when base stations meet terminals, and some results beyond

PowerPoint-PräsentationVodafone Chair Mobile Communications Systems, Prof. Dr.-Ing. Dr. h.c. G. Fettweis
Gerhard P. Fettweis – Vodafone Chair Professor, TU Dresden with Vinay Suryaprakash, Ana Belen Martinez, Ines Riedel, Michael Grieger, and many more
When Base Stations Meet Terminals, And Some Results Beyond
Introduction
Numerous works have demonstrated the benefits of using spatial point processes to model
wireless networks and the use of a homogeneous Poisson process to model base stations is
validated in [Andrews et al. 2011]. 22
−20 −15 −10 −5 0 5 10 15 20 −20
−15
−10
−5
0
5
10
15
Fig. 1. Poisson distributed base stations and mobiles, witheach mobile associated with the nearest BS. The cell boundaries are shown
and form a Voronoi tessellation.
1 2 3 4 5
1
2
3
4
5
Base stations: big dots. Mobiles: little dots.
Fig. 2. A regular square lattice model for cellular base stations with one tier of eight interfering base stations. The base stations are
marked by circles and the active mobile user in the tagged cell by a cross.
(a) Poisson distributed base stations and mobiles.
23
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−15
−10
−5
0
5
10
15
X coordinate (km)
m )
Fig. 3. A 40× 40 km view of a current base station deployment by a major service provider in a relatively flat urban area, with cell
boundaries corresponding to a Voronoi tessellation.
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Coverage probability for α = 4
Grid N=8, SNR=10 Grid N=24, SNR=10 Grid N=24, No Noise PPP BSs, SNR=10 PPP BSs, No Noise
PPP
Square Grid
Fig. 4. Probability of coverage comparison between proposed PPP base station model and square grid model withN = 8, 24 and
α = 4. The no noise approximation is quite accurate, and it can be seen there is only a slightly lower coverage area with 24 interfering
base stations versus 8.
(b) A 40×40 km view of a current base station de-
ployment by a major service provider in a relatively
flat urban area.
Some Limits of Stochastic Geometry
Downtown of a major EU city center sectorization!
5/21/2015 Gerhard Fettweis Slide 2
Dresden Field Trial: (Uplink) Coordinated Multipoint Works!
Sectorization & CoMP
CoMP & Sectorization
N: #sectors DPC: CoMP with dirty paper coding WF: CoMP with Wiener filtering NC: no cooperation HPBW: half power beam width 5/21/2015 Gerhard Fettweis Slide 5
6-Fold Sectors: Dresden Field Trial Setup
6-fold sectorization with overlapping antennas for 120° sectors!
6-Fold Versus 3-Fold Sectorization
C: CoMP cluster size
CoMP Outcome
Good News: Overlapping 6-fold sectors! Sectors do not play a role Each site can be treated as omni
Bad News: Now intra-cluster interference occurs, even for sector-wise orthogonal signaling, as e.g. OFDMA
Vodafone Chair Mobile Communications Systems, Prof. Dr.-Ing. Dr. h.c. G. Fettweis
When Base Stations Meet Mobile Terminals, and Some Results Beyond
Gerhard Fettweis Vinay Suryaprakash
Objectives
Develop models to understand the behavior of interference in homogeneous and
heterogeneous networks while taking load or network traffic into account.
Compute Key Performance Indicators (KPIs), i.e. probability of coverage and spectral
efficiency (spatially averaged rate), for these networks.
Ensure that the expressions obtained for the KPIs are easy to use in other optimization
problems such as those dealing with energy efficiency or deployment cost.
If not, find suitable approximations.
May 20, 2015 Gerhard Fettweis , Vinay Suryaprakash Slide 2
Model 1: A simple extension of [Andrews et al. 2011].
− Incorporates the average number of users connected to a base station while deriving
expressions for probability of coverage and spatially averaged rate.
− Relevant publications: [Suryaprakash et al. 2012a], [Suryaprakash et al. 2012b].
Framework for Model 1
Macro base stations are modeled by a homogeneous Poisson process Φb ⊂ R2, intensity
λb > 0.
Users are modeled by a homogeneous Poisson process Φu ⊂ R2, intensity λu > λb.
Cell definition:
}
(z) + W ) }
; h – fading, l (|z − xi |) – pathloss.
Macro base stations are modeled by a homogeneous Poisson process Φb ⊂ R2, intensity
λb > 0.
Users are modeled by a homogeneous Poisson process Φu ⊂ R2, intensity λu > λb.
Cell definition:
}
(z) + W ) }
; h – fading, l (|z − xi |) – pathloss.
Model 1: Interference Limited Scenario
Theorem
For a path loss exponent β > 2 and a threshold T , the probability of coverage is
pc (λb, λu,T , β) = β − 2
(β − 2) + 2λuT 2F1
λb
) λb
,
where 2F1 (·) is the Gaussian hypergeometric function. If β = 4, the spatially averaged rate is given by
RΦb (λb, λu) =
λb
] dy .
Model 1: Scenario with Interference and Noise
Theorem
For β = 4 and a threshold T , the probability of coverage is given by
pc ( λb, λu,T ,PTx, σ
2 W
where K = π 2
} . From which, the
approximated closed form expression for the spatially averaged rate is obtained as
RΦb (λb, λu,PTx, σ
Applications of results obtained using Model 1
Additional power density required versus a unit increase in user demands.
3−4 Mbps 4−5 Mbps 5−6 Mbps 6−7 Mbps 7−8 Mbps 0
1
2
3
4
5
6
7
8
A dd
iti on
al p
ow er
d en
si ty
r eq
ui re
Comparison of energy management strategies.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0
20
40
60
80
100
120
140
160
P ow
er s
av ed
Using bandwidth variation
Motivation for further exploration
[Andrews et al. 2011] derives tractable expressions for the probability of coverage and
spatially averaged rate in homogeneous networks.
⇒ Model 1 only extends this by introducing a dependence between the interference
and user as well as base station intensities.
[Dhillon et al. 2012] derives similar expressions for heterogeneous networks with many
different types of base stations.
However, these works assume that the point processes used to model each network
component are independent of one another.
In reality, base stations are deployed where ever a large number of users tend to be present
and smaller base stations are deployed when the macro (main) base station is unable to
satisfy user demands.
Motivation for further exploration
[Andrews et al. 2011] derives tractable expressions for the probability of coverage and
spatially averaged rate in homogeneous networks.
⇒ Model 1 only extends this by introducing a dependence between the interference
and user as well as base station intensities.
[Dhillon et al. 2012] derives similar expressions for heterogeneous networks with many
different types of base stations.
However, these works assume that the point processes used to model each network
component are independent of one another.
In reality, base stations are deployed where ever a large number of users tend to be present
and smaller base stations are deployed when the macro (main) base station is unable to
satisfy user demands.
Model 2: Homogeneous networks using a Neyman-Scott Process
− In this model, users are clustered around base stations based on a particular distribution.
− Relevant publication: [Suryaprakash et al. 2013].
Macro base stations (or cluster centers) are modeled by a stationary Poisson process
Φc ⊂ R2 with intensity λc > 0.
Conditioned on Φc , the users (or cluster members) are modeled by an inhomogeneous
Poisson process Φu ⊂ R2 with intensity function
ρ(y) = λu ∑ x∈Φc
f (y − x), y ∈ R2,
where λu > 0 is a parameter and f is a continuous density function.
Note that Φu (not conditioned on Φc ) is stationary with intensity λcλu .
The interference at a given location z ∈ R2 is given by
IΦ (z) = ∑ xj∈Φc
f (y − x), y ∈ R2,
f (y − x), y ∈ R2,
Illustration of Model 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
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1
Figure 2: Cluster members generated using a zero-mean radially symmetric Gaussian density f with variance 0.05.
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1
Model 2: Coverage in Homogeneous Networks
Theorem
For a given distance ‘r ’ between the user and base station, pathloss exponent ‘β’, and threshold
‘T ’, the conditional probability of coverage in a homogeneous network with users clustered
around base stations is
−λc ∫ R2
1− exp
−λu 1−
Conditional probability of coverage vs. distance
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Distance between user and base station (r km)
P ro
ba bi
lit y
of c
ov er
ag e
Conditional probability of coverage vs. threshold
−15 −12 −9 −6 −3 0 3 6 9 12 15 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
λ c = 1/km2, σ2 = 0.5, β = 4, r = 0.3 km
Threshold (T dB)
Conditional probability of coverage vs. cluster variance
0 0.2 0.4 0.6 0.8 1.0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
λ c = 1/km2, T = −9dB, β = 4, r = 0.03 km
Variance in cluster distribution (σ2)
P ro
ba bi
lit y
of c
ov er
ag e
Figure 4: User to base station distance, r = 0.03 km.
Conditional probability of coverage vs. cluster variance
0 0.2 0.4 0.6 0.8 1.0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
λ c = 1/km2, T = −9dB, β = 4, r = 0.3 km
Variance of the cluster distribution (σ2)
P ro
ba bi
lit y
of c
ov er
ag e
Figure 5: User to base station distance, r = 0.3 km.
Theorem
The probability of coverage in a homogeneous network with users clustered around base
stations, pathloss exponent ‘β’, and threshold ‘T ’ is
p (λc , λu , f (·),T , β) u
∫ R+
exp
g(r)dr .
where the continuous density function of R (the distance between a user and a base station)
g(r) = − d dr
v(r) = exp
( −λc ∫ R2
[ 1− exp
Probability of coverage
Base station intensity ( λ c )
P ro
ba bi
lit y
of c
ov er
ag e
0 25
T = −9dB, σ2 = 0.5, β = 4
User intensity ( λ u )
Model 3: Heterogeneous networks using a stationary Poisson Cluster Process
− An alternative to [Dhillon et al. 2012] in which there are only two types of base stations
and micro base stations are clustered around macro base stations.
− Relevant publication: [Suryaprakash et al. 2014].
Base stations are modeled using a stationary Poisson cluster process Φ ⊂ R2.
Conditioned on Φc , the micro base stations (or cluster members) are modeled by an
inhomogeneous Poisson process Φm ⊂ R2 with intensity function
ρ(y) = λm ∑ x∈Φc
f (y − x), y ∈ R2,
where λm > 0 is a parameter and f is a continuous density function.
Note that Φm (not conditioned on Φc ) is stationary with intensity λcλm.
Hence, the base stations, i.e., the superposition Φ = Φc ∪ Φm, form a stationary Poisson
cluster process with intensity λ = λc (1 + λm).
IΦ (z) = ∑ xj∈Φ
f (y − x), y ∈ R2,
f (y − x), y ∈ R2,
f (y − x), y ∈ R2,
f (y − x), y ∈ R2,
f (y − x), y ∈ R2,
Illustration of Model 3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
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0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.1
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1
Model 3: Coverage in Heterogeneous Networks
Theorem
For a given distance ‘r ’ between the user and base station, pathloss exponent ‘β’, and threshold
‘T ’, the conditional probability of coverage in a heterogeneous network with micro base stations
clustered around macro base stations is derived as
p (λc , λm, f (·),T , β | r) =
exp
Theorem
The probability of coverage in a heterogeneous network with micro base stations clustered
around macro base stations, pathloss exponent ‘β’, and threshold ‘T ’ is
p (λc , λm, f (·),T , β) =∫ R+
exp
g(r) dr ,
where the continuous density function of the distance between the user and the base station
g(r) = − d dr
v(r) = exp
−λc ∫ R2
−λm ∫ b(o,r)
Conditional probability of coverage
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P ro
ba bi
lit y
of c
ov er
ag e
1 2 3 4 5 6 7 8 9 10 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Micro base station intensity (λ m
)
Conditional probability of coverage
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P ro
ba bi
lit y
of c
ov er
ag e
1 2 3 4 5 6 7 8 9 10 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Micro base station intensity (λ m
) P
Comments about expressions obtained using Models 2 and 3
The expressions, shown in the previous slides, are easily evaluated using commercially
available computational software.
However, they are rather large and cumbersome which prevents easy re-use in other
optimization problems (which need more than the final value obtained by evaluating the
expressions numerically by fixing certain parameters).
Hence, we investigate other suitable approximations for describing the interference which
could allow easy re-use.
Comments about expressions obtained using Models 2 and 3
The expressions, shown in the previous slides, are easily evaluated using commercially
available computational software.
However, they are rather large and cumbersome which prevents easy re-use in other
optimization problems (which need more than the final value obtained by evaluating the
expressions numerically by fixing certain parameters).
Hence, we investigate other suitable approximations for describing the interference which
could allow easy re-use.
Asymptotic Behavior of the Interference
Define an estimator S(r) of the interference, where the distance between the user and base
station pair is r .
Wn,r ≡Wn b(o, r) =
Sn(r) = 1
|Wn,r | ∑ x∈Φ
1Wn,r (x)1H(x,r)(Φ− δx ),
where, for a set A, 1A(·) is its indicator function and H(x , r) are the sets of point
configurations which are not r -close to x .
Create a centered random variable using the estimator over the windows, which is given by
Zn(r) = |Wn,r |1/2 (Sn(r)− E [Sn(r)]) .
More Details
station pair is r .
Sn(r) = 1
|Wn,r | ∑ x∈Φ
Zn(r) = |Wn,r |1/2 (Sn(r)− E [Sn(r)]) .
More Details
station pair is r .
Sn(r) = 1
|Wn,r | ∑ x∈Φ
Zn(r) = |Wn,r |1/2 (Sn(r)− E [Sn(r)]) .
More Details
Theorem
Var [Zn(r)] = σ2 λ(r) > 0, we have
Zn(r) D−−−→
λ(r))
where N (0, σ2 λ(r)) is a zero-mean Gaussian distribution with variance σ2
λ(r) and D denotes convergence in distribution.
Proof The proof is derived along lines similar to those used in [Heinrich 1988].
Therefore, interference in clustered (and highly correlated) networks can be
approximated by a Gaussian random variable with mean E [Sn(r)] and variance
|Wn,r |σ2 λ(r). Mean Variance
It also implies that the influence of the transmit power, fading, and pathloss can be
observed solely in the mean and variance of the interference.
Theorem
Var [Zn(r)] = σ2 λ(r) > 0, we have
Zn(r) D−−−→
λ(r))
where N (0, σ2 λ(r)) is a zero-mean Gaussian distribution with variance σ2
λ(r) and D denotes convergence in distribution.
Proof The proof is derived along lines similar to those used in [Heinrich 1988].
Therefore, interference in clustered (and highly correlated) networks can be
approximated by a Gaussian random variable with mean E [Sn(r)] and variance
|Wn,r |σ2 λ(r). Mean Variance
It also implies that the influence of the transmit power, fading, and pathloss can be
observed solely in the mean and variance of the interference.
Verification of the results for Model 3
−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0
20
40
60
80
100
120
Interference values Gaussian fit from theory
Figure 8: Histogram for simulated interference values and the Gaussian density (from theory) with the mean and variance using the equations derived. Note that the mean has been subtracted to center both the histogram and the theoretic curve.
Plots of mean and variance
Verification of Expressions
Mean
5 6 7 8 9 10 11 12 13 14 15 10
−3
)
Variance
5 6 7 8 9 10 11 12 13 14 15 10
−7
) V
Conclusions
Reality: Base station and terminal distributions are correlated processes Generic Results: intra-cell interference generic expressions for HetNets derived Open HetNet Challenge: correlated distribution of mobiles to base stations, and correlated distribution of mobiles to micro/small cells
Open Stochastic Geometry Challenge: finding simpler approximations
References
Jeffrey G. Andrews, Francois Baccelli, Radha Krishna Ganti, A Tractable Approach to Coverage and Rate in Cellular Networks, in IEEE Transactions on Communications, vol. 59, pp. 3122 - 3134, 2011.
Dhillon, H.S. and Ganti, R.K. and Baccelli, F. and Andrews, J.G. Modeling and analysis of K-tier downlink heterogeneous cellular networks in IEEE Journal on Selected Areas in Communications, vol. 30 pp. 550 - 560, 2012.
Heinrich, L. Asymptotic behaviour of an empirical nearest-neighbour distance function for stationary poisson cluster processes in Mathematische Nachrichten, vol. 136, no. 1, pp. 131 - 148, 1988.
Heinrich, L. Stable limit theorems for sums of multiply indexed m-dependent random variables in Mathematische Nachrichten, vol. 127, no. 1, pp. 193 - 210, 1986.
References
Vinay Suryaprakash, Albrecht Fehske, Andre Fonseca dos Santos, Gerhard. P. Fettweis, On the Impact of Sleep Modes and BW Variation on the Energy Consumption of Radio Access Networks, in the proceedings of the 75th IEEE Vehicular Technology Conference (VTC Spring), 2012, 2012.
Vinay Suryaprakash, Andre Fonseca dos Santos, Albrecht Fehske, Gerhard. P. Fettweis, Energy Consumption Analysis of Wireless Networks using Stochastic Deployment Models, in the proceedings of the IEEE Global Communications Conference (GLOBECOM), 2012, 2012.
Vinay Suryaprakash, Gerhard. P. Fettweis, A stochastic examination of the interference in heterogeneous radio access networks, in the proceedings of the 11th International Symposium on Modeling Optimization in Mobile, Ad Hoc Wireless Networks (WiOpt), 2013, pp. 68 - 74, 2013.
Vinay Suryaprakash, Jesper Møller, Gerhard. P. Fettweis, On the Modeling and Analysis of Heterogeneous Radio Access Networks using a Poisson Cluster Process, in the IEEE Transactions on Wireless Communications, 2014.
Thank You!
Back up
The asymptotic behavior of the interference is studied using an increasing sequence of
compact sampling windows (Wn)n≥1 in Rd and eroded sets
Wn,r ≡Wn b(o, r) = {x ∈Wn : b(x , r) ⊆Wn}, which satisfy the Regularity Condition.
Regularity Condition : There exist a sequence of subsets of Rd satisfying
(a) each Wn is convex and compact;
(b) Wn ⊂Wn+1;
(c) sup{r ≥ 0 : B(x , r) ⊂Wn for some x} → ∞ as n→∞.
Define an estimator S(r) of the interference where the user and base station pair is r .
Then, the estimator Sn(r) over eroded sampling windows can be defined as
Sn(r) = 1
|Wn,r | ∑ x∈Φ
Zn(r) = |Wn,r |1/2 (Sn(r)− E [Sn(r)]) .
Back
Proof of Asymptotic Behavior of the Interference
Introduce a truncated Poisson cluster process, Φρ whose cluster center process is still Φc
but the process of cluster members Φmρ consists of atoms of Φm which are located in the
sphere b(0, ρ), where ρ > r , i.e., Φmρ({x}) > 0 if Φm({x}) > 0 and ||x || ≤ ρ. For A ∈ Bd0 ,
Snρ(r ,A) = 1
|Wn,r | ∑ x∈Φρ
1A∩Wn,r (x)1H(x,r)(Φρ − δx )
which implies that the centered random variable can be written as
Znρ(r) = (|Wn,r |)1/2 ( Snρ(r ,Wn,r )− E [Snρ(r ,Wn,r )]
) .
Define a set Ez = [z1 − 1, z1)× · · · × [zd − 1, zd ) for any { z ∈ Un ⊂ Zd
} where
Zd = {z = (z1, · · · , zd ) : zi = 0,±1,±2, · · · ; i = 1, · · · , d} and
Un = d × i=1
Xnz (r) = (|Wn,r |)1/2 (Snρ(r ,Ez )− E [Snρ(r ,Ez )])
Var [Znρ(r)] =
This implies Xnz (r) forms an m-dependent random field.
From [Heinrich 1986], we know that
Znρ(r)
since the following conditions are satisfied for every ε > 0.
(i) ∑ z∈Un
] ≤ C (ε) <∞,
σ2,
where C(ε) is a positive constant that changes with ε and σ > 0.
Back
Then, we show that
sup n≥1
Var ( Zn(r)− Znρ(r)
) = 0, ∀r ≥ 0.
For the functional limit theorem to hold, the tightness of Zn must be proven. This is done
by determining bounds on E [ Zn(t)− Zn(s)
]4 , ∀ 0 ≤ s ≤ t ≤ R by means of the fourth- and
second-order cumulants.
From Lemma 2 of [Heinrich 1988], the bounds are given by
E [ Zn(t)− Zn(s)
] ,
Back
Mean of the Estimator of the Interference
The mean of the estimator is
E [Sn(r)] = 1
ξM(r) ≡ |Wn,r |E [Sn(r)] = E ∑ x∈Φc
1Wn,r (x)1H(x,r)(Φ− δx ) + ∑
1Wn,r (y)1H(y,r)(Φ− δy )
. Using the Slivnyak-Mecke Theorem first for Φ and then for Φ
(x) m (after conditioning on both
Φ− Φ (x) m and Φ
(x) m ), we get
Φ (x) m ∈ H(y , r)
) f (x − y) dy dx .
Mean of the Estimator of the Interference
Thereby, we get
E [Sn(r)] = λcv(r)
v(r) = exp
−λc ∫ Rd
−λm ∫ b(o,r)
.
and the probability of Φ (x) m not being r -close to y is
P (
) = exp
Variance of the Estimator of the Interference
The variance of the estimator is given by
σ2 λ(r) =
] ,
ξCM(r) = λcλm
∫ z>r
ξCCM(r) = λ2 cλm
∫∫ z−w>r z>r w>r
|Wn,r ∩ (Wn,r + z)| v(z, r) um(z, r) um(z,w , r) f (w)dz dw ,
Variance of the Estimator of the Interference
ξCMM(r) = λcλ 2 m
∫∫ z>r w>r z−w>r
|Wn,r ∩ (Wn,r + w − z)| v(w − z, r) um(w − z,w , r)f (z) f (w) dz dw ,
ξCCMM(r) = λ2 cλ
z′>r
um(w ,−z ′, r) f (z) f (z ′)dw dz dz ′,
and
Conclusions & Future Work
- Models for homogeneous and heterogeneous networks have been developed.
- Expressions for the interference and the relevant KPI’s in these scenarios have been
derived.
Future Work:
- Though current insights are useful, finding good approximations for some of the more
unwieldy expressions could help easier reuse in other optimization problems.
- In order to improve upon the models presented in this work, alternatives in which the
degree of heterogeneity as well as the extent of correlation between locations of
different network components are adjustable can be explored.
Investigate triply stochastic point process models wherein users are clustered
around micro base stations, and micro base stations are in turn clustered around
macro base stations.
Conclusions & Future Work
- Models for homogeneous and heterogeneous networks have been developed.
- Expressions for the interference and the relevant KPI’s in these scenarios have been
derived.
Future Work:
- Though current insights are useful, finding good approximations for some of the more
unwieldy expressions could help easier reuse in other optimization problems.
- In order to improve upon the models presented in this work, alternatives in which the
degree of heterogeneity as well as the extent of correlation between locations of
different network components are adjustable can be explored.
Investigate triply stochastic point process models wherein users are clustered
around micro base stations, and micro base stations are in turn clustered around
macro base stations.
Dresden Field Trial:(Uplink) Coordinated Multipoint Works!
When Base Stations Meet Terminals,And Some Results Beyond
Dresden Field Trial:(Uplink) Coordinated Multipoint Works!
Conclusions

when base stations meet terminals, and some results beyond

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