An information system with multi-channel receiving system and mutual in uence of channels is considered. Signal and interference are spatially concentrated sources of radio emission. The mutual in uence of channels is researched by calculating the matrix of mutual impedances for thin vibrators. The magnitude of the load impedances in individual channels in uences the resulting signal-to-interference ratio. Also, the value of the load impedances depends on correlation properties of the signal and interference at the input of the spatial channels. It complicates the problem of optimal matching. The problem of nding the optimal load impedances as a function of the signal-interference ratio is solved. The solution of the optimization problem is carried out by a numerical analysis using the MatLab or, in a particular case, by the analytical method. The optimal load impedances are compared depending on the degree of mutual in uence of the channels of the information system, spatial structure, and interference parameters. Comparing the optimum load impedances, spatial structure and parameters of interferences is carried out depending on a degree of mutual in uence of the information system channels. The output signal-interference ratio behavior at di erent values of the load impedances is researched. The gain of optimal matching versus the load impedances is calculated in the absence of mutual in uence and in the case of applying non-optimal mutual impedances when there is the mutual in uence.
Evolution of the integral semiconductor technology, microwave solid state high
e ciency ampli ers, and super high frequency digital circuitry stimulate design
of radio systems with signal processing in spatial channels of antenna array (AA)
[1 { 3]. Often, characteristics of the AA are obtained without mutual in uence
of the elements. It is obtained with multiplying the beam pattern of one element
by the factor AA. If there is a signi cant mutual in uence of the elements, the
characteristics of a real antenna can di er from the calculated characteristics.
In paper [
The antenna array includes N omnidirectional elements arranged in a certain way in space. In space, there are signal source of variance DS and M interference of variance Dm; m = 1; : : : ; M . The spatial position of signal sources and interference is de ned by the vectors V S and V m; m = 1; : : : ; M respectively. The spatial correlation matrices of the signal and interference in the elements of the antenna array are equal to RS = DS V S V SH; RJ = the sign H means Hermitian transpose.
The mutual in uence of the antenna array elements a ects to the
characteristics of the signal and interference at output. The value of mutual in uence
is characterized by a matrix of mutual impedances Z [
In addititon to external interference, there are internal noise in the system caused by losses from the diagram-making circuit and frontend of receivers of the active antenna. The total noise power associated to AA elements load is equal to M P DmV mV Hm; where m=1 Pn = kT f
Re ( R11
ZL
2) ZL
ZL + Z11 + N F ! 1 ; where k = 1; 38 10 23J=K is the Boltzmann constant, T is the temperature (degrees Kelvin), f is the bandwidth, R11 is the active component of the antenna element impedance equal to the emitting resistance, ZL is the load impedance, N F is the frontend noise factor. Power of the signal in the load of the antenna element is: PS = Re n jUj2 o. As a result, the signal-to-noise ZL ratio q = PS =Pn depends on the load impedance. Next, it should be taken into account the load impedance optimization. In the antenna array, we assume that the noises in the impedance load are not correlated to each other and have a diagonal correlation matrix RT with signal power vector:
P T = fPn; n = 1; : : : ; N g on the main diagonal.
The correlation matrix of the interference, which emitted in the load impedance is equal to RIL = U LU LH, where U L = U pRe f1=V Lg = = UnpGn; n = 1; : : : ; N is the normalized value of interference at the load impedance, Gn; n = 1; : : : ; N , is the active conductivity of the load. As a result, the correlation matrix of noise and noise at the output of AA elements with taking into account their mutual in uence is equal to
R1 =
Z 1RJ Z 1H pRe f1=V Lg pRe f1=V Lg
T
+ RT :
The signal at the load impedance is equal to S L = (Z 1S ) pRe f1=V Lg. The
signal-to-interference ratio is obtained as a result of optimal processing the
signal from the output of the elements AA. It is equal to [
There is calculating of the signal-to-noise ratio depending of the receiving system
spatial structure with di erent modes of the load matching. Suppose that the
elements of AA are equidistant and that the AA is spatially linear. The
calculation is carried out for antenna elements in the form of thin vibrators, which are
located vertically. The value of the mutual impedance between two any vibrators
is equal to [
n Z1 = 30 2Ei(jk0d) 2Eih
jk0(pd2 + L2 + L)i where Ei( jx) = Ci(x) jSi(x) is the integral exponential function. It is calculated on the basis of the functions of the integral cosine and integral sine with k = 2 as the wave number, d the distance between the vibrators, L the length of vibrator. With a large spatial separation of the vibrators AA, the mutual in uence becomes insigni cant and the matrix of mutual impedances is Z = I (73:1 + j42:5) :
Optimization of the load impedance is a di cult calculational problem. Commonly, it solved by numerical methods. The optimal load impedances also depend on the angular position of the signal and the interference sources.
Consider an example of an analytical solution of the problem of optimal matching for an antenna array with two elements (N = 2). The eld of each antenna element is inphase. This ensures the symmetry of the problem. With a large distance between the elements, the mutual coupling is negligible. The output resistance of a thin vibrator is equal to (73:1 + j42:5) . Optimal value of each of the antenna elements loads impedance ZL = (73:1 + j42:5) : An equivalent circuit that shows the e ect of the mutual impedance of the AA elements is given in Fig. 1.
In the example, the currents in the circuit of each antenna element are the same and equal I = Zl+E2Z12 : The power in the load of one of the elements is P1 =
E2RL jZl+2Z12j2 . The maximum value of the power is achieved with compensation of the reactive impedance in the denominator ImZL = 2ImZ12 ImZ11: Solution of the optimization problem max(RL)P1(RL) gives the values of the optimum load resistance RLopt = ReZ11 + 2ReZ12:
In the more general case of several antennas, optimization of the load impedances is performed by numerical method (using the fminsearch function in the MatLab). It should be noted that in order to obtain achievable values of the load impedances, it is necessary to introduce constraints. The real part of the impedances be nonnegative. To nding an optimum close to the global optimum, the value obtained at the previous step of calculating the dependence is chosen as the initial value of the impedances.
Optimization of the load impedance was carried out for various values of load impedances { without the mutual in uence of antenna elements; for all elements impedance is equal to ZL0 = 73:1 j42:5 ; the solid line graphic in Fig. 2; { with the mutual in uence of the antenna elements in the form of thin vibrators; for all elements the impedances is equal to ZL0 = 73:1 j42:5 ; the dashed line graphic in Fig. 2; { with the mutual in uence of the antenna elements in the form of thin vibrators, the impedances were obtained as a result of optimization by the numerical method in the MatLab; the dotted line graphic in Fig. 2.
a) N F = 0:4 dB b) N F = 10 dB dB. Angular interference positions were 350; 200; 700, direction of the arrival of the signal is perpendicular to the plane of the AA elements placement.
Bene ts 5 : : : 8 dB of the gain in relation to the signal interference were obtained as a result of optimization of load impedances. The bene ts from the load impedances optimization are especially noticeable for small distances between AA elements d= = 0:1; : : : ; 0:3: With an increase of the frondend noise factor, the gain from optimizing the spatial structure decreases and it is signi cant (Fig.2.) With a decrease in the frontend noise gure, the fminsearch optimization procedure is unstable (there is not well-de ned optimum). For the technical implementation of the optimal matching principle, it is necessary to know the dependence of the load impedances on the operating conditions of the AA. Figure 3 shsows the dependence of the norm of the di erence on the load impedance vector and the value ZL0 the distance between the AA (N F = 0:4 dB). Increasing the noise gure N F > 10 dB enhances optimization process stability. The value of load impedance approaches to ZL0. The multi channel information system with active antenna array is investigated. As a result of the research, it was established that the matching impedance of the load has a great in uence onto increasing the noise immunity of signal processing in the antenna array with mutual in uence of the elements. The calculations for the antenna array in the form of thin vibrators (with small distances between) show a gain from the optimization of the load impedance of 5 : : : 8 dB. This should be taken into account when performing optimization of the spatial structure of the multi channel information system involving of heavily lled antenna arrays. 5
The research was supporeted by the Ministry of education and science of the Russian federation project no. 8.2810.2017 in the Ryazan State Radio Engineering University.