1、 Recommendation ITU-R P.1057-4 (07/2015) Probability distributions relevant to radiowave propagation modelling P Series Radiowave propagation ii Rec. ITU-R P.1057-4 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, 2015 ITU 2015 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.1057-4 1 RECOMMENDATION ITU-R P.1057-4 Probability distributions relevant to radiowave propagation modelling (1994-2001-2007-2013-2015) Scope This Recommendation describes the various probability distributions relevant to radio
9、wave propagation modelling and predictions. The ITU Radiocommunication Assembly, considering a) that the propagation of radio waves is mainly associated with a random medium which makes it necessary to analyse propagation phenomena by means of statistical methods; b) that, in most cases, it is possi
10、ble to describe satisfactorily the variations in time and space of propagation parameters by known statistical distributions; c) that it is therefore important to know the fundamental properties of the probability distributions most commonly used in statistical propagation studies, recommends 1 that
11、 the statistical information relevant to propagation modelling provided in Annex 1 should be used in the planning of radiocommunication services and the prediction of system performance parameters; 2 that the step-by-step procedure provided in Annex 2 should be used to approximate a complementary cu
12、mulative distribution by a log-normal complementary cumulative distribution. Annex 1 Probability distributions relevant to radiowave propagation modelling 1 Introduction Experience has shown that information on the mean values of the signals received is not sufficient to characterize the performance
13、 of radiocommunication systems. The variations in time, space and frequency also have to be taken into consideration. The dynamic behaviour of both wanted signals and interference plays a decisive role in the analysis of system reliability and in the choice of system parameters such as modulation ty
14、pe. It is essential to know the extent and rapidity of signal fluctuations in order to be able to specify such parameters as type of modulation, transmit power, protection ratio against interference, diversity measures, coding method, etc. 2 Rec. ITU-R P.1057-4 For the description of communication s
15、ystem performance it is often sufficient to observe the time series of signal fluctuation and characterize these fluctuations as a stochastic process. Modelling of signal fluctuations for the purpose of predicting radio system performance, however, requires also knowledge of the mechanisms of intera
16、ction of radio waves with the atmosphere (neutral atmosphere and the ionosphere). The composition and physical state of the atmosphere is highly variable in space and time. Wave interaction modelling, therefore, requires extensive use of statistical methods to characterize various physical parameter
17、s describing the atmosphere as well as electrical parameters defining signal behaviour and the interaction processes via which these parameters are related. In the following, some general information is given on the most important probability distributions. This may provide a common background to th
18、e statistical methods for propagation prediction used in the Recommendations of the Radiocommunication Study Groups. 2 Probability distributions Stochastic processes are generally described either by a probability density function or by a cumulative distribution function. The probability density fun
19、ction, here denoted by p(x) for the variable x, is such that the probability of x taking a value in the infinitesimal interval x to x + dx is p(x) dx. The cumulative distribution function, denoted by F(x), gives the probability that the variable takes a value less than x, i.e. the functions are rela
20、ted as follows: )()( xFdxdxp or: xcttpxF d)()( where c is the lowest limit of the values which t can take. The following distributions are the most important: normal or Gaussian distribution; log-normal distribution; Rayleigh distribution; combined log-normal and Rayleigh distribution; Nakagami-Rice
21、 distribution (Nakagami n-distribution); gamma distribution and exponential distribution; Nakagami m-distribution; Pearson 2 distribution. Rec. ITU-R P.1057-4 3 3 Normal distribution This distribution is applied to a continuous variable of any sign. The probability density is of the type: p(x) = eT
22、(x) (1) T(x) being a non-negative second degree polynomial. If as parameters we use the mean, m, and the standard deviation, , then p(x) is written in the usual way: 221e x p21)( mxxp(2) hence: x mxtmtxF2e r f121d21e x p21)(2 (3) with: z t tz0 de2)(erf 2 (4) The solid lines in Fig. 1 represent the f
23、unctions p(x) and F(x) with m equal to zero and equal to unity. The cumulative normal distribution F(x) is generally tabulated in a short form for the same conditions. Table 1 gives the correspondence between x and F(x) for a number of round values of x or F(x). TABLE 1 x 1 F(x) x 1 F(x) 0 0.5 1.282
24、 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 the purpose of practical calculations, F(x) can be represented by approximate functions, for example the following which is valid for positive x with a r
25、elative error less than 2.8 103: 4 Rec. ITU-R P.1057-4 51.5339.0661.02)2/(e x p)(122xxxxF (5) A normal distribution is mainly encountered when values of the quantity considered result from the additive effect of numerous random causes, each of them of relatively slight importance. In propagation mos
26、t of the physical quantities involved (power, voltage, fading time, etc.) are essentially positive quantities and cannot therefore be represented directly by a normal distribution. On the other hand this distribution is used in two important cases: to represent the fluctuations of a quantity around
27、its mean value (scintillation); to represent the logarithm of a quantity. We then obtain the log-normal distribution which is studied later. Diagrams in which one of the coordinates is a so-called normal coordinate are available commercially, i.e. the graduation is such that a normal distribution is
28、 represented by a straight line. These diagrams are very frequently used even for the representation of non-normal distributions. 4 Log-normal distribution This is the distribution of a positive variable whose logarithm has a normal distribution. It is possible therefore to write directly the probab
29、ility density and the cumulative density: 2ln21e x p121)( mxxxp(6) 2lne r f121dln21e x p121)(20mxtmttxF x (7) However, in these relations m and are the mean and the standard deviation not of the variable x but of the logarithm of this variable. The log-normal distribution is very often found in conn
30、ection with propagation, mainly for quantities associated either with a power or field-strength level or a time. Power or field-strength levels are generally only expressed in decibels so that sometimes reference is made to a log-normal distribution simply as a normal distribution. This usage is not
31、 recommended. In the case of time (for example fading durations), the log-normal distribution is always used explicitly because the natural variable is the second or the minute and not their logarithm. Since the reciprocal of a variable with a log-normal distribution also has a log-normal distributi
32、on, this distribution is sometimes found in the case of rates (reciprocals of time). For example, it is used to represent rainfall rate distributions. In comparison with a normal distribution, it can be considered that a log-normal distribution means that the numerical values of the variable are the
33、 result of the action of numerous causes of slight individual importance which are multiplicative. When considered in numerical terms, a log-normal distribution is extremely asymmetrical, unlike the normal distribution. In particular, the mean value, the median value and the most probable value (oft
34、en called the mode) are not identical (see the dashed lines in Fig. 1). Rec. ITU-R P.1057-4 5 The characteristic quantities of the numerical 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 p2
35、e x p 22 m . FIGURE 1 Normal and log normal distributions P . 1 0 5 7 - 0 13 2 1 0 1 2 3 4 5 6x00 . 10 . 20 . 30 . 40 . 50 . 60 . 70 . 80 . 91Probability p x( ), n o rm alF x( ), n o rm alp x( ), l o g -n o rm alF x( ), l o g -n o rm alM o d eM ea nM ed i an0 0 05 Rayleigh distribution The Rayleigh
36、distribution applies to a positive continuous variable. It is linked with the normal distribution as follows. Given a two-dimensional normal distribution with two independent variables y and z of mean zero and the same standard deviation , the random variable 22 zyx (8) has a Rayleigh distribution.
37、The most probable value of x is equal to . The Rayleigh distribution represents the distribution of the length of a vector which is the sum of a large number of vectors of similar amplitudes whose phases have a uniform distribution. The probability density and the cumulative distribution are given b
38、y: 222 2e xp)( xxxp(9) 6 Rec. ITU-R P.1057-4 222e xp1)( xxF(10) Figure 2 gives examples of these functions p(x) and F(x) for three different values of b. FIGURE 2 Rayleigh distribution p(x) is shown as solid lines and F(x) as dashed lines for three different values of b: blue b = 1; red b = 2; green
39、 b = 4 P . 1 0 5 7 - 0 2x000 . 20 . 40 . 60 . 810 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4P()or()xFx4 .5 5 5 .5 60 .10. 30. 50. 70. 9The characteristic values of the variable are as follows: most probable value: 2; median value: bb 833.02ln ; mean value: 2 0.886b; root mean square value: b; standard deviation:
40、 bb 463.041 . The Rayleigh distribution is often only used near the origin, i.e. for low values of x. In this case we have: () 22 (11) This expression can be interpreted as follows: the probability that the random variable X will have a value of less than x is proportional to the square of this valu
41、e. If the variable in question is a voltage, its square represents the power of the signal. In other words, on a decibel scale the power decreases Rec. ITU-R P.1057-4 7 by 10 dB for each decade of probability. This property is often used to find out whether a received level has a Rayleigh distributi
42、on at least asymptotically. It should be noted, however, that other distributions can have the same behaviour. In particular the Rayleigh distribution occurs for scatter from independent, randomly-located scatterers for which no scattering component dominates. Footnote: b = .2 6 Combined log-normal
43、and Rayleigh distribution In some cases the distribution of a random variable can be regarded as the resultant of a combination of two distributions, i.e. a log-normal distribution for long-term variations and a Rayleigh distribution for short-term variations. The distribution of instantaneous value
44、s is obtained by considering a Rayleigh variable whose mean (or mean square) value is itself a random variable having a log-normal distribution. The probability density function of the combined log-normal and Rayleigh distribution is: uumumukxkxxl d222e x pe x p2p22 (12) and the cumulative distribut
45、ion of the combined log-normal and Rayleigh distribution is: 1() = 12 22( +) 22 (13) where m and are used to designate the mean and the standard deviation of the normal distribution The value of k depends on the interpretation of and . 1) If and are the standard deviation and mean of the natural log
46、arithm of the most probable value of the Rayleigh distribution, then = 1/2; 2) if and are the standard deviation and mean of the natural logarithm of the median value of the Rayleigh distribution, then = ln2; 3) if and are the standard deviation and mean of the natural logarithm of the mean value of
47、 the Rayleigh distribution, then = /4; and 4) if and are the standard deviation and mean of the natural logarithm of the root mean square value of the Rayleigh distribution, then = 1. The mean, root mean square, standard deviation, median, and most probable value of the combined Rayleigh log-normal distribution are: 8 Rec. ITU-R P.1057-4 Mean value, E: = 2 22( +)2( +)22 0 = 2( +22 ) Root mean square value, RMS: = 22 22( +)2( +)22 0= 1exp ( +2) Standard deviation, SD: = 12(+2) 42( +22 ) = 1( +22 )(2)4 Median value: The median value i
