1、 Recommendation ITU-R P.1057-5 (12/2017) Probability distributions relevant to radiowave propagation modelling P Series Radiowave propagation ii Rec. ITU-R P.1057-5 Foreword The role of the Radiocommunication Sector is to ensure the rational, equitable, efficient and economical use of the radio-freq
2、uency spectrum by all radiocommunication services, including satellite services, and carry out studies without limit of frequency range on the basis of which Recommendations are adopted. The regulatory and policy functions of the Radiocommunication Sector are performed by World and Regional Radiocom
3、munication Conferences and Radiocommunication Assemblies supported by Study Groups. Policy on Intellectual Property Right (IPR) ITU-R policy on IPR is described in the Common Patent Policy for ITU-T/ITU-R/ISO/IEC referenced in Annex 1 of Resolution ITU-R 1. Forms to be used for the submission of pat
4、ent statements and licensing declarations by patent holders are available from http:/www.itu.int/ITU-R/go/patents/en where the Guidelines for Implementation of the Common Patent Policy for ITU-T/ITU-R/ISO/IEC and the ITU-R patent information database can also be found. Series of ITU-R Recommendation
5、s (Also available online at http:/www.itu.int/publ/R-REC/en) Series Title BO Satellite delivery BR Recording for production, archival and play-out; film for television BS Broadcasting service (sound) BT Broadcasting service (television) F Fixed service M Mobile, radiodetermination, amateur and relat
6、ed satellite services P Radiowave propagation RA Radio astronomy RS Remote sensing systems S Fixed-satellite service SA Space applications and meteorology SF Frequency sharing and coordination between fixed-satellite and fixed service systems SM Spectrum management SNG Satellite news gathering TF Ti
7、me signals and frequency standards emissions V Vocabulary and related subjects Note: This ITU-R Recommendation was approved in English under the procedure detailed in Resolution ITU-R 1. Electronic Publication Geneva, 2017 ITU 2017 All rights reserved. No part of this publication may be reproduced,
8、by any means whatsoever, without written permission of ITU. Rec. ITU-R P.1057-5 1 RECOMMENDATION ITU-R P.1057-5 Probability distributions relevant to radiowave propagation modelling (1994-2001-2007-2013-2015-2017) Scope This Recommendation describes the various probability distributions relevant to
9、radiowave propagation modelling and prediction methods. Keywords Probability distributions, normal, Gaussian, log-normal, Rayleigh, Nakagami-Rice, gamma, exponential, Pearson The ITU Radiocommunication Assembly, considering a) that the propagation of radio waves is mainly associated with a random me
10、dium that makes it necessary to analyse propagation phenomena by means of statistical methods; b) that, in most cases, it is possible to describe satisfactorily the variations in time and space of propagation parameters by known statistical probability distributions; c) that it is important to know
11、the fundamental properties of the probability distributions most commonly used in statistical propagation studies, recommends 1 that the statistical information relevant to propagation modelling provided in Annex 1 should be used in the planning of radiocommunication services and the prediction of s
12、ystem performance parameters; 2 that the step-by-step procedure provided in Annex 2 should be used to approximate a complementary cumulative probability distribution by a log-normal complementary cumulative probability distribution. Annex 1 Probability distributions relevant to radiowave propagation
13、 modelling 1 Introduction Experience has shown that information on the mean values of received signals is not sufficient to accurately characterize the performance of radiocommunication systems. The variations in time, space, and frequency should also be considered. 2 Rec. ITU-R P.1057-5 The dynamic
14、 behaviour of both wanted signals and interference plays a significant role in the analysis of system reliability and in the choice of system parameters such as modulation type. It is essential to know the probability distribution and rate of signal fluctuations in order to specify parameters such a
15、s modulation type, transmit power, protection ratio against interference, diversity measures, coding method, etc. For the description of communication system performance, it is often sufficient to observe the time series of signal fluctuation and characterize these fluctuations as a stochastic proce
16、ss. Modelling of signal fluctuations for the purpose of predicting radio system performance requires knowledge of the mechanisms of the interaction of radio waves with the neutral atmosphere and ionosphere. The composition and physical state of the atmosphere is highly variable in space and time. Wa
17、ve interaction modelling, therefore, requires extensive use of statistical methods to characterize various physical parameters describing the atmosphere as well as electrical parameters defining signal behaviour and the interactive processes which these parameters are related. In the following, some
18、 general information is given on the most important probability distributions. This may provide a common background to the statistical methods for propagation prediction used in the Recommendations of the Radiocommunication Study Groups. 2 Probability distributions Probability distributions of stoch
19、astic processes are generally described either by a probability density function (PDF) or by a cumulative distribution function (CDF). The probability density function of the random variable X, denoted by p(x), is the probability of X taking a value of x; and the cumulative distribution function of
20、the random variable X, denoted by F(x), is the probability of X taking a value less than or equal to x. The PDF and CDF are related as follows: )()( xFdxdxp (1a) or: xcttpxF d)()( (1b) where c is the lower limit of integration. The following probability distributions are the most important for the a
21、nalysis of radiowave propagation: normal or Gaussian probability distribution; log-normal probability distribution; Rayleigh probability distribution; combined log-normal and Rayleigh probability distribution; Nakagami-Rice (Nakagami n) probability distribution; gamma probability distribution and ex
22、ponential probability distribution; Nakagami m probability distribution; Pearson 2 probability distribution. Rec. ITU-R P.1057-5 3 3 Normal probability distribution The normal (Gaussian) probability distribution of a propagation random variable is usually encountered when a random variable is the su
23、m of a large number of other random variables. The normal (Gaussian) probability distribution is a continuous probability distribution in the interval from = to+. The probability density function (PDF), (), of a normal distribution is: p(x) = k eT(x) (2) where T(x) is a non-negative second degree po
24、lynomial of the form ( )2, where m and are the mean and standard deviation, respectively, of the normal probability distribution, and is selected so () = 1 . Then p(x) is: () = 122 exp (12( )2) (3) The cumulative distribution function (CDF), (), of a normal probability distribution is: () = 122 exp(
25、12( )2) (3a) = 12 exp(122) (3b) = 121+erf (2) (3c) where: erf() = 2 exp (2)0 (3d) The complementary cumulative distribution function (CCDF), Q(), of a normal probability distribution is: Q() = 122 exp (12( )2) (4a) = 12 exp (122) (4b) = 12erfc(2) (4c) where: erfc() = 2 exp (2) (4d) Note that ()+Q()
26、= 1 and erf()+erfc() = 1. The inverse cumulative distribution function = 1() is the value of such that () = ; and the inverse complementary cumulative distribution function = Q1() is the value of such that Q() = . The solid lines in Fig. 1 represent the functions p(x) and F(x) with m = 0 and = 1, an
27、d Table 1 shows the correspondence between x and 1() for various example values of x or 1(). 4 Rec. ITU-R P.1057-5 TABLE 1 x 1 F(x) x 1 F(x) 0 0.5 1.282 101 1 0.1587 2.326 102 2 0.02275 3.090 103 3 1.350 103 3.719 104 4 3.167 105 4.265 105 5 2.867 107 4.753 106 6 9.866 1010 5.199 107 5.612 108 For p
28、ractical calculations, the following simple approximation for Q() = 1() is valid for any positive x and has a relative approximation error of less than 2.8 103: Q() = 1() = exp (22 )2(0.661+0.3392+5.51) (5) Most modern mathematical software packages include the (), Q(), erf (), and erfc(x) functions
29、. In propagation, most of the physical quantities involved (power, voltage, fading time, etc.) are essentially positive quantities and cannot be represented directly by a normal probability distribution. The normal probability distribution is used in two important cases: to represent the fluctuation
30、s of a random variable around its mean value (e.g. scintillation fades and enhancements); to represent the fluctuations of the logarithm of a random variable, in which case the variable has a log-normal probability distribution (see 4). Diagrams in which one of the coordinates is a so-called normal
31、coordinate, where a normal cumulative probability distribution is represented by a straight line, are available commercially. These diagrams are frequently used even for the representation of non-normal probability distributions. 4 Log-normal probability distribution The log-normal probability distr
32、ibution is the probability distribution of a positive random variable X whose natural logarithm has a normal probability distribution. The probability density function, (), and the cumulative distribution function, (), are: 2ln21e x p121)( mxxxp(6) 2lne r f121dln21e x p121)(20mxtmttxF x (7) Rec. ITU
33、-R P.1057-5 5 where m and are the mean and the standard deviation of the logarithm of X (i.e. not the mean and the standard deviation of X). The log-normal probability distribution is very often found in propagation probability distributions associated with power and field-strength. Since power and
34、field-strength are generally expressed in decibels, their probability distributions are sometimes incorrectly referred to as normal rather than log-normal. In the case of probability distributions vs. time (e.g. fade duration in seconds), the log-normal terminology is always used explicitly because
35、the natural dependent variable is time rather than the logarithm of time. Since the reciprocal of a variable with a log-normal probability distribution also has a log-normal probability distribution, this probability distribution is sometimes found in the case of the probability distribution of the
36、rate of change (e.g. fading rate in dB/s or rainfall rate in mm/hr). In comparison with a normal probability distribution, a log-normal probability distribution is usually encountered when values of the random variable of interest results from the product of other approximately equally weighted rand
37、om variables. Unlike a normal probability distribution, a log-normal probability distribution is extremely asymmetrical. In particular, the mean value, the median value, and the most probable value (often called the mode) are not identical (see the dashed lines in Fig. 1). The characteristic values
38、of the random variable X are: most probable value: exp (m 2); median value: exp (m); mean value: 2exp 2m ; root mean square value: exp (m + 2); standard deviation: 1)(e x p2e x p 22 m . 6 Rec. ITU-R P.1057-5 FIGURE 1 Normal and log-normal probability distributions P . 1 0 5 7 - 0 1 6 4 2 0 2 4 6x00
39、. 10 . 20 . 30 . 40 . 50 . 60 . 70 . 80 . 91PxFx()or()N = 0 = 1o rma l ( ) ,p x N = 0 = 1o rma l ( ) ,F x L o g -n = 0 = 1o rma l ( ) ,p x L o g -n o = 0 , = 1rma l ( ) F x L o g -n mo d eo rma lL o g -n me d i ano rma lL o g -n me ano rma lN me an , me d i an , mo d eo rma l5 Rayleigh probability d
40、istribution The Rayleigh probability distribution is a continuous probability distribution of a positive-valued random variable. For example, given a two-dimensional normal probability distribution with two independent random variables y and z of mean zero and the same standard deviation , the rando
41、m variable 22 zyx (8) has a Rayleigh probability distribution. The Rayleigh probability distribution also represents the probability distribution of the length of a vector that is the vector sum of a large number of constituent vectors of similar amplitudes where the phase of each constituent vector
42、 has a uniform probability distribution. The probability density function and the cumulative distribution function of a Rayleigh probability distribution are given by: 222 2e xp)( xxxp(9) 222e xp1)( xxF(10) Figure 2 provides examples of p(x) and F(x) for three different values of b. Rec. ITU-R P.105
43、7-5 7 FIGURE 2 Rayleigh probability distribution P . 1 0 5 7 - 0 2x000 . 20 . 40 . 60 . 810 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4PxFx()or()4 .5 5 5 .5 60 .10. 30. 50. 70. 9p x b( ), = 1F x b( ), = 1p x b( ), = 2F x b( ), = 2p x b( ), = 4F x b( ), = 4Defining = 2, the characteristic values of the random vari
44、able X are: most probable value: 2; median value: bb 833.02ln ; mean value: 2 0.886b; root mean square value: b; standard deviation: bb 463.041 . The Rayleigh probability distribution is often applicable for small values of x. In this case, the cumulative distribution function, (), can be approximat
45、ed as: () 22 (11) This approximate expression can be interpreted as follows: the probability that the random variable X will have a value less than x is proportional to the square of x. If the variable of interest is a voltage, its square represents the power of the signal. In other words, on a deci
46、bel scale the power decreases by 10 dB for each decade of probability. This property is often used to determine whether a received level has an asymptotic Rayleigh probability distribution. Note, however, that other probability distributions can have the same behaviour. 8 Rec. ITU-R P.1057-5 In radi
47、owave propagation, a Rayleigh probability distribution occurs in the analysis of scattering from multiple, independent, randomly-located scatterers for which no single scattering component dominates. 6 Combined log-normal and Rayleigh probability distribution In some cases, the probability distribut
48、ion of a random variable can be regarded as the combination of two probability distributions; i.e. a log-normal probability distribution for long-term (i.e. slow) variations and a Rayleigh probability distribution for short-term (i.e. fast) variations. This probability distribution occurs in radiowave propagation analyses when the inhomogeneities of
