improving the system capacity of a cellular network with partial power equalization

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




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.




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;'

in Proc. of Global Teleconlmunications Conference

(GLOBECOM),




[2] B. Rimoldi, L. Duan and Li, "Rate loss due to power

equalization in cellular communications," in Proc. IEEE

Int. Conf. Personal Wirelles COl1ununication, pp. 311

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[3] B. Rimoldi and R. Urbanke, "ARate-Splitting Approach

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[4] B. Rimoldi and R. Urbanke, "On the structure of thedominant face of multiple-access channels," in Proc.

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[5] A. Grant, B. Rimoldi , R. Urbanke, P. Whiting, "Rate

splitting multiple access for discrete memoryless chan

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[6] B. Rimoldi , "Generalized time sharing: a low

complexity capacity-achieving multiple-access tech

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[7] R. Ahlswede, "Multi-way communication channels," in

Proc. 2nd Int. SYlnp. Information Theory, Armenian

SSR, pp. 23-52, 1971.

[8] H. Liao, "Multiple Access Channels," Ph.D. disserta

tion, Dept. Elec. Eng., Univ. Hawaii, Honolulu 1972.

[9] T. M. Cover and J. A. Thomas, Ele111ents of Infor/nation

Theory. New York: Wiley, 1991.

[10] J. D. Parsons, Mobile Radio Propagation Channel. 2nd

edition Chichester, UK: Wiley, 2001.

[11] M. Abramowitz and A. Stegun, Handbook ofMathe-

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