mpact of Mobility and Closed-Loop Power Control to Received Signal Statistics in Rayleigh Fading Channels Pekka Pirinen University of Oulu Telecommunication Laboratory and Centre for Wireless Communications P.O. Box 5 (Tutkijantie E), FN-9 University of Oulu E-mail: pekka.pirinen@ee.oulu.fi ABSTRACT mpact of closed-loop power control to received signal statistics is studied. Results in this paper are based on single-user link level simulations with a fixed-step power control. Main interest is on the Rayleigh fading channel correlation, amplitude and phase changes in the system due to the closed-loop power control. Power control dependent changes in the complex channel coefficient correlation are needed, e.g., in the Kalman or Wiener filter based channel estimation. Key adjustable parameters in the simulations are mobile speed, power control step size and power control command error rate. Simulation results indicate that power control can compensate channel variations at low velocities. When the fading is fast enough compared to the power control rate power control can increase fluctuation in the received signal. Wellworking power control reduces channel correlation slightly.. ntroduction Power control is an important component of direct-sequence CDMA systems. Main tasks of power control (PC) are to combat the near-far problem and to reduce the transmitted power levels [,, 3]. The near-far effect is an inherent CDMA property where the near base station users overwhelm more distant simultaneous users due to the non-orthogonality of these transmissions. Savings in transmitted powers decrease mutual interference and extend battery life, which is crucial for the lightweight mobile units. As far as the power control is close to perfect the system capacity can be high [, 5]. However, it is also well known that CDMA capacity is very sensitive to imperfections in power control [, 6, 7, 8]. These errors are usually modeled by a lognormal distribution. The conventional fixed-step PC algorithm is very simple. t requires measurement of the received signal power (S-based PC) and/or signal-to-interference level (SRbased PC). The measured value is compared to the desired target value. Depending on the comparison result a power control command of one step up or down is sent over a feedback channel and executed at the transmitting unit. Fixed-step algorithms cannot fully react to fast changes in the signal level (deep fades) at reasonable update rates. Therefore adaptive step-size and predictive approaches have also been proposed, e.g., [, 9]. mpact of power control becomes a complicated issue when multiple users contribution in one or multiple cells is to be studied. The reason for this complexity is that every user s power control commands change interference statistics observed in other links. This leads to a challenging SR target optimization problem where targets should be adaptive depending on the network load and transmitted data rates [5, ]. n this paper the focus is on the conventional fixed-step closed-loop power control. Only a single-user link with a feedback power control loop is modeled. Data, modulation, coding, interleaving, etc. link details have been omitted in the system model. nteresting statistics are the received signal magnitude and phase plus mean value, variance and correlation of the received signal power. These statistics are compared with and without power control. Additionally, sensitivity of these statistics to mobile speed, power control step size and error rate variations is investigated. Results of this study can be used in the research of advanced receiver structures for wireless CDMA systems.. System Model.. Preliminaries Theoretical channel autocorrelation values have been calculated from the Clarke s model [] that can be represented as ρ ( τ) - ( π τ) ()
where ( - ) is the zeroth order Bessel function of the first kind, Y () is the maximum Doppler spread, v is the speed of the mobile, F is the speed of light, is the carrier frequency and τ is the delay (sampled at power control command rate, / 7 ). n the frequency domain the corresponding power spectrum density (classical Jakes Doppler spectrum []) is of the form 6 ( ) π.. Simulation Model F,, >. (3) Fig. shows a block diagram of the simulation model used in this paper. Thick solid lines represent complex signals divided into in-phase () and quadrature (Q) branches in the model. SGNAL SOURCE AMP POWER CONTROL FADNG CHANNEL NTERFERENCE SOURCE SGNAL STATSTCS - AMPLTUDE AND PHASE - AUTOCORRELATON - POWER SPECTRAL DENSTY + AWGN. Fig.. Simulation model block diagram. Signal source generates a constant, complex signal (equally strong normalized and Q components). Data and modulation are neglected in the model. The amplification block adjusts the transmitted power up or down according to power control commands. The fading channel model is multiplicative complex noise whose magnitude depends on the desired distribution (Rayleigh in this case). Additionally, the signal waveform is filtered in order to include the Doppler spectrum characteristics, e.g., classical Jakes spectrum. Signal statistics (channel complex envelope and phase, correlation, power spectral density, received power mean and variance) are recorded at the output of the Rayleigh fading channel. Additive white Gaussian noise (AWGN) is summed according to the desired signal-tonoise ratio. Desired signal power is estimated by taking the square of the complex envelope magnitude. nterference caused by other users is modeled as a Gaussian random variable. The complex channel coefficient is of the form F ( ( ) $ W H θ( ) $ ( cos θ( + M$ ( sin θ( where $ ( $ ( + $ ( is the magnitude of the complex coefficient that can be divided to in-phase and quadrature components $ ( and $ ( and θ ( is the phase of the coefficient. The averaged and normalized channel sample autocorrelation function is calculated as () * τ ρ τ * F( F ( W ) F( L) F ( L N) ( W, ) Re Re (5) Q F ( F( L) where, in this paper, the delay index k ranges from to 9. n the simulations the averaging is made over n samples of channel coefficients. 3. Numerical Results Key simulation parameters and variables are presented in Table. Bold parameters are nominal values. t should be noted that the signal-to-noise ratio in the AWGN block is so high that the channel can be treated as noiseless. Hence, the flat Rayleigh fading channel statistics purely causes the channel distortions. Table. Simulation parameters Carrier frequency 9 MHz Channel model -path Rayleigh (Classical Doppler spectrum) Mobile speed 3,,, 3, 5,, km/h PC algorithm S-based PC command rate (/ 7 ) 6 Hz PC loop delay 7.65 ms PC step size ( S) PC command error rate (BER PC ) PC dynamic range S target Signal-to-Noise Ratio.5,, db,.,.5,. ± db db db Figs. - illustrate how well the power control algorithm can track the fading channel variations. At low speed (Fig., v 3 km/h) the fading is so slow that power control operates almost perfectly. Only the deepest fades cause some additional variation to the step-size dependent
ripple. At slightly higher velocities (Fig. 3, v km/h) deep fades occur more often, and consequently, power control fails to track these notches. At the mobile speed v 3 km/h (Fig. ) the complex envelopes of the received signal with or without power control become quite similar. Hence, the power control command rate is too slow at high velocities. v 3 km/h 5 5 5 3 35 5 5 Fig.. Received envelope (v 3 km/h). Channel autocorrelation and power spectral density (PSD) characteristics are illustrated in Figs. 5-7. Fig. 5 shows the channel autocorrelation function values for τ,, 9 7 ( - 5.65 ms) at slow mobile speeds. mpact of power control is most significant at km/h velocity. n majority of the cases the power control decreases correlation less than %. Without power control the theoretical values and simulations agree well. Fig. 6 depicts autocorrelation properties at higher speeds. For velocities above 3 km/h it is difficult to distinguish between the curves with or without PC. Due to symmetry only one-sided channel power spectral density is plotted in Fig. 7. n PSD simulation the Bartlett method was used with the FFT length of. Without PC the classical U-shaped spectrum is received. With PC the power levels are slightly higher. Additionally, peaks in PSD are observed at lowest simulated speeds. These smaller Doppler peaks can be explained by the successful PC (-> higher probability for lower Doppler frequencies). Channel autocorrelation function v 3 km/h.95.9.85.8.75 (simulation) (simulation) (theoretical) v km/h v km/h v km/h 5 5 5 3 35 5 5 Fig. 3. Received envelope (v km/h). v 3 km/h 5 5 5 3 35 5 5 Fig.. Received envelope (v 3 km/h)..7 3 5 6 7 8 9 Delay, T p Fig. 5. Channel autocorrelation at low speeds..9.8.7.6.5..3.....3. v km/h Channel autocorrelation function v 5 km/h v km/h (simulation) (simulation) (theoretical) v 3 km/h.5 3 5 6 7 8 9 Delay, T p Fig. 6. Channel autocorrelation at high speeds.
PSD [db] 5 5 5 One sided Doppler spectrum, v 3 km/h, v km/h, v 3 km/h high mobile speeds (v > 5 km/h) the changes in correlation became more arbitrary. PC step size of db showed the best performance at v km/h and was a good compromise at other speeds as well. Received signal statistics were barely affected by moderate PC command error rates. Acknowledgment 5 5, v 3 km/h 3...3..5.6.7.8.9 Normalized frequency Fig. 7. Channel power spectral density. Some statistics of the received signals are presented in the Appendix (Table ). At the mobile speeds higher than km/h the power control only causes extra variance to the signal. Table also demonstrates how the correlation between two consecutive samples of complex channel coefficients, during one PC interval of a τ, varies with and without power control. All simulation results have been averaged over samples. Simulation results and theoretical correlation values agree perfectly at very low mobile speeds. Small deviations are seen for higher speeds (3 - km/h). According to these simulation results the impact of PC at low velocities could be modeled by multiplying () with a constant a.9935. For high velocities (v > 5 km/h) the impact of PC to autocorrelation seems to diverge from the trend noticed at lower velocities. n the Appendix the received mean power and variance are compared at variable PC step sizes (Table 3) and at variable PC command error rates (Table ). According to Table 3 the db step size seems to be a good compromise at moderate velocities. PC command error rates less than 5 % cause only a minor degradation in the statistics. The variance is roughly doubled at low velocities when BER PC grows from to %.. Conclusions mpact of closed-loop power control to Rayleigh fading channel parameters was studied. Power control was noticed to be able to compensate relatively slow fading. However, in fast fading channels power control only introduced additional channel variations and the received mean signal power exceeded the target level. Power control decreased the correlation between consecutive channel samples at low mobile speeds (v < 5 km/h). This reduction could be compensated in the Clarke s model by multiplying the Bessel function with a correction constant. For The author would like to thank Prof. Savo Glisic for the valuable discussions and comments during the preparation of the manuscript. References [] C.-C. Lee and R. Steele, Closed-loop power control in CDMA systems, nst. Elect. Eng. Proc.-Commun., vol. 3, pp. 3-39, Aug. 996. [] S. Ariyavisitakul and L. F. Chang, Signal and interference statistics of a CDMA system with feedback power control, EEE Trans. Commun., vol., pp. 66-63, Nov. 993. [3] A. Chockalingam, P. Dietrich, L. B. Milstein and R. R. Rao, Performance of closed-loop power control in DS- CDMA cellular systems, EEE Trans. Vehic. Tech., vol. 7, pp. 77-789, Aug. 998. [] K. S. Gilhousen et al., On the capacity of a cellular CDMA system, EEE Trans. Vehic. Tech., vol., pp. 33-3, May 99. [5] B. Hashem and E. S. Sousa, Reverse link capacity and interference statistics of a fixed-step power-controlled DS/CDMA system under slow multipath fading, EEE Trans. Commun., vol. 7, pp. 95-9, Dec. 999. [6] P. Newson and M. R. Heath, The capacity of a spread spectrum CDMA system for cellular mobile radio with consideration of system imperfections, EEE J. Select. Areas Commun., vol., pp. 673-68, May 99. [7] R. Prasad, M. G. Jansen and A. Kegel, Capacity analysis of a cellular direct sequence code division multiple access system with imperfect power control, ECE Trans. Commun., Vol. E76-B, pp. 89-95, Aug. 993. [8] E. Kudoh and T. Matsumoto, Effects of power control error on the system capacity of DS/CDMA cellular mobile radios, ECE Trans. Commun., Vol. E75-B, pp. 5-59, June 99. [9] M. L. Sim, E. Gunawan, B.-H. Soong and C.-B. Soh, Performance study of close-loop power control algorithms for a cellular CDMA system, EEE Trans. Vehic. Tech., vol. 8, pp. 9-9, May 999. [] S. Ariyavisitakul, Signal and interference statistics of a CDMA system with feedback power control - part, EEE Trans. Commun., vol., pp. 597-65, Feb./Mar./Apr. 99. [] S. Qu and T. Yeap, A three-dimensional scattering model for fading channels in land mobile environment, EEE Trans. Vehic. Tech., vol. 8, pp. 765-78, May 999.
Appendix Table. Mean value, variance and autocorrelation comparison with and without power control Case Mean E[S] Variance E[S ] ρ (,) (simulated) ρ() (theoretical) v 3 km/h,.999.68.9999.9999 v 3 km/h,..985.993 (-.65 %) - v km/h,.99.987.9988.9988 v km/h,..7.99 (-.66 %) - v km/h,.986.978.995.995 v km/h,.39.77.9887 (-.63 %) - v 3 km/h,..9975.9889.989 v 3 km/h,.55 3.3.985 (-.65 %) - v 5 km/h,.3..9685.9698 v 5 km/h,.59 3.67.967 (-.6 %) - v km/h,.3.3.877.888 v km/h,.69.658.8758 (-. %) - v km/h,..5.558.5689 v km/h,.65.6.576 (+.8 %) - Table 3. mpact of PC step size to received signal statistics ( ) Case S.5 db S db S db S.5 db S db S db v 3 km/h.3..3.8.985.37 v km/h.36..898.87.7.737 v km/h..39.6.3.77.58 v 3 km/h.55.55.653.535 3.3. v 5 km/h.9.59.879.7 3.67 6.67 v km/h.53.69.3.975.658.57 v km/h.5.65.996.77.6.5 Table. mpact of variable PC command error rates to received signal statistics ( S db) Case..5...5. v 3 km/h....6.985.6.3.755 v km/h..8.7.6.7.55.66.3 v km/h.39..37.98.77.88.9 3.93 v 3 km/h.55.536.57.68 3.3.97 3.63.3 v 5 km/h.59.65.6.655 3.67 3.93.5.598 v km/h.69.7.7.7.658.799.88 5.6 v km/h.65.658.665.69.6..35.6