The paper introduces the brief results of modeling an antenna array designed to be used as part of anti-jamming satellite navigation equipment. The study describes the procedure for calculating the weight coefficients of the antenna array based on the orthogonalization of signal and jamming spaces. According to the weight coefficients found, the resultant antenna patterns are plotted. Findings of the research show the results of how the depth of sample accumulation influences the effectiveness of adaptation to the jamming environment
Представлены краткие результаты моделирования антенной решетки, предназначенной для применения в составе помехозащищенной аппаратуры спутниковой навигации. Описан порядок вычисления весовых коэффициентов антенной решетки на основе ортогонализации пространств сигналов и помех. По найденным весовым коэффициентам построены результирующие диаграммы направленности антенной решетки. Приведены результаты влияния глубины накопления выборок на эффективность адаптации к помеховой обстановке
Satellite radio navigation systems are widely used in such areas as transport, geodesy, telecommunications, and other fields. As a result of large distances between GPS, GLONASS, Galileo, and other navigation satellites and the consumer equipment, the power spectral density of the wanted signals becomes lower than noise power in the antenna. Low power of the received satellite navigation signals leads to situations when jamming of natural or artificial origin, commensurable in power with the wanted signal, may present a danger for navigation communications integrity.
In case of receiving navigation signals against the background of powerful electromagnetic jamming, a substantial increase of signal-to-jamming ratio can only be achieved by way of spatial filtration of signals through application of antenna arrays (AA) [1-3]. There are two fundamental methods of controlling AA radiation pattern (RP). The first one consists in maximising power received from the wanted signal’s source direction, while minimising that arriving from other directions. The second method implies maintaining the received power level equal for all directions, but minimising it in the unwanted signal direction.
For the case of navigation signals reception, the first method of RP control is less efficient, since for solving a navigation task, signals from at least four navigation satellites have to be received. In that case, satellite positions may be unknown in advance. Also, minimisation of the power of signals from directions other than that of the wanted signal can be insufficient for elimination of jamming component from the navigation receiver’s input signal.
The other method is more efficient, since for its implementation it is not required to know the positions of wanted signal sources. Blind control of the AA RP can only be provided based on the jamming environment data. For AAs of the satellite navigation system operation band, an RP control criterion is suppression of all signals in the working frequency band whose power exceeds noise power in the antenna elements.
This paper gives a brief description of a software model of an adaptive AA designed for receiving satellite navigation signals of the LI frequency band. Adaptation to jamming environment is exhibited in formation of deep nulls of the RP in directions corresponding to disturbance signal incidence. Fig. 1 shows a block diagram of AA model.
Fig. 1. Block diagram of AA model:x – input signal; АE – antenna element; LNA – low noise amplifier; BPF – bandpass filter; MIX – mixer; G – heterodyne; IFF – intermediate frequency filter; IFA – intermediate frequency amplifier; ADC – analogue-to-digital converter; DSP – digital signal processing; w – weight coefficient; y – output signal
A software model of adaptive AA is constructed in such a way so that reception of navigation signals and outside jamming could be considered in the best possible approximation to a real device. Listed below are specific features taken into account.
The information and disturbance signals come to the AA sheet at different angles (φ, θ) of spherical coordinate system. In different antenna elements signals appear with delays, depending on the AA angle of incidence. Propagation of each of the signals is specified by wave vector ks [
where λ - wavelength;
θs - elevation angle for signal incidence direction;
φs - azimuth angle for signal incidence direction.
If AA configuration is known, phase distribution vector of the received signal is assigned for all AEs based on the wave vector [
where u1...uN - vectors of AE coordinates;
N - quantity of AEs;
unx, uny, unz - coordinates of the и-th AE. Expressions for wave vector k, and phase distribution vector Vp of disturbance signals have a view similar to formulas (1) and (2).
A sum signal in each AE is defined by the expression [5, 6]
where M - quantity of information signals;
sm max - amplitude of the m-th information signal;
- phase multiplier of the m-th information signal in AE;
L - quantity of disturbance signals;
Pl max - amplitude of the z-th interference signal;
- phase multiplier of disturbance signal in AE;
n(t) - AE self-noise.
The obtained complex signal samples contain a mixture of wanted signals, noise, and disturbance signals. Adaptive processing of the received signals implies accumulation of signal samplings in buffer delay lines, so as to be used for calculating signal cross-covariance matrix [
Rxx = XXH, (4)
where X = [x1,x2, x3, ..., xN]T- column matrix of the samplings of signals received from all AA antenna elements;
Xn = [x1,х2,x3,...,xk] - signal samplings received from the n -th antenna element; к - sampling number;
k - sampling number;
XH - Hermitially conjugated matrix X.
It is presumed that information signals, disturbance signals, and noise are statistically independent, therefore, for the cross-covariance matrix, the following equation is true [
where Rs, Rp, Rn - cross-covariance matrix components corresponding to the information signals, disturbance signals, and noise;
Ps - information signal power;
νs, νp - signal and disturbance distribution vectors;
Pp - disturbance power; σn - noise dispersion;
I - unity matrix.
Numerical values in the cross-covariance matrix describe the signal space, which contains the data of disturbance and wanted signals/ noise mixture, namely, power and propagation direction. These data can be extracted through eigenvector expansion of the cross-covariance matrix [
Rxx = EΛEH, (6)
where Λ - diagonal matrix, consisting of eigenvalues λ1...λN of matrix R XX;
E - matrix with size N x M, whose columns are eigenvectors of matrix R XX;
EH - Hermitially conjugated matrix E.
Eigenvector matrix [
E = [e1, e2, e3, ..., eN], (7)
where en = [e1,e2, e3, ..., eN]T - eigenvector column of cross-covariance matrix Rxx, corresponding to eigenvalue λn.
The eigenvectors are determined after calculation of eigenvalues λ1...λn.. Matrix eigenvalues can be computed by different methods. In case of small quantity of cross-covariance matrix elements, an approximate calculation of the eigenvalues can be performed in accordance with the methodology given in [
First, the highest eigenvalue λ1 is computed [
where α1 = Rxx - initial cross-covariance matrix;
р - certain power to which matrix α is raised.
The higher the value of p in expression (8), the more accurate the result of eigenvalue calculation. In this case the requirements for computational resources are increasing. The power value should be selected with account of permissible calculation error.
To determine the second and subsequent eigenvalues, it is required each time to recalculate the matrix that has to be raised to power [
Then matrix α2 is substituted in expression (8) instead of α1, and the second eigenvalue λ2 is determined. After that, similarly to expression (9), matrix α3 is calculated, then eigenvalue λ3, and so on, until all eigenvalues are found.
Finding the eigenvectors is done by solving a system of linear algebraic equations composed of the cross-covariance matrix and the eigenvalues. Solving a system of equations consists in determining coefficients [e1,e2, e3, ..., eN] [
where Rxx (n, n) - cross-covariance matrix elements;
en - eigenvector elements; λ - eigenvalue of matrix Rxx.
To determine all eigenvectors, all eigenvalues λ1...λn are sequentially substituted in the system of equations (10).
A specific feature of eigenvectors e1...eN is their orthogonality to one another:
Each eigenvalue and its related eigenvector represent an individual signal subspace, or a separate degree of freedom of the AA. The maximum quantity of the degrees of freedom is equal to N - 1, i.e. one short of the quantity of the AEs. Each of the subspaces is orthogonal to the rest ones too.
In the presence of L powerful jamming signals, it is possible to discriminate jamming subspace Pp and subspace of noise mixed with information signals Pc+H [
The greater the eigenvalue, the higher the amplitude of the signal it corresponds to. The wanted signal is below noise level, so in the absence of a powerful disturbance signal the cross-covariance matrix eigenvalues will not exceed the threshold value, which is equal to the mean noise level of the antenna elements. Within the framework of the constructed model, conclusions on disturbance detection were made in those cases when the maximum eigenvalue exceeded noise medium power 5-10-fold.
Let us consider two instances of the eigenvalues of cross-covariance matrices calculated for an AA containing five AEs. In either case, five information signals with amplitude of 0.07 μν, at noise amplitude of 10 μν, are involved. In the first case there is no disturbance signal, and in the second case, harmonic disturbance with frequency of 1600 MHz and 10 μν amplitude is received.
The orthogonality property of spaces makes it possible to suppress disturbance in the AA output signal. For that purpose the output signal should be multiplied by the sum of eigenvectors representing the subspace of noise and wanted signals [
where y (t) - output samplings matrix with size N x 1;
Pc+H - matrix with size N x N, representing the subspace of wanted signals and noise (14);
x(t) - column matrix with the values of input signal samplings.
Given a correct selection of eigenvectors, the output signal obtained in expression (15) will not contain a disturbance component, but will still be below the noise level. Signals are received by the AA through formation of partial RPs (beams), evenly dividing the space region observed by an AA of N elements into N sectors [
A resultant output signal, for which filtering and amplifying were performed, is determined with consideration of expression (15) [
where νПД - signal phase distribution in AE, corresponding to the required partial pattern.
Weight coefficients obtained by means of the adaptive algorithm can be determined by the formula [
Let us consider model operation through examples of a linear 5-element AA observing the upper half-space with the constant azimuth angle of φ = 90°. The first example is AA operation under the influence of a harmonic disturbance with frequency 1600 MHz and amplitude of 100 μν (-67 dBm on antenna load of 50 Ω) incident from the direction of 5° relative to the normal. The wanted signals are specified with amplitudes of 0.07 μν (-130 dBm) under different directions of incidence on the AA plane. The antenna noise has an amplitude of 10 μν (-87 dBm).
Fig. 2 shows normalised AA RPs Ffi) built on the base of weight coefficients according to expression (17), using one degree of AA freedom for jamming compensation. Incidence directions of the wanted signals are shown by solid lines, and those of the jamming signals - by dashed lines. The wanted signals are denoted as S1, S2, and so on. The jamming signal is denoted as I1. Jamming attenuation in the incidence direction is about -28 dB. The difference between noise and jamming levels is 20 dB. Hence, the AA has a certain margin for suppressing jamming and detecting wanted signals. The navigation signals are received by partial beams. With five partial beams possible, the active ones are four of them (see Fig. 2). The central beam, crossing the direction of jamming incidence, turns out a suppressed one.
Fig. 2. Normalised five-element AA RPs under harmonic disturbance of 10 µV
In the second example there are two jamming signals: harmonic disturbance on 1600 MHz frequency with amplitude of 10 μΥ, received at an angle of 5° to the normal, and narrowband noise disturbance within a range of 1600...1610 MHz with amplitude of 10 μΥ, acting in the direction of -40°. For jamming compensation, three degrees of AA freedom are introduced. Based on the simulation results, patterns given in Fig. 3 were obtained. In the specified jamming incidence directions, nulls are formed, where jamming attenuation is set at the levels of -20 dB for harmonic disturbance (I1) and -30 dB for noise disturbance (I2) (see Fig. 3). In this way, the greater the received jamming signal power, the higher the attenuation it undergoes.
Fig. 3. Normalised five-element AA RPs in the presence of two jamming signals
The third example is an influence of phase- shift jamming signal with amplitude of 10 μΥ (-87 dBm), i.e. the excess above the level of wanted signals was about 40 dB. Jamming signal was received from the direction of 60° to the AA normal. The period of BPSK jamming signal phase shift was set at 5 μΥ. Fig. 4 shows the RPs obtained in the course of simulation. In this case, with one degree of AA freedom used, jamming attenuation (dashed line I4) is equal to about -35 dB. The rightmost beam, crossing jamming incidence direction, turns out a suppressed one. The wanted signal S5, located nearby, also becomes attenuated by about -8 dB.
Fig. 4. Normalised five-element AA RPs in the presence of phase-shift jamming signal with 10 µV amplitude
The fourth example is similar to the previous one, but the amplitude of phase-shift jamming signal is increased to 100 μΥ (-67 dBm), the excess above the wanted signals is about 60 dB. In this situation, one degree of AA freedom is not enough to form a sufficiently deep null on the RP, therefore, two eigenvectors of the signal cross-covariance matrix, corresponding to the maximum eigenvalues, were allocated for the jamming subspace. Fig. 5 shows patterns calculated for this case.
Fig. 5. Normalised five-element AA RPs in the presence of phase-shift jamming signal with 100 µV amplitude
When two degrees of freedom are used, attenuation in the jamming incidence direction is set at about -35 dB. It can also be seen in Fig. 5 that apart from the null in the jamming direction, a secondary null has formed, with mirrored location relative to the AA normal. In this way, the outermost AA beams turn out suppressed.
Calculation of the cross-covariance matrix (4) is performed on the basis of accumulated AE signal samplings. An analysis of jamming compensation process in the AA under different initial data has shown that the number of memorised signal samplings tells upon the range of suppressed frequencies in the AA output signal and stability of the weight coefficients. Fig. 6 shows spectra of the output signals of an adaptive AA after transfer to the IF region of 0...20 MHz and harmonic disturbance compensation.
Fig. 6. AA output signal spectrum under harmonic disturbance compensation with different number of accumulated input signal samplings: а - 10, b - 100, c - 1000; 1 - disturbance suppression range; 2 - satellite signals
Use of a small number of samplings in the signal cross-covariance matrix calculation can eliminate harmonic disturbance in the AA output signal, but in this case the suppressed frequency region becomes extremely wide (see Fig. 6). Such widening is capable of suppressing the wanted signals along with the jamming. Increasing the number of processed samplings leads to narrowing of jamming suppression region, and information integrity of the wanted signals is then free from jamming influence (Fig. 6, b, c).
Another problem arising with accumulation of a small number of samplings is time instability of the weight coefficients (Fig. 7). The shown time dependencies of the weight coefficients were obtained for the case of broadband noise disturbance compensation. Under accumulation of 10 samplings, the absolute weight coefficient values in a five-element AA turn out noise-contaminated (Fig. 7, a). Accumulation of a large number of samplings (Fig. 7, b) makes it possible to obtain weight coefficients with practically constant values.
Fig. 7. Time dependencies of weight coefficients w1...w5 under adaptation to noise disturbance of 100 µV with the number of accumulated input signal samplings: а – 10, b – 100
Increasing the number of processed samplings leads to increased computation time. However, the result obtained in this case is more accurate and stable, i.e. the procedure of adaptation to jamming environment can be performed with certain periodicity, skipping part of the input samplings for the sake of running the computations on a medium-performance processor device.
Proceeding from the results of the presented study, the following conclusions can be made.
The demonstrated methodology of spatial filtering on the basis of orthogonalisation of signal spaces allows to effectively compensate for jamming of satellite navigation signals. Its specific feature is adaptation of the AA DP to jamming environment in the absence of a priori information on the position of the sources of wanted and unwanted signals.
The built software model of AA, in combination with the analogue/digital segment of an adaptive receiving device, can be used in working with radio-frequency hardware equipment under laboratory conditions when engaging in further research into and development of anti-jamming satellite positioning receivers.
Subsequent work in this direction implies further refinement of the software model up to a finished software receiver of satellite navigation signals and its eventual transfer onto a hardware platform.
The authors declare that there are no conflicts of interest present.