NEW DISTRIBUTED POWER CONTROL ALGORITHMS FOR MOBILE COMMUNICATIONS

Most distributed power control algorithms have been proposed assuming constant interference and constant path gain. These considerations may result in lower performance gains in fast time-varying channel conditions. In this work, we present two algorithms that address this problem efficiently outperforming the classical Distributed Power Control (DPC) algorithm.


INTRODUCTION
In cellular wireless systems, good communication can be efficiently provided by ensuring just a minimum signal quality for individual connections.
By appropriately adjusting transmission power levels, minimum link quality requirements can be attained without incurring in unnecessary interference generation.This technique is called power control.
However. the employment of this technique is not trivial when we strive with a multipath environment, where fast fading occurs, since SINR (Signal-to-Interference plus Noise Ratio) depends on the path gain and the co-channel interference, which are influenced by fast fading.Fast fading, also called short-term fading or multipath fading, is the phenomenon that describes the rapid amplitude fluctuations of a radio signal over a short period of time or travel distance.These rapid fluctuations cause degradation in the action of power control [1].
, Some papers have studied the performance of power control in fast fading environment.In [2], a new algorithm is derived from the classical Distributed Power Control (Ope) .. algorithm [3], considering a time-varying path gain.In [4], a neural network is used to predict the future channel R. A. de Oliveira Neto, F. de S. Chaves, F. R. P. Cavalcanti and T. F. Maciel are with Grupo de Pesquisa em Telecomunica90es Sem-Fio (GTELl.Universidade Federal do Ceara (UFC).(E-mails: {neto, fabiano.rodrigo, maciel}@gtel.ufc.br)65 conditions.Similarly, [5] and [6] use adaptive filters, with tap weights updated by least-mean-square (LMS) and recursive least-square (RLS) algorithms, respectively, in order to predict the future path gain.
In this work, two new distributed power control algorithms are presented, outperforming the classical DPC algorithm [3] and Lee algorithm [2] when a time-varying channel is considered.These new algorithms differ from the DPC in that its deduction assumes both path gain and interference to be time-varying functions.

DPC ALGORITHM
OPC algorithm can be deducted from a differential dynamic that makes the SINR of each user evolve towards a target SINR value by an amount proportional to the offset from its current SINR.This dynamic can be expressed by the following time-continuous differential equation [3]: (1) where the upper symbol' in p,(t) denotes the derivative with respect to time, (3 is a positive constant between 0 and I which controls the convergence rate of the algorithm, PI is the target SINR and Pi (t) is the instantaneous SINR of the ill, link expressed by: where g,(t) is the path gain, Pi(t) is the transmitter power and lilt) is the total interference (including noise).The total interference Ii (t) is defined as: where v is the thermic additive noise.
Considering only the SINR to be time-variant, that is, assuming that the path gain and interference do not vary between two consecutive iterations, the algorithm becomes: In [3], it is proved that when {J = 1, the convergence speed is higher.So, setting j3 = 1, eq. ( 4) becomes: When we use this algorithm in a time-varying channel, according to eq. ( 2), the SINR in instant k + 1 is: .(/,: 1) _ 9;(1.' + 1) Ii(k) P, + -Pt ' 9;(1.')'1;(1.'+1)'(6) As seen in eq. ( 6), the result of DPC algorithm is disturbed by path gain and interference variations.In order to overcome this problem, we derive new algorithms that can be considered evolutions of the DPC algorithm.

NEW ope ALGORITHMS
In this section, we propose two new distributed power control algorithms that can be considered improvements of the classical DPC algorithm [3].
So, the obtained SINR in discrete time I.. + 1 is: Pt .

ALGORITHM 2
The discrete-time SINR P; (1.') of a link is given by: 9i(k) .pi(k) So. the instantaneous transmit power necessary to balance this link for Pi (k) = PI, for all instants k, satisfies the following equation: We do not predispose of the values of I; (1.') and 9; (k).because these are instantaneous values.In order to solve this problem, we propose a simple prediction method based on Taylor's Series.
Taylor's Series is used to expand continuous functions f(x) in the following form [7]: where the term P")(x) represents the nth derivative of f(x) with respect to x. Due to (:1' -xo)f] and n!, when.r and Xo are adjacent values, the higher order terms can be neglected.Thus, keeping only the first two terms of the series, we have: Now, we transform (14) into a discrete time equation.For this, we assume that xo is the current discrete time instant k and x the next instant k + 1.Further, l' (xo) is substituted by In this way, we obtain: Therefore, we can use (IS) in order to predict the path gain and interference: So, using (12), ( 16) and (17), the transmit power at instant (k + 1) is expressed by the following proposed algorithm: .' _ .1, The obtained SINR in discrete time 1.' + 1 is: Note that when the estimates tend to correct values, that is, g(1.: + 1) ::::: g(l, + 1) and i(k + 1) ::::: 1(1.: + 1), the SINR tends to PI.

RELATIONSHIP BETWEEN PROPOSED ALGORITHMS AND DPe
It is interesting to compare the DPe algorithm with algorithms I and 2. Algorithm 1 (eq.9) comes directly from eq. (5) using eqs.(7) and (8).The obtained SINR is expressed in (10).
Analyzing eq.(10), we can conclude that algorithm 1 works better when the two sequences gi(1.: -1), gi(I.:),gi(k + 1) and Ii(k -1), Ii(k), Ii(k + 1) vary similarly to a geometric progression.This can be seen in eqs.(7) and (8), where according to these approximations, the term of time k is the geometric mean of terms of time k -1 and k + 1.As in algorithm 2, it is possible to interpret eqs.(7) and (8) like predictions of the terms gi (k + 1) and Ii (k + 1): With respect to algorithm 2, it has the same format of DPC, in spite of the fact that DPC is deducted from a differential equation.The update equation of DPC can be written simply as: gi(1i: while the proposed algorithm can be expressed as: The difference comes from the fact that the standard DPC algorithm is based on past information about path gain and interference, causing performance degradation. The obtained SINR by algorithm 2 is expressed in eq. ( 19).Expanding this equation we have: Analyzing the above equation, we can conclude that the algorithm 2 works better when the two sequences gi(k -1), gi (k), gi (I.: + 1) and Ii (k -1), Ii (k), Ii (k + 1) vary similarly to an arithmetic progression.This can be seen in eqs.( 16) and (17), where according to these approximations the term of time k is the arithmetic mean of terms of time k -1 and 1.:+1.

SIMULATION RESULTS
We now illustrate the performance of the proposed algorithms by simulations in the context of a single cell CDMA (Code Division Multiple Access) system in the uplink direction.

SIMULATION MODEL
A snapshot simulation model is assumed where mobile stations are uniformly distributed over the cell area.The actuation period of the power control algorithm is 1 ms, unless otherwise stated.Other simulation parameters are set as follows.
The cell radius is set to 1.5 km and omnidirectional antennas are considered.
A simplified path loss model is used, where P L (d) . The distance d is expressed in kilometers and represents the distance between mobile and base stations.Shadowing standard deviation is assumed 6 dB.Fast fading is implemented following Jakes model [8] with three different Doppler spreads: 18.5 Hz, 55.5 Hz and 92.5 Hz.It is assumed that the measurements gi (t) and Ii (t) are exact for each algorithm.
The processing gain of the simulated COMA system is assumed 21 dB and the noise power is set to -110 dBm.The target SINR PI for all algorithms is set to -12 dB resulting in a target bit energy per interference spectral density ratio, (Eb/No)l, of 9 dB after despreading.Maximum mobile station transmit power is limited to 21 dBm and the initial transmit power is set to the minimum transmit power (-70 dBm).

LINK-LEVEL RESULTS
Fig. 1 shows a sample of the Eb/No evolution achieved by a given user in a typical snapshot for DPC, Lee algorithm, algorithms 1 and 2. In this case, ten mobile stations are placed in the cell and the same system configuration and fading realizations are used for all algorithms, with Doppler spread 55.5 Hz.
From fig. 1, it is clearly observable that the proposed algorithms are able to stabilize the E b / No around the target (Eb/No)t better than DPC algorithm and Lee algorithm.In other words, the mean squared error (MSE) between the actual and the target Eb/No is smaller for the proposed algorithms than for DPC and Lee algorithm.This behavior was observed for all snapshots and other values of Doppler spreads, as it is shown in table 1, in which it is also presented the MSE for Doppler spreads 18.5 Hz and 92.5 Hz, for the same configurations used in fig. 1 , ------------------------  by larger variability of the interference.Similar conclusions Table 1.MSE in dB between the actual and the target E b / No. are valid for the path gain prediction (eqs.16 and 20).
A sample of how algorithms I and 2 perform with respect to interference prediction is shown in fig. 2 for a single snapshot.This figure presents the behavior of the tracking performance of the interference prediction for the Doppler spread 18.5 Hz, using eqs.( 17) and (21).It can be observed that prediction of both algorithms achieves good performance for the interference.The same behavior was observed typically in all other snapshots.Evidently, as the Doppler spread increases, it occurs a performance loss caused

SYSTEM-LEVEL RESULTS
Now, we illustrate how the superior tracking capability tJ' of the proposed algorithms translates into a system-level advantage.In practical systems, it is difficult to keep the Eb/N o exactly at the target value, especially for high speeds [1].Therefore, we assume an Eb/N o margin below the target (Eb/Nol t in which signal quality is assumed acceptable.We simulated 5000 snapshots for several system loads and calculate the average fraction of time in which the achieved Eb/NO is below the target (Eb/NO)t by margins of 0.3 dB, 0.5 dB and I dB.The simulated maximum load N max was determined by the pole capacity equation given by [9]: ."rrlecrl'7'l-t-+----1eJ -110 --0-Interference prediction (eq.21) ..  Comparison between actual and predicted interference using the proposed prediction methods, for the Doppler spread 18.5 Hz. (25 where PG is the processing gain. Table 2 shows the average fraction of time in which the achieved Eb/NO is below the target (Eb/NO)t for each simulated margin and Doppler spread 18.5 Hz.It can be observed that the employment of the new algorithms allows for a significant capacity gain when compared to the ope and Lee algorithm, for all simulated margins.For each margin and load system, the smallest value is highlighted, showing that the proposed algorithms outperform the DPC and Lee algorithm.Moreover, we can see that the DPC and Lee algorithm are more sensible to margin variations than the other algorithms.In other words, the proposed algorithms are more robust.Tables 3 and 4 show the performance of the algorithms when higher Doppler spreads are considered.The Doppler spreads are 55.5 Hz and 92.5 Hz.As the channel variation rate increases, a performance decrease for all algorithms is expected.However, it can be observed that the proposed algorithms still outperform the DPC algorithm and Lee algorithm.
With regard to the sensitivity of the algorithms in relation to Doppler spread, we can conclude that proposed algorithms, DPC and Lee algorithm present similar sensitivity, however the proposed algorithms are more efficient.

EFFECT OF ACTUATION PERIOD
In section 3, for each proposed algorithm an assumption was done for the good working of them.This assumption consists of "the rate of power control commands is higher than the channel fading rate".In this subsection we will illustrate the limits of the actuation rate of the power control in which those assumptions are valid.Therefore, we simulate the proposed power control algorithms for some actuation rates and we compare them to DPC and Lee algorithm in  order to estimate the superior limit for the actuation.periodin which the proposed algorithms outperform the DPC and Lee algorithm.
In fig.3(a), it can be observed that the proposed algorithms make the Eb/N o to converge for the (Eb/No)t better than DPC and Lee algorithm, for an actuation period of 2 ms.
In this case, the protection margin is 1 dB and the Doppler spread is 18.5 Hz.In fig.3(b), for an actuation period of 4 ms, DPC begins to outperform the new algorithms for some system loads.Therefore, we can infer that the superior limit for the actuation period of the proposed algorithms is between 2 and 4 ms, for the simulated conditions.From this on, the assumed assumptions are not valid any longer and the performance of the algorithms will decrease.When we increase the Doppler spread for 55. 5 Hz with actuation period of 2 ms and 3 ms, it is expected that the performance decreases, as it can be seen in figs.4(a) and 4(b).In this case the limit is then between 2 and 3 ms.For higher Doppler For each margin and load system, the smallest value is highlighted.
.. For each margin and load system, the smallest value is highlighted.
spreads, the actuation period of 1 ms can be considered the limit as it was shown in table 4. It is important to remember that in 30 systems such as WCOMA (Wideband COMA), the actuation period of the power control is 0.67 illS [1], that is, below the value of I ms used in the simulation results of this paper.

CONCLUSIONS AND PERSPECTIVES
This work has presented new algorithms for power control in mobile communication systems.The proposed algorithms work well in fast time-variant channels, since they track both fast fading and interference variations.We deducted analytically their equations and observed through simulations that they are superior to the conventional DPC and Lee algorithm, for actuation period of the power control of 1 ms, thus resulting in potential capacity gains in mobile cellular systems.As future work we will investigate theoretical aspects about the convergency of the proposed algorithms in slow fading environments.Furthermore, a more complete system-level study, including dynamic simulations in a multi-cellular network, should be accomplished.

Figure 1 .
Figure 1.Sample of Eb/N o evolution for the evaluated power control algorithms, with Doppler spread 55.5 Hz.

Figure 2 .
Figure 2.Comparison between actual and predicted interference using the proposed prediction methods, for the Doppler spread 18.5 Hz.

Figure 3 .
Figure 3. Averaged fraction of time in which Eb/No is I dB below (Eb/NO)t for actuation periods of the power control of 2 ms and 4 ms, with Doppler spread 18.5 Hz.

Figure 4 .
Figure 4. Averaged fraction of time in which E b / No is I dB below (Eb / No) t for actuation periods of the power control of 2 illS and 3 ms.with Doppler spread 55.5 Hz.

Table 2 .
Averaged fraction of time in which Eb/N o is below (Eb/No)t for several margins, with Doppler spread 18.5 Hz.

Table 3 .
Averaged fraction of time in which E b / No is below (E b / No)t for several margins, with Doppler spread 55.5 Hz.For each margin and load system, the smallest value is highlighted.

Table 4 .
Averaged fraction of time in which Eb/NO is below (Eb/NO)t for several margins, with Doppler spread 92.5 Hz.
5s I dB below (Eb / No) t for actuation periods of the power control of 2 illS and 3 ms.withDopplerspread 55.5Hz.Raimundo Abreu de Oliveira Neto was born in Fortaleza, Ceara.Brazil in 1977.He received his Bachelor and Master Degree in Electrical EngineeIing from Federal University of Ceara (UFO, Brazil, at 2001 and 2004.respectively.Since 2002 he has been with the Wireless Telecommunications Research Group (GTEL) located at the Teleinformatics Engineering Department (DETl), UFC.Presently, he is a researcher of the same research group.He is also member of the Brazilian Telecommunications Society (SBrt).His research interests are radio resource management and multi-access networks.Fabiano de Sousa Chaves was born in Fortaleza, Ceara, Brazil in 1978.He received his Bachelor and Master Degrees in Electrical Engineering from Federal University of Ceara (UFO, Brazil, in 2003 and 2005, respectively.Since 2002 he has been with the Wireless Telecommunications Research Group (GTEL) located at the Teleinformatics EngineeIing Department (DETI), UFC.He is also member of the IEEE and of the Brazilian Telecommunications Society (SBrt).His research interests are multiobjective optimization.noncooperative game theory, interplays