How many users are needed for non-trivial
performance of random beamforming in highlydirectional 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
5G Key Technologies
5G Requirements
Average rate (bits/s/active user)
Average area rate (bits/s/km2)
Active devices (per km2)
Energy efficiency (bits/Joule)
Key Technologies
Cellular band
3 GHz
(argos ant.)
Massive MIMO
10~100x
1000x
10-100x
1000x
Focus of this talk
Mm-wave band
30 GHz
300 GHz
(Prof. Heath’s Fig)
Mm-wave MIMO
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)
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
cf.
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
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!
A Challenging Issue: Channel Estimation
An efficient method: Multi-resolution Hierarchical approach
1. Divide full-range of the BS into two regions
and transmit training beams to each of them
2. 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. 2014.
A Challenging Issue: Channel Estimation
An efficient method: Multi-resolution Hierarchical approach
2nd stage
1. Divide full-range of the BS into two regions
and transmit training beams to each of them
2. After feedback, the BS chooses the better
region
3. and further divides the chosen region
into two sub-regions.
A. Alkhateeb, O. E. Ayach, G. Leus, and R. W. Heath, “Channel estimation and hybrid precoding for millimeter wave cellular systems,” IEEE JSTSP, Oct. 2014.
A Challenging Issue: Channel Estimation
An efficient method: Multi-resolution Hierarchical approach
3rd stage
1. Divide full-range of the BS into two regions
and transmit training beams to each of them
2. After feedback, the BS chooses the better
region
3. and further divides the chosen region
into two sub-regions.
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. 2014.
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?
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!
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. 2005.
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
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
Path gain
AoD
Steering vector
UR-SP Channel Model
• UR-SP Channel Model
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
Singe Beam Case
• The BS transmits a randomly directional training beam to receivers
in the direction of a random angle
.
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)
.
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
Fejer kernel of order M
and .
Fejer Kernel of Order M: Observation
Order of 1/M
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?
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
signal power
where
Proof of Lemma 1: omitted
Theorem 1 – Asymptotic Rate of
• Based on Lemma 1, we can show the following theorem.
Theorem 1
For
Corollary 1
and any given
where
, we have
.
Proof of Theorem 1: omitted
𝟏
𝒒 = 𝟐 is the performance transition point !
For
Rate when perfect CSI
is available
Theorem 1 – Asymptotic Rate of
Corollary 1
For
Rate when perfect CSI
is available
𝟏
𝒒 = 𝟐 is the performance transition point !
Theorem 1 – Asymptotic Rate of
Need more training beams!
𝟏
𝒒 = 𝟐 is the performance transition point !
Effect of Training
Multiple Training Beams:
where
Remark:
is an offset angle.
Theorem 2 – Asymptotic Rate of
Corollary 2
Theorem 2
For
,
such that
and any
For
, we have
where
.
Proof of Theorem 2: omitted
𝟏
ℓ + 𝒒 = 𝟐 is the performance transition point !
Rate when perfect CSI
is available
Theorem 2 – Asymptotic Rate of
Corollary 2
For
Rate when perfect CSI
is available
𝟏
ℓ + 𝒒 = 𝟐 is the performance transition point !
Theorem 2 – Asymptotic Rate of
𝟏
ℓ + 𝒒 = 𝟐 is the performance transition point !
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!
Simulation Results
M=1000
Single beam case
Multi-beam case
Extension to the Multi-User Selection
• Multi-user selection with multi-beam
Random beamforming (RBF)
1. Each user k feeds back the maximum SINR
and the corresponding beam index.
2. For each beam, the BS chooses the user that
has the maximum SINR
3. 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.
Theorem 3 – Asymptotic Rate of
Theorem 3
For
,
with
where
and
for
, we have
.
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.
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.
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.