**2. The analysis of noise immunity radio-line at usage mode intermittent of radiation**

#### **2.1 Error probably of signal reception, use factor of radio-line when usage intermittent radio communication channel with different types of fading**

The main parameter that characterizes the noise immunity of digital communication systems is the probability of error. Because in considered channels of communication signal propagation suffers fading, the probability of error is a random variable, and the quality of communication is estimated by the mean probability of error, which is usually defined as mathematical expectation this random variable. Using the known expression, determining the probability of error for noncoherent reception of orthogonal signals in channels with Gaussian noise and the probability density function of the signal / noise ratio (SNR), was

found analytic expressions for the dependence of the average probability of error from the mean SNR and the level of threshold's interruption for the most commonly used when describing the fading distribution laws of random variables:

$$P\_{nc}(\boldsymbol{\chi}\_{0}) = \frac{1}{2 \cdot \eta(\boldsymbol{\chi}\_{0})} \prod\_{\boldsymbol{\chi}\_{\ell}}^{n} \exp(-a\boldsymbol{\chi}) \cdot \frac{1}{\boldsymbol{\chi}\_{0}} \cdot \exp\left(-\frac{\boldsymbol{\chi}}{\boldsymbol{\chi}\_{0}}\right) d\boldsymbol{\chi} = \frac{1}{2 \cdot \eta(\boldsymbol{\chi}\_{0})} \frac{\exp\left[-a\boldsymbol{\chi}\_{\ell}\left(1 + \frac{1}{a\boldsymbol{\chi}\_{0}}\right)\right]}{1 + a\boldsymbol{\chi}\_{0}},\tag{1}$$

$$P\_{nc}\left(\boldsymbol{\chi}\_{0}\right) = \frac{1}{2\eta\left(\boldsymbol{\chi}\_{0}\right)}\int\_{\boldsymbol{\chi}\_{1}}^{\boldsymbol{a}} \exp\left(-a\boldsymbol{\chi}\right) \frac{1}{\boldsymbol{\chi}\_{0} - \boldsymbol{\chi}\_{a}} \cdot \exp\left(-\frac{\boldsymbol{\chi} + \boldsymbol{\chi}\_{a}}{\boldsymbol{\chi}\_{0} - \boldsymbol{\chi}\_{a}}\right) \cdot I\_{0}\left(\frac{2 \cdot \sqrt{\boldsymbol{\chi} \cdot \boldsymbol{\chi}\_{a}}}{\boldsymbol{\chi}\_{0} - \boldsymbol{\chi}\_{a}}\right) d\boldsymbol{\upgamma}\tag{2}$$

$$P\_{nc}\left(\boldsymbol{\gamma}\_{0}\right) = \frac{1}{4\eta\left(\boldsymbol{\gamma}\_{0}\right)\sqrt{\pi}}\int\_{\boldsymbol{\gamma}\_{\ell}}^{\infty} \frac{1}{\sqrt{\ln\boldsymbol{\gamma}\_{0} - \ln\boldsymbol{\gamma}\_{0}}\boldsymbol{\gamma}\_{0}} \exp\left[-\frac{\left(\ln\sqrt{\frac{\boldsymbol{\gamma}}{\boldsymbol{\gamma}\_{0}}\boldsymbol{\gamma}\_{\ell}}\right)^{2}}{\ln\boldsymbol{\gamma}\_{0} - \ln\boldsymbol{\gamma}\_{0}\boldsymbol{\gamma}\_{\ell}}\right] \cdot e^{-\alpha\boldsymbol{\gamma}}d\boldsymbol{\gamma}\,,\tag{3}$$

$$P\_{nc}\left(\boldsymbol{\gamma}\_{0}\right) = \frac{1}{2\eta\left(\boldsymbol{\gamma}\_{0}\right)}\frac{1}{\Gamma\left(m\right)}\left[\frac{m}{\boldsymbol{\gamma}\_{0}\cdot\eta\left(\boldsymbol{\gamma}\_{0}\right)}\right]^{m}\int\_{\boldsymbol{\gamma},\eta\left(\boldsymbol{\gamma}\_{0}\right)}^{\alpha}\rho^{\gamma=1}\exp\left[-\left(\frac{m}{\boldsymbol{\gamma}\_{0}\cdot\eta\left(\boldsymbol{\gamma}\_{0}\right)} + \alpha\right)\boldsymbol{\gamma}\right]d\boldsymbol{\gamma} = \frac{1}{2}\frac{1}{\Gamma\left(m\right)\Gamma\left(m\right)}\left[1 + \alpha\left(\boldsymbol{\gamma}\_{0}\right)\frac{\boldsymbol{\gamma}\_{0}\cdot\eta}{\boldsymbol{\gamma}\_{0}}\rho^{\gamma=1}\exp\left(-\left(\frac{m}{\boldsymbol{\gamma}\_{0}\cdot\eta}\right)\boldsymbol{\gamma}\_{0}\right)\right] \tag{4}$$

$$= \frac{1}{2}\frac{1}{\Gamma\left(m\right)\Gamma\left(m\right)}\left[1 + \alpha\frac{\boldsymbol{\gamma}\_{0}}{m}\frac{\boldsymbol{\Gamma}\left(m\right)}{\boldsymbol{\Gamma}\left(m\right)}\right]^{-m}\Gamma\left[m\boldsymbol{\gamma}\_{0}\frac{\boldsymbol{\gamma}\_{1}}{\boldsymbol{\gamma}\_{0}} + \alpha\boldsymbol{\gamma}\_{t}\frac{\boldsymbol{\Gamma}\left(m\right)}{\boldsymbol{\Gamma}\left(m\right)}\right] \tag{5}$$

where analytical expressions (1-4) define mean values probability of error under the laws accordingly Rayleigh, the generalized Rayleigh, lognormal and Nakagami, m-parameter of fading and *m>* 0.5, *a* - the constant coefficient equal 0.5 and 1 for accordingly frequency and phase demodulations, *y, yo, Yt* - accordingly current, average and threshold value SNR, *TJ(Yo)* - the parameter named use factor (utilization factors or fill factor) of a radio-line, 00 00 r(m)= **f** *tm-le-tdt* and *r(m,n)=* **f** *tm-le-tdt* accordingly gamma-function and incomplete 0 *n* 

gamma-function. Here and below value a will make 1.

Integral in the formula (4) is tabular, defined by expression

$$\int\_{\mu}^{\nu} \varkappa^{\nu - 1} \exp(-\mu x) dx = \frac{\Gamma(\nu, \mu \cdot \nu)}{\mu^{\nu}}.\tag{5}$$

Value *Ya* correspond SNR for the regular component of a signal when a signal fading down under laws of the generalized Rayleigh (formula 2). When a signal fading under the lognormal law (formula 3) value *yo\_ref* correspond SNR in the nonperturbed environment.

Random changes of an envelope of a signal of millimeter wave range at a lognormal fading, unlike a fading under Rayleigh, the generalized Rayleigh and Nakagami laws, are caused not by its total interference components, and fluctuation of dielectric transmittivity &, a wave refraction index *n* owing to turbulence of troposphere.

Expression *77(yr)* defines use factor radio-line at intermittent communication (the relation of transmission time of the data to the general time of a communication session) and its lowering corresponds to reduction of spectral efficiency, accordingly for the above-stated types of a fading will make, according to analytical expressions (6-9)

$$\log(\chi\_0) = \prod\_{\mathcal{Y}\_t}^{\infty} \frac{1}{\mathcal{Y}\_0} \cdot \exp\left(-\frac{\mathcal{Y}}{\mathcal{Y}\_0}\right) d\mathcal{Y} = \exp\left(-\frac{\mathcal{Y}\_t}{\mathcal{Y}\_0}\right),\tag{6}$$

$$\eta(\boldsymbol{\gamma}\_{0}) = \bigcap\_{\boldsymbol{\gamma}\_{t}}^{\boldsymbol{\alpha}} \frac{1}{\boldsymbol{\gamma}\_{0} - \boldsymbol{\gamma}\_{a}} \cdot \exp\left(-\frac{\boldsymbol{\gamma} + \boldsymbol{\gamma}\_{a}}{\boldsymbol{\gamma}\_{0} - \boldsymbol{\gamma}\_{a}}\right) \cdot I\_{0}\left(\frac{2 \cdot \sqrt{\boldsymbol{\gamma} \cdot \boldsymbol{\gamma}\_{a}}}{\boldsymbol{\gamma}\_{0} - \boldsymbol{\gamma}\_{a}}\right) d\boldsymbol{\gamma} \,, \tag{7}$$

$$\eta(\boldsymbol{\gamma}\_{0}) = \left| \frac{1}{2\sqrt{\pi}} \frac{1}{\boldsymbol{\gamma}} \cdot \frac{1}{\sqrt{\ln \boldsymbol{\gamma}\_{0} - \ln \boldsymbol{\gamma}\_{0}} \boldsymbol{\exp} \left[ -\frac{\left(\ln \sqrt{\frac{\boldsymbol{\gamma}}{\boldsymbol{\gamma}\_{0}}\right)^{2}}}{\ln \boldsymbol{\gamma}\_{0} - \ln \boldsymbol{\gamma}\_{0}} \right] \right|\_{\boldsymbol{\gamma}} = \frac{1}{2} \cdot \text{erfc} \left( \frac{1}{2} \cdot \frac{\ln \boldsymbol{\gamma}\_{1} - \ln \boldsymbol{\gamma}\_{0\\_ \text{ref}}}{\sqrt{\ln \boldsymbol{\gamma}\_{0} - \ln \boldsymbol{\gamma}\_{0\\_ \text{ref}}}} \right), \quad \text{(8)}$$

$$\eta\left(\boldsymbol{\gamma}\_{0}\right) = \frac{1}{\Gamma\left(m\right)} \left(\frac{m}{\boldsymbol{\gamma}\_{0}}\right)^{m} \int\_{\boldsymbol{\gamma}\_{t}}^{\boldsymbol{m}} \boldsymbol{\gamma}^{m-1} \exp\left(\frac{\boldsymbol{\gamma}\cdot\boldsymbol{m}}{\boldsymbol{\gamma}\_{0}}\right) = \frac{\Gamma\left(m\right)m\frac{\boldsymbol{\gamma}\_{t}}{\boldsymbol{\gamma}\_{0}}}{\Gamma\left(m\right)},\tag{9}$$

**2** *a:,*  where *erfc* ( *x)* = r **f** exp (-t<sup>2</sup> ) *dt* - error function complement. Integral in the formula (9) is '\Jlr X tabular (5). Analytical expression (6) is a special case of the formula (9) at m=l.

Let's accept *Yt =kyo,* (O<k<oo) and value k is fixed, then the use factor of a radio- line in case of the Rayleigh fading is fixed, doesn't depend from *yo.* In cases of a fading of a signal under laws of the generalized Rayleigh, lognormal and Nakagami it depends accordingly on ratios k, *ya, yo;* k, *yo\_ref, yo* and k, *m,* and in two last cases the value of the specified coefficient will be defined by formulas

$$\eta(\boldsymbol{\gamma}\_{0}) = \frac{1}{2} \cdot \text{erfc}\left[\frac{1}{2} \cdot \left(\frac{\ln k}{\sqrt{\ln \frac{\boldsymbol{\gamma}\_{0}}{\boldsymbol{\gamma}\_{0}}}} + \sqrt{\ln \frac{\boldsymbol{\gamma}\_{0}}{\boldsymbol{\gamma}\_{0}}}\right)\right]; \ \eta(m) = \frac{\Gamma(m, k\cdot m)}{\Gamma(m)}.$$

Analyzing dependences of probability of errors on mean value SNR, it is necessary to consider that at constant instantaneous power of the transmitter at intermittent

communication energy of bit of a signal decreases proportionally to use factor of a radioline. Therefore for, the comparative analysis of noise immunity at intermittent communication and without it, it is expedient in the first case at calculation of probability of error accordingly to reduce level of a threshold and an average of value SNR and that is showed in analytical expression (4). In the absence of it, for correct comparing of the specified probabilities of errors, it is necessary to admit that instantaneous power transmitters at intermittent communication comparing communication without interruption is increased in inverse proportion to use factor of radio-line that is illustrated in formulas (1-3).

Application algorithm of intermittent communication reduces probabilities of errors in the channel with a fading and, especially with growth of mean value SNR (ro) at some lowering of spectral efficiency (use factor of a radio-line). For example, for the Rayleigh channel (1) at *Yt =ro; TJ(ro )=1/* e, (6) therefore

$$P\_{nc}(\chi\_0) = \frac{e}{2} \int\_{\gamma\_0}^{\eta} \exp(-a\gamma) \cdot \frac{1}{\gamma\_0} \cdot \exp\left(-\frac{\gamma}{\gamma\_0}\right) d\gamma = \frac{1}{2} \frac{\exp\left(-a\gamma\_0\right)}{1+a\chi\_0} = \frac{1}{2} \frac{1}{1+a\chi\_0} \cdot \exp\left(-a\chi\_0\right) \dots$$

Thus, application of intermittent communication with a threshold equal *ro* reduces probability of error incoherent reception in comparing probability of errors in the Rayleigh channel without interruption exponentially, in *expay* <sup>0</sup>time.

#### **2.2 Comparing of energetic effectiveness usage of intermittent communication and noise immunity coding**

The comparative analysis of energetic efficiency of the radio-line using intermittent communication in comparison with a radio-line, using the noise immunity code for the channel with a fading of Nakagami is carried out. Variation of coefficient *k* we will select for m = 1.4 such value of a threshold *(Yt =kro)* that for them T/ (9) it was equal 0.5.

Comparing of effectiveness of communication is expedient for carrying out on energetic coding gain and energetic gain of intermittent communication at the given probabilities of errors. Energetic coding gain for intermittent communication (m =1.4; q=0.5) and the convolutional code of speed 0.5 are presented to tab. 1. At realization of intermittent communication, as well as transmission of an encoded signal, energy of the bit (average power) at the fixed instantaneous power of the transmitter will decrease twice, the width of spectra of both signals thus will double, and spectral efficiency accordingly twice will decrease.


Table 1.

The data presented to tab. 1 show that intermittent communication, is more effective comparing convolutional code usage (at equal values of use factor of radio-line and speed of coding). Besides, in the channel with a fading for possibility of realization of decoding of the convolutional code, originating burst errors necessary to convert in single-error. Normally it is fulfilled by operations of interleaving/ de-interleaving which are implemented with additional hardware expenses that causes additional signal delays. At deeper fading of a signal when 0.5 *<m* <1 energetic gain in application of intermittent communication will increase.
