RU2696084C1 - Method of estimating radial velocity of an object - Google Patents

Abstract

FIELD: measurement technology.

SUBSTANCE: invention relates to methods of estimating radial velocity of objects along axis X, perpendicular to trajectory of flight of side-looking radar (SLR) – Y-axis. Assessment is carried out based on radar images (RI) of the area formed in a SLR during probing of ground and water surfaces using synthesized antenna arrays (SAA). Essence of the invention consists in the fact that according to the results of the locating, the radio holograms are formed and stored and the initial RI is formed when processing is matched with the speed of the carrier V_car. On the RI, objects are detected and their position X₀__obi Y₀__obi are determined and_. Range of measured speeds is specified, the interval of unambiguous estimation of radial speed is determined ΔV and divide the range of measured radial speeds into intervals of unambiguous determination of radial velocity. For each interval, using radio holograms, two fragments of radar images are formed, during synthesis of which two different shifts of values of radial velocity of the carrier along the axis X_car__{x_1} and V_{car_x_2}. Beginning of Y_{beg_1i} and Y_{beg_2i} and ending Y_{end_1i} and Y_end__2i synthesis of each fragment of radar map for i-th object along Y axis is determined, considering shift Y₀__obi in the presence of object speed in range from V_r__min up to V_{r_max} and due to input V_{car_x_1} and V_{car_x_2}. Then, for each uniqueness interval, matching of objects of two RIs is established, for each i object it is determined its position X_{max_1i} and Y_{max_1i} at first and X_max__2i and Y_max__2i on second radar image and find offset ΔX_{ob_12i} as the difference of its coordinates along the X axis on two radar images ΔX_{ob_12i}=X_{max_1i}-X_{max_2i}. Evaluating the radial velocity of the objects according to the corresponding formula. There are also determined possible order of operation when expanding the radar map of an object, an expression for determining the single-valued interval and a rule for forming a frequency grid for unambiguous determination of speed.

EFFECT: high accuracy of estimates of the radial velocity of SLR objects, which do not depend on the values of these velocities and on knowledge of the azimuths of the objects themselves.

8 cl, 7 dwg

Description

1 Field of the invention

The present invention relates to the field of radar, in particular, to methods for assessing the radial velocity of objects along the X axis perpendicular to the flight path of the side-scan radar carrier (RL BO) - the Y-axis. Evaluation is carried out using radar images (RLI) of the terrain formed in the RL BO during sounding of the earth and water surface using synthesized antenna arrays (ATS).

At present, synthetic aperture radar (SAR) radars and, in particular, side-view radars (RL BO) are increasingly used. A fairly complete development of the theory of the use of synthesized antenna arrays (ATS) has been carried out and real SARs with high characteristics have been created, which are used to monitor the terrain, pipelines, reconnaissance of ice conditions, detection of objects, fires, other emergency situations, etc.

However, some SAR tasks remain difficult to implement, and SAR characteristics do not meet practical requirements. Such tasks, in particular, include the detection of moving objects and an estimation of their speed.

2 prior art

As analogues of the proposed method for estimating the speed of a located object, one can consider the methods of selection of moving targets (SDC), their detection and speed estimation, discussed in G.S. Kondratenkova and A.Yu. Frolova “Radiovision”, Moscow, Publishing House “Radio Engineering”, 2005 p. 302-330.

This book describes a method for selecting and measuring the speed of moving targets in systems with a stopped phase center (the book "Radio-vision", G. Kondratenkov, A.Yu. Frolov, Moscow, the publishing house "Radio Engineering", 2005, pp. 309-312 ) The method is based on the use of three receiving channels, the antennas of which are spaced along the flight path of the carrier. In this case, the distance between the phase centers of the receiving antennas of these channels and the speed of the SAR carrier are coordinated in such a way that during the sensing period the phase centers will shift exactly by the distance between the two antennas. In this case, some antennas will occupy the positions of others in space, and the distance from them to the target will not change. Therefore, applying, for example, subtracting the received signals of these channels, they achieve compensation of signals from stationary objects. Assessing the speed of an object depends on the presence of signal residues from stationary targets, noise, and frequency resolution.

Of the main disadvantages of this method, it is possible to note the presence of "blind speeds", suppression of signals from objects with a low radial speed, the ambiguity of the measurement "azimuth - radial velocity", etc. To eliminate the ambiguity, an additional channel is necessary.

Another analogue can be the method of SDC and speed measurement in SAR with a monopulse antenna, considered in the same book "Radio Vision" (p. 312-318). The method is based on the input of the second channel and the formation of an equal signal direction (RSN). In this case, signals from stationary objects located on the RSN are compensated, and from moving objects, even when the Doppler frequencies of the stationary and moving objects coincide, they are not compensated, since the signals from moving objects have angles of arrival in azimuth other than the RSN. Having a sufficiently high efficiency of suppressing background signals, the method does not work at low radial velocities of objects and in the presence of only two channels (total and difference) it is not possible to unambiguously measure the radial speed and azimuth of a moving target (ibid., P. 318).

The prototype is a method for selecting moving targets and estimating their radial speed by Doppler filtering of signals, considered in the same book, "Radio Vision" (p. 304-309). To do this, using the radar detector BO with a synthesized aperture, they locate the terrain along the Y axis, form and store complex envelopes of the input signals (radio holograms), and form the initial radar image (RIR) using the processing of the path signal (ibid., P. 137), coordinated with carrier speed V _{nose_y} and with motionless terrain and objects on it. To distinguish moving objects using the difference of the Doppler frequencies of moving and stationary objects. The signals of stationary objects are filtered out, and the signals from moving objects are selected (extracted) in the system for processing trajectory signals (ibid., P. 306). With simplified, consistent processing, this is done through frequency filtering. A moving target is detected when the threshold is exceeded in a Doppler filter tuned to the frequency of the target signal. This provides an estimate of the Doppler frequency f _dts . To determine the radial velocity of the target _Vcr, it is necessary to additionally measure (in any way) the direction of arrival of the signal (azimuth of the target) _βc relative to the direction of the axis of the radiation pattern of the SAR antenna θ _n . The angle θ _{n is} measured relative to the direction of the velocity vector of the SAR carrier V and is considered known. Having estimates of the azimuth and Doppler frequency, it is possible to determine the radial velocity of the target (ibid., P. 308)

where λ is the wavelength of the SAR. In the particular case for the radar radiation protection θ _n = 90 deg.

Among the disadvantages of the prototype, in addition to the need to measure the azimuth of the target for an unambiguous assessment of its radial velocity, it should also be noted the dependence of the error in determining the velocity on the errors in determining the azimuth, the possibility of confusion when determining moving and stationary targets, in particular, when the signal from the stationary object significantly exceeds the background signal, method at low target speeds, etc.

The technical result in the application of the proposed method is to provide estimates of the radial velocity of the radar radar objects located, independent of the values of these speeds and the knowledge of the azimuths of the objects themselves and, as a result, not requiring the formation of additional antenna channels.

3 Disclosure of invention

The essence of the invention lies in the fact that they use the known method of estimating the radial speed of a moving object V _{ob_x} directed along the X axis perpendicular to the Y axis, which coincides with the flight path of the side-scan radar carrier (RL BO) with a synthesized aperture, in which the band is located using the RL BO area along the Y axis is formed and stored Radioholograms form the original radar image (RLI) in agreement with the processing speed V _{nos_u} carrier and fixed on the terrain and ektami thereon detected thereon objects and their measured radial velocity, characterized in that for all i-th detected objects that require evaluation of their radial velocity,

- determine their position X _{0_obi} and Y ₀ _ _obi on the original radar,

- set the minimum V _{r_min} and maximum V _r _ _{max the} radial velocities of the objects, determining the range of its measurement,

- determine the interval of unambiguous estimates of the radial velocity ΔV,

- divide the range of measured radial velocities into intervals of unambiguous determination of the radial velocity

and for each interval, using radio holograms, two radar fragments are formed, during the synthesis of which two different displacements of the radial velocity of the carrier along the axis XV _{nose_x_1} and V _{nose_x_2 are introduced} by an amount lying within the unambiguity interval ΔV,

wherein Y _{nach_1i} beginning and ending Y _{nach_2i} and Y and Y _{kon_1i kon_2i} synthesizing radar image for each piece of i-th object determined on the Y axis, given _{0_obi} displacement Y in the presence of an object velocity in the range of V _{r_min} to V _r _ _max and iz for input V _{nose_x_1} and V _{nose_x_2} ,

for each interval of uniqueness, the correspondence of the objects of two _{radar images} is established, for each i-th object its position X _{max_1i} and Y _{max_1i} on the first and X _max _ _2i and Y _max _ _2i on the second radar are determined and the offset ΔX _{ob_12i is found} as the difference of its coordinates X-axis on two radar

ΔX _{rev_12i} = X _{max_1i} -X _{max_2i}

and then evaluate their radial velocity according to the formula:

where D _synt is the interval for synthesizing radar images.

At the same time, the beginning of the formation of each radar fragment for the ith object Y _{start_1i} and Y _{start_2i} along the Y axis is selected offset from Y _{0_obi} , so

where λ is the wavelength of the emitted radar signal BO, k is a constant,

and forming the end of each fragment for RLI i-th object _{kon_1i} Y and Y _con _ _2i determined by a displaced relative to the axis Y Y _{0_obi} so that

In this case, based on the required completeness of the radar image of the located object on the generated radar fragment, k choose more than 0.5.

For each interval of unambiguity, the correspondence of the radar image of objects on two fragments of the radar image to one object is preferable to establish by checking their alignment along the Y axis when the first radar image is shifted relative to the second along the Y axis by ΔY _i , determined by the expression

When determining ΔX _{ob_12i} for each i-th object, its position X _{max_1i} on the first and X _{max_2i} on the second X- _ray fragment is determined as the maximum cross-section of the X- _ray object on the X axis, corresponding to the center of this X- _ray object on the Y axis.

To maintain the accuracy of the estimation of radial velocity, regardless of its value, the interval of unambiguous determination of speed is determined based on the expression

where R _max is the removal of the far boundary of the synthesized radar image from the path of the carrier.

In this case, based on the practical implementation of the method, it is advisable when the carrier velocity displacements during the formation of two radar images for each interval correspond to the boundary values of the velocities of this interval and for adjacent intervals these radars are common, and the initial radar is used as one of the boundary radar images.

Also, based on simplify practical implementation of the method, the beginning and ending of the formation of radar data for all objects of one interval is selected uniform, though the top equal to the minimum of the Y _nach _ _1i and Y _{nach_2i,} and the end is the maximum value of Y _{kon_1i} and Y _con _ _2i for all i.

Thus, the proposed method for estimating the radial velocity of an object in a radar beam detector based on the construction of two (or more) radar images when introducing an offset (error) in the velocity vector of the radar carrier radar detector does not require the introduction of additional antennas and angular channels. In addition, the accuracy of estimating the speed is determined by the accuracy of estimating the displacement of two images of the object on different radar images, and this does not depend on the value of the speed itself and is determined by the resolution and step of calculating the radar image. Therefore, the instrumental accuracy of measuring the radial velocity of an object can be very high, which is confirmed by the results of mathematical modeling below.

4 List of drawings

FIG. 1 shows the radar images (responses) of the first (right) and second (left) objects, and FIG. 2 - the third (right) and fourth (left) objects obtained at a zero value of the offset of the radial velocity of the carrier, i.e. with V _{nose_x1} = 0.

FIG. 3 and FIG. 4, show the corresponding responses calculated in the presence of a displacement V _{nose_x2} = 5 m / s,

FIG. 5 and FIG. 6 - correspond to responses at V _{nose_x3} = 10 m / s.

FIG. 7 reflects the expansion of the response with large differences in the velocities of the object and the carrier.

5 The implementation of the invention

The basis of the proposed method for measuring the radial (relative to the flight path of the radar carrier BO carrier) speed of the located object V _{rev_x} is the _radar displacement of this object having a radial speed when it is formed without taking into account this speed.

In accordance with the article Bryzgalova A.P. “The effect of the speed of an object on its radar image formed in a side-scan radar with a synthesized aperture”, published in the Digital Signal Processing journal No. 2 of 2017, this shift along the X axis (normal to the carrier’s flight path) is determined by the expression

This is the X-ray radar shift (taking into account the movement of the positioned object and the carrier along the X axis during the carrier’s flight along the Y axis when forming a radar fragment for the ith object, and also taking into account the radar displacements along the Y axis due to radial disregarded speeds) is used in the proposed method for the technical implementation of the assessment of the radial speed of the radar object being radiated.

The implementation of the method and its capabilities can be illustrated by the example of the formation of radar images when using correlation-filter matching processing in SAR (see the article Bryzgalova A.P., Karaulova E.V., Khnykina A.V. “Analog-digital data processing in radars with synthesized aperture using ultra-wideband signals with linear frequency modulation. ”, published in the journal“ Digital Signal Processing ”No. 4 for 2004). In this case, the received input signal is multiplied by the reference local oscillator signal, which is matched with the signal generated in the transmitter and tuned to certain parameters of this signal associated with the signal passing through its propagation path, which is determined by the relative spatial position of the aircraft (RS) carrier - RSA carrier and the location of the site and their mutual speeds. In particular, the local oscillator signal can be configured to receive a signal reflected from a selected point on the ground.

When multiplying, they form two quadrature channels. In each quadrature channel, the signals are digitized and then processed in N range channels. Each range channel is tuned to a specific point on the ground. For this, the signal after the multiplier is processed in N range channels, taking into account the difference in the signal parameters of each channel (delay and time transformation coefficient p = 1-2 V _r / c, where V _r is the radial component of the PCA carrier velocity relative to the location, c is the speed of light ) from the parameters of the local oscillator signal (see the above article, formulas (10) - (19)]. In this case, N samples of the output signal of the receiver are generated for the given terrain points, thereby forming the radar data of the located terrain.

In the formation of the initial _{radar image} , it is believed that the presence and value of the velocity V _r is due only to the speed of the carrier V _{nose_y} , its projection onto the carrier-object line. To assess the radial velocity of the object, _{radar images} are additionally formed when an additional (erroneous) radial component of the carrier velocity V _{nose_x is} entered into V _r .

The model of the above processing was used to confirm the operability of the proposed method. The modeling of signal processing from four point reflectors with the formation of responses (RLI) from each of them was carried out. The motion of the radar carrier BO was carried out along the Y axis with a velocity V _{nose_y} = 100 m / s. The first reflector was chosen motionless with coordinates X ₁₀ = 10000 m and Y ₁₀ = 0. The second, third and fourth reflectors had the same zero initial position on the Y axis, i.e. Y ₂₀ = Y ₃₀ = Y ₄₀ = 0, and are slightly shifted relative to the first and each other along the X axis: X ₂₀ = 9999 m, X ₃₀ = 9998 m and X ₄₀ = 9997 m. In addition, the reflectors 2 ... 4 moved along X axis, i.e. in the direction radial to the carrier flight path: V _{2_x} = 3 m / s, V _{3_x} = 5 m / s and V _{4_x} = 7 m / s. The image synthesis interval D _synt = 1000 m. Moreover, since a linear frequency modulated signal with a deviation of 200 MHz was used, the range resolution was high (about 1 m).

The processing results, presented for clarity, not by luminance radar, but in the form of three-dimensional responses, are shown in FIG. 1-6. Moreover, FIG. 1 shows the responses of the first (right) and second (left) objects, and FIG. 2 - the third (right) and fourth (left) objects obtained at a zero value of the offset of the radial velocity of the carrier, i.e. with V _{nose_x1} = 0. The following pair of figures: FIG. 3 and FIG. 4, show the corresponding responses calculated in the presence of a displacement V _{nose_x2} = 5 m / s, and FIG. 5 and FIG. 6 - with V _{nose_h3} = 10 m / s.

, наблюдается снижение амплитуды отклика и расширение его основного лепестка. Это объясняется отсутствием при обработке, согласованной с лоцированием неподвижного объекта, радиальных скоростей выше ΔV. Для рассматриваемого случая ΔV=5 м/с. При этом максимум отклика остается достаточно отчетливо выражен даже при ΔV_p=7 м/с. И только при ΔV_p=10 м/с (правый отклик на Фиг. 5 - РЛИ неподвижного объекта при V_{нос_х3}=10 м/с) появляется заметное расщепление главного лепестка, затрудняющее нахождение его положения.As can be seen from the figures, in accordance with the previously mentioned article Bryzgalova A.P. “The influence of the speed of an object on its radar image ...” the movement of an object radially to the path of the radar carrier of the radar detector leads to a displacement and distortion of its response to the synthesized radar image. The ideal response corresponding to the theory, located exactly at the point where the object was located at the time of the start of synthesis, is formed in the case of equality of the object velocity V _{j_x} and the displacement of the carrier velocity V _{nose_xi} (see the right response in Fig. 1 - REL of a stationary object obtained for V _{nose_x1} = 0 , - and the right response in Fig. 4 - the _{radar image of the} third object with V _{3_x} = 5 m / s with V _{nose_x2} = 5 m / s). The remaining responses are shifted along both axes, and to a greater extent along the Y axis. For example, in FIG. 1 and 2, the response of the second object is located at the point with coordinates X _{max_2} = 10019.2 m, Y _{max_2} = -300.4 m, of the third - X _{max_3} = 10046.4 m, Y _{max_3} = -501.6 m and of the fourth - X _{max_4} = 10057.8 m, Y _{max_4} = -702.4 m. In addition, when the difference between the object’s speed and the carrier’s speed offset ΔV _p exceeds a certain interval of unambiguous determination of speed

, there is a decrease in the amplitude of the response and the expansion of its main lobe. This is explained by the absence of radial velocities higher than ΔV during processing, consistent with the location of a stationary object. For the case under consideration, ΔV = 5 m / s. In this case, the response maximum remains quite clearly expressed even at ΔV _p = 7 m / s. And only at ΔV _p = 10 m / s (the right response in Fig. 5 is the _{radar image of a} stationary object with V _{nose_x3} = 10 m / s) does a noticeable splitting of the main lobe appear, making it difficult to find its position.

If we now use the relation given in paragraph 1 of the claims and estimate the speed of objects, we will obtain the following results:

for the 1st object, using the _{radar data} calculated for V _{nose_x1} = 0 and V _{nose_x2} = 5 m / s (Fig. 1 and Fig. 3), we have: X _{max_11} = 10000 m, Y _max _ ₁₁ = 0, X _{max_21} = 9971.4 m, Y _max _ ₂₁ = 500.6 m, Y _{beg_11} = -320 m, Y _{beg_21} = 180 m, X ₀ _ _vol1 = 10000 m, V _{nose_y} = 100 m / s and D _synth = 1000 m , from here V _{rev_x1} = 0.014 m / s,

for the 2nd object, using the same _{radar data as} for the first, we have V _{about_x2} = _3.021 m / s,

for the 3rd object, using again the _{radar data} calculated for V _{nose_x1} and V _{nose_x2} (Fig. 2 and Fig. 4), we have V _{ob_x3} = 4.996 m / s,

for the 4th object, using pairwise _{radar images} calculated for all three V _{nose_xi} (Fig. 2, Fig. 4 and Fig. 6), we obtain:

_{Ob_x4_1} V = 7.013 m / s (V RLI at _nose _ _x1 and V _{nos_h3)}

V _{about_x4} _ ₂ = 7.011 m / s ( _{radar images} with V _{nose_x1} and V _{nose_x2} ),

V _{about_x4_ 3} = 7.014 m / s ( _X-rays with V _{nose_x2} and V _{nose_x3} ).

Thus, all estimates of the radial velocity of objects with high accuracy correspond to true velocities.

In this case, the instrumental accuracy of the radial velocity estimate is quite high, since in accordance with the expression for this estimate it is determined mainly by the accuracy of the determination of two parameters X _{max_1i} and X _max _ _2i . The remaining parameters included in this formula are either specified or have little effect on the estimation of speed. In practice, the estimation of parameters can be difficult due to the length of the located object, the application of background, noise, etc. However, to reduce the influence of these and other factors on finding ΔX _{rev_12i} = X _{max_1i} -X _{max_2i,} you can use special treatments, for example, described below.

It should also be noted that the value of the measured speed does not affect the accuracy of its estimation. However, to evaluate large values of the radial velocity of the object, it is necessary to increase the offset of the radial velocity of the carrier so that their difference does not exceed ΔV. To confirm this, an assessment was made of the radial velocity of an object moving with V _{rev_x} = 30 m / s. For this, two treatments were simulated for the formation of an object's radar image at two values of the offset of the radial velocity of the carrier: 30 m / s and 35 m / s. When using these radar data in accordance with the claims, the object velocity was estimated at 29.8754 m / s, i.e. the estimation accuracy is quite high, and the error is mainly related to the evaluation of the response position along the X axis. In FIG. _Figure 7 shows the _{radar image} for the considered case with V _{nose_x} = 35 m / s. The radar object has expanded. But this expansion occurs along the Y axis, and to find X _max , for example, it is possible to determine the maximum cross section of the response along X corresponding to the center of this response along the Y axis.

An example with a measurement of the radial velocity of 30 m / s confirms the possibility of using boundary radar data calculated with a step equal to the unambiguity interval (items 6, 7 of the formula), and an example with 4 objects with V _{1_x} = 0, V _{2_x} = 3 m / s, V _{3_x} = 5 m / s and V _{4_x} = 7 m / s (Fig. 1-6) corresponds to paragraph 8 when using two radar data for measuring the speeds of all objects included in the same unambiguity interval.

Claims (27)

1. A method for estimating the radial speed of an object V _{rev_x} directed along the X axis perpendicular to the Y axis, coinciding with the flight path of the side-scan radar carrier (RL BO) with a synthesized aperture, in which using the RL BO they locate the terrain along the Y axis, form and store radio holograms , form the initial radar image (RLI) during processing, consistent with the speed of the carrier V _{nose_y} and with motionless terrain and objects on it, detect objects on it and measure their radial speed, excellent being that for all i-th detected objects that require radial velocity estimates, - determine their position X ₀ _ _obi and Y ₀ _ _obi on the original radar, - set the minimum V _{r_min} and maximum V _{r_max} radial velocities of objects that determine the range of its measurement, ... - determine the interval of unambiguous estimates of the radial velocity ΔV, - divide the range of measured radial velocities into intervals of unambiguous determination of the radial velocity and for each interval, using radio holograms, two fragments of radar images are formed, during the synthesis of which two different offsets of the radial velocity of the carrier are introduced along the axis XV _{nose_x_1} and V _{nose_x_2} by an amount lying within the unambiguity interval ΔV, wherein the top Y _{nach_1i} and Y _nach _ _2i and ending Y _{kon_1i} and Y _con _ _2i synthesizing each fragment radar data for i-th object on Y axes define a given offset Y ₀ _ _obi in the presence of an object velocity in the range of V _r _ _min to V _r _ _max and due to input V _{nose_x_1} and V _{nose_x_2} , for each interval of uniqueness, the correspondence of the objects of two _{radar images} is established, for each i-th object its position X _{max_1i} and Y _{max_1i} on the first and X _{max_2i} and Y _max _ _2i on the second radar are determined and the offset ΔX _{ob_12i is found} as the difference of its coordinates along the axis X on two radar ΔX _{rev_12i} = X _{max_1i} - X _{max_2i} and then evaluate their radial velocity according to the formula:

where D _synt is the interval for synthesizing radar images. 2. The method according to p. 1, in which the beginning of the formation of each fragment of the _{radar image} for the i-th object Y _{start_1i} and Y _{start_2i} along the Y axis is chosen offset relative to Y ₀ _ _obi , so that

where λ is the wavelength of the emitted radar signal BO, k is a constant, the end of the formation of each radar fragment for the i-th object Y _{kon_1i} and Y _{kon_2i} is determined to be offset along the Y axis relative to Y ₀ _ _obi , so that

3. The method according to p. 2, in which choose k more than 0.5. 4. The method according to p. 1, in which for each interval of uniqueness, the correspondence of the radar image of objects on two fragments of the radar image to one object is established by checking their alignment along the Y axis when the first radar image is displaced relative to the second along the Y axis by ΔY _i determined by the expression

5. The method according to p. 1, in which for each i-th object determine its position X _{max_1i} on the first and X _{max_2i} on the second X- _ray fragment as the maximum cross-section of the X- _ray object on the X axis, corresponding to the center of this X- _ray object on the Y axis. 6. The method according to p. 1, in which the interval of unambiguous determination of speed is determined based on the expression

where R _max is the removal of the far boundary of the synthesized radar image from the path of the carrier. 7. The method according to claim 1, wherein the carrier velocity offsets during the formation of two radar images for each interval correspond to the boundary values of the velocities of this interval and for adjacent intervals these boundary radar images are common, and the initial radar image is used as one of the boundary radar data. 8. The method of claim. 1 wherein the beginning and ending of the formation of radar data for all objects of the same selected interval uniform, though the top equal to the minimum of Y and Y _{nach_1i nach} _ _2i, and the end is the maximum value of Y and Y _{kon_1i con} _ _2i for all i.

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