Generator of Time Series of Rain Attenuation: Results of Parameter Extraction

32 M. GRÁBNER, U.-C. FIEBIG, V. KVICERA, GENERATOR OF TIME SERIES OF RAIN ATTENUATION: RESULTS... Generator of Time Series of Rain Attenuation: Results of Parameter Extraction Martin GRÁBNER1, Uwe-Carsten FIEBIG2, Václav KVIČERA1 1Dept. of Microwave Communications, TESTCOM, Hvožďanská 3, 148 01 Praha 4, Czech Republic 2Institute for Communication and Navigation, DLR Germany, D-82230 Wessling, PO Box 1116, Germany [email protected], [email protected], [email protected] Abstract. Rain attenuation has a significant impact on the availability of millimeter wave communication systems. In order to dynamically simulate such radio systems, several generators of artificial time series of rain attenuation have been developed. This paper briefly describes the DLR channel model and presents the results of model parameter extraction from time series measured on terrestrial microwave paths in the Czech Republic. Keywords Microwave communication, propagation, millimeter wave, rain attenuation, channel model. 1. Introduction Rain fading events may cause severe outages in millimeter wave communication systems. Different fade mitigation techniques are being developed in order to overcome such propagation impairments. Recently, several time series generators have been developed [1], [2]. They produce artificial time series of rain attenuation for the purpose to simulate fade mitigation techniques under realistic channel conditions. Apparently, the generated time series should be similar to the measured ones in terms of their statistical and dynamical characteristics. In this paper, the DLR channel model [3], [4] is briefly described. So far the DLR channel model has been applied successfully to satellite microwave paths which are subject to rain fading. In this paper, the DLR channel model is applied to three terrestrial microwave paths in the Czech Republic. 2. DLR Channel Model The DLR channel model is based on a mix of a 2nd order Markov chain and Gaussian random variables. These define transition probabilities of the Markov chain. The generated time-discrete time series of attenuation is denoted by yi = y(iTs), where i = 1,2,… and 1/Ts is the sampling rate. Ts is typically in the order of 1 minute. The next value of attenuation yi+1 is obtained as the outcome of a single experiment of a Gaussian random variable ni with distribution N( µ i, σ i2), see Fig. 1. Power level time Last generated sample yi Likelihood of next sample yi+1 Fig. 1. Principle of generating subsequent samples of an artificial rain attenuation time series with the DLR channel model. The parameters µ i, σ i depend on the previous value of the chain yi as well as on the attenuation trend described by a segment type si: () () ( ()() ) ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ −< ≤ > =Λ−Λ= === = − + dB 1for dB 1for dB 1for , ,,,,,N 2 1 xU xC xD xyys sysyn ny Kiii iiiiiiiii ii σσµµσµ (1) where D, C, and U denote down, constant and up segment type, respectively. K is a positive constant determining the duration of a segment relative to 1/Ts, where typically K=1 is chosen. In the continuous-valued model described above, the model parameters are the functions µ (y, s), σ (y, s). The discrete-valued version of the DLR channel model applies an internal quantization procedure and assigns different discrete attenuation levels to the states of a Markov chain, see Fig. 2. RADIOENGINEERING, VOL. 14, NO. 1, APRIL 2005 33 Down Up 4 dB 5 dB 6 dB 7 dB 8 dB 9 dB ii-1 Constant i+1 Down Up 4 dB 5 dB 6 dB 7 dB 8 dB 9 dB 4 dB 5 dB 6 dB 7 dB 8 dB 9 dB 4 dB 5 dB 6 dB 7 dB 8 dB 9 dB ii-1 Constant i+1 Fig. 2. DLR channel model as a discrete-valued 2nd order Markov chain. Grouping of transitions allows for a reduction of the number of transition probabilities. A discrete-valued 2nd order Markov chain over M states is completely defined by its M3 transition probabilities. With a quantization step size of 1 dB and M=40 a total of 64,000 transition probabilities has to be derived. This is very cumbersome to handle. However, grouping the 2nd order transitions into three groups reduces the number of transition probabilities to 3M2. Furthermore, all M transition probabilities emerging from a particular state can be very accurately described by a Gaussian distribution which results in only 3M distributions to describe the channel model, see Fig. 3. The use of Gaussian distributions was observed empirically to be valid for large datasets. Down Up 4 dB 5 dB 6 dB 7 dB 8 dB 9 dB ii-1 Constant i+1 Down Up 4 dB 5 dB 6 dB 7 dB 8 dB 9 dB 4 dB 5 dB 6 dB 7 dB 8 dB 9 dB ii-1 Constant i+1 Fig. 3. DLR channel model: Description of the transition probabilities by Gaussian distributions. A Gaussian distribution is fully described by its mean value and its standard deviation. Thus, to fully describe the DLR channel model, a total of 3M mean values and 3M standard deviations have to be determined. The set of these 3M mean values and 3M standard deviations represents the specific channel model parameters (SCMPs) of the quantized version of the DLR channel model: Mj U j U j C j C j D j D jK,1,,,,,, = σµσµσµ . (2) These parameters can be extracted from the measured time series of rain attenuation by means of a straightforward model parameter extraction process. 3. Model Parameter Extraction This chapter describes the procedure of the DLR model parameters extraction. 3.1 Measured Time Series and Preprocessing TESTCOM has been carried out long-term measurements of the received signal level on terrestrial microwave paths in several frequency bands. A model parameter extraction procedure was applied on measured data summarized in Tab. 1. Path Frequency (MHz) Length (km) Data amount (years) TESCO – Chloumek 19 370.00 26.2 2 Strahov – TESTCOM 38 319.75 9.3 4 Úvaly – TESTCOM 38 491.25 15.2 3 Tab. 1. Overview of available measured time series of attenuation. The sampling time of the measured data is 0.1 second. One-second block averaging is performed prior to parameter extraction in order to obtain a time series with a sampling time Tr = 1 s. A zero attenuation level is determined so that it corresponds to a monthly median value of the measured received signal level. Then, a moving averaging with a window size of 60 seconds is applied to remove fast fluctuations caused by a scintillation effect. The resulting sequence is denoted xn = x(nTr), n = 1,2,… and has a sampling time of 1s. Filtering is employed, because the DLR model is intended to generate only slow variations of a signal caused by rain fading events. 3.2 Extraction Process An extraction process is applied on the filtered time series xn = x(nTr), n = 1,2,… in order to derive the SCMPs. Note that the measured time-series has a sampling rate 60 times larger than the sampling rate of the channel model. First, the time series xn, n = 1,2,… is mapped onto a discrete-valued time-series dn, where the elements of dn can take on integers ranging from 1 to M representing attenuation levels. Then, three matrices HD, HC and HU of size M×M assigned to the corresponding 3 types of segments are determined. An element hkl of Hb, b ∈ {D,C,U} gives the number of transitions from attenuation level k to attenuation level l 34 M. GRÁBNER, U.-C. FIEBIG, V. KVICERA, GENERATOR OF TIME SERIES OF RAIN ATTENUATION: RESULTS... between samples dn and dn+K for K = 60 and for all n; i.e. only samples separated by 60 seconds are considered; matrix Hb contains those transitions which are obtained for the case that b = Λ(dn − dn−K) with K = 60. A convenient determination of these matrices is to go sample-wise through the whole time series dn and to increment the elements hkl of Hb according to above rules. The rows of matrices HD, HC, HU provide distributions for every state. These distributions describe the probability of the transition from one state to another under the condition of the trend Λ which is either down, constant or up. Finally, the parameters described by eqn. (2) are extracted from the distributions. The received signal fluctuates around the free space level which represents zero attenuation. That is why negative attenuation values may occur. In order to process these values properly, we considered in our investigation attenuation values ranging from -5 to +45 dB. 4. Results This chapter presents the results of both the model parameter extraction and the simulation with the DLR channel model. The examples of extracted parameters and of generated time series of attenuation are shown. The first and the second order statistics of measured and simulated time series are compared. 4.1 Extracted Parameters Fig. 4 shows the DLR model parameters extracted from the attenuation time series obtained on the path TESCO - Chloumek in 2003. Since the parameters describe transition probabilities, mean values are almost linearly dependent on an attenuation level. It means, the next attenuation level will have a similar value as the previous one for most of the time. However, significant relative differences are obvious between the curves for down, constant and up segments. Standard deviation slightly increases with an attenuation level and is minimal around the zero level (5 dB because of the shift reasoned in the last paragraph of the section 3.2). Polynomial fitting of the model parameters from Fig. 4 was performed in order to provide one quantitative example of the continuous DLR channel model. The resulted polynomials are as follows: () [] )3( 00784.1,23473.1,00399.0 y⋅= −−y D µ () [] , )3( 46872.0,94223.0,00046.0 y⋅=y C µ () [] , )3( 70162.1,69399.0,00410.0 y⋅=y U µ () [ , )4( 00537.0,32907.0,00867.0,00005.0 y⋅= −−y D σ ] ( ) [ ] , )4( 51906.0,03033.0,00112.0,00002.0 y⋅= −y C σ ( ) [ ] , )4( 32198.1,06743.0,00745.0,00011.0 y⋅= −−y C σ [ ] .1,,,,:where 21 )( T nn nyyy K −− =y (3) Note that the discrete-valued model was actually used for calculations presented in the next sections. 0 10 20 30 40 50 0 10 20 30 40 50 mean and sigma (dB) attenuation ( dB ) mean, down mean, const mean, up sigma, down sigma, const sigma, up Fig. 4. Extracted parameters µ and σ as functions of the attenuation level and of segment type, path TESCO - Chloumek, year 2003. 4.2 Generated Time Series The DLR channel model has been set up with above extracted parameters. It generates a time series with a sampling time of 1 minute. The generation followed eqn. (1) with parameters Ts = 60 s and K = 1. The duration of all generated time series of attenuation has been 1000 days. Fig. 5 shows an example of an artificial time series of attenuation generated by the DLR channel model with a parameter set extracted from the TESCO - Chloumek measured data. The received signal level relative to the free space level is depicted. The simulated time-series looks very similar to those which have been measured. 4.3 Statistics Comparisons of statistics are presented in this section. First order statistics are represented by cumulative distributions of attenuation (CDA). The second order statistics are represented by fade duration distributions. 4.3.1 Cumulative Distributions Figs. 6 through 8 show the comparison of the cumulative distributions of attenuation (CDA) calculated from measured time series (“meas”) and from generated ones (“simul”). Note that model parameters were always extracted from the same measured data that was used for comparison. RADIOENGINEERING, VOL. 14, NO. 1, APRIL 2005 35 -50 -40 -30 -20 -10 0 50 100 150 200 250 300 350 400 received signal level (dB) time ( minute ) Fig. 5. Artificial time series of received signal level generated by DLR model with parameter set extracted from path TESCO - Chloumek, year 2003. 0 10 20 30 40 50 1e-005 0.0001 0.001 0.01 0.1 1 attenuation (dB) p robabilit y ( - ) simul meas Fig. 6. CDA for path TESCO - Chloumek, year 2003. 0 5 10 15 20 25 30 35 40 45 1e-005 0.0001 0.001 0.01 0.1 1 attenuation (dB) p robabilit y ( - ) simul meas Fig. 7. CDA for path Strahov - TESTCOM, year 2002. 4.3.2 Fade Duration Statistics Figs. 9 through 11 show the fade duration statistics of measured and generated time series. These statistics represent the probability that an attenuation sample at an arbitrary time instant is part of a fade which exceeds a given attenuation level for a time duration which is longer than the value on the abscissa. 0 10 20 30 40 50 60 1e-005 0.0001 0.001 0.01 0.1 1 attenuation (dB) p robabilit y ( - ) simul meas Fig. 8. CDA for path Úvaly - TESTCOM, year 2003. 1e-005 0.0001 0.001 0.01 0.1 1 10 100 1000 10000 probability (-) fade duration ( sec ) sim 10dB meas 10dB sim 20dB meas 20dB sim 30dB meas 30dB sim 40dB meas 40dB Fig. 9. Fade duration statistics obtained from measured and simulated time series for path TESCO - Chloumek, year 2003. 4.4 Discussion on the Results The figures reveal a very good fit of measured and simulated time-series in terms of the first order statistics (cumulative distributions of attenuation). The best fit was obtained for data from the TESCO - Chloumek path, see Fig. 6. Differences at higher attenuation values are generally larger than at lower attenuation values. The fit of the second order statistic is not as good as of the first order statistics. Figs. 9 through 11 indicate that the generated fade events can be shorter than the measured ones. These differences are smallest for the Strahov – TESTCOM path. These differences are generally not so large when the DLR channel model is applied to a satellite link. One of the reasons for these differences might be that spurious multipath fading which often occurs burst-like (many short and deep fades within a short amount of time) influence the SCMPs such that long fades are less likely to occur. These effects cannot be simulated by the DLR channel model, since it does not follow fast fluctuations. 36 M. GRÁBNER, U.-C. FIEBIG, V. KVICERA, GENERATOR OF TIME SERIES OF RAIN ATTENUATION: RESULTS... 1e-005 0.0001 0.001 0.01 0.1 1 10 100 1000 10000 probability (-) fade duration ( sec ) sim 10dB meas 10dB sim 20dB meas 20dB sim 30dB meas 30dB sim 40dB meas 40dB Fig. 10. Fade duration statistics obtained from measured and simulated time series for path Strahov - TESTCOM, year 2002. 1e-005 0.0001 0.001 0.01 0.1 1 10 100 1000 10000 probability (-) fade duration ( sec ) sim 10dB meas 10dB sim 20dB meas 20dB sim 30dB meas 30dB sim 40dB meas 40dB Fig. 11. Fade duration statistics obtained from measured and simulated time series for path Strahov - TESTCOM, year 2003. 5. Conclusions Rain fading generators are intended to support the simulation of microwave or millimeter wave radio systems that are impaired by the rain attenuation. The results of the DLR channel model parameter extraction were presented in this paper. Time series of received signal level measured on the three terrestrial microwave paths were processed to obtain the parameters for the model. In contrast to satellite paths, at which the DLR model has proven excellent performance, terrestrial paths are more likely to be subject to multipath propagation. It seems that this effect may influence the extraction process and cause poorer results with respect to fade duration statistics. Acknowledgements The Ministry of Education supported this research under the grant No. OC 280.001 (TESTCOM task No. 9331-1960) “Time Variability, Dynamic and Space Features of Characteristics of Electromagnetic Waves for Advanced Telecommunication Systems”. References [1] FIEBIG, U.-C., CASTANET, L., LEMORTON, J., MATRICCIANI, E., PÉREZ-FONTÁN, F., RIVA, C., WATSON, R. Review of propagation channel modelling. In Proceedings of 2nd International workshop of COST280 Action. May 2003, p. 153 – 164. [2] CASTANET, L., LEMORTON, J., LACOSTE, F., RIVA, C., MATRICCIANI, E., FIEBIG, U.-C., VAN DE KAMP, M., MARTELLUCCI, A. Development and validation of time series synthesizers for Ka-band satellite communication systems. In Proceedings of 10th Ka-Band Utilization Conference. Vicenza (Italy), October 2004. [3] FIEBIG, U.-C. Modelling rain fading with a time-series generator considering seasonal and diurnal variations. In Proceedings of the 8th Ka-Band Utilization Conference. Baveno/Strese (Italy), 2002. [4] FIEBIG, U.-C. Satellite channel modelling for rain fading. In Proceedings of AAIA International Communications Satellite System Conference. Yokohama (Japan), 2003. About Authors... Martin GRÁBNER received the B.S. and M.S. degrees in Electrical Engineering from the Czech Technical University in Prague in 1998 and 2000, respectively. He has been with the Department of Microwave Communications, TESTCOM. His work is focused on the propagation of radio waves and on the quality of terrestrial microwave systems. Uwe-Carsten FIEBIG received his Diploma degree in Electrical Engineering from the Technical University of Munich. In 1988 he joined the Institute for Communications and Navigation of DLR (German Aerospace Research) as a member of the research staff. In 1993 he received his PhD from the University of Kaiserslautern. In 1994 he became head of the Department “Communications Systems”. From 1996 till 1998 he spent several months at the Communications Research Center (CRC), Ottawa, Canada, at the Yokohama National University, Japan, and at the University of Pretoria, South Africa. His interests are in the field of satellite navigation and communications and mobile radio. He lectures satellite communications at the University of Ulm, Germany, and the University of Linz, Austria. Václav KVIČERA received the M.S. and Ph.D. degrees in Electrical Engineering from the Czech Technical University in Prague in 1971 and 1986, respectively. He is a senior researcher at the Department of Microwave Communications, TESTCOM. His work is aimed at electromagnetic wave propagation in frequency bands of fixed links, especially at the influence of hydrometeors in frequency bands above 10 GHz.