=Paper= {{Paper |id=Vol-2534/17_short_paper |storemode=property |title=Signal Processing in the Receiving System with Spatially Distributed Receiving Elements |pdfUrl=https://ceur-ws.org/Vol-2534/17_short_paper.pdf |volume=Vol-2534 |authors=Darya N. Zima,Darya O. Sokolova,Alexander A. Spector }} ==Signal Processing in the Receiving System with Spatially Distributed Receiving Elements== https://ceur-ws.org/Vol-2534/17_short_paper.pdf 
Signal Processing in the Receiving System with Spatially Distributed
                         Receiving Elements
                              Darya N. Zima; Darya O. Sokolova; Alexander A. Spector

            Novosibirsk State Technical University, Novosibirsk, Russia Federation, zima.darya@mail.ru




              Abstract. The issues of signal processing and noise suppression in receiving systems
              with spatially distributed elements are considered. Processing algorithms are based on the
              theory of Markovian processes. The processing takes into account the broadband nature
              of the observable spatial-temporal signals. The signal with linear frequency modulation
              was taken as the useful signal.

              Keywords: spatially distributed receiving elements; wideband signal; chirp signal;
              Markovian process.


1        Introduction
   The task of detecting a wideband signal (in terms of space-time) by spatially distributed elements seems to be
laborious, since only in some cases can be divided temporal and spatial filtering [1]. Also, applying the Bayesian
optimal signal detection criterion requires reversing the correlation noise matrices of order corresponding to the
square of the product of the number of receiving spatially distributed elements π‘˜π‘Ž and the number of time
moments π‘˜π‘‘ . This requires laborious calculations, leads to errors, and at small angles of arrival of the noise, the
correlation matrix is close to a degenerate matrix. These problems can be solved by using the Markovian random
process model to describe the noise on spatially distributed receiving elements.


2        Spatial Time Process Model
    The problem of detecting a broadband signal in the space-time sense against a background of noise is considered
for the case of a line of spatially distributed antenna elements. A signal with linear frequency modulation (1) in radar
is often used as a useful signal.
                                                                                         𝜏   2
                               𝑠𝑖 (𝑑) = 𝑆0 cos [πœ”0 (𝑑 βˆ’ (𝑖 βˆ’ 1)𝜏0 + 𝑏 (𝑑 βˆ’ (𝑖 βˆ’ 1)𝜏0 βˆ’ 𝑝 ) + πœ‘)],                     (1)
                                                                                         2
where πœ”0 is the carrier frequency of the radio signal, 𝑏 is the rate of change of frequency with linear frequency
modulation, πœπ‘ is the duration of the pulse signal, 𝜏0 is the relative delay between adjacent elements of the spatially
distributed receiving system, depending on the angle of arrival of the signal Ξ˜π‘† and the distance between the antenna
elements, 0 ≀ 𝑑 ≀ πœπ‘ . The matched filter of the time part of the space-time filter is based on this signal.
    The space-time process with fixation 𝑑 is transformed into a spatial fluctuation with a harmonic character. Figure 1
illustrates a model of this fluctuation. It is agreed that the source of noise is located at a considerable distance from
the receiving antenna elements. We have a flat phase wave front and observe the same fluctuation on all linearly
distributed antenna elements, which is delayed by 𝜏0 between adjacent elements.
    We believe that the noise in the space-time representation has the form presented below.
                            𝑒𝑖 (𝑑) = Ξ(𝑑 βˆ’ (𝑖 βˆ’ 1)𝜏0 ) cos[πœ”0 (𝑖 βˆ’ 1)𝜏0 + Ξ¨(𝑑 βˆ’ (𝑖 βˆ’ 1)𝜏0 )]
    This noise signal with fixed 𝑑 and ΠΈ Ξ˜π‘› is an oscillatory random function of the antenna element number, and the
parameter in equation (2) determines the average normalized frequency of spatial fluctuations:
                                                                   𝑑 sin ΘП
                                                          πœ”Π½ = 2πœ‹           .                                          (2)
                                                                     𝑐

   Thus, Figure 2 shows a graph of spatial fluctuations on the elements of a linear antenna without taking into
account fluctuations caused by modulations Ξ(𝑑 βˆ’ (𝑖 βˆ’ 1)𝜏0 ) and Ξ¨(𝑑 βˆ’ (𝑖 βˆ’ 1)𝜏0 ).




Copyright Β© 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0
International (CC BY 4.0).
                              Figure 1. The formation of spatial-temporal fluctuations.




                       Figure 2. Graph of spatial fluctuations on the line of antenna elements.

   In this case, when different sections of the broadband signal are sequentially hit the line of spatially distributed
elements, only one frequency is visible in the discrete spectrum of the spatial signal, regardless of the presence of
modulation. This property is convenient when there are several noise signals with different directions of arrival. This
character is continued when considering the spatial-temporal fluctuations. Figure 3 presents the spatial-temporal
spectrum of one interfering fluctuation after it passes through a matched filter with a complex frequency response
corresponding to (1).
                    Figure 3. Spatial-temporal spectrum of noise on the line of antenna elements.
It should be noted that the signal wideband in time is narrowband in space, that is, the base of the spatial signal is
close to 1.


3        Markovian Model of noise
   The model is based on the autoregressive model, that is, the linear prediction model [2]:
                                                𝑒 𝑖 = βˆ‘πΎπ‘˜=1 π‘Žπ‘˜ π‘’π‘–βˆ’π‘˜ + πœˆπ‘– ,                                           (3)
Since it is necessary to describe quasiharmonic processes, the minimum order of the Markovian model for the
formation of the spatial fluctuation of the noise is the second order.
                                                𝑒𝑖 = π‘Ž1 π‘’π‘–βˆ’1 + π‘Ž2 π‘’π‘–βˆ’2 + πœˆπ‘– ,                                        (4)
where πœˆπ‘– is the prediction noise, which is the information equivalent of the initial interference, π‘Žπ‘– are the prediction
coefficients. The method of moments is used to determine these prediction coefficients. This method is based on the
use of relations connecting the desired parameters with the moments of the observed processes [4].
   As an indicator of the correspondence of the noise of the Markovian model, Figure 4 shows the temporary
implementation of the noise before and after the decorrelation procedure (4). After applying the processing, it can be
seen from Figure 4 that the frequency of the signal crossing the zero level has increased, which can serve as a sign of
a decrease in the correlation between neighboring samples.




                                    Figure 4. Temporary implementation of noise.

Figure 5 gives the normalized auto correlation functions of the interference signal before and after decorrelation,
where it can be seen that the auto correlation function of the resulting information noise is closer to the auto
correlation function of white Gaussian noise (tending to a delta-shaped form).
                                Figure 5. Auto correlation functions of the noise signal.
Figure 6 shows the auto correlation functions of the initial interference, interference after decorrelation according to
(4) when we use two prediction coefficients, and interference after decorrelation according to (3) when we use ten
prediction coefficients. The graph displays that the processing quality increases with an increase in the number of
prediction coefficients and the sample size of the interference.




     Figure 6. . Auto correlation functions of the noise signal with a different number of prediction coefficients.

If a signal is received from a known direction against a background of white noise, then its processing can always be
divided into spatial and temporal [1,5]. Then the joint distribution of the noise samples on all spatially distributed
elements for a model with two prediction coefficients has the form:
                                  π‘Š0 (π‘ˆ) = π‘Š0 (𝑒1 , 𝑒2 , … , 𝑒𝑁 ) = π‘Š0 (𝑒1 , 𝑒2 ) βˆπ‘
                                                                                   𝑖=3 πœ‹(𝑒𝑖 |π‘’π‘–βˆ’1 , π‘’π‘–βˆ’2 ),              (5)
The decisive statistics for (4) and (5), taking into account [3], will have the form:
                                      𝑍(π‘ˆ) = π‘ˆ22 𝑇 𝑅2 βˆ’1 𝑆22 + βˆ‘π‘        𝑇    βˆ’1          𝑇    βˆ’1
                                                                  𝑖=3[π‘ˆ3𝑖 𝑅3 𝑆3𝑖 βˆ’ π‘ˆ2𝑖 𝑅2 𝑆2𝑖 ],                         (6)
where π‘ˆ2𝑖 , π‘ˆ3𝑖 , 𝑆2𝑖 , 𝑆3𝑖 are the shortened vectors from the samples of the signal and noise oscillations at fixed 𝑑 and
ΠΈ Ξ˜π‘› , 𝑅2 and 𝑅3 are the correlation matrices of the shortened vectors. In a theoretical study, the quality criterion of
the proposed method was to improve the signal-to-noise ratio (SNR) after spatial processing to the input signal-to-
noise ratio. In Figure 7 is shown the improvement of the SNR depending on the direction of arrival of one noise
fluctuation and with a fixed direction to the useful signal.




              Figure 7. The dependence of the SNR improvement from the direction of the noise arrival.

Figure 8 shows the improvement of the SNR depending on the number of antenna elements in the line of spatially
distributed elements in the presence of one noise fluctuation and a fixed direction to the useful signal.
                Figure 8. The dependence of the SNR improvement on the number of antenna elements.

   We turn to the spatial-temporal detection algorithm, provided that the set of samples at the given time 𝑑 is
independent, that is, we have the vector:
                                                                                     𝑇
                                               π‘ˆ βŒ©π‘‘βŒͺ = ‖𝑒1 βŒ©π‘‘βŒͺ , 𝑒2 βŒ©π‘‘βŒͺ , … , 𝑒𝑁 βŒ©π‘‘βŒͺ β€– .                               (7)
    The decisive statistics according to (6) and (7) has the form:
                                                                          βŒ©π‘‘βŒͺ   βŒ©π‘‘βŒͺ
                                                     𝑍(π‘ˆ) = βˆ‘π‘€    𝑑=1 𝑍 (π‘ˆ ),
                                                 𝑇                             𝑇             𝑇
                     𝑍(π‘ˆ) = βˆ‘π‘€  𝑑=1 {π‘ˆ22
                                         βŒ©π‘‘βŒͺ
                                             𝑅2 βˆ’1 𝑆22 βŒ©π‘‘βŒͺ + βˆ‘π‘
                                                              𝑖=3 [π‘ˆ3𝑖
                                                                       βŒ©π‘‘βŒͺ
                                                                           𝑅3 βˆ’1 𝑆3𝑖 βŒ©π‘‘βŒͺ βˆ’ π‘ˆ2𝑖 βŒ©π‘‘βŒͺ 𝑅2 βˆ’1 𝑆2𝑖 βŒ©π‘‘βŒͺ ]}.
Thus, temporary accumulation of the above-described spatial processing occurs.
   Signal processing in the presence of several noise requires a larger number of prediction coefficients in (3).


4         Conclusion
    The paper considers the issues of signal processing against the background of noise in reception systems with
spatially distributed antenna elements. Processing algorithms are built on the model of Markovian random processes,
which makes it possible to factorize spatial-temporal processing, and, therefore, leads to a simplification of the
implementation of the algorithm. Using the spatial spectrum allows to determine the amount and direction of arrival
of active interference.


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