IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 4, JULY 1999 1047
Area Spectral Efficiency of Cellular
Mobile Radio Systems
Mohamed-Slim Alouini, Member, IEEE, and Andrea J. Goldsmith, Member, IEEE
Abstract— A general analytical framework quantifying the
spectral efficiency of cellular systems with variable-rate trans-
mission is introduced. This efficiency, the area spectral efficiency,
defines the sum of the maximum average data rates per unit
bandwidth per unit area supported by a cell’s base station. Ex-
pressions for this efficiency as a function of the reuse distance for
the worst and best case interference configurations are derived.
Moreover, Monte Carlo simulations are developed to estimate the
value of this efficiency for average interference conditions. Both
fully loaded and partially loaded cellular systems are investigated.
The effect of random user location is taken into account, and the
impact of lognormal shadowing and Nakagami multipath fading
is also studied.
Index Terms—Adaptive transmission systems, cellular systems
capacity, cochannel interference, spectral efficiency.
I. INTRODUCTION
THE RADIO spectrum available for wireless data ser-vices and systems is extremely scarce, while demand
for these service is growing at a rapid pace [1]. Spectral
efficiency is therefore of primary concern in the design of
future wireless data communications systems. This efficiency
is partly achieved by cellular systems which exploit the
power falloff with distance of signal propagation to reuse the
same frequency channel at spatially separated locations [2].
However, while frequency-reuse provides more efficient use of
the limited available spectrum, it also introduces unavoidable
cochannel interference [3]–[8], which ultimately determines
the bit error rates (BER’s) available to each user. Thus, there is
a tradeoff between the system spectral efficiency, measured in
[b/s]/[Hz m ] or Erlangs/[Hz m ], and the communication
link quality, measured in terms of the BER provided to the
users [5], [8].
Another technique to increase spectral efficiency is to use
multilevel modulation, such as M-QAM, which increase link
Manuscript received May 9, 1997; revised April 8, 1998. This work was
supported in part by the NSF CAREER Development Award NCR-9501452
and by Pacific Bell. This is an expanded version of work which was presented
at the IEEE Vehicular Technology Conference VTC’97, Phoenix, AZ, May
1997, and at the IEEE International Conference on Communications ICC’97,
Montreal, P.Q., Canada, June 1997.
M.-S. Alouini was with the Communication Group, Department of Electri-
cal Engineering, California Institute of Technology (Caltech), Pasadena, CA
91125 USA. He is now with the Department of Electrical and Computer
Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail:
alouini@ece.umn.edu).
A. J. Goldsmith was with the Communications Group, Department of Elec-
trical Engineering, California Institute of Technology (Caltech), Pasadena, CA
91125 USA. She is now with the Department of Electrical Engineering, Stan-
ford University, Stanford, CA 94305 USA (e-mail: andrea@ee.stanford.edu).
Publisher Item Identifier S 0018-9545(99)05742-4.
spectral efficiency, measured in [b/s]/Hz, by sending multiple
bits per symbol [9]. However, wireless channels are subject
to severe propagation impairment which results in a serious
degradation in the link carrier-to-noise ratio (CNR), and even if
efficient fading compensations techniques are used, multilevel
schemes will still require higher power levels than binary
modulations for a specified BER. Therefore, to keep the
cochannel interference at an acceptable level it is necessary
to increase the frequency-reuse distance (or equivalently the
cluster size) which in turns leads eventually to a lower
system efficiency. This problem has been recently addressed
by Haas and Belfiore [10], who showed that there is a tradeoff
between the system and link spectral efficiencies. This was also
confirmed by Morinaga et al. [11], who claimed that 4-QAM is
the optimum multilevel modulation for high-capacity cellular
systems, and that opting for higher modulation levels will
just reduce the system spectral efficiency. This is essentially
due to the fact that fixed modulation systems are designed
relative to the worst case interference/fading conditions. How-
ever, adapting certain parameters of the transmitted signal
relative to the CNR leads to better link and system spectral
efficiencies. The basic concept of variable-rate transmission
is real-time balancing of the link budget through adaptive
variation of the symbol time duration, constellation size,
coding rate/scheme, or any combination of these parameters
[12]–[16]. Thus, without wasting power, increasing cochannel
interference, or sacrificing BER, these schemes provide a much
higher average spectral efficiency by taking advantage of the
“time-varying” nature of the wireless channel and interfer-
ence conditions: transmitting at high speeds under favorable
interference/channel conditions and responding to an increase
in interference and/or channel degradation through a smooth
reduction of their data throughput. Since buffering/delay of the
input data may be required in this process, adaptive systems
are to be used for applications which are, to some extent,
bursty in nature and are therefore best suited to high-speed
wireless data transmission.
Previous studies of the system spectral efficiency for cellular
systems assumed that the data rate is constant and equal for all
users, regardless of interference conditions and channel quality
[4], [5], [8], [10], [17]–[19]. In these papers, spectral efficiency
calculations was based on a criterion introduced by Hatfield
[17] and defined as the ratio of the carried traffic per cell
(in Erlangs) to the product of the total system bandwidth and
area supported by a base station. This criterion is not suitable
for data systems since Erlangs are just a measure of traffic
loading rather than throughput intensity. A more pertinent
0018–9545/99$10.00  1999 IEEE
1048 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 4, JULY 1999
measure of spectral efficiency in cellular data systems is the
total throughput, i.e., number of [b/s]/[Hz m ] associated
with each base station (BS).
The aim of this paper is to investigate the theoretical limits
of this spectral efficiency limits for cellular data systems
where mobile users continuously adapt their rate relative to
their fading and interference conditions. This efficiency, called
the area spectral efficiency (ASE), is defined as the sum of
the maximum average data rates per unit bandwidth per unit
area for a specified BER. The ASE definition captures the
tradeoffs between a cellular system’ spectral efficiency, the
users’ link spectral efficiency, and the communication link
quality provided to these users. We take into account the
effect of the users random location in their respective cells and
study the ASE under the impact of shadowing, and multipath
fading superimposed on path loss, with propagation parameters
for both macrocells and microcells. Although both shadowing
and fading will typically be superimposed on path loss, we
first consider these two phenomena separately to assess their
respective impact. We then study their combined effect on the
ASE. For our analyses and simulations, we regard only the
uplink (mobile to BS) of systems using frequency-division
multiple access (FDMA) or time-division multiple access
(TDMA). We first consider fully loaded cellular systems. We
then generalize our analysis to partially loaded systems, and
determine the effect of traffic loading on the ASE.
The remainder of this paper is organized as follows. The
next section describes in more detail our propagation, cochan-
nel interference, adaptive transmission system, and user’s
random location models. Section III introduces the concept
of ASE for fully loaded cellular systems. Section IV presents
analyses, computer simulations, and numerical results for
the ASE when only path loss is considered. The impact
of shadowing and multipath fading on the ASE are studied
in Section V and VI, respectively. Section VII considers the
combined effect of shadowing and multipath fading on the
ASE. Section VIII deals with the ASE of partially loaded
systems. Finally, the paper concludes with a review of the
main results.
II. CHANNEL AND SYSTEM MODELS
In this section, we first outline the models for the different
propagation impairments affecting cellular systems. We then
present our assumptions for the cochannel interference and
the adaptive communication system under consideration. We
finally describe the random location model used for the users’
positions.
A. Propagation Models
It is well known that signal propagation in a radio mobile
environment is affected by three independent phenomena: 1)
deterministic path-loss variation with distance; 2) random slow
shadowing; and 3) random fast multipath fading.
Path loss is due to the decay of the intensity of a propagating
radio-wave. In both our analyses and simulations, we use the
two-slope path-loss model [20] to obtain the average received
power as function of distance. According to this model, the
average received signal power [W] is given by
(1)
where is a constant, [m] is the distance between the mobile
and the BS, is the basic path-loss exponent (approximately
two), is the additional path-loss exponent (ranges from two to
six), and [W] is the transmitted signal power. The parameter
[m] is called the break point of the path-loss curve and is
given by , where [m] is the BS antenna
height, [m] is the mobile antenna height, and [m] is
the wavelength of the carrier frequency. We use the following
typical values [21]: m, m for microcells
and m for macrocells. The resultant break points
for 900-MHz systems are m for microcells, and
m for macrocells. For 2-GHz systems, the break
points increase to m for microcells, and
m for macrocells.
In urban microcells systems, the link quality is also af-
fected by the shadowing of the line-of-sight path from terrain,
buildings, and trees. The shadowing is generally modeled as
lognormal distributed [22, Sec. 2.4]. The probability density
function (PDF) of the slowly varying received signal power is
thus given by the standard lognormal expression
(2)
where , is the area
(logarithmic) mean power which is related to path loss and
which is expressed in decibels (dB), and is the shadow
(logarithmic) standard deviation in decibels.
Mobile radio systems are also subject to fast (relative to
the shadowing effect) multipath fading due to the combination
of randomly delayed reflected, scattered, and diffracted signal
components. We consider slowly varying flat-fading multipath
channels. The slowly varying condition holds when the chan-
nel fading changes at a rate much slower than the data rate,
so the channel remains constant over hundreds of symbols.
Flat fading occurs when the symbol time duration is much
greater than the delay spread so the signal is just affected
by a degradation in its strength without a distortion in its
shape. We assume that the multipath fading environment is
characterized by a Nakagami- distribution such as the PDF
of the amplitude of the received signal is given by [23, eq.
(11)]
(3)
where is the local mean received
power which is related to path loss and shadowing, is the
Nakagami fading parameter ( ), and is the gamma
function defined by [24, p. 942, eq. (8.310.1)]
(4)
ALOUINI AND GOLDSMITH: AREA SPECTRAL EFFICIENCY OF CELLULAR MOBILE RADIO SYSTEMS 1049
Thus, the PDF of the received signal power is a
gamma distribution given by
(5)
The amount of fading of a channel is defined as the
variance of the received power to the square of the mean of the
received energy [25]. For the Nakagami- distribution,
. The Nakagami- distribution therefore spans a range
of fading environments via the parameter. For instance, it
includes the one-sided Gaussian distribution ( , which
corresponds to worst case fading) and the Rayleigh distribution
( ) as special cases. In addition, when , a one-to-
one mapping between the Rician factor and the Nakagami
parameter allows the Nakagami- distribution to closely
approximate the Rice distribution [23]. As increases, the
fading amount decreases, and in the limit as ,
, and the Nakagami fading channel converges to
an additive white Gaussian noise (AWGN) channel. Finally,
and perhaps most importantly, the Nakagami distribution often
gives the best fit to urban [26], [27] and indoor [28] multipath
propagation.
Although path loss, shadowing, and multipath fading si-
multaneously affect a radio mobile communication link, we
first analyze their effects separately to quantify their respective
impact on the ASE. We then study their combined effect on the
ASE in Section VII. This separation is valuable since in many
cases lognormal or Nakagami distributions give the best fit for
the overall fading process. For instance, Suzuki [26] concludes
that the distribution of the path strengths within a site tends
to follow a lognormal or a Nakagami distribution. In addition,
Abbes and Sheikh [29] recently showed that both lognormal
and Nakagami distributions fit the overall fading data for
LOS microcells, and argued that the segregation of slow and
fast fading components (e.g., lognormal and Nakagami) is
therefore unnecessary.
B. Interference and the Adaptive System Models
To simplify the analyses the following assumptions have
been made in the cochannel interference model. First, we
consider interference-limited systems in which the thermal
noise power is negligible relative to the cochannel interference
power [18]. Thus, the ratio of carrier power to noise plus
interference power reduces to the carrier-to-interference power
ratio (CIR). We also neglect cochannel interferers outside the
first dominant tier of interfering cells and all interchannel
interference [18]. Therefore, the desired user CIR, , can be
written as
(6)
where [W] is the received power level from the desired
mobile at a distance from its BS, [W] is the total
interfering power, and [W] is the received power level
from the th interfering mobile at a distance from the
desired mobiles’ BS. We assume throughout our study that
the cochannel interfering signals add up incoherently since
this leads to a more realistic assessment of the cochannel
interference in cellular systems [5]. In (6), is the number
of active cochannel interferers in the first tier, and is
the maximum number for . For example, for
nonsectorized cellular systems, whereas for 120
cell sectorized systems or linear highway cellular systems.
The interference model for the uplink of a fully loaded
nonsectorized ( ) cellular system is shown in Fig. 1.
Since the signal powers of both the desired and interfering
mobiles experience fluctuations due to multipath fading, shad-
owing, and the random location of users in their respective
cells, is also a random variable which depends on the
distribution of the and . Accurate techniques for “real-
time” estimation of these variations in the CIR are available
[30], and we assume throughout this paper that the fluctuations
in the CIR are tracked perfectly by the BS receiver. We also
assume that this information is sent back to the transmitting
mobiles via an error-free feedback path. The time delay in
this feedback path is also assumed to be negligible compared
to the rate of the channel/interference variations. All these
assumptions, which are reasonable slowly varying channels,
allow the mobiles to adapt their transmission rate relative to the
actual CIR state. A block diagram of the adaptive transmission
system model is shown in Fig. 3.
C. Users’ Random Location Model
For analytical convenience, we assume that the cell shape is
approximated by a circle of radius . All the mobiles (desired
and interfering users) are assumed to be mutually independent
and uniformly distributed in their respective cells. Thus, the
PDF of the mobiles polar coordinates ( ) relative to their
BS’s are
(7)
(8)
corresponds to the closest distance the mobile can be
from the BS antenna, and is approximately equal to 20 m
for microcellular systems and 80 m for macrocellular systems.
Note that we do not use mobility correlation statistics in
this model because they do not affect the adaptive model
(instantaneous adaptation) and its corresponding ASE.
III. AREA SPECTRAL EFFICIENCY
In this section, we introduce the concept of area spectral
efficiency for fully loaded systems in which the cell’s resource
(serviced channels) are fully used and the number of interferers
is constant and equal to . We generalize the ASE to partially
loaded systems in Section VIII.
Define the reuse distance [m] to be the distance between
two BS’s using the same set of frequencies. The ASE of a
cell is defined as the sum of the maximum bit rates/Hz/unit
area supported by a cell’s BS. Since frequencies are reused at
a distance , the area covered by one of these partitions is
roughly [m ]. The ASE, [b/s/Hz/m ], is therefore
1050 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 4, JULY 1999
Fig. 1. Cochannel interference on the uplink at a desired BS. In fully loaded nonsectorized cellular systems, there are six primary cochannel interfering mobiles.
approximated by
(9)
where is the total number of active serviced channels per
cell, [b/s] is the maximum data rate of the th user, and
[Hz] is the total allocated bandwidth per cell. We define
the maximum rate to be the Shannon capacity of the
th user in the cell, which depends on , the received CIR
of that user, and , the bandwidth allocated to that user.
The Shannon capacity formula assumes that the interference
has Gaussian characteristics. With FDMA or TDMA, there is
usually only a few dominant interferers coming from the first
ring of interfering cells, so the central limit theorem does not
apply and the Gaussian assumption for the interference signal
may not be valid. However, for capacity calculations, Gaussian
interference is a worst case noise assumption [31], [32],
and under this assumption the capacity-achieving transmit
spectrum for all users (i.e., desired and interfering users) is
Gaussian [33]. In addition, the Shannon’s formula indicates
an arbitrarily small BER, so our ASE is not parameterized
by BER. The ASE thus quantifies the tradeoff between the
increased system efficiency induced by a small frequency reuse
and the decreased capacity of each user resulting from the
corresponding increase in cochannel interference. In particular,
if we shrink the reuse distance then the denominator of (9) is
reduced. However, decreasing the reuse distance increases in-
tercell interference, thereby reducing the CIR of each user, and
its corresponding channel capacity. Therefore, there should be
an optimal reuse distance which maximizes (9).
ALOUINI AND GOLDSMITH: AREA SPECTRAL EFFICIENCY OF CELLULAR MOBILE RADIO SYSTEMS 1051
For constant, is given by Shannon’s formula:
. However, is not constant in our system
since both the interference and signal power of the th user
will vary with mobiles locations and propagation conditions.
When varies with time, equals the average channel
capacity of the th user [34], [35], given by
(10)
where is the PDF of the th user’s CIR. The av-
erage capacity intrinsically assumes that the users’ rate is
continuously adapted relative to their CIR (i.e., interference
conditions) in such a manner that the BER goes to zero
asymptotically. We therefore define the average area spectral
efficiency [b/s/Hz/m ] as the sum of the maximum average
data rates/Hz/unit area for the system, given by (9), with
replaced by . In (10) assuming that all users are assigned
the same bandwidth, becomes the same for all users,
and can therefore be written as
(11)
where is the normalized reuse distance, .
Consider first FDMA systems, where all users are allocated
the same bandwidth . Substituting in (11)
yields
(12)
In TDMA systems, the total bandwidth is allocated to only one
active user per time slot ( and ). Substituting
this in (11) we see that TDMA systems have the same ASE
as FDMA systems, so
(13)
IV. EFFECT OF PATH LOSS
In this section, we study the ASE of fully loaded systems,
ignoring the effects of shadowing and multipath fading. We
obtain the reuse distance which maximizes the ASE and
also determine the impact of the cell size, carrier frequency,
propagation parameters, and cell sectorization.
A. Analyses
Recall that due to the random location of users in their
respective cells, is a random variable depending on the
random positions of the desired and interfering
mobiles. To simplify our analysis, we reduce the problem
from dimensions to one dimension by computing
and the corresponding ASE for the worst case ( ) and best
case ( ) interference configurations. Without power control,
the worst case interference configuration corresponds to the
case where all the cochannel interferers are on the near
boundary of their respective cells, at a distance
[m] from the desired mobile’s BS. On the other hand, the best
case interference configuration corresponds to the case where
all the cochannel interferers are on the far boundary of
their cells, at a distance [m] from this BS. The
worst case and the best case interference configurations are
illustrated in Fig. 1. Assuming that the transmitted power of
all users is the same and substituting (1) into (6) yields
(14)
Note that is function of , so the desired user capacity is
(15)
for given by (14). Substituting (15) in (11) yields the ASE
conditioned on the desired mobile position , for a fully loaded
system. Integrating (15) over the desired user’s position PDF
(7) yields the average ASE for the two extreme interference
configurations as
(16)
In (16) and in what follows, the brackets denotes the operation
of averaging over the desired user’s position PDF (7). In
practice, when each interferer is uniformly distributed at
a distance between and , the average ASE will be
between these bounding values, as we will confirm by Monte
Carlo simulations.
B. Monte Carlo Simulations
The exact analytical value of , averaged over the
random positions of the desired and interfering users, requires
a numerical -fold integration, which is not only
computationally burdensome but also subject to roundoff and
stability problems. Rather than computing this integral, we
instead opted for a Monte Carlo simulation to estimate it. The
simulation algorithm is composed of the following steps.
1) The position of the desired user is randomly picked
according to (7) as follows.
a) Generate a pseudorandom number uniformly
distributed in [0, 1].
b) Deduce the user’s position according to (7) using
the percentile transformation method [36, p. 226]
(17)
2) The polar coordinates of the cochannel
interferers are randomly picked according to (7) and (8)
as follows.
a) Generate pseudorandom, pseudoindependent
numbers and uniformly distributed in [0, 1].
b) Deduce the polar coordinates as
(18)
(19)
1052 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 4, JULY 1999
Fig. 2. Geometry of the problem.
Fig. 3. Adaptive transmission system model (CIR).
3) The distance from each cochannel interferer to the
considered BS is calculated as (see Fig. 2)
(20)
4) The average received signal power of the desired
user and interfering mobiles ( and ’s) are calculated
using the two-slope path model (1).
5) The CIR of the desired user is then obtained accord-
ing to (6).
6) The ASE is calculated as
(21)
Repeating the above process [steps 1)–6)] 10 000 times, we
can estimate the value of by taking the average of all
the observations of the ASE as given by (21). After 10 000
computations, converges to within a three-digit accuracy.
C. Numerical and Simulation Results
In this section, we compute the effects of propagation
parameters, the cell size, and carrier frequency on the ASE
of fully loaded cellular systems. The ASE for the worst case
( ) and the best case ( ) interference configura-
tions are numerically computed for specific system param-
eters. Based on the Monte Carlo simulation described in
ALOUINI AND GOLDSMITH: AREA SPECTRAL EFFICIENCY OF CELLULAR MOBILE RADIO SYSTEMS 1053
(a)
(b)
(c)
Fig. 4. Comparison of the average uplink area spectral efficiency versus the normalized reuse distance for different values of the additional path exponent
b. (Fully loaded system with N
I
= 6; cell radius R = 200 m; R
o
= 20 m; antenna heights: 10-m BS, 2-m mobile; carrier frequency f
c
= 900
MHz; basic path exponent: a = 2.) (a) b = 2, (b) b = 4, and (c) b = 6.
Section IV-B, the exact value of is also estimated for
the same system parameters of interest.
Fig. 4 depicts the effect of path-loss propagation parameters
on the ASE. Our computer simulations confirm our analysis,
since the simulated values always lie between the predicted
theoretical bounds corresponding to the two extreme inter-
ference configurations. As expected the spectral efficiency
improves as the additional path-loss exponent increases,
since the interfering signals are more attenuated.
Fig. 5 shows plots of the ASE versus the normalized reuse
distance for typical microcellular systems [(a) m and
(b) m] and macrocellular systems [(c) km]. We
see from this figure that the spectral efficiency is increased by
decreasing the cell size. This observation is further investigated
in Fig. 6, where we plot the ASE as a function of the cell
radius. The “ ” points in Fig. 6 correspond to simulation
results, whereas the curves correspond to the best fit in the
mean-square-error sense of these simulated values. We found
that the ASE decreases as an exponential of a fourth-order
polynomial relative to the cell size. Thus, we have exactly
quantified how spectral efficiency of cellular systems increases
as cell size decreases.
Figs. 4 and 5 both indicate that, based on the worst case
interference configuration curves, the optimal reuse distance to
maximize the ASE is four. In fact, the actual optimum occurs
for a reuse distance of about three. However, our statement is
based on the fact that is constrained, by definition, to be
an even number. On the other hand, the best case interference
configuration and the average interference configuration (sim-
ulation) curves do not show an ASE maximum. This implies
that, for typical interference conditions, the spectral efficiency
is maximized by a reuse distance of two (i.e., frequencies
are reused every cell), and is monotonically decreasing for
. Note that because our ASE is defined in terms of
Shannon capacity, this increase in spectral efficiency is not
obtained at the expense of a higher BER. In fact, smaller
frequency reuse results in a higher BER only if the data rate
of the system is not adapted to compensate for the resulting
higher level of cochannel interference [4], [5], [8]. We will see
in the following sections that all these results still hold when
shadowing and fading are taken into account.
The ASE is increased if interference can be reduced while
maintaining the same number of users per cell and the same
reuse distance. Cell sectorization [37] is commonly employed
to accomplish this, whereby directional antennas are used
at the BS. In fully loaded systems, three-sector antennas
(or 120 cell sectorization) reduces the number of primary
cochannel interferers from to . Fig. 7 shows
the improvement in ASE when a 120 cell sectorization is
employed for a m microcellular system. Note the
1054 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 4, JULY 1999
(a)
(b)
(c)
Fig. 5. Comparison of the average uplink area spectral efficiency versus the normalized reuse distance for different cell sizes. (Fully loaded system with
N
I
= 6; carrier frequency f
c
= 900 MHz; propagation parameters: a = b = 2.) (a) R = 200 m, (b) R = 800 m, and (c) R = 5 km.
Fig. 6. Average uplink area spectral efficiency versus cell radius for different reuse distances [(a) R
u
= 4, (b) R
u
= 6, and (c) R
u
= 8] and
carrier frequencies [() f
c
= 900 MHz and () f
c
= 2 GHz]. (Fully loaded system with N
I
= 6; R
o
= 20 m; antenna heights: 10-m BS,
2-m mobile; propagation parameters: a = b = 2.)
ALOUINI AND GOLDSMITH: AREA SPECTRAL EFFICIENCY OF CELLULAR MOBILE RADIO SYSTEMS 1055
Fig. 7. Average uplink area spectral efficiency versus normalized reuse distance with 120 cell sectorization. (Fully loaded system with N
I
= 6; cell radius
R = 200 m; R
o
= 20 m; antenna heights: 10-m BS, 2-m mobile; carrier frequency f
c
= 900 MHz; propagation parameters: a = b = 2.)
ASE increase in both the upper and lower dashed curves
compared to the omnidirectional case (solid curves). Further
improvement on the ASE can be obtained with 60 cell
sectorization (six-sector antennas) since the number of primary
cochannel interferers is reduced in that case to .
V. EFFECT OF SHADOWING
In this section, we consider the ASE of a fully loaded
cellular system when both the desired and interfering users are
affected by lognormal shadowing superimposed on path loss.
This case applies when the BS radio receiver is able to average
out the fast multipath fading, in which case the adaptive system
need only react to lognormal channel variations. This is also
the scenario when an efficient antenna diversity-combining
system is used at the BS to eliminate the effects of multipath
fading.
A. Analyses
The desired user’s signal is assumed to be lognormally
shadowed according to (2) with area mean power and
standard deviation . There are mutually independent
lognormally shadowed interferers, each with mean and
standard deviation . The interferers are assumed to be
statistically identical so that
(22)
(23)
We will refer to all these assumptions from now on as
the independent identically distributed (i.i.d.) hypothesis on
the . Assumptions (22) and (23) hold when all the
interferers are constrained to be on a circle of radius
from the considered BS. Note that these assumptions are not
essential in our derivation of the ASE. Our analyses can be
easily generalized for interferers which are not i.i.d., but this
assumption makes our analyses more tractable.
1) PDF of the Desired User CIR: The total interference
power is the sum of i.i.d. lognormally distributed RV’s.
Although no exact closed-form expression for the PDF of the
sum of lognormally distributed RV’s is known, it is widely
accepted that such a sum can be accurately approximated
by another lognormal distribution [22, Sec. 3.1]. Several
methods have been proposed to find the mean and variance
of the resulting lognormal PDF [38], [39]. A very thorough
description and comparison of these methods is available in
[22, Sec. 3.1]. Here, we use the Fenton–Wilkinson method
[38] for its relative simplicity. According to this method,
the logarithmic mean and the logarithmic variance
of can be found by matching the first- and second-order
moments, which yields
(24)
1056 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 4, JULY 1999
(25)
Since the ratio of two lognormal RV’s is also a lognormal
RV, the desired user CIR, , is also lognormally distributed
with the following logarithmic mean and logarithmic
variance :
(26)
(27)
The PDF of can therefore be written as
(28)
2) Desired User Average Capacity: Inserting (28) in (10),
the average capacity of the desired user can be written as
(29)
Using the following inequality in (29):
(30)
we obtain an upper bound on the desired user average capacity
given by
(31)
On the other hand, using the following inequalities in (29):
(32)
where when and zero otherwise, we
obtain a lower bound on the desired user average capacity
given by
(33)
where is the Gaussian integral function defined as
(34)
3) ASE for the Two Extreme Interference Configurations:
Substituting (29) in (11) yields the ASE for the two extreme
interference configurations in a shadowing environment as
(35)
Similarly substituting (31) and (33) in (11) yields an upper
and a lower bound on the ASE as given by (35) for the two
extreme interference configurations
(36)
Note that , , and are computed using
(37)
and given by (27). Since the expressions in (35) and
(36) are conditioned on the desired mobile position, they
are averaged over the user’s position PDF (7), to obtain the
overall average ASE for the two extreme interference
configurations.
B. Numerical and Simulation Results
Fig. 8 compares the closed-form upper and lower bounds
(36) averaged over (7) with the exact value found by
averaging (35) over (7) with dB for the two
extreme interference configurations. The bounds are very tight
for the best case interference configuration. As increases,
the bounds become tighter. On the other hand, the bounds
are quite loose for the worst interference configuration when
. This is because the inequalities (30) and (32) are
not tight when is small. However, the bounds improve
as increases and become very tight for . The
“ ” points in Fig. 9 correspond to simulation results obtained
using an algorithm similar to the one described in Section IV-
B except for step 4), which is changed to incorporate the
effects of shadowing. The changes in the simulation algorithm
are described in detail in Appendix A-1. Note again that
the simulated values corresponding to average interference
configurations always lie between the predicted theoretical
values for the two extreme interference conditions.
Fig. 9 compares the ASE with and without shadowing. We
see in all cases that the ASE curves for different interference
configurations have the same relative shape. However, the
ASE with shadowing is always smaller. We will see in the
next section that multipath fading also reduces the ASE.
VI. EFFECT OF MULTIPATH FADING
In this section, we consider the ASE of fully loaded cellular
systems with Nakagami fading superimposed on path loss.
Specifically, we study how the ASE is affected by different
amount of fading on the desired and interfering users. This is
ALOUINI AND GOLDSMITH: AREA SPECTRAL EFFICIENCY OF CELLULAR MOBILE RADIO SYSTEMS 1057
Fig. 8. Comparison of the bounds and the exact values for the average uplink area spectral efficiency with shadowed users () with
d
=
I
= 4
dB. (Fully loaded system with N
I
= 6; cell radius R = 200 m; R
o
= 20 m; antenna heights: 10-m BS, 2-m mobile; carrier frequency f
c
= 900
MHz; propagation parameters: a = b = 2.)
Fig. 9. Comparison of the average uplink area spectral efficiency versus normalized reuse distance with nonshadowed users () and shadowed users
() with
d
=
I
= 4 dB. (Fully loaded system with N
I
= 6; cell radius R = 200 m; R
o
= 20 m; antenna heights: 10-m BS, 2-m mobile;
carrier frequency f
c
= 900 MHz; propagation parameters: a = b = 2.)
of particular interest for microcellular environments in which
the desired signal typically experiences less severe fading then
the cochannel interfering signals.
A. Analyses
The fading of the desired user’s signal has a Nakagami
parameter and a local mean power . There are inter-
ferers, each with mutually independent fading, with Nakagami
fading parameter and mean power . The interferers are
assumed to be statistically identical so that the ’s are i.i.d.
(38)
(39)
Assumption (38) holds when all the interferers are con-
strained to be on a circle of radius from the considered
1058 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 4, JULY 1999
BS. This assumption is convenient for deriving the ASE in
the worst case and best case interference configurations.
1) PDF of the Desired User CIR: Let denote the
characteristic function (CF) of the received power of the th
interferer. As a CF of a gamma RV, is given by
(40)
Since we assume that the are i.i.d., the CF of is just
raised to the power
(41)
Thus, Nakagami interferers are equivalent to a single
Nakagami interferer with parameter and with local
mean power . Therefore, is gamma distributed with
a PDF given by
(42)
Since and are independent, the PDF of is given by
[36, p. 138, eq. (6–43)]
(43)
Substituting (5) and (42) in (43), the PDF of is found using
[24, p. 364, eq. (3.381.4)]
(44)
where is the beta function defined by [24, p. 957, eq.
(3.380.1)] as
(45)
and related to the gamma function by
(46)
Defining the parameters and
, we can rewrite (44) in a more compact form
as
(47)
Note that for the special case where and are restricted
to be positive integers, reduces to
(48)
2) Desired User Average Capacity: Inserting (48) in (10),
the average capacity of the desired user can be written as
(49)
The integral in (49) can be viewed as the th-order generalized
Stieltjes transform (GST) [40, p. 213], , of the
function . Thus, can be written as
(50)
The GST as well as some of its properties that will be useful in
our analyses are reviewed in Appendix B. We restrict ourselves
in this section to integer values of , but we do not put any
constraint on . The evaluation of is derived
in Appendix C and is given in (82). Using that result, we can
rewrite as
(51)
where is the Gauss’ hypergeometric function de-
fined as [24, p. 1065, eq. (9.100)]
(52)
and denotes the binomial coefficient.
For the special case of [i.e., ],
using [24, p. 589, eq. (4.293.14)] reduces to
(53)
where is the Euler’s psi function defined by [24, p. 952,
eq. (8.360)]
(54)
Note that in this special case, (53) applies even if
is not restricted to be an integer [contrary to (51)]. If
is restricted to be an integer, we can apply [24, p. 954, eq.
(8.365.7)] to (53) to get a very simple expression for as
(55)
ALOUINI AND GOLDSMITH: AREA SPECTRAL EFFICIENCY OF CELLULAR MOBILE RADIO SYSTEMS 1059
(a) (b) (c)
Fig. 10. Effect of the desired users’ amount of fading on the average uplink area spectral efficiency. (Fully loaded system with N
I
= 6; cell radius
R = 200 m; R
o
= 20 m; antenna heights: 10-m BS, 2-m mobile; carrier frequency f
c
= 900 MHz; propagation parameters: a = b = 2.) (a)
m
d
= 1; m
i
= 1, (b) m
d
= 2; m
i
= 1, and (c) m
d
= 3; m
i
= 1.
3) ASE for the Two Extreme Interference Configurations:
Substituting (51) in (11), we get the ASE averaged over
the multipath fading and conditioned on the desired mobile
position as
(56)
where
(57)
Integrating (56) over the user’s position PDF (7) yields the
overall average ASE for the two extreme interference
configurations.
B. Numerical and Simulation Results
Figs. 10 and 11 show the ASE for the two extreme interfer-
ence configuration computed numerically for different values
of the Nakagami fading parameter. These figures also show
the exact value of based on a Monte Carlo simulation
algorithm using the same system and fading parameters.
The simulation algorithm is similar to the one described
in Section IV-B except for step 4), which is changed to
incorporate the effects of Nakagami fading. The changes in
the simulation algorithm are described in detail in Appendix
A-2.
First, note again that our computer simulations confirm our
analyses, since the simulated values always lie between the
predicted theoretical bounds. Comparing these ASE results
with our ASE results without fading we see that both sets of
ASE curves have the same relative shape, although the ASE
with fading is always smaller. Recall that the same behavior
was observed for shadowing.
Fig. 10 shows how the desired user’s affects
ASE by fixing (Rayleigh fading) and setting to one
[Fig. 10(a)], two [Fig. 10(b)], or three [Fig. 10(c)]. We see that
the ASE curves with fading (dashed lines) approach the ASE
curves without fading (solid lines) as increases. Hence,
as the channel quality between the user and its BS improves
(i.e., severity of fading decreases), the system average ASE
increases.
Fig. 11 shows how the interferers affects
the ASE by fixing to be three and setting to one
[Fig. 11(a)], two [Fig. 11(b)], or three [Fig. 11(c)]. The inter-
ferers’ seems to have little impact, since the average ASE
is about the same (in fact, slightly decreasing with ) for
all cases in Fig. 11. Hence, the ASE is predominantly affected
by the channel quality of the desired users, rather than by the
fading parameters of the interferers.
VII. COMBINED EFFECT OF
SHADOWING AND MULTIPATH FADING
In this section, we consider the ASE of fully loaded cellular
systems in a shadowed/Nakagami fading environment, consist-
1060 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 4, JULY 1999
(a) (b) (c)
Fig. 11. Effect of the cochannel interferers’ amount of fading on the average uplink spectral efficiency. (Fully loaded system with N
I
= 6; cell radius
R = 200 m; R
o
= 20 m; antenna heights: 10-m BS, 2-m mobile; carrier frequency f
c
= 900 MHz; propagation parameters: a = b = 2.) (a)
m
d
= 3; m
i
= 1, (b) m
d
= 3; m
i
= 2, and (c) m
d
= 3; m
i
= 3.
ing of Nakagami multipath fading superimposed on lognormal
shadowing and two-slope path loss. This is typically the
scenario in congested downtown areas with a high number of
slow-moving pedestrians and vehicles. Under these conditions,
the system does not average out the envelope fading due to
multipath, but rather adapts to the instantaneous composite
shadowed/faded signal power.
A. Analyses
The desired user’s signal is assumed to follow a Nakagami
distribution with parameter and a slowly varying local
mean power , which is itself lognormally distributed with
area logarithmic mean and logarithmic standard deviation
. There are i.i.d. Nakagami-faded interferers, each with
parameter and a lognormally distributed local mean power
with area logarithmic mean and logarithmic variance .
Combining (2) and (5), we see that the composite shad-
owed/faded received signal power for all users follows a
gamma/lognormal PDF given by [22, p. 92, eq. (2.187)]
(58)
This integral form of the PDF requires numerical techniques
for solution and becomes computationally burdensome when
further analysis is required. Fortunately, Ho and Stu¨ber [22,
p. 92] showed that the composite gamma/lognormal PDF
can be accurately approximated by another lognormal PDF
with logarithmic mean and logarithmic variance . These
parameters are obtained by matching the first two moments of
(58) with the first two moments of a lognormal approximation
giving [22, p. 106]
(59)
(60)
where is the Euler’s psi function defined in (54), and
is the generalized Reimann’s zeta function defined by
[24, p. 1101, eq. (9.521.1)]
(61)
Therefore, all the analyses of Section V applies in this case,
with the substitution in all the expressions of the ’s by ’s
and the ’s by ’s, for the desired user as well as for the
cochannel interferers.
B. Numerical and Simulation Results
The combined effect of shadowing and Nakagami fading
on the ASE for the two extreme/average interference config-
urations is shown in Figs. 12 and 13 with different sets of
shadowing and fading parameters. The ASE for the average
interference configuration is determined using a simulation
algorithm similar to the one used previously except for the
generation of the desired ( ) and interfering ( ) signal
powers, which are obtained as described in Appendix A-3.
Comparing these results with Figs. 10 and 11, we see
that the ASE curves still conserve the same relative shape.
ALOUINI AND GOLDSMITH: AREA SPECTRAL EFFICIENCY OF CELLULAR MOBILE RADIO SYSTEMS 1061
(a) (b)
Fig. 12. Combined effect of shadowing (
d
=
I
= 4 dB) and Nakagami fading on the average uplink area spectral efficiency. (Fully loaded system
with N
I
= 6; cell radius R = 200 m; R
o
= 20 m; antenna heights: 10-m BS, 2-m mobile; carrier frequency f
c
= 900 MHz; propagation parameters:
a = b = 2.) (a) = 4 dB; m
d
= 1; m
i
= 1 and (b) = 4 dB; m
d
= 3; m
i
= 1.
(a) (b)
Fig. 13. Combined effect of shadowing (
d
=
I
= 6 dB) and Nakagami fading on the average uplink area spectral efficiency. (Fully loaded system
with N
I
= 6; cell radius R = 200 m; R
o
= 20 m; antenna heights: 10-m BS, 2-m mobile; carrier frequency f
c
= 900 MHz; propagation parameters:
a = b = 2.) (a) = 6 dB; m
d
= 1; m
i
= 1 and (b) = 6 dB; m
d
= 3; m
i
= 1.
However, simulations show clearly that for the same amount
of multipath fading on the desired and interfering mobiles,
shadowing further reduces the system ASE. It is reported in
[5], [19], and [39] that the Fenton–Wilkinson method loses
its accuracy for dB and leads to optimistic results
for the cochannel interference calculations in that case. This,
combined with the fact that the Ho and Stu¨ber approximation
increases the shadow standard deviation to incorporate the
effect of Nakagami fading (60), explains the slightly high
theoretical predictions of the ASE under the combined impact
1062 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 4, JULY 1999
of shadowing and fading. More accurate ASE predictions can
be obtained for high shadow standard deviation if the Schwartz
and Yeh’s method [39] as reviewed by Prasad and Arnbak [19]
is used. This remark also apply for Section V when dB.
Note, however, that all our analyses for Sections V and VII
would still apply in that case except that (24) and (25) should
be recomputed according to the Schwartz and Yeh’s recursive
technique [19], [39].
VIII. PARTIALLY LOADED SYSTEMS
In Section III, we have introduced the ASE for fully loaded
cellular systems in which: 1) the cell’s serviced channels
are all used and 2) the number of interferers is constant and
equal to . In this section, we consider partially loaded
systems in which the cell’s serviced channels and the number
of interferers are random variables depending on the traffic
loading. In what follows, we first briefly describe a fixed
channel assignment scheme. We then study the effect of
traffic loading on the ASE when this fixed channel assignment
scheme is employed.
A. Channel Assignment Scheme
Each cell has a dedicated and constant number of serviced
channels with the same bandwidth [Hz].
In this scheme, if free channels are available in the cell,
the BS allocates randomly one of its free channels to a new
connection request or handoff attempt. On the other hand, if
all serviced channels are busy, any new/handoff connection is
blocked or dropped. We assume that the calls are not dynam-
ically ordered/rearranged in order to minimize the number of
cochannel interference channels. We also assume that queuing
is not provided for connection requests which have been
initially blocked or dropped. This channel allocation scheme is
obviously quite simple. Besides making our analyses tractable,
this schemes offers the advantage of being “instantaneous” (no
delay in calls rearrangement), fully distributed, and of very
low complexity.
The serviced channels are considered to be independent
and active with the same probability . Hence, the block-
ing/dropping probability is given by . Equiv-
alently, we have . Thus, the probability that
serviced channels are active has a binomial distribution,
given by
(62)
where denotes the binomial coefficient.
For a cell with active users, let the number of active
cochannel interferes of the th desired user be denoted by ,
and let be the vector . We assume
that the are mutually independent. Hence, the joint
distribution of the , , is given by
(63)
In addition, the are assumed to be independent from ,
and the traffic loading conditions are considered to be uniform
so that the blocking/dropping probability is the same for all
the cells (desired cell and cochannel interfering cells). Hence,
the are identically distributed according to a binomial
distribution given by [3]
(64)
B. Analyses
Recall that in partially loaded systems the number of active
users and the number of active cochannel interferers
are random variables. Let [b/s/Hz/m ] denote the
ASE conditioned on and and given by
(65)
Averaging (65) over the joint distribution of the ,
, yields
(66)
Substituting (63) in (66) yields
(67)
Using the fact that all users are assigned the same bandwidth,
and that the are independent from and are identically
distributed according to , (67) simplifies to
(68)
Averaging (68) over the binomial distribution of (62) yields
the average ASE for a partially loaded system as a function
of , , and as
(69)
Integrating (69) over the user’s position PDF (7) yields the
overall average ASE, , for a partially
loaded system and for the two extreme configurations. Recall
that the traffic loading is directly proportional to the blocking
probability . In the limit of heavy-loaded traffic conditions,
, so that , where denotes
the Kro¨necker delta function which is equal to one for
and zero otherwise. As expected the ASE in this
case , corresponding to
the ASE of a fully loaded system.
ALOUINI AND GOLDSMITH: AREA SPECTRAL EFFICIENCY OF CELLULAR MOBILE RADIO SYSTEMS 1063
Fig. 14. Comparison of the average uplink area spectral efficiency versus the normalized reuse distance for different traffic loading conditions. (Interferers
N
I
= 6; cell radius R = 200 m; R
o
= 20 m; antenna heights: 10-m BS, 2-m mobile; carrier frequency: 900 MHz; propagation parameters: a = b = 2.)
C. Numerical and Simulation Results
The effect of system loading (as parameterized by ) on the
ASE for the two extreme/average interference configurations
is shown in Fig. 14 for a microcellular radio system with a cell
radius m. Since the effect of shadowing and fading
has been thoroughly studied in previous sections, only path
loss has been taken into account in computing the desired
user average capacity in (69). These results can be
easily generalized to include the effects of shadowing and
fading by recomputing in (69) according to (29) and
(51), respectively. The “ ” points in Fig. 14 correspond to
simulation results obtained using an algorithm similar the one
described in Section IV-B except that the number of cochannel
interferers is randomly picked following (64). Partially
loaded systems have a lower spectral efficiency than fully
loaded systems since the ASE increases as the blocking prob-
ability increases. Therefore, although the reduced interference
to the fewer desired users allows these users to achieve higher
rates in partially loaded systems, this effect is more than offset
by the fact that with fewer active desired users, only a fraction
of the cell allocated bandwidth is used.
IX. CONCLUSION
We have presented a general analytical framework to quan-
tify the overall spectral efficiency of cellular systems in which
mobile users continuously adapt their rate relative to the
variation in their respective carrier-to-interference ratio. This
efficiency, named the area spectral efficiency, and defined
as the sum of the maximum average data rates/unit band-
width/unit area supported by a cell’s base station, captures
the tradeoff between a cellular system’ spectral efficiency, the
link spectral efficiency, and the communication link quality
provided to the users. We have calculated this efficiency for
FDMA and TDMA systems as a function of the reuse distance
for the worst case and best case interference configurations. We
have also developed Monte Carlo simulations to estimate the
value of this efficiency for average interference conditions.
Our theoretical analyses are in good agreement with the
corresponding simulated values.
We haven take into account the effect of the users random
location in their respective cells and have presented numerical
results showing the impact of propagation parameters, cell
size, carrier frequency, and cell sectorization on the spectral
efficiency. Results indicate that, based on the worst case
interference configuration, the optimal reuse distance is ap-
proximately four. However, this optimal reuse distance is two
for the best case and the average interference configurations
(i.e., frequencies should be reused every cell). Moreover, sim-
ulation show that area efficiency decreases as an exponential
of a fourth-order polynomial relative to the cell size.
We have also studied the area spectral efficiency under the
influence of lognormal shadowing and Nakagami multipath
fading superimposed on path loss. Results show that both
shadowing and fading reduce the area spectral efficiency, but
do not affect its general behavior relative to the reuse distance.
In addition, the spectral efficiency is predominantly affected
by the fading parameters of the desired users, rather than by
the fading parameters of the interferers.
This paper also presented initial steps toward the determina-
tion of the area spectral efficiency of partially loaded systems.
Results reveal that heavier traffic loading (i.e., higher blocking
probability) leads to a higher area spectral efficiency when a
simple fixed channel allocation scheme. A study with more
sophisticated channel allocation schemes is being conducted
so as to assess the impact on the area spectral efficiency.
1064 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 4, JULY 1999
Our results are useful for the prediction of the spectral
efficiency of cellular systems with variable-rate transmission
under various “realistic” conditions. Furthermore, they provide
system engineers with valuable input information for the
efficient design, planning, and dimensioning of such systems.
In particular, these results give guidelines for optimizing reuse
distance and cell size.
APPENDIX A
MONTE CARLO SIMULATIONS
A-1. Shadowing
Step 4) of the algorithm described in Section IV-B is
changed as follows to incorporate the effect of shadowing.
• The area mean powers ( and ’s) at the
considered BS are calculated using the two-slope path
model (1).
• The instantaneous received power from the desired and
interfering mobiles ( and ’s) are randomly generated
according to a lognormal distribution (2) with area means
and ’s, respectively, and standard deviations and
, respectively.
Since we need to average out the additional effect of shad-
owing a higher number of iterations is required (typically
100 000) than in Section IV-B for the same degree of accuracy
in the estimation of .
A-2. Multipath Fading
Step 4) of the algorithm described in Section IV-B is
changed as follows to incorporate the effect of Nakagami
multipath fading.
• The local mean powers ( and ’s) at the
considered BS are calculated using the two-slope path
model (1).
• The instantaneous received powers from the desired and
interfering mobiles ( and ’s) are randomly generated
according to a gamma distribution (5) with fading param-
eters and , respectively, and local mean powers
and ’s, respectively.
The same number of iterations as in the shadowing case
is typically required to average out the additional effect of
multipath and to estimate within a three-digit accuracy.
A-3. Combined Effect of Shadowing and Multipath Fading
Step 4) of the algorithm described in Section IV-B is
changed as follows to incorporate the combined effect of
shadowing and Nakagami multipath fading.
• The area mean powers ( and ’s) at the
considered BS are calculated using the two-slope path
model (1).
• The slowly varying local mean powers from the desired
and interfering mobiles ( and ’s) are randomly gen-
erated according to a lognormal distribution (2) with area
means and ’s, respectively, and standard deviations
and , respectively.
• The instantaneous received power from the desired and
interfering mobiles ( and ’s) are randomly generated
according to a gamma distribution (5) with fading param-
eters and , respectively, and local mean powers
and ’s, respectively.
Since we need to average out the effect of both shadowing
and Nakagami fading a higher number of iterations is required
(typically 300 000 iterations) than in the previous cases for the
same degree of accuracy in the estimation of .
APPENDIX B
SOME PROPERTIES OF THE GENERALIZED
STIELTJES TRANSFORM (GST)
In this Appendix, we review some basic properties of the
GST. We restrict our review to the properties that are useful in
our derivation of the ASE in Nakagami fading environment.
A more detailed summary of the GST and its properties can
be found in [40, ch. XIV].
The GST of order of the function is defined by
(70)
where and can be complex variables. The relationship
between and can also be expressed symbolically
by a GST transform pair as
(71)
Theorem 1—Differentiation Property [40, p. 233, eq. (5)]:
Assuming (71) then
(72)
Theorem 2—Multiplication by Property [40, p. 233, eq.
(2)]: Assuming (71) then
(73)
Theorem 3—A Useful GST Pair [40, p. 233, eq. (9)]: For
and assuming (71) then
If then
(74)
where is Gauss’ hypergeometric function defined
in (52).
APPENDIX C
EVALUATION OF
In this Appendix, we evaluate the GST of
. To simplify our notations, let us denote
by . First, note that
(75)
ALOUINI AND GOLDSMITH: AREA SPECTRAL EFFICIENCY OF CELLULAR MOBILE RADIO SYSTEMS 1065
so that
(76)
Using the differentiation property of the GST (72), note that
the second term vanishes so that
(77)
Using the GST transform pair (74) applied (for ) to (76),
then substituting the result in the left-hand side of (77), yields
(78)
where is defined in (52). Note that since
and , we have
and therefore (74) always applies with no singularities.
As a second step of our derivation, we apply the property
(73) to . This provides a difference equation given by
(79)
Then it can be show by induction that for positive integer
(80)
is solution for the difference equation (79) where
denotes the binomial coefficient given by
(81)
Finally, inserting (78) in (80), we find as
(82)
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Mohamed-Slim Alouini (S’94–M’99) was born
in Tunis, Tunisia. He received the “Diploˆme
d’Inge´nieur” degree from the Ecole Nationale
Supe´rieure des Te´le´communications (TELECOM
Paris), Paris, France, and the “Diploˆme d’Etudes
Approfondies (D.E.A.)” degree in electronics from
the University of Pierre & Marie Curie (Paris VI),
Paris, both in 1993. He received the M.S.E.E. degree
from the Georgia Institute of Technology (Georgia
Tech), Atlanta, in 1995 and the Ph.D. degree in
electrical engineering from the California Institute
of Technology (Caltech), Pasadena, in 1998.
While completing his D.E.A. thesis, he worked with the optical submarine
systems research group of the French national center of telecommunications
(CNET-Paris B) on the development of future transatlantic optical links.
While at Georgia Tech, he conducted research in the area of K
a
-band
satellite channel characterization and modeling. From June 1998 to August
1998, he was a Post-Doctoral Fellow with the Communications Group
at Caltech carrying out research on adaptive modulation techniques and
on CDMA communications. He joined the Department of Electrical and
Computer Engineering, University of Minnesota, Minneapolis, in September
1998, where his current research interests include statistical modeling of
multipath fading channels, adaptive modulation techniques, diversity systems,
and digital communication over fading channels.
Dr. Alouini is the recipient of a National Semiconductor Graduate
Fellowship Award and the Charles Wilts Prize for outstanding independent
research in electrical engineering leading to a Ph.D. degree at Caltech. He is
a member of the IEEE Communications and Vehicular Technology Societies.
Andrea J. Goldsmith (S’94–M’95) received
the B.S., M.S., and Ph.D. degrees in electrical
engineering from the University of California,
Berkeley, in 1986, 1991, and 1994, respectively.
From 1986 to 1990, she was with Maxim
Technologies, where she worked on packet radio
and satellite communication systems. From 1991 to
1992, she was with AT&T Bell Laboratories, where
she worked on microcell modeling and channel
estimation. She was an Assistant Professor of
Electrical Engineering at the California Institute
of Technology, Pasadena, from 1994 to 1998 and is currently an Assistant
Professor of Electrical Engineering at Stanford University, Stanford, CA.
Her research includes work in capacity of wireless channels, wireless
communication theory, adaptive modulation and coding, joint source and
channel coding, and resource allocation in cellular systems.
Dr. Goldsmith is the recipient of the National Science Foundation CAREER
Development Award, the ONR Young Investigator Award, two National
Semiconductor Faculty Development Awards, an IBM Graduate Fellowship,
and the David Griep Memorial Prize from the University California at
Berkeley. She is an Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS
and the IEEE PERSONAL COMMUNICATIONS MAGAZINE.




























































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































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