IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 49, NO. 3, MAY 2000
965
Truncated Power Control in Code-Division
Multiple-Access Communications
Sang Wu Kim, Senior Member, IEEE, and Andrea J. Goldsmith, Senior Member, IEEE
Abstract—We analyze the performance of truncated power
control in a code-division multiple-access (CDMA) communication
system. This power control scheme compensates for propagation
gain above a certain cutoff fade depth: below the cutoff level,
data transmission is suspended. We assume a channel with fast
Rayleigh fading and slow lognormal shadowing plus path loss,
where truncated power control is applied to the shadowing plus
path loss and a RAKE receiver combines the Rayleigh fading
multipath components separated by more than a chip time. We
find that truncated power control exhibits both a power and
capacity gain. These performance improvements result from
the fact that with truncated power control, less power is wasted
compensating for deep fading, and mobiles on the cell boundaries
cause less interference to adjacent cells. We find that truncated
power control is most effective for channels with large power
fluctuations or with large background noise.
In this paper, we consider a truncated power control scheme
which mitigates these undesirable effects. Truncated power control compensates for fading above a certain cutoff fade depth
so that
(1)
is the desired received power when
. The
where
is adjusted to meet the target bit error probability
value of
, there
given the current interference conditions. For
at the mobile
is no transmission. The average transmit power
handset corresponding to truncated power control (1) is given by
Index Terms—CDMA, power control, Rayleigh fading.
(2)
I. INTRODUCTION
I
T IS WELL known that code-division multiple-access
(CDMA) systems that employ correlation receivers suffer
from the near–far problem: the problem of a strong signal
masking out a weaker signal, causing unreliable detection of
the latter. In addition, the channel power gain for each mobile
fluctuates due to multipath fading and shadowing [1], which
gives rise to an additional performance degradation. A solution
to the near–far problem and the channel gain fluctuation is the
use of power control, which ensures that the received signal
strength of all users is equal [2].
The conventional way of maintaining a desired signal
strength at the receiver is that each transmitter estimates
at time
and inverts the
the channel power gain
as
channel power gain by adjusting the transmit power
, where
is the target received signal
power. This power control ensures that all mobile signals are
and is done in current CDMA
received with the same power
cellular systems using both open- and closed-loop techniques
[2]–[4]. However, this conventional power control scheme
requires a large average transmit power to compensate for deep
fades. It also exacerbates the interference caused by boundary
mobiles to adjacent cells, since they typically must transmit at
a very high power to compensate for path loss and shadowing
effects.
Manuscript received December 22, 1997; revised October 16, 1998.
S. W. Kim is with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea (e-mail:
swkim@san.kaist.ac.kr).
A. J. Goldsmith is with the Department of Electrical Engineering, Stanford
University, Stanford, CA 94305-9515 USA (e-mail: andrea@ee.stanford.edu).
Publisher Item Identifier S 0018-9545(00)02572-X.
is the expectation of
over
.
where
Truncated power control reduces to conventional power control
.
for a cutoff value of
Truncated power control has been shown to increase the spectral efficiency of nonspread-spectrum single-user systems in [5].
In this paper, we consider the use of truncated power control in
CDMA systems with slow lognormal shadowing, path loss, and
fast Rayleigh fading. Since fast fading is typically difficult to
track, we assume that our truncated power control scheme tracks
the lognormal shadowing and path loss, but not the Rayleigh
fading. We therefore use a RAKE receiver to mitigate the effects of the fast fading.
The motivation for applying truncated power control to these
systems is as follows. First, mobiles on the cell boundaries
are more likely to have buildings blocking their line-of-sight
to the base station and are therefore more likely to experience
deep shadowing. We thus expect truncated power control to
reduce the interference caused by mobiles on cell boundaries,
since these mobiles will often be in outage and therefore will
not contribute to adjacent channel interference. In addition,
truncated power control reduces the required average transmit
power, since it need not compensate for deep fading. We
will see these effects result in a capacity increase relative to
conventional power control. We will also see that truncated
power control provides a power gain over conventional power
control for our channel model; this model consists of a Rayleigh
fading channel with RAKE combining superimposed onto a
lognormally shadowed channel, with the power adapted to
the shadow fading and path loss. The power gain translates
into a reduction of the required transmit power for a given
performance. The required transmit power for CDMA systems
is often neglected since the system is assumed to be interference
0018–9545/00$10.00 © 2000 IEEE
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 49, NO. 3, MAY 2000
limited; thus scaling the average transmit power of all users has
no effect, since both the signal and interference power scale
accordingly. However, the noise power must also be factored
into the performance calculation, and we will see that when
both interference and noise are taken into account, truncated
power control reduces the average transmit power required to
achieve a given performance. However, this reduction comes at
the expense of an increase in truncation probability, defined as
the probability that the channel is below the cutoff fade depth
.
We assume that the transmission rate during active (i.e., nonsuch that the
truncated) conditions is increased relative to
data rate averaged over both active and nonactive conditions is
the same for all . Thus, changing the truncation threshold
does not impact this average data rate, which corresponds to the
average rate experienced by a user moving throughout the coverage region.
The remainder of this paper is organized as follows. The
system model is described in Section II. In Section III, we
describe the truncated power control scheme for single-cell
systems and derive its outage probability as well as its power
gain relative to conventional power control. In Section IV, we
investigate the outage probability and the capacity gain provided by the truncated power control in multiple-cell systems.
Our conclusions are presented in Section V.
II. SYSTEM MODEL
We consider a direct sequence CDMA system with binary
phase shift-keyed (BPSK) signaling. We begin by considering a
single cell, which can also serve as a model for a satellite system
whose cell site is a single hub. In Section IV, we will extend this
analysis to multicell systems. We assume that there are users
at time consists of
in the system. The channel power gain
, where
is the gain astwo components,
is the propagation
sociated with fast Rayleigh fading and
gain associated with slow lognormal shadowing and path loss
[1]. We model the path loss as proportional to the th power
is estimated perfectly and the
of distance. We assume that
transmit power is adjusted to fully compensate for this compochanges
nent of the channel gain. We further assume that
sufficiently fast so that the transmitter cannot compensate for it.
changes much slower than
.
Normally,
and
are effectively constant over a
We assume that
block of symbols (slowly fading), so that the random processes
and
can be replaced by random variables and ,
respectively. This model is appropriate when the transmitter, receiver, and all reflecting surfaces are slowly moving relative to
the carrier wavelength and symbol rate. The propagation gain
is generally modeled as
(3)
where
constant that depends on the antenna gains, the signal
wavelengths, etc. ( is normalized to unity in this
paper);
path-loss exponent;
zero-mean Gaussian random variable with variance ;
normalized distance between the base station and mobile.
We assume that a mobile’s location is uniformly distributed
within a cell. This leads to a probability density function (pdf)
of given by
.
It is well known that the pdf of
(4)
is given by
(5)
.
where
We assume that the channel is frequency selective with respect to the spreading bandwidth and exhibits slow Rayleigh
at the base station can be repfading. The received signal
resented by
(6)
where
number of users in a cell;
number of resolvable paths;
propagation gain for user ;
exponential random variable representing the multipath fading for user on the th path with
for all and .
are statistically independent for
We will assume that
all and , and all the Rayleigh paths of each user have the
is the transmit power of
same shadowing and path loss.
user when the propagation gain associated with shadowing
, and
and
are the data signal
and path loss is
and the spreading signal for user , respectively. We assume
and
are either 1 or 1 at any time (rectanthat
gular pulse shape) and that BPSK modulation is employed.
represents the path delay for user on the th path, and
is
represents the zero-mean white Gaussian
the carrier phase.
. We assume
noise with two-sided power spectral density
an -branch RAKE combiner with maximal-ratio combining
of the Rayleigh fading paths.
III. TRUNCATED POWER CONTROL
In this section, we determine the outage probability and
power gain of truncated power control applied to single-cell
systems.
A. Power Gain
With truncated power control, the transmit power is adjusted
to compensate for the shadowing and path loss above a certain
cutoff fade depth . In implementing this technique, the th
transmitter estimates the propagation gain ( ) associated with
slow lognormal shadowing and path loss and adjusts its transmit
such that
power
(7)
KIM AND GOLDSMITH: TRUNCATED POWER CONTROL IN CODE-DIVISION MULTIPLE-ACCESS COMMUNICATIONS
967
where
is the received signal power when
and the
is adjusted to meet the target bit error probability.
value of
is given by
The average transmit power
(8)
where
(9)
(10)
with
Fig. 1. Power gain G versus the truncation probability 1
values of ; m = 4.
0 p( ) for several
In (9), we assume that random variables and are independent. Since the transmitter does not transmit (i.e., no service) if
, the activity factor (or fraction of time during which
is given by
the mobile transmits)
(11)
depends on the cutoff fade depth
Since the activity factor
and the received signal power
can be controlled either by
or , the transmit power can be traded
the transmit power
with the activity factor in maintaining a desired received signal
power.
bits/s
In order to maintain a constant average data rate
during
independent of the activity factor, the bit duration
must be set to
, since the data transmisis zero. As a result, the processing
sion rate when
, where
is the chip duration, changes as
is
gain
changed. We will assume that is fixed in this paper. Then, the
when
is
average received energy per bit
(12)
. It follows from (12) and the fact that
that the power gain
over the conventional
) is
power control (
where
(13)
Fig. 2. Power gain G versus the truncation probability 1
values of m; = 8 dB.
0 p( ) for several
(14)
Notice that as is increased, i.e., as the shadow fading varies
over a wider range, the power gain is higher. This indicates that
the truncated power control scheme is most effective for channels with large power fluctuations. This result is expected since
channels with large power fluctuations will experience deep
fades more often, and conventional power control uses a great
deal of power to compensate for these deep fades. Figs. 1 and
versus the truncation proba2 are plots of the power gain
for several values of and . When is 8 dB,
bility
a typical value in terrestrial cellular systems [3], Fig. 1 indicates
that the power gain is 1.3–4.5 dB for the truncation probability
in the range of 10
10 . It should also be noticed
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 49, NO. 3, MAY 2000
from Fig. 2 that the power gain is higher for larger
path loss is higher.
, i.e., as the
B. Outage Probability
A coherent correlation receiver recovering the signal of user
on the th path forms a decision statistic
, given by
when the number of resolvable paths is one. However, user ’s
is also a random variable due to the slow variations in
and . Therefore, for a given target BER , by inverting (20)
such that user ’s BER does not exceed
we can find
BER as long as
(21)
When condition (21) is not met, the system is considered to be
in outage [1]. Let
if
if
(15)
where is the data bit duration and is the data bit for user ,
taking values of 1 or 1. The first term in (15) is the desired
represents the multipath intersignal term. The second term
ference of the user of interest, and the third term
.
(22)
Then
and
(24)
(16)
is a random
represents the other users’ interference, where
. Because
is typvariable, uniformly distributed over
, especially for
,
will be
ically much smaller than
, when averaged
ignored in what follows. The variance
over path delays, channel induced phase, and random spreading
codes, is given by [6]
is the processing gain with the conventional
where
. Therefore, the outage probpower control scheme
is given by
ability
(25)
(26)
(17)
(27)
The fourth term in (15) represents the background white
. The RAKE
Gaussian noise of mean zero and variance
receiver uses maximal ratio combining, thereby forming the
of user given by
decision statistic
(18)
(28)
where
(29)
Therefore, the average (over fast fading) bit energy-to-equivaat the RAKE receiver
lent noise spectral density ratio
output for user is given by
(19)
The bit error rate (BER) of user is obtained by averaging his
BER in AWGN over the fast fading distribution given average
. For example, if BPSK modulation is assumed and the
fast fading is Rayleigh, then the average BER is given by [7]
BER
(20)
Note that the outage is caused by either the truncation of the
desired user [the first term in (27)] or the multiple-access interference exceeding a threshold [the second term in (27)]. If we
assume that call arrivals occur randomly at Poisson-distributed
intervals with an average rate of calls/s and that the average
s, then
is a Poisson-distributed random
call duration is
. Also, the random varivariable with the occupancy rate
is a Poisson-distributed random variable with
able
[8]. Thus, the outage probability
the occupancy rate
is
(30)
KIM AND GOLDSMITH: TRUNCATED POWER CONTROL IN CODE-DIVISION MULTIPLE-ACCESS COMMUNICATIONS
969
Fig. 3. Hexagonal cell boundaries and distances.
(31)
where
station. Consequently, the average received power
) is
to a user at coordinate (
due
is the integer part of .
(34)
IV. MULTIPLE-CELL SYSTEMS
and
are independent.1 If we let
and
, where
is the distance from the user at coordinates
to
is the distance between the
the nearest base station and
and the given cell’s base station, then
user at
where we assumed that
In this section, we investigate the outage probability and the
capacity gain that the truncated power control provides over
the conventional power control in multiple-cell systems. We assume a multiple-cell system with hexagonal cell, as illustrated
in Fig. 3. The capacity gain is evaluated under the assumption
of a large number of users, so that the interference is nearly at
its average value by the weak law of large numbers [8], [9].
We assume a uniform density of users throughout all cells, and
the handoff occurs exactly at the boundary between cells (hard
handoff). Power control attempts to equalize users’ received
signal power at a given cell’s base station for all users controlled
by that base station. But interference also arrives from users controlled by other cells’ base stations.
The average received power at the given cell’s base station
due to the users in the given cell, given users per cell, is
(35)
and
(36)
Therefore, the average received power
other cells is
due to all users in all
(32)
(33)
(37)
(38)
The received power due to a user in another cell located at a distance from the given cell’s base station (the base station in the
shaded cell in Fig. 3) is
, where
is the propagation
to the given cell’s base
gain for the user at coordinates
1If we assume that L and L are correlated, then a lower other-cell interference I (thus a higher capacity) would be obtained.
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 49, NO. 3, MAY 2000
where
(39)
is the entire region outside the given cell, which consists
and
is the density of users
of all other cells. The factor
[users/unit area] for the hexagonal shape of the normalized cell.
at the decorreTherefore, the equivalent noise density
lator output is
(40)
is the chip rate, i.e., the total bandwidth, and a factor
where
of 1/3 is included to account for the random path delays and
the pulse shape. Since the desired signal power during
is
and the bit duration during
is
, the bit energy
is
. Consequently, the
is
average bit energy-to-equivalent noise density ratio
Fig. 4. Capacity gain G versus the truncation probability 1
multiple-cell system; m = 4.
0 p( ) in
(41)
(42)
where (42) is valid for interference-limited systems. Then, the
capacity in terms of number of users supported is
(43)
is the average
required to meet the
where
target BER after averaging over fast fading, as discussed in
Section III. The concept of the average bit energy-to-equivalent noise density ratio, presented in (41), and the capacity in
(43), makes sense in an environment where the user population
is large enough so that the interference is nearly at its average
value by the weak law of large numbers. Therefore, the trunof
cated power control provides a capacity gain
(44)
).
over the capacity using conventional power control (
Fig. 4 is a plot of this capacity gain versus the truncation probfor several values of . When
dB, a
ability
typical value in terrestrial cellular systems, Fig. 4 indicates that
the capacity gain is 1.2–1.8 (i.e., 20–80% capacity increase) for
0
Fig. 5. Capacity versus the truncation probability 1 p( ) in multiple-cell
system; m = 4, R = 5 M, R = 10K, and (E =N ) = 6 dB.
the truncation probability in the range of 10
10 . Fig. 5 is a
plot of the capacity versus the truncation probability for several
values of . We see that the capacity decreases, but the capacity
gain (truncated versus conventional) increases as is increased.
If we employ soft handoff, then the other cell interference is reduced, and thus a higher capacity (than shown in Fig. 5) will be
obtained.
A. Outage Probability
We determine in (42) that users controlled by other cell base
stations introduce interference power into a given base station
, where
with an average value of
is the other cell spillover factor. Since the total contribution from
, instead of deterthese additional terms has a mean value of
mining the complete distribution of each term, we model their
KIM AND GOLDSMITH: TRUNCATED POWER CONTROL IN CODE-DIVISION MULTIPLE-ACCESS COMMUNICATIONS
Fig. 6. Outage probability P ( ) versus occupancy rate =; = 8 dB,
(E =N ) = 7 dB, S T =N = 30 dB, N = 128, and m = 4.
effect as that of many additional users [8]. Note that these additional users also transmit intermittently with probability
and all cells have the same Poisson occupancy distribution as
is also a Poisson varithose in the given cell. Thus,
. With this model, the
able with parameter
outage probability is
(45)
Fig. 6 is a plot of the outage probability versus the occupancy
in multicell systems. At low-occupancy rates, the
rate
is
outage probability increases as the activity factor
decreased. This is because the outage probability is dominated
[the first term in (27)]
by the truncation probability
when the occupancy rate is low. However, at high-occupancy
rates, the outage probability decreases as the activity factor is
is increased. This
decreased, i.e., as the cutoff threshold
is because the multiple-access interference (MAI) is reduced
by decreasing the activity factor, and the outage probability is
dominated by the probability of MAI exceeding a threshold
[the second term in (27)] when the occupancy rate is high.
V. CONCLUSIONS
We have analyzed the performance of truncated power control in CDMA systems that adapt the transmit power relative
to lognormal shadowing and path loss and use RAKE reception to compensate for Rayleigh fading. Truncated power control compensates for fading above a certain cutoff fade depth
and increases the data rate during transmissions so that the average data rate is independent of the cutoff depth and its associated activity factor. We compare the performance of truncated
971
power control with that of conventional power control for both
single-cell and multicell systems. For single-cell systems, we
find that our scheme exhibits a power gain of 1.3–4.5 dB relative to conventional power control for truncation probabilities of
10
10 , assuming an 8-dB standard deviation for the lognormal shadowing and a path-loss exponent of four. The power
gain is higher for a larger standard deviation and path-loss exponent, indicating that truncated power control is most effective
for channels with large power fluctuations.
For multicell systems, we find that truncated power control
exhibits a capacity gain of 1.2–1.8 (i.e., 20–80% capacity
10 , when
increase) for truncation probabilities of 10
the lognormal shadowing has a standard deviation of 8 dB,
the path-loss exponent is four, and hard handoff is employed.
As the truncation probability is increased, a higher power
gain and capacity gain can be obtained. When the occupancy
rate (traffic) is low, the outage probability increases as the
truncation probability increases. However, when the occupancy
rate is high the outage probability decreases as the truncation
probability increases. That is because the outage probability
is dominated by the truncation probability at low occupancy
and by the MAI (which decreases as truncation probability
increases) at high occupancy.
REFERENCES
[1] T. S. Rappaport, Wireless Communications: Principles and Practice. Englewood Cliffs, NJ: Prentice-Hall, 1995.
[2] A. J. Viterbi and R. Padovani, “Implications of mobile cellular CDMA,”
IEEE Commun. Mag., pp. 38–41, Dec. 1992.
[3] K. S. Gilhousen et al., “On the capacity of a cellular CDMA systems,”
IEEE Trans. Veh. Technol., vol. 40, pp. 303–312, May 1991.
[4] A. J. Viterbi, A. M. Viterbi, and E. Zehavi, “Performance of power-controlled wideband terrestrial digital communication,” IEEE Trans.
Commun., vol. 41, pp. 559–569, Apr. 1993.
[5] A. J. Goldsmith and S. G. Chua, “Variable-rate variable-power MQAM
for fading channels,” IEEE Trans. Commun., vol. 45, pp. 1218–1230,
Oct. 1997.
[6] N. Kong and L. B. Milstein, “Error probability of multicell CDMA over
frequency selective fading channels with power control error,” IEEE
Trans. Commun., vol. 47, pp. 608–617, Apr. 1999.
[7] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1989.
[8] A. J. Viterbi, Principles of Spread Spectrum Communication. Reading, MA: Addison-Wesley, 1995.
[9] R. Kohno, R. Meidan, and L. B. Milstein, “Spread spectrum access
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58–67, Jan. 1995.
Sang Wu Kim (M’88–SM’99) received the Ph.D. degree in electrical engineering from the University of
Michigan, Ann Arbor, in 1987.
He is currently with the Korea Advanced Institute
of Science and Technology (KAIST), Taejon, Korea,
where he is a Professor of Electrical Engineering. His
research interests include spread-spectrum communications, adaptive wireless communications, and error
correction coding. During 1996–1997, he was a Visiting Associate Professor at the California Institute of
Technology, Pasadena.
Dr. Kim is an Editor for the Journal of Communications and Networks.
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Andrea J. Goldsmith (S’94–M’95–SM’99) received the B.S., M.S., and Ph.D. degrees in electrical
engineering from the University of California at
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, and 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,
cellular system design and resource allocation, and ad hoc wireless networks.
She is an Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS and the
IEEE PERSONAL COMMUNICATIONS MAGAZINE.
Dr. Goldsmith is a Terman Faculty Fellow at Stanford University and the
recipient of the National Science Foundation CAREER Development Award,
the Office of Naval Research Young Investigator Award, the National Semiconductor Faculty Development Award, the Okawa Foundation Award, and the
David Griep Memorial Prize from the University of California at Berkeley.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 49, NO. 3, MAY 2000