INTRODUCTION
Code Division Multiple Access (CDMA) is interference limited Network. CDMA Network capacity depends significantly on the nearfareffect where a mobile terminal is instructed by the base station to either increase or reduce its transmitter power. The transmitter power affects the link signal quality and the interference environment in a wireless system. However adjusting the transmitter power to improve the link performance is not a trivial problem. If a terminal with a low SIR increases its transmitter power, the SIR is momentarily increased. The increase in transmitter power will on the hand increase the interference in the other links in the system, causing these terminals to increase their powers which results into power competition. If a mobile terminal’s transmitter power is lowered, this will decrease interference to the other links, but could jeopardize its own link. In CDMA systems, many terminals will communicate with the same access port through a common frequency channel (Glisic, 2003). Many researchers have worked on different perspectives of power control algorithms during the recent decades (Rosberg and Zander, 1998; Bambos, 1998; Hanly, 1999 for reviews on power control). Power control in cellular radio systems, especially, has drawn much attention since (Zander’s 1992) work on centralized and distributed SIR balancing. SIR balancing was further investigated by Grandhi et al. (1993). Foschini and Miljanic (1993) considered a more general and realistic model, in which a positive receiver noise and a respective target SIR were taken into account. Foschini’s and Miljanic’s distributed algorithm (FMA) was shown to converge either synchronously or asynchronously to a fixed point of a feasible system. The convergence rate of power control is especially important when propagation and traffic conditions are changing rapidly. Jung et al. (2006) worked on prioritized data services under power constraints. It is expected that Next Generation Wireless Networks will be dominated by bursty traffic than today’s voicedominated traffic. With bursty traffic, slow algorithms will perhaps not even be able to converge before the data burst ends. To track these changes, the power control algorithm must converge quickly. For instance, in a WCDMA system, the interference situation can change drastically from frame to frame due to changes in the traffic load. In this study we introduced the effect of fading channels on Distributed power control.
SYSTEM MODEL
Assume that there are Q transmitters assigned to the channel c_{0}, where transmitter j uses a transmission power p_{j}. By using the following vector notation to describe all transmission powers of the transmitters (terminals).
In the uplink case, the value p_{j} means the transmission power of
terminal j. However in the downlink, it denotes the transmission power dedicated
to terminal j by the access port to which terminal j is connected.
The expression for the SIR in the receiver i on the channel can be derived:
where:
g_{ii} 
= 
Link gain between receiver i and transmitter i. 
g_{ij} 
= 
Link gain between receiver i and transmitter j. 
p_{i} 
= 
Transmission power of terminal i. 
P_{j} 
= 
Transmitter power of terminal j. 
n_{i} 
= 
Noise power at receiver i. 
Proposition: A transmitter (terminal) is said to be supported by an Access Point (AP) if it has the SIR satisfying:
Γ_{i} ≥ γ_{0} where, γ_{o} is a target SIR threshold.
Substituting this value in the Eq. 2, we get:
Equation 3 shows the minimal power that a Mobile Terminal
(MT) i should use to achieve the target SIR, assuming the other transmitters’
powers are fixed.
Define the QχQ (nonnegative) normalized link gain matrix H_{ij} = (h_{ij}) such that:
Let the normalized noise vector η = (η_{i}) such that.
With the matrix notation, the Q linear inequalities (Γ_{i} ≥
γ_{o}, for all i) can be described by:
Where, I denotes the identity matrix and the power P should be nonnegative.
Definition: The target SIR γ_{0} is said to be achievable if there exists a nonnegative power vector P such that Γ_{i} ≥ γ_{o} for all i.
Proposition 1: The target SIR γ_{0} is achievable if the largest eigenvalue of the matrix H, denoted by ρ(H) is less than or equal to one. The case of ρ(H) will make the γ_{0} achievable only when the receiver noise is zero.
DISTRIBUTED POWER CONTROL AND ITERATIONS
From Eq. 2 it can be concluded that link gain matrix G should
be known. This assumption requires that a centralized measurement mechanism
should be employed which in turn will result into very heavy signaling between
access ports and terminals. Hence the practicability of designing such a system
is not trivial. In this study, we directed our focus into how to avoid such
a centralized control by using the Distributed Power Control (DPC) algorithms.
DPC algorithms were first suggested by (Foschini and Milianic, 1993).
In this study it is assumed that the receiver noise is not negligible and that
there exists a unique and nonnegative power vector P* that solves Eq.
7. In other words, ρ(H) < 1 so that the matrix (I–H) = η
is nonsingular and;
Every element in the matrix H is hardly available in practical systems; hence
methods such as Gaussian elimination method for solving systems of linear equations
cannot be used here. A general iterative method used for power control algorithms
are derived from numerical linear algebra. A general iterative method proposed
to solve to solve (8) is given by:
Where, M and N are matrices of appropriate sizes such that:
The vector P^{(n)} represents the power level at iteration n. When
M and N are appropriately selected, the iterative method in (9) can converge,
that is,
Let M = I and N = H, a power control algorithm can be constructed as:
Hence for each transmitter i, the iterative power P_{i} becomes:
Where, γ_{i}^{(n)} and p_{i}^{(n)} denote
the received SIR and transmission power of transmitter i at iteration n, respectively.
Convergence of the iterative method: In the general iterative method
in Eq. 9, let α_{1}, α_{2} be the
eigen values of the iteration matrix, M‾^{1} N and define ρ(M‾^{1}
N) = max_{k}α_{k}. If we define the vector
error by
From (9), the error vector ∈^{(n)} can be expressed as:
According to proof Zander et al. (2001), in order for (15) to converge
to zero vector, ρ(M‾^{1} N) < 1 should hold. From (15)
it can be proved that the power vector converges to a fixed point with a geometric
rate.
Hence for DPC
As stated in proposition 1 that ρ(H) < 1 when the target SIR is achievable
and receiver noise is positive. Therefore DPC will converge to P* whenever the
given target is achievable.
Convergence speed of iterative method: Convergence speed of power control
is important characteristic by which we can determine the practical applicability
of a given power control algorithm. It has always been assumed that the link
gain matrix is invariable during the power control process. However, in practical
systems the values of the gain matrix and the size of the matrix are changing
continuously due to mobile movement and the propagation condition change. A
good power control algorithm should quickly converge to the state where the
system supports as many users as possible. Hence the smaller ρ(M‾^{1}
N) is, the faster the convergence.
SIMULATION AND METHODOLOGY
The simulation setup is described in this section. We simulated the distributed power control (DPC) algorithm. The path loss model used for the simulations is COST231.
Present assumptions and default parameter values are stated.
Path loss model: We considered path loss and shadowing in our path model. Fading affects the signal strength measurements and transmit power values. The path loss was modeled using the COST231Hata model (Mogensen et al., 1991). The signal from the BS to the MT is assumed to decay at the rate of 4th power of the distance. The signal received by a MT from all other BSs except the one that is serving the MT is treated as interference. Considering only path loss, the interference power from each interfering BS j to a MT i is:
Where, d_{ij} is the distance between the BS j and the MT i. The constant
c corresponds to the intercept in the path loss model and is assumed to be 28.5
dB when distance is in meters (Mogensen et al., 1991). The slow shadow
fading is modeled by independent lognormal variables. To account for the spatial
correlation of the shadows, we assume the model proposed by Gudmundson (1991)
where lognormal shadowing was modeled as a Gaussian white noise process that
is filtered by a firstorder lowpass filer:
Where, Ψ_{l(dB)} is the meansquared envelope expressed in decibels,
that is experienced at location is
a zeromean Gaussian random variable with the standard deviation of 8 dB and
ζ is a parameter that controls the spatial correlation of the shadows.
After every given time interval T in seconds, the spatial correlation factor
ζ for a mobile that is traveling with velocity v is calculated as:
Where, ζ_{D} represents a shadow correlation between two points
separated by a spatial distance of Dm. In present simulation ζ_{D}
is set to 0.82 for a distance of 100 m, based on the experiments by Gudmundson
(1991). Taking into account the shadowing, the interference power received from
an interfering BS j by a MT i at location l is:

Fig. 1: 
Generation of MTs in various BSs 
Simulation parameter assumptions: To simulate a very large cellular
network, (Lin and Mak, 1994) recommend a wraparound topology. This approach
eliminates the boundary effects in an unwrapped topology. Hence we simulated
our network using a wrapped mesh topology with 81 hexagonal cells. Each cell
is surrounded by two rings of BSs so that a significant fraction of interference
is captured. We make the following assumptions in our simulations.
• 
The mobile terminals move based on a twodimensional random
walk model, that is, the mobiles can travel in any direction in a plane
with an equal probability. The speed of a mobile is chosen randomly below
the maximum speed. We set the maximum speed to 120 km h‾^{1},
unless otherwise stated. Mobile terminals (MT) are generated randomly and
uniformly across the cells and can appear anywhere with an equal probability
(Fig. 1). 
• 
The target SIR = 6 dB, number of iterations = 20, the default diameter
of a cell is 1 km and all the BSs are assumed to use the same transmission
power of 15 W. The spread bandwidth is 3.84 MHz and the thermal noise is
set to 105 dBm, derived from (WEA, 2000). 
RESULTS AND DISCUSSION
Figure 2 and 3 demonstrate the scenario
where two mobile terminals behave under the influence of DPC algorithm. The
scenario shows the power required by either terminal to reach a point of convergence.
When an appropriate target SIR is attained, the terminals will converge.

Fig. 2: 
Convergence of Distributed Power Control (DPC) when SIR =
1 dB 

Fig. 3: 
Convergence of Distributed Power Control (DPC) when SIR =
6 dB 

Fig. 4: 
Convergence of Distributed Power Control (DPC), SIR = 1 dB 

Fig. 5: 
Convergence of Distributed Power Control (DPC), SIR = 6 dB 
In Fig. 1, the SIR was set to a different value other than
the intended target SIR of 6 dB. As can be observed no proper convergence was
attained by the system. In Fig. 2, the SIR value was set to
6 dB which is the target SIR and this has resulted into convergence.
Figure 4 and 5 show the number of iterations
against the normalized Euclidean error for SIR values of 1 dB and 6 dB, respectively.
It is observed that, the Euclidean error is better for an SIR of 1 dB than 6
dB which is our target SIR.
CONCLUSIONS
In this research Distributed Power Control (DPC) algorithm has been investigated. CDMA systems are power and interference limited; hence, controlling the transmission power of Mobile Terminals in a cell is crucial to enhance the overall capacity of the network. The advantage of DPC algorithm is that the transmission power of MTs is not centrally controlled. This saves a lot of capacity in terms of massive signaling.