Radar target angle measurement method under main lobe interference based on null-steering algorithm

Abstract

This invention discloses a radar target angle measurement method under main lobe interference based on a joint air-pole cancellation algorithm. The implementation steps are as follows: receiving the echo signal from a dual-polarized radar antenna array; estimating the autocorrelation matrix of the auxiliary antenna using the echo signal; performing multi-directional beamforming on the main antenna echo signal, calculating the cross-correlation matrix of the main and auxiliary antennas using the beamforming results, and calculating adaptive weights using the minimum mean square error criterion; calculating the joint air-pole cancellation output using the adaptive weights, and using the scanning angle corresponding to the maximum magnitude in the joint air-pole cancellation output as the target angle estimate. The angle measurement method designed in this invention significantly improves the accuracy of target angle estimation while maintaining low computational load under main lobe interference conditions, and has high engineering application value.

Description

Technical Field

[0001] This invention belongs to the field of radar technology, and more specifically relates to a radar target angle estimation method based on a joint air-pole cancellation algorithm under main lobe interference in the field of radar signal processing technology. This invention can be applied to radar systems operating under main lobe interference conditions. It utilizes the joint air-pole cancellation algorithm to adaptively suppress main lobe interference within the scanning angle range, and estimates the target angle by searching for the optimal value of the joint air-pole cancellation result corresponding to the scanning angle. Background Technology

[0002] Estimating the target angle of radar echo signals under main lobe interference is a major challenge in the radar field. Suppressing main lobe interference using only spatial filtering algorithms is not ideal. Polarization, as a crucial characteristic of electromagnetic waves, offers significant potential. If radar arrays can sensitively detect the polarization information of electromagnetic signals, and a joint spatial-polarization domain algorithm can be developed to estimate the target angle using the differences between the target and interference signals in the polarization domain, it would further improve and enhance the radar's target angle measurement performance.

[0003] In their paper "A Novel Anti-Interference Method Based on Spatial-Pole Joint Estimation" (Electronic Information Countermeasures Technology, 2021, 36(06), 18-22), Tang Jia et al. proposed a method to suppress interference signals by constructing a notch filter in the polarization domain using the angle information of the interference signal in the echo, and then estimating the target angle using the sum-difference single-pulse angle measurement method. The implementation process of this method is as follows: the covariance matrix of the echo signal is eigenvalued to obtain the noise subspace, the spectral peak search is performed on the noise subspace matrix to obtain the spatial-pole joint estimation information, the angle information of the interference signal is used to construct a notch filter in the polarization domain to suppress the main lobe interference of the echo signal, and then the sum-difference single-pulse angle measurement method is used to estimate the target angle. Although this method can complete the estimation of the target angle, it still has two shortcomings. First, since only the angle information of the interference signal is used when constructing the notch filter, the degree of freedom is limited. Therefore, when suppressing the main lobe interference, interference residue will be caused, resulting in poor interference cancellation performance. Secondly, the angle identification curve established when using sum-difference single pulse angle measurement does not utilize interference information, and the angle identification curve deviates from the actual situation, further resulting in poor target angle measurement accuracy and low reliability. Summary of the Invention

[0004] The purpose of this invention is to address the shortcomings of the existing technologies by proposing a radar target angle measurement method based on the air-pole joint cancellation algorithm under main lobe interference. This method aims to solve the problems in the existing technologies where interference remains after the main lobe interference is suppressed, resulting in poor radar interference cancellation performance, and poor target angle estimation accuracy due to the fact that the construction of the angle discrimination curve does not consider the main lobe interference information during the target angle measurement process.

[0005] To achieve the above objectives, the present invention employs an adaptive cancellation algorithm in the spatial domain, incorporating the polarization characteristics of the target signal and interference signal in the polarization domain, to perform spatial-polar joint cancellation of the echo signal within the scanning angle range. This suppresses interference signals by finding the optimal value of the spatial-polar joint cancellation result and using the corresponding scanning angle as the target angle estimate. This addresses the problems of residual interference after suppressing main lobe interference and the further reduction in target angle estimation accuracy caused by the lack of consideration for main lobe interference information when constructing the angle discrimination curve for target angle estimation. The rationale is that, under main lobe interference conditions, the angle difference between the target signal and interference signal is very small, making it difficult to accurately extract the target angle in the spatial domain. Therefore, the spatial-polarization domain joint cancellation algorithm, constructed using the difference between the target signal and interference signal in the polarization domain, can adaptively suppress interference signals, offering higher degrees of freedom and more complete interference suppression. Furthermore, the use of beam scanning angle information during spatial-polar joint cancellation avoids the need for establishing an angle discrimination curve, overcoming the problems of poor interference suppression performance and low angle estimation accuracy under main lobe interference conditions.

[0006] To achieve the above objectives, the technical solution adopted by the invention includes the following steps:

[0007] Step 1: Receive the echo signal from the radar dual-polarized antenna array;

[0008] Step 2: Estimate the autocorrelation matrix of the auxiliary antenna using the echo signal;

[0009] Step 3: Perform beamforming on the echo signal from the main antenna in multiple directions, calculate the cross-correlation matrix of the main and auxiliary antennas based on the beamforming results, and calculate the adaptive weights using the minimum mean square error criterion.

[0010] Step 4: Calculate the output of the empty pole joint cancellation using adaptive weights, and take the scanning angle corresponding to the maximum modulus value in the output of the empty pole joint cancellation as the angle estimate of the target.

[0011] Compared with the prior art, the present invention has the following advantages:

[0012] First, because this invention incorporates the information of the target signal and the interference signal in the polarization domain into the adaptive cancellation algorithm in the spatial domain to perform joint spatial-polarity cancellation, it adaptively suppresses the main lobe interference by utilizing the differences between the target signal and the interference signal in the polarization domain. This overcomes the shortcomings of existing technologies, which have limited degrees of freedom due to only utilizing the angle information of the interference signal, resulting in residual interference after interference cancellation. This allows the invention to achieve better interference suppression performance, improve the signal-to-interference-plus-noise ratio of the target, and enhance the accuracy of target angle estimation.

[0013] Secondly, since the present invention utilizes scanning angle information when performing air-pole joint cancellation, it finds the optimal interference suppression result within the scanning angle range and uses the angle corresponding to the optimal interference suppression result as the target angle estimate. Therefore, it avoids the disadvantage of low angle measurement accuracy caused by the lack of interference information when establishing the angle discrimination curve in the prior art. This makes the present invention improve the reliability of target angle estimation while reducing the time cost of the target angle measurement process, and enhances the application value of the present invention in engineering. Attached Figure Description

[0014] Figure 1 This is a flowchart of the present invention;

[0015] Figure 2 This is a comparison chart of the results of the present invention and the prior art in canceling the main lobe interference of the echo signal in simulation experiment 1 of the present invention;

[0016] Figure 3 This is a comparison chart of the root mean square error of angle estimation between the present invention and the prior art within the range of detection signal-to-noise ratio variation in simulation experiment 2 of the present invention;

[0017] Figure 4 This is a comparison chart of the root mean square error of angle estimation between the present invention and the prior art within the range of noise-to-interference ratio variation in simulation experiment 3 of the present invention. Detailed Implementation

[0018] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0019] Reference Figure 1 The implementation steps of the embodiments of the present invention will be described in further detail below.

[0020] Step 1: Receive the echo signal from the radar dual-polarized antenna array;

[0021] The expression for the echo signal is as follows:

[0022]

[0023] Where X represents the echo signal of size M×L, M represents the total number of array elements in the main antenna and auxiliary antenna, M=N1+N2, N1 represents the number of array elements in the main antenna, N2 represents the number of array elements in the auxiliary antenna, and L represents the total number of snapshots of the signal received by the dual-polarized antenna array. H This represents the echo signal from the main antenna, N1×L. X represents the main antenna echo signal of the l-th snapshot. V This represents the echo signal from the N2×L auxiliary antenna. This represents the echo signal from the auxiliary antenna in the l-th snapshot. The polarization of the main antenna and the auxiliary antenna are orthogonal. αTH a represents the initial value of the target echo relative to the complex envelope of the main antenna. H (θ t ) indicates that the target direction of the main antenna is θ. t The target guidance vector, α TV a represents the initial value of the target echo relative to the complex envelope of the auxiliary antenna. V (θ t ) indicates that the target direction of the auxiliary antenna is θ. t The target guidance vector, s t Let s represent the complex envelope vector of the target. t =[γ1,γ2,...,γ r ,...,γ L ],1≤r≤L,γ r Let s represent the complex envelope of the target, where s's value indicates that the target exists in the r-th snapshot. t The remaining L-1 elements are all 0, indicating that none of these snapshots contain a target. JH a represents the initial value of the main lobe interference relative to the complex envelope of the main antenna. H (θ i The main antenna main lobe interference angle is θ. i The main lobe interference steering vector, α JV a represents the initial value of the main lobe interference of the auxiliary antenna relative to the complex envelope. V (θ i The main lobe interference angle of the auxiliary antenna is θ. i The main lobe interference steering vector, Let represent the complex envelope of the g-th snapshot of the main lobe interference, n represent the E×S noise matrix, E represents the number of rows in the noise matrix n, which is equal to the total number of elements in the radar dual-polarized antenna array, S represents the number of columns in the noise matrix n, which is equal to the total number of snapshots of the echo signal received by the array, j represents the imaginary unit sign, d represents the element spacing, λ represents the wavelength, [] T This indicates the transpose operation.

[0024] The radar auxiliary antenna is added around the main antenna or in the same position as the main antenna, and the radar antenna array can be of any configuration. In this embodiment of the invention, a dual-polarized antenna uniform linear array is selected.

[0025] Step 2: Estimate the autocorrelation matrix of the auxiliary antenna using the echo signal:

[0026]

[0027] in, Let X represent the N2×N2 auxiliary antenna autocorrelation matrix, Q represent the N2×u auxiliary antenna echo selection matrix, N2 represent the number of rows in matrix Q, which is equal to the total number of auxiliary antenna elements, and u represent the number of columns in matrix Q, which is equal to the number of columns in the auxiliary antenna echo signal X. V The number of snapshots to be selected from the total number of snapshots, max(N1,N2)≤u≤L, and the index of each selected snapshot is not equal to the index of the snapshot containing the target. max() represents the maximum value operation. [] H This indicates the conjugate transpose operation.

[0028] According to the minimum mean square error criterion, the adaptive weights need to be calculated using the autocorrelation matrix of the auxiliary antenna and the cross-correlation matrix of the main and auxiliary antennas. Therefore, the cross-correlation matrix of the main and auxiliary antennas is calculated in step 3, and then the adaptive weights are calculated using the autocorrelation matrix of the auxiliary antenna and the cross-correlation matrix of the main and auxiliary antennas.

[0029] Step 3: Perform beamforming on the echo signal from the main antenna in multiple directions, calculate the cross-correlation matrix of the main and auxiliary antennas based on the beamforming results, and calculate the adaptive weights using the minimum mean square error criterion.

[0030] The beamforming result of the main antenna is obtained by the following formula:

[0031] K = A H P

[0032] Where K represents the main antenna beamforming result, and A represents the steering matrix composed of F scanning beam steering vectors, A=[a H (θ1),a H (θ2),...,a H (θ f ),...,a H (θ F )], θ f The scanning angle is determined by the main lobe of the radar detection beam. The target angle must be included within the scanning angle range. P represents the N1×q main antenna echo selection matrix, where N1 represents the number of rows in matrix P, which is equal to the total number of array elements in the radar main antenna. q represents the number of columns in matrix P, which is equal to the number of columns in the main antenna received from the main antenna. H The number of snapshots to be selected from the total number of snapshots, max(N1,N2)≤q≤L, and the sequence number of each selected snapshot is not equal to the sequence number of the snapshot containing the target, q=u.

[0033] There are multiple ways to select F angle values ​​from the scanning angle range. In this embodiment of the invention, F angle values ​​are selected uniformly within the scanning angle range.

[0034] The cross-correlation vector of the main and auxiliary antennas is obtained by the following formula:

[0035]

[0036] in, This represents the N2×F cross-correlation matrix between the primary and secondary antennas.

[0037] The adaptive weights are obtained from the following formula:

[0038]

[0039] in, Let N2×F represent the adaptive weights, [] -1 This indicates the inverse operation.

[0040] Step 4: Calculate the output of the vacuous pole joint cancellation using adaptive weights, and take the scanning angle corresponding to the maximum modulus value in the vacuous pole joint cancellation output as the target angle estimate.

[0041]

[0042] in, This represents the estimated angle of the target. The main antenna echo signal, representing the number of snapshots (r) indicating the presence of the target. The target exists in the auxiliary antenna echo signal with a snapshot number of r, θ∈[θ1,θ2,...,θ F ], This represents the angle at which the modulus of the combined empty pole cancellation output is maximized within the scanning angle range, and |·| represents the modulus taking operation.

[0043] The effects of this invention will be further illustrated below with simulation experiments:

[0044] 1. Simulation experimental conditions.

[0045] The hardware platform for the simulation experiment of this invention is: an Intel i5-12500 CPU with a main frequency of 2.5GHz and 16GB of memory.

[0046] The software platform for the simulation experiment of this invention is: Windows 11 operating system and MATLAB R2022b.

[0047] 2. Simulation Content and Result Analysis

[0048] The simulation experiments of this invention use the present invention and existing technologies to estimate the angle of radar echo signals under main lobe interference and without main lobe interference. There are a total of three simulation experiments.

[0049] Simulation Experiment 1 compares the results of the present invention and the prior art in canceling the main lobe interference of the echo signal. Simulation Experiment 2 compares the root mean square error of angle estimation between the present invention and the prior art within the range of signal-to-noise ratio variation. Simulation Experiment 3 compares the root mean square error of angle estimation between the present invention and the prior art within the range of interference-to-noise ratio variation.

[0050] The existing technology refers to the method proposed by Tang Jia et al. in their published paper "A new anti-interference method based on joint estimation of space poles" (Electronic Information Countermeasures Technology, 2021, 36(06), 18-22).

[0051] Simulation Experiment 1 is a simulation of the main lobe interference cancellation results of the echo signal by the present invention and existing technologies.

[0052] The simulation experiment 1 of this invention uses radar with the following array elements: main antenna N1 = 20, auxiliary antenna N2 = 20, interference-to-noise ratio (INR) = 30 dB, number of snapshots L = 1000, number of sampling points u = 800, wavelength λ = 1 m, main antenna element spacing d1 = 0.5 m, auxiliary antenna element spacing d2 = 0.5 m, detection signal-to-noise ratio (SNR) SNR = 20 dB, and target angle θ. t =0.5°, the target's snapshot number t=100, the target's polarization phase descriptor Interference signal angle θ i =2o, the polarization phase descriptor of the disturbance

[0053] Simulation Experiment 1 of this invention plots the results of main lobe interference cancellation of echo signals using the present invention and existing technologies, as well as the results of beamforming echo signals (i.e., without interference suppression), as shown in the figure below. Figure 2 The three curves shown.

[0054] Simulation Experiment 2 is a simulation of the root mean square error of angle estimation within the range of signal-to-noise ratio variation of the present invention and existing technology.

[0055] The simulation experiment 2 of this invention uses radar with the following array elements: main antenna N1 = 20, auxiliary antenna N2 = 20, interference-to-noise ratio (INR) = 30 dB, number of snapshots L = 1000, number of sampling points u = 800, wavelength λ = 1 m, main antenna element spacing d1 = 0.5 m, auxiliary antenna element spacing d2 = 0.5 m, detection signal-to-noise ratio range SNR = [10, 40] dB, and target angle θ. t =0.5°, the target's snapshot number t=100, the target's polarization phase descriptor Interference signal angle θ i =2o, the polarization phase descriptor of the disturbance Target polarization angle scan range The Monte Carlo experiment was conducted 1000 times.

[0056] Simulation Experiment 2 of this invention plots the root mean square error of angle estimation for the detection signal-to-noise ratio (SNR) range of SNR∈[10,40]dB under main lobe interference and without interference, as shown in the figure. Figure 3 The three curves shown.

[0057] Simulation Experiment 3 simulates the root mean square error of angle estimation within the range of interference-to-noise ratio (INR) variation of the present invention and existing technologies. The parameters used in Simulation Experiment 3 are the same as those in Simulation Experiment 2, with a detection signal-to-noise ratio (SNR) of 20 dB and an INR range of [10, 40] dB.

[0058] Simulation Experiment 3 of this invention plots the root mean square error of angle estimation for the interference-to-noise ratio (IR) range of INR∈[10,40]dB for both the present invention and existing technologies under main lobe interference, but without interference, as shown in the figure. Figure 4 The three curves shown.

[0059] The effects of the present invention will be further described below with reference to simulation diagrams.

[0060] Figure 2 The horizontal axis represents snapshot time, and the vertical axis represents power. Figure 2 The solid lines represent the results of existing methods in canceling main lobe interference, the dotted lines represent the results of beamforming the echo signal, and the dashed lines represent the results of the present invention in canceling main lobe interference.

[0061] from Figure 2 As can be seen, compared with the beamforming results, both existing methods and the method of this invention have the effect of suppressing main lobe interference. Both methods can detect the target peak at the snapshot number 100 where the target is located. However, the interference suppression effect of the method of this invention is better than that of the existing methods. Figure 2 It is evident that the method of this invention exhibits better interference suppression performance under main lobe interference conditions.

[0062] Figure 3 The horizontal axis represents the detection signal-to-noise ratio, and the vertical axis represents the root mean square error. Figure 3 The solid line represents the result of the root mean square error of the existing method changing with the detection signal-to-noise ratio under main lobe interference; the dotted line represents the result of the root mean square error of the present invention changing with the detection signal-to-noise ratio under main lobe interference; and the dashed line represents the result of the root mean square error of the present invention changing with the detection signal-to-noise ratio when there is no interference.

[0063] from Figure 3It can be seen that the root mean square error (RMSE) of angle estimation in both the present invention and existing methods decreases with increasing detection signal-to-noise ratio (SNR). However, the present invention's method exhibits a significantly smaller RMSE for angle estimation under main lobe interference compared to existing methods. Furthermore, as the SNR increases, the RMSE of angle estimation in the present invention under main lobe interference tends to converge with the RMSE of angle estimation in the absence of main lobe interference (i.e., the ideal case). Figure 3 As can be seen, under main lobe interference, the angle estimation accuracy and reliability of the method of the present invention are higher, and it has obvious advantages.

[0064] Figure 4 The horizontal axis represents the interference-to-noise ratio, and the vertical axis represents the root mean square error. Figure 4 The solid line represents the result of the root mean square error of the existing method changing with the detection signal-to-noise ratio under main lobe interference; the dotted line represents the result of the root mean square error of the present invention changing with the detection signal-to-noise ratio under main lobe interference; and the dashed line represents the result of the root mean square error of the present invention changing with the detection signal-to-noise ratio when there is no interference.

[0065] from Figure 4 It can be seen that the root mean square error of angle estimation by the method of the present invention remains basically consistent with that under main lobe interference and without interference, while the root mean square error of angle estimation by the existing method is higher under main lobe interference. Figure 4 It is evident that the method of the present invention can maintain high angle estimation accuracy and more stable performance when the noise-to-interference ratio changes.

[0066] The simulation experiments above demonstrate that the radar target angle measurement method based on the spatial-polarity joint cancellation algorithm provided in this invention, under main lobe interference, constructs the spatial-polarity joint cancellation algorithm by introducing the polarization characteristics of the target signal and the interference signal in the polarization domain into the adaptive cancellation algorithm in the spatial domain. This algorithm adaptively suppresses main lobe interference within the scanning angle range, and the scanning angle corresponding to the optimal value of the spatial-polarity joint cancellation result is used as the target angle estimate. This invention significantly improves the accuracy of target angle estimation while maintaining low computational load under main lobe interference conditions, and has high practical value in engineering.

Claims

1. A radar target angle measurement method under main lobe interference based on a space-pole joint cancellation algorithm, characterized in that, The main lobe interference is adaptively suppressed using a combined spatial-pole cancellation algorithm. The target angle is estimated by searching for the scanning angle corresponding to the optimal value of the combined spatial-pole cancellation result. The steps of this angle measurement method are as follows: Step 1: Receive the echo signal from the radar dual-polarized antenna array; Step 2: Estimate the autocorrelation matrix of the auxiliary antenna using the echo signal; Step 3: Perform beamforming on the echo signal from the main antenna in multiple directions, calculate the cross-correlation matrix of the main and auxiliary antennas based on the beamforming results, and calculate the adaptive weights using the minimum mean square error criterion. Step 4: Calculate the output of the empty pole joint cancellation using adaptive weights, and take the scanning angle corresponding to the maximum modulus value in the output of the empty pole joint cancellation as the angle estimate of the target.

2. The radar target angle measurement method under main lobe interference based on the air-pole joint cancellation algorithm according to claim 1, characterized in that, The expression for the echo signal mentioned in step 1 is as follows: ； in, express echo signal, This indicates the total number of array elements for the main antenna and auxiliary antennas. , This indicates the number of array elements in the main antenna. This indicates the number of elements in the auxiliary antenna array. This indicates the total number of snapshots of the signal received by the dual-polarized antenna array. express The main antenna echo signal, , Indicates the first The main antenna echo signal of the second snapshot express The auxiliary antenna echo signal, , Indicates the first The echo signal from the auxiliary antenna in the second snapshot has orthogonal polarization between the main antenna and the auxiliary antenna. This represents the initial value of the target echo relative to the complex envelope of the main antenna. Indicates the target direction of the main antenna is The target guidance vector, This represents the initial value of the target echo relative to the complex envelope of the auxiliary antenna. Indicates the target direction of the auxiliary antenna. The target guidance vector, The complex envelope vector of the target. , , The complex envelope of the target is represented by a value indicating that the target exists in the first position. A quick snapshot, The rest If all elements have a value of 0, it means that none of these snapshots contain a target. This represents the initial value of the main lobe interference relative to the complex envelope of the main antenna. The main antenna main lobe interference angle is indicated as... The main lobe interference steering vector, This represents the initial value of the main lobe interference relative to the complex envelope of the auxiliary antenna. The auxiliary antenna main lobe interference angle is indicated as... The main lobe interference steering vector, , , The first part representing the main lobe interference The complex envelope of the next quick shot express The noise matrix, Representing the noise matrix The number of rows is equal to the total number of elements in the radar dual-polarized antenna array. Representing the noise matrix The number of columns is equal to the total number of snapshots in the array that receive echo signals. The symbol representing the imaginary unit. Indicates the spacing between array elements. Indicates wavelength. This indicates the transpose operation.

3. The radar target angle measurement method under main lobe interference based on the space-pole joint cancellation algorithm according to claim 2, characterized in that, The estimation of the auxiliary antenna autocorrelation matrix in step 2 is achieved by the following formula: ； in, express The autocorrelation matrix of the auxiliary antenna, express The auxiliary antenna echo selection matrix, Representation matrix The number of rows is equal to the total number of auxiliary antenna elements. Representation matrix The number of columns, whose value is equal to the echo signal from the auxiliary antenna. The number of times selected from the total number of snapshots. Furthermore, the sequence number of each selected snapshot is not equal to the sequence number of the target snapshot. This indicates the operation of retrieving the maximum value. This indicates the conjugate transpose operation.

4. The radar target angle measurement method under main lobe interference based on the space-pole joint cancellation algorithm according to claim 3, characterized in that, The beamforming result of the main antenna in step 3 is obtained by the following formula: ； in, This indicates the beamforming result of the main antenna. Indicates by The guidance matrix is ​​composed of the guidance vectors of the scanning beams. , This represents the scanning angle determined by the main lobe of the radar detection beam; the target angle must be included within the scanning angle range. express The main antenna echo selection matrix, Representation matrix The number of rows is equal to the total number of elements in the radar's main antenna. Representation matrix The number of columns, whose value is equal to the signal received from the main antenna. The number of times selected from the total number of snapshots. Furthermore, the sequence number of each selected snapshot is not equal to the sequence number of the target snapshot. .

5. The radar target angle measurement method under main lobe interference based on the space-pole joint cancellation algorithm according to claim 4, characterized in that, The cross-correlation vector of the primary and secondary antennas mentioned in step 3 is obtained by the following formula: ； in, express The cross-correlation matrix of the primary and secondary antennas.

6. The radar target angle measurement method under main lobe interference based on the space-pole joint cancellation algorithm according to claim 5, characterized in that, The adaptive weights mentioned in step 3 are obtained by the following formula: ； in, express Adaptive weights, This indicates the inverse operation.

7. The radar target angle measurement method under main lobe interference based on the space-pole joint cancellation algorithm according to claim 6, characterized in that, The target angle estimate in step 4 is obtained by the following formula: ； in, This represents the estimated angle of the target. The number of snapshots indicating the existence of the target is . The main antenna echo signal, The number of snapshots containing the target is The auxiliary antenna echo signal, , This represents the angle at which the magnitude of the output result of the empty pole joint cancellation is maximized within the scanning angle range. This indicates a modulo operation.

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