Broadband coherent signal source DOA estimation method based on three orthogonal dipoles

Abstract

The invention discloses a broadband coherent signal source DOA estimation method based on three orthogonal dipoles. The method comprises the following steps: obtaining an array output matrix by using a single three orthogonal dipole antenna; carrying out FFT (Fast Fourier Transform) on the output matrixes of the three antennas to obtain an array frequency domain output matrix; if the signal is a frequency agility signal, taking out an amplitude-phase peak value, and accumulating frequency domain amplitude-phase peak values of a plurality of frequency points; if the signal is a linear frequency modulation signal, taking an amplitude-phase value in a certain range from a frequency spectrum; rearranging the frequency domain amplitude and phase values of the plurality of frequency points; forming a frequency shift data set; performing surface stacking on the frequency shift data set to obtain a third-order tensor; realizing third-order tensor decomposition by using a trilinear alternating least square method to obtain a steering vector matrix; and calculating an azimuth angle, a pitch angle, a polarization auxiliary angle and a polarization phase difference according to the steering vector matrix. According to the method, a small array aperture, a small number of array elements and low operation complexity are used, DOA estimation of a plurality of coherent signals is realized by using information of a plurality of frequency points, and polarization information estimation is given.

Description

Technical Field

[0001] The invention belongs to the technical field of array signal processing, and relates to a method for estimating the direction of arrival of a coherent signal source, and in particular to a method for estimating the DOA of a broadband coherent signal source based on three orthogonal dipoles. Background Art

[0002] Direction of arrival (DOA) estimation is one of the important directions in the field of radar signal processing. After years of development, there are a large number of perfect DOA estimation algorithms such as MUSIC and ESPRIT. These algorithms generally use a single frequency point in a certain frequency band for measurement and estimation. However, with the development of radar, broadband systems such as frequency agile radar (FAR) and linear frequency modulation (LFM) are often used in military radar technology to deal with blocking interference and electronic countermeasures (ECCM), and decoy devices are often deployed to transmit coherent signals to protect radars. This causes the high-order statistics received by the above algorithm array to have rank deficiency problems, and the above traditional algorithms will not be able to perform effective statistical processing.

[0003] In order to effectively estimate the DOA of coherent signals, spatial smoothing, Toeplitz matrix reconstruction and other methods are generally used to rank compensate the statistics of coherent signals to construct a full-rank decoherence covariance matrix, thereby realizing the direction of arrival estimation of coherent signals. However, it still requires a large number of sensors and a larger spatial distribution, and only uses the information of a single frequency point. Summary of the invention

[0004] Purpose of the invention: In order to overcome the deficiencies in the prior art, a method for DOA estimation of a broadband coherent signal source based on three orthogonal dipoles is provided, which uses a smaller array aperture, a smaller number of array elements, a lower computational complexity, and utilizes information from multiple frequency points to achieve DOA estimation of multiple coherent signals, and provides polarization information estimation.

[0005] Technical solution: To achieve the above object, the present invention provides a broadband coherent signal source DOA estimation method based on three orthogonal dipoles, comprising the following steps:

[0006] S1: Array output matrix obtained using a single three-orthogonal dipole antenna;

[0007] S2: Perform FFT on the output matrices of the three antennas to obtain the array frequency domain output matrix;

[0008] S3: Determine the signal type according to the frequency domain output matrix of the array. If it is a frequency agile signal, extract the amplitude and phase peaks, wait for the incident signal to be frequency-converted, and repeat steps S1 to S3 to accumulate the frequency domain amplitude and phase peaks of multiple frequency points;

[0009] If it is a linear frequency modulation signal, a certain range of amplitude and phase values ​​are taken from the spectrum;

[0010] S4: rearrange the frequency domain amplitude and phase values ​​of multiple frequency points from small to large according to the frequency;

[0011] S5: taking multiple frequency samples of the same length to form a frequency shift data set;

[0012] S6: stack the frequency shift datasets to obtain a third-order tensor;

[0013] S7: Use trilinear alternating least squares method to achieve third-order tensor decomposition to obtain the steering vector matrix;

[0014] S8: Calculate the azimuth angle, elevation angle, polarization auxiliary angle, and polarization phase difference according to the steering vector matrix.

[0015] Furthermore, the array output matrix in step S1 is expressed as follows:

[0016] X n (t) = AS n (t)+N n (t)(1) Where S n (t) is the incident signal sampling matrix, N n (t) is the array element circular space-time-polarization white noise, and the incident signal is statistically independent of the noise; n = 1, 2, 3, ..., N, N is the total number of frequency points; A is the polarization-space steering vector matrix, expressed as:

[0017]

[0018] Where K is the total number of incident signals, and the azimuth and elevation angles of the kth signal are and θ k , the polarization auxiliary angle and polarization phase difference are γ k and η k , for

[0019]

[0020] Furthermore, the array frequency domain output matrix in step S2 is expressed as follows:

[0021] X n (ω)=AS n (ω)N n (ω) (4)

[0022] Furthermore, the frequency conversion signal in step S3 is expressed as follows:

[0023]

[0024] Among them, f nis the signal frequency, which changes at equal intervals over time according to a certain sequence, and T is the frequency modulation period.

[0025] Furthermore, the linear frequency modulation signal in step S3 is expressed as follows:

[0026]

[0027] Where R is the linear frequency modulation slope, T is the frequency modulation period, and j is the imaginary unit.

[0028] Furthermore, the complex numbers corresponding to the frequency domain peak values ​​of the multiple frequency points rearranged in step S4 are recorded as a matrix Z, which is expressed as:

[0029] Z=[X1(ω1),X2(ω2),...,X n (ω n )], n=1,2,3,...,N (7)

[0030] Among them, X n (ω n ) is the complex number corresponding to the frequency domain peak value of the nth frequency point.

[0031] Furthermore, the method for constructing the frequency shift data set in step S5 is: there are a total of N frequency samples, and at this time, samples of (N-P+1) frequency points are sequentially taken from n=1 to form P frequency shift data sets Z p ,p=1,2,...,P,Z p It is a 3×N-P+1 dimensional matrix.

[0032] Furthermore, the expression of the third-order tensor in step S6 is as follows:

[0033]

[0034] Here the planes are stacked into a three-dimensional matrix.

[0035] Furthermore, the expression of the trilinear alternating least squares method decomposition of the third-order tensor in step S7 is as follows:

[0036] The trilinear alternating least squares method can decompose the three-dimensional matrix into three two-dimensional matrices, that is, decomposing G into The sizes of are 3×K, P×K, (N-P+1)×K respectively, where This is the estimate of the steering vector.

[0037] Furthermore, the expression of calculating the estimated angle using the steering vector matrix in step S8 is as follows:

[0038] Definition: |||| represents the Frobenius norm, R[] represents the real part, I[] represents the imaginary part, and ∠{} represents the phase angle.

[0039] First define Then, by decomposing Vector Estimated to be:

[0040]

[0041] Establish the following five real nonlinear equations:

[0042]

[0043] Combining equations (13) and (14) we can obtain

[0044]

[0045] Combining equations (10), (11), (12) and (15) and simplifying again yields

[0046]

[0047] Combining equations (12) and (16) we can obtain

[0048]

[0049] Combining equations (10), (11), (13), (14) and (15) and simplifying them again yields

[0050]

[0051] There are two estimation angles in total. When you finally get the result, you need to substitute it into the equation to determine which result it is.

[0052] The three orthogonal dipole antennas of the present invention have no spatial phase difference between the three antennas, and present the same estimation accuracy for signals in the entire frequency band, which is a good hardware carrier for utilizing multi-frequency point information. The present invention provides a new method for DOA estimation of broadband coherent signal sources using three orthogonal dipole antennas.

[0053] Beneficial effects: Compared with the prior art, the present invention has the following advantages:

[0054] 1. The present invention uses only a single three-orthogonal dipole antenna, has a small number of channels, occupies a small space for the device, and has a low cost.

[0055] 2. The present invention utilizes the fact that the frequency domain values ​​of broadband signals at different frequency points all contain the same steering vector matrix. By decomposing the third-order tensor formed by frequency domain stacking, the steering vector matrix can be uniquely obtained, thereby estimating the angle and polarization parameters. Compared with the subspace-type algorithm, the number of estimated targets must be less than the number of array elements. The number of signals simultaneously measured by the method of the present invention can be higher than the number of array elements. It utilizes the information of multiple frequency points, breaks through the limitation of the number of array elements, and can measure more than 3 coherent incident signals.

[0056] 3. The computational complexity of the present invention is relatively low, and only requires FFT computation and one third-order tensor decomposition. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] Figure 1 Flow chart of the method of the present invention.

[0058] Figure 2 Array model diagram of the method of the present invention.

[0059] Figure 3 This is the distribution diagram of the arrival angle estimation results of the frequency-agile signal.

[0060] Figure 4 This is the distribution diagram of the polarization angle estimation results of the frequency agile signal.

[0061] Figure 5 This is a graph showing how the DOA estimation accuracy of frequency agile signals changes with the signal-to-noise ratio.

[0062] Figure 6 The DOA estimation accuracy of a single target linear frequency modulation signal varies with the number of sampling points in the frequency domain.

[0063] Figure 7 This is the distribution diagram of the arrival angle estimation results of the dual-target linear frequency modulation signal.

[0064] Figure 8 This is the distribution diagram of the polarization angle estimation results of dual-target linear frequency modulation signals.

[0065] Fig. 9 This is the distribution diagram of the arrival angle estimation results of four target linear frequency modulation signals.

[0066] Fig.10 This is the distribution diagram of the polarization angle estimation results of four target linear frequency modulation signals. DETAILED DESCRIPTION

[0067] The present invention is further explained below in conjunction with the accompanying drawings and specific embodiments. It should be understood that these embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention. After reading the present invention, various equivalent forms of modifications to the present invention by those skilled in the art all fall within the scope defined by the claims attached to this application.

[0068] like Figure 1 As shown, the present invention provides a broadband coherent signal source DOA estimation method based on three orthogonal dipoles, comprising the following steps:

[0069] S1: Using a single three-orthogonal dipole antenna to obtain the array output matrix, the array output matrix is ​​expressed as follows:

[0070] X n (t) = AS n (t)+N n (t) (1)

[0071] Among them, S n (t) is the incident signal sampling matrix, N n (t) is the array element circular space-time-polarization white noise, and the incident signal is statistically independent of the noise; n = 1, 2, 3, ..., N, N is the total number of frequency points; A is the polarization-space steering vector matrix, expressed as:

[0072]

[0073] Where K is the total number of incident signals, and the azimuth and elevation angles of the kth signal are and θ k , the polarization auxiliary angle and polarization phase difference are γ k and η k , for

[0074]

[0075] S2: Perform FFT on the output matrices of the three antennas to obtain the array frequency domain output matrix. The array frequency domain output matrix is ​​expressed as follows:

[0076] X n (ω)=AS n (ω)+N n (ω) (4)

[0077] S3: Determine the signal type according to the array frequency domain output matrix. If it is a frequency agile signal, extract the amplitude and phase peaks. After the incident signal is frequency-converted, repeat steps S1 to S3 to accumulate the frequency domain amplitude and phase peaks of multiple frequency points. The frequency agile signal is expressed as follows:

[0078]

[0079] Among them, f n is the signal frequency, which changes at equal intervals over time according to a certain sequence, and T is the frequency modulation period;

[0080] If it is a linear frequency modulation signal, a certain range of amplitude and phase values ​​are taken from the spectrum. The linear frequency modulation signal is expressed as follows:

[0081]

[0082] Where R is the linear frequency modulation slope, T is the frequency modulation period, and j is the imaginary unit.

[0083] S4: Rearrange the frequency domain amplitude and phase values ​​of multiple frequency points from small to large according to the frequency, and record the complex numbers corresponding to the frequency domain peak values ​​of the rearranged multiple frequency points as matrix Z, which is expressed as:

[0084] Z=[X1(ω1),X2(ω2),...,X n (ω n )], n=1,2,3,...,N (7)

[0085] Among them, X n (ω n ) is the complex number corresponding to the frequency domain peak value of the nth frequency point.

[0086] S5: Take multiple frequency samples of the same length to form a frequency shift data set. The method for constructing the frequency shift data set is: there are a total of N frequency samples. At this time, starting from n=1, samples of (N-P+1) frequency points are taken in sequence to form P frequency shift data sets Z p ,p=1,2,...,P,Z p It is a 3×N-P+1 dimensional matrix.

[0087] S6: The frequency shift data set is stacked to obtain a third-order tensor, which is expressed as follows:

[0088]

[0089] Here the planes are stacked into a three-dimensional matrix.

[0090] S7: Use trilinear alternating least squares method to achieve third-order tensor decomposition to obtain the steering vector matrix;

[0091] The expression of the trilinear alternating least squares decomposition of the third-order tensor is as follows:

[0092] The trilinear alternating least squares method can decompose the three-dimensional matrix into three two-dimensional matrices, that is, decomposing G into The sizes of are 3×K, P×K, (N-P+1)×K respectively, where This is the estimate of the steering vector.

[0093] S8: Calculate the azimuth angle, elevation angle, polarization auxiliary angle, and polarization phase difference according to the steering vector matrix.

[0094] The expression for calculating the estimated angle using the steering vector matrix is ​​as follows:

[0095] Definition: |||| represents the Frobenius norm, R[] represents the real part, I[] represents the imaginary part, and ∠{} represents the phase angle.

[0096] First define Then, by decomposing Vector Estimated to be:

[0097]

[0098] Establish the following five real nonlinear equations:

[0099]

[0100] Combining equations (13) and (14) we can obtain

[0101]

[0102] Combining equations (10), (11), (12) and (15) and simplifying again yields

[0103]

[0104] Combining equations (12) and (16) we can obtain

[0105]

[0106] Combining equations (10), (11), (13), (14) and (15) and simplifying them again yields

[0107]

[0108] There are two estimation angles in total. When you finally get the result, you need to substitute it into the equation to determine which result it is.

[0109] In this embodiment, the above solution is applied as an example, specifically:

[0110] In this embodiment, the array elements are arranged as follows: Figure 2 As shown, it is assumed that there is a fixed distance between the radar and the decoy, that is, there is a fixed delay τ when the two coherent signals are received at any frequency. And due to the equipment differences between the radar and the decoy, there is a fixed phase difference Δφ between the two coherent signals at any frequency. And the signal amplitude of the decoy is 10% higher than that of the radar. Ignore the noise mixed in the signal in the air, and the white noise is only set at the receiving end of the array element. The number of snapshots is denoted as M.

[0111] First, according to formula (1), the array will output a 3-row M-column matrix, which represents the signal data received by each of the three antennas.

[0112] Then, if the received signal is a frequency-agile signal, FFT operation is performed on the data received by the three antennas to obtain a complex matrix X(ω), and the frequency domain amplitude and phase peak value X1(ω1) of the first frequency point is taken out. X1(ω1) is a matrix of 3 rows and 1 column, which represents the complex frequency domain peak values ​​of the three antennas.

[0113] When the frequency of the signal changes, repeat the above operation to obtain X2(ω2)...X N (ω N ), combined to form a new 3-row N-column matrix Z;

[0114] However, if the received signal is a linear frequency modulation signal, the data received by the three antennas are subjected to FFT operation to obtain a complex matrix X(ω), and a certain number of values ​​are directly taken at equal intervals within the frequency spectrum range of X(ω) to form Z;

[0115] Decompose the matrix Z, take its 1st column to (N-P+1)th column to form the first frequency shift data set Z1, take the 2nd column to (N-P+2)th column to form the second frequency shift data set Z2, and continue in sequence until the Pth to Nth columns are taken to form the last frequency shift data set Z p .

[0116] Z p is a 3×N-P+1 dimensional matrix, so it can be stacked to form a third-order tensor. Then perform a third-order tensor decomposition on G to obtain the steering vector Estimates.

[0117] Finally, the azimuth angle, elevation angle, polarization auxiliary angle, and polarization phase difference are calculated using formulas (9) to (18).

[0118] In order to verify the effectiveness and effect of the estimation method of the present invention, a variety of simulation experiments are performed in this embodiment, as follows:

[0119] Experiment 1: Set up two coherent incident signals, one for frequency-agile radar and the other for decoy.

[0120] Radar signal: azimuth angle 30°, elevation angle 70°, polarization auxiliary angle 45°, polarization phase difference 90°.

[0121] Decoy signal 1: azimuth 140°, elevation 20°, polarization auxiliary angle 35°, polarization phase difference 60°, amplitude relative radar signal gain 1.1, fixed delay 1.667μs, initial phase difference 60°.

[0122] The two signals have the following common parameters: center frequency 8 GHz, number of frequency points 10, frequency hopping interval 50 MHz, number of sampling points 128, signal-to-noise ratio 15 dB, number of FFT points 1024, number of frequency shift data sets P 6, and Monte Carlo 100 times.

[0123] DOA estimates the angular distribution as Figure 3 As shown, the polarization angle is estimated as Figure 4 shown.

[0124] Experiment 2: Increase the signal-to-noise ratio of Experiment 1 from 0 to 20 dB in steps of 5 dB, and perform Monte Carlo analysis 1000 times. Figure 5 The radar signal DOA estimation accuracy is shown as a function of the signal-to-noise ratio.

[0125] All simulation results use the rms definition

[0126]

[0127] h represents the hth valid estimate, M c is the sequence length.

[0128] Experiment 3: Setting up a linear frequency modulation signal.

[0129] Linear frequency modulation signal: azimuth angle 50°, elevation angle 30°, polarization auxiliary angle 45°, polarization phase difference 90°.

[0130] The bandwidth is 100MHz, the frequency modulation time is 10μs, the sampling frequency is 500MHz, the FFT points are 1024, the signal-to-noise ratio is 15dB, the number of frequency shift data sets P is 6, and the Monte Carlo is repeated 1000 times. The DOA estimation accuracy varies with the number of frequency domain sampling points as shown below: Figure 6 As shown:

[0131] It can be found that the more frequency points used, the higher the accuracy.

[0132] Experiment 4: Set up two coherent incident signals, one for linear frequency modulation radar and one for decoy.

[0133] Radar signal: azimuth angle 30°, elevation angle 70°, polarization auxiliary angle 45°, polarization phase difference 90°.

[0134] Decoy signal 1: azimuth angle 140°, elevation angle 20°, polarization auxiliary angle 45°, polarization phase difference 60°, amplitude relative radar signal gain 1.1, fixed delay 1μs, initial phase difference 60°.

[0135] The two signals have the following parameters in common: bandwidth 100MHz, frequency modulation time width 10μs, sampling frequency 500MHz, FFT points 2048, signal-to-noise ratio 15dB, number of frequency shift data sets P 6, frequency domain sampling points 50, and Monte Carlo 100 times.

[0136] DOA estimates the angular distribution as Figure 7 As shown, the polarization angle is estimated as Figure 8 shown.

[0137] Experiment 5: Set up four coherent incident signals, one for linear frequency modulation radar and three for decoys.

[0138] Radar signal: azimuth angle 30°, elevation angle 70°, polarization auxiliary angle 45°, polarization phase difference 90°.

[0139] Decoy signal 1: azimuth angle 60°, elevation angle 30°, polarization auxiliary angle 45°, polarization phase difference 60°, amplitude relative radar signal gain 1.10, fixed delay 1μs, initial phase difference 60°.

[0140] Decoy signal 2: azimuth angle 130°, elevation angle 60°, polarization auxiliary angle 70°, polarization phase difference 80°, amplitude relative radar signal gain 1.15, fixed delay 2μs, initial phase difference 20°.

[0141] Decoy signal 3: azimuth 150°, elevation 20°, polarization auxiliary angle 20°, polarization phase difference 80°, amplitude relative radar signal gain 1.20, fixed delay 3μs, initial phase difference 30°.

[0142] The four signals have the following common parameters: bandwidth 100 MHz, frequency modulation time width 10 μs, sampling frequency 500 MHz, FFT points 2048, signal-to-noise ratio 15 dB, number of frequency shift data sets P 6, number of frequency domain sampling points 50, and Monte Carlo 100 times.

[0143] DOA estimates the angular distribution as Fig. 9 As shown, the polarization angle is estimated as Fig.10 shown.

[0144] It can be seen that the estimation method of the present invention can break through the limitation of array elements and can measure the number of coherent signals exceeding the number of antennas.

Claims

1. A broadband coherent signal source DOA estimation method based on three orthogonal dipoles, characterized in that: The steps include: S1: Array output matrix obtained using a single three-orthogonal dipole antenna; S2: Perform FFT on the output matrices of the three antennas to obtain the array frequency domain output matrix; S3: Determine the signal type according to the frequency domain output matrix of the array. If it is a frequency agile signal, extract the amplitude and phase peaks, wait for the incident signal to be frequency-converted, and repeat steps S1 to S3 to accumulate the frequency domain amplitude and phase peaks of multiple frequency points; If it is a linear frequency modulation signal, a certain range of amplitude and phase values ​​are taken from the spectrum; S4: rearrange the frequency domain amplitude and phase values ​​of multiple frequency points from small to large according to the frequency; S5: taking multiple frequency samples of the same length to form a frequency shift data set; S6: stack the frequency shift datasets to obtain a third-order tensor; S7: Use trilinear alternating least squares method to achieve third-order tensor decomposition to obtain the steering vector matrix; S8: Calculate the azimuth angle, elevation angle, polarization auxiliary angle, and polarization phase difference according to the steering vector matrix.

2. The method for estimating DOA of a broadband coherent signal source based on three orthogonal dipoles according to claim 1, characterized in that: The array output matrix in step S1 is expressed as follows: X n (t)=AS n (t)+N n (t) (1) Among them, S n (t) is the incident signal sampling matrix, N n (t) is the array element circular space-time-polarization white noise, and the incident signal is statistically independent of the noise; n = 1, 2, 3, ..., N, N is the total number of frequency points; A is the polarization-space steering vector matrix, expressed as: Where K is the total number of incident signals, and the azimuth and elevation angles of the kth signal are and θ k , the polarization auxiliary angle and polarization phase difference are γ k and η k , for 3. The method for estimating DOA of a broadband coherent signal source based on three orthogonal dipoles according to claim 2, characterized in that: The array frequency domain output matrix in step S2 is expressed as follows: X n (ω)=AS n (ω)+N n (oh) (4).

4. The method for estimating DOA of a broadband coherent signal source based on three orthogonal dipoles according to claim 1, characterized in that: The frequency conversion signal in step S3 is expressed as follows: Among them, f n is the signal frequency, which changes at equal intervals over time in a sequence, and T is the frequency modulation period.

5. The method for DOA estimation of a broadband coherent signal source based on three orthogonal dipoles according to claim 1, characterized in that: The linear frequency modulation signal in step S3 is expressed as follows: Where R is the linear frequency modulation slope, T is the frequency modulation period, and j is the imaginary unit.

6. The method for DOA estimation of a broadband coherent signal source based on three orthogonal dipoles according to claim 1, characterized in that: The complex numbers corresponding to the frequency domain peak values ​​of the multiple frequency points rearranged in step S4 are recorded as a matrix Z, which is expressed as: Z=[X1(ω1),X2(ω2),...,X n (oh n )],n=1,2,3,...,N (7) Among them, X n (ω n ) is the complex number corresponding to the frequency domain peak value of the nth frequency point.

7. The method for estimating DOA of a broadband coherent signal source based on three orthogonal dipoles according to claim 6, characterized in that: The method for constructing the frequency shift data set in step S5 is: there are N frequency samples in total, and at this time, starting from n=1, samples of (N-P+1) frequency points are sequentially taken to form P frequency shift data sets Z p ,p=1,2,...,P,Z p It is a 3×N-P+1 dimensional matrix.

8. The method for estimating DOA of a broadband coherent signal source based on three orthogonal dipoles according to claim 7, characterized in that: The expression of the third-order tensor in step S6 is as follows: The planes are stacked into a three-dimensional matrix.

9. The method for estimating DOA of a broadband coherent signal source based on three orthogonal dipoles according to claim 8, characterized in that: The expression of the third-order tensor decomposition by the trilinear alternating least squares method in step S7 is as follows: The trilinear alternating least squares method can decompose the three-dimensional matrix into three two-dimensional matrices, that is, decomposing G into The sizes of are 3×K, P×K, (N-P+1)×K respectively, where This is the estimate of the steering vector.

10. The method for estimating DOA of a broadband coherent signal source based on three orthogonal dipoles according to claim 9, characterized in that: The expression of calculating the estimated angle using the steering vector matrix in step S8 is as follows: Definition: || || represents the Frobenius norm, R[ ] represents the real part, I[ ] represents the imaginary part, and ∠{} represents the phase angle; First define Then, by decomposing Vector Estimated to be: Establish the following five real nonlinear equations: Combining equations (13) and (14) we can obtain Combining equations (10), (11), (12) and (15) and simplifying again yields Combining equations (12) and (16) we can obtain Combining equations (10), (11), (13), (14) and (15) and simplifying them again yields There are two estimation angles in total. When you finally get the result, you need to substitute it into the equation to determine which result it is.

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