P.1.4. The laws of distribution of random variables used in mathematical models.

Characteristic of cellular mobile communication systems is multiple reflection, scattering, absorption of radio waves and the absence of deterministic patterns observed for the conditions of direct visibility between the base and mobile stations. At the receiving point (both in the case of BTS and MS), the resulting value of the field strength vector is defined as the geometric sum of a set of vectors (multipath propagation of radio waves), the amplitudes and phases of which vary randomly.
In addition, the cellular construction of the network leads to the mutual influence of emitters having equal carrier frequencies (co-channel interference), as well as noise leads to an additional change in the conditions for reliable reception of information.
Therefore, in order to assess the quality of radio communications in a cellular mobile communication system, it is necessary to use probabilistic characteristics.
Particularly important for the construction of mathematical models of radio channels are the laws of the distribution of random variables. In the theory of information transfer with statistically changing channel parameters, the following distribution laws are used:

- Rayleigh distribution (to describe fast decay);
- Gaussian distribution (used to describe noise and interference from random signal sources);
- normal logarithmic (or log-normal) distribution (used to describe slow attenuation);
- Reiss-Nakagami distribution (used to describe the propagation of radio waves in media with diffuse scattering on random inhomogeneities).
Rayleigh distribution
The propagation of radio signals in media with fading (more precisely, in environments where diffuse multipath processes occur and the field at the receiving point represents the interference of an infinitely large number of elementary rays) can be described by the Rayleigh probability density function of the form:
-X2
'202 (P. 1.16)
W (x) = 4-e2
o2
where X is a random variable; A2 = A - variance, random variable; while the value of X can take many values ​​in the range from 0 to oo, that is:

In fig. Section 1.4 shows the Rayleigh distribution W (X). By comparing the derivative to zero dW / dX = 0, we determine the most probable Hnd value for the Rayleigh law: Xn = a, with W (X = a) = 0.606 / a.


The average value (mathematical expectation) in the case of the Rayleigh distribution is determined by the formula:

(x) = jx2 w (x) dx = -e2o2 that is, 1.25 times the rms value.
The integral distribution function for the Rayleigh law is:
P (x) = jw (X) dX = - • jx-c2aldx = 1-е2 ° 2, OsZsoo. (P.1.19)
If P (X) * 0.5, then the median value of the random variable is obtained:
Xm = 1.177a = 0.94 (J0- (P.1.20)
The Rayleigh law gives a statistical description of changes in the length of the resultant vector with the geometric addition of a sufficiently large number of vectors with randomly varying amplitudes (modules) with statistical independence of amplitudes and phases and a uniform distribution of the phase. To estimate the proximity of distributions to the Rayleigh law, the ratio of values ​​of X, corresponding to probabilities of 0.9 and 0.1, is often used. From the formula (P. 1.19) it can be determined that this ratio is 4.66, which corresponds to 13.37 dB.
Completing the consideration of the Rayleigh distribution, it should be noted that the random variable X is understood as the random variable of the field strength of the radio wave (either E or E). Therefore, in the sections devoted to the consideration of the Rayleigh model of multipath propagation of radio waves in mobile communication systems, instead of X, it is necessary to substitute E (or H).
Generalized Rayleigh distribution law
If the law describes the distribution of the length of the resultant vector with geometric addition of vectors with a constant wavelength and constant phase and the sum of random vectors distributed according to the Rayleigh law, this distribution law is called the generalized Rayleigh distribution law. The distribution density for this law is expressed by the formula:

If the length of the constant vector is very small and b0, then in the formula (P. 1.21) the limit transition to the Rayleigh distribution will be obtained, which is natural.
If b is so large that bX / o2 »1, then for 10 (bX / o2) you can use the asymptotic value:

Gaussian distribution law (normal distribution law)
The Gaussian probability density function is a function used to describe noise and sources of random signals.
The Gaussian probability density function is determined by the formula:
W (x) = - == e2 ° 2, (P.1.27)
l / 2l: about
where X is a random variable, (X) is the mean value, o is the standard deviation. In fig. Section 1.5 shows the Gaussian function W (X).
As follows from the figure, the function W (X) is symmetric with respect to (X) the average value, and the value of X can vary from -oo to + oo. We determine the most probable value Xn: since dW / dX = 0 exists only for (X), then Xn = (X).

Fig. P.1.6. The emissions from the integral distribution function for the Gaussian law is determined from the formula:
Р (ЛГ) = - = - fe 2 ° 2 dX, (A.1.28)
V2jta xi „
By replacing § = (X- (X)) / a, the formula (P. 1.28) can be reduced to the form:
* - <*)
1 ° g = 1
P (x) - == Je2dl = ed (X), (P.1.29)
V 2n _00
where erf (Z) is the probability integral (tabulated), the error function.
If X = - (X), then P ((X)) = 0.5, that is, for the Gaussian law, the median value of the random variable Hm = (X) = Chi.
To assess the proximity of statistical distributions obtained from the experiment, the Gaussian law is usually applied the so-called Gaussian scale along the probability axis, in it P (X) linearly varies from X, passing through the median X = Xmrp R (XM) = 0.5, with this slope of this straight line is determined by the value of the standard deviation of o. As noted in [P. 1.1], outliers in a can be up to ± 4а and ± 5а, that is, with a Gaussian distribution, the change in Р (Х) is within ± 5а (Fig. P. 1.6):
- at Р (Х) = 0.01 —X = 2.32 a,
- at Р (Х) = 0.1 —Х = 1.28а,
- with P (X) = 0.99 - X = 2.32 a,
- at Р (Х) = 0,9999 ... —Х * 5о.
Log-normal distribution law
The logarithmically normal distribution law occurs when the random variable X itself is not distributed according to the normal law, but its logarithm (with any basis in principle). If you take the base 10, and enter the notation:

From this formula it follows that it is valid not only for the logarithm of a random variable, but also the product of this logarithm by any constant value. Therefore, the log-normal law corresponds to the normal distribution of a random variable expressed in decibels.
The numerical characteristics of such a distribution: (X) = (Xm) l, oL are also expressed in decibels relative to the selected level.
To estimate the proximity of the distributions found from the experiment to the logarithmically normal law, one can use the Gaussian scale on the abscissa axis, and on the ordinate axis, plot the values ​​of a random variable in decibels on a uniform scale.
For example, with P (XL) = 0.99, XL = 2.32 aL, dB.
Reiss-Nakagami distribution
When considering the diffusely scattered electromagnetic field Ena inhomogeneities of the type of forests, that is, outside the mirror directions, the average value of the field strength in this case (E) => 0, and the Es field itself consists of fields scattered by different areas of inhomogeneities [P. 1.2].
If we assume that the field at the receiving point is equal to the sum of the quadrature components:
Е = Ег +] ЕЪ (П.1.31)
then the values ​​of the components E \ and E2 can be represented as a sum of a large number of independent values ​​of Ezi E2h:
E, ^ Eu, E2 ^ E2g (P.1.32)
t-1 t-1
In accordance with the central limit theorem, the distribution of a random variable, which is the sum of N independent random variables, with N-> oo approaches the normal distribution law, regardless of which distribution each of the terms complies with.
If we assume that each term in the quadrature representation of E, that is, in the formulas (P. 1.30) and (P. 1.31), the components are independent random variables, then in the direction where the field consists of coherent (E) and incoherent components, the expression (P. 1.32).
Assuming that the coherent field is written in the form:
(E) = Aqei4 °, (A.1.33)
and count the phase so that gr0 = 0, then the incoherent field can be represented as a difference: Es = Е- (Е). (P. 1.34)
Given the formula (P. 1.28), you can write it down in the form of a large number of terms:
Es = E \ -L) +) E2.
where Ex-A, = ^ Eii E2 =. (P.1.35)
Using the central limit theorem, the probability density function is written as:
(e, -A)) 2 + e22
e 2a ', (P.1.36)
where o] is the variance (Ex-Aq) and E2.
Then the probability density distribution of the amplitude is determined by the Reiss-Nakagami formula [A1.2]:
2l
W, (A) = (Ej, E2) A-dip = (A / o2s) exp [- (A2 + A ^) / 2o2s} l0 \ (A0 A) / os]. (P.1.37)
It should be noted that for electromagnetic waves scattered in forest areas (densely overgrown forest), taking into account wind fluctuations of forest vegetation, the distribution (P.1.3.7) at t2 <1 will be written as [P.1.4]:
(P. 1.38)
where t2 = S2IPq is the ratio of the power of a stable signal to the average power of a signal from moving reflectors (wind movement), P is a random function (in this case, the amplitude distribution of field strength A), / 0 is a modified zero-order Bessel function.
When / 0 = 0, this distribution becomes Rayleigh.
When t2 = 0, the distribution (A.1.38) turns into a typical exponential function:
W (p) dp = e (p) - & g, (P)
which expresses the distribution of the field created by the scattering of electromagnetic waves on a conglomerate of random reflectors alone.