x0 is the current discrete time instant k and x the next instant k + 1. Further, f (x0) is ... So, using (2), (6) and (7), the transmit power at instant (k + 1) is expressed.
A New Distributed Power Control Algorithm Based on a Simple Prediction Method Raimundo A. de O. Neto, Fabiano de S. Chaves, Francisco R.P. Cavalcanti, and Tarcisio F. Maciel Wireless Telecommunications Research Group - GTEL Federal University of Cear´ a, Fortaleza, Brazil {neto, fabiano, rod, maciel}@gtel.ufc.br http://www.gtel.ufc.br
Abstract. 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. The algorithm presented herein addresses this problem efficiently and it is based on a simple prediction method, utilizing Taylor’s Series. In each iteration, the proposed algorithm predicts both the path gain and the interference and after that, adjust the transmit power.
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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 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 algorithm (DPC) [3], considering a time-varying path gain. In [4], a neural network is used to predict the future channel 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, a new distributed power control algorithm is presented, outperforming the classical Distributed Power Control algorithm (DPC) [3] when a J.N. de Souza et al. (Eds.): ICT 2004, LNCS 3124, pp. 431–436, 2004. c Springer-Verlag Berlin Heidelberg 2004
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time-varying channel is considered. This new algorithm differs from the DPC in that its deduction assumes both path gain and interference to be time-varying functions and it predicts this variations through the Taylor’s Series.
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The New Algorithm
The discrete-time SINR ρi (k) of a link is given by: ρi (k) =
gi (k) · pi (k) Ii (k)
(1)
So, the instantaneous transmit power necessary to balance this link for ρi (k) = ρt , for all instants k, is such that: ρt =
gi (k) · pi (k) ρt · Ii (k) ⇒ pi (k) = Ii (k) gi (k)
(2)
We do not predispose of values of Ii (k) and gi (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 following form [7]: ∞ f (n) (x0 ) · (x − x0 )n (3) f (x) = f (x0 ) + n! n=1 where the term f (n) (x) represents the nth derivative of f (x) with respect to x. Due to (x − x0 )n and n!, when x and x0 are adjacent values, the higher order terms can be neglected. Thus, keeping only the first two terms of the series, we have: f (x) ≈ f (x0 ) + f (x0 ) · (x − x0 ) (4) Now, we transform (4) into a difference equation. For this, we assume that x0 is the current discrete time instant k and x the next instant k + 1. Further, f (x0 ) is approximated by f (k) − f (k − 1). In this way, we obtain: f (k + 1) ≈ 2 · f (k) − f (k − 1)
(5)
Therefore, we can use (5) in order to predict the path gain and interference: gˆi (k + 1) = 2 · gi (k) − gi (k − 1)
(6)
Iˆi (k + 1) = 2 · Ii (k) − Ii (k − 1)
(7)
So, using (2), (6) and (7), the transmit power at instant (k + 1) is expressed by the following proposed algorithm: pi (k + 1) = ρt ·
Ii (k + 1) 2 · Ii (k) − Ii (k − 1) = ρt · gi (k + 1) 2 · gi (k) − gi (k − 1)
(8)
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The obtained SINR in discrete time k + 1 is: ρi (k + 1) = ρt ·
Ii (k + 1) gi (k + 1) · Ii (k + 1) gi (k + 1)
(9)
Note that when the estimations tend to correct values, that is, gˆ(k + 1) ≈ ˆ + 1) ≈ I(k + 1), the SINR tends to ρt . g(k + 1) and I(k
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Simulation Results
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We now illustrate the performance of the proposed algorithm by simulations utilizing a simulator consisting of a co-channel set of trisectorized base station in downlink direction. The co-channel sector set size comprises one layer of interferes. Base stations are localized on the corner of sectors. The sector antenna radiation pattern employed is ideal. The main-lobe gain is 0 dBi and the gain outside sector is -200 dBi. A snapshot simulation model is assumed where mobile stations are uniformly distributed over the cell area. In each snapshot, up to 600 iterations of the power control algorithm are performed, in intervals of 1 ms. Other simulation parameters are set as follows. The cell radius is set to 1.5 km. A simplified path loss model is used, where P L(d) = 120 + 40log10 (d) [dB]. 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 the Jakes’ model [8] with two different Doppler spreads: 18.5 Hz and 92.5 Hz. The target SINR ρt for both algorithms is set to 8 dB. Maximum base station transmit power is limited to 35 dBm and the initial transmit power is set to the minimum transmit power (-70 dBm). The noise power is set to -110 dBm.
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Fig. 2. Sample of SINR evolution for the evaluated power control algorithms, with Doppler spread 18.5 Hz and reuse pattern 1/3.
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Fig. 3. Comparison between actual and predicted path gain using the proposed prediction method, for a Doppler spread 18.5 Hz.
Figs. 1 and 2 show a sample of the SINR evolution achieved by a given user in a typical snapshot for DPC and the proposed algorithm. In this case, seven co-channel base stations are considered and the same system configuration and fading realizations are used for both algorithms. The simulated reuse pattern for each figure is 3/9 and 1/3, respectively, with Doppler spread 18.5 Hz. From figs. 1 and 2, it is clearly observable that the proposed algorithm is able to stabilize the SINR around the target SINR better than DPC algorithm. In other words, the mean squared error (MSE) between the actual and the target SINR is smaller for the proposed algorithm than for DPC. This behavior was observed for all snapshots. A sample of how the proposed algorithm performs with gain and interference prediction is shown in figs. 3 and 4 for the same snapshot in figs. 1 and 2. Figs.
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Fig. 4. Comparison between actual and predicted interference using the proposed prediction method, for a Doppler spread 18.5 Hz.
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Fig. 5. Averaged fraction of time in which SINR is 1 dB below ρt , for two different Doppler spreads.
3(a) and 3(b) present the behavior of the path gain and the tracking performance of the path gain prediction for reuse patterns 3/9 and 1/3, respectively, with the Doppler spread 18.5 Hz. Equivalently, figs. 4(a) and 4(b) show interference and its prediction. It can be observed that prediction based on Taylor’s Series achieves good performance for both path gain and interference. The same behavior was observed in all snapshots. In fig. 5, we illustrate how the superior tracking capability of the proposed algorithm translates into a system-level advantage. In practical systems, it is difficult to keep the SINR exactly at the target value, especially for high speeds [1]. Therefore, we assume an SINR margin below the target SINR in which signal quality is assumed acceptable. We simulated 5000 snapshots for several system loads with reuse pattern 3/9 and 1/3 and calculate the average fraction of time in which the achieved SINR is below the target SINR by a margin of 1 dB.
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The simulated maximum load is equivalent to seven co-channel cell, that is, the central cell and an interferer ring with six co-channel cells. In fig. 5(a), it can be observed that the employment of the new algorithm allows for a significant capacity gain when compared to the DPC, when a Doppler spread of 18.5 Hz is considered. Fig. 5(b) shows the performance of the algorithm with a Doppler spread 92.5 Hz. As the channel variation rate increases, it is expected a performance decrease for both algorithms. However, it can be observed that the proposed algorithm still outperform the DPC algorithm for this Doppler spread.
4
Conclusions
This work has presented a new algorithm for power control in wireless communications systems. The proposed algorithm works well in fast time-varying channels, since they predicts both fast fading and interference variations. The prediction method is based on Taylor’s Series and it has low complexity. We demonstrated through simulations that the proposed algorithm is superior to the conventional DPC algorithm, thus resulting in potential capacity gains in mobile communications systems. Acknowledgments. This work is supported by a grant from Ericsson of Brazil - Research Branch under ERBB/UFC.07 technical cooperation contract. Fabiano de S. Chaves is supported by CNPq. Tarcisio F. Maciel is supported by FUNCAP-CE.
References 1. Toskala, A. and Holma, H., “WCDMA for UMTS - Radio Access for Third Generation Mobile Communications”, Wiley, England, 2001. 2. Lee, G. J. and Miljanic, Z., “Distributed Power Control in Fading Channel”, IEE, Elec. Letters, vol. 38, pp. 653-654, Jun. 2002. 3. Foschini, G. J. and Miljanic, Z., “A Simple Autonomous Power Control Algorithm and its Convergence”, IEEE Trans. Veh. Technol., vol. 42, pp. 641-646, Nov.1993. 4. Visweswaran, B. and Kiran, T., “Channel Prediction Based Power Control in WCDMA Systems”, 3G Mobile Communication Technologies, Conference Publication no 471, pp. 41-45, 2000. 5. Evans, B. G., Gombachica, H. S. H. and Tafazolli, R., “Predictive Power Control for S-UMTS Based on Least-Mean-Square Algorithm”, 3G Mobile Communication Technologies, Conference Publication no 489, pp. 128-132, 2002. 6. Lau, F. C. M. and Tam, W. M., “Novel Predictive Power Control in a CDMA Mobile Radio System”, Vehicular Technology Conference, vol. 3, pp. 1950-1954, May 2000. 7. Apostol, T. M., “Calculus”, 2nd ed. Editorial Revert´e, vol. 1, 1967. 8. Jakes, W. C., “Microwave mobile communications”, 2nd ed. Wiley, New York, 1974. 9. Lee, J. S. and Miller, L. E., “CDMA Systems Engineering Handbook”, Artech House Publishers, 1997.
