The paper deals with investigation of in uence of a lowfrequency additive noise and interference acting in the communication channel. The considered channel performs reception and transmission of a narrowband signal in systems of unmanned vehicles groups, as well as the system of Internet of Things. For the purpose of energy-e cient usage of the receiving and transmitting equipment, a low-power narrowband signal is used and a low-speed data channel is organized. The fractal measures of received signal are used to compensate the action of low-frequency icker noise, as well as the fractal Brownian motion model for the statistical description of low-frequency icker noise is substantiated. Parameters of the communication channel are calculated under the action of low-frequency interference. A maximum likelihood algorithm for detecting signals against the background of additive fractal jamming has been developed. It is established that the usage of fractal models makes it possible to improve the e ciency of signal processing against background noise in cases where there are no other di erences between them. Approaches are suggested to integration of the signal processing techniques against low-frequency noise with a fractional character of the spectral power density.
Under conditions of high frequency density of signals in the air, ensuring
reliable transmission of information with minimal distortion is an actual task of
radio engineering. When creating a communication channel of a large volume,
the broadband signals with a large base value and a high carrier frequency are
used. At the same time, a small amount of information is often required, for
example, sensor readings, telemetry measurements, and so on. In this case, a fairly
e ective solution is the use of broadband and low-power signals. New access
technologies targeting IoT, such as enhanced Machine Type Communications
(eMTC), Narrow-Band IOT (NB-IoT), and the 5th generation mobile networks
(5G) are currently under development. Massive numbers of MTC services require
low-cost devices with low power consumption pro les [1{3]. For this purpose, in
the paper algorithms for optimal processing the signals based on
probabilistic models are used [
Fractal Brownian motion (FBM) is used as a model of fractal interference [
M f(X(t2)
X(t1)) (X(t4)
X(t3))g t4)2H = 0:5 2 (t2 t1)2H + (t2 t3)2H + (t1 (t1 t3)2H ; (5) where the matrix R, which contains correlations of all possible increments M = 0:5 N (N 1) has the size of M M and is formed by given N samples: xm = x(ti) x(tj ); i = 1; : : : ; N; j = 1; : : : ; i; m = 1; : : : ; M: Consider noncorrelated samples of FBM in spectral eld. In such cases, the correlation equals M f Xi Xj g = ij DX ti2H , where ij is the Kronecker delta, i = 1; : : : ; N 1. In this case, the matrix R is diagonal and its determinant equals detR =
N 1 DXN Q t2nH .
n=1
Figure 1 represents the FFT power spectrum of signal (solid line) and icker noise (dashed line) spectra for H = 0:5, amount of samples K = 800. It is proved, that increasing the signal duration leads to spectrum transfer to low frequency area with high intensity of icker noise spectrum. It leads to decreasing noise resistance.
In many cases, vector of observed signal samples is represented as x(t) = V s(t)+ n(t), where n(t) is a vector of icker noise samples with zero mean and correlation function Rn, s(t) is a vector of signal samples, V is a transformation matrix. Suppose that random signal s is observed in the known vector x. The random signal and icker noise vector n are described as a model of fractal Brownian motion with di erent Hurst exponent. This parameter is used as a detection statistics for signal detection at the background of noise. Consequently, correla, tion matrices of signal and noise are as follows:
R = qF 2 [jn1j2H + jn2j2H jn1 n2j2H ]; n1; n2 = 1; : : : ; N; where n1; n2 are the counting numbers of signal and noise samples.
As it follows from the optimal processing theory of determined signal at the background of Gaussian noise, the optimal impulse response equals w opt = S T RN1. As FBM increments have Gaussian distribution law and noise samples are additive and independent, it is possible to calculate the signal estimate and correlation matrix of errors s^ = (Rx 1 + Rn 1) 1Rn 1Y = (RnRx 1 + I) 1Y
Rs^ = (Rx 1 + Rn 1) 1 = (I + RxRn 1)Rx , where Rx is the correlation matrix of the observation vector, I is the unity matrix, 1 is the operation of matrix inversion. Results of calculation are presented at Figures 2-4. In the case of xed value of signal power and optimal processing, the output signal-noise ratio increases monotonically with increasing the signal duration. Along with this, it is proved that increasing the output signal-noise ratio is limited by the xed value. This is determined by the fact that spectrum of longer signal is focused in the area of low frequencies, where the icker noise spectral power density increases. Processing quality is estimated by the signal-noise ratio calculated in spectral eld
M q = X : 4
As a result of evaluation, it is shown that methods of the theory of optimal statistical solutions can be successfully applied, also, to processing the fractal signals against the background of additive fractal noise. The basis for the e ectiveness of statistical methods is the irregular character, as well as the relatively large amount of the observable data. Under these conditions, the statistical description of fractal signal is produced by various methods: the use of a one-dimensional and two-dimensional fractal Brownian motion model, and a statistical description of distances between vectors in a pseudo-phase space. This approach allows us to obtain processing algorithms based on the theory of optimal statistical solutions for solving various problems: detection, discrimination, delineation of boundaries, estimation of parameters, and analysis of the processing e ciency. At the same time, the statistical description is not obtained for all fractal signals and their characteristics. This makes important to continue research in this direction. Optimal signal processing at the background of icker noise provides the predetermine .quantity of information in the case of ultra-low-power signal of IoT. To increase the quantity of the information IoT, the optimal waveform is needed. The matched lter is not optimal at the background of the icker noise. When the matched lter is used, the optimal signal width may be evaluated. Note that a continuous waveform observes at the background of a fading, which is caused multipath waveform propagation. Therefore, the MIMO technology have to be used for IoT applications. The power e ciency depends on many conditions not only on receiver sensitivity. Therefore, simpli cation of signal processing algorithm has to be performed when the optimal processing is developed. 5
The research has been carried out under the support of the project 8.2810.2017 in Ryazan State Radio Engineering University funded by the Ministry of education and sci-ence of the Russian Federation.