Introduction. Safety of navigation and work in underwater mineral deposits require high-quality detection of underwater objects and tracking their movements [1]. Hydroacoustic sensors are used in underwater surveillance systems. They capture the signal reflected from the object and provide for the calculation of its location using the trilateration method [2]. In this process, each sensor provides information about the time interval of the probing signal reflected from the object. In the case of using several sensors [3], it becomes possible to solve the problem of spatial resection and determine the coordinates of the observed object [4]. To increase the accuracy of measurements, it seems promising to use stereo sensors with an ultrashort base as a receiver [5], which allow obtaining phase information [6] and determining the bearing on an object [7]. The possibilities of using sonar data on remote underwater targets, as well as on the bearing of underwater objects [8] through the use of phase [9] or frequency [10] information of reflected probing signals have been studied.

At the same time, it is known that the signal sampling procedure causes errors in determining the parameters of the trajectory of underwater objects [11]. This applies to both distance measurements and bearing determination procedures. Digitization of the analog sensor signal, which is required for further digital processing, introduces the so-called “sampling error”. The error caused by the signal amplitude quantization is estimated by the number of quantization levels of the analog-to-digital converter. The error caused by time sampling is proportional to the range of the quantization time interval and the stress rate. Therefore, the sampling frequency of the signal is sought to be as high as possible above the upper frequency of the signal itself. However, increasing the sampling rate in a number of cases is limited by the capabilities of recording equipment: processor, analog-to-digital converter, data transmission channels, data storage devices. The limited possibilities of sampling in time when performing trajectory measurements do not allow for accurate recording of the extreme values of the trajectory (range and bearing). As a result, the accuracy of measurement results decreases, and outliers appear on the fixed trajectories. To eliminate this disadvantage, it is necessary to study the potential of oversampling the digitized sonar signal. There are no publicly available publications dedicated to solving this problem. The materials of this article fill in the existing gap.

The objective of the presented study is to evaluate the possibilities of determining the bearing on an object using phase information, and to develop a phase method of bearing using the resampling of a digitized sonar signal.

Materials and Methods. An algorithm for determining coordinates using the direction-finding phase method is considered. The results of its testing for an object moving along a circular trajectory in an experimental pool are presented.

Setting up an experiment. During the experiments, an experimental assembly created by V.A. Shirokov and V.N. Milich at the Udmurt Federal Research Center of the Ural Branch of the RAS was used to determine the location of an object in the hydroenvironment with the direction finding by the phase method [12]. This is a laboratory measuring complex with a linear aqua medium in the form of an extended cylindrical tank (hydro waveguide) and a basin equipped with a system for generating test sonar signals. The installation elements used in the experiments are described in detail below.

1. Hydroacoustic signal emitter with known coordinates S (Fig. 1). An amplitude-modulated signal is used in the work [13].

2. Reflecting object. In the experiment, an extended cylindrical object is used as a test object — a copper wire whose diameter (0.15 mm) is significantly less than the acoustic wave length (1.5 mm). In the presented work, the problem is solved on a plane; therefore, for the compilation of computational procedures for calculating coordinates, this object can be considered a point.

3. Stereo sensor that converts a sonar signal into an electrical one. The coordinates of each of the two receivers of the stereo sensor are known (see st1, st2 in Fig. 1).

4. Hardware-softwarecomplex that performsamplification,digitizationandprocessing of sensorsignals.

The measurement result is a recording file of the received two-channel signal.

Algorithm for processing measurement results. Using a stereo sensor with a small base (30 mm) compared to the distance to the object (≈ 800–900 mm) allows us to apply simplified formulas.

Now then, we calculate the coordinates of the object under study in the system x'y', formed by the plane of the stereo sensor and the normal to it. To do this, we use the algorithm that includes five stages.

1. Preprocessing:

High-frequency filtering is used in the work.

2. Selecting informative fragments imp1, imp2 (in two channels) containing pulses reflected from the object under study. To do this, thresholding is performed according to amplitude A of the signal. The threshold is assumed to be equal to p×max(A) (p = 0.5).

This processing allows for determining the leading edge of the pulse (BeginIndex), reflected from the object. Sections [BeginIndex–0.5×LenSignal; BeginIndex+LenSignal] are cut from two signal channels. Here, LenSignal — length of the input signal. As a result, we get two signals imp1 and imp2 with a length of 1.5×LenSignal. Each of them contains the pulse reflected from the object.

3. Calculation of distanceD(O; st) between the object and the stereo sensor.

. (1)

4. Determination of bearing α on object [14]:

(2)

where λ — wavelength; F — signal frequency; dt — sampling interval; m — difference in the arrival of the reflected pulse to two stereo sensors (in counts); d — base of the stereo sensor (in the presented study λ = 1.5 mm, F = 1 MHz, dt = 0.2 µs, d = 30 mm).

5. Calculation of the coordinates of the analyzedobjectin the systemformed by the plane of the stereosensoranditsnormal(Fig.1),according to the formulas: x’= D(O, st) ×cos(α); y’= D(O, st) ×sin(α).

Determining the bearing on an object. Assume the condition of the smallness of the stereo base compared to the distance from the sensor to the object. In this case, the stereo sensor allows us to determine the bearing on the object using information about the phase difference of the received pulses previously reflected from the object, as well as the difference in the time of arrival of the leading edges of the pulses received by the stereo sensor (Fig. 2).

To implement this approach, the following procedures are performed.

1. Identification of the moments of arrival of the leading edges of stereo signals using thresholding [15] with threshold p×max(A). In this paper, coefficient p is assumed to be 0.5 [16].

2. Calculation of difference ds (number of intervals dt) between the leading edges of the pulses registered in two receivers of the stereo sensor.

3. Calculation of bearing α by formula (1). We take value d for

With this method of calculating the phase shift, the leading edge of the signal is the main indicator of the pulse reflected from the object. The internal structure of the signal is not taken into account, which may be crucial for the correct determination of the bearing on the object. We take into account the similarity of the received signals and consider another approach to calculating the bearing — based on the cross-correlation function (CCF) of stereo signals. It is logical to determine the phase shift by the CCF maximum. The elements of the bearing calculation algorithm using CCF are described below.

1. Calculation of the CCF pulses arrived at two receivers of one stereo sensor: r= xcorr(imp1, imp2). The value of function xcorr at the shift of m of the second signal relative to the first is calculated as follows:

Here, N — length imp1 and imp2, –N<m<N.

2. Determination of the maximum CCF and the corresponding shift dc (in counts).

3. Calculation of bearing α from formula (1) taking into account that value dc is taken as m.

Research Results

Testing the bearing determination methods. The presented methods for determining the bearing on an object (by the difference in the time of arrival of the pulse-leading edges and by the CCF maximum) were tested for an object moving along a circular trajectory in the experimental pool. A harmonic signal with a duration of 7 periods and a frequency of 1 MHz was used. Digitized stereo signals were recorded at 256 points of the object's trajectory. The digitization frequency was 5 MHz.

Figure 3 shows the change in bearing values as the object moves around the circle for the two proposed approaches (difference of the leading edges, CCF maximum).

When using the signal-leading edge (Fig. 3, red marker), small outliers of values are observed almost along the entire curve of the bearing on the object. When using the CCF maximum (Fig. 3, blue marker), only in some areas there are outliers of bearing values, but they are quite large. Perhaps this is due to the peculiarities of the object's movement in the area where the probing and reflected signals cross the zone of turbulence caused by the object's movement. When finding the phase shift of stereo signals using the CCF maximum, additional errors occur, which is then reflected in the calculated trajectory of the object. Also, outliers in the calculated bearing values may be due to an insufficiently high sampling rate. The values of these errors are significantly higher than the emission values observed for the method using the pulse-leading edge.

Figure 4 a presents the trajectories of the object, calculated using the global CCF maximum and its neighboring local maxima.

Using aprior information about the smoothness of the movement of the experimental object, it is possible to select points that correspond to a smoother and more valid trajectory. This approach is used in other works [17]. In Figure 4 a, at x’ϵ[ 800;870], y’>0, the correct values are on the curve for the trajectory calculated from the left local CCF maximum. There are also erroneous points. If they are replaced by the points of the trajectory obtained as a result of using the right local CCF maximum, the correct value of the trajectory point can be calculated (x’≈800, y’<0 in Fig. 4 а). This graphical method of error correction can be implemented programmatically.

Thus, at some points of the trajectory, the correct phase shift of the stereo signals required for calculating the bearing may correspond not to the global CCF maximum, but to one of the neighboring local extremes (Fig. 4 b, c). This shift of the CCF maxima may be due to an insufficiently high sampling rate of the signals. This also explains the stepwise nature of the resulting trajectory.

Taking into account the quasi-harmonicity of the signal, it is proposed to use interpolation of rare measurements of the original signal with frequent calculated values. This virtual increase in the sampling rate (oversampling) makes it possible to fix intermediate values in the digitized source data. Interpolation of the signal values by a cubic spline allowed us to obtain 20 points for 1 period of the signal instead of 5 points in the original version. Figure 5 shows the calculated trajectories of the object, formed with preliminary resampling of the signals.

The red line is the object’s trajectory, obtained using the CCF maximum to calculate the bearing from pre-interpolated data. It seems to be more correct compared to a similar trajectory obtained using the difference in the leading edges of reflected pulses (blue line).

Discussion and Conclusion. Analysis of the features of determining the bearing on an underwater object using the mutual phase information of the differential stereo sensor signals makes it possible to draw the following conclusions.

Thus, the investigated possibilities of refining the phase direction finding method (resampling of digitized signals and using the CCF maximum) make it possible to effectively assess the trajectory of an underwater object.

The conditions for obtaining a smooth correct trajectory of the tracked object are described. The oversampling of interpolated quasi-harmonic stereo signals digitized with insufficient sampling rate serves this goal. At the same time, the bearing calculation method is used with the maximum of the cross-correlation function of the stereo sensor signals.

The data in Figure 5 confirm that the direction-finding problem can be solved with the accuracy required for practical application. Taking into account the factor of smoothness and continuity of the object's trajectory allows for the unique adjustment of the selection of the maximum of the cross-correlation function of the stereo sensor signals (Fig. 4).

The proposed methods are of great importance for the development of underwater vision systems.