CN109425875B - Satellite signal separation processing device and method - Google Patents

Satellite signal separation processing device and method Download PDF

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CN109425875B
CN109425875B CN201710792191.3A CN201710792191A CN109425875B CN 109425875 B CN109425875 B CN 109425875B CN 201710792191 A CN201710792191 A CN 201710792191A CN 109425875 B CN109425875 B CN 109425875B
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崔晓伟
张佳琪
石钦
陆明泉
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Tsinghua University
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    • GPHYSICS
    • G01MEASURING; TESTING
    • G01SRADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S19/00Satellite radio beacon positioning systems; Determining position, velocity or attitude using signals transmitted by such systems
    • G01S19/01Satellite radio beacon positioning systems transmitting time-stamped messages, e.g. GPS [Global Positioning System], GLONASS [Global Orbiting Navigation Satellite System] or GALILEO
    • G01S19/13Receivers
    • G01S19/24Acquisition or tracking or demodulation of signals transmitted by the system
    • G01S19/28Satellite selection
    • GPHYSICS
    • G01MEASURING; TESTING
    • G01SRADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S19/00Satellite radio beacon positioning systems; Determining position, velocity or attitude using signals transmitted by such systems
    • G01S19/01Satellite radio beacon positioning systems transmitting time-stamped messages, e.g. GPS [Global Positioning System], GLONASS [Global Orbiting Navigation Satellite System] or GALILEO
    • G01S19/13Receivers
    • G01S19/35Constructional details or hardware or software details of the signal processing chain
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Abstract

The application provides a satellite signal separation processing device and a satellite signal separation processing method. The satellite signal separation processing apparatus includes: an antenna array; and the processor is used for adjusting the shape of a synthesized directional diagram of the array elements, forming directional gain in the coming direction of the satellite signal to be separated and forming attenuation in the coming direction of other satellite signals, so that the satellite signal to be separated is separated. The number of required antenna elements is greatly reduced, and the size and the cost of the antenna array are greatly reduced.

Description

Satellite signal separation processing device and method
Technical Field
The present application relates to a satellite signal separation processing apparatus and method.
Background
With the wide application of Global Navigation Satellite System (GNSS) in the military and civilian fields, the research on navigation countermeasure around GNSS system is also gaining attention.
Under a complex transponder type deception jamming scene, received satellite signals need to be separated, and therefore set delay and Doppler are respectively introduced into each path of satellite signals. Thus, the target can be interfered to a designated position, and compared with direct forwarding without signal separation, the method has higher concealment and controllability.
The disclosed civil signal may be signal separated based on pseudo-code. However, for unknown pseudorandom code grant signals, there are two main spatial processing methods currently used for signal separation. The first method uses multiple parabolic antennas, and adjusts the azimuth and pitch angles of the antennas to point and track a satellite to be separated. The second method is to use an antenna array to form a gain beam to be directed to the satellites to be separated. In the prior art, the two satellite signal separation methods only consider the incoming information of the satellite signal to be separated, and thus, in order to achieve sufficient spatial resolution, the main lobe width of the antenna and the main beam of the array need to be sufficiently narrow to meet the requirement of the spatial resolution, which means a larger antenna aperture and a larger number of array elements. A set of 3m diameter parabolic antennas is typically required to separate the multiple navigation satellite signals. To achieve the same spatial resolution, a half-wavelength planar antenna array requires at least hundreds of elements. Both approaches have high hardware cost and implementation complexity.
Disclosure of Invention
The invention aims to provide a satellite signal separation processing device and a satellite signal separation processing method, which are low in hardware cost and simple to implement.
According to an aspect of the present application, there is provided a satellite signal separation processing apparatus including: an antenna array; and a processor, wherein the antenna array comprises a plurality of array elements, and the processor adjusts the shape of a synthetic directional diagram of the plurality of array elements to form directional gain in the coming direction of the satellite signals to be separated and attenuation in the coming direction of other satellite signals.
According to an aspect of the present application, there is provided a satellite signal separation processing method, including: the method comprises the steps of receiving satellite signals from a plurality of satellites through an antenna array comprising a plurality of array elements, adjusting the shape of a synthetic directional diagram of the plurality of array elements, and forming directional gain in the coming direction of the satellite signals to be separated and attenuation in the coming direction of other satellite signals.
According to the embodiment of the application, when satellite signals are separated, the incoming information of all satellite signals is considered, the guidance vectors of the antenna array for all satellites are utilized, the weighting vectors of the antenna array for the satellite signals to be separated are calculated, directional gains are formed in the incoming directions of the satellite signals to be separated, meanwhile, attenuation is formed in the incoming directions of other satellite signals, and therefore the satellite signals to be separated are separated, the number of required antenna array elements is greatly reduced, and the size and the cost of the antenna array are greatly reduced.
Drawings
Fig. 1 shows a satellite signal separation processing apparatus according to an embodiment of the present application.
Fig. 2 shows a satellite signal separation processing apparatus according to another embodiment of the present application.
Fig. 3 is a schematic diagram illustrating an array element arrangement and satellite signal directions of an antenna array according to an embodiment of the present application.
Fig. 4 is a schematic diagram illustrating the shape of a synthetic pattern of a plurality of array elements obtained by a satellite signal separation processing method according to an embodiment of the present application.
Detailed Description
The satellite signal separation processing apparatus and method based on an antenna array disclosed in the present application will be described in detail with reference to the accompanying drawings. For the sake of simplicity, the same or similar reference numerals are used for the same or similar devices in the description of the embodiments of the present application.
In the following, taking the GPS satellite navigation system as an example, the spatial distribution characteristics of the satellites are analyzed from the number of visible satellites and the angles between the satellites.
According to the reference configuration of the GPS, the GPS constellation consists of 24 working satellites, which are respectively located in 6 geocentric orbital planes, with 4 satellites in each orbital plane. A sufficiently dispersed satellite geometry ensures good observability for all users around the world. The number of visible satellites varies from a minimum of 4 to a maximum of 11 at any location on the earth's surface. Further, it is general that the minimum angle between the satellites is not less than 10, and the angle between the satellites is more than 20 degrees in most cases.
From the above analysis of the satellite spatial distribution, it can be seen that an effective signal separation method should be able to extract any given satellite signal from up to 11 received satellite signals with a minimum included angle of 10 °. Considering the sensitivity range of a general GPS receiver, the attenuation of other satellite signals should be not less than 20dB to 30 dB. At present, the satellite signal separation method in the prior art only considers the incoming information of the satellite signal to be separated, so that the number of antenna array elements required by signal separation depends on the spatial resolution of the antenna array, and a very narrow beam must be formed by a large number of antenna array elements to effectively attenuate signals of other satellites. According to the relation between the main lobe width of the uniform linear array beam pattern and the number of the array elements, at least 12 antenna array elements are usually needed to achieve the main lobe width of 10 degrees. If two-dimensional beam scanning is performed on the azimuth angle and the pitch angle of the incoming direction of the satellite signal, a planar array is required, and the directional diagram function of a general rectangular array is the product of directional diagram functions of two linear arrays, that is, at least 12 × 12 antenna elements are required to achieve a spatial resolution of 10 °.
According to one embodiment of the application, a satellite signal separation processing device and a satellite signal separation processing method are provided, wherein satellite signals from a plurality of satellites are received through an antenna array comprising a plurality of array elements, the shape of a synthetic directional diagram of the plurality of array elements is adjusted, directional gain is formed in the coming direction of the satellite signals to be separated, and attenuation is formed in the coming direction of other satellite signals. Thus, all the incoming information of the satellite signals is utilized, and the number of array elements of the required antenna array depends on the number of satellites to be separated, not the spatial resolution.
Fig. 1 shows a satellite signal separation processing apparatus according to an embodiment of the present application. As shown in fig. 1, the satellite signal separation processing apparatus 10 includes an antenna array 100 and a processor 200. The antenna array 100 is capable of receiving satellite signals from M (M >1) satellites, and the antenna array 100 further includes N (N >1) array elements 110. The processor 200 can adjust the shape of the composite pattern of the plurality of array elements 110 of the antenna array 100 to form directional gain in the incoming direction of the satellite signal to be separated and attenuation in the incoming direction of other satellite signals, thereby separating the satellite signal to be separated.
Any one satellite signal can be selected as the satellite signal to be separated, so that the M satellite signals can be separated one by one only through the antenna array with M antenna elements. Of course, in order to achieve a more suitable antenna array arrangement in engineering applications, the number of antenna elements may also be greater than M, for example, M +1, M +2, or M +3 antenna elements may be used. In this way the size and cost of the antenna array will be greatly reduced.
Fig. 2 shows a satellite signal separation processing apparatus according to another embodiment of the present application. As shown in fig. 2, processor 200 includes a determination module 210 and a calculation module 220. The determination module 210 determines a steering vector for the antenna array for each satellite. The calculation module 220 calculates a weighting vector of the antenna array for the satellite to be separated according to the steering vector of the antenna array for each satellite, so as to adjust the shape of the composite directional diagram of the plurality of array elements of the antenna array.
According to one embodiment of the present application, M satellite signals are separated by an antenna array having N array elements, and a signal model received by the array may be expressed as:
Figure BDA0001399502610000041
wherein,
x (t) is an N x 1 dimensional observation data vector, and each row corresponds to a signal received by an array element;
sj(t) represents a jth satellite signal;
aj(j ═ 1, … M) denotes the steering vector of the antenna array for the jth satellite signal, depending on the geometry of the antenna array and the incoming wave direction of the satellite signal;
n (t) is an N × 1 dimensional noise signal vector, each row corresponding to additive white gaussian noise at its respective array element.
By constructing a weighting vector for the antenna array to weight and sum the received signals of each array element, the gain of the antenna array for different incoming signals, i.e., the shape of the composite pattern of multiple array elements, can be controlled. The output signal of the array can be expressed as:
Figure BDA0001399502610000042
in order to realize satellite signal separation, a weighting vector is constructed for a satellite to be separated by utilizing the incoming information of all satellite signals so as to weight and sum array received signals, so that directional gain is formed in the incoming direction of the satellite signal to be separated, and attenuation is formed in the incoming direction of other satellite signals, thereby separating the satellite signal to be separated.
Without loss of generality, assuming that a satellite with j ═ 1 is the satellite signal to be separated, the received signal can be expressed as:
Figure BDA0001399502610000051
for the satellite j to be separated to be 1, the output signal of the array with the weighting vector w is:
Figure BDA0001399502610000052
according to this embodiment, the processor 200 determines the steering vector a of the antenna array for each satellitejAccording to the steering vector a of the antenna array for each satellitejCalculating a weight vector w for a plurality of elements of the antenna array such that in the output signal, the satellite signal s to be separated1(t) forming directional gain in the incoming direction while in other satellite signals sj(t) the incoming direction of the beam forms an attenuation. In this way, only the satellite signal to be separated will be included in the output signal, while the other satellite signals have been attenuated into noise components.
According to an embodiment of the present application, the determining module 210 of the processor 200 may determine the steering vector a of the antenna array for each satellite based on an array synthesis methodj
For a given satellite, the steering vector is related to the direction of arrival of the satellite signal, the attitude and geometry of the array, the antenna elements, and the non-ideality of the rf channels. After the antenna array is calibrated, the non-ideal factors of the antenna elements and the radio frequency channels can be ignored, and the steering vector can be calculated according to the relative positions of the satellite and the array.
Fig. 3 shows an array element arrangement manner of an antenna array of a satellite signal separation processing apparatus and a satellite signal arrival direction diagram according to an embodiment of the present application. Assume that the incident direction vector of a satellite signal is:
Figure BDA0001399502610000053
wherein θ represents the elevation angle of the direction vector on the X-Y plane of the array,
Figure BDA0001399502610000054
Indicating the azimuth angle to the X-axis. For a typical receiver, the direction of the visible satellite is defined on the ENU local coordinate system, and the coordinates in the antenna coordinate system of the incident direction can be calculated by multiplying the coordinates in the ENU coordinate system by a rotation matrix under the condition that the attitude of the antenna array is known.
The geometrical configuration of the array, that is, the arrangement of the antenna elements, can be represented by a matrix:
P=[p1,p2,…pN]T
wherein N is the number of array elements, pk=[xk,yk,zk]TIndicating the position of the kth element in the antenna coordinate system.
The steering vector a of the antenna array for each satellitejCan be expressed as:
Figure BDA0001399502610000061
whereinλThe wavelength of the carrier signal representing the satellite signal,
Figure BDA0001399502610000062
indicating the incident direction vector of the jth satellite signal.
According to an embodiment of the present application, the determining module 210 of the processor 200 may further determine the steering vector a of the antenna array for each satellite based on a signal tracking methodj. For example, satellite navigation systems typically broadcast both civilian and military signals, and signal tracking-based methods can determine the steering vector of the military signal by tracking the civilian signal.
Compared with the method based on array synthesis, the method based on signal tracking does not need a large amount of prior information and a complicated calibration process, and the sum of the amplitudes of the received satellite signals is directly extracted from different array element channelsPhase, steering vector a of antenna array for each satellitejCan be expressed as:
Figure BDA0001399502610000063
wherein v isjkjkRespectively representing the amplitude and the carrier phase of the jth satellite signal S extracted from the kth array element channel.
For each visible satellite, the amplitude and phase of the satellite signal received by each of the N array elements may be extracted by the determining module 210 to form a steering vector. The method for determining the steering vector does not need prior knowledge of the satellite direction and the geometric configuration of the array, and all non-ideal factors of the array and the radio frequency front end are contained in the extracted amplitude and phase, so the method is self-calibration and is more convenient and effective.
After obtaining the steering vector a of the antenna arrayjThen, the calculating module 220 of the processor 200 calculates a weighting vector w of the antenna array, and adjusts the shape of the composite directional diagram of the plurality of array elements of the antenna array to separate the satellite signals to be separated.
Assuming that the satellite signals are statistically independent and uncorrelated with each other and noise, the power of the output signal y (t) of the antenna array can be expressed as:
Figure BDA0001399502610000071
wherein, PjIs the power, P, of the jth satellite signalnIs the power of the noise signal.
The weight vector w is calculated such that the gain of the satellite signals to be separated is increased, for example, such that the incoming direction of the signals to be separated has a directional gain, i.e.,
wHa1=α1=1。
meanwhile, when the weight vector is calculated, other satellite signals are regarded as interference, and the constraint of attenuation is set upwards,
wHaj=αj(j=2,...,M)。
make the attenuation of other satellite signals to a predetermined threshold alphaj(j ═ 2.., M). The predetermined threshold may be set according to the receiver sensitivity, for example, the level of the residual signal after weighted combination may be at least lower than the receiver sensitivity.
Furthermore, as can be seen from the calculation formula of the output signal power, the noise signal power may be amplified by the modulus of the weighting vector, and in order to ensure a specific and sufficient signal-to-noise ratio of the desired signal for subsequent deceptive signal synthesis, the weighting vector should be constructed such that the noise is as small as possible, i.e., the modulus of the weighting vector should be as small as possible.
Therefore, according to one embodiment of the present application, the signal separation problem is transformed into the following optimization problem based on the multi-linear constraint:
Figure BDA0001399502610000072
wherein w represents the weighting vector of the antenna array for the satellite signals to be separated, aj(j 1.. M) denotes a steering vector of the antenna array for each satellite, M denotes a total number of visible satellites, i denotes a satellite to be separated, and α denotes a total number of satellites to be separatedjRepresents a predetermined threshold for attenuation, min represents minimization, and s.t. represents a constraint.
Thus, according to the embodiment of the application, when satellite signals are separated, the incoming information of all satellite signals is considered, the guidance vectors of the antenna array for all satellites are utilized, the weighting vectors of the antenna array for the satellite signals to be separated are calculated, the directional gain is formed in the incoming direction of the satellite signals to be separated, meanwhile, the attenuation is formed in the incoming direction of other satellite signals, and therefore the satellite signals to be separated are separated, the number of required antenna array elements is greatly reduced, and the size and the cost of the antenna array are greatly reduced.
Without loss of generality, a satellite i to be separated can be made to be 1, and in order to solve the optimal weighting vector, a lagrangian multiplier method is adopted to construct a lagrangian function:
Figure BDA0001399502610000081
the optimal weight vector w should satisfy:
Figure BDA0001399502610000082
as can be seen from the first equation, the optimal weighting vector can be expressed as a linear combination of the steering vectors of the antenna array for each satellite:
Figure BDA0001399502610000083
substituting it into the latter two constraint equations to solve λ, μj(j ═ 2.., M). It is demonstrated that the optimal weighting vector can be written as follows:
wopt=A(AHA)-1b,
wherein,
A=[a1,a2,…aM]an array flow pattern matrix composed of the steering vectors of the antenna array for all satellites;
b=[1,α2,…αM]Tand is an M × 1-dimensional column vector composed of directional gain and attenuation coefficients.
Thus, the output signal of the receiver can be expressed as:
Figure BDA0001399502610000091
the weighting vector of the antenna array for the satellite to be separated is obtained through the satellite gain multi-linear constraint optimization calculation, so that the gain of the satellite signal to be separated is ensured in the output signal of the antenna array, meanwhile, the level of the residual signal of other satellite signals after weighted synthesis is lower than the sensitivity of a receiver, and the noise is as small as possible, so that the receiver can successfully receive the satellite signal to be separated, and the other satellite signals are regarded as interference, thereby realizing the separation of the signal to be separated.
Here, a satellite with j ═ 1 is schematically taken as the satellite to be separated. It will be appreciated that the satellites to be separated may be any one of the satellites in view. Generally, in practical application, all visible satellite signals need to be separated, the optimization problem is solved for each satellite signal based on the known guide vector information of each satellite, and weighting synthesis is performed on array received signal vectors respectively by using the weighting vectors of the results, so that all visible satellite signals can be separated one by one.
According to another aspect of the present application, there is also provided a satellite signal separation processing method, including: the method comprises the steps of receiving satellite signals from a plurality of satellites through an antenna array comprising a plurality of array elements, adjusting the shape of a synthetic directional diagram of the plurality of array elements, and forming directional gain in the coming direction of the satellite signals to be separated while forming attenuation in the coming direction of other satellite signals. The satellite signal separation processing method further comprises the following steps: determining a steering vector of the antenna array for each satellite; and calculating a weighting vector of the antenna array for the satellite to be separated according to the steering vector of the antenna array for each satellite, adjusting the shape of a synthetic directional diagram of a plurality of array elements of the antenna array through the weighting vector, and forming directional gain in the coming direction of the satellite signal to be separated and attenuation in the coming direction of other satellite signals.
According to one embodiment, the number of elements of the antenna array may be determined according to the maximum number of visible satellites. The steering vector of the antenna array for each satellite signal may be determined according to the incoming wave direction of the satellite signal to be separated and other satellite signals, and the arrangement of the plurality of array elements of the antenna array. Alternatively, the steering vector of the antenna array for each satellite may be determined by tracking known signals from the satellite to be separated and other satellites, and from the amplitude and phase of the tracked known signals.
According to one embodiment, the weighting vector of the antenna array for the satellite to be separated is calculated based on the satellite signal gain multi-linear constraint optimization according to the obtained steering vector of the antenna array for each satellite signal. Wherein the optimization condition based on the satellite signal gain multi-linearity constraint comprises that the weighting vector is calculated so that the incoming direction of the signal to be separated has a directional gain, the incoming direction signals of other satellite signals are attenuated to be lower than a preset threshold, and the modulus of the weighting vector is minimum.
For example, the weight vector may be calculated based on the following satellite signal gain multi-linear constraint optimization:
Figure BDA0001399502610000101
wherein w represents the weighting vector of the antenna array for the satellite signals to be separated, aj(j 1.. M) denotes a steering vector of the antenna array for each satellite, M denotes a total number of visible satellites, i denotes a satellite to be separated, and α denotes a total number of satellites to be separatedjRepresents a predetermined threshold for attenuation, min represents minimization, and s.t. represents a constraint.
Fig. 4 is a schematic diagram illustrating the shape of a synthetic pattern of a plurality of array elements obtained by a satellite signal separation processing method according to an embodiment of the present application. For ease of explanation, it is assumed here that the incoming medial pitch of all visible satellite signals is fixed, with the azimuth angle varying over [0 °,360 ° ]. After signal separation processing, the shape of the synthesized directional diagram of the multiple array elements of the antenna array is shown in fig. 4, a solid line in the azimuth represents the incoming wave direction of the satellite signal to be separated, and a dotted line in the azimuth represents the incoming direction of other satellite signals. Therefore, directional gain is formed in the incoming direction of the satellite signal to be separated, and meanwhile attenuation is formed in the incoming direction of other satellite signals, so that the satellite signal to be separated is separated.
The application provides a device and a method for separating satellite signals based on an antenna array by using known satellite space distribution, which are used for separating authorization signals with unknown pseudo code generation modes. Furthermore, it can be understood that the satellite signal separation processing apparatus and method proposed in the present application are also applicable to unauthorized signals whose pseudo code generation manner is known.
Exemplary embodiments of the present application are described above with reference to the accompanying drawings. It will be appreciated by those skilled in the art that the above-described embodiments are merely exemplary for purposes of illustration and are not intended to be limiting, and that any modifications, equivalents, etc. that fall within the teachings of this application and the scope of the claims should be construed to be covered thereby.

Claims (10)

1. A satellite signal separation processing apparatus comprising:
an antenna array comprising a plurality of array elements; and
a processor, comprising:
a determination module that determines a steering vector for each satellite for the antenna array; and
the calculation module is used for calculating a weighting vector of the antenna array for the satellite to be separated based on satellite signal gain multi-linear constraint optimization according to the steering vector of the antenna array for each satellite, and adjusting the shape of a synthetic directional diagram of a plurality of array elements of the antenna array through the weighting vector, so that the incoming direction of the signal to be separated has directional gain, incoming direction signals of other satellite signals are attenuated to be lower than a preset threshold, and the modulus value of the weighting vector is minimum.
2. The apparatus of claim 1, wherein the number of elements of the antenna array is determined by a maximum number of visible satellites.
3. The apparatus of claim 1, wherein the determining module determines the steering vector of the antenna array for each satellite according to the incoming wave directions of the satellite signal to be separated and other satellite signals and the arrangement of the plurality of elements of the antenna array.
4. The apparatus of claim 1, wherein the determining module tracks known signals from the satellite to be separated and other satellites, and determines the steering vector of the antenna array for each satellite based on the amplitude and phase of the tracked known signals.
5. The apparatus of claim 1, wherein the calculation module calculates the weight vector based on the following satellite signal gain multi-linear constraint optimization:
Figure FDA0002957672830000011
wherein w represents the weighting vector of the antenna array for the satellite signals to be separated, aj(j 1.. M) denotes a steering vector of the antenna array for each satellite, M denotes a total number of visible satellites, i denotes a satellite to be separated, and α denotes a total number of satellites to be separatedjRepresents a predetermined threshold for attenuation, min represents minimization, and s.t. represents a constraint.
6. A satellite signal separation processing method, comprising:
receiving satellite signals from a plurality of satellites via an antenna array comprising a plurality of elements, determining a steering vector for the antenna array for each satellite,
and calculating a weighting vector of the antenna array for the satellite to be separated based on satellite signal gain multi-linear constraint optimization according to the steering vector of the antenna array for each satellite, and adjusting the shape of a synthetic directional diagram of a plurality of array elements of the antenna array through the weighting vector, so that the incoming direction of the signal to be separated has directional gain, incoming direction signals of other satellite signals are attenuated to be lower than a preset threshold, and the modulus value of the weighting vector is minimum.
7. The method of claim 6, wherein the number of elements of the antenna array is determined according to the maximum number of visible satellites.
8. The method as claimed in claim 6, wherein the steering vector of the antenna array for each satellite signal is determined according to the incoming wave directions of the satellite signal to be separated and other satellite signals and the arrangement of the plurality of array elements of the antenna array.
9. The method of claim 6, wherein known signals from the satellite to be separated and other satellites are tracked, and a steering vector for each satellite is determined by the antenna array based on the amplitude and phase of the tracked known signals.
10. The method of claim 6, wherein the weight vector is calculated based on the following satellite signal gain multi-linear constraint optimization:
Figure FDA0002957672830000021
wherein w represents the weighting vector of the antenna array for the satellite signals to be separated, aj(j 1.. M) denotes a steering vector of the antenna array for each satellite, M denotes a total number of visible satellites, i denotes a satellite to be separated, and α denotes a total number of satellites to be separatedjRepresents a predetermined threshold for attenuation, min represents minimization, and s.t. represents a constraint.
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