1、 E 4855232 0.523372 b55 Rec. ITU-R PN.1057 53 RECOMMENDATION ITU-R PN. 1057 PROBABILITY DISTRIBUTIONS RELEVANT TO RADIOWAVE PROPAGATION MODELLING (1994) The ITU Radiocommunication Assembly, considering a) analyse propagation phenomena by means of statistical methods; that the propagation of radio wa
2、ves is mainly associated with a random medium which makes it necessary to b) parameters by known statistical distributions; that, in most cases, it is possible to describe satisfactorily the variations in time and space of propagation c) commonly used in statistical propagation studies, that it is t
3、herefore important to know the fundamental properties of the probability distributions most recommends 1. of radiocommunication services and the prediction of system performance parameters. that the statistical information relevant to propagation modelling provided in Annex 1 be used in the planning
4、 ANNEX 1 Probability distributions relevant to radiowave propagation modelling 1. Introduction Experience has shown that information on the mean values of the signais received is not sufficient to characterize the performance of radiocommunication systems. The variations in time, space and frequency
5、 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 type. It is essential to know the extent and rapidity of signal fluctuations
6、in order to be able to specify such parameters as type of modulation, 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 cha
7、racterize 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 interaction of radio waves with the atmosphere (neutral atmosphere and the ionosphere). The composition
8、 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 parameters describing the atmosphere as well as electrical parameters defining signal behaviour and the in
9、teraction 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 the statistical methods for propagation prediction used in the Recommendations of the ITU-R Study G
10、roups. COPYRIGHT International Telecommunications Union/ITU RadiocommunicationsLicensed by Information Handling Services54 Rec. ITU-R PN.1057 2. Probability distributions Stochastic processes are generally described either by a probability density function or by a cumulative distribution function. T
11、he probability density function, 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
12、.e. the functions are related as follows: or: where c is the lowest limit of the values which r can take. The following distributions are the most important: - normal or Gaussian distribution, - log-normal distribution, - Rayleigh distribution, - - Nakagami-Rice distribution (Nakagami n-distribution
13、), - gamma distribution and exponential distribution, - Nakagami m-distribution, - Pearson x2 distribution. combined log-normal and Rayleigh distribution, 3. Gaussian or normal distribution This distribution is applied to a continuous variable of any sign. The probability density is of the type: p(x
14、) = e-T(X) (1) T(x) being a non-negative second degree polynomial. If as parameters we use the mean, m, and the standard deviation, 6, then p (x) is written in the usual way: hence: X F(x) = a* -00 exp - i (?y dt = i i + erf (%)I with: (3) The cumulative normal distribution F(x) is generally tabulat
15、ed in a short form, i.e. with m taken to be zero and o equal to unity. Table 1 gives the correspondence between x and F(x) for a number of round values of x or F(x). COPYRIGHT International Telecommunications Union/ITU RadiocommunicationsLicensed by Information Handling Services 4855232 0523374 428
16、= X Rec. ITU-R PN.1057 TABLE 1 1 - F(x) 5.000 O0 x 10-1 1.586 55 x 10- 2.275 O1 x 1W2 1.349 90 x 3.167 12 x 2.866 52 x 9.865 87 x lo- X 1.281 55 2.326 35 3.090 23 3.719 O1 4.264 89 4.753 42 5.199 34 5.612 O0 55 1 - F(x) lo- 10-2 1 0-3 10-4 10-5 1 0-6 1 0-7 1 0-8 For the purpose of practical calculat
17、ions, F(x) can be represented by approximate functions, for example the following which is valid for positive x with a relative error less than 2.8 x lP3: exp (-x2/2) (.66, x + 0.3394m) 1 - F(x) = A Gaussian distribution is mainly encountered when values of the quantity considered result from the ad
18、ditive effect of numerous random causes, each of them of relatively slight importance. In propagation most of the physical quantities involved (power, voltage, fading time, etc.) are essentially positive quantities and cannot therefore be represented directly by a Gaussian distribution. On the other
19、 hand this distribution is used in two important cases: - - to represent the fluctuations of a quantity around 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
20、-called Gaussian coordinate are available commercially, Le. the graduation is such that a Gaussian distribution is represented by a straight line. These diagrams are very frequently used even for the representation of non-Gaussian distributions. 4. Log-normal distribution This is the distribution of
21、 a positive variable whose logarithm has a Gaussian distribution. It is possible therefore to write directly the probability density and the cumulative density: COPYRIGHT International Telecommunications Union/ITU RadiocommunicationsLicensed by Information Handling Services- Lt8552L2 0523375 364 56
22、Rec. ITU-R PN.1057 However, in these relations m and o are the mean and the standard deviation not of the variable n but of the - most probable value: exp (rn - 02) logarithm of this variable. The characteristic quantities of the variable x can be derived without difficulty. We find: - median value:
23、 exp (m) - meanvalue: exp (m + :) - - standard deviation: exp root mean square value: exp (m + 02) Unlike the Gaussian distribution, a log-normal distribution is extremely asymmetrical. In particular, the mean value, the median value and the most probable value (often called the mode) are not identi
24、cal (see Fig. 1). FIGURE 1 Nom1 and log-nom1 distributions 1.0 O. 8 O. 6 ,x .e 2 a 0.4 0.2 O i- +. -3 -2 -1 O 1 2 3 4 5 6 X : normal : log-normal - - - - - - - - The log-normal distribution is very often found in connection with propagation, mainly for quantities associated either with a power or fi
25、eld-strength level or a time. Power or field-strength levels are generally only expressed in decibels so that it is more usual to refer to a normal distribution of levels. In the case of time (for example fading durations), the log-normal distribution is used explicitly because the natural variable
26、is the second or the minute and not their logarithm. COPYRIGHT International Telecommunications Union/ITU RadiocommunicationsLicensed by Information Handling Services4855232 0523376 2TO Rw. ITU-R PN.1057 57 Since the reciprocal of a variable with a log-normal distribution also has a log-normal distr
27、ibution, this distribution is sometimes found in the case of rates (reciprocals of time). For example, it is used to represent rainfall rate distribution at least for low and medium rainfall rates. In comparison with a Gaussian distribution, it can be considered that a log-normal distribution means
28、that the numerical values of the variable are the result of the action of numerous causes of slight individual importance which are mu1 tiplicative. 5. Rayleigh distribution The Rayleigh distribution applies to a non-limited positive continuous variable. The probability density and the cumulative di
29、stribution are given by: Figure 2 represents these functions p(x) and F(x). FIGURE 2 Rayleigh distribution 0.6 h Y .3 - 3 % 0.4 0.2 O O 0.5 1 1.5 2 2.5 3 3.5 I II X “O02 COPYRIGHT International Telecommunications Union/ITU RadiocommunicationsLicensed by Information Handling Services 4855232 0523377
30、137 W 58 Re ITU-R PN.1057 The characteristic values of the variable are as follows: - most probable value: q - median value: q$zz = 1.18 q - mean value: qqn2 = 1.25 q - root mean square value: q $ = 1.41 q - standard deviation: qc- 2 - - 0.655 The Rayleigh distribution is often only used near the or
31、igin, i.e. for low values of x. In this case we have: This 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 value. If the variable in question is a voltage, its square represents the power of the signai.
32、In other words on a decibel scale the power decreases by 10 dB for each decade of probability. This property is often used to find out whether a received level has a Rayleigh distribution at least asymptotically. It should be noted however that other distributions can have the same behaviour. The Ra
33、yleigh distribution is linked with the Gaussian distribution as follows. Given a two-dimensional Gaussian distribution with two independent variables x and y of mean zero and the same standard deviation a, the random variable has a Rayleigh distribution and the most probable value of r is equal to a
34、. As r represents the length of a vector joining a point in a two-dimensional Gaussian distribution to the centre of this distribution, it may be deduced that the Rayleigh distribution represents the distribution of the length of a vector which is the sum of a large number of vectors of low amplitud
35、es whose phases have a uniform distribution. In particular the Rayleigh distribution occurs in scattering phenomena. 6. Combined log-normal 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
36、-normal distribution for long-term variations and a Rayleigh distribution for short-term variations. The distribution of instantaneous values is obtained by considering a Rayleigh variable whose mean (or mean square) value is itself a random variable having a log-normal distribution. If m and a are
37、used to designate the mean and the standard deviation of the Gaussian distribution associated with the log-normal distribution, the following distribution is obtained: +- 1 - F(x) = - J exp- ge-20 - du 2 27t -00 In this formula the standard deviation a is expressed in nepers. If o is used to designa
38、te its value in decibels, we have: O = 0.115 0 (13) Figure 3 shows a graph of this distribution for a number of values of the standard deviation, the value of m being taken to be equal to zero. The distribution occurs mainly in propagation via inhomogeneities of the medium when the characteristics o
39、f the latter have non-negligible long-term variations, as for example in the case of tropospheric scatter. COPYRIGHT International Telecommunications Union/ITU RadiocommunicationsLicensed by Information Handling Services4855232 0523378 073 M 20 IO O - IO s. a, 3 Y -20 .e - 30 - 40 - 50 - 60 Rec. IW-
40、R PN.1057 FIGURE 3 Combined log-normal and Rayleigh distributiom (with standard deviation of the log-normal distribution as parameter) I 1 1030 50 70 90 99 99.9 99.99 99.999 o. 1 *r 10.01 Percentage probability that ordinate will be exceeded, (1 - F(x) x 100 (%) 0.001 E? : 7. Nakagami-Rice distribut
41、ion (Nakagami n-distribution)* The Nakagami-Rice distribution is also derived from the Gaussian distribution and it generalizes the Rayleigh distribution. Given a two-dimensional Gaussian distribution with two independent variables x and y and with the same standard deviation, the length of a vector
42、 joining a point in the distribution to a fixed point different from the centre of the distribution will have a Nakagami-Rice distribution. This distribution can therefore also be considered as the distribution of the length of a vector which is the sum of a fixed vector and of a vector whose length
43、 has a Rayleigh distribution. * Not to be confused with the Nakagami m-distribution. COPYRIGHT International Telecommunications Union/ITU RadiocommunicationsLicensed by Information Handling Services4855232 0523377 TOT 60 Rec. ITU-R PN.1057 If a is used to designate the length of the fixed vector and
44、 (T the most probable length of the Rayleigh vector, the probability density is given by: where io is a modified Bessel function of the first kind and of zero order. This distribution depends on two parameters but for the purposes of propagation problems it is necessary to choose a relation between
45、the amplitude a of the fixed vector and the root mean square amplitude o 4 of the random vector. This relation depends on the application envisaged. The two main applications are as follows: a) For studies of the influence of a ray reflected by a rough surface, a represents the field of the direct r
46、ay which Power in the fixed vector is constant, but the total power in fixed and random components varies. can be considered constant and which is taken as a reference by writing: p=- a or, in dB, R = 20 log p (15) p representing the relative amplitude of the reflected ray. The mean power received i
47、s then equal to (1 + p2) a2. b) For the purpose of studying multipath propagation through the atmosphere, it can be considered that the sum of the power carried by the fixed vector and the mean power carried by the random vector is constant since the power carried by the random vector originates fro
48、m that of the fixed vector. If the total power is taken to be unity, one then has: Total power in the fixed and random components is constant, but both components vary. where: and: a = COST (17) The fraction of the total power carried by the random vector is then equal to sin2 cp. If X is used to de
49、signate the instantaneous amplitude of the resultant vector and x a numerical value of this amplitude, we find that the probability of having an instantaneous level greater than x is given by: Prob(X x) = 1 - F(x) = 2exp(-l/tan2cp) rvexp(-v2) dv (19) tan cp xl sin cp Figure 4 shows this distribution for different values of the fraction of power carried by the random vector (sin2 cp). The Nakagami-Rice distribution is shown in Fig. 4, but for the purpose of practical applications use has been made of a decibel scale for the amplitudes, and for the probabi
