Pixel-by-pixel scanning frequency hopping signal extraction method and device based on time-frequency graph region marking

By adopting a pixel-by-pixel scanning method based on time-frequency graph area marking in frequency hopping signal detection, the problem of inaccurate noise floor estimation and signal clustering is solved, and the accurate detection of frequency hopping signals and the improvement of time resolution are achieved.

CN120151669AActive Publication Date: 2025-06-13成都玖锦科技有限公司
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Patent Information

Application Number
CN202510631207.7
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2025-05-16
Publication Date
2025-06-13
Estimated Expiration
2045-05-16

AI Technical Summary

Technical Problem

The prior art has problems in the detection of frequency hopping signals, such as difficulty in estimating noise floors and inaccurate clustering of time-frequency graph signals, and has failed to effectively detect the signal duration and obtain the jump speed parameters.

Method used

Using a pixel-by-pixel scanning method based on time-frequency graph area marking, the FFT mode square spectrum is calculated and the rectangular smooth window convolution is performed, the marking matrix is ​​initialized, and the pixel-by-pixel scanning is used for signal extraction, realizing effective detection of the frequency hopping signal.

Benefits of technology

It improves the accuracy and time resolution of signal detection, can effectively detect the start and end time of the signal, and obtain complete information of the frequency hopping signal, including the left and right frequency point index and time index.

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Abstract

The invention relates to a pixel-by-pixel scanning frequency hopping signal extraction method and device based on time-frequency graph region marking, and belongs to the field of signal processing.The signal extraction method comprises the steps that S1, an FFT modular square spectrum of baseband IQ data subjected to down-conversion and AD sampling is calculated and put into a two-dimensional matrix, convolution processing is conducted on the matrix through a rectangular smooth window, and a frequency hopping signal is obtained; calculating a high threshold and a low threshold; s2, performing convolution processing on the matrix through the rectangular smooth window again to obtain a time-frequency graph, and initializing a mark matrix; and S3, taking the time-frequency graph as an image, and carrying out pixel-by-pixel scanning on the time-frequency graph to carry out signal extraction. According to the method, the accuracy of the starting time and the ending time is improved, the time frequency area occupied by the detected signal is marked as the occupied area, the occupied area is skipped during subsequent detection, and the operation efficiency is improved.
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Description

Technical Field

[0001] The present invention relates to the field of signal processing, and in particular, to a method and device for extracting a frequency-hopping signal by pixel-by-pixel scanning based on time-frequency map region marking. Background Art

[0002] Frequency-hopping signal detection and parameter measurement play a crucial role in electronic reconnaissance. There are mainly two difficulties in frequency-hopping signal detection: one is the noise floor estimation problem. Common estimation methods include the frequency point average method, the continuous mean removal method, the morphological method, etc. Each method has its applicable conditions. The second difficulty is the signal clustering problem on the time-frequency map. Common methods include the 8-connected region clustering method, the edge detection method of image processing, etc.

[0003] Currently, the main idea of the existing method is to divide the time-frequency map matrix into different groups according to time, and calculate the average within the group to obtain the smoothed time-frequency map. Subsequent detection only performs signal detection on the smoothed time-frequency map. However, this method has the following deficiencies: 1. There is blindness when dividing groups, and there is a situation where a signal spans two groups, resulting in inconsistent detection results for the same signal in different groups; 2. The noise floor estimation uses morphological processing, and morphological processing requires knowing the maximum bandwidth of the signal, which brings certain limitations to practical applications; 3. The signal time domain is not processed, the signal duration cannot be detected, and the hopping speed parameter cannot be obtained. Summary of the Invention

[0004] The purpose of the present invention is to overcome the shortcomings of the prior art, and provides a method and device for extracting a frequency-hopping signal by pixel-by-pixel scanning based on time-frequency map region marking, which solves the deficiencies existing in the prior art.

[0005] The purpose of the present invention is achieved through the following technical solutions: A method for extracting a frequency-hopping signal by pixel-by-pixel scanning based on time-frequency map region marking, the signal extraction method includes: S1. Calculate the FFT magnitude squared spectrum of the baseband IQ data after down-conversion and AD sampling, and put it into a two-dimensional matrix. Perform convolution processing on the matrix through a rectangular smoothing window, and then calculate the high threshold and the low threshold; S2. Perform convolution processing on the matrix through a rectangular smoothing window again to obtain a time-frequency map, and initialize a marking matrix; S3. Regard the time-frequency map as an image, and perform pixel-by-pixel scanning on it for signal extraction.

[0006] The specific content of S1 includes: S11. Calculate the FFT magnitude squared spectrum of the baseband IQ data of continuously Nco1 groups after down-conversion and AD sampling, and put it into a two-dimensional matrix pad_Matrix. This matrix has NFFT rows and Ncol columns. NFFT represents the number of points of the Fourier transform, and Ncol represents the number of groups of the Fourier transform; S12. Convolve each row of the matrix psd_Matrix with a rectangular smoothing window of length N_win_th, find the minimum value for each row of the resulting matrix, obtain an array min_table containing NFFT elements, and calculate the high threshold and low threshold through the array min_table.

[0007] The calculation of the high threshold includes: th_max = K1 × min_table; the calculation of the low threshold includes: th_min = K2 × min_table, where K1 and K2 are the high threshold coefficient and low threshold coefficient respectively.

[0008] The specific steps of S2 include: S21. Convolve each row of the matrix psd_Matrix with a rectangular smoothing window of amplitude 1 / N_win and length N_win, discard the first to N_win - 1 columns after convolution, and obtain the time-frequency diagram psd_smooth after time domain smoothing. S22. Initialize the flag matrix flag_matrix as a logical data type with all 1s, having NFFT rows and Ncol columns.

[0009] The specific steps of S3 include: S31. Treat the time-frequency diagram as an image and perform pixel-by-pixel scanning. If the flag_matrix(fi, ti) of the current pixel (fi, ti) is true, then execute step S32; otherwise, do nothing. Here, fi is the loop variable for frequency points as the inner loop, and ti is the loop variable for time as the outer loop. S32. If both psd_Matrix(fi, ti) and psd_smooth(fi, ti) of the current pixel are greater than the high threshold th_max(fi), then execute step S33; otherwise, do nothing. S33. Search downward from the current frequency point fi for the current column of psd_smooth until conti_Num consecutive spectral values are less than the low threshold. Denote the row index of the first time below the low threshold as the lower endpoint f_down of the signal spectrum. Use the same method to search upward from the current frequency point fi to obtain the upper endpoint f_up of the signal spectrum, and complete the bandwidth detection with the smoothed time-frequency diagram. S34. Calculate the sum value of the low threshold th_min within the range of f_down and f_up, and denote it as th_min_win. Search backward from the current ti for the start time and end time of the current signal, that is, calculate the sum value of the frequency points from f_down to f_up in the ti-th column of psd_Matrix column by column. If the current sum value is greater than th_min_win and the sum value of the previous column is less than th_min_win, then the current column number is the start time of the signal, denoted as t_start. After detecting the start time, enter the detection of the decline process. The condition for detecting the decline time is that the sum value of the frequency points from f_down to f_up in the search column of psd_Matrix is less than th_min_win. Then record the previous moment as the end time of the signal, denoted as t_end. If the start time and end time are detected, execute step S35; otherwise, do nothing; S35. Save the start frequency point f_down, end frequency point f_up, start time, and end time t_start and t_end detected in step S34 to the detected signal list, and set the column elements of flag_matrix belonging to the rows from f_down to f_up and columns from t_start to t_end to 0; S36. Repeat steps S31 - S35 in a loop until the loop ends.

[0010] A device for extracting frequency-hopping signals by pixel-by-pixel scanning based on time-frequency graph region marking, the device includes: a first convolution processing module, a second convolution processing module, and a signal extraction module; The first convolution processing module: is configured to calculate the FFT magnitude squared spectrum of the baseband IQ data after down-conversion and AD sampling, and put it into a two-dimensional matrix. Perform convolution processing on the matrix through a rectangular smoothing window, and then calculate the high threshold and low threshold; The second convolution processing module: is configured to perform convolution processing on the matrix again through a rectangular smoothing window to obtain a time-frequency graph, and initialize the marking matrix; The signal extraction module: is configured to regard the time-frequency graph as an image and perform pixel-by-pixel scanning to extract signals.

[0011] The first convolution processing module specifically includes the following content: Calculate the FFT magnitude squared spectrum of Nco1 consecutive groups of baseband IQ data after down-conversion and AD sampling, and put it into a two-dimensional matrix pad_Matrix. This matrix has NFFT rows and Ncol columns. NFFT represents the number of Fourier transform points, and Ncol represents the number of Fourier transform groups; Perform convolution on each row of the matrix psd_Matrix using a rectangular smoothing window with a length of N_win_th, find the minimum value for each row of the resulting matrix after convolution, obtain an array min_table containing NFFT elements, and calculate the high threshold and low threshold through the array min_table.

[0012] The second convolution processing module specifically includes the following: Perform convolution on each row of the matrix psd_Matrix using a rectangular smoothing window with an amplitude of 1 / N_win and a length of N_win, discard the first to N_win - 1 columns after convolution, and obtain the time-frequency diagram psd_smooth after time-domain smoothing. Initialize the marker matrix flag_matrix as a logical data type with all 1s, having NFFT rows and Ncol columns.

[0013] The signal extraction module specifically includes the following: A1. Regard the time-frequency diagram as an image and perform pixel-by-pixel scanning. If the marker matrix flag_matrix(fi, ti) of the current pixel (fi, ti) is true, then execute step A2; otherwise, do nothing. A2. If both psd_Matrix(fi, ti) and psd_smooth(fi, ti) of the current pixel are greater than the high threshold th_max(fi), then execute step A3; otherwise, do nothing. A3. Search downward from the current frequency point fi for the current column of psd_smooth until conti_Num consecutive spectral values are less than the low threshold. Denote the row index of the first time below the low threshold as the lower endpoint f_down of the signal spectrum. Use the same method to search upward from the current frequency point fi to obtain the upper endpoint f_up of the signal spectrum, and complete the bandwidth detection with the smoothed time-frequency diagram. A4. Calculate the sum value of the low threshold th_min within the range of f_down and f_up and denote it as th_min_win. Search backward from the current ti to find the start time and end time of the current signal, that is, calculate the sum value of the frequency points from f_down to f_up of the ti column of psd_Matrix column by column. If the current sum value is greater than th_min_win and the sum value of the previous column is less than th_min_win, then the current column number is the start time of the signal, denoted as t_start. After detecting the start time, enter the detection descent process. The condition for detecting the descent time is that the sum value of the frequency points from f_down to f_up of the search column of psd_Matrix is less than th_min_win, then record the previous moment as the end time of the signal, denoted as t_end. If the start time and end time are detected, then execute step A5; otherwise, do nothing. A5. Save the start frequency point f_down, end frequency point f_up of the signal detected in step A4, as well as the start time and end time t_start, t_end to the detected signal list, and set the column elements of flag_matrix that belong to the rows from f_down to f_up and the columns from t_start to t_end to 0; A6. Repeat steps A1 - A5 in a loop until the loop ends.

[0014] The present invention has the following advantages: A method and device for extracting frequency-hopping signals by pixel-by-pixel scanning based on time-frequency diagram region marking. The smoothing is implemented by convolution. When detecting signals, the smoothed time-frequency diagram and the instantaneous time-frequency diagram are combined, and the time resolution is not affected by the time window; when calculating the threshold, the method of finding the minimum value for each row of the smoothed time-frequency diagram is used. The threshold eliminates the fixed-frequency signals and retains the frequency-hopping signals, and is not affected by the bandwidth of the detected signals; the method of point-by-point scanning is used to obtain the complete information of each hop, including the left and right frequency point indexes, as well as the start and end time indexes. Description of the Drawings

[0015] Figure 1 is the flowchart of step 5 of the present invention; Figure 2 is the flowchart of the start time and end time of step 5 of the present invention. Detailed Embodiments

[0016] To make the objectives, technical solutions, and advantages of the embodiments of the present application clearer, the technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. Usually, the components of the embodiments of the present application described and illustrated herein can be arranged and designed in various different configurations. Therefore, the detailed description of the embodiments of the present application provided below with reference to the accompanying drawings is not intended to limit the protection scope of the present application that is required to be protected, but only represents the selected embodiments of the present application. Based on the embodiments of the present application, all other embodiments obtained by those skilled in the art without creative efforts belong to the protection scope of the present application. The present invention will be further described below with reference to the accompanying drawings.

[0017] The present invention specifically relates to a method for extracting a hopping frequency signal by pixel-by-pixel scanning based on time-frequency diagram region marking. It utilizes the sparsity characteristics of the signal, with a high threshold as the triggering condition for signal discovery, and then determines the bandwidth by the method of increasing the bandwidth on both sides until it is lower than the low threshold. The signal start time and end time detection method is: comparing the sum of energies within the bandwidth with the sum within the threshold bandwidth, which improves the accuracy of the start time and end time. Mark the time-frequency region occupied by the detected signal as the occupied region, and skip the occupied region during subsequent detections, improving the operating efficiency; specifically including the following content: Step 1: Calculate the FFT (Fast Fourier Transform) magnitude squared spectrum of Ncol consecutive groups of baseband IQ data after down-conversion and AD sampling, and put it into a two-dimensional matrix psd_Matrix. This matrix has NFFT rows and Ncol columns, where NFFT represents the number of Fourier transform points and Ncol represents the number of Fourier transform groups.

[0018] Step 2: Convolve each row of psd_Matrix with a rectangular smoothing window of length N_win_th. Find the minimum value of each row of the matrix obtained after convolution to get an array min_table containing NFFT numbers.

[0019] Among them, the calculation of the high threshold is: th_max = K1 × min_table; the calculation of the low threshold is: th_min = K2 × min_table, where K1 and K2 are the high threshold coefficient and the low threshold coefficient, such as 20 and 5.

[0020] Step 3: Convolve each row of psd_Matrix with a rectangular smoothing window with an amplitude of 1 / N_win and a length of N_win. After convolution, discard the first to N_win - 1 columns to obtain the time-domain smoothed time-frequency diagram psd_smooth.

[0021] Step 4: Initialize the marking matrix flag_matrix as a logical data type of all "1" with NFFT rows and Ncol columns.

[0022] Step 5: Regard the two-dimensional time-frequency diagram as an image, perform pixel-by-pixel scanning on the time-frequency diagram. The outer loop is the column of the time-frequency diagram, and the loop range is 3 to Ncol - N_win; the inner loop is the frequency point, and the loop range is 1 to NFFT. Denote the values of the two loop variables as (fi, ti).

[0023] As Figure 1 and Figure 2 shown, perform the following steps a to d: Step a: If both psd_Matrix(fi, ti) and psd_smooth(fi, ti) of the current pixel are greater than the high threshold th_max(fi), then proceed to Step b; otherwise, do nothing.

[0024] Step b: Search downward from the current frequency point fi in the current column of psd_smooth until conti_Num consecutive spectral values are less than the low threshold. Denote the row index of the first time below the low threshold as the lower endpoint f_down of the signal spectrum. Use the same method to search upward from the current frequency point fi to obtain the upper endpoint f_up of the signal spectrum. This step completes the bandwidth detection using the smoothed time-frequency diagram. Proceed to Step c.

[0025] Step c: Calculate the sum value of the low threshold th_min within the range of f_down and f_up and denote it as th_min_win. Search backward from the current ti for the start time and end time of the current signal. That is, calculate the sum value of the frequency points from f_down to f_up in the ti column of psd_Matrix for each subsequent column. If the current sum value is greater than th_min_win and the sum value of the previous column is less than th_min_win, then the current column number is the start time of the signal and is denoted as t_start. After detecting the start time, enter the detection descent process. The condition for detecting the descent time is that the sum value of the frequency points from f_down to f_up in the search column of psd_Matrix is less than th_min_win, then record the previous moment as the end time of the signal and denote it as t_end. If the start time and end time are detected, then proceed to Step d; otherwise, do nothing.

[0026] Step d: Save the detected start frequency point f_down, end frequency point f_up, start time, and end time t_start and t_end of the signal to the detected signal list. Set the column elements of flag_matrix belonging to the rows from f_down to f_up and columns from t_start to t_end to "0". Continue the loop in Step 5 until the loop ends. Continue the loop in Step 5 until the loop ends.

[0027] The above is only the preferred embodiment of the present invention. It should be understood that the present invention is not limited to the form disclosed herein, should not be regarded as excluding other embodiments, but can be used in various other combinations, modifications, and improvements, and can be changed within the scope of the concept described herein through the above teachings or the technology or knowledge in related fields. And any changes and variations made by those skilled in the art without departing from the spirit and scope of the present invention should fall within the protection scope of the appended claims of the present invention.

Claims

1. A method for extracting frequency hopping signals by scanning pixel by pixel based on time-frequency graph region marking, characterized in that: The signal extraction method comprises: S1, calculate the FFT modulus square spectrum of the baseband IQ data after down-conversion and AD sampling, put it into a two-dimensional matrix, perform convolution processing on the matrix through a rectangular smoothing window, and then calculate the high threshold and the low threshold; S2, convolve the matrix again through a rectangular smoothing window to obtain a time-frequency diagram and initialize the labeling matrix; S3. The time-frequency diagram is regarded as an image and is scanned pixel by pixel to extract signals.

2. The method for extracting frequency hopping signals by scanning pixel by pixel based on time-frequency graph region marking according to claim 1, characterized in that: The S1 specifically includes: S11, calculate the FFT modulus square spectrum of Nco1 consecutive groups of baseband IQ data after down-conversion and AD sampling, and put them into a two-dimensional matrix pad_Matrix, which has NFFT rows and Ncol columns, where NFFT represents the number of Fourier transform points and Ncol represents the number of Fourier transform groups; S12. Convolve each row of the matrix psd_Matrix with a rectangular smoothing window of length N_win_th, find the minimum value of each row of the matrix obtained after the convolution, obtain an array min_table containing the number of NFFTs, and calculate the high threshold and the low threshold through the array min_table.

3. The method for extracting frequency hopping signals by scanning pixel by pixel based on time-frequency graph region marking according to claim 2, characterized in that: The calculation of the high threshold includes: th_max=K1×min_table; the calculation of the low threshold includes: th_min= K2×min_table, wherein K1 and K2 are the high threshold coefficient and the low threshold coefficient respectively.

4. The method for extracting frequency hopping signals by scanning pixel by pixel based on time-frequency graph region marking according to claim 1, characterized in that: The S2 specifically includes: S21, convolve each row of the matrix psd_Matrix using a rectangular smoothing window with an amplitude of 1 / N_win and a length of N_win, discard the 1st to N_win-1 columns after convolution, and obtain the time-frequency diagram psd_smooth after time domain smoothing; S22. Initialize the flag matrix flag_matrix to all-1 logical data with NFFT rows and Ncol columns.

5. The method for extracting frequency hopping signals by scanning pixel by pixel based on time-frequency graph region marking according to claim 1, characterized in that: The S3 specifically includes: S31, treat the time-frequency graph as an image and scan it pixel by pixel. If the flag matrix flag_matrix(fi,ti) of the current pixel (fi,ti) is true, execute step S32, otherwise do not perform any operation, fi is the loop variable of the frequency point for the inner loop, ti is the loop variable of the time for the outer loop; S32, if the psd_Matrix(fi,ti) and psd_smooth(fi,ti) of the current pixel are both greater than the upper threshold th_max(fi), execute step S33, otherwise do not perform any operation; S33, starting from the current frequency point fi, searching downwards for the current column of psd_smooth until conti_Num consecutive spectrum values ​​are less than the lower threshold, recording the row index that is first less than the lower threshold as the lower endpoint f_down of the signal spectrum, searching upwards with the current frequency point fi in the same way to obtain the upper endpoint f_up of the signal spectrum, and completing the bandwidth detection with a smoothed time-frequency diagram; S34, calculate the sum of the lower threshold th_min within the range of f_down and f_up, and record it as th_min_win, start from the current ti and search backward for the start time and end time of the current signal, that is, calculate the sum of the frequency point f_down to the frequency point f_up after the ti column of psd_Matrix column by column, if the current sum is greater than th_min_win, and the sum of the previous column is less than th_min_win, then the current column number is the signal start time and is recorded as t_start, after the start time is detected, enter the detection descent process, the condition for detecting the descent time is that the sum of the frequency point f_down to the frequency point f_up of the search column of psd_Matrix is ​​less than th_min_win, then record the previous time as the signal end time, recorded as t_end, if the start time and the end time are detected, execute step S35, otherwise do not perform any operation; S35, save the signal start frequency f_down, end frequency f_up, and start time t_start, t_end detected in step S34 to the detected signal list, and set the column elements of flag_matrix belonging to rows f_down to f_up and columns t_start to t_end to 0; S36. Repeat steps S31-S35 in a loop until the loop ends.

6. A device for extracting frequency hopping signals by scanning pixel by pixel based on time-frequency graph area marking, characterized in that: The device comprises: a first convolution processing module, a second convolution processing module and a signal extraction module; The first convolution processing module is configured to calculate the FFT modulus square spectrum of the baseband IQ data after down-conversion and AD sampling, put it into a two-dimensional matrix, perform convolution processing on the matrix through a rectangular smoothing window, and then calculate the high threshold and the low threshold; The second convolution processing module is configured to perform convolution processing on the matrix again through a rectangular smoothing window to obtain a time-frequency diagram and initialize a labeling matrix; The signal extraction module is configured to regard the time-frequency diagram as an image and perform pixel-by-pixel scanning on the image to extract signals.

7. The device for extracting frequency hopping signals by scanning pixel by pixel based on time-frequency graph area marking according to claim 6, characterized in that: The first convolution processing module specifically includes the following contents: Calculate the FFT modulus square spectrum of Nco1 consecutive groups of baseband IQ data after down-conversion and AD sampling, and put them into the two-dimensional matrix pad_Matrix, which has NFFT rows and Ncol columns. NFFT represents the number of Fourier transform points, and Ncol represents the number of Fourier transform groups. Use a rectangular smoothing window with a length of N_win_th to convolve each row of the matrix psd_Matrix, find the minimum value of each row of the matrix obtained after the convolution, and get an array min_table containing the number of NFFTs. The high threshold and low threshold are calculated through the array min_table.

8. The device for extracting frequency hopping signals by scanning pixel by pixel based on time-frequency graph area marking according to claim 6, characterized in that: The second convolution processing module specifically includes the following contents: Use a rectangular smoothing window with an amplitude of 1 / N_win and a length of N_win to convolve each row of the matrix psd_Matrix. After the convolution, discard the 1st to N_win-1 columns to obtain the time-frequency diagram psd_smooth after time domain smoothing. Initialize the flag matrix flag_matrix to all-1 logical data with NFFT rows and Ncol columns.

9. The device for extracting frequency hopping signals by scanning pixel by pixel based on time-frequency graph area marking according to claim 6, characterized in that: The signal extraction module specifically includes the following contents: A1. Treat the time-frequency graph as an image and scan it pixel by pixel. If the flag matrix flag_matrix(fi,ti) of the current pixel (fi,ti) is true, execute step A2, otherwise do not perform any operation. The loop variable with fi as the frequency point is the inner loop, and the loop variable with ti as the time is the outer loop. A2. If the psd_Matrix(fi,ti) and psd_smooth(fi,ti) of the current pixel are both greater than the upper threshold th_max(fi), then execute step A3, otherwise, do not perform any operation; A3. Search downward from the current frequency point fi for the current column of psd_smooth until conti_Num consecutive spectrum values ​​are less than the lower threshold. The row index that is first less than the lower threshold is recorded as the lower endpoint f_down of the signal spectrum. The same method is used to search upward from the current frequency point fi to obtain the upper endpoint f_up of the signal spectrum, and the bandwidth detection is completed by smoothing the time-frequency diagram. A4, calculate the sum of the lower threshold th_min within the range of f_down and f_up, and record it as th_min_win, start from the current ti and search backward for the start time and end time of the current signal, that is, calculate the sum of the frequency point f_down to the frequency point f_up after the ti column of psd_Matrix column by column, if the current sum is greater than th_min_win, and the sum of the previous column is less than th_min_win, then the current column number is the signal start time and is recorded as t_start, after detecting the start time, enter the detection decline process, the condition for detecting the decline time is that the sum of the frequency point f_down to the frequency point f_up of the search column of psd_Matrix is ​​less than th_min_win, then record the previous time as the signal end time, recorded as t_end, if the start time and the end time are detected, execute step A5, otherwise do not perform any operation; A5, save the signal start frequency f_down, end frequency f_up, and start time t_start, t_end detected in step A4 to the detected signal list, and set the column elements of flag_matrix belonging to rows f_down to f_up and columns t_start to t_end to 0; A6. Repeat steps A1-A5 until the loop ends.

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