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Aly
El-Osery received his B.S. degree in electrical engineering in
1997, M.S. degree in electrical engineering in 1998, and is
currently pursuing his Ph.D. in electrical engineering all at the
University of New Mexico. He is currently working at the Autonomous
Control Engineering Center at the University of New Mexico. His
research interests are in the areas of wireless communications,
control systems, and soft computing.
Dr. Chaouki Abdallah
received his B.E. degree in electrical engineering in 1981 from
Youngstown State University, his M.S. degree in 1982 and the Ph.D.
in electrical engineering in 1988 from Georgia Tech. Between 1983
and 1985 he was with SAWTEK. He joined the department of electrical
and computer engineering at the University of New Mexico in 1988
and was promoted to associate professor in August 1994. His
research interests are in the areas of wireless communications,
robust control, and adaptive and nonlinear systems. Dr. Abdallah is
a senior member of IEEE.
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Code Division Multiple Access (CDMA) is a
viable cellular system alternative to both frequency-division
multiple access (FDMA) and time-division multiple access (TDMA)
technologies.
Although there are different types of CDMA
schemes, this article concentrates on direct sequence (DS) CDMA.
DS-CDMA offers numerous advantages, including:
- Universal one-cell frequency
reuse
- Narrow band interference rejection
- Inherent multipath diversity
- Soft hand-off capability
- Soft capacity limit.
Increased interference caused by other users can hinder these
advantages. 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 (the probability that a call will
have to be dropped due to inadequate SIR level).
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 battery life—if the mobile station were to
continuously transmit at a power higher than that needed to
maintain an acceptable SIR, the battery lifetime is 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 that operate in
the absence of a base station, we will concentrate on the
base-mobile station 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, and trees. 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.
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 scenario
eliminates the possibility of using complicated algorithms in the
typical mobile-radio environment.
Some of the early work in power control was reviewed in
Zander.
In Kim, Wu, and Grandhi, 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, Foschini suggested a
convergent distributed-power-control algorithm to compute the
required transmission power of each mobile station.
Jeantti presents a second-order constrained power
control (CSOPC) algorithm.
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 Foschini
and was shown to converge at a faster rate. Convergence analysis of distributed power-control
algorithms is investigated in Huang.
Yates presented a framework for uplink power
control in cellular radio systems.
Our review to solving the power control problem
will be within such framework.
The remainder of this article reviews the concept of centralized
power control, but concentrates on the general class of distributed
control algorithms as they become more realistic when the number of
mobiles grows. Although we only review the uplink (mobile to base)
control, you may apply all of the results to the downlink (base to
mobile) case. Finally, we remind the reader that our approach is
applicable in the case of ad-hoc networks.
Centralized Control
In this section, we discuss the power-control problem 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.
Figure 1: The gain of the communication link
Assuming that mobile station i is communicating with base
station k, the SIR for mobile i, denoted 
i, is defined as (note that this model
does not yet include system noise)
|
(1) |
where
|
(2) |
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
|
(3) |
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
|
(4) |
where the transmission power is constrained as follows:
|
(5) |
and 
is the maximum transmission power of mobile i.
You analytically solve the LBP as follows: The largest achievable SIR level, 
, is related to the matrix W, by, =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 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.
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
SIR. Let us then re-consider the LBP problem, and assume that * is the desired SIR, and that each mobile station,
i, has receiver noise ni. Equation 3 may be
re-written as:
|
(6) |
The goal now is to find the transmission power of mobile i such
that the following inequality is satisfied:
|
(7) |
where Yi(P) is
known as the interference function.
Since it is desired to use the minimum transmission power
possible, inequality (Equation 7) becomes an equality, and
an iterative method for power control could be formulated as
|
(8) |
Given inequality (Equation 5), the constrained iterative
power control algorithm in Equation 8 becomes
|
(9) |
where i(n) is the SIR of mobile i at iteration
n. Convergence of Equations 8 and 9 is proven in Yates.
Equation 9 is a first-order power-control command since
pi(n+1) depends only on pi(n). Jeantti presents a second-order algorithm. This algorithm
is known as the constrained second-order power control and is given
by
|
(10) |
where a(n) is a decreasing sequence such that 
. As an example, Jeantti has the following a(n) sequence
|
(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.
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 window 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 addresses both problems.
Linear Quadratic Power Control
(LQPC)
Borrowing on results from modern control theory, we present a
state-space formulation and linear quadratic control
as a viable design methodology for power control.
Our approach is to view each mobile-to-station connection as a
separate subsystem
, described by
|
(12) |
where
and by definition,
si(n)=pi(n)/Ii(n). The 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
make each si track a desired SIR *. Although our algorithm allows for
different SIR levels for different mobiles, 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, a new state is added to the system. This new state is the integrator of the error,
ei(n)=si(n)-*
, which, in the discrete-time case, is nothing
more than a summation of the previous values. Therefore, the new
state is zi(n), where
|
(13) |
Let us define xi(n) =[zi(n) si(n)]', where (
)' denotes transpose. Then each subsystem can be expressed as a
second-order linear state-space system by
|
(14) |
We then choose the feedback controller, which includes tracking
signal, as
|
(15) |
If we choose the appropriate feedback gains, the steady-state
state, si(n), will go to the desired SIR * driving ei(n) to zero as n
goes to infinity.
To find the optimal feedback control for the linear system in
Equation 14, we define quadratic penalties on the state and
control variables, and hence the name linear quadratic control
(LQR). In order to use the LQR theory we define the quadratic
performance measure as
|
(16) |
where the term x'(n)Qx(n) is a weight on the control accuracy
and v'(n)Rv(n) is a measure of control effort. Q and R are chosen
to be
|
(17) |
The gain matrix K=(kz ks) is found by solving a
Riccati equation. Q and R are chosen in such a way that the
inequality (Equation 5) and the properties of the standard
interference function are satisfied. Once K is found, the
new power command can be computed as follows
|
(18) |
You can verify Equation 18 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.
Signal-to-Interference Estimation
Since most power-control algorithms need access to the SIR, our
approach can calculate this important term. In the reverse link we
don't have the luxury of transmitting a pilot signal, making it
necessary to have a non-coherent demodulation scheme. Consequently,
the structure presented here assumes that the transmitted symbols
are mapped using orthogonal codes. Figure 2 shows a block diagram for the estimation of the
SIR for one signal path. We assume establishment of
synchronization. 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(f) in Figure 2 are the received in-phase and quadrature
components, after synchronization. The subscript l indicates that
you can track more than one path and, in that case, we will have
one of the above blocks to determine the SIR for each path being
tracked. The SIR level comes from the ratio of the power of the
signal most likely to have been sent divided by the power of the
interference (which is the total power minus the power of the
signal that most likely has been sent). A simulation of the block
in Figure 2 is shown in Figure 3. In this case, the
averaging window is 12 bits.
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 that we are
only averaging over 12 bits.
Simulation
We ran simulations to compare the CSOPC and the LQPC. We chose
the CSOPC algorithm because it was shown in Jeantti to be superior to other distributed power control
algorithms. The system was simulated with two different maximum
transmission powers, 1 W and 5 W. The outage probability was used
as a measure for comparing the CSOPC and the LQPC approach. We
computed and plotted outage probability versus the number of
iterations, and versus the number of mobile stations in each cell.
Since, in reality, the users are randomly dispersed, you obtain
each point on the curves shown in Figures 5 and 6 after
simulating the system 100 times and averaging the results.
Figure 4 shows a seven-cell cluster created for the
simulation. The mobile stations are allocated randomly.
Figure 4: The seven-cell configuration
As seen from the simulation results in Figure 5 for =1 W, 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 three iterations versus
five iterations for the CSOPC algorithm. For =5 W, 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, the outage probability
for 26 mobile stations per cell goes to zero in seven iterations.
In seven iterations the outage probability using CSOPC is
approximately 19%. 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. 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.
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 =1W.
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 =5W.
Conclusions
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. We created a simulation environment to
compare our approach to alternative techniques. The simulation
shows the effectiveness of using state-space and linear quadratic
control to determine the necessary power control command in few
iterations.
Currently, there is an unavoidable delay in the estimation of
the interference level due to the computation requirements. In
future work, we will design a predictor to estimate future changes
in the signal-to-interference level to obtain a more accurate
power-control algorithm. We will also design and incorporate a
removal algorithm. Another important future addition will be
designing an algorithm to dynamically determine the threshold
level, thus taking advantage of the soft capacity inherent in
DS-CDMA.
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