Power Control in CDMA Cellular Systems
Aly El-Osery
Research Assistant
Autonomous Control Engineering Center
University of New Mexico
EECE Building
Albuquerque, New Mexico 87131
elosery@unm.edu
Chaouki Abdallah
Professor
Electrical Engineering Department
University of New Mexico
EECE Building
Albuquerque, New Mexico 87131
chaouki@eece.unm.edu
Abstract
In any multiple access system, the need for power control is essential. In DS-CDMA systems,
multiple users share the same bandwidth and are separated by pseudorandom codes;
consequently, each user contributes a degree of interference to all others. The role of power
control is to command the mobile station to transmit at the lowest power possible while
maintaining the desired signal quality, thereby minimizing the total interference. In this paper,
we review some of the different research directions in power control, then present our own
approach to the problem. We will also include our approach to computing the signal-tointerference ratio, which is an important parameter in any power control scheme but does not
seem to be explicitly discussed in most papers.
1
Introduction
Code Division Multiple Access (CDMA) has been recognized as a viable alternative to both
frequency division multiple access (FDMA) and time division multiple access (TDMA)
[reference] I will add the references at the end when we finalize the changes, so that the order
will be right. Although there are different types of CDMA schemes, we will concentrate on
direct sequence (DS) CDMA. DS-CDMA has many advantages such as: i) universal one-cell
frequency reuse, ii) narrow band interference rejection, iii) inherent multipath diversity, iv)
soft hand-off capability, and v) soft capacity limit [reference]. But, these advantages can be
hindered by the increased interference caused by other users. Since all signals in a DS-CDMA
system are sharing the same bandwidth and overlapping in time, it is essential to exercise
some kind of control to maintain acceptable signal-to-interference ratio (SIR) for all users,
hence maximizing the system capacity by minimizing the outage probability which is the
probability that a call will have to be dropped due to inadequate SIR level[1].
One critical problem with DS-CDMA is the near-far problem. This problem occurs in the
absence of power control: If all mobiles were to transmit at the same power level, the mobile
closest to the base station will overpower all others (since the signal power drops
exponentially with the distance). Yet, another reason for power control is the battery life
time: If the mobile station were to continuously transmit at a power higher than that to
1
maintain an acceptable SIR, the battery lifetime will be reduced. Using power control, each
mobile station may transmit using the minimum power needed for maintaining the required
SIR ratio, thus conserving its battery life.
In a typical mobile-radio environment, a moving mobile station is in communication with a
fixed base station. While our power control scheme may be adopted for ad-hoc networks,
which operate in the absence of a base station, we will concentrate on the base-mobile stations
scenario in the remainder of this paper. The movement of the mobile is usually in such a way
that the direct line between the mobile station and the base station is obstructed by various
objects such as buildings, cars, trees, etc. Therefore, the mode of propagation of the
electromagnetic energy from the transmitter to the receiver will be largely by way of
scattering, reflection from flat sides of the obstacles, or by diffraction around such obstacles
[reference]. This energy will consequently vary tremendously, and any power control
algorithm has to be fast enough to accommodate these rapid changes in the channel. This
then eliminates the possibility of using complicated algorithms in the typical mobile-radio
environment.
Some of the early work in power control was reviewed in [2]. Aly: Reference Viterbi. In
[3-5] centralized power control was studied, and due to the complexity of the system,
centralized power control was suggested only for providing theoretical limits. When all users
could be accommodated with acceptable signal-to-interference ratios, [6] suggested a
convergent distributed power control algorithm to compute the required transmission power
of each mobile station. In [7] a second-order constrained power control (CSOPC) algorithm
was presented. This approach uses the current and past power values to determine the
necessary transmission power of each mobile. CSOPC was compared with the algorithm
presented in [6] and was shown to converge at a faster rate. Convergence analysis of
distributed power control algorithms is investigated in [8]. In [9] a framework for uplink
power control in cellular radio systems was presented. Our review to solving the power
control problem will be within such framework.
In the remainder of this paper, we will review the idea of centralized power control but
concentrate on the general class of distributed control algorithms as they become more
realistic when the number of mobiles grows. Also, only the uplink (mobile to base) control
will be reviewed, but all results may be applied to the downlink (base to mobile) case. Finally,
we remind the reader that our approach is applicable in the case of ad-hoc networks.
2
Centralized Control
In this section, the power control problem is discussed from the link balance problem point of
view. Figure 1 shows a simplified diagram of the communication link. A mobile i uses a base
station A which is closest to it for communication purposes. The mobile transmits at a power
level pi and the communication gain between base station A and mobile station i is denoted by
GAi. Thus the power that reaches base station A is GAi pi.
2
Figure 1: The gain of the communication link
Assuming that mobile station i is communicating with base station k the signal-tointerference ratio for mobile i, denoted i, is defined as
i 
(1)
pi
Q
p w
j
ij
j i
where
 G kj

wij   G ki
 0

i j
(2)
i j
and Q is the total number of mobiles in the system .
Using Equation (1), the link-balance problem (LBP) is formulated as follows:
find the power level pi such that
i 
pi
 *
Q
p w
j
(3)
ij
j i
where * is the desired threshold below which the signal quality is unacceptable.
Centralized power control assumes that all information about the link gains is available to all
mobiles, then, in one step, the maximum achievable SIR level is computed. In fact, let

3
Note that this model does not yet include system noise.
 0

 w21
W


 wQ1

w12  w1Q 


w2Q 

 

  0 
QQ
 p1 


P  


 pQ 
(4)
where the transmission power is constrained as follows:
0  pi  pi
(5)
and p i is the maximum transmission power of mobile i.
The LBP may be analytically solved as follows : The largest achievable SIR level, ˆ , is
related to the matrix W, by ˆ  1 / * where * is the largest real eigenvalue of matrix W. The
power vector, P*, achieving this maximum level is given by the eigenvector corresponding to
* . In the case that ˆ is less than the desired SIR, some calls will have to be dropped. The
power control problem is thus reduced in this case to a general eigenvalue problem. The main
limitation of such an approach is exactly the fact that it is centralized: To compute the power
for a given mobile station i, the data for all other mobile stations has to be available. From a
practical point of view, as the number of mobiles grows, this approach becomes unfeasible.
3
Distributed Power Control
As opposed to centralized power control, distributed power control will be able to iteratively
adjust the power levels of each transmitted signal using only local measurements. Then,
within a reasonable time all users will achieve and maintain the desired signal-to-interference
ratio. Let us then re-consider the LBP problem, and assume that * is the desired signal-tointerference ratio, and that each mobile station, i, has receiver noise ni. Equation (3) may be
re-written as
i 
pi
p w
j
(6)
 *
Q
ij

j i
ni
G ki
The goal now is to find the transmission power of mobile i such that the following inequality
is satisfied:

p i  i (P)   * 


Q

p j wij 
j i
ni
G ki




(7)
where i (P) is known as the interference function.
Since it is desired to use the minimum transmission power possible, inequality (7) becomes an
equality, and an iterative method for power control could be formulated as [9]
pi (n  1)  i (P)
Given inequality (5), the constrained iterative power control algorithm in Equation (8)
becomes
4
(8)


*


p i (n  1)  min p i , i (P)  min  p i ,
p i ( n) 


  i ( n)

(9)
where i(n) is the signal-to-interference ratio of mobile i at iteration n. Convergence of
Equations (8) and (9) is proven in [10].
Equation (9) is a first-order power control command since pi(n+1) depends only on pi(n), , in
[7] a second-order algorithm is presented. This algorithm is known as the constrained secondorder power control and is given by the following equation.



*




p i (n  1)  min  p i , max 0, a(n)
p i (n)  1  a(n) p i (n  1)

(
n
)



i




(10)
where, a(n) is a decreasing sequence such that lim a(n)  1 . As an example, the following
n
a(n) sequence used in [8]:
a(n)  1 
1
1.5
n
, n  1,2, , l
(11)
where l is the total number of iterations. Equation (10) determines the necessary power using
the current and the past power value, which accounts for the terminology of “second-order”.
Note that in the case when a(n)=1, Equation (10) simply reduces to Equation (9).
At the beginning of simulation of this approach, * is set to a certain level. During simulation
if the SIR levels of the mobiles are within 1dB of *, the simulation is stopped.
What is the problem with this algorithm? Why go to LQPC?
There are two main problems with this approach. Assume that the achievable SIR level is
6dB, but in the simulation * is set to 7dB. In this case, the SIR levels will converge to low
values making the outage probability very high despite the tolerance widow of 1dB set in the
simulation. At the same time, if * is set to 6dB in the first place, the outage probability
would have been zero. The second problem is the fact that the above approach does not allow
for incorporation of measurement noise. Our approach, which will be presented in the next
section, eliminates these problems.
4
Linear Quadratic Power Control (LQPC)
Borrowing on the results from modern control theory, we present a state-space formulation
and linear quadratic control [11] as a viable design methodology for power control. Our
approach is to view each mobile-to-station connection as a separate subsystem [12], described
by
s i (n  1) 

p i ( n)  u i ( n)
 s i ( n)  v i ( n )
I i ( n)
(12)
ni
, vi(n)=ui(n)/Ii(n), and by definition, si(n)=pi(n)/Ii(n). The
G ki
input, ui(n), to each subsystem depends only on the total interference produced by the other
users and the noise in the system. The goal is to find the right control command that will
where I i (n) 
5
Q
j i
p i wij 
make each si track a desired signal-to-interference ratio *. For simplicity and without loss of
generality, we will assume that * is the same for all mobile stations. To accomplish such a
task, and to eliminate any steady-state errors [11], a new state is added to the system. This is
that of the integrator of the error, ei(n)=si(n)-* [13], which, in the discrete-time case, is
nothing more than a summation of the previous values. Therefore, the new state is i(n),
where
 i (n  1)   i (n)  ei (n)   i (n)  si (n)   *
(13)
Aly: note definition of e_I is not what you use in (13). I had a sign mistake. Let us define
xi(n) =[i(n) si(n)]', where ( )' denotes transpose. Then each subsystem can be expressed as a
second-order linear state-space system by
  (n  1)   1 1
 0
1
  
 x i (n)   v i (n)    *
xi (n  1)   i
1
 0
 s i (n  1)   0 1
(14)
We then choose the feedback controller
vi (n)  [k k s ]xi (n)  k s *
(15)
If we choose the appropriate feedback gains, the steady-state state, si(n), will go to the desired
signal-to-interference ratio * .
In order to use LQ control theory, we choose the following quadratic performance measure

J

x' (n)Qx (n)  v' (n) Rv(n)
(16)
n 0
where the term x'(n)Qx(n) is a weight on the control accuracy, v'(n)Rv(n) is a measure of
control effort, and they are chosen to be Explain why you do this: why is the last entry of Q
zero? I chose the values of Q based on trial an error. These are the values that gave me the
best results.
 200
Q  
 0
0
,
0 
R  0. 1
(17)
Aly: We still have a problem here: This is not a stabilization problem unless you use error
signals. The gain matrix K=(k ks) is found by solving a Riccati equation. Q and R are chosen
in such a way that the inequality (5) and the properties of the standard interference function
are satisfied. Once K is found, the new power command can be computed as follows
pi (n  1)  min pi , si (n  1) I i (n)
(18)
Why is this the answer? Also give an interpretation.
Equation (18) can be verified by examining the way we defined our state si(n+1) in Equation
(12). The ‘min’ operator in Equation (18) ensures that the power levels will not exceed the
maximum transmission power of the mobile.
6
5
Signal-to-Interference Estimation
Since most power control algorithms need access to the SIR, we include here our approach to
calculate this important term. Due to the fact that in the reverse link we don’t have the luxury
of transmitting a pilot signal, it is necessary to have a noncoherent demodulation scheme.
Consequently, the structure presented here assumes that the transmitted symbols are mapped
using orthogonal codes [explain and reference]. Figure 2 shows a block diagram for the
estimation of the SIR for one signal path. It is assumed that synchronization has been
established [reference].
Figure 2: SIR estimation. The signal processor determines the likelihood of a symbol being
sent and uses averagers that determine the power of the signal of interest as well as the total
signal power, which includes noise and interference due to other users as well as multipath
signals.
The input signals r(t) in Figure 2 are the received inphase and quadrature components, after
synchronization. The subscript l is used to indicate that more than one path can be tracked and
in that case we will have l of the above blocks to determine the SIR for each path being
tracked. The SIR level is computed by taking the ratio of the power of the signal most likely
to have been sent to the power of the interference (which is the total power minus the power
of the signal that most likely have been sent). The block shown in Figure 2 has been
simulated and the results are shown in Figure 3. In this case the averaging widow is 12 bits.
7
Figure 3: SIR estimation using the block diagram shown in Figure 2.
As shown in Figure 3 our algorithm is able to accurately track the changes in the SIR level.
The error in the estimation of the SIR level is due to the noise in the system, and the fact that
we are only averaging over 12 bit.
6
Simulation
The CSOPC and the LQPC were simulated for comparison. We chose the CSOPC algorithm
because it was shown in [7] to be superior to other distributed power control algorithms. The
system was simulated with two different maximum transmission powers, 1 and 5 watts. The
outage probability (define this) was used as a measure for comparing the CSOPC and the
LQPC approach. Outage probability versus the number of iterations and versus the number of
mobile stations in each cell was computed and plotted. Since in reality, the users are randomly
dispersed, each point on the curves shown in Figures 5 and 6 (what curves?????) is obtained
after simulating the system 100 times, and averaging out the results. Figure 4 shows a 7-cell
cluster created for the simulation. The mobile stations are allocated randomly (give
parameters).
8
Figure 4: The seven-cell configuration
As seen from the simulation results in Figure 5 for p i =1 watts, the difference between the
CSOPC and LQPC approaches is not very pronounced. Nevertheless, as shown in Figure 5a,
for 18 mobile stations per cell, the LQPC approach reaches zero outage probability in 3
iterations versus 5 iterations for the CSOPC algorithm. For p i =5 watts, the difference is
more noticeable as shown in Figure 6. With higher maximum transmission power, the system
can accommodate more mobile stations. By comparison, the new approach is more effective
in handling a larger number of mobile stations in the system. In Figure 6a, it can be seen that
the outage probability for 26 mobile stations per cell goes to zero in 7 iterations. In 7
iterations the outage probability using CSOPC is approximately 19 percent. This does not
mean that CSOPC can not accommodate 26 mobiles, but rather that the CSPOC requires more
iterations in order to converge to the right solution. There were no removal algorithms
incorporated in either approach that minimizes the outage probability by optimally removing
the right mobiles (EXPLAIN!). It is also important to note that our approach can handle 26
mobile stations with zero outage probability as opposed to 21 using CSOPC, as shown in
Figure 6b.
(a)
(b)
Figure 5: The outage probability as a function of: (a) the number iterations with 18 mobile
stations per cell, and (b) the number of mobile stations per cell, for p i =1.
9
(a)
(b)
Figure 6: The outage probability as a function of: (a) the number iterations with 26 mobile
stations per cell, and (b) the number of mobile stations per cell, for p i =5.
7
Conclusion
Due to the movement of the mobile station with respect to the base station, the power levels
reaching the base station undergo constant changes. . The problem is compounded in the
case of DS-CDMA by the near-far problem, and the need to prolong the battery lifetime. To
overcome these obstacles, (inexpensive) power control is essential. In this paper we reviewed
two main directions in power control, namely, centralized and distributed power control. We
also presented our approach, which falls under the category of distributed power control. A
simulation environment is created to compare our approach to the others. The simulation
shows the effectiveness of using state-space and linear quadratic control to determine the
necessary power control command in a few numbers of iterations.
Mention remaining problems and future work
Currently, there is an unavoidable delay in the estimation of the interference level due to the
computation requirements. In future work a predictor to estimate future changes in the signalto-interference level, will be designed in order to have a more accurate power control
algorithm. Also, a removal algorithm will be designed and incorporated. Another important
future addition will be designing an algorithm to dynamically determine the threshold level,
and hence, taking advantage of the soft capacity inherited in DS-CDMA.
References
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Algorithm,” Electronics Letters, Vol. 35, no. 17, pp. 1425-1427.
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[4] Kim, D. (1999) “Simple Algorithm for Adjusting Cell-Site Transmitter Power in CDMA
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[5] Wu, Q. (1999) “Performance of Optimum Transmitter Power Control in CDMA Cellular
Mobile Systems,” IEEE Transactions on Vehicular Technology, VT-48, no. 2, pp. 571575.
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[6] Grandhi, S. A., Vijayan, R. and Goodman, D. J. (1993) “Centralized Power Control in
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