CN110426670B - Super-resolution DOA estimation method for external radiation source radar based on TLS-CS - Google Patents

Abstract

The invention discloses a TLS-CS-based super-resolution DOA estimation method for an external radiation source radar. The method mainly solves the problem that compressed sensing super-resolution DOA estimation angle measurement precision and target resolution are reduced due to array amplitude-phase errors in the prior art. It includes: obtaining echo received by an array antenna and direct wave received by a reference antenna; utilizing the direct wave and the multipath interference signal in the delay suppression echo of the direct wave to perform distance-Doppler two-dimensional correlation processing on the echo signal after clutter suppression to obtain a complex vector signal S; then adding amplitude-phase disturbance to S to obtain complex vector signal with amplitude-phase error

Constructing a guide vector D of the whole observation space, and solving the guide vector D to obtain a guide vector after correcting the amplitude-phase error

By using

And

and carrying out sparse reconstruction on the azimuth information of the plurality of targets to obtain the azimuth of the targets. The invention reduces the influence of array amplitude-phase errors on the steering vectors, improves the angle measurement precision and the resolution performance of the target, and can be used for positioning the target.

Description

Super-resolution DOA estimation method for external radiator radar based on TLS-CS

technical field

The invention belongs to the technical field of radar signal processing, and in particular relates to an external radiation source radar super-resolution DOA estimation method, which can be used for target positioning.

Background technique

External radiation source radar refers to a radar system that does not actively emit electromagnetic waves and relies on the existing third-party non-cooperative radiation source signals in the target reflection environment, such as FM radio, TV signals, and mobile phone signals to detect, locate and track the radar system. The radiation source signal used by this system of radar has a low frequency band, faces the ground, and has the ability to stealth and detect low-altitude targets, so it has received extensive attention.

In an external radiator radar system, DOA estimation of the direction of arrival of the array signal is a very important link in the target positioning process. Usually, the energy of the echo signal reflected from the target received by the array antenna is much lower than the strong direct wave from the radiation source and the multi-path clutter and noise signal reflected by the ground and buildings, so it is difficult to achieve direct direction finding of the target. In order to estimate the DOA of the target in the external radiator radar, the clutter cancellation algorithm is used to suppress the strong direct wave and multipath clutter in the echo signal received by the array antenna; The processing improves the signal-to-noise ratio of the received target echo signal; finally, the azimuth information of multiple targets is sparsely reconstructed by compressed sensing on the range-Doppler unit where the target is located to achieve super-resolution DOA estimation. However, using compressed sensing can achieve good performance only if the array is free of errors. In the actual external radiation source radar system, the amplitude and phase gains of each channel of the array are usually inconsistent, that is, the array has amplitude and phase errors, which will cause steering vector mismatch. In the super-resolution DOA estimation method based on compressive sensing, the sensing matrix is composed of Steering vector composition, the mismatch of the steering vector will cause the super-resolution performance to drop rapidly.

To sum up, when there are amplitude and phase errors in the array, the angle measurement accuracy and target resolution performance of the above existing methods drop sharply, and the super-resolution DOA estimation of the target cannot be effectively achieved.

SUMMARY OF THE INVENTION

The purpose of the present invention is to propose a method for super-resolution DOA estimation of external radiation source radar based on total least squares-compressed sensing TLS-CS in view of the above-mentioned deficiencies of the prior art, so as to improve the target super-resolution DOA estimation accuracy. The angle measurement accuracy and target resolution performance can effectively realize the super-resolution DOA estimation of the target.

The idea of realizing the purpose of the present invention is to solve the overall least squares TLS signal model with amplitude and phase errors by a singular value decomposition method, obtain a steering vector after correcting the amplitude and phase errors, use the corrected steering vector as a perception matrix, and use greedy iteration. The tracking and matching algorithm sparsely reconstructs the orientation information of the target by compressed sensing, and realizes super-resolution DOA estimation.

According to the above-mentioned ideas, the implementation scheme of the present invention includes the following:

(1) respectively obtain the echo signal S _ech received by the array antenna and the direct wave signal S _ref in the direction of the radiation source received by the reference antenna;

(2) Utilize the reference antenna to receive the direct wave signal in the direction of the radiation source, adopt the extended cancellation algorithm to suppress the direct wave and multipath interference in the echo signal, and obtain the echo signal S _sur after clutter suppression;

(3) Perform range-Doppler two-dimensional correlation processing on the clutter-suppressed echo signal S _sur to obtain a complex vector signal S;

S=A*S _tar +Z <1>

_Star represents the target echo signal, A represents the steering vector corresponding to the target, and Z represents the noise signal;

为存在幅相误差的复矢量信号：(4) Add the amplitude and phase disturbance parameter ΔA to the complex vector signal S to obtain the TLS signal model with amplitude and phase errors,

is a complex vector signal with amplitude and phase errors:

(5) Set the number of rows of the steering vector of the entire observation space as the number of array elements M, and the number of columns as the number of divisions of the observation space N, and construct an M×N-dimensional matrix as the steering vector D of the observation interval;

中，并利用奇异值分解方法对其进行求解，得到修正幅相误差后的导向矢量

(6) Take the steering vector D of the entire observation interval as the steering vector with amplitude and phase disturbance, and substitute it into the complex vector signal

, and use the singular value decomposition method to solve it, and get the steering vector after correcting the amplitude and phase errors

作为测量矢量，修正幅相误差后的导向矢量

作为感知矩阵，利用贪恋迭代追踪算法对处于同一距离-多普勒单元的多个目标的方位信息进行压缩感知稀疏重构，得到多个目标的方位信息。(7) Convert the complex vector signal

As the measurement vector, the steering vector after correcting the amplitude and phase errors

As a perception matrix, the orientation information of multiple targets in the same range-Doppler unit is sparsely reconstructed by compressive sensing using the greedy iterative tracking algorithm, and the orientation information of multiple targets is obtained.

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

1. In the present invention, since the influence of the amplitude and phase error on the echo signal is considered, the amplitude and phase error is added to the echo signal model, and the singular value decomposition method is used to solve and correct the amplitude and phase error, and then the greedy iterative matching and tracking algorithm is used to detect multiple targets. The azimuth information of the target is sparsely reconstructed to obtain the azimuth of the target, which improves the angular measurement accuracy and resolution performance of the target.

2. The present invention only needs an array antenna for receiving echo signals and a reference antenna for receiving direct wave signals, no additional calibration antenna is required, and the implementation is simple.

Description of drawings

In order to explain the embodiments of the present invention or the technical solutions in the prior art more clearly, the following briefly introduces the accompanying drawings that need to be used in the description of the embodiments or the prior art. Obviously, the accompanying drawings in the following description are only These are some embodiments of the present invention. For those of ordinary skill in the art, other drawings can also be obtained according to these drawings without creative efforts.

Figure 1 is a schematic diagram of the application scenario of the external radiation source radar;

Fig. 2 is the realization flow chart of the present invention;

Fig. 3 is the simulation experiment result diagram of traditional method and the inventive method when there is no amplitude-phase error;

FIG. 4 is a graph showing the simulation results of the traditional method and the method of the present invention when there is an amplitude-phase error.

Detailed ways

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, but not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

First of all, for ease of understanding, the following will be based on the application scenario of the external radiation source radar shown in Figure 1. The application scenarios of the external radiation source radar are introduced as follows:

As shown in Figure 1, the third-party non-cooperative external radiation source is placed in the far field of the radar receiving station of the external radiation source as a transmitting station to transmit electromagnetic wave signals, and the electromagnetic wave signals are reflected by the targets in the airspace covered by the array antenna. The signal is called the target echo signal, and the electromagnetic wave that directly irradiates the reference antenna without reflection is called the direct wave signal, also known as the reference signal. The external radiation source radar receives the target echo signal through the array antenna, receives the direct wave signal through the reference antenna, and then uses the radar signal processing algorithm to process the target echo signal and the direct wave signal, and then obtains the target distance, speed and azimuth information. It can be seen from Figure 1 that in addition to receiving the target echo signal, the radar array antenna of the external radiation source will inevitably receive the strong direct wave signal from the direction of the transmitting station and the multi-path interference signal reflected by different obstacles. The energy of the target echo signal is much lower than the direct wave and multipath interference signal, so it is necessary to use the direct wave received by the reference antenna to eliminate the direct wave and multipath interference signal received by the array antenna and improve the target echo through range-Doppler processing. energy of.

Referring to Fig. 2, an embodiment of the present invention provides an external radiation source radar super-resolution DOA estimation method based on overall least squares-compressed sensing, and its implementation steps include the following:

Step 1: respectively obtain the echo signal S _ech received by the array antenna and the direct wave signal S _ref in the direction of the radiation source received by the reference antenna;

Suppose that the array antenna is composed of a uniform linear array with an array element spacing of half wavelength, the number of array elements is M, M≥11, and receives M echo signals S _ech , wherein each echo signal S _ech contains a strong straight line from the direction of the transmitting station. Arrival waves, multipath interference signals and noise signals reflected by multipath;

It is assumed that the reference antenna consists of a narrow-beam antenna that points to the direction of the transmitting station alone, and receives the direct wave signal S _ref from the direction of the radiation source.

Step 2: Use the reference antenna to receive the direct wave signal S _ref in the direction of the radiation source, and use the extended cancellation algorithm to suppress the direct wave and multipath interference signals in the echo signal S _ech to obtain the echo signal S after clutter suppression _sur .

Since the echo signal S _ref received by the array antenna includes not only the echo signal reflected by the target, but also the direct wave signal from the radiation source and the multipath clutter signal reflected by buildings and roads, the energy of which is far greater than that of the target echo signal. The target echo is submerged in the clutter signal, and the target cannot be detected, so the clutter signal must be suppressed. By projecting the echo signal onto the orthogonal subspace V composed of the direct wave signal S _ref and its time delay, the clutter signal will have a set of projection coefficients that are not all zero in the orthogonal subspace V, which represent According to the intensity of various clutter signals, the clutter existing in the echo signal can be eliminated by solving this group of projection coefficients that are not all zero, and the echo signal S _sur after clutter suppression can be obtained.

The specific implementation of this step is as follows:

(2a) Use the direct wave S _ref and its time delay to construct the clutter orthogonal subspace V:

Among them, G is the data length of the direct wave signal, C is the clutter cancellation order, the first column of the matrix represents the direct wave signal, the second column represents the multipath signal with one delay unit, and the Nth column represents the delay The multipath signal whose unit is C;

(2b) Project the echo signal S _ech into the clutter orthogonal subspace V, and solve the set of projection coefficients W that are not all zero on this subspace:

W=(V ^H *V) ^-1 V ^H S _ech , <3> where V ^H represents the conjugate transpose of the matrix V, and (V ^H *V) ^-1 represents the inversion of the matrix product result;

(2c) According to the echo signal S _ech , the clutter orthogonal subspace V and the non-zero projection coefficient W, the residual echo signal S _sur after clutter suppression is obtained:

S _sur =S _ech -V*W. <4>

Step 3: Perform range-Doppler two-dimensional correlation processing on the residual echo signal S _sur to obtain a complex vector signal S.

After clutter suppression, the clutter signal contained in the echo signal S _ech has been eliminated, but the energy of the target echo signal in the remaining echo signal S _sur is still lower than the noise signal, so the distance-Doppler two-dimensional The correlation processing increases the energy of the target echo signal, and obtains the signal model under ideal conditions. S is the complex vector signal obtained after range-Doppler two-dimensional correlation processing.

The specific implementation of this step is as follows:

(3a) Dot-multiply the residual echo signal S _sur after clutter suppression and the time-delayed conjugate reference signal S _ref ^* [g-τ] to obtain the complex vector signal S _m after range dimension correlation processing:

where τ represents the distance of the moving target, and G is the length of the residual echo signal S _sur ;

(3b) After performing Doppler-dimensional correlation accumulation on the complex vector signal S _m after range-dimensional correlation processing, the target signal S _tar after Doppler-dimensional correlation accumulation is obtained:

(3c) Add the steering vector A and the noise Z to the target signal S _tar to obtain the range-Doppler two-dimensional correlation processing complex vector signal S:

S=A* _Star +Z. <7>

为存在幅相误差的复矢量信号：。Step 4: Add the amplitude and phase disturbance parameter ΔA to the complex vector signal S to obtain the TLS signal model with amplitude and phase errors,

is a complex vector signal with amplitude and phase errors: .

为存在幅相误差的复矢量信号：The complex vector signal S in step 3 is derived under ideal conditions, but in the actual operation of the external radiation source radar, the complex vector signal S is inevitably affected by the amplitude and phase error, so the amplitude and phase error ΔA is added to the formula <7>. , get the TLS model,

is a complex vector signal with amplitude and phase errors:

Step 5: Construct the steering vector D for the entire observation space.

Let the number of rows of the steering vector of the entire observation space be the number of array elements M, and the number of columns to be the number of divisions of the observation space N, and construct an M×N-dimensional matrix as the steering vector D of the observation interval:

Among them, θ ₁ is the starting azimuth of the observation interval, θ _N is the end azimuth of the observation interval, d is the distance between the antenna elements, and λ is the wavelength of the emitted electromagnetic wave.

Step 6: Take the steering vector D of the entire observation interval as the steering vector with amplitude and phase disturbance, replace A+ΔA in the formula <6>, and solve it by the singular value decomposition method, and obtain the steering vector after correcting the amplitude and phase error.

The specific implementation of this step is as follows:

其中

为M×1维存在幅相误差的复矢量信号，D为M×N维修正幅相误差后的导向矢量；(6a) Construct M×(N+1) dimensional extended matrix

in

is the complex vector signal with amplitude and phase error in M×1 dimension, D is the steering vector after M×N repairing the positive amplitude and phase error;

(6b) Calculate the singular value decomposition of the extended matrix B:

B=UΣV ^H ,

where Σ is an M×MM×M dimensional diagonal matrix Σ=(diag(σ ₁ ,σ ₂ ,…,σ _M ),0), 0 is an M×(N-M+1) dimensional matrix, and its elements are is 0; diag(σ ₁ ,σ ₂ ,…,σ _M ) is an M×M-dimensional matrix whose main diagonal elements are σ ₁ ,σ ₂ ,…,σ _M , and V ^H represents the conjugate transpose of matrix V;

(6c) Find a mutation element σ _p from the main diagonal elements σ ₁ , σ ₂ ,…,σ _M , when σ _p satisfies σ _p >σ _M +ξ≥σ _p+1 ≥…≥σ _M , ξ= max[(σ ₁ -σ ₂ ),(σ ₂ -σ ₃ ),…,(σ _M-1 -σ _M )], using the subscript P as the effective rank order, construct (N+1)×(N +1) dimensional correction matrix: E=[I _p ,0] ^T , I _p is a p×p dimensional unit matrix, and 0 is a zero matrix;

(6d) Add the correction matrix E to formula <7>, and multiply the diagonal matrix Σ to the left to obtain the best approximation matrix of the augmented matrix B

进行拆分，将

的第2列至第N+1列作为修正幅相误差后的导向矢量

为M×N维矩阵。(6e) For the best approximation matrix

split,

Columns 2 to N+1 of , are used as the steering vector after correcting the amplitude and phase errors

is an M×N dimensional matrix.

作为测量矢量，修正幅相误差后的导向矢量

作为感知矩阵，利用贪恋迭代追踪算法对处于同一距离-多普勒单元的多个目标的方位信息进行压缩感知稀疏重构，得到多个目标的方位信息。Step 7: Convert the complex vector signal

As the measurement vector, the steering vector after correcting the amplitude and phase errors

As a perception matrix, the orientation information of multiple targets in the same range-Doppler unit is sparsely reconstructed by compressive sensing using the greedy iterative tracking algorithm, and the orientation information of multiple targets is obtained.

The specific implementation of this step is as follows:

作为初始输入，记作e₀；(7a) will measure the vector

As the initial input, denoted as e ₀ ;

中筛选与e₀内积绝对值最大的一列，并表示为

(7b) From the perception matrix

Filter the column with the largest absolute value of the inner product with e ₀ and express it as

(7c) According to the results of (7a) and (7b), calculate the residual value e ₁ :

表示e₀与

的内积；in

means e ₀ with

The inner product of ;

中与测量矢量

最相关的K个矢量

K的范围小于等于7，本实例取值为7；(7d) Use the residual value calculated in (7c) as the new input of (7a), and repeat (7b) and (7c) K times to obtain the perception matrix

Neutral and measure vector

Most relevant K vectors

The range of K is less than or equal to 7, and the value in this example is 7;

位于感知矩阵

中的第n列，则目标的方位θ为：(7e) When the correlation vector

in the perception matrix

In the nth column, the orientation θ of the target is:

θ=(nN/2)*(θ _N -θ ₁ )/N, n∈1,2,...,N

Among them, N is the total number of columns in the entire space, θ ₁ is the starting orientation of the observation interval, and θ _N is the ending orientation of the observation interval.

The effect of the present invention can be further illustrated by the following simulation experiments:

1. Experimental conditions:

In the experiment of the invention, the FM broadcast signal FM is used as the radiation source, the frequency is 93.1MHz, the bandwidth is 200kHz, the sampling rate is 200kHz, and the accumulation time is 1s; , and the array antenna receives two target echo signals located in the same range-Doppler unit. The performance of the method of the present invention is described below through two sets of simulation experiments, and the simulation parameters of the target are shown in Table 1.

Table 1. Simulation parameters of the target

2. Simulation content and results

Simulation 1: Under the condition that there is no amplitude and phase error in the array, the traditional method and the method of the present invention are used to perform compressive sensing sparse reconstruction simulation on the orientation information of the two targets, and the orientations of the two targets are obtained. The results are shown in Figure 3 ,in:

Figure 3(a) is the orientation information obtained by using the traditional compressed sensing super-resolution DOA estimation method;

Figure 3 (b) is the orientation information obtained by the method of the present invention;

It can be seen from Figure 3 that both the traditional method and the method of the present invention can effectively distinguish two targets, and the detected target orientation is [5°, 14°], which is consistent with the assumed target orientation.

Simulation 2, under the condition that the amplitude and phase error of the array is -30dB, the traditional method and the method of the present invention are used to carry out compressive sensing sparse reconstruction simulation on the orientation information of the two targets, and the orientations of the two targets are obtained. The results are shown in Figure 4. shown, including:

Figure 4(a) shows the azimuth information obtained by using the traditional compressed sensing super-resolution DOA estimation method when the amplitude and phase error of the array is -30dB;

Figure 4(b) is the azimuth information obtained by the method of the present invention when the amplitude and phase error of the array is -30dB;

It can be seen from Fig. 4 that when the amplitude and phase error of the array is -30dB, the azimuths of the two targets detected by the method of the present invention are respectively 5° and 14°, indicating that the method can be used when the array has amplitude and phase errors. Suppressing the influence of amplitude and phase errors on the steering vector, and effectively distinguishing the two targets, which are consistent with the assumed target direction, while using the traditional method to detect multiple false azimuth information, which are inconsistent with the assumed target azimuth information, The two targets cannot be effectively distinguished.

The above are only specific embodiments of the present invention, but the protection scope of the present invention is not limited thereto. Any person skilled in the art can easily think of changes or substitutions within the technical scope disclosed by the present invention. should be included within the protection scope of the present invention. Therefore, the protection scope of the present invention should be based on the protection scope of the claims.

Claims (5)

1. A TLS-CS-based super-resolution DOA estimation method for an external radiation source radar is characterized by comprising the following steps:

(1) respectively acquiring echo signals S received by array antennas _ech And a reference antenna for receiving the direct wave signal S in the radiation source direction _ref ；

(2) Utilizing a reference antenna to receive a direct wave signal in the radiation source direction, and adopting an extended cancellation algorithm to suppress the direct wave and multipath interference in an echo signal to obtain an echo signal S after clutter suppression _sur ；

(3) Echo signal S after clutter suppression _sur Performing distance-Doppler two-dimensional correlation processing to obtain a complex vector signal S;

S＝A*S _tar +Z <1>

S _tar representing a target echo signal, A representing a steering vector corresponding to a target, and Z representing a noise signal;

(4) adding the complex vector signal S into the amplitude-phase disturbance parameter delta A to obtain a TLS signal model with amplitude-phase errors,

for complex vector signals with amplitude and phase errors:

(5) setting the number of rows of a guide vector of the whole observation space as the number M of array elements and the number of columns as the number N of division of the observation space, and constructing an MxN dimensional matrix as a guide vector D of an observation interval;

(6) taking the guide vector D of the whole observation interval as a guide vector with amplitude-phase disturbance instead of the guide vector D<2>A + delta A of the formula, and solving by using a singular value decomposition method<2>After obtaining the corrected amplitude-phase errorGuide vector of

(7) Will complex vector signal

As a measurement vector, a guide vector after correcting an amplitude-phase error

As a sensing matrix, compressed sensing sparse reconstruction is performed on the orientation information of a plurality of targets in the same range-Doppler unit by using a greedy iterative tracking algorithm to obtain the orientation information of the plurality of targets, and the implementation is as follows:

(7a) will measure the vector

As an initial input, note e ₀ ；

(7b) From the perception matrix

Middle screening with e ₀ One column with the largest absolute value of inner product is shown as

(7c) Calculating residual value e according to the results of (7a) and (7b) ₁ ：

Wherein

Denotes e ₀ And

inner product of (d);

(7d) taking the residual value obtained by the calculation of (7c) as a new input of (7a), and repeatedly executing (7b) and (7c) for K times to obtain a perception matrix

Neutralization measurement vector

Most relevant K vectors

(7e) When the vector is correlated

Is located in the sensing matrix

Column n in (d), the azimuth θ of the target is:

θ＝(n-N/2)*(θ _N -θ ₁ )/N，n∈1,2,…,N

where N is the total number of columns in the entire space, θ ₁ To the starting position of the observation interval, θ _N Is the cut-off orientation of the observation interval.

2. The method of claim 1, wherein the step (2) is implemented as follows:

(2a) using direct waves S _ref And constructing a clutter orthogonal subspace V by time delay:

wherein G is the data length of the direct wave signal, C is the clutter cancellation order, the first column of the matrix represents the direct wave signal, the second column represents the multipath signal with the time delay unit of one, and the Nth column represents the multipath signal with the time delay unit of C;

(2b) echo signal S _ech Projecting into a clutter orthogonal subspace V, solving the set of projection coefficients W which are not all zero on the subspace:

W＝(V ^H *V) ^-1 V ^H S _ech ， <3>

wherein V ^H Represents a conjugate transpose of the matrix V; (V) ^H *V) ^-1 Expressing the inversion of the matrix product result;

(2c) according to the echo signal S _ech Obtaining a clutter suppressed residual echo signal S by using a clutter orthogonal subspace V and a projection coefficient W which is not completely zero _sur ：

S _sur ＝S _ech -V*W <4>。

3. The method of claim 1 wherein step (3) is practiced with respect to clutter suppressed echo signal S _sur And (3) performing distance-Doppler two-dimensional correlation processing, which is realized as follows:

(3a) the residual echo signal S after clutter suppression _sur With time-delayed conjugate reference signal S _ref ^* [g-τ]Point multiplication to obtain complex vector signal S after distance dimension correlation processing _m ：

Wherein τ represents the distance of the moving object, f _d Representing the Doppler induced by relative motion between the moving object and the array antenna, G being the residual echo signal S _sur Length of (d);

(3b) the complex vector signal S after the distance dimension correlation processing _m After Doppler dimension correlation accumulation is carried out, a target signal S after Doppler dimension correlation accumulation is obtained _tar ：

(3c) For the target signal S _tar Adding a guide vector A and a noise Z to obtain a distance-Doppler two-dimensional correlation processing complex vector signal S:

S＝A*S _tar +Z <7>。

4. the method of claim 1, wherein the observation space steering vector D in step (5),

is represented as follows:

wherein M is the number of array elements, N is the total column number of the whole space, and theta ₁ To the starting position of the observation interval, θ _N D is the distance between the antenna elements, and lambda is the wavelength of the emitted electromagnetic wave.

5. The method according to claim 1, characterized in that said step (6) is implemented as follows:

(6a) construction of M × (N +1) -dimensional extended matrix

Complex vector signals with amplitude-phase errors exist in MxN 1 dimensions, and D is a guide vector after the M xN maintenance of positive amplitude-phase errors;

(6b) calculating the singular value decomposition of the expansion matrix B:

B＝UΣV ^H ， <8>

wherein Σ is M × MM × M dimensional diagonal matrix ═ i (diag (σ) ₁ ,σ ₂ ,…,σ _M ) 0), 0 is an M × (N-M +1) -dimensional matrix, the elements of which are all 0; diag (sigma) ₁ ,σ ₂ ,…,σ _M ) Is that the main diagonal element is sigma ₁ ,σ ₂ ,…,σ _M M × M dimensional matrix of (a); v ^H Conjugate transitions of the representation matrix VPlacing;

(6c) from the principal diagonal element σ ₁ ,σ ₂ ,…,σ _M Search for a mutant element sigma _p When σ is _p Satisfy sigma _p ＞σ _M +ξ≥σ _p+1 ≥…≥σ _M When xi is max [ (sigma) ₁ -σ ₂ ),(σ ₂ -σ ₃ ),…,(σ _M-1 -σ _M )]Taking P as an effective rank order, constructing a (N +1) × (N +1) -dimensional correction matrix: e ═ I _p ,0] ^T ,I _p The matrix is a p multiplied by p dimension unit matrix, 0 is a zero matrix, and the elements of the zero matrix are 0;

(6d) adding the correction matrix E to the equation<8>Left-hand multiplication of the diagonal matrix sigma to obtain the optimal approximation matrix of the augmented matrix B

(6e) For the best approximation matrix

Carrying out the resolution of

As a guide vector after correcting the amplitude-phase error

Is an M × N dimensional matrix.

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