CN105158749A - High-frequency radar sea-clutter amplitude statistical distribution test method - Google Patents

Abstract

A high-frequency radar sea clutter amplitude statistical distribution inspection method belongs to the field of high-frequency radar signal processing. The existing methods for analyzing the statistical characteristics of sea clutter amplitude have high requirements for theoretical modeling, and it is difficult to quantitatively analyze the statistical distribution of parameterized results theoretically. A high-frequency radar sea clutter amplitude statistical distribution inspection method, using the SVD method to extract the main component of the sea clutter to be detected, and performing a segmented adaptive regression model modeling on the extracted sea clutter main component; using the obtained model Calculate the high-order cumulants of clutter principal components based on the order and segmented data; set the inspection threshold according to the confidence level; calculate the test statistics based on the calculation results of the high-order cumulants of the clutter principal components, and compare the test statistics with the inspection threshold ,get conclusion. The invention can directly test the statistical characteristics of the sea clutter amplitude under the high-frequency radar single propagation system and the mixed propagation system, and has strong applicability and application range.

Description

Test Method for Statistical Distribution of Sea Clutter Amplitude in High Frequency Radar

technical field

The invention relates to a method for inspecting the statistical distribution of the amplitude of high-frequency radar sea clutter.

Background technique

High-frequency (3-30MHz) radar works in the short-wave band, and utilizes the characteristics of small diffraction and propagation loss of vertically polarized electromagnetic waves along the coastal surface. The detection distance is not limited by the curvature of the earth, and can complete ultra-long-distance detection of targets. Sea clutter, as the main component of the ocean echo spectrum, constitutes the main clutter background for target detection. Theoretically, the sea clutter amplitude of shore-based high-frequency ground wave radar obeys Gaussian distribution, and this conclusion essentially determines the selection of most existing clutter suppression algorithms and the design of constant false alarm detectors.

However, with the rapid development of high-frequency radar in recent decades, its propagation mode (sky-wave transmission-ground wave reception, sky-wave transmission-shipboard platform reception), layout mode (dual/multi-base) and working mode (MIMO) are all different. With great changes, the statistical characteristics of sea clutter amplitude will also deviate from the theoretical distribution to varying degrees. In practical engineering, we need to obtain the statistical characteristic distribution of sea clutter in time, so as to change the corresponding signal processing methods. At present, in order to obtain the real amplitude statistical characteristic distribution of sea clutter under complex conditions, there are two main analysis methods: (1) Through the method of theoretical modeling, the influence of other factors is parametrically expressed in the analysis of sea clutter echo spectrum In the formula, the amplitude characteristics of the sea clutter echo in the actual detection environment can be described as much as possible in theory; (2) Statistical inspection is carried out directly according to the measured echo data sampled by the radar. Although the former method avoids the requirement of manpower and material resources for the acquisition of measured data, this method has higher requirements for theoretical modeling, especially in some complex cases, it is difficult to obtain accurate parametric representation results, and even if it can be obtained , the final echo signal will also be a multi-parameter high-composite expression form, and it is very difficult to quantitatively analyze its statistical distribution in theory for such a form. For the latter method based on measured data, since the length of data obtained by the radar in one accumulation period is often relatively large, the computational complexity required for the actual statistical inspection of clutter is relatively high. But even so, because this method mainly uses measured data, it essentially avoids the analysis of other complex disturbance parameters, and the analysis method is more flexible and stable.

Contents of the invention

The purpose of the present invention is to solve the problem that the existing sea clutter amplitude statistical characteristic analysis method has high requirements for theoretical modeling, and it is difficult to quantitatively analyze the statistical distribution of parameterized results theoretically, and proposes a high-frequency radar sea clutter Wave amplitude statistical distribution test method.

A high-frequency radar sea clutter amplitude statistical distribution inspection method, the high-frequency radar sea clutter amplitude statistical distribution inspection method is realized by the following steps:

Step 1. Perform distance and azimuth processing on the echo data collected by each array element of the antenna to obtain a three-dimensional data matrix of the echo to be detected, and complete the echo data preprocessing process;

Step 2: Use the SVD method to extract the principal components of the sea clutter to be detected, and perform segmented adaptive regression model modeling on the extracted principal components of the sea clutter to obtain the model order and the length of the segmented data; The SVD method is a singular value decomposition method; wherein, the SVD method is a singular value decomposition method;

Step 3, using the model order and segmented data calculated in step 2 to calculate the high-order cumulant of the principal component of the clutter;

Step 4, setting the inspection threshold according to the confidence level;

Step 5. Combining the calculation result of the high-order cumulant of the clutter principal component in step 3 to calculate the test statistic, and compare the test statistic with the test threshold, so as to determine the subsequent processing means of the echo data.

The beneficial effects of the present invention are:

The invention provides a method for detecting the statistical characteristics of sea clutter amplitude in a high-frequency radar system. The method can quantitatively give the difference between the statistical characteristics of sea clutter amplitude in multiple detection distance units and the theoretical Gaussian distribution under a certain confidence probability condition. , so as to provide reference for subsequent estimation of clutter covariance matrix, selection of clutter suppression method and construction of target detector. Mainly for the measured data after the distance compression processing, through the establishment of the segmented autoregressive model, the length of the data sample used for the statistical test of the clutter amplitude is greatly reduced, thereby reducing the amount of calculation to a large extent, and combined with high Finally, the rapid detection of the amplitude statistical characteristic distribution of Shanghai clutter in multiple range units is realized. A method for detecting the statistical characteristics of the sea clutter amplitude in a high-frequency radar system is provided. The method can quantitatively give the difference between the statistical characteristics of the sea clutter amplitude of multiple detection range units and the theoretical Gaussian distribution under a certain confidence probability condition, so that It provides reference for subsequent estimation of clutter covariance matrix, selection of clutter suppression method and construction of target detector.

Moreover, the validity verification of the first-order sea clutter extraction algorithm of the method of the present invention is shown in Fig. The proposed method can accurately determine the actual distribution range of the first-order sea clutter. Compared with the previous method of determining the distribution range of first-order sea clutter by fixing the clutter Doppler region, this method has strong adaptability. This method is still effective for complex phenomena such as spectral peak splitting and translation.

Description of drawings

Fig. 1 is the flowchart of the inventive method;

Fig. 2 is the Doppler spectrum of the echo data of a range unit after the measured data is preprocessed;

Fig. 3 is the distribution range of the first-order spectral peak extremum points determined after the secondary smoothing process;

Figure 4 shows the comparison between the spectral peak division results and the actual first-order sea clutter distribution range;

Figure 5 Validation of the first-order sea clutter extraction algorithm based on SVD;

Fig. 6 is a flow chart of the first-order sea clutter amplitude statistical characteristic inspection method;

Fig. 7 is the distance Doppler map of the measured sea clutter echo data on a certain beam unit;

Figure 8 shows the test results of multiple distance units based on the statistical characteristics of the first-order sea clutter amplitude based on the third-order cumulant;

Figure 9. The basic process of data preprocessing of the high-frequency radar system;

Detailed ways

Specific implementation mode one:

The high-frequency radar sea clutter amplitude statistical distribution inspection method of the present embodiment, in conjunction with the flowchart shown in Figure 1, the high-frequency radar sea clutter amplitude statistical distribution inspection method is realized by the following steps:

Step 1. Perform distance and azimuth processing on the echo data collected by each array element of the antenna to obtain a three-dimensional data matrix of the echo to be detected, and complete the echo data preprocessing process;

Step 2: Use the SVD method to extract the principal components of the sea clutter to be detected, and perform segmented adaptive regression model modeling on the extracted principal components of the sea clutter to obtain the model order and the length of the segmented data; The SVD method is a singular value decomposition method;

Step 3, using the model order and segmented data calculated in step 2 to calculate the high-order cumulant of the clutter principal component; wherein, the high-order cumulant includes a third-order cumulant and a fourth-order cumulant;

Step 4, setting the inspection threshold according to the confidence level;

Step 5. Combining the calculation result of the high-order cumulant of the clutter principal component in step 3 to calculate the test statistic, and compare the test statistic with the test threshold, so as to determine the subsequent processing means of the echo data.

Specific implementation mode two:

The difference from the first embodiment is that in the method for testing the statistical distribution of the amplitude of high-frequency radar sea clutter in this embodiment, the process of extracting the main component of the sea clutter to be detected by using the SVD method in step two is as follows:

Generally speaking, sea clutter components include first-order sea clutter components, second-order sea clutter components and higher-order sea clutter components, among which the first-order sea clutter components are dominant and have the greatest impact on the detection of slow targets such as ships. The first-order sea clutter component is determined as the main component of the sea clutter to be detected. According to this characteristic, the main component of the sea clutter to be detected is extracted by using the SVD method.

Step 21. For the preprocessed echo data z(n) and (1<n<N) in step 1, construct a matrix H with C columns and N-C+1 rows (where H is a Hankel matrix):

Among them, N represents the data length; C=3r, r represents the number of sinusoidal signals in the preprocessed echo data z(n), and the value of C directly affects the calculation amount of the SVD method and the effect of clutter extraction, through For the method of entropy spectrum analysis, C is generally selected as N/6, which can ensure a better extraction effect;

Step 22: Perform SVD decomposition on the matrix H to get:

H＝U·S·V(2)

Among them, U and V represent orthogonal or unitary matrices respectively, and satisfy U∈C ^{(N-C+1)×(N-C+1)} , V∈C ^C×C , C represents complex space; S represents singular matrix of values.

Specific implementation mode three:

The difference from the specific embodiment 1 or 2 is that the method for inspecting the statistical distribution of the amplitude of high-frequency radar sea clutter in this embodiment,

In Step 2, the extracted sea clutter principal component is modeled with a segmented adaptive regression model, and the process of obtaining the model order and segmented data length is as follows. According to the principle of SVD decomposition, each singular value corresponds to the preprocessed A principal component echo signal in the echo data z(n), through the Doppler distribution range of the first-order sea clutter to judge and identify the corresponding singular value of the first-order sea clutter, in the actual environment, the first-order sea clutter Sea clutter is easily affected by factors such as sea state, radar layout, and external electromagnetic environment, and the spectral peak will undergo complex changes such as broadening, splitting, and translation. In order to accurately determine the Doppler distribution range of the main component of the sea clutter to be detected, we The following steps are to be taken:

First, perform FFT transformation on the main component of the sea clutter to be detected to obtain the Doppler spectrum, and perform three-point smoothing processing on the obtained Doppler spectrum, which is equivalent to low-pass filtering processing to eliminate spectral peak splitting, glitches, etc. the impact of the phenomenon;

Second, take the logarithm of the smoothed Fourier spectrum, and perform differential processing of adjacent points to obtain the difference spectrum, and perform secondary smoothing on the obtained difference spectrum to further avoid the possibility of extreme value misjudgment caused by local spikes and burrs ;

Third, according to the local dominance of the Doppler domain of the main component of the sea clutter to be detected, the distribution range of the maximum and minimum points in the difference spectrum after the previous step processing is determined as the sea clutter Doppler distribution range of principal components;

Fourth, according to the division results of the above steps, determine the singular value range corresponding to the main component of sea clutter, and return to zero other singular values that are not in this range; and assume that the singular value matrix after setting zero is S ₁ , according to the singular value The value matrix S ₁ obtained after reconstruction is:

H ₁ =U ₁ ·S ₁ ·V ₁ (3)

According to formula (3), the time-domain echo data z ₁ (n) containing only the first-order sea clutter signal is obtained:

z ₁ (n)=(∑H _1(i,j) )/M(4)

In formula (4), i+j-1=n, M is the number of H1 _{(i, j)} meeting i+j-1=n;

Fifth, the time-domain echo data z ₁ (n) containing only the first-order sea clutter signal is modeled with an autoregressive model through the method of the segmented AR model, and the P-order AR model is obtained, and the P-order AR model is determined The minimum non-redundant delay domain is:

${\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; )</span>\stackrel{\mathrm{<span\; class="notranslate">\; \&Delta;</span>}}{<span\; class="notranslate">\; =</span>}<span\; class="notranslate">\; \{</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; ,</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; \&le;</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; ...</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

Modeling the AR model through the method of the segmented AR model reduces the size of the non-redundant delay domain on the one hand, and on the other hand, with the reduction of the data length, it also greatly reduces the amount of computation and improves the new method. detection efficiency;

Sixth, according to the above obtained time-domain echo data z ₁ (n) containing only the first-order sea clutter signal, combined with the non-redundant delay domain and model order determined in the previous step, further calculate the size of the high-order cumulant ( , a large number of experiments have shown that the highest calculation to the fourth-order cumulant can meet the test requirements)

Among them, the third-order cumulant calculation formula is:

$\begin{array}{cc}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}\underset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>& <span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Element;</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}^{\mathrm{<span\; class="notranslate">\; N</span>}}\end{array}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

The formula for calculating the fourth-order cumulant is:

$\begin{array}{c}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}\underset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span>\\ \begin{array}{cc}<span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>& <span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Element;</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{\mathrm{<span\; class="notranslate">\; N</span>}}\end{array}\end{array}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

in, Indicates the non-redundant delay domain, which is determined by the model parameter results in the previous step, for example And the second-order cumulant calculation formula in formula 7 is:

${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}\underset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span>$

Specific implementation mode four:

Different from the third specific embodiment, in the method for testing the statistical distribution of the amplitude of high-frequency radar sea clutter in this embodiment, the process of calculating the test statistic as described in step 5 and comparing the test statistic with the test threshold is as follows:

Step 51. According to the calculation result of the previous step, redefine the test statistic d _G as:

The formula for the test statistic based on a third-order cumulant is: $d_{G,3}\stackrel{\mathrm{\&Delta;}}{=}N{\left({\hat{c}}_3^{z_1}\right)}^T{\hat{S}}_{c3}^{-1}{\hat{c}}_3^{z_1}---\left(9\right)$

The formula for the test statistic based on the fourth-order cumulant is: $d_{G,4}\stackrel{\mathrm{\&Delta;}}{=}N{\left({\hat{c}}_4^{z_1}\right)}^T{\hat{S}}_{c4}^{-1}{\hat{c}}_4^{z_1}---\left(10\right)$

Combined with the judgment conclusion under the determined confidence probability, the calculation result of the high-order cumulant of the Gaussian variable is zero; where, is the high-order cumulant covariance matrix in the statistical sense; Represents the third-order cumulant covariance matrix estimate; Indicates the estimated value of the covariance matrix of the fourth-order cumulant; ( ) ^T indicates the matrix transposition operation;

Step 52. When the confidence level is α, obtain the χ ² distribution inspection threshold t _g by consulting the χ ² distribution table, and obtain the judgment conclusion under a certain confidence probability as follows:

${\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; G</span>}}\begin{array}{c}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\\ <span\; class="notranslate">\; ></span>\\ <span\; class="notranslate">\; <</span>\\ {\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\end{array}{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; g</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&chi;</span>}}_{{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 11</span>}<span\; class="notranslate">\; )</span>$

Step 53. Let H ₀ represent the event that the test result obeys the Gaussian distribution, and H ₁ represent the event that the test result does not obey the Gaussian distribution. Under the condition of H ₀ , the test statistic converges to the degree of freedom as N _c =3P(3P+1 )/2 chi-square distribution, namely: Further combined with the judgment threshold under the determined confidence probability, the judgment conclusion is obtained:

If the test statistic is less than the test threshold, it indicates that the main component of the clutter obeys the Gaussian distribution, then choose the method based on the second-order statistical information of the clutter to construct the covariance matrix, and complete the subsequent corresponding signal processing processes such as clutter suppression and target detection;

If the test statistic is greater than the test threshold, indicating that the main component of the clutter does not obey the Gaussian distribution, you should select higher-order clutter statistics and continue to complete the subsequent echo data processing;

So far, the inspection process of the statistical characteristics of the amplitude of the principal component of sea clutter is completed, and the third-order cumulant is taken as an example. The flow chart is shown in Figure 3.

Among them, the symbol χ2 represents chi ^- square distribution.

Specific implementation mode five:

The difference from specific implementation modes 1, 2 or 4 is that in the method for testing the statistical distribution of the amplitude of high-frequency radar sea clutter in this embodiment, the estimated value of the third-order cumulant covariance matrix and the fourth-order cumulant covariance matrix described in step 51 The calculation process of the matrix estimate is,

According to the measured data samples, the estimated value of the high-order cumulant covariance matrix based on the sample data is obtained, namely:

Estimated third-order cumulant covariance matrix:

$\begin{array}{c}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; S</span>}}}_{\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; \&ap;</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; P</span>}}}_{\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; \&ap;</span>\frac{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; R</span>}}\underset{\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; R</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}<span\; class="notranslate">\; \&lsqb;</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span><span\; class="notranslate">\; \&times;</span><span\; class="notranslate">\; \&lsqb;</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span>\end{array}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 12</span>}<span\; class="notranslate">\; )</span>$

and fourth-order cumulant covariance matrix estimates:

$\begin{array}{c}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; S</span>}}}_{\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; \&ap;</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; \&CenterDot;</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; P</span>}}}_{\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; \&ap;</span>\frac{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; R</span>}}\underset{\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; R</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}<span\; class="notranslate">\; \&lsqb;</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span><span\; class="notranslate">\; \&times;</span><span\; class="notranslate">\; \&lsqb;</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span>\end{array}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 13</span>}<span\; class="notranslate">\; )</span>$

in, ${\stackrel{\&OverBar;}{c}}_3^{z_1}(l_1,l_2)=(1/R){\mathrm{\&Sigma;}}_{r=1}^R{\hat{c}}_3^{z_1\left(r\right)}(l_1,l_2),{\stackrel{\&OverBar;}{c}}_4^{z_1}(l_1,l_2,l_3)=(1/R){\mathrm{\&Sigma;}}_{r=1}^R{\hat{c}}_4^{z_1\left(r\right)}(l_1,l_2,l_3);$ R represents the number of range gates corresponding to the echo data used.

Example 1:

Step 1. Perform distance and azimuth processing on the echo data collected by each array element of the antenna to obtain a three-dimensional data matrix of the echo to be detected, and complete the echo data preprocessing process;

Step 2, using the SVD method to extract the main component of the sea clutter to be detected, and performing a segmented adaptive regression model modeling on the extracted main component of the sea clutter to obtain the model order and the length of the segmented data;

Step 21. For the preprocessed echo data z(n) and (1<n<N) in step 1, construct a matrix H with C columns and N-C+1 rows (where H is a Hankel matrix):

Wherein, N represents the data length; C=3r, r represents the number of sinusoidal signals in the preprocessed echo data z(n), and the value of C is selected as N/6;

Step 22: Perform SVD decomposition on the matrix H to get:

H＝U·S·V(2)

Among them, U and V represent orthogonal or unitary matrices respectively, and satisfy U∈C ^{(N-C+1)×(N-C+1)} , V∈C ^C×C , C represents complex space; S represents singular matrix of values.

First, perform FFT transformation on the main component of the sea clutter to be detected to obtain the Doppler spectrum, and perform 3-point smoothing on the obtained Doppler spectrum;

Second, take the logarithm of the smoothed Fourier spectrum, and perform adjacent point difference processing to obtain the difference spectrum, and perform secondary smoothing processing on the obtained difference spectrum;

Thirdly, the distribution range of the maximum point and the minimum point in the difference spectrum processed in the previous step is determined as the Doppler distribution range of the main component of sea clutter;

Fourth, according to the division results of the above steps, determine the singular value range corresponding to the main component of sea clutter, and return to zero other singular values that are not in this range; and assume that the singular value matrix after setting zero is S ₁ , according to the singular value The value matrix S ₁ obtained after reconstruction is:

H ₁ =U ₁ ·S ₁ ·V ₁ (3)

According to formula (3), the time-domain echo data z ₁ (n) containing only the first-order sea clutter signal is obtained:

z ₁ (n)=(∑H _1(i,j) )/M(4)

In formula (4), i+j-1=n, M is the number of H1 _{(i, j)} meeting i+j-1=n;

According to the above method, Fig. 2 shows the first-order sea clutter extraction results of a certain range gate echo signal in the measured data. It can be seen that the method proposed in this paper can accurately determine the actual distribution range of the first-order sea clutter. Compared with the previous method of determining the distribution range of first-order sea clutter by fixing the clutter Doppler region, this method has strong adaptability. This method is still effective for complex phenomena such as spectral peak splitting and translation.

Figure 2-5 First-order sea clutter Doppler distribution range division and verification of extraction method.

Fifth, the time-domain echo data z ₁ (n) containing only the first-order sea clutter signal is modeled with an autoregressive model through the method of the segmented AR model, and the P-order AR model is obtained, and the P-order AR model is determined The minimum non-redundant delay domain is:

${\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; )</span>\stackrel{\mathrm{<span\; class="notranslate">\; \&Delta;</span>}}{<span\; class="notranslate">\; =</span>}<span\; class="notranslate">\; \{</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; ,</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; \&le;</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; ...</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ;</span>$

Sixth, according to the time-domain echo data z ₁ (n) obtained above containing only the first-order sea clutter signal, combined with the non-redundant delay domain and model order determined in the previous step, further calculate the size of the high-order cumulant, A large number of experiments show that the highest calculation to the fourth-order cumulant can meet the inspection requirements;

Among them, the third-order cumulant calculation formula is:

$\begin{array}{cc}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}\underset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>& <span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Element;</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}^{\mathrm{<span\; class="notranslate">\; N</span>}}\end{array}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

The formula for calculating the fourth-order cumulant is:

$\begin{array}{c}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}\underset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span>\\ \begin{array}{cc}<span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>& <span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Element;</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{\mathrm{<span\; class="notranslate">\; N</span>}}\end{array}\end{array}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

in, Indicates the non-redundant delay domain, which is determined by the model parameter results in the previous step; and the second-order cumulant calculation formula in Equation 7 is:

${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}\underset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span>$

Step 3, using the model order and segmented data calculated in step 2 to calculate the high-order cumulant of the clutter principal component; wherein, the high-order cumulant includes a third-order cumulant and a fourth-order cumulant;

Step 4, setting the inspection threshold according to the confidence level;

Step 5. Combining the calculation results of the high-order cumulant of the clutter principal component in step 3 to calculate the test statistic, and compare the test statistic with the test threshold to draw a conclusion:

First, according to the measured data samples, the estimated value of the high-order cumulant covariance matrix based on the sample data is obtained, that is, the estimated value of the third-order cumulant covariance matrix:

$\begin{array}{c}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; S</span>}}}_{\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; \&ap;</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; \&times;</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; P</span>}}}_{\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; \&ap;</span>\frac{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; R</span>}}\underset{\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; R</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}<span\; class="notranslate">\; \&lsqb;</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span><span\; class="notranslate">\; \&times;</span><span\; class="notranslate">\; \&lsqb;</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span>\end{array}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 12</span>}<span\; class="notranslate">\; )</span>$

and fourth-order cumulant covariance matrix estimates:

$\begin{array}{c}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; S</span>}}}_{\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; \&ap;</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; \&times;</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; P</span>}}}_{\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; \&ap;</span>\frac{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; R</span>}}\underset{\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; R</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}<span\; class="notranslate">\; \&lsqb;</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; l</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span><span\; class="notranslate">\; \&times;</span><span\; class="notranslate">\; \&lsqb;</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}^{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; j</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span>\end{array}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 13</span>}<span\; class="notranslate">\; )</span>$

in, ${\stackrel{\&OverBar;}{c}}_3^{z_1}(l_1,l_2)=(1/R){\mathrm{\&Sigma;}}_{r=1}^R{\hat{c}}_3^{z_1\left(r\right)}(l_1,l_2),{\stackrel{\&OverBar;}{c}}_4^{z_1}(l_1,l_2,l_3)=(1/R){\mathrm{\&Sigma;}}_{r=1}^R{\hat{c}}_4^{z_1\left(r\right)}(l_1,l_2,l_3);$ R represents the number of range gates corresponding to the echo data used;

Second, according to the calculation results of the previous step, redefine the test statistic d _G as:

The formula for the test statistic based on a third-order cumulant is: $d_{G,3}\stackrel{\mathrm{\&Delta;}}{=}N{\left({\hat{c}}_3^{z_1}\right)}^T{\hat{S}}_{c3}^{-1}{\hat{c}}_3^{z_1}---\left(9\right)$

The formula for the test statistic based on the fourth-order cumulant is: $d_{G,4}\stackrel{\mathrm{\&Delta;}}{=}N{\left({\hat{c}}_4^{z_1}\right)}^T{\hat{S}}_{c4}^{-1}{\hat{c}}_4^{z_1}---\left(10\right)$

Combined with the judgment conclusion under the determined confidence probability, the calculation result of the high-order cumulant of the Gaussian variable is zero; where, is the high-order cumulant covariance matrix in the statistical sense; ( ) ^T represents the matrix transposition operation; when the confidence level is α, the χ ² distribution test threshold t _g can be obtained by consulting the χ ² distribution table, so that in The expression of the judgment conclusion under the certain confidence probability is as follows:

${\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; G</span>}}\begin{array}{c}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\\ <span\; class="notranslate">\; ></span>\\ <span\; class="notranslate">\; <</span>\\ {\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\end{array}{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; g</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&chi;</span>}}_{{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 11</span>}<span\; class="notranslate">\; )</span>$

Third, step 53. Let H ₀ represent that the event is that the test result obeys Gaussian distribution, and H ₁ represent that the event is that the test result does not obey Gaussian distribution. Under the condition of H ₀ , the test statistic converges to N _c =3P( The chi-square distribution of 3P+1)/2, namely: Further combined with the judgment threshold under the determined confidence probability, the judgment conclusion is obtained:

If the test statistic is less than the test threshold, it indicates that the main component of the clutter obeys the Gaussian distribution, then choose the method based on the second-order statistical information of the clutter to construct the covariance matrix, and complete the subsequent corresponding signal processing processes such as clutter suppression and target detection;

If the test statistic is greater than the test threshold, indicating that the main component of the clutter does not obey the Gaussian distribution, you should select higher-order clutter statistics and continue to complete the subsequent echo data processing;

So far, the inspection process of the statistical characteristics of the amplitude of the principal component of sea clutter is completed, and the third-order cumulant is taken as an example. The flow chart is shown in Figure 6.

Validation of test methods based on measured data:

In order to test the effectiveness of the method, we tested some actual echo data. First, range and azimuth preprocessing is performed on the echo data, and the sea clutter signal on a certain beam unit is taken for the processed data, and the result after Doppler processing is shown in Fig. 7, and then the sea clutter signal on some of the range units is The clutter signal is inspected by the above-mentioned method for amplitude statistical characteristics, and the inspection results are shown in Figure 8. It can be seen that the inspection results on multiple range gates do not exceed the inspection threshold (when the confidence probability is 95%, the inspection threshold is obtained by looking up the table It is known as 70), which shows that the first-order sea clutter amplitude of this batch of echo data obeys the Gaussian distribution, and this result is also consistent with the theoretical analysis in the background technology about the sea clutter amplitude of the shore-based high-frequency ground wave radar obeying the Gaussian distribution be consistent.

Claims (5)

1. A statistical distribution test method for sea clutter amplitude of a high-frequency radar is characterized by comprising the following steps: the statistical distribution test method of the high-frequency radar sea clutter amplitude is realized by the following steps:

step one, carrying out distance and direction processing on echo data acquired by each array element of an antenna to obtain a three-dimensional data matrix of an echo to be detected, and finishing the echo data preprocessing process;

extracting main components of the sea clutter to be detected by using an SVD (singular value decomposition) method, and performing segmented adaptive regression model modeling on the extracted main components of the sea clutter to obtain a model order and a segmented data length;

thirdly, calculating clutter principal component high-order cumulant by using the model order and the segmented data obtained by calculation in the second step; wherein the high order cumulants include a third order cumulant and a fourth order cumulant;

setting a detection threshold according to the confidence level;

and step five, calculating test statistic by combining the clutter principal component high-order cumulant calculation result in the step three, and comparing the test statistic with a test threshold so as to determine a subsequent processing means of the echo data.

2. The statistical distribution test method for the amplitude of the high-frequency radar sea clutter according to claim 1, wherein: step two, the process of extracting the sea clutter main components to be detected by using the SVD method comprises the following steps:

step two, aiming at the echo data z (N) after the pretreatment in the step one and (1< N < N), constructing a matrix H (H is a Hankel matrix) with C columns and N-C +1 rows:

wherein N represents a data length; c is 3r, r represents the number of sinusoidal signals in the preprocessed echo data z (N), and the value of C is selected to be N/6;

secondly, carrying out SVD decomposition on the matrix H to obtain:

H＝U·S·V(2)

wherein, U and V respectively represent orthogonal or unitary matrix and satisfy U epsilon C^{(N-C+1)×(N-C+1)}，V∈C^C×CC represents a complex space; s represents a singular value matrix.

3. The statistical distribution test method for amplitude of high-frequency radar sea clutter according to claim 1 or 2, wherein:

step two, the process of carrying out the segmentation self-adaptive regression model modeling on the extracted sea clutter principal components to obtain the model order and the segmented data length is as follows:

firstly, performing FFT (fast Fourier transform) on a main component of a sea clutter to be detected to obtain a Doppler spectrum, and performing 3-point smoothing on the obtained Doppler spectrum;

secondly, taking logarithm of the Fourier spectrum after smoothing, carrying out adjacent point difference processing to obtain a difference spectrum, and carrying out secondary smoothing on the obtained difference spectrum;

thirdly, determining the distribution range of the maximum value point and the minimum value point in the difference spectrum after the last step of processing as the Doppler distribution range of the sea clutter main component;

fourthly, judging a singular value range corresponding to the sea clutter principal component according to the division result of the step, and simultaneously returning other singular values which are not in the range to zero; and assuming that the singular value matrix after zero setting is S₁From a matrix S of singular values₁The reconstructed matrix is obtained as follows:

H₁＝U₁·S₁·V₁(3)

obtaining time domain echo data z only containing first-order sea clutter signals according to the formula (3)₁(n)：

z₁(n)＝(∑H_1(i,j))/M(4)

In the formula (4), i + j-1 ═ n, and M is H satisfying i + j-1 ═ n_1(i,j)The number of (2);

fifthly, time domain echo data z only containing first-order sea clutter signals₁(n) performing autoregressive model modeling by using a segmented AR model method to obtain a P-order AR model, and determining the minimum non-redundant time delay domain of the P-order AR model as follows:

<math> <mrow> <msub> <mi>I</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mn>3</mn> <mi>P</mi> <mo>)</mo> </mrow> <mover> <mo>=</mo> <mi>&Delta;</mi> </mover> <mo>{</mo> <mo>|</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>|</mo> <mo>,</mo> <mo>|</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>|</mo> <mo>&le;</mo> <mn>3</mn> <mi>P</mi> <mo>,</mo> <mo>-</mo> <mi>P</mi> <mo>&le;</mo> <msub> <mi>l</mi> <mi>i</mi> </msub> <mo>&le;</mo> <mn>2</mn> <mi>P</mi> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </math>

sixthly, obtaining time domain echo data z only containing first-order sea clutter signals according to the method₁(n) and combining the non-redundant time delay domain and the model order determined in the last step, further calculating the high-order cumulant magnitude (a large number of experiments show that the highest cumulant to the fourth-order cumulant can meet the inspection requirement)

Wherein, the third-order cumulant calculation formula is:

<math> <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mn>3</mn> <msub> <mi>z</mi> <mn>1</mn> </msub> </msubsup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> </mrow> </munderover> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>)</mo> <mo>&Element;</mo> <msubsup> <mi>I</mi> <mn>3</mn> <mi>N</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow> </math>

the fourth order cumulant calculation formula is:

<math> <mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mn>4</mn> <msub> <mi>z</mi> <mn>1</mn> </msub> </msubsup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> </mrow> </munderover> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <msub> <mi>l</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mn>2</mn> <msub> <mi>z</mi> <mn>1</mn> </msub> </msubsup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mn>2</mn> <msub> <mi>z</mi> <mn>1</mn> </msub> </msubsup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>l</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mn>2</mn> <msub> <mi>z</mi> <mn>1</mn> </msub> </msubsup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mn>2</mn> <msub> <mi>z</mi> <mn>1</mn> </msub> </msubsup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mn>2</mn> <msub> <mi>z</mi> <mn>1</mn> </msub> </msubsup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mn>2</mn> <msub> <mi>z</mi> <mn>1</mn> </msub> </msubsup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>3</mn> </msub> <mo>)</mo> <mo>&Element;</mo> <msubsup> <mi>I</mi> <mn>4</mn> <mi>N</mi> </msubsup> </mrow> </mtd> </mtr> </mtable> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow> </math>

wherein, representing a non-redundant time delay domain, and determining by the result of the model parameter in the last step; and the second order cumulant calculation in equation 7 is:

<math> <mrow> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mn>2</mn> <msub> <mi>z</mi> <mn>1</mn> </msub> </msubsup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> </mrow> </munderover> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>+</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>&GreaterEqual;</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow> </math>

4. the statistical distribution test method for the amplitude of the high-frequency radar sea clutter according to claim 3, wherein: step five, the process of calculating the test statistic and comparing the test statistic with the test threshold is that,

step five, redefining the test statistic d according to the calculation result of the previous step_GComprises the following steps:

the test statistic formula based on the third-order cumulant is as follows:

the test statistic formula based on the fourth order cumulant is as follows:

combining a decision conclusion under the determined confidence probability to enable a high-order cumulant calculation result of the Gaussian variable to be zero; wherein,is a statistical high-order cumulant covariance matrix;representing the third-order cumulant covariance matrix estimation value;representing a fourth-order cumulant covariance matrix estimate; (.)^TRepresenting a matrix transposition operation;

step five and step two, when the confidence level is alpha, by consulting chi²Obtaining χ of distribution table²Distribution check threshold t_gThe decision conclusions at a certain confidence probability are expressed as follows:

step five and step three, setting H₀The representative event is that the test result follows Gaussian distribution, H₁The representative event is that the test result does not obey Gaussian distribution, and is in H₀Under the condition, the test statistic converges to the degree of freedom N_cChi-square distribution of 3P (3P +1)/2, i.e.:and further combining the determined judgment threshold under the confidence probability to obtain a judgment conclusion:

if the test statistic is smaller than the test threshold, indicating that the clutter principal component obeys Gaussian distribution, selecting a method based on clutter second-order statistical information to construct a covariance matrix, and completing subsequent corresponding signal processing processes such as clutter suppression, target detection and the like;

if the test statistic is larger than the test threshold, indicating that the clutter principal components do not obey Gaussian distribution, selecting higher-order clutter statistical information and continuing to complete subsequent echo data processing;

and finishing the sea clutter principal component amplitude statistical characteristic test process.

5. The statistical distribution test method for the amplitude of the high-frequency radar sea clutter according to claim 1, 2 or 4, wherein: step fifthly, the calculation process of the third-order cumulant covariance matrix estimation value and the fourth-order cumulant covariance matrix estimation value is,

according to the measured data sample, obtaining a high-order cumulant covariance matrix estimation value based on the sample data, namely:

third-order cumulant covariance matrix estimation:

<math> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>S</mi> <mo>^</mo> </mover> <mrow> <mi>c</mi> <mn>3</mn> </mrow> </msub> <mo>&ap;</mo> <mi>N</mi> <mo>&CenterDot;</mo> <msub> <mover> <mi>P</mi> <mo>^</mo> </mover> <mrow> <mi>c</mi> <mn>3</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>j</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&ap;</mo> <mfrac> <mi>N</mi> <mi>R</mi> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>R</mi> </munderover> <mo>&lsqb;</mo> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mn>3</mn> <mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mover> <mi>c</mi> <mo>&OverBar;</mo> </mover> <mn>3</mn> <msub> <mi>z</mi> <mn>1</mn> </msub> </msubsup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>&times;</mo> <mo>&lsqb;</mo> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mn>3</mn> <mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>j</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>j</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mover> <mi>c</mi> <mo>&OverBar;</mo> </mover> <mn>3</mn> <msub> <mi>z</mi> <mn>1</mn> </msub> </msubsup> <mrow> <mo>(</mo> <msub> <mi>j</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>j</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow> </math>

and fourth order cumulant covariance matrix estimate:

<math> <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>S</mi> <mo>^</mo> </mover> <mrow> <mi>c</mi> <mn>4</mn> </mrow> </msub> <mo>&ap;</mo> <mi>N</mi> <mo>&CenterDot;</mo> <msub> <mover> <mi>P</mi> <mo>^</mo> </mover> <mrow> <mi>c</mi> <mn>4</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>j</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&ap;</mo> <mfrac> <mi>N</mi> <mi>R</mi> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>R</mi> </munderover> <mo>&lsqb;</mo> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mn>4</mn> <mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mover> <mi>c</mi> <mo>&OverBar;</mo> </mover> <mn>4</mn> <msub> <mi>z</mi> <mn>1</mn> </msub> </msubsup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>&times;</mo> <mo>&lsqb;</mo> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mn>4</mn> <mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>j</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>j</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>j</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mover> <mi>c</mi> <mo>&OverBar;</mo> </mover> <mn>4</mn> <msub> <mi>z</mi> <mn>1</mn> </msub> </msubsup> <mrow> <mo>(</mo> <msub> <mi>j</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>j</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>j</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow> </math>

wherein, <math> <mrow> <msubsup> <mover> <mi>c</mi> <mo>&OverBar;</mo> </mover> <mn>3</mn> <msub> <mi>z</mi> <mn>1</mn> </msub> </msubsup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>/</mo> <mi>R</mi> <mo>)</mo> </mrow> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>R</mi> </msubsup> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mn>3</mn> <mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> <msubsup> <mover> <mi>c</mi> <mo>&OverBar;</mo> </mover> <mn>4</mn> <msub> <mi>z</mi> <mn>1</mn> </msub> </msubsup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>/</mo> <mi>R</mi> <mo>)</mo> </mrow> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>R</mi> </msubsup> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mn>4</mn> <mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </math> r represents the number of range gates corresponding to the echo data used.