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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 5, 2006
A Multivariate Autoregressive Model of Rain Attenuation on Multiple Short Radio Links Gamantyo Hendrantoro, Member, IEEE, Indrabayu, Titiek Suryani, and Achmad Mauludiyanto
Abstract—A multivariate autoregressive (AR) model for rain attenuation on a network of radio links is presented. Underlying assumptions are discussed, including the stationarity of rainfall rate in space and time within the region enclosing the links. Formulation of the model is described, together with some considerations for its application in assessing performance of a network of radio links. An example of use of the model for simulation of rain attenuation on two short converging links operating at 30 GHz is given. Index Terms—Diversity, modeling, rain attenuation, simulation.
I. INTRODUCTION
R
AIN attenuation exhibits a devastating impact on the performance of a radio link at frequencies above 10 GHz. Mitigation of rain attenuation has long been studied with a goal of providing techniques to achieve reliable, high-quality radio links at these frequencies even under rainy conditions ([1], [2]). Performance evaluation of adaptive fade mitigation techniques, such as adaptive power control, adaptive coding, or switched diversity among a number of links, necessitates computer generation of time sequences of correlated rain attenuation on multiple links. The lognormal approximation of rain rate and attenuation has been found from measurements in the past (e.g., [3]), with the multivariate form for rain on multiple points or links having recently been discussed in [4]. Accordingly, a set of time series of log attenuation occurring on multiple links can be generated through synthesis of a multichannel autoregressive (AR) process under some assumptions. Following this brief introduction, the underlying assumptions of the model are stated and the model itself mathematically formulated. Some practical considerations for application of the model are subsequently discussed. Finally, a numerical example of the model application is given and some conclusions are drawn. II. ASSUMPTIONS AND MODEL FORMULATION It is assumed at the outset that point rainfall rate in mm/h and rain attenuation in dB experienced by a radio link are each lognormally distributed. In addition, the region that contains all Manuscript received October 11, 2005; revised December 8, 2005. G. Hendrantoro, T. Suryani, and A. Mauludiyanto are with Jurusan Teknik Elektro, Institut Teknologi Sepuluh Nopember, Kampus ITS Sukolilo, Surabaya 60111, Indonesia (e-mail:
[email protected];
[email protected];
[email protected]). Indrabayu is with Jurusan Teknik Elektro, Universitas Hasanuddin, Kampus Unhas Tamalamarea, Makassar 90245, Indonesia (e-mail:
[email protected]). Digital Object Identifier 10.1109/LAWP.2006.870362
of the links of interest is assumed to be shrouded by single rain events. To allow such an assumption, links must be sufficiently short, which is usually the case for terrestrial millimeterwave wireless systems like local multipoint distribution services (LMDS) where links are 4 km or shorter to assure line-of-sight situation. Moreover, these rain events must be temporally and spatially stationary so that the same statistical characteristics apply homogeneously throughout the time and space domain of the events. The spatial stationarity of the rain event does not necessarily mean homogeneity of rainfall intensity within the rainy area. Rather, it implies that point rainfall intensity (mm/h), or likewise, specific attenuation (dB/km), is identically lognormally distributed at all points throughout the rainy area with locationinvariant spatial autocorrelation function. Attenuation (dB) is the path-long integral sum of specific attenuation (1) with (2) where coefficients and depend on the frequency, polarization, and the raindrop size distribution. Since the sum of correlated lognormal random variables is well approximated by a lognormal random variable [5], the spatial stationarity assumption implies that attenuation on a link of a given length (km) is approximately lognormal and location-invariant, i.e., the distribution is fixed regardless of the actual location of the link within the rainy area. Also, by the spatial stationarity assumption, for a given multilink structure the interlink correlation coefficient of attenuation will be location-invariant (see, for instance, equafixed). Therefore, we tions (9)–(11) in [6] with function can be sure that the distribution and correlation statistics of attenuation on a multilink structure do not change both spatially and temporally. auThe proposed model involves synthesizing a set of toregressive processes that behave like time sequences of natural logarithmic value of rain attenuation on radio links, where and denotes the sampling period, according to the discrete-time cross-covariance functions of log attenuation on the th . and th links, with The functions can be obtained directly from a series of radar measurements whenever available (e.g., [7]). Oth, where erwise, it must be assumed that
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HENDRANTORO et al.: MULTIVARIATE AUTOREGRESSIVE MODEL OF RAIN ATTENUATION
is the normalized temporal autocorrelation of log attenis the normalized correlation of instantaneous uation and log attenuations on the th and th links [8]. The correlation coefficient can be estimated by firstly measuring the spatial autocorrelation function of rainfall intensity, and subsequently applying the Kanellopoulos–Koukoulas (K–K) method [6] to derive the correlation coefficient for rain attenuation on links of equal lengths. The K–K method can be applied to various configurations of radio links, including those in serial, parallel, and converging configuration. For the spatial autocorrelation of rainfall rate, such models as those proposed by Morita–Higuti [9] or Lin [3] can be adopted. Coefficients for links of different lengths can be obtained by applying a modified version of the K–K formula. Temporal autocorrelation of rain attenuation on individual links can be obtained through direct measurement of rain attenuation on an experimental radio link. For the special case, where the link length is 1 km or less, along which the rain rate can be assumed to be approximately homogeneous, it can be easily shown through (2) that the autocovariance is basically identical for log rain rate and log attenuation. A vector sequence of zero-mean normally distributed , where , is generated recursively as follows: (3) where with of the AR processes
are
.. .
..
.
matrix coefficients
.. .
(4)
denotes the order of the processes that depends whereas on the maximum time lag of the autocorrelation function, a vector sequence of independent white zero-mean unit-variance Gaussian random numbers, and a scaling matrix that determines covariances of the . Once vector elements of Gaussian vector sequence sequence is obtained, the rain attenuation sequence can be obtained (5) From the discrete-time covariance functions (or equivalently, the correlation functions of matrices can be formed .. .
..
.
.. .
of ), a set of
(6)
. For real-valued AR processes, the AR where matrix coefficients can subsequently be obtained from [10] (7)
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and dimension, while
where both are of the
matrix
.. .
Taking matrix
.. .
..
.
.. .
(8)
and , the scaling is obtained through Cholesky decomposition (9)
III. PRACTICAL CONSIDERATIONS In applying the AR model, parameters of the lognormal distribution for the attenuation on each link must be determined. This can be accomplished by computing maximum-likelihood estimation of the parameters using real measurements of attenuation. In the absence of real measurements, rough estimates of these parameters can be found by applying linear regression on some percentiles of the attenuation [11]. A number of computation issues also have to be dealt with prior to applying the model. Due to high correlation of rain attenuation for typical sampling intervals, both in space and time, correlation matrix is generally not well-conditioned. We propose to take the following simple strategy. We let the computer generate a number of sequences of rain attenuation on multiple links based on the AR model, evaluate the auto- and cross-correlation properties of each generated set, and subsequently select and use only those that show correlation properties close to the desired. In evaluation of a fade mitigation scheme, it is commonly desired to assess the error rate or capacity performance of a multiple-link configuration employing the scheme during rainy periods only. Time-inefficient evaluation might happen herein since the probability of rain event is small, only 1%–5% on average per year. Therefore, we recommend that after the whole rain attenuation sequences are generated, only portions of time in which rain events occur are examined for the system performance analysis. All statistics obtained from the performance evaluation are hence conditional upon the occurrence of rain. For simulation purposes we define the occurrence of rain to be an event in which the rain rate exceeds the value associated with probability of occurrence specified in the ITU-R Rec. P.837 [12]. The starts and ends of rain events can be determined by linearly interpolating the attenuation series and locating upward and downward “threshold”-crossing points. In addition, due to the slow variation of rain attenuation relative to the transmission rate, we recommend the adoption of quasianalytical simulation technique for error rate evaluation of a fade mitigation scheme. Values of signal-to-noise power ratio are obtained at sampling times and subsequently translated to error rates using an analytical equation or an empirically derived curve for the modulation scheme employed.
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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 5, 2006
Fig. 1. The evaluated system: Converging links of lengths L and L with separation angle between two receivers (BS1 and BS2) and a transmitter (ST).
Fig. 3. Simulated and desired complementary cumulative distributions of rain attenuation on a 1-km 30-GHz link.
Fig. 4. Simulated and desired temporal autocorrelations of rain attenuation on a 1-km 30-GHz link.
Fig. 2. Computer-generated rain attenuation. (a) Rare event with spikes of nearly 10 dB followed half an hour later by another event of lower attenuation. (b) Rare event as “observed” on two converging links separated by 180 .
IV. NUMERICAL EXAMPLE In this section, sequences of rain attenuation on two converging links at 30 GHz links are generated from the AR model. For the sake of exemplification, we arbitrarily take the temporal autocorrelation of rainfall rate obtained in Barcelona [13]. We
also compute the interlink correlation coefficients using the K–K method and assume Morita–Higuti spatial correlation model [6]. Fig. 1 illustrates the configuration of the converging links to and , with Fig. 2 be evaluated with exemplifying the computer-generated time series of rain attenuation on both links. Figs. 3 and 4 show the closeness of distribution and temporal autocorrelation of the computer-generated rain attenuation on a single 1-km 30 GHz link to the desired curves. The interlink average correlation coefficients of attenuation are invariably found from the simulation to differ from the prescribed values by less than 0.01. For instance, coefficients of 0.9563 and 0.8954 for two 1-km links with 45 and 180 separations, respectively, computed based on [6], are used in generating rain attenuation sequences, for which the corresponding values are found to be 0.9577 and 0.8998.
HENDRANTORO et al.: MULTIVARIATE AUTOREGRESSIVE MODEL OF RAIN ATTENUATION
V. CONCLUSION The multichannel AR model provides an alternative method to simulate the effects of rain on a network of radio links. With complete statistical knowledge of rainfall rates both in time and space, multiple sequences of rain attenuation occurring on a number of links, be they convergent, parallel, or connected in tandem, can be simulated and the performance of the entire multilink network can be assessed. Use of the model, however, must consider that some underlying assumptions concerning the statistical properties of rain events, both in the time and space domain, are fulfilled and that some limitations might cause inaccuracies in the results. In [14], an example of application of the model in assessment of a fade mitigation scheme involving adaptive modulation and site diversity is presented.
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[4] A. Paraboni and F. Barbaliscia, “Multiple site attenuation prediction models based on the rainfall structures (Meso- or Synoptic-Scales) for advanced TLC or broadcasting systems,” presented at the 27th General Assembly Int. Union Radio Sci., Maastricht, The Netherlands, Aug. 2002. [5] A. A. Abu-Dayya and N. C. Beaulieu, “Outage probabilities in the presence of correlated lognormal interferers,” IEEE Trans. Veh. Technol., vol. 43, no. 1, pp. 164–173, Feb. 1994. [6] J. D. Kanellopoulos and S. G. Koukoulas, “Analysis of the rain outage performance of route diversity systems,” Radio Sci., vol. 22, no. 4, pp. 549–565, Jul.–Aug. 1987. [7] C. J. Gibbins and K. S. Paulson, “Durations of rain events and rain attenuations at millimetric wavelengths,” presented at the Millennium Conf. Antennas and Propagation (AP 2000), Davos, Switzerland. [8] A. D. Panagopoulos, G. Fikioris, and J. D. Kanellopoulos, “Rain attenuation power spectrum of slant path,” Electron. Lett., vol. 38, no. 20, pp. 1220–1222, Sep. 2002. [9] K. Morita and I. Higuti, “Prediction methods for rain attenuation distributions of micro- and millimeter waves,” Rev. Electr. Commun. Lab., vol. 24, no. 7–8, pp. 651–668, Jul.–Aug. 1976. [10] J. H. Michels, P. K. Varshney, and D. D. Weiner, “Synthesis of correlated multichannel random processes,” IEEE Trans. Signal Process., vol. 42, no. 2, pp. 367–375, Feb. 1994. [11] C.-Y. Chu and K. S. Chen, “Effects of rain fading on the efficiency of the Ka-band LMDS system in the Taiwan area,” IEEE Trans. Veh. Technol., vol. 54, no. 1, pp. 9–19, Jan. 2005. [12] “Characteristics of Precipitation for Propagation Modeling,” ITU-R, Rec. P.837-4, 2003. [13] A. Burgueno, E. Vilar, and M. Puigcerver, “Spectral analysis of 49 years of rainfall rate and relation to fade dynamics,” IEEE Trans. Commun., vol. 38, no. 9, pp. 1359–1366, Sep. 1990. [14] G. Hendrantoro and Indrabayu, “A multichannel autoregressive model of rain attenuation on multiple radio links and its application in assessment of fade mitigation schemes in fixed wireless systems above 10 GHz,” presented at the 28th General Assembly Int. Union Radio Sci, New Delhi, India, Oct. 2005.
