improving the system capacity of a cellular network with partial power equalization
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8/14/2019 Improving the System Capacity of a Cellular Network With Partial Power Equalization
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IMPROVING THE SYSTEM CAPACITY OF A CELLULAR NETWORK WITH PARTIAL
POWER EQUALIZATION
Ninoslav Marina*
UNIK - University Graduate Center
University of OsloInstituttveien 25
NO-2027 Kjeller
Norway
ABSTRACT
We derive an expression for the system capacity of a cellu
lar network in which power equalization (PE) is applied only
to the mobile stations (MS) that have stronger channel, while
there is no power equalization for those with weaker channel.
Power equalization means adjustment of the transmit power
of each MS in order to have the sanle received power at thebase station (BS). Systems that use matched filter receiver
need power equalization. In general, however, if another re
ception strategy is used, the system capacity is higher if there
is no power equalization. Although this strategy is optimal
for the sum rate capacity, it will give an advantage to users
staying very close to the BS, and is not fair. In this paper
we consider a combined model in which the power equaliza
tion is done partially, i.e. , stations that are closer to the BS
equalize their transmit powers, while the others do not. For
that reason, we define a cut-offrate beyond which noMS can
transmit. This corresponds to an equivalent cut-off radius in
the cell within which the transmit powerof the mobile stations
has to be scaled down such that all of them have the same receive power. Outside the cut-off radius all users can transmit
with their maximal power (i.e. no PE is applied to them),
such that the overall system capacity is increased. We derive
a closed form expression in terms of the hypergeometric func
tions for the system capacity when partial power equalization
is applied and the channel is Gaussian.
Index Terms- Interference cancellation, power equal
ization, spectral efficiency, cellular communications.
1. INTRODUCTION
The main purpose of this paper is like 2] to study the pos
sibility of using the rate splitting multiple access (RSMA)[3, 4, 5, 6] on cellular communications. From theoretical
point of view maximum achievable rates of multiple access
*This work has been supported by Nokia Research Centre, Helsinki and
was submitted while N. Marina was at University of Hawaii at Manoa.
Olav Tirkkonen
Helsinki University of Technology
Communications LaboratoryOtakaari 5A
FIN-02015 Espoo
Finland
channels have been well understood for long time [7, 8, 9].
The transmit power is affected by large scale path loss, shad
owing and small scale path loss (fast fading). Power equal
ization (PE) is an operation that adjusts the transmit power of
mobile units in such a way that the mean received poweris the
same for all units. It should not be mixed with power control
(PC), that is an operation that allocates transmit power to the
transmit antennae according to the channel state information
obtained from the training sequence, in order to maximize
some objective function such as the throughput of the result
ing channel. In this paper we consider only power equaliza
tion. Note that in both cases, PE and PC, the transmitter must
have information about the channel, but for the PE this infor
mation is much smaller (a pilot signal might be used for that)
than for the power control case where real channel estimation
is needed (training sequence). For systems that use nlatch fil
ter receiver, perfect power equalization is necessary. In gen
eral, however, if we use another reception strategy, the system
capacity is higher if there is no power equalization at all [I].
In [I] authors analyze a cellular system by comparing its spec
tral efficiency of a spread spectrum multiple access (SSMA)
scheme and of the ideal interference cancellation multiple ac
cess scheme. The latter gives the theoretical upper bound on
the maximal sum-rate. Authors conclude that there is a big
gap between the two schemes. They also observe that another
huge improvement could be obtained if power equalization is
not used. Although this strategy is optimal for the sunl rate, it
will give an advantage to users staying very close to the BS,
and is not fair, and therefore not applicable in practice. To
that end we propose a combined model in which users that
are close to the BS are equalized but those who are further are
not. We define a cut-off rate beyond which no user can trans
mit. This corresponds to an equivalent (since there is a ran
dom fading) cut-off radius in the cell within which the transmit power of users has to be scaled such that all of thenl have
the same receive power. Outside the cut-off radius all users
can transmit with their maximal power, such that the overall
system capacity is maximized. A pure case without power
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equalization, although theoretically more efficient, does not
have practical importance. The paper is organized as follows.
In Section 2 we consider the circular cell model, in Section
3 we propose the combined model in which for the closer
users, power equalization is applied and for the far users it is
not. Section describes the combined hexagonal cell model
and Section concludes the paper.
where in order comparison to be fair, Px is chosen such that
the total radiated power within the cell is equal as in the case
with power equalization, that is,
i27r
fC i27r /c(rlc)l3rdrdd> Pxrdrd¢.o Ci 0 . Ci
Solving this we get
2. CIRCULAR CELL MODEL
where No is the sing le-sided power spectral density of
the thermal noise, TV is the frequency bandwidth, and
PBI(NolV). All logari thms in this paper are base 2.
Observe that as increases loge bps/Hz. The
maximum spectral efficiency if the interference cancellation
scheme is used in a system with power equalization is
As a system model for the uplink system capacity we use the
Gaussian Multiple Access Channel. As it was presen ted in
[1], for the SSMA (also known asCDMA) system, since there
is power equalization, the received power from all the users
in the cell is received with the same power. In this case, if we
denote the transmit power of the user at the cell boundary by
P, the total received power at the base station is
PB N P c - f 3 ~where is the total number of users in the cell, (3 is the path
loss coefficient [10], and c is the cell radius. Note that here
we use general while in The transmit power
from a user at distance r is Pt (r) P (ric) f3. Therefore, themaximum spectral efficiency in bps/Hz for an SSMA system
is
where
f3 - 2
2 - 1
k 2 - k-P
k 2 - 1
Now since > atking a limit of the last expression when
we get
. k2
1Inn v -k
2log k.
p12 - 1
log (1 + v( ) "
2 k2 - k-
Px /3 2 ' k 2 - 1 '
Note that for (3 2, v is monotonically increasing function
of ;3 for any k and
4k 2 2 - k(3-2 - 1
+ 2 - 2
In practice, >> 1 and we have
4k /3- 2
v (32 _
where k ci is the ra tio between the outer and the inner
radius of the cell. Now
In the case where users t ransmit the ir available power and
there is no power equalization, the maximum spectral effi
ciency achieved by interference cancellation becomes at least
(Pc-(3 )
Nlog 1 (N _ 1)Pc-/3 N o ~ VNlo (1 (IN )
g (N - N+
(NPc-(3)
= log 1+ NoW =log( l+( ) .
In this case C grows without a bound if the number of users
increases. Assume now a system with no power equalization
and that the weakest user in the cell transmit at powerP. That
means P is the maximal t ransmitted power coming from a
user at the cell boundary. The transmit power from all users
in the cell is cons tant and equal to Px . Assuming uniform
distribution on an annulus with outer radius c and inner radius
Ci, the density of users is given by N -12
- -1 .
Therefore, the total received power at the base station is given
by
j'27r j'C( .2 _ .2) . Px r-
f3rdrd¢
c c 0 . Ci
2NPx c;-,8 - c2- 13
(3 - c2 - c
in f == lirn lirn == 1./3>2 11 /312r> l
Therefore, for any meaningful (3 > 2 and > 1, it follows
that v > which gives > That means we always
get an improvement if there is no power equalization. Notice
that the improvement is
= log
and increases with the total received (in the PE case) signal
to-noise ratio ( == NPc-/] I(NolV). For (big enough
A . 4log 2) log + log {32 _ 4'
This means there is a b ig improvement if there is no power
equalization, For example for 4, we get an improvement
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of approximately log k bps/Hz without power
equalization. There is a slight difference in the comparison of
the two scenarios with [1]. There authors assume that that the
transmitted power from all users is and therefore the radi-
ated powers within the whole cell is not equal in both cases.
Their approach is also understandable since they assume a
system in which users can transmit with their entire available
power. This possibil ity for improvement is not real istic in
practice since without power equalization, the closer the user
is to the base station the bigger its part in the total sum rate
is. Imagine a system in which all but one user are close to the
cell boundary and only one of them is approaching the base
station. In that case the sum rate is getting basically equal to
the rate of the close user. This of course is not fair and does
not make sense to construct a practical system in which most
of the sum rate will be given by the sum rate of the users with
strong channel. Therefore, in the next sect ion we propose a
combined model in which power equal izat ion is applied to
users with strong channel (close users) and there is no power
equalization for the users with weak channel (far users).
3. COMBINED MODEL
As we showed in the previous sect ion although there is a big
potential for increase of the spectral efficiency by not apply-
ing power equal izat ion, such a system will not make sense
from practical point of view. Here we propose a combined
model in which for those users in the cell that are within a dis-
tance from the base stat ion, power equal izat ion is applied
and for those between distances Crn and C it is not (Fig. 1).
Here the transmit power is
/
Fig. 1. The equivalent circular cell model for the combined
system.
where in order to have the same anlount of radiated power in
the cell as in the previous two cases should be the same in
order to have a fair C0I11parison
From the last equation, we get
where == Here the total received power at the base
station is
Hence, for the combined model we get for the maximal spec-
tral efficiency
C == 10g(1
where
For 1 :S q ::; k the above expression is monotonically increas-
ing in and it follows
where the lower bound, obtained for q == is the case with
power equalization, and the upper bound, obtained for q ==
is the case with no power equalization. Here, also for /3 > 2
and :S q :S k, v > so we have a clear improvement overthe case with power equalization. Indeed,
So, we also get an improvement over the power equaliza-
t ion case. This improvement is smaller than in the case with
no power equalization, however here we have more realistic
system in which users that have weak channel , can also get
chance to transmit. In other words, this system is more effi-
cient than the first one and fairer than the second one.
4. HEXAGONAL CELL MODEL
Ci
C < r ::; c.
In this sect ion we repeat the analysis from the previous two
sections, but here we analyze the hexagonal cell model.
Assume the combined model in which close users (within the
region defined by are equalized and far users are not.
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where
An alternative formula to calculate Jb i s g iven by
~ . ( ~ b . ~ ) .2f (1 - ~ 2 2 2
/6 ,3 ( c n (/1
.n/6 . r3
12P . ' ./., 2 C ( ) ~ ( ) 3
./0 . : ~ : J ~ :3
k2 [ ~ 12(q2 -
h· '1 - 1 q'1 k /32 _ '2
./0
is the inconlplete beta function. In this cas e t he received
power at the base station is
Fig. 2. The equivalent ceJl nlodel for the conlbined hexagonal
systenl.
where k == (J In conclusion, even for the hexagonal cell
model there is an inlprovement in the spectral efficiency if the
interference cancellation is used for the conlbined
nlodeL that is.
Again we assunle lV users distributed over the area defined
by (J and The density of users is /) == 21\-/ (:3
is the power transnlitted by the most distant user in the cell.
namely the one at the ver tices. For the far use rs the constant
transnlit power is def ined again by the assumption that
the radiated power within the cell is th e s ame for the non
equalized. equalized and the combined nlodel. Then fn)nl
Fig. 2 we have
(1 log( 1
where
It is eas ily seen tha t > 1. Since in practice >> 1
12 I'.( )
1 '2 - 1)~ J,)Ii-- - 4 --
• 11 \' 3 .
12 rir/hf' 2(.'0:' 12 n/
(I ' ( (J . . . ()
( ' ( ) ~ .
5. CONCLUSION
where
where == alam and
ii i
1(1 /2. 2 112: :3,/2: 1/'
,)
and solving it we get
J+ 1
8· :3:r
H - 2(-j+2(.-j
1 --1-_-q-'1- . P
In thi s paper we provide a simplif ied analysis t o show that
a system without power equalization perfornls better than a
systenl with power equalized users. Since a sys tenl tha t has
no power equalization at a ll does not nlake sense we propose
a conlbined nlodel in which users with strong channels, in our
case close users, are equalized and the users with weak chan-
nels are not. The proposed nlodel improves a lo t t he perf or
nlance of the equalized systenl and at the san le t in le is fai re r
for the users. In our Inodel we note that theequalized and non-
equalized systenls are special cases of the proposed conlbined
nlodel.
is the Gaussian hypergeometric function Il l . and
dt .()
6. REFERENCES
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Splitting Multiple Access on cel lular comlTIunications;'
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(GLOBECOM),
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