achieved. The base-station location, its pilot-signal power (which
determines the size of the cell), and the transmission power of the
mobiles all affect the received SIR. In addition, because of the need
for power control in CDMA networks, large cells can cause a lot of
interference to adjacent small cells, posing another constraint to
design. In order to maximize the network capacity associated with
a design, we develop a methodology to calculate the sensitivity of
capacity to base-station location, pilot-signal power, and transmis-
sion power of each mobile. To alleviate the problem caused by dif-
ferent cell sizes, we introduce the power compensation factor, by
which the nominal power of the mobiles in every cell is adjusted.
We then use the calculated sensitivities in an iterative algorithm
to determine the optimal locations of the base stations, pilot-signal
powers, and power compensation factors in order to maximize ca-
pacity. We show examples of how networks using these design tech-
niques provide higher capacity than those designed using tradi-
tional techniques.
Index Terms—Capacity optimization, cell design, code division
multiple access (CDMA), location design, power compensation.
I. I
NTRODUCTION
T
HE reverse link capacity of a single cell in a cellular code
division multiple access (CDMA) network depends on the
interference of users within that cell (intracell interference), as
well as on the interference of users in adjacent cells (intercell
interference) [1]–[3]. Thus, the number of simultaneous users
that can be handled within one cell depends on the number of
simultaneous users in all the cells in the network. This inter-
ference limitation makes the cell placement design in CDMA
networks particularly difficult: the problem of placing cells in a
region with a given user profile would require the calculation of
the intercell interference, which depends on the cell geometry,
the transmit power levels of the users, and the number of users in
adjacent cells. This problem is not present in networks that use
fixed channel assignment algorithms, wherein cochannel inter-
ference is eliminated by using different frequency sets in ad-
jacent cells, thereby separating the problem of cell placement
and cell capacity. In such networks, the design rule-of-thumb is
to place cells so that each will have a constant demand. Thus,
Manuscript received November 30, 1999; revised July 26, 2000. This work
was supported in part by the McDonnell Foundation and in part by the National
Science Foundation under Grant NCR 9706545.
R. G. Akl, M. V. Hegde, and P. S. Min are with the Department of Electrical
Engineering, Washington University, St. Louis, MO 63130 USA.
M. Naraghi-Pour is with the Department of Electrical and Computer Engi-
neering, Louisiana State University, Baton Rouge, LA 70803 USA.
Publisher Item Identifier S 0018-9545(01)03959-7.
smaller cells are used in areas of high demand while larger
cells are used in areas of low demand. In CDMA networks,
differing usage results in differing intercell interference, which
suggests the possibility of more efficient topology design. In this
paper, we describe a new network topology design technique for
CDMA networks that outperforms traditional techniques.
In [3]–[12], the authors calculate the capacity of a single
CDMA cell or a CDMA network by restricting analysis to re-
verse link capacity. These papers concede that CDMA networks
are interference- and reverse-link limited in that the system ca-
pacity for the reverse link will be lower than that of the for-
ward link. In [1], the authors analyze both forward- and re-
verse-link capacity. They have shown that for an ideal power
control and hard handoff case, reverse-link capacity limits the
system capacity; however, the difference between forward- and
reverse-link capacities is not large. In [13], the authors show that
reverse-link capacity is increased considerably by soft handoff,
but, at the same time, imperfect power control reduces it and
compensates for the increase. On the other hand, forward-link
capacity is decreased due to soft handoff, and the reduction is
shown to be more than the difference between reverse- and for-
ward-link capacities. They believe that the overload in forward
link limits the capacity of the system. From the above discus-
sion, it is evident that among researchers, a consensus does not
exist on whether the CDMA system capacity is reverse- or for-
ward-link limited. However, the majority of the literature pub-
lished on the subject is of the former view. In light of this, in
this paper we consider the reverse-link capacity only. In the rest
of this paper, when we refer to capacity, we actually mean the
reverse-link capacity.
Despite the abundant literature on CDMA, few contributions
address the cell design problem. The main emphasis to date
has been on capacity analysis through simulation with uniform
cells. In [14], the performance of IS-95-based CDMA systems
[15] is studied in nonuniform and uniform traffic distribution
with equal cell sizes. The elements that are investigated are
the pilot channel chip energy to received signal power spectral
density ratio, the forward link bit energy to noise power spectral
density ratio, and the reverse link bit energy to noise power
spectral density ratio. Monte Carlo simulations are carried out
under a hexagonal omnicell configuration to obtain statistics
of capacity, handoff percentage, and signal and interference
levels.
In [16], the problem of nonuniform traffic’s decreasing of the
system capacity in microcell CDMA systems is investigated.
The authors propose a scheme whereby every base station adap-
tively controls the required transmission power level of the mo-
biles based on minimizing the difference between the target
signal-to-interference ratio (SIR) and the observed SIR on the
0018–9545/01$10.00 © 2001 IEEE
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 50, NO. 3, MAY 2001
reverse link at the base station. The authors evaluate the ef-
fectiveness of their proposed method with regard to the signal
quality and the outage probability through computer simula-
tion. The results show that the proposed method is effective
against localized high-density traffic, but the effectiveness be-
comes smaller as the traffic load in the vicinity of congested
cells increases.
In [8] and [9], an iterative algorithm for cell design that mini-
mizes the disparity in communication quality between base sta-
tions is proposed. To equalize the SIR in order to maximize ca-
pacity, every base station adjusts both the pilot-signal power and
the desired transmission power level based on the difference be-
tween the average SIR and the observed SIR on the reverse link.
The authors confirm the effectiveness of the proposed algorithm
through computer simulation.
In [11], the problem of adaptive cell sectorization to increase
capacity in CDMA systems is investigated. The authors pro-
pose to minimize the total received power and the total transmit
power of the mobiles in order to reduce intercell interference
while retaining acceptable quality of service. The results show
that under nonuniform traffic conditions, the optimum arrange-
ment of the sector boundaries is quite different from uniform
cell sectorization.
In a CDMA network, the near–far problem necessitates power
control, whereby the transmit power of mobiles is proportional
to distance (from the base station) raised to the path-loss ex-
ponent. Typically, the power control scheme used in CDMA
networks is signal level based, i.e., the power control equal-
izes the received power from the mobiles at the base station.
We assume such a power control scheme in this paper. In par-
ticular, SIR-based power control is not investigated [17] [18].
When large cells are adjacent to small cells, users at the bound-
aries of large cells cause a lot of interference to users in small
cells. This causes a significant reduction in the capacity of the
small cells. To alleviate this problem, we propose to adjust the
nominal power of the mobiles in every cell by a power com-
pensation factor (PCF) [19]. Since CDMA is interference lim-
ited, any decrease in the amount of interference translates into
a capacity gain. Increasing the pilot-signal power of a base sta-
tion increases the coverage region of that cell and thus increases
the number of users and the intracell interference in that cell.
However, it will decrease the number of users in the adjacent
cells, thus decreasing the intercell interference on this base sta-
tion. In addition, changing the location of a base station changes
the coverage region of that cell and the coverage regions of the
adjacent cells. Thus, by controlling the transmitted pilot-signal
power and adjusting the location of the base stations, the cov-
erage region of each cell is controlled, which in turn controls
the intracell and intercell interference. Given a fixed configura-
tion of user distribution, we try to place a given number of cells
in order to maximize capacity. We evaluate the capacity of the
entire network as a function of all the PCFs, the base-station lo-
cations, and the transmitted pilot-signal powers and present an
optimization framework that allows us to maximize capacity.
We develop design rules that apply to general user configura-
tions (uniform or with hot spots). We validate our design rules
by presenting comparative capacity results for networks that are
designed by our method versus those designed with traditional
rules.
The remainder of this paper is organized as follows. In Sec-
tion II, we calculate the relative average intercell interference.
In Section III, we define network capacity. In Section IV, we
study the sensitivity of the network capacity with respect to
base-station locations, pilot-signal powers, and power compen-
sation factors. In Section V, the optimization of capacity is per-
formed, and numerical results are presented in Section VI. Sec-
tion VII concludes this paper.
II. R
ELATIVE
A
VERAGE
I
NTERCELL
I
NTERFERENCE
Consider two cells and . We assume that each user is always
communicating with and is power controlled by the base station
that has the highest received power at the user. Let
denote the
region where the received pilot-signal power from base station
is the highest among all base stations. A user located at coordi-
nates
is at distance
from base station . Let
be
the number of users in cell
and
= Area(
), the area of cell
. It is assumed that the power-control mechanism overcomes
both large-scale path loss and shadow fading. It does not, how-
ever, overcome the fast fluctuations of the signal power associ-
ated with Rayleigh fading [1]. The propagation loss of a user in
cell is modeled as the product of the
th power of distance and
a log-normal component representing shadowing losses. Now
let
denote the Rayleigh random variable that represents the
fading on the path from this user to cell . The average of
is the log-normal fading on that path, i.e.,
[20], where
is the decibel attenuation due to shadowing and is
a Gaussian random variable with zero mean and standard devi-
ation
. Consequently, the relative average interference at cell
caused by all users in cell
is given by [21]
(1)
The expectation is calculated as follows:
E
E E
E
E
E
(2)
Let
, where
is a Gaussian random variable with zero
mean and variance equal to 2
since
and
are independent.
Substituting in (2), we get
E
E
(3)
where
. Substituting the result back into (1)
(4)
AKL et al.: MULTICELL CDMA NETWORK DESIGN
713
Let
denote the per-user intercell interference factor of cell
to cell , i.e.,
. Note that in our model,
equals
zero.
Equation (4) is used to calculate the relative average inter-
cell interference for a uniform user distribution, i.e., when the
relative user density at
in cell
is
. For a nonuni-
form user distribution, let
be the relative user density at
. A hot spot is a region of a cell with a higher relative user
density than the rest of the cell. The relative average intercell in-
terference at cell caused by all users from cell for the general
case becomes
(5)
The
s will be used to determine the reverse-link capacity. A
closed-form expression can be derived for
for the case of
a uniform user distribution and a specific cell geometry (e.g.,
hexagonal). However, since for the case of nonuniform user dis-
tribution we evaluate
in (5) numerically, we have used the
same approach for the uniform case and have not obtained the
closed-form solution for (4).
III. CDMA N
ETWORK
C
APACITY
Consider a multicell CDMA network with spread signal
bandwidth of
, information rate of
bits/s, voice activity
factor of
, and background noise spectral density of
.
Assuming a total of
cells with
users in cell , the bit
energy to interference density ratio in cell is given by [22]
for
(6)
To achieve a required bit error rate we must have
for some constant . Thus, rewriting (6), the number of users in
every cell must satisfy
for
(7)
A set of users
satisfying the above equations
is said to be a feasible user configuration, i.e., one that satisfies
the
constraint. The right-hand side of (7) is a constant,
determined by system parameters and by the desired maximum
bit error rate, and can be regarded as the total number of effec-
tive channels
available to the system. As can be seen from
(7), the capacities of the CDMA cells in a network must be con-
sidered jointly. Thus the notion of capacity in a CDMA network
is that of a capacity region, which indicates all tradeoffs in ca-
pacity between the cells in the network.
We define equal capacity as the requirement that all cells have
an equal number of users, i.e.,
for all . For the equal
capacity case, the network capacity
is equal to
, where
(8)
In general, for a given fixed configuration of user distribution,
a feasible user configuration yields a network capacity
that
is the solution to the following optimization problem
subject to
for
(9)
We should point out that the pilot-signal strengths and the base-
station locations affect the values of the set of factors
and
thereby affect the decision variables
in (9).
The optimization problem in (9) is an integer programming
(IP) problem. One technique to solve the IP problem is based
on dividing the problem into a number of smaller problems in
a method called branch and bound [23]. Branch and bound is a
systematic method for implicitly enumerating all possible com-
binations of the integer variables in a model. The number of sub-
problems and branches required can become extremely large.
If the integer variables
,
, are relaxed and as-
sumed to be continuous variables, then (9) becomes a linear pro-
gramming (LP) problem whose solution can be obtained by any
general LP technique, e.g., the simplex method [24]. We should
point out that (9) is a convex optimization problem, and there-
fore, the approach described above does converge to a global
optimal solution. It should also be noted that although the op-
timization problem in (9) maximizes the total network capacity
and provides an improvement over that obtained from (8), the
capacity of individual cells that results from (9) may vary sig-
nificantly.
In Section VI, we provide results to the continuous relaxation
of (9), which give an upper bound on the optimal value. We
give the rounded-down solution (rounding down the solution
of the continuous relaxation problem), which in general is not
the optimal solution of the IP problem. We also solve the IP
problem and provide the optimal solution as well as the number
of branches required to arrive at the solution.
IV. S
ENSITIVITY
A
NALYSIS
Having formulated the optimization problem that calculates
network capacity, we now investigate the effect of changing the
transmission power of the mobiles, the pilot-signal powers, and
the locations of the base stations on this capacity. Consider a
network with a large cell adjacent to a small cell , and consider
two users
and
both located at the boundary of these two
cells as shown in Fig. 1. User
will cause a lot of interference to
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 50, NO. 3, MAY 2001
Fig. 1.
Effect of a large cell adjacent to a small cell.
user
because mobiles’ transmit power is proportional to their
distance from the base station raised to the path-loss exponent.
Thus, a small cell adjacent to a large cell will experience a great
deal of interference—which causes a significant reduction in the
capacity of the small cell.
This problem can be resolved by adjusting the nominal power
of the mobiles in every cell by the PCF in order to make the SIR
in small cells comparable to that in large cells. If cell
has a
PCF
, its mobiles’ signal powers have increased by a factor
of
. Thus, the new relative average intercell interference of
cell
to cell
becomes
(10)
As a result, the new intercell interference factor per user be-
comes
. Thus, once the original intercell interference fac-
tors have been calculated (for PCFs equal to one), changing the
PCFs for the cells does not require recalculation of the original
since
is linear in
.
Let
be the received nominal power at base station without
PCF. Then,
. With PCF, the received power at base
station is
. Now,
. The ratio of bit energy to
interference density is
(11)
To achieve a required bit error rate, we must have
,
yielding
(12)
Rearranging terms
(13)
Let
for
(14)
Assuming that the variables
,
, are relaxed to be
continuous variables, the derivative of
with respect to
is
if
if
.
(15)
The above derivatives will be used in the solution to the opti-
mization problems (31)–(34).
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