**3. Complexation of intermittent communication and diversity reception**

#### **3.1 Substantiation of expediency of complexation of intermittent communication and diversity reception of signals**

It is known that at use of diversity reception fading in the channel becomes less deep, due it increases the noise immunity of signal reception. At complexation intermittent communication with diversity reception, gain comparing to diversity reception without intermittent communication will be less than that of an intermittent connection, comparatively the continuous reception in a single. And this gain decreases with magnification of number of branches of diversity as with growth of the last depth of a fading decreases. However usage of intermittent communication at restricted number of branches of the diversity, normally put into practice, leads to noticeable magnification of noise immunity. Besides, application of diversity reception at intermittent communication increases use factor of radio-line that also should raise noise immunity of reception and spectral efficiency of data transfer.

Value *T/M,* named use factor of radio-line, essentially influences noise immunity of reception of signals. At complexation of intermittent communication and diversity reception in communication paths for calculation probability of error reception it is necessary to define use factor of radio-line at various number of branches of diversity (M) depending on J1J.

#### **3.2 Analysis of noise immunity at complexation of intermittent communication and diversity reception with a combination branches of diversity on algorithm of an auto select**

It is known that the probability density function at *M* - the multiple diversity reception with an ideal auto select at an independent homogeneous fading can be defined from expression

$$\mathcal{W}\_{\mathcal{M}}\left(\boldsymbol{\gamma}\right) = \frac{d\left(\int\_{0}^{\boldsymbol{\gamma}} \boldsymbol{f}\_{\boldsymbol{\gamma}} \, d\boldsymbol{\gamma}\right)^{\mathcal{M}}}{d\boldsymbol{\gamma}},\tag{10}$$

where *M* is number of branches of diversity, *fr* - probability density SNR at various types of a fading.

At complexation of intermittent communication and diversity reception with a combination of branches of diversity on algorithm of an auto select, use factor of radio-line at various number of branches of diversity (M) depending on J1J are defined according to expression

$$\eta\_M(\boldsymbol{\gamma}\_0) = \bigcap\_{\boldsymbol{\gamma}\_1}^{\infty} \mathcal{W}\_M(\boldsymbol{\gamma}) \, d\boldsymbol{\gamma} = \int\_{\boldsymbol{\gamma}\_1}^{\infty} \frac{d \left( \int\_{\boldsymbol{\gamma}}^{\boldsymbol{\gamma}} \boldsymbol{d}\boldsymbol{\gamma} \right)^m}{d\boldsymbol{\gamma}} \, d\boldsymbol{\gamma} \, \,. \tag{11}$$

Use factor of radio-line at complexation of intermittent communication and diversity reception with a combination of branches of diversity on algorithm of an auto select at a signal fading under accordingly Rayleigh (12), the generalized Rayleigh (13), lognormal (14) and Nakagami (15) laws will be defined according to (11) expressions

$$\eta\_M(\boldsymbol{\chi}\_0) = \int\_{\boldsymbol{\chi}\_{\boldsymbol{\epsilon}}}^{\infty} \frac{d \left[ 1 - \exp\left( -\frac{\boldsymbol{\chi}}{\boldsymbol{\chi}\_0} \right) \right]^m}{d \boldsymbol{\chi}} d\boldsymbol{\gamma} = 1 - \left[ 1 - \exp\left( -\frac{\boldsymbol{\chi}\_{\boldsymbol{\epsilon}}}{\boldsymbol{\chi}\_0} \right) \right]^M,\tag{12}$$

$$\eta\_M(\boldsymbol{\gamma}\_0) = \int\_{\boldsymbol{\gamma}\_1}^{\boldsymbol{\alpha}} \frac{d \left[ \frac{1}{\boldsymbol{\gamma}\_0 - \boldsymbol{\gamma}\_a} \cdot \exp\left(-\frac{\boldsymbol{\gamma} + \boldsymbol{\gamma}\_a}{\boldsymbol{\gamma}\_0 - \boldsymbol{\gamma}\_a}\right) \cdot I\_0\left(\frac{2\sqrt{\boldsymbol{\gamma}\_a \cdot \boldsymbol{\gamma}}}{\boldsymbol{\gamma}\_0 - \boldsymbol{\gamma}\_a}\right) d\boldsymbol{\gamma}\right]^M}{d\boldsymbol{\gamma}} d\boldsymbol{\gamma} \tag{13}$$

$$\begin{split} \eta\_{M}\left(\boldsymbol{\gamma}\_{0}\right) &= \int\_{\boldsymbol{\gamma}\_{1}}^{\boldsymbol{\alpha}} \frac{d\left[\frac{1}{2} + \frac{1}{2} \operatorname{erf}\left(\frac{1}{2} \cdot \frac{\ln\boldsymbol{\gamma} - \ln\boldsymbol{\gamma}\_{0\dots\text{ref}}}{\sqrt{\ln\boldsymbol{\gamma}\_{0} - \ln\boldsymbol{\gamma}\_{0\dots\text{ref}}}}\right)\right]^{M} d\boldsymbol{\gamma} = \\ &= 1 - \left[\frac{1}{2} + \frac{1}{2} \cdot \operatorname{erf}\left(\frac{1}{2} \cdot \frac{\ln\boldsymbol{\gamma}\_{t} - \ln\boldsymbol{\gamma}\_{0\dots\text{ref}}}{\sqrt{\ln\boldsymbol{\gamma}\_{0} - \ln\boldsymbol{\gamma}\_{0\dots\text{ref}}}}\right)\right]^{M} \end{split} \tag{14}$$

$$\begin{split} \boldsymbol{\eta}\_{M} \left( \boldsymbol{\gamma}\_{0} \right) &= \frac{M}{\Gamma\left(m\right)} \left( \frac{m}{\boldsymbol{\gamma}\_{0}} \right)^{m} \Bigg\limits\_{\boldsymbol{\gamma}\_{1}} \Bigg[ \frac{\Gamma\left(m, \frac{\boldsymbol{\gamma}}{\boldsymbol{\gamma}\_{0}} m\right)}{\Gamma\left(m\right)} \Bigg]^{M-1} \exp\left( -\frac{\boldsymbol{\gamma}}{\boldsymbol{\gamma}\_{0}} m \right) \boldsymbol{\gamma}^{m-1} d\boldsymbol{\gamma} = \\ &= 1 - \left[ \frac{\Gamma\left(m\right) - \Gamma\left(m, \frac{\boldsymbol{\gamma}\_{1}}{\boldsymbol{\gamma}\_{0}} m\right)}{\Gamma\left(m\right)} \right]^{M} \end{split} \tag{15}$$

2 *<sup>X</sup>* where *erf(x)* = r= f exp(-t <sup>2</sup> )dt errorfunction. v1r o

Analytical expression (12) is a special case of the formula (15) at *m,* equal 1. Dependence of use factor of radio-line on number of branches of diversity for a fading under the law of Nakagami, at *rt=}1)* and m equal 0.7, under the formula (15), is presented on fig. 1.

Fig. 1. Dependence of use factor of radio-line from *M* for a fading under the law of Nakagami, at *y i=yo* and m equal 0.7.

Similarly, methods of calculation for a single reception and intermittent communication is defined depending on the average error probability of noncoherent reception in the complexation intermittent links and diversity reception of the average SNR (Yo) for the cases of the fading signal differential binary phase shift key (D-BPSK), respectively, under the laws of the Rayleigh (16) [], the generalized Rayleigh (17) [], log-normal (18) [] and Nakagami (19)

$$P\_{\mathcal{M}}(\boldsymbol{\gamma}\_{0}) = \frac{1}{2\eta\_{\mathcal{M}}} \cdot \int\_{\boldsymbol{\gamma}\_{1} \cdot \eta\_{\mathcal{M}}(\boldsymbol{\gamma}\_{0})}^{\circ} \frac{d\left[1 - \exp\left(-\frac{\boldsymbol{\gamma}}{\boldsymbol{\gamma}\_{0}}\right)\right]^{\circ \mathrm{M}}}{d\boldsymbol{\gamma}} \cdot \exp(-a\boldsymbol{\gamma})d\boldsymbol{\gamma} =$$

$$\frac{1}{2\eta\_{\mathcal{M}}(\boldsymbol{\gamma}\_{0})} \cdot M \cdot \exp\left[-a\boldsymbol{\gamma}\_{i} \cdot \eta\_{\mathcal{M}}(\boldsymbol{\gamma}\_{0})\right] \cdot \left[\sum\_{k=1}^{M} C\_{\mathcal{M}-1}^{k-1} \cdot \frac{(-1)^{k-1} \cdot \exp\left(-\frac{k\boldsymbol{\gamma}\_{i}}{\boldsymbol{\gamma}\_{0}}\right)}{a\boldsymbol{\gamma}\_{0} \cdot \eta\_{\mathcal{M}}(\boldsymbol{\gamma}\_{0}) + k}\right]^{\prime} \tag{16}$$

$$P\_{\mathcal{M}}(\boldsymbol{\gamma}\_{0}) = \frac{1}{2\eta\_{\mathcal{M}}(\boldsymbol{\gamma}\_{0})} \prod\_{\mathcal{I}\_{1}}^{\circ} \exp(-a\boldsymbol{\gamma}) \frac{d\left[\frac{1}{\nu}\frac{1}{\boldsymbol{\gamma}\_{0} - \boldsymbol{\gamma}\_{a}} \cdot \exp\left(-\frac{\boldsymbol{\gamma} + \boldsymbol{\gamma}\_{a}}{\boldsymbol{\gamma}\_{0} - \boldsymbol{\gamma}\_{a}}\right) \cdot I\_{0}\left(\frac{2\boldsymbol{\gamma}\_{\mathcal{I}} - \boldsymbol{\gamma}}{\boldsymbol{\gamma}\_{0} - \boldsymbol{\gamma}\_{a}}\right)\right]}{d\boldsymbol{\gamma}} d\boldsymbol{\gamma} \tag{17}$$

$$P\_{M}(\boldsymbol{\chi}\_{0}) = \frac{1}{\eta\_{M}} \int\_{\boldsymbol{\chi}\_{M}}^{\boldsymbol{\chi}} \int\_{\boldsymbol{\chi}\_{M}} \left| \frac{\left[\frac{1}{2} + \frac{1}{2} \cdot \text{erf}\left(\frac{1}{2} \cdot \frac{\ln\gamma - \ln\gamma\_{0\dots\eta}\eta\_{M}}{\sqrt{\ln\gamma\_{0} - \ln\gamma\_{0\dots\eta'}}}\right)\right]^{\mathsf{M}-1}}{2\sqrt{\pi}} \cdot \frac{\exp\left[-\frac{1}{4} \frac{\left(\ln\gamma - \ln\gamma\_{0\dots\eta'}\eta\_{M}\right)^{2}}{\ln\gamma\_{0} - \ln\gamma\_{0\dots\eta'}}\right]}{\boldsymbol{\chi} \cdot \sqrt{\ln\gamma\_{0} - \ln\gamma\_{0\dots\eta'}}} \right| \cdot \frac{\exp(-\alpha\boldsymbol{\chi})}{2} d\boldsymbol{\chi}\prime \tag{18}$$

$$P\_M(\boldsymbol{\gamma}\_0) = \frac{M}{2\eta\_M(\boldsymbol{\gamma}\_0)} \left[\frac{m}{\boldsymbol{\gamma}\_0 \eta\_M(\boldsymbol{\gamma}\_0)}\right]^w \frac{1}{\Gamma(m)} \int\_{\boldsymbol{\gamma} \eta\_M(\boldsymbol{\gamma}\_0)}^{\boldsymbol{\gamma}} \left\{1 - \frac{\Gamma\left[m, \frac{m\boldsymbol{\gamma}}{\boldsymbol{\gamma}\_0 \eta\_M(\boldsymbol{\gamma}\_0)}\right]}{\Gamma(m)}\right\}^{M-1} \boldsymbol{\gamma}^{w-1} \exp\left\{-\left[\frac{m}{\boldsymbol{\gamma}\_0 \eta\_M(\boldsymbol{\gamma}\_0)} + \alpha\right] \boldsymbol{\gamma}\right\} d\boldsymbol{\gamma} \tag{19}$$

Dependences of probability of errors at a signal fading under the law of Nakagami, and complexation of intermittent communication and diversity reception with a combination of branches of diversity on algorithm of an auto select, at r1= *yo* and at m equal 0.7 in comparison with continuous diversity reception are shown in Fig. 2.

Fig. 2. Probabilities of errors at complexation of intermittent communication and diversity reception, with a combination of branches of diversity on algorithm of an auto select for a signal fading under the law of Nakagami at *Yt* = *yo* and at m equal 0.7 in comparison with continuous diversity by reception (continuous curves) at various number of branches of diversity (M).

Dependence of probability of an error at a lognormal fading of a signal (18) is obtained for a non stationary case, when the dispersion of the lognormal process defined by a ratio at fixed *Yo\_ref* isn't a constant

$$\log \sigma\_{\chi}^{2} = \ln \left\langle X^{2} \right\rangle = \ln \left\langle \chi\_{0} - \ln \chi\_{0\\_ref.} \right\rangle = \ln \frac{\mathcal{Y}\_{0}}{\mathcal{Y}\_{0\\_ref.}!} \,. \tag{20}$$

In practice, at calculation of noise immunity of communication lines the dispersion of lognormal process normally is the fixed value. For this case it is expedient to define analytical expressions of probability density SNR (21), use factor of radio-line (22, 23) and probabilities of error reception of incoherent signal differential D-BPSK (24) in the conditions of a lognormal fading at complexation of intermittent communication and

diversity reception with a combination branches of diversity on algorithm of an auto select. The diagrams illustrating specified dependences are presented accordingly on fig. 3, 4, 5. Analytical expression of probability density SNR for the specified type of a fading is defined by a method of symbolic mathematical modeling in the environment of MathCAD 14

Fig. 3. Probability densities SNR diversity reception of signals in the channel with a lognormal fading at combination of branches of diversity on algorithm of an auto select, at the fixed dispersion (o/) and mean value SNR (yo).

$$\mathcal{W}\_{M}(\boldsymbol{\gamma}) = \frac{M}{2} \cdot \frac{1}{\sqrt{2\pi\sigma\_{\boldsymbol{\chi}\boldsymbol{\gamma}}^{2}\boldsymbol{\gamma}}} \cdot \left[\frac{1}{2} + \frac{1}{2} \cdot \text{erf}\left(\frac{\sqrt{2}}{2} \cdot \frac{\ln\sqrt{\frac{\boldsymbol{\gamma}}}{\boldsymbol{\gamma}\_{0}} + \sigma\_{\boldsymbol{\chi}}^{2}}{\sigma\_{\boldsymbol{\chi}}}\right)\right]^{M-1} \cdot \exp\left[-\frac{\left(\ln\sqrt{\frac{\boldsymbol{\gamma}}{\boldsymbol{\gamma}\_{0}} + \sigma\_{\boldsymbol{\chi}}^{2}}\right)^{2}}{2\sigma\_{\boldsymbol{\chi}}^{2}}\right] = $$

$$\frac{M}{2} \cdot \frac{1}{\sqrt{2\pi\sigma\_{\boldsymbol{\chi}\boldsymbol{\gamma}}^{2}\boldsymbol{\gamma}}} \cdot \left[\Phi\left(\frac{\ln\sqrt{\frac{\boldsymbol{\gamma}}{\boldsymbol{\gamma}\_{0}} + \sigma\_{\boldsymbol{\chi}}^{2}}}{\sigma\_{\boldsymbol{\chi}}}\right)\right]^{M-1} \cdot \exp\left[-\frac{\left(\ln\sqrt{\frac{\boldsymbol{\gamma}}{\boldsymbol{\gamma}\_{0}} + \sigma\_{\boldsymbol{\chi}}^{2}}\right)^{2}}{2\sigma\_{\boldsymbol{\chi}}^{2}}\right].\tag{21}$$

the use factor of radio-line will thus be defined under the formula

$$\eta\_M(\gamma\_0) = \bigcap\_{\gamma\_1}^{\alpha} \mathcal{W}\_M(\gamma) \, d\gamma = 1 - \left[ \frac{1}{2} + \frac{1}{2} \, erf\left(\frac{1}{\sqrt{2}} \, \frac{\ln\sqrt{\frac{\mathcal{V}\_t}{\mathcal{V}\_0}} + \sigma\_\chi^{-2}}{\sigma\_\chi} \right) \right]^M = 1 - \left[ \Phi\left(\frac{\ln\sqrt{\frac{\mathcal{V}\_t}{\mathcal{V}\_0}} + \sigma\_\chi^{-2}}{\sigma\_\chi} \right) \right]^M \tag{22}$$

At *Yt =yo* the use factor of radio-line depends only on number branches of diversity, fig. 4

$$\eta\_M = 1 - \left[\frac{1}{2} + \frac{1}{2}\operatorname{erf}\left(\frac{\sigma\_\chi}{\sqrt{2}}\right)\right]^M = 1 - \left[\Phi\left(\sigma\_\chi\right)\right]^M,\tag{23}$$

where ¢(x) is normal cumulative distribution function.

The probability of error reception for signal D-BPSK will be defined thus according to expression

Fig. 4. Dependence of use factor of radio-line from number branches of diversity (M) at *r1=ro*  and the fixed dispersion *(az2).* 

In the fading signal under the laws of Nakagami and lognormal use factor of the radio line increases, respectively, with increasing parameter of fading (m) and lowering variance (( *az2).*  The probabilities of errors are reduced under these conditions.

#### **3.3 The noise immunity analysis at complexation of intermittent communication with diversity reception and optimal addition of branches diversity**

Let's consider a variant of diversity reception of signal D-BPSK when combination of branches of diversity is carried out by a method of optimal addition for which the equality defined by expression is valid

$$P\_M\left(\boldsymbol{\gamma}\_{\Sigma}\right) = \frac{1}{2} \exp\left(-\sum\_{i=1}^{M} \boldsymbol{\gamma}\_i\right),\tag{25}$$

where value SNR of optimally added signal (n·) will be to equally arithmetical total SNR of all of i channels.

It is possible to show that the average probability of an error after optimal addition will be defined by the formula

(24)

$$P\_M = \frac{1}{2} \stackrel{\alpha}{\underset{0}{\cdots}} \cdots \stackrel{\alpha}{\underset{0}{\text{exp}}} \exp\left(-\sum\_{i=1}^{M} \gamma\_i \right) \mathcal{W}\_M \left(\gamma\_1 \cdots \gamma\_M \right) d\gamma\_1 \cdots d\gamma\_M \,. \tag{26}$$

where WM(r1 ···rM) -is a combined M-dimensional density of probabilities .

Fig. 5. Probabilities of errors of diversity reception of signals in the channel with a lognormal fading at combination of branches of diversity on algorithm of an auto select (continuous curves) and probabilities of errors at complexation of intermittent communication and diversity reception at combination of branches of diversity on algorithm of an auto select for *Yt* = *yo, a}* =0.5 and M=l, 2, 4, 8.

Using the fact that the independent fading diversity branches in the combined probability density is the product of probability densities in separate branches of reception according to the analytical expression

$$P\_M = \frac{1}{2} \prod\_{i=1}^{M} \int \exp(-\gamma\_i) W\_i(\gamma\_i) d\gamma\_i \,\,\,\,\,\tag{27}$$

and with homogeneous channels in branches of diversity the probability of error reception on an output of the demodulator signal D-BPSK after operation of optimal addition will be described by the formula

$$P\_M = \frac{1}{2} \left[ \int\_0^\alpha \exp(-\gamma) \mathcal{W}(\gamma) d\gamma \right]^M. \tag{28}$$

Formulas (27, 28) are valid at any distribution laws.

The average probability of error at complexation of the intermittent communication which are carried out in branches of diversity with the subsequent optimal addition and incoherent demodulation of signal D-BPSK at a fading of an envelope of a signal under laws Rayleigh and Nakagami will be defined accordingly by expressions (29, 30)

$$\begin{split} P\_{\mathcal{M}}\left(\boldsymbol{\chi\_{0}}\right) &= \frac{1}{2} \Bigg[ \frac{1}{\eta\left(\boldsymbol{\chi\_{0}}\right)\_{\boldsymbol{\chi},\boldsymbol{\eta}\left(\boldsymbol{\chi\_{0}}\right)}} \Bigg[ \exp\left(-a\boldsymbol{\chi}\right) \cdot \frac{1}{\boldsymbol{\chi\_{0}}} \cdot \exp\left(-\frac{\boldsymbol{\chi}}{\boldsymbol{\chi\_{0}}}\right) d\boldsymbol{\eta} \Bigg]^{\mathcal{M}} = \\ &= \frac{1}{2} \Bigg[ \frac{1}{1 + \boldsymbol{\chi\_{0}}\eta\left(\boldsymbol{\chi\_{0}}\right)} \Bigg]^{\mathcal{M}} \exp\left[-M\boldsymbol{\chi\_{i}}\eta\left(\boldsymbol{\chi\_{0}}\right)\right] \end{split} \tag{29}$$

$$P\_M(\boldsymbol{\gamma}\_0) = \frac{1}{2} \left\langle \frac{1}{\eta(\boldsymbol{\gamma}\_0)} \frac{1}{\Gamma(m)} \bigg[\frac{m}{\boldsymbol{\gamma}\_0 \cdot \eta(\boldsymbol{\gamma}\_0)}\right]^m \int\_{\boldsymbol{\gamma}, \eta(\boldsymbol{\gamma}\_0)}^{\boldsymbol{\gamma}} \boldsymbol{\gamma}^{m-1} \exp\left[-\left(\frac{m}{\boldsymbol{\gamma}\_0 \cdot \eta(\boldsymbol{\gamma}\_0)} + \alpha\right) \boldsymbol{\gamma}\right] d\boldsymbol{\gamma}\right\rangle^M = \frac{1}{2} \left\langle \frac{1}{\Gamma(m)\eta(\boldsymbol{\gamma}\_0)} \frac{1}{\boldsymbol{\gamma}\_0 \cdot \eta(\boldsymbol{\gamma}\_0)} \bigg[\frac{m}{\boldsymbol{\gamma}\_0 \cdot \eta(\boldsymbol{\gamma}\_0)} + \alpha\right] = \frac{1}{2} \left\langle \frac{1}{\eta(\boldsymbol{\gamma}\_0)} \frac{1}{\boldsymbol{\gamma}\_0 \cdot \eta(\boldsymbol{\gamma}\_0)} \right\rangle^M \tag{30}$$
 
$$= \frac{1}{2} \left\langle \frac{1}{\Gamma(m\_r m \frac{\boldsymbol{\gamma}\_1}{\boldsymbol{\gamma}\_0})} \frac{\Gamma\left[m\_r m \frac{\boldsymbol{\gamma}\_1}{\boldsymbol{\gamma}\_0} + \alpha \boldsymbol{\gamma}\_r \eta(\boldsymbol{\gamma}\_0)\right]}{\Gamma\left[1 + \alpha \frac{\boldsymbol{\gamma}\_0}{m} \eta(\boldsymbol{\gamma}\_0)\right]^m} \right\rangle^M,\tag{31}$$

where *77(yo)* - the use factor of radio-line in a separate branch of diversity, is defined for the specified types of a fading accordingly expressions (6, 9), thus level of a threshold of interruption in various branches of diversity is identical. Analytical expression (29) is a special case of the formula (30) at m equal 1.

The probability of presence of a signal in diversity branches, under condition of their homogeneity and independence, is defined binomial by the distribution law, therefore the use factor of radio-line on an output of the circuit of optimal addition will be defined by expression (31)

$$\eta\_M(\boldsymbol{\gamma}\_0) = 1 - \left[\mathbb{1} - \eta\left(\boldsymbol{\gamma}\_0\right)\right]^M. \tag{31}$$

The diagrams illustrating dependence of probability of an error from mean value SNR at various number of branches of diversity, according to analytical expression (30), at the complexation of intermittent communication, considered above, and diversity reception for a fading of an envelope of a signal under the law of Nakagami, at *Yt* = *yo* and m equal 0.6 are presented on fig. 6.

At application of intermittent communication after performance of operation of optimal addition it is more convenient to define probability density SNR on an output of the circuit of optimal addition by a method of characteristic functions, and then on the obtained probability density to calculate use factor of radio-line and probability of error.

Fig. 6. Probabilities of errors at diversity reception (continuous curves) and probabilities of errors at complexation of intermittent communication carried out in branches of diversity with their subsequent optimal addition, at *M* equal 1,2,4 and 8, m equal 0,6 and y1 = yo.

Characteristic function from probability density SNR at a fading an envelope of a signal under the law of Nakagami will be defined by expression

$$S(\nu) = \bigcap\_{0}^{\upsilon} f(\gamma) \exp(i\nu \gamma) d\gamma = \frac{m^m}{\Gamma(m)} \frac{1}{\gamma\_0^m} \prod\_{0}^{\upsilon} \gamma^{m-1} \exp\left[-\left(\frac{m}{\gamma\_0} - i\nu\right)\gamma\right] d\gamma. \tag{32}$$

Integral in the formula (32) tabular, according to expression

$$\int\_0^\alpha \mathbf{x}^{\nu - 1} \exp(-\mu \mathbf{x}) \, d\mathbf{x} = \frac{\Gamma(\nu)}{\mu^\nu} \,, \tag{33}$$

therefore characteristic function will be defined by the formula

$$S(\nu) = \frac{m^m}{\varkappa\_0^m} \frac{1}{\left(\frac{m}{\varkappa\_0} - i\nu\right)^m} \,. \tag{34}$$

Probability density SNR in the channel with diversity reception at combination of independent homogeneous channels on algorithm of optimal addition will be defined by inverse transformation from characteristic function in raise to power of number branches of diversity

$$f\_M(\boldsymbol{\gamma}) = \int\_{\boldsymbol{\gamma}}^{\boldsymbol{\alpha}} \frac{1}{2\pi} \mathbb{E}\left[S(\boldsymbol{\nu})\right]^M \exp(-i\boldsymbol{\nu}\boldsymbol{\gamma})d\boldsymbol{\nu} = \frac{1}{2\pi} \frac{m^{m\mathcal{M}}}{\boldsymbol{\gamma}\_0^m} \int\_{\boldsymbol{\alpha}}^{\boldsymbol{\alpha}} \frac{\exp(-i\boldsymbol{\nu}\boldsymbol{\gamma})}{\left(\frac{m}{\boldsymbol{\gamma}\_0} - i\boldsymbol{\nu}\right)^{m\mathcal{M}}} d\boldsymbol{\nu} \,. \tag{35}$$

Integral in the formula (35) tabular, defined by to expression

$$\int\_{a}^{a} (\beta - i\mathbf{x})^{\mathsf{T}'} \exp(-ip\mathbf{x}) d\mathbf{x} = \frac{2\pi \cdot p^{\mathsf{v} - 1} \exp(-\beta p)}{\Gamma(\mathsf{v})},\tag{36}$$

therefore final expression of probability density SNR at optimal addition branches of diversity will be defined as

$$f\_M(\boldsymbol{\gamma}) = \frac{m^{m\mathcal{M}}}{\Gamma(m\mathcal{M})} \frac{\boldsymbol{\mathcal{Y}}^{m\mathcal{M}-1}}{\boldsymbol{\gamma}\_0^{m\mathcal{M}}} \exp\left(-\frac{\boldsymbol{\mathcal{Y}}}{\boldsymbol{\gamma}\_0} m\right). \tag{37}$$

On fig. 7 probability density SNR is presented at optimal addition of branches of diversity, at m equal 0.6; 1.4 and 7 in the absence of the diversity, doubled and quadrupled diversity receptions.

Fig. 7. Probability densities SNR at optimal addition of branches of diversity, at m equal 0.6; 1.4 and 7, accordingly red, dark blue and brown curves, in the absence of diversity; doubled and quadrupled diversity receptions, accordingly continuous, dashed and dash-dotted curves.

Use factor of radio-line, at complexation of diversity reception with optimal addition of branches of diversity and the intermittent communication which are carried out after that addition, in channels with a fading under laws Rayleigh and Nakagami are described accordingly by formulas (38, 39)

$$\eta\_M\left(\gamma\_0\right) = \left(\frac{1}{\gamma\_0}\right)^M \frac{1}{\left(M-1\right)!} \prod\_{\mathcal{I}\_1}^{\infty} \gamma^{M-1} \exp\left(-\frac{\mathcal{I}}{\mathcal{I}\_0}\right) = \frac{\Gamma\left(M, \frac{\mathcal{I}\dagger}{\mathcal{I}\_0}\right)}{\Gamma\left(M\right)},\tag{38}$$

$$\eta\_{\rm M} \left( \gamma\_0 \right) = \left( \frac{m}{\gamma\_0} \right)^{mM} \frac{1}{\Gamma \left( mM \right)} \int\_{\gamma\_t}^{\infty} \nu^{mM - 1} \exp \left( -\frac{\mathcal{I}}{\gamma\_0} m \right) = \frac{\Gamma \{ mM, \frac{J \cdot \mathcal{I}}{\gamma\_0} m \}}{\Gamma \{ mM \}} \,. \tag{39}$$

Integrals in expressions (38) and (39) are tabular (5). Analytical expression (38) is a special case of the formula (39) at *m,* equal 1.

Dependences probability of error at incoherent reception of a signal at complexation of diversity reception with optimal addition of branches of diversity and the intermitten t communication, which are carried out after that addition, from mean value SNR (yo), under formulas (40, 41) for cases of a fading of signal D-BPSK under accordingly laws Rayleigh and Nakagami will be defined by averaging of probability of errors in Gaussian noise according to the specified fading at values SNR above the given threshold level (r1)

$$P\_{M}\left(\boldsymbol{\gamma}\_{0}\right) = \frac{1}{2\eta\_{M}\left(\boldsymbol{\gamma}\_{0}\right)} \frac{1}{\boldsymbol{\gamma}\_{0}^{M}\left(M-1\right)!} \stackrel{\text{\tiny $\boldsymbol{\gamma}^{M}$ }}{\boldsymbol{\gamma}\_{0}} \boldsymbol{\gamma}^{M-1} \exp\left[-\left(\frac{1}{\boldsymbol{\gamma}\_{0}} + \alpha\right)\boldsymbol{\gamma}\right] = \frac{1}{2} \left(\frac{1}{1+\alpha\boldsymbol{\gamma}\_{0}}\right)^{M} \frac{\Gamma\left(M,\frac{\boldsymbol{\gamma}\_{1}}{\boldsymbol{\gamma}\_{0}} + \alpha\boldsymbol{\gamma}\_{1}\right)}{\Gamma\left(M,\frac{\boldsymbol{\gamma}\_{1}}{\boldsymbol{\gamma}\_{0}}\right)}.\tag{40}$$

$$P\_{M}\left(\boldsymbol{\gamma}\_{0}\right) = \frac{1}{2\eta\_{M}\left(\boldsymbol{\gamma}\_{0}\right)} \left[\frac{m}{\boldsymbol{\gamma}\_{0}\eta\_{M}\left(\boldsymbol{\gamma}\_{0}\right)}\right]^{mM} \frac{1}{\Gamma\left(mM\right)} \stackrel{\text{\tiny $\boldsymbol{\gamma}^{M}$ }}{\boldsymbol{\gamma}\_{0}\eta\_{\left(\boldsymbol{\gamma}\_{0}\right)}} \exp\left\{-\left[\frac{m}{\boldsymbol{\gamma}\_{0}\eta\_{\left(\boldsymbol{\gamma}\_{0}\right)}} + \alpha\right]\right\} =$$

$$= \frac{1}{2} \left[1 + \alpha \frac{\boldsymbol{\gamma}\_{0}}{m} \frac{\Gamma(mM,\frac{\boldsymbol{\gamma}\_{1}}{\boldsymbol{\gamma}\_{0}} + \alpha)}{\Gamma(mM)}\right]^{mM} \underbrace{\frac{\Gamma(mM,\frac{\boldsymbol{\gamma}\_{1}}{\boldsymbol{\gamma}\_{0}} + \alpha)}{\Gamma(mM)}\right]}\_{\Gamma\left(mM,\frac{\boldsymbol{\gamma}\_{1}}{\boldsymbol{\gamma}\_{0}} + \alpha)$$

The diagrams illustrating dependence of probability of error from mean value SNR at complexation of diversity reception with optimal addition of branches of diversity and intermittent communication, carried out after that additions, for a fading of an envelope of a signal under the law of Nakagami, at r1 = yo and m equal 0.6 is presented on fig. 8.

Dependences use factor of radio-line from number branches of diversity (M) at complexation of diversity reception with optimal addition branches of diversity and intermittent communication , which are carried out after that addition, and use factor of radio-line at complexation of intermittent communication, which carried out in branches of diversity with their subsequent optimal addition, is presented on fig. 9.

The use factor of radio-line at complexation of diversity reception with optimal addition of branches of diversity and the intermittent communication which are carried out after that addition, increases from magnification of branches of diversity (M) faster, than use factor of radio-line at communication of diversity reception and the intermittent communication which

Fig. 8. Probabilities of errors at diversity reception (continuous curves) and probabilities of errors at complexation of diversity reception with optimal addition branches of diversity and the intermittent communication which are carried out after that addition, at *M* equal 1,2,4 and 8, m equal 0.6 and *Yt* = *yo ..* 

Fig. 9. Dependences of use factor of radio-line from number branches of diversity at m equal 0.6 and *Yt* = *yo,* at complexation of intermittent communication and diversity reception when interruption is carried out on an output of the circuit of optimal addition (a line 1) and when interruption is carried out in diversity branches, with their subsequent optimal addition (a line 2).

are carried out in branches of diversity with their subsequent optimal addition (fig. 9). Values last are commensurable with use factor of radio-line at complexation of intermittent communication and diversity reception and combine of branches of diversity on algorithm of an auto select (fig. 1), even at smaller value *m.* Accordingly, spectral efficiency thus increases.

Probabilities of errors at complexation of diversity reception and the intermittent communication, which are carried out in branches of diversity with their subsequent optimal addition, decrease with magnification of mean value SNR faster probabilities of errors at complexation of diversity reception with optimal addition branches of diversity and intermittent communication, which are carried out after that addition.

Unlike complexation of intermittent communication and diversity reception with combination branches of diversity on algorithm of an auto select when the receiver selects a peak signal from all branches of diversity, at optimal addition the receiver should accept a signal from all branches for the subsequent handling. For this purpose it is necessary to use some antennas and to provide orthogonality of transmittable signals.

#### **4. Complexation some methods adaptation of radio-lines**

Let's consider a combination of an intermittent of radiation with constant average power and transmitter's speed automatic adjustment. We will suppose that information transfer rate changes depending on signal level in such a manner that SNR on a receiving device input remains constant. In this case at signal level changes under the Rayleigh law distribution density of probability of the random variable which is transmission rate will be defined by expression

$$\mathcal{W}(o) = \frac{r}{o\_0} \exp\left(-\frac{or}{o\_0}\right),\tag{42}$$

where *m* is information transfer rate, *0-0* is a constant for the given system value, r - fixed valueSNR

Average rate of information transfer is defined as mathematical expectation of value *m* 

$$V\_{cp} = \bigcap\_{0}^{\alpha} \frac{r}{\alpha\_0} \exp\left(-\frac{\alpha r}{\alpha\_0}\right) d\alpha = \frac{\alpha\_0}{r} \,. \tag{43}$$

Median value of speed will be equal

$$V\_M = \ln 2 \frac{\alpha\_0}{r} \,. \tag{44}$$

Let's enter designation mi meaning threshold value of the instant transmission rate at which the transmitter is shut down. We will find a number of the elementary signals which are transferred at speed exceeding threshold value. For this purpose we will consider integral

$$N = \int\_{\alpha\_l}^{\alpha} \phi \frac{r}{\alpha\_0} \exp\left(-\frac{\alpha r}{\alpha\_0}\right) d\phi = \frac{\alpha\_0}{r} \exp\left(-\frac{\alpha\_l r}{\alpha\_0}\right) \left(\frac{r}{\alpha\_0} \alpha\_l + 1\right). \tag{45}$$

Having divided it into value of average rate from (43) we will obtain a share of characters transferred at excess of threshold value ~

$$\frac{N}{V\_{cp}} = \exp\left(-\frac{\alpha\_t r}{\alpha\_0}\right) \left(\frac{r}{\alpha\_0}\alpha\_t + 1\right). \tag{46}$$

To transfer all characters which would be transferred without transmitter shut-down it is necessary to increase average rate which we will find, having divided (43) on (46)

$$V\_{cp}^{\prime} = \frac{\alpha\_0}{r} \frac{\alpha\_0}{r o\_{\parallel} + o\_{\parallel}} \exp\frac{o\_{\parallel}r}{o\_{\parallel}}\,. \tag{47}$$

The use factor of radio-line in this case is equal

$$\eta(\alpha\_t) = \exp\left(-\frac{\alpha\_t r}{\alpha\_0}\right). \tag{48}$$

From ratios (46) and (48) follows, that at the same level r, speed can be increased in comparison with a transmitter's speed automatic adjustment by value

$$
\Delta \left( \alpha\_t \right) = 1 + \frac{r \alpha\_t}{\alpha\_0} = 1 + \frac{\alpha\_t}{V\_{cp}} \,. \tag{49}
$$

Possible gain at various values of threshold level of speed are presented in table 2


Table 2.
