Power control of CDMA systems with successive interference cancellation using the knowledge of battery power capacity.

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Power Control of CDMA Systems with Successive Interference Cancellation Using
the Knowledge of Battery Power Capacity*

Yan Wang

ASTRI
Hong Kong
wangyan@astri.org


Chi-Ying Tsui, Roger S. Cheng, Wai Ho Mow

Dept. of Electrical & Electronic Engineering
The Hong Kong University of Science & Technology,
Clear Water Bay, Kowloon, Hong Kong
eetsui@ust.hk, eecheng@ust.hk, eewhmow@ust.hk

Abstract— Successive interference cancellation (SIC) has been
used in multi-rate code division multiple access (CDMA)
systems to eliminate the co-channel interference. In SIC, the
required transmission power of a certain user can be reduced if
it is detected late in the cancellation order as the interferences
caused by the users that are detected before have been
cancelled. In this work, we study the power control strategy
based on different orderings for the SIC given the knowledge of
the battery capacity of the mobiles. In particular we propose
different ordering schemes for SIC for different optimization
objectives such as minimizing the total power, maximizing the
minimum transmission/connection time for a group of users
within the same cell and maximizing the total amount of data
transmitted for a group of users. Experimental results show that
significant improvement can be achieved by using the proposed
ordering methods.
1. INTRODUCTION
Power control is an important vehicle for performing
several network operations such as admission control, link
QoS maintenance, channel probing, resource allocation, and
hand-offs for wireless systems [1]. In addition, power control
is essential for optimizing the energy consumption of
wireless system especially in minimizing the transmitter
power and maximizing the battery life of the mobile nodes
[2].
Multi-rate code division multiple access (CDMA) systems
[3] are becoming popular for the next generation wireless
communication systems. One of the limiting factors for the
performance and capacity of these systems is the interference
seen by a user, which is caused by the other users in the
systems. To reduce this interference, successive interference
cancellation has been proposed [4]. In perfect SIC, for a
particular user, only those users that are detected after him
would generate interference for its detection. So the later the
user is detected, the less transmission power is required as
less interference is seen. Various power control strategies
based on different interference cancellation ordering schemes
can be used to satisfied different optimization objectives. In
[5] an ordering scheme has been proposed to minimize the
total transmission power of a group of users within the same
cell.
However, few attentions have been given to the issues of
power control considering the mobile nodes’ battery capacity.
In this paper, using the uplink of a single cell multi-rate code
division multiple access (CDMA) systems with perfect
successive interference cancellation (SIC) as our reference
system, we present a framework of power control for
different optimization objectives based on different ordering
schemes for SIC with the knowledge of the battery capacity
of the mobiles. One obvious objective is to minimize the
total power consumption of all the users. In other
applications, we may want to have all the users connected at
the same time as long as possible. Here if the base-station
finds that one of the users is about to run out of battery, it can
change the detecting order of the SIC so that this user’s signal
is the last one to be detected so to reduce the required
transmission power to extend the transmission time. In this
way, the minimum transmission time of the whole group is
extended.
For other applications, we may want to maximize the total
number of transmission data of a group of users before all of
them run out of battery. Assuming that all of them have the
same channel loss and Quality of Service requirement,
ranking the users in the ascending order of their battery
capacity allows those users with less battery capacity run out
of battery first so that they will generate less interference to
others. The total number of transmission data can be
increased in this way. Instead of maximizing the total
number of transmission data, we can also maximize the sum
of the transmission time of each user weighted by a rate. This
is equivalent to maximizing the total revenue if the system
charges the users based on the transmission rate and the
transmission time.
The rest of the paper is organized as follows. We first
introduce SIC, the system specification and the notation in
session 2. Then we will present different optimization
strategies for SIC in section 3. Experimental results for each
optimization strategy will be presented. Finally, session 4
gives the conclusions.
2. SIC AND NOTATION
2.1. Successive Interference Cancellation
The performance and the capacity of the CDMA system
can be significantly degraded due to the co-channel
interference and near-far effect [3]. To overcome these
problems, interference cancellation (IC) techniques have been
proposed to increase the capacity by canceling the multiple
access interference [4]. The idea of IC is to cancel the inter-
user interference based on the previously detected symbols
from other users to make more reliable symbol
demodulations. IC can be classified into two types. They are
*This work was supported in part by Hong Kong RGC CERG under
Grant HKUST6214/03E and HKUST HIA02/03.EG03.

Page 2

successive interference cancellation (SIC) and parallel
interference cancellation (PIC), respectively.
The structure of the SIC is shown in Fig. 1. The received
signal power of each user is measured after the matched filter
and the users are ranked in certain order. There are K
Demodulation & Signal Recovery Units (DSRUs) where K is
the number of simultaneously communicating users. Each
DSRU performs matched decision (dispreading), channel
estimation, tentative symbol decision, and replica generation
of each user. The interference replicas of the first i-1 users
are subtracted from the received spread signal y(t) before
sending to the ith DSRU.
2.2. System specification and notation

In this work, we consider the uplink of a single cell multi-
rate CDMA system with perfect SIC, i.e., perfect channel
state information (CSI) is assumed and the users’ signals can
be decoded and regenerated correctly so that the interference
to other users is totally eliminated when the signal is
cancelled. Different services with different data rates and
QoS requirements are supported. The QoS requirement is
usually specified in the form of bit error rate (BER) or frame
error rate (FER). We assume the QoS requirement is
specified by an equivalent received Eb/No requirement.
Different rates are achieved by using different processing
gains with a fixed chip rate. Also the total bandwidth is used
by all users.
To formulate the optimization problem, we define the
notations for the following useful parameters:

N: the number of users in the cell.
W: the bandwidth of the system
E: the battery capacity vector of the users [
V: the normalized charge vector of the users [
R: the rate requirement vector of the users [R1, R2, …, RN].
Q: the QoS requirement vector of the users [Q1, Q2, …, QN].
P: the transmit power vector of the users [P1, P2, …, PN].
h: the channel power gain vector of the users [h1, h2, …, hN].
Eb: bit energy.
No: Additive White Gaussian Noise (AWGN) with one-side
power spectral density.
The expression of the Eb/No of the ith user is given by [4]

W
N
i
i
o


∑
+=
]
N
E
,
EE
,,,
V
2
V
1
K
,
2

]
N
V
,
1
K

Ni
WNPh
Ph
R
E
N
ij
ojj
iib
K
, 1
1
=
+
=




(1)
3. DIFFERENT SIC ORDERING STRATEGIES FOR DIFFERENT
OPTIMIZATION OBJECTIVES
3.1. Minimizing the total transmission power
Given the rates and the QoS requirements of the users, the
objective of the power control is to minimize

∑
=
i
N
iP
1
(2)
subject to the constraints

NiQ
N
E
i
i
o
b
,...,1
=≥






(3)

From [5], it has been known that ranking the users in the
descending order of the channel gain (h) can minimize the
total transmission power of the users, regardless of the users’
data rate and QoS requirements. We call this ordering scheme
the Gain ordering scheme.


Fig.1 The structure of SIC

3.2. Maximizing the minimum transmission time
Given the rate and the QoS requirement, and the battery
capacity of each user, the objective of this power control
scheme is to maximize

min






N
N
P
E
P
EL
1
1
(4)
subject to the constraints of (3). Here we propose two
schemes. The first is a static ordering scheme which assumes
the order cannot be changed once it is determined. The
second one is a dynamic ordering scheme in which the
ordering can be changed from time to time.

3.2.1. Static ordering scheme
From [5], it is shown that the required transmission power
under a certain detecting order is given by
QR
Wh
ij
i
1


For static ordering scheme, we assume that the detecting
order is decided at the beginning of transmission and it
cannot be changed before one of the users run out of battery.
Let A and B be the users i and i+1, respectively, (i.e., two
consecutive users) rank according to the decoding order of
the SIC scheme. Keeping the decoding order of all other
users unchanged, there are two possible orderings for these
two users. If user A's signal is cancelled before detecting user
B's signal, the required transmitting power of A and B are

=
A
QR
h
respectively. If user B's signal is cancelled before detecting
user A's signal, the corresponding transmission power values
become
Ni
W
QR
W
o
NP
N
jj
ii
i
, , 11
1
K
=




+=
∏
=+

(5)













+=






+





+
1
∏
=
ij
∏
=
ij
Q
+
+
N
jj
BB
W
B
oB
N
jj
BB
W
AA
W
oA
W
R
WNP
W
QR
QRQR
h
WNP
2
2
1
1
1
1
(6)
Channel
Ranking
Informatio
Demodulation & Signal
-
D
+
User
Demodulation & Signal
-
D
+
User
Demodulation & Signal
-
D
+
User #K-
Demodulation & Signal
User #1
User #2
User #K-1
User #K
Received
y
( ) t

Page 3














+





+
1
=





R

+=
∏
=
ij
∏
=
ij
+
+
N
jj
AA
W
BB
W
B
oB
N
jj
AA
W
A
oA
W
QR
QQR
h
WNP
W
QR
QR
h
WNP
2
'
2
'
1
1
1
1
(7)

Then, it’s easy to find that

It means that for the users whose signals are detected before
these two consecutive users, they see the same total
interference created by these two users, regardless of their
ordering. For the users whose signals are detected after these
two users, they do not experience any interference from these
two users, as their signals have already been cancelled
completely. Therefore, the ordering of two consecutive users
in the SIC scheme does not affect other users' transmission
power.
Let
AB
BA
under these two orderings, respectively, we have




NBA
PPPP
1

From (6) and (7), we can find that

'
,,
AA
PPP
E
P
′
first. Otherwise user A should be detected first.
From (6) and (7), we can derive that
E
P
AA
QR
Let
(
ii
ii
QR
the user. We can conclude that if there are two consecutive
users whose time factors are not in descending order, we can
switch their decoding orders in the SIC scheme to increase
the minimum transmission time of them.
By simple induction, we can show that one of the optimal
ordering to maximize the minimum transmission time if the
ordering can not be changed during transmission is to rank
their detecting orders according to their time factors in
descending order. We call this static ordering scheme.

3.2.2. Dynamic ordering scheme
When the decoding order can be changed during the
transmission time, the static ordering scheme may not be
optimal. This can be shown by the following argument.
Assume that all the users suffer the same channel loss, have
the same battery capacity and the same transmission rate. In
the beginning, we can see that the minimum transmission
time under any ordering scheme is the same. However after
some time, the remaining battery capacities of the users are
no longer the same since the transmission power required for
each user is different according to the detection order. If we
can re-order them and let those users with less battery
capacity be detected later to reduce their required
transmission power, the minimum transmission time can be
extended.

'
BB
'
AABBAA
PhPhPhPh
+=+
(8)
T and
T represent the minimum transmission time






′
A
′
=



=
N
NA
B
B
BA
NBA
AB
P
E
P
E
P
E
P
E
T
EEEE
T
,,,min,,,, min
1
11
LL
(9)
'
'
B
B
B
BAA
B
′
BAA
P
EE
and
EE
PPPP
><<>
.
If
B
B
A
A
P
E
≤
, then
BA AB
TT
≤
and user B should be detected
⇒
′
≤
B
B
A
A
P
E
()()
BB
BBBBAAAA
QR
QRWEhQRWEh
+
≤
+

(10)
)
ii
i
QRW
hE
G
+=
, and we call it the time factor of

Actually, if we find that under the current order, the user
that will run out of battery first has an instantaneous time
factor less than that of one of the users’ who is detected after
him, we can extend the minimum transmission time of the
group of users by simply swapping their detecting orders.
Based on this, we propose the following dynamic ordering
scheme
Step1: Compute the time factor of each user within the group
and arrange them in the descending order of time factor.
Step2: Compute the required transmitting power of each user
under this ordering (
battery first. Let it be the user m among N users and there are
m-1 users who are decoded after it. User m would run out of
E
) under the current decoding
iP). Find the user that will run out of
battery after time
1t ∆ (=
i
i
P
order.
Step3: Compute the time ( t ∆ ) when at least one of the other
m-1 user’s instantaneous time factors is not less than user m’s
instantaneous time factor, i.e.

)min(
10
−
=∆
tt
K
where
(
(
mm
mm
QRWhQRE
t
+

Step4: If
tt
∆>∆1
, after time
capacity information of each user and go back to step 1.
Otherwise keep on transmitting till one of the users runs out
of battery.
However we can see that if t ∆ is too small, a lot of re-
ordering will be needed. To reduce the number of re-
ordering, a threshold τ is introduced. If the value from (11)
is less than τ , the value of t ∆ is set to τ .

3.2.3. Performance comparison
In this sub-section, we compare the averaged minimum
transmission time, cell outage probability, total transmission
power and number of re-ordering required for different SIC
ordering schemes, namely the dynamic ordering, the static
ordering, the gain ordering and the conventional rate ordering
[3] in which the users are arranged according the descending
order of their rate requirement. Path loss, shadowing and
Rayleigh fading models are included in the comparison.
We carried out simulation to find the maximum minimum
transmission time for a group of users for different schemes.
Here we consider a system with three classes of services. The
rate and QoS requirements of the services are summarized in
Table 1. Users are randomly picked within a uniformly
distributed circular cell. Peak power constraints are imposed
on the users. For those users whose required transmission
powers exceed the peak power constraint, it is assumed that
they fail to meet the QoS requirement. The outage probability
of the cell is defined as the probability that at least one user in
the cell cannot meet the QoS requirement. The parameters
used in the simulation are shown in Table 2.
mt
(11)
)
)
()
()
(
(
)
)
(
(
W
)
)
iimmiimmmiim
ii
Q
mmii
h
mmmiim
i
ii
ii
iiiimimm
RQRPQRWhQRP
QRWQRhE
QRW
QR
ht PE
QRW
QR
htPE
+−
+−+
=
+
−
=+
−

t ∆ , update the battery

Page 4

The channel gain of the user in db is given by

⋅−=
h
0 . 1
( )
d






++






⋅⋅+
γ
σs
X
d
d
nPL
0
100
log10
(12)
where
( )
PL






⋅=
2
2
0
2
100
16
G
log10
λ
r
π
tG
d
d
(13),
s
Xσ is lognormal shadowing with standard deviation
γ is the amplitude of a Rayleigh fading channel whose real
and imaginary parts have normal distributions with zero
s
σ,
Table 1. Required Performance of Multi-class services
Information bit rate Required BER
384k
192k
96k

Required SNR
10.6dB
9.6dB
8.4dB
1
2
3
10-6
10-5
10-4


Table 2. Parameters of the multirate CDMA system
Item
Bandwidth
AWGN spectral density
Path loss exponent
Shadowing standard deviation
Rayleigh fading minimum gain
Carrier frequency
Radius of the cell
Close-in distance
Maximum allowable received

mean and variance ½. The minimum gain of rayleigh fading
is -30db, i.e., 0.001. We also assume unity gain for the
transmit and receive antenna, i.e.
In this simulation, we use the three classes of services in
Table 1. The corresponding peak power is 0.226w, 0.113w
and 0.056w. The ratio of the number of type 1, 2 and 3 users
is equal to 2, 2 and 2, respectively. We ran simulations for 6,
12, 18 and 24 users, respectively.
For every simulation, we randomly generate the location of
each user (d ) which is uniformly distributed between 50m to
1000m, the battery capacity ( E ) which is uniformly
distributed between 0 to 1.
The averaged minimum transmission time, the outage
probability, the total transmission power and the number of
re-ordering required are summarized in tables 3 to 6.
For comparison, we normalize the minimum transmission
time of all the schemes by the value obtained for the rate-
ordering scheme. Table 3 summarizes the average results
over 1000 different simulations.
We can see that the average normalized minimum
transmission time increases when there are more and more
users in the cell. Given 24 users in the cell, the minimum
time can be extended more than 30 times by using static
ordering. With dynamic ordering, an additional 17%
improvement can be obtained when there are 24 users in the
cell. For smaller number of users such as 6, the improvement
is reduced to about 3%. Comparing with gain ordering which
minimized the total power consumption, both static and
dynamic order schemes have significant improvement in
extending the minimum transmission time.
Symbol
W
N0
N
σs
C
fc
dcell
d0
Qpeak
Value
5MHz
38. 1
3
5dB
0.001
900MHz
500m
50m
50dB
300
23⋅
−
e

0
NEb

1
==
rt
GG
.
Table 4 shows the average outage probability. It can be
shown that the outage probability has been reduced using
dynamic, static and gain ordering compared with that using
rate ordering.
Table 5 summarizes the total transmission power of all the
users. The power consumption of both dynamic and static
time ordering schemes are higher than that of using gain
ordering scheme which has the minimum transmission
power. The differences are from 8 to 10%.

Table 3. Average normalized minimum transmission time
No. of users 6
Dynamic 1.95
Static 1.91
Gain 1.60
Rate 1

Table 4. Average outage probability
No. of users 6 12
Dynamic 4.9e-5 1.09e-4
Static 4.9e-5 1.09e-4
Gain 4.9e-5 1.09e-4
Rate 2.0e-4 1.51e-3

Table 5. Average total transmission power in dbm
No. of users 6
Dynamic 2.54
Static 2.52
Gain 2.32
Rate 4.64

Table 6. Average number of re-ordering required
6
Dynamic
1.23

Table 6 shows the number of re-ordering required for the
dynamic ordering scheme.
For comparison, it is better to have a bound of the
optimum minimum transmission time. However, it is quite
hard to find a tight bound of the maximum minimum
transmission time when a group of users suffer from different
channel losses and have different rate, QoS requirement and
initial battery capacity.
To show the effectiveness of our proposed dynamic
ordering scheme, we assume that all the users suffer from the
same channel loss and have the same rate QoS requirement
and initial battery capacity.
The total transmission powers of different decoding orders
are the same under this situation. The optimal ordering
scheme should let all the users run out of battery at almost the
same time. A tight bound thus can be obtained by dividing
the total battery of the users with the total transmission
power.
Simulations were carried out where we assume that all the
users are transmitting at 192kbps and suffering from the same
channel loss and have the same initial battery capacity.
The simulation results are shown in Tables 7 and 8 where I
represent the tight bound of the minimum transmission time.
We can see that no extension of minimum transmission time
can be obtained using static time and gain ordering scheme.
The dynamic schemes are more effective and there is only
about 20% degradation comparing to the tight bound. At the
12
5.16
4.92
2.90
1
18
14.72
13.58
4.93
1
24
37.35
33.00
6.77
1
18
1.5e-4
1.5e-4
1.5e-4
7.4e-3
24
7.2e-4
2.2e-4
2.1e-4
2.8e-2
12
7.18
7.15
6.57
11.04
18
11.99
11.91
10.80
16.63
24
18.08
17.87
16.28
22.34
12
1.55 2.21
18 24
3.47

Page 5

same time, the number of re-ordering required increases
significantly compared with that shown in table 6.

Table 7. Normalized minimum transmission time under the same initial
condition
6
I 1.86
Dynamic 1.73
Static/gain/rate 1

Table 8. Numb of re-ordering required under the same initial condition
6
Dynamic 6.98

3.3. Maximize the total transmission revenue
3.3.1. A Simple ordering schemes
Given the rate, battery power capacity and QoS
requirements of all the users, we find an ordering scheme to
maximize the total revenue which is given by
(
N
i
1

=
subject to the constraints of (3), given that the battery
capacity of user i is
12
3.20
2.83
1
18
4.69
3.98
1
24
6.23
4.82
1
12
14.74
18
22.23
24
24.65
)
NN
N
ii
PV
,
EPVEfVt
,,,,max
111
L
=



∑
(14)
i E its charge per minute is
i V and user i
iP until
will keep on transmitting with a transmitting power
it runs out of battery. From (7), the revenue coming from user
i before he runs out of battery is

VE
ii
=
0
11

N
W
QR
QR
hVE
P
jj
N
ij
ii
iii

i
⋅




⋅
+Π
=
⋅⋅
⋅⋅⋅
+
(15)
where ∏
the users decoded after him.
When all the users have the same R and Q, (15) is
simplified to

VE
ii
=
+=






+
N
ij
jj
W
QR
1
1
represents the interference coming from
0
11

N
W
E
QR
QR
hVE
P
N
Π
=
ij
iii
i
⋅ 

V
⋅



⋅
+⋅⋅
⋅⋅⋅
+
(16)
It is clear that for those users with large
them later can have more gain in terms of the total revenue
than detecting those users with small
R and Q of each user are not the same, from table 1, we can
find that the higher the R, the higher the Q. To reduce the
interference among a group of users, we should detect the
users with high transmitting rate first. As the user with high R
may have large
iii
hVE
⋅⋅
at the same time, we need different
trade off schemes to solve this problem.
Based on the above argument, we propose the following 6
schemes and use simulation to test their effectiveness in
increasing the total revenue. These schemes can be divided
into two types, P1 to P3 do not use the knowledge of the
battery capacity of each user while E1 to E3 use these
knowledge.
P1: Let
iii
VhG
⋅=
, the users are detected in descending
order of rate. For those users with the same rate, they are
ordered according to the ascending order of
iii
h
⋅
, detecting
iii
hVE
⋅⋅
later. When
i
G .
P2: Let
ii
ii
i
RQ
Vh
G =
, the users are detected according to the
ascending order of
P3: Let
G
i
G
Vh
i
+
()3
1
WQRRQ
iiii
i
i
=
, the users are detected
according to the ascending order of
E1: Let
iii
hEG
⋅=
descending order of rate. For those users with the same rate,
they are ordered according to the ascending order of
i
G
i
V
⋅
, the users are detected according the
i G.
E2: Let
ii
iii
i
RQ
VhE
G =
, the users are detected according to the
ascending order of
i
h
G
E
i
+
E3: Let
()3
1
WQRRQ
V
G
iiii
ii
i
=
, the users are detected
according to the ascending order of
When using these ordering schemes, we also assume that
re-ordering would be done only when one of the users runs
out of battery, so at most N-2 re-ordering are required if there
are N users in the cell.

4. SIMULATION RESULTS
We use a system with similar parameters as that described
in section 3.2 except that the ratio of the type 1, 2, and 3 users
is 3, 2 and 1.
For comparison, we normalize the total revenue of these
schemes with that obtained from a random ordering.

4.1. Maximizing the total transmitting time
We first assume that the normalized charge per minute for
type 1, 2 and 3 users are all 1. Thus
i
G
i
i
i
ii
P
E
P
VE
=
⋅
, and the
optimization problem is actually equal to maximizing the
total transmitting time with the knowledge of each user’s
battery capacity.
Table 9 shows the total transmitting time of all the users
under different ordering scheme, respectively. Comparing
with the results of using random ordering, it can be seen that
using scheme P1 and E1 have the biggest improvement. This
means that when the charge per minute of each user is the
same, reducing the interference coming from those users with
high transmission rate can have more gain in total revenue
compared with other schemes. The users with high data rate
should be detected first.

4.2. Maximizing the total transmitting data
Next we assume that the normalized charges per minute of
type 1, 2 and 3 users are 4, 2 and 1, respectively. It is just the
same as the ratio of their transmission rate. The problem is
equal to maximizing the total number of transmitted data with
the knowledge of battery capacity of each user.
Tables 10 summarizes the total transmitting data of the
users under different ordering schemes, respectively.
Comparing with the results using random ordering, it can be
seen that using schemes P3 and E3 have the biggest
improvement in revenue. This means that to maximize the
total transmitting data, the channel power gain of the user

Page 6

should be weighted with the amount of the interference
generated by this user to the others.

4.3. Maximizing the total revenue
Now we assume that the normalized charges per minute of
type 1, 2 and 3 users are 16, 4 and 1, respectively. The
problem is to maximize the total revenue with the knowledge
of battery power of each user.
Tables 11 summarize the total transmitting data of the users
under different ordering schemes, respectively. Comparing
with the results using a random ordering, it can be seen that
using scheme P2 and E2 have the biggest improvement in
revenue. This means that as the charge per minute for the
high data rate user is increasing, the influence of its
interference to others on the ordering is reduced.

4.4. The influence of the user’s battery capacity on the total
revenue
From the above simulation results, we can see that only a
few percentage improvements are achieved with the
knowledge of battery capacity of the users. To study the
effect of the battery capacity, the user type and the channel
gain of each user on the total revenue, we assume the charge
per minute of each user is the same.
We first assume that all the users are of type 2 and the
channel gain of each user is -20db. The ordering scheme P1
is just the same as random ordering, and the ordering scheme
E1 becomes that the users are detected according to the
increasing order of
We can see that almost 40% increase in the total transmitting
time is obtained using the knowledge of the battery capacity
of the users.
Next, we assume that the ratio of the number of type 1, 2
and 3 users are 2, 2 and 2, respectively, and the channel gain
of each user is still the same. The ordering scheme P1
becomes that the users are detected according to the
decreasing order of the rate. The ordering scheme E1 is just
the same as the users are detected according to the decreasing
order of the rate and the increasing order of
are summarized in Table 13. We can see that the difference
between using E1 and P1 is quite small.
Finally we assume that all the users are of type 2 while the
channel gain of each user is not the same. They are generated
according the scheme used in previous sub-section. Using
scheme P1, the users are detected according to the increasing
order of their channel gain. Using scheme E1, the users are
detected according to the increasing order of the channel gain
weighted with the battery capacity.


Table 9. Total transmitting time of users
U O
P1 E1
P2
6 1
1.09 1.10
1.06
12 1
1.15 1.16
1.10
18 1
1.18 1.19
1.12
24 1
1.20 1.21
1.14





i E . The results are shown in Table 12.
i E . The results
E2
1.07
1.12
1.15
1.17
P3
1.08
1.13
1.16
1.19
E3
1.09
1.14
1.18
1.20
Table 10. Total transmitting data of users
O P1 E1 P2
1 1.03 1.03 1.04
1 1.05 1.06 1.07
1 1.06 1.08 1.10
1 1.07 1.09 1.11
U
6
12
18
24
E2
1.05
1.09
1.12
1.14
P3
1.04
1.08
1.11
1.13
E3
1.05
1.10
1.13
1.16

Table 11. Total revenue under case 2
O P1 E1
1 0.98 0.99
1 0.98 0.99
1 0.98 0.99
1 0.98 0.99
U
6
12
18
24
P2
1.06
1.11
1.15
1.17
E2
1.07
1.13
1.17
1.20
P3
1.04
1.10
1.13
1.16
E3
1.06
1.12
1.16
1.19

Table 12. Total transmitting time of users with the same R, Q and h
User P1
6 1
12 1

Table 13. Total transmitting time of users with the same h
User Non
6 1
12 1

Table 14. Total transmitting time of users with the same R, Q
User Non
6 1
12 1

The results are summarized in Table 14. We can again see
that the difference between using E1 and P1 is not large.
From the above results, we can see that the extra gain in total
revenue is only a few percent with the knowledge of the
battery capacity of the user when the rate and channel gain of
the users are not the same. However, significant improvement
in total revenue can be obtained if the channel gain and
battery capacity of each user are the same.
E1
1.12
1.40
P1
1.19
1.42
E1
1.20
1.44
P1
1.10
1.18
E1
1.12
1.21
4. CONCLUSION
In this paper, we have studied different optimization
objectives for the power control of the up-link of CDMA
system with perfect SIC. Different ordering schemes, which
utilize the knowledge of the battery capacity of the users are
proposed. It is shown that the minimum transmitting time of
a group of users can be extended significantly using these
ordering schemes. We also presented ordering schemes
which can increase the total revenue of the system.

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[1] M. Zorzi and R.R Rao, “Energy-Constrained Error Control for
Wireless Channels”, IEEE Personal communications, Vol. 4, issue
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[4] Sergio Verdu, “Multiuser detection”, imprinted by Cambridge
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[5] Siu Man Shum and Roger S. Cheng, “Power control for CDMA
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