distributed power control of

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AbstractA novel distributed power control (DPC) scheme forcellular networks in the presence of radio channel uncertainties

such as path loss, shadowing, and Rayleigh fading is presented.

Since these uncertainties can attenuate the received signal strengthand can cause variations in the received Signal-to-Interference ratio

(SIR), the proposed DPC scheme maintains a target SIR at the

receiver provided the uncertainty is slowly varying with time. The

DPC estimates the time varying nature of the channel quickly and

uses the information to arrive at a suitable transmitter power value.

Further, the standard assumption of a constant interference during

a links power update used in other works in the literature is

relaxed.

A CDMA-based cellular network environment is used to compare

the proposed scheme with earlier approaches. The results show

that our DPC scheme can converge faster than others by adapting

to the channel variations. The proposed DPC scheme can render

outage probability of 5 to 30% in the presence of uncertainties

compared with other schemes of 50 to 90% while consuming low

power per active mobile user. In other words, the proposed DPCscheme allows significant increase in network capacity while

consuming low power values even when the channel is uncertain.

KeywordsWireless Cellular Networks, Fading Channels, DistributedPower Control, Signal-to-Interference Ratio (SIR), Outage Probability.

I. INTRODUCTIONhe objectives of transmitter power control includeminimizing power consumption while increasing thenetwork capacity, and prolonging the battery life of mobile

units, by managing mutual interference so that each mobile unitcan meet its signal-to-interference ratio (SIR) and other quality

1 This work was supported in part by the University of MissouriResearch Board Award and National Science Foundation AwardECS#0296191.

of service (QoS) requirements. Early work on power control [1][2] focused on balancing the signal-to-interference (SIR) ratioson all radio links using centralized power control. Laterdistributed SIRbalancing schemes [3][4] were developed to

maintain quality of service requirements of each link. Inparticular, Foschini and Miljanic [5] proposed a more generaand realistic model in which a positive receiver noise and a user-specific target SIR were taken into account. This distributedalgorithm was proven to converge either synchronously [5] orasynchronously [6] to a fixed point of a feasible system. Basedon this, Grandhi and Zander suggested a distributed constrainedpower control (DCPC) algorithm [7], in which a ceiling wasimposed on the transmit power of each user. Another distributedalgorithm was proposed by Bambos et al. [8], which aimed atprotecting the active links from performance degradation whennew users try to access the channel. In [9] and [10], second-order power control (CSOPC), state space-based distributed

power control design (SSCD) and its optimization arerespectively proposed to efficiently manage the userinterferences so that the network capacity can be increased.

However, the radio channel ultimately places fundamentalimitations on wireless communication systems since the pathbetween the transmitter and the receiver can vary from simpleline-of-sight to one that is severely obstructed by buildings,mountains, and foliage. Unlike wired networks, radio channeuncertainties in a wireless network for instance path loss,shadowing, Rayleigh fading can attenuate the power of thetransmitted signal at the receiver and thus cause variations in thereceived SIR and therefore degrading the performance of anydistributed power control (DPC) scheme. Low SIR at the

receiver will result in high bit error rate (BER), which isunsatisfactory for 3G systems, where voice and video will betypically transmitted.

Though the earlier DPC schemes may be accurate in arrivingat a power value, they all neglect the changes observed in the

DISTRIBUTED POWER CONTROL OFCELLULAR NETWORKS IN THE PRESENCE

OF RAYLEIGH FADING CHANNEL1

S. Jagannathan Maciej Zawodniok and Q. ShangEmbedded Systems and Networking Lab Embedded Systems and Networking Lab.

Department of Electrical and Computer Engineering Department of Electrical and Computer EngineeringUniversity of Missouri-Rolla University of Missouri-Rolla

1870 Miner Circle, Rolla MO 65409, USA. 1870 Miner Circle, Rolla MO 65409, [email protected] [email protected]

T

0-7803-8356-7/04/$20.00 (C) 2004 IEEE IEEE INFOCOM 2004


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radio channel. In fact, they all assume that: 1) path losscomponent is only present, 2) no other channel uncertainty existsand 3) the mutual interference among users is held constantduring power update for each user. Moreover, the slower rate ofconvergence of these algorithms and the associated powerupdates is of an issue in a highly dynamic wireless environmentas shown in this paper. The proposed work overcomes these

limitations.In this paper, we propose a novel DPC scheme for the next

generation wireless networks in the presence of radio channeluncertainties. The rate of convergence of the proposed DPCscheme to attain a target SIR is faster compared with the existingschemes. The proposed algorithm estimates the time varyingchannel accurately, and uses this information in the powerupdate so that a desired SIR is maintained at the receiverprovided the channel variation is slow compared to itsestimation. The algorithms being highly distributive in naturedoesn't require inter-link communication, centralizedcomputation, and reciprocity assumption as required in acentrally controlled wireless environment. In addition, the

proposed DPC scheme converges to any target SIR value in thepresence of channel uncertainties. As the necessity of inter-linkcommunication is eliminated, network capacity increasessignificantly and easy controlled recovery from error events ispossible. Also, a comparison of the proposed DPC with certainearlier algorithms is included.

The organization of the paper is as follows. Section IIdescribes wireless channel with uncertainties. In Section III, asuite of DPC schemes is studied and a novel DPC scheme isproposed along with convergence proofs. Section IV presentsthe simulations whereas Section V carries the conclusions.

II. RADIOCHANNELWITHUNCERTAINTIESThe radio channel places fundamental limitations on wireless

communication systems. Unlike wired channels that arestationary and predictable, radio channels involve manyuncertain factors such as path loss, the shadowing, and Rayleighfading, so they are extremely random and do not offer easyanalysis. These channel uncertainties can attenuate the power ofthe signal at the receiver and thus cause variations in the receivedSIR and therefore degrading the performance of any DPCscheme. Therefore we focus our effort on these main channeluncertainties before the DPC development. They are discussednext.

A. Path Loss

If only path loss is considered, the power attenuation istaken to follow the inverse fourth power law [11]:

n

ij

ijd

gg = (1)

where gis a constant usually equal to 1 andijd is the distance

between the transmitter of thejth link to the receiver of the ith linkand n is the path loss exponent. A number of values for n havebeen proposed for different propagation environments

depending on the characteristics of the communication mediumA value of n=4 is taken in our simulations, which is commonlyused to model path loss in an urban environment. Furtherwithout user mobility,

ijg is a constant.

B. Shadowing

High building, mountains and other objects block thewireless signals. Blind area is often formed behind a highbuilding or in the middle of two buildings. This is often seenespecially in large urban areas. The term 1.010 is often used tomodel the attenuation of the shadowing to the received power[12][20], where is assumed to be a Gaussian random variable.

C. Rayleigh Fading

In mobile radio channels, the Rayleigh distribution iscommonly used to describe the statistical time varying nature ofthe received envelope of a flat fading signal, or the envelope ofan individual multipath component. The Rayleigh distributionhas a probability density function (pdf) give by [11]

0 is the thermal noise at its receiver

node. For each link i there is a lower SIR threshold i.Therefore, it is clear that

( ) *iii

tR , (5)

for every i = 1,2,3,n . The lower threshold value for all linkscan be taken equal to for convenience, reflecting a certain

QoS the link has to maintain in order to operate properly.

An upper SIR limit is also set, in order to decrease theinterference due to its transmitter power at other receiver nodes.In the literature, several DPC schemes have been proposed and itis very difficult to present all of them. The most recent andimportant work include DPC with active link protection [8],CSOPC [9], SSCD and optimal [10][16][18] and they arediscussed next.

A. DPC Scheme with Link Protection [8]

When equation (5) is not satisfied (i.e. ( ) ii tR < ), each link

independently increases its power if its current SIR is below its

targeti , and decreases it otherwise using the power update

[8][13]

)(

)()1(

lR

lPlP

i

iii

=+ (6)

where l=(1,2,3), lTt= and T is the sampling interval. When

max)1( PlPi >+ , the new link is not added. Otherwise when

min)1( PlPi


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where by definition)(

)()(

lI

lPlR

i

ii =

, and interference

+=

n

ij ii

i

ii

ijji

gg

glPlI

)()( , with n is the number of active links.

The feedback input ( )lvi for each link should only depend upon

the total interference produced by the other users. To maintainthe SIR of each link above a desired targetiand to eliminate

any steady-state errors, the error in SIR, iii lRlx = )()( , has to

be minimal by appropriately selecting a power update for the thi link. The closed-loop system is expressed as

)()()1( lvlxlx iii +=+ (12)

The closed-loop system (12) can now be used to select a DPCscheme. First of all, each transmitter-receiver pair now isrepresented as a closed-loop system (12), which is subsequentlyused to design a DPC scheme. Moreover, analytically, theconvergence can be demonstrated unlike other power controlschemes.

Theorem 1[10]:Given the closed-loop SIR system in (12), and ifthe feedback for the ith transmitter is chosen as

iii lxklv += )()( 1 with 1k representing the feedback

gain, i is the protection margin, and the power is updated as

( ) ( ) ( ) ( )lIlvlPlP iiii +=+1 , the closed loop SIR system for each

link is stable, and the actual SIRs converge to theircorresponding target values.

Proof: See [16].

Remark: The transmission power is subject to the

constraint maxmin PPP i where minP is the minimum needed to

transmit,maxP is the maximum allowed power and ( )lPi is the

transmission power of the mobile i.

D. Optimal Control Algorithm

Performance criterion, ( )

=

+

1iii

T

ii

T

i vQvxTx, is used to arrive at

an optimal DPC, which can be stated as follows:

Theorem 2 (Optimal Control)[10]: Given the hypothesispresented in the previous theorem for DPC, with the feedbackselected as

iiii lxklv += )()( , where the feedback gains are

taken as

i

T

iii

T

ii GSHTHSHk

+=

1)( (13)

where Si is the unique positive definite solution of the Algebraic

Ricatti Equation (ARE)

ii

T

iiii

T

iiii

T

ii QGSHTHSHHSSGS ++= ])([ 1 (14)

Then the resulting time invariant closed loop SIR systemdescribed by

)()()1( lxkIlx iiii =+ (15)

is asymptotically stable.

Proof: See[10]. Only path loss component is considered both in SSCD andoptimal DPC schemes.

E. Proposed DPC AlgorithmAs indicated earlier, in the previous DPC schemes, only path

loss uncertainty is considered. Moreover, the DPC algorithmsappear to be slow in convergence and the outage probability isquite high in the presence of channel uncertaintiesConsequently, the performance of these DPC schemes fails torender satisfactory performance as shown in our simulations.The proposed work is aimed at demonstrating the performance inthe presence of several channel uncertainties.

In the time domain, however, the channel is time-varyingwhen channel uncertainties are considered and therefore ( )tgij

is

not a constant. In [15], a new DPC algorithm is presentedwhere ( )tgii is treated as a time-varying function due to Raleigh

fading by assuming that the interference ( )tIi is held constant

Since this is a strong assumption, in this paper, a novel DPCscheme is given where both ( )tgii and the interference ( )tIi are

time-varying, and channel uncertainties are considered for all themobile users. In other words, in all existing works[8][9][10][18], both ( )tgij and ( )tPj are considered to be held

constant, whereas in our work, this assumption is relaxed.

Considering SIR from (4) where the powerattenuation ( )tgij is taken to follow the time-varying nature of the

channel and differentiating (4) to get

( )( ) ( )( ) ( ) ( ) ( )( ) ( )

( )tI

tItPtgtItPtgtR

i

iiiiiii 2

''' = (16)

where ( )'tRi is the derivative of ( )tRi and ( )'

tIi is the derivative

of ( )tIi .

To transform the differential equation into the discrete timedomain, ( )tx ' is expressed using Eulers formula as

( ) ( )T

lxlx +1 , where T is the sampling interval. Equation (16)

can be expressed in discrete time as



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( )( ) ( )( ) ( ) ( ) ( )( ) ( )

( )

( )( ) ( )( ) ( ) ( ) ( )( ) ( )lIlPlglIlPlg

lI

lI

lIlPlglIlPlglR

iiiiiii

i

i

iiiiii

i

''

2

2

'''

[1

+=

=

( ) ( )( ) ( ) ( ) ( ) ]'

+

tlPlglPlg iij

jijiii (17)

In other words,( ) ( )

( )( ) ( )[ ]

( )( )

( )( ) ( )[ ]

( ) ( )

( )

( ) ( )( )

( ) ( )( )

++

+

++

+=

+

ijij

jj

j

ijij

i

iii

iiii

i

iiiii

i

ii

lgT

lPlPlP

T

lglg

lI

lPlg

T

lPlPlg

lIlP

T

lglg

lI

T

lRlR

11

1111

1

2

(18)

Canceling T on both sides and combining to get

( )( ) ( )

( )

( ) ( )[ ] ( ) ( ) ( )[ ] ( )

( )

( ) ( )( )

( )lI

lPlglR

lI

lglPlPlPlglg

lg

lglglR

i

iiii

i

ijijjjjijij

ii

iiii

i

1

111

1

++

+++

+

=+

(19)Now, define

( )

( ) ( )

( )

( ) ( )[ ] ( ) ( ) ( )[ ] ( )

( )

( )

( )

( ) ( ) ( ) ( )

( )lI

lglPlPlg

lg

lg

lI

lglPlPlPlglg

lg

lglg

l

i

ijijjjij

ii

ii

i

ijijjjjijij

ii

iiii

i

+

=

+++

+

=

111

,

(20)where

( ) ( )lgl iii = , (21)

and

( )( )

( )lIlP

lvi

ii

1+= (22)

Equation (19) can be expressed as( ) ( ) ( ) ( ) ( )lvllRllR iiiii +=+1 (23)

with the inclusion of channel noise, equation (19) is written as

( ) ( ) ( ) ( ) ( ) ( ) ( )llrlvllRllR iiiiiii ++=+1 (24)

where ( )li is the zero mean stationary stochastic channel noise

with ( )lri is its coefficient.

The SIR of each link at time instant, l, is obtained using (24)Carefully observing (24), it is clear that the SIR at the time

instant, 1+l , is a function of channel variation from time instantl to 1+l . The channel variation is not known beforehand andthis makes the DPC scheme development difficult andchallenging. Since is not known, it has to be estimated forDPC development. Note available DPC scheme [1-13,15-19]ignore the channel variations and therefore they renderunsatisfactory performance.

Now define ( ) ( )kRky ii = , then equation (24) can be expressed

as( ) ( ) ( ) ( ) ( ) ( ) ( )llrlvllylly iiiiiii ++=+1 (25)

The DPC development is given in two scenarios.

Case 1:i , i and ir are known. In this scenario, one can

select feedback as( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]lekllrlylllv iviiiiii ++=

1 (26)

where the error in SIR is defined as ( ) ( ) = lRle ii . This results

in

( ) ( )lekle ivi =+1 (27)By appropriately selecting

vk via placing the eigenvalues

within a unit circle, it is easy to show that the closed-loop SIRsystem is asymptotically stable [21] in the mean orasymptotically stable, ( ){ } 0lim =

leE il

. This implies that

( ) lyi .

Case 2:i , i and ir are unknown. In this scenario, equation

(25) can be expressed as

( ) ( ) ( )[ ]( )

( )( ) ( )

( ) ( ) ( ) ( )lvlll

lvll

lylrlly

iii

T

i

ii

i

i

iii

+=

+

=+1 (28)

where ( ) ( ) ( )[ ]lrll iii = is a vector of unknown parameters

and ( )( )

( )

=

l

lyl

i

i

i

is the regression vector. Now selecting

feedback for DPC as

( ) ( ) ( ) ( ) ( )

++=

lekllllv iviiii 1 (29)

where ( )li

is the estimate of ( )li , then the SIR error system isexpressed as

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )lllek

lllllekle

i

T

iiv

i

T

ii

T

iivi

~

1

+=

+=+

(30)

where ( ) ( ) ( )lll iii

= ~

is the error in estimation. From (30)

it is clear that the closed-loop SIR error system is driven bychannel estimation error. If the channel uncertainties are



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properly estimated and the transmitter power is selected toovercome these uncertainties, then estimation error tends to zero.In this case, equation (30) becomes (27). In the presence of errorin estimation, only a bound on the error in SIR can be shown. Inother words, the power control scheme will ensure that the actualSIR is close to its target or bounded. In other words, the actualSIR is within the protective margin provided the channel

uncertainties are properly estimated. Figure 1 illustrates theblock diagram representation of the proposed DPC wherechannel state estimation and power selection are part of thereceiver. To proceed further, the Assumption 1 is required.

*TPC: Transmitter power control

Figure 1. Block diagram representation of DPC.

Assumption 1: The channel changes slowly compared to theparameters updates.

Remark: The channel variation has to be slower than the speedof estimation. In fact, the estimation scheme is computationallysimple that it can be executed extremely fast.

Theorem 3: Given the DPC scheme above with channel

uncertainties (Path loss, Shadowing and Rayleigh fading) areestimated accurately (no estimation error), if the feedback fromDPC scheme is selected as (29), then the mean channelestimation error along with the mean SIR error converges to zeroasymptotically, if the parameter updates are taken as

( ) ( ) ( ) ( )11 ++=+

lelll Tiiii (31)

provided

( ) 12


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The system is simulated with different DPC algorithms with 100users.

A.Performance Metrics

Outage probability and total power consumption are chosenas metrics to evaluate the performance of the power controlscheme. Outage probability is defined as the probability of

failing to achieve adequate reception of the signal due to co-channel interference. It is defined as the ratio of the number ofdisconnected or handed over users to that of the total number ofusers in the system. The total power consumed by the mobileswill be the sum of all the powers of mobiles in the network.Given these metrics, the power per active mobile can be derived.

B. Results

In the first few simulations, the users in each cell are placedat random and they are considered stationary for convenience toevaluate the proposed DPC. Later, the users are mobile.

Stationary Users: Case I: Constant and Abruptly Changing

Channel

Figure 2. Channel fluctuations.

In this scenario, we select the parameters as 01.0=vk and

01.0= . Figure 2 shows the change ofiig with time as a result

of channel fluctuation, which obeys the path loss, Rayleighfading and shadowing behaviors. Though channel changessharply at every ten time units,

iig is changed once in every 10

time units and it is held constant otherwise or between changes.

Figure 3 illustrates the actual SIR of a randomly selectedmobile user. From this figure it is clear that the proposed DPCscheme is the only scheme that maintains the target SIR in thepresence of channel variations, while the others [8,9,10,18] failto meet the target. Moreover, using the proposed scheme theactual SIR converges faster (about 2 time units) compared to

CSOPC and optimal (10 time units). Though the DPC in [8]converges similar to the proposed, it is unable to maintain theactual SIR to its target.

Figure 3. SIR of a randomly selected user.

Figure 4. Total power consumed.

Figure 4 presents the total power consumed by all themobiles in the network. The result shows that all the schemesconsume about 90mw. From Figure 5 the outage probabilityusing the proposed DPC scheme is significantly less (near zero)when compared to all the other schemes (about 85%), i.e., ourapproach can accommodate more number of mobile usersrendering high channel utilization or capacity in the presence ofchannel variations. This together with similar total powerconsumption suggests that the average power consumed peractive user is less using the proposed DPC compared to otherschemes.

The incorporation of channel state estimator with theproposed DPC scheme allows the mobiles to consume lowpower. The channel state estimator was able to accurately obtain



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an assessment of the prevailing channel state and its variation

with background noise, which is specified in terms of i (k).The proposed DPC uses the channel state estimate and the errorin the SIR to arrive at a more appropriate transmitter power valuefor the next time instant (or for the packet) whereas the otherDPC schemes do not. Consequently, the user interferenceexperienced by each mobile is higher when other DPC schemes

are employed (due to large power values for each mobile)whereas it is low with the proposed DPC.

Figure 5. Outage probability.

Higher transmitter power of a mobile user can cause largeinterference to others resulting in an increase in transmitterpowers of all mobiles until an equilibrium state is reached for theentire system. With the proposed DPC, an equilibrium statewhere the transmitter powers of all mobiles will be lower thanwhen other DPC schemes are used, will be achieved. Therefore,

a good channel estimator working in conjunction with the DPCis critical. Moreover, as per Assumption 1, when the channelcondition changes suddenly at time instants 10, 20, 30 and 40 ofFigure 2, the channel state estimator is unable to keep up.Though this is a disadvantage, but for most situations, theproposed DPC with the channel state estimator is able to performsuperior than other available DPC schemes.

Case II: Slowly Varying Channel

In this scenario, 01.0=vk and 3= . In this case, though thechannel changes every ten-time units and satisfies the path loss,Rayleigh fading and shadowing behavior, the channel variationis considered as a linear function between the changes (see

Figure 6 of a randomly selected mobile). In this case also, theproposed DPC scheme renders a low outage probability of about30% as observed in Figure 9 while consuming less power peractive mobile and converging faster since the proposed DPCscheme maintains the SIR of each link closer to its target

compared to others as displayed in Figure 7. Other schemesattain an outage probability of 85%. The low outage probabilityfor the proposed DPC scheme is the result of faster convergencebetter estimation of the channel state and using this informationto select an appropriate transmitter power to maintain a low SIRerror.

Figure 6. Channel variation with time.

Figure 7. SIR of a randomly selected user.

The performance of the channel state estimator can be easilyobserved by comparing the outage probability plots for the caseswhen the channel changes abruptly but remains constant betweenchanges and the case where it changes smoothly. In the case ofchannel changing abruptly, the estimator was unable to obtain anaccurate assessment at the instant when the channel condition

changes whereas in other instants, it was able to estimate thechannel condition accurately and subsequently the proposedDPC was able to select a suitable transmitter power as presentedin Figure 5.



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Figure 8: Total power consumed.


On the other hand, as expected in the case of smooth channelvariation, the outage probability changes with the variation in thechannel state (See Figure 9). This is due to the delay in thechannel estimation and the selection of the transmitter powervalue using this estimate (the channel state changes by the timethe power is calculated and feedback, which results in amismatch). Also, this is reflected in the error in SIR plot shownin Figure 7 for a randomly selected mobile. In any case, theoutage probability presented in Figure 9 is significantly lower forthe proposed DPC when compared with the other schemes due tothe inclusion of the channel state estimator. Note also that the

performance of the channel estimator depends upon the rate ofchange of the channel state. The change in the channel conditionillustrated in Figure 6 is slightly higher (slope of the curve inFigure 6) than the scenario presented in Assumption 1.

For much slower change in the channel conditions (resultsnot shown), the estimator performance is much superior whencompared to the case in Figure 6. Consequently a much loweroutage probability was observed. This clearly demonstrates thatthe addition of a channel state estimator is critical for any DPCscheme even though there could be feedback delays.

Performance Evaluation with Number of Users



When the total number of mobile users in the cellularnetwork varies, the total power consumed and the resultingoutage probability was compared. In this simulation scenario

the number of mobile users trying to gain admission increases by25 from the previous value. Figures 10 and11 display theperformance of the DPC in terms of total power consumed andoutage probability respectively when the channel changesslowly.As expected, the proposed scheme renders significantly



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low outage probability while ensuring low power per activemobile user compared to other schemes.

As expected, a linear increase in power consumption withusers is observed with all the DPC schemes though. theproposed DPC maintains a low outage probability consistentlyunder 40% when compared with other schemes. Similar totalpower consumption together with low and consistent outage

probability indicates that the proposed DPC was able to select asuitable transmitter power value thus lowering mutualinterference between users even in the presence of channeluncertainties.

Mobile User Scenario

Figure 12 illustrates the initial location of 100 mobile userswhereas Figure 13 presents their final location. Users in thecellular network try to move in any one of the 8 pre-defineddirections chosen at random at the beginning of the simulation. Auser can move a maximum of 0.01 km per time unit. Since thetime unit is small, the 0.01 km is a considerable amount ofdistance for any mobile user. The channel state is also changed

slowly for each mobile user. Figures 14 and 15 illustrate thetotal power consumed and the corresponding outage probability.

Figure 12. Initial state of mobile placement.

As evident from the result, the proposed DPC renders a loweroutage probability (average of about 30%) compared to others(about 90%) while consuming low power per active mobile evenin the presence of user mobility. The variation in outageprobability displayed in Figure 15 with respect to time units isexplained in the case of stationary user scenario with slowlyvarying channel.

As explained earlier, the mobility can be expressed as part ofthe uncertainty. In other words, the channel state now is afunction of user mobility and can be included as part of theparameter estimation process. As a result, the proposed DPC

was able to assess the accurate transmitter power for each mobilewithout the knowledge of its location based on the channel statethus lowering the mutual interference. This results in loweroutage probability as displayed in Figure 15.

Figure 13. Final state of mobile placement.





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V. CONCLUSIONSIn this paper, a novel DPC schemes is presented for cellular

and can be potentially extended to ad hoc networks capturing theessential dynamics of power control. It was observed that theproposed DPC scheme allows fully distributed power and hasrendered better performance in the presence of radio channeluncertainties. The inclusion of channel state estimator is very

critical in the design of the DPC. The simulation results showthat the proposed scheme converges faster than others, maintainsa desired target SIR value for each link and can adapt to the radiochannel variations better. In the presence of channeluncertainties, the proposed scheme can render lower outageprobability (an improvement of about 50%), using significantlyless transmitter power per active user compared to other DPCschemes. As a result, the proposed DPC scheme offers asuperior performance in terms of convergence and it maximizesthe network capacity compared to the available ones in theliterature. Simulation results justify theoretical conclusion.

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[17] M. Rintamaki and H. Koivo, "Adaptive robust power control for WCDMAsystems",Proc. of IEEE Vehicular Technology Conference, Vol.1, pp.62 66, 2001.

[18] A. l-Osery and C. Abdallah, "Distributed power control in CDMA cellularsystems", IEEE Antennas and Propagation Magazine, Vol. 42, No.4August 2000.

[19] Kin Kwong Leun, Chi Wan Sung, Wing Shing Wong and Tat Ming Lok,"Convergence theorem for a general class of power control algorithms",

IEEE International Conference on Communications, Vol.3, pp.811 -8152001.

[20] B. Hashem and E. Sousa, "Performance of cellular DS/CDMA systemsemploying power control under slow Rician/Rayleigh fading channels"

Proc. of the International Symposium on Spread Spectrum Techniques and

Applications, Vol.2, pp.425 -429, Sep. 1998.[21] K.J.strm and B.Wittenmark,Adaptive control, Addison-Wesley

Company, Reading, Massachusetts, 1989.[22] T. S. Rappaport, Wireless Communications: Principles and Practice

Prentice Hall, 1999.

APPENDIX

Proof of Theorem 3: Define the Lyapunov function candidate

( ) ( ) ( ) ( )

+= llleleJ i

Tii

T

ii

~~1

(A.1)

whose first difference is

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

+++

++=

+=

llll

lelelele

JJJ

iT

iiT

i

i

T

ii

T

i

~~~~

21

111

11

(A.2)

Consider1J from (35) and substituting (30) to get

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )lelellleklllek

leleleleJ

i

T

ii

Tiiv

T

i

Tiiv

i

T

ii

T

i

+

+=

++=

~~

1

11(A.3)

Taking the second term of the first difference from (A.2) andsubstituting (31) yields

( ) ( ) ( ) ( )

( )[ ] ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

+

++

=

++=

llleklllekll

lllllllek

llllJ

i

Tiiv

T

i

Tiivi

T

i

i

Ti

T

i

Tii

Ti

T

iv

iT

iiT

i

~~

~~~

~~~~

2

22

111

(A.4)Combining (A.3) and (A.4) to get



12/12

( ) ( ) ( )( ) ( )

( ) ( ) ( )[ ] ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( )( ) ( )

( )

( )( )

2

2

2~

222

max

~~

~

1

11

1

2

1

lekl

lll

llek

llllll

lllekll

lekkllIleJ

iv

i

i

i

Ti

iiv

i

Ti

T

i

Tii

T

i

i

Ti

T

ivi

T

i

iv

T

vi

T

i

T

i

+

+=

(A.5)

where is given after (33). Taking now expectations on bothsides yields

( ) ( ) ( ) ( )(

( ) ( )( )

( )( ) )

1

11(

2

2

2~

222max

lekl

lll

llekEJE

iv

i

i

i

Ti

iiv

+

(A.6)

Since ( ) 0>JE and ( ) 0JE , this shows the stability in the

mean via sense of Lyapunov provided the conditions (32) and

(33) hold, so ( )[ ]leE i and ( )

lE i

~

(and hence ( )

lE i ) are

bounded in the mean if ( )[ ]0leE i and ( )

0

~

lE i are bounded in a

mean. Sum both sides of (A.6) and take limits ( )JEl

lim , the

SIR error ( )[ ] 0leE i .

Proof for Theorem 4: Define a Lyapunov function candidate asin (A.1) whose first difference is given by (A.2). The first term

1J and the second term 2J can be obtained respectively as

( ) ( ) ( )[ ] ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( )[ ] ( ) ( ) ( ) ( ) ( )TiT

ii

TT

iv

T

ii

T

i

Ti

i

T

i

T

iviv

T

v

T

i

lelelelllek

llllll

llleklekkleJ

++

+

+

+=

22

2

~~

~

1

(A.7)

( )[ ] ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )llll

llekllllell

llleklllekll

lllllllekJ

T

i

T

i

T

ivi

T

i

T

ii

T

i

i

Tiiv

T

i

Tiivi

T

i

i

Ti

T

i

Tii

Ti

T

iv

+

+

+

++

=

212

22

~~

~~~

2

(A.8)

Using (A.8) and completing the squares for ( ) ( )ll iT

i ~

yields

( ) ( )( )

( )

( )( ) ( ) ( ) ( )( )

( ) ( )( )

2

2

2~

2

2

2max

2max

2

max22max

11

11

21

llekl

llll

kle

k

lklekJ

iv

i

i

i

Tii

N

v

iN

v

iv

iv

+

(A.9)

with is given after (33). Taking expectations on both sides toget

( ) ( )

( )( )

( )

( )( ) ( ) ( ) ( )( )

( ) ( )( ) )1

1

11

2

1(

2

2~

2

2

2max

2max

2

max2

2max

llekl

llll

kle

k

lkle

kEJE

iv

i

i

i

Tii

N

v

iN

v

iv

i

v

+

+

(A.10)as long as (32) and (33) hold, and

( )[ ] ( ) ( ) +> max2max11

vN

v

i kkleE (A.11)

This demonstrates that ( )JE is negative outside a compact set

U. According to a standard Lyapunov extension, the SIR error

( )[ ]leE i is bounded for all 0l . It is required to show that ( )li

or equivalently ( )li~

is bounded. The dynamics in error in the

parameters are

( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ]TiviiiT

ii llekllllIl +=+~~

1 (A.12)

where SIR error, ( )lei

, is bounded and estimation error, ( )l , is

bounded. Applying the persistency of excitation ( )l [21], one

can show that ( )li~

is bounded.

distributed power control of

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