5G Key Technologies Average rate (bits/s/active user) 10~100x
5G Requirements
Average rate (bits/s/active user) ~100x
Average area rate (bits/s/km2) x
Active devices (per km2) x
Energy efficiency (bits/Joule) x
Key Technologies
Focus of this talk
Cellular band
Mm-wave band
3 GHz
30 GHz
300 GHz
(Prof. Heath’s Fig)
(argos ant.)
Mm-wave MIMO
Massive MIMO

2. Mm-wave Channel Characteristics
Quasi-optical nature of propagation
Very few multi-path components
Channel sparsity
Channels are sparse
Many literatures use
geometric channel model
Ex)
cf.
S. Sun, T. S. Rappaport, “Wideband mmWave Channels: Implications for Design and Implementation of Adaptive Beam Antennas ,” IEEE 2014 Intl. Microwave Symp. (IMS), June 2014, Tampa, Fl

3. Mm-wave Channel Characteristics
Large path-loss
High noise power due to large-bandwidth
(Friis’ law)
Mm-wave noise BW
microwave noise BW
G. R. MacCartney, M. K. Samimi, and T. S. Rappaport, "Omnidirectional Path Loss Models in New York City at 28 GHz and 73 GHz,“ IEEE 2014 PIMRC.
Massive antennas
Exploiting large-array antenna gain

4. A Challenging Issue: Channel Estimation
Channel sounding also requires highly-directional training beams due to large path-loss and high noise power.
Furthermore, there are few multi-path components in channels
Full-training method
Long training period is required!

5. A Challenging Issue: Channel Estimation
An efficient method: Multi-resolution Hierarchical approach
Divide full-range of the BS into two regions and transmit training beams to each of them
After feedback, the BS chooses the better region
1st stage
feedback
A. Alkhateeb, O. E. Ayach, G. Leus, and R. W. Heath, “Channel estimation and hybrid precoding for millimeter wave cellular systems,” IEEE JSTSP, Oct

6. A Challenging Issue: Channel Estimation
An efficient method: Multi-resolution Hierarchical approach
Divide full-range of the BS into two regions and transmit training beams to each of them
After feedback, the BS chooses the better region
and further divides the chosen region into two sub-regions.
2nd stage
A. Alkhateeb, O. E. Ayach, G. Leus, and R. W. Heath, “Channel estimation and hybrid precoding for millimeter wave cellular systems,” IEEE JSTSP, Oct

7. A Challenging Issue: Channel Estimation
An efficient method: Multi-resolution Hierarchical approach
Divide full-range of the BS into two regions and transmit training beams to each of them
After feedback, the BS chooses the better region
and further divides the chosen region into two sub-regions.
3rd stage
Properties of the method
Training period can be significantly reduced.
However, power consumption for training is very large at early stages.
This approach is useful for single-user case.
A. Alkhateeb, O. E. Ayach, G. Leus, and R. W. Heath, “Channel estimation and hybrid precoding for millimeter wave cellular systems,” IEEE JSTSP, Oct

8. A Fundamental Question
We have seen some results of singe-user systems.
Then, what about multi-user systems?
Is long training period still needed to obtain reasonable performance for multi-user systems?

9. Some Insights
Let’s assume the BS transmits a highly directional training beam to a random angle direction.
Now we ask what happens if there are many users in the cell.
Intuitively, we can expect the single random beam performs well when the number of users is large.
Then how many users are needed? Specify the # of users!

10. Multi-User Diversity
In fact, many works on the multi-user diversity (opportunistic beamforming) have been conducted in the Rayleigh fading channel model which is usually suitable for the cellular band.
i.i.d.
Ex. Random beamforming (RBF), Semi-orthogonal user selection (SUS)
However, there are no any analysis results on multi-user diversity in the mm-wave band.
M. Sharif and B. Hassibi, “On the capacity of MIMO broadcast channels with partial side information,” IEEE Trans. Inf. Theory, vol. 51, pp. 506–522, Feb
T. Yoo and A. Goldsmith, “On the optimality of multiantenna broadcast scheduling using zero-forcing beamforming,” IEEE J. Sel. Areas
Commun., vol. 24, pp. 528–541, Mar. 2006

11. Exploring Multi-User Diversity in Mm-wave
System Model
MU-MISO downlink
BS with ULA of antennas
single-antenna users
Channel Model
Uniform-Random Single-Path (UR-SP)
Channel vector of user k
AoD
Path gain
Steering vector

12. UR-SP Channel Model UR-SP Channel Model When LoS exists
Considering not only LoS environment but also one dominant path in NLoS environment
When LoS exists
When LoS does not exist
One dominant NLOS path
The different path gain between LoS and NLoS components can be captured by the assumption

13. Singe Beam Case
The BS transmits a randomly directional training beam to receivers in the direction of a random angle

14. Singe Beam Case
The BS transmits a randomly directional training beam to receivers in the direction of a random angle
Then, each user feeds back the signal power to the BS.
After the feedback is over, the BS selects the user that has the maximum signal power, and transmits a data stream with the beamforming vector .
(single beam rate)

15. Fejer Kernel of Order M
Since mm-wave systems use many antennas, we adopt asymptote.
Beam pattern
The value of beam pattern w.r.t. the difference btw and .
Fejer kernel of order M

16. Fejer Kernel of Order M: Observation
Order of 1/M

17. Singe Beam Rate
Based on the above facts, we have the following observation.
Observation
When for all k, and for fixed , we have
(trivial performance)
If we can find a user k such that almost surely, we have
(non-trivial performance)
Q) How many users K as a function of M are needed to obtain non-trivial performance?

18. Lemma 1
To explicitly derive it, we assume for simple explanation and provide a lemma related to signal power as follows. (The effect of is fully considered in the paper, but is trivial so we omitted it in this presentation.)
Lemma 1
For any constant and sufficiently large M, we have
where
signal power
Proof of Lemma 1: omitted

19. Theorem 1 – Asymptotic Rate of
Based on Lemma 1, we can show the following theorem.
Theorem 1
Corollary 1
For and any given , we have
where
For
Rate when perfect CSI is available
Proof of Theorem 1: omitted
𝒒= 𝟏 𝟐 is the performance transition point !

20. Theorem 1 – Asymptotic Rate of
Corollary 1
For
Rate when perfect CSI is available
𝒒= 𝟏 𝟐 is the performance transition point !

21. Theorem 1 – Asymptotic Rate of
Need more training beams!
𝒒= 𝟏 𝟐 is the performance transition point !

22. Effect of Training Multiple Training Beams: Remark:
where is an offset angle.
Remark:

23. Theorem 2 – Asymptotic Rate of
Corollary 2
For , and any
such that , we have
where
For
Rate when perfect CSI is available
Proof of Theorem 2: omitted
ℓ+𝒒= 𝟏 𝟐 is the performance transition point !

24. Theorem 2 – Asymptotic Rate of
Corollary 2
For
Rate when perfect CSI is available
ℓ+𝒒= 𝟏 𝟐 is the performance transition point !

25. Theorem 2 – Asymptotic Rate of
ℓ+𝒒= 𝟏 𝟐 is the performance transition point !

26. Theorem 2 – Asymptotic Rate of
When the number of users is too small to achieve non-trivial performance, Theorem 2 specify how much training is required to achieve it!

27. Simulation Results
M=1000
Single beam case
Multi-beam case

28. Extension to the Multi-User Selection
Multi-user selection with multi-beam
Random beamforming (RBF)
Each user k feeds back the maximum SINR and the corresponding beam index.
For each beam, the BS chooses the user that has the maximum SINR
and transmits data streams to the chosen users through the corresponding beams at the same time.
Rate of a selected user
where
G. Lee, Y. Sung, and J. Seo, “Randomly directional beamforming in millimeter wave multi-user MISO downlink,” to appear in IEEE Tran. Wireless Commun.

29. Theorem 3 – Asymptotic Rate of
For , with and , we have
where for
Sum Rate
As ,
(the optimal rate)
G. Lee, Y. Sung, and J. Seo, “Randomly directional beamforming in millimeter wave multi-user MISO downlink,” to appear in IEEE Tran. Wireless Commun.

30. Asymptotic Results of RBF in UR-SP
Linear sum rate scaling w.r.t. M can be achieved, if K increases linearly w.r.t. M.
Furthermore, the optimal sum rate can be achieved if K increases linearly w.r.t. M.
This result is contrary to the existing result in the Rayleigh fading channel model where linear sum rate scaling w.r.t. M can be achieved, if K increases exponentially w.r.t. M.
As ,
G. Lee, Y. Sung, and J. Seo, “Randomly directional beamforming in millimeter wave multi-user MISO downlink,” to appear in IEEE Tran. Wireless Commun.

31. Conclusion
(Single beam case)
There exists a performance transition point in the number of users (relative to the number of antennas) for non-trivial performance.
(Single-user selection with multi-beam case)
We specify how much training is required for obtaining non-trivial performance.
(Multi-user selection with multi-beam case)
Furthermore, the performance of random beamforming can achieve the optimal sum rate if K increases linearly w.r.t. M.