Audio forensics method based on power grid frequency harmonic information fusion

Abstract

The present invention discloses an audio evidence collection method based on the fusion of power grid frequency harmonic information, and the steps are as follows: the main steps include: 1) extracting sample data of multi-order harmonics of the power grid frequency from the target audio; 2) pre-processing the sample data of step 1; 3) calculating the de-fragmentation rate of each harmonic and determining the main harmonic and optional harmonic; 4) determining the complementation interval of the main harmonic; 5) in each complementation interval, judging whether there is a fragment of the optional harmonic; 6) calculating the longest complementation fragment of each complementation interval and splicing it to the main harmonic; 7) matching the result obtained in the previous step with the real power grid frequency signal in the reference database, selecting a reference signal with the largest correlation coefficient, and the corresponding time is considered to be the recording time of the target audio. The present invention uses more power grid frequency information brought by power grid harmonics to make the fused signal closer to the reference signal, thereby obtaining more accurate results in audio evidence collection.

Description

Technical Field

[0001] The present invention belongs to the technical field of signal processing, and in particular relates to an audio evidence collection method based on power grid frequency harmonic information fusion. Background Art

[0002] With the development of science and technology, today's society is highly interconnected, and various multimedia contents are spread through the Internet, affecting all aspects of our daily lives. The rapid development of multimedia technology has made its application more and more popular. A large amount of digital information is recorded in common ways such as audio and video, which has brought great convenience and fun to people's lives. However, while digital multimedia brings convenience to people, there is also the risk of posing a security risk to people. Since the recorded digital content can be edited, such as cutting out audio clips or inserting external audio clips, if such multimedia content is used as evidence in courts, cases and investigations, it will have a very bad impact. Therefore, multimedia evidence collection is a very important task carried out to protect the rights and interests of citizens and social security.

[0003] Grid frequency is a naturally occurring random frequency fluctuation that occurs during power grid operation. In China's power grid, it fluctuates around 50 Hz. Since grid frequency exhibits the same fluctuations throughout the interconnected system, a reference grid frequency can be obtained from any power outlet in the network. Grid frequency fluctuations are embedded in recordings containing power supply electromagnetic fields or acoustic power supply hum, allowing the grid frequency signal to be extracted from audio files. By leveraging the randomness of the grid frequency, a match is performed against a grid frequency reference database to obtain the reference signal with the highest correlation coefficient. The time corresponding to this reference signal is then considered the time of the audio recording. This explains the principle of audio timestamp verification, one of the most popular forensic techniques.

[0004] Timestamp verification requires that the matching grid frequency is relatively accurate, otherwise it is easy to get an erroneous matching result. However, general audio files always contain a large amount of other frequency information, such as environmental noise, human voices, etc., which makes the extracted grid frequency signal often contaminated by noise. In addition, since some recording equipment will automatically remove frequency signals below 100Hz, the fundamental frequency signal of the grid frequency cannot be extracted. Fortunately, the harmonics of the grid frequency can provide more useful information. However, most of the existing technologies only extract and process a certain order of harmonics of the grid frequency, and do not fully utilize the information provided by other harmonics. Therefore, there is an urgent need for a method to fuse the useful information from multiple harmonics to obtain a grid frequency signal that is closer to the reference signal. Summary of the Invention

[0005] In order to solve the above problems, the purpose of the present invention is to provide an audio forensics method based on the fusion of power grid frequency harmonic information, which can solve the problem of obtaining more accurate power grid frequency from audio.

[0006] This invention primarily utilizes an audio forensics method based on the fusion of power grid frequency harmonic information. This method utilizes the useful information from multiple harmonics of the power grid frequency, particularly the 2nd through 7th harmonics, to fuse the resulting power grid frequency signal, which more closely approximates the reference signal. Furthermore, this invention provides accurate results in audio forensics applications such as timestamp verification and tamper detection, effectively addressing security risks in digital multimedia forensics.

[0007] The technical solutions provided by the present invention are as follows:

[0008] An audio evidence collection method based on power grid frequency harmonic information fusion includes the following steps:

[0009] Step 1: extracting sample data of at least two-order multi-order harmonics of the power grid frequency from the target audio;

[0010] Step 2: Preprocess the sample data from step 1, including: enhancement, short-time Fourier transform (STFT) and removal of noise fragments;

[0011] Step 3: Calculate the defragmentation rate of each harmonic and determine the main harmonic and optional harmonics;

[0012] Step 4: Determine the compensable interval of the main harmonics;

[0013] Step 5: In each compensable interval, determine whether there is a segment of the optional harmonic; if so, record the compensable range of the compensable harmonic; if not, record it as blank;

[0014] Step 6: Calculate the longest complementable segment of each complementable interval and splice it to the main harmonic;

[0015] Step 7: Match the harmonic fusion signal with the grid frequency signal in the reference database, select the reference signal with the largest correlation coefficient, and consider the corresponding time as the recording time of the target audio.

[0016] Furthermore, in step 1, a comb filter is used to filter the read audio. The specific implementation is as follows:

[0017] For the target audio file, first read it to obtain the sample data y and sampling rate Fs of the audio file. Usually the sampling rate of audio files is 8000Hz and 44100Hz. In order to reduce the amount of calculation, a lower sampling frequency fs is used for downsampling. The fundamental frequency of the power grid frequency is around 50Hz. In order to obtain its multi-order harmonics, the present invention sets fs to 800Hz, and the downsampled sample data is recorded as y'. Finally, considering that the power grid frequency and its harmonics are narrowband signals, out-of-band noise will affect the effect of the invention, so a comb filter is used to filter y', where the length of the comb filter is 256 and the amplitude response is 0. With the same passband width, the multi-order harmonic time domain sample data input is obtained after filtering.

[0018] Furthermore, in step 1, the multi-order harmonics include 2nd to 7th order harmonics.

[0019] Furthermore, in step 2, the enhanced algorithm is used to enhance the filtered time domain sample data input, and the enhanced data is processed using the algorithm for removing noise fragments. The specific implementation method is as follows:

[0020] Since the time domain sample data input retains not only the harmonic components but also the in-band noise, direct harmonic fusion does not produce good results. The sample data input needs to be preprocessed. The specific steps are as follows:

[0021] First, HRFA ​​is used to enhance the sample data input to reduce noise interference. Then, STFT conversion is performed to obtain multi-order harmonic frequency domain signals f from the time domain data input. e_n , n is the order. Then, the noise detection method based on the grid frequency is a very effective algorithm for removing noise fragments. Using this algorithm, the value of the noise fragment is set to 0, and the result after multi-order harmonic removal is obtained. d_n .

[0022] Furthermore, in step 3, the defragmentation rate is used as a criterion to select the main harmonics and optional harmonics. The specific implementation method is as follows:

[0023] For the result f obtained in step 2 d_n , n is the order, the present invention designs a parameter - the defragmentation rate n =length(f d_n_e ) / length(f d_n ), where rate n represents the desegmentation rate of the nth order harmonic, length(f d_n_e ) represents the effective segment length of the nth order harmonic, length(f d_n) represents the total length of the nth-order harmonic. The present invention uses the de-fragmentation rate to measure the quality of harmonics. It is believed that the effective segments retained by harmonics with lower de-fragmentation rates are closer to the grid frequency of the reference signal and have less noise.

[0024] For multiple harmonics, calculate the defragmentation rate for each order and designate the harmonic with the lowest defragmentation rate as the primary harmonic. Record the order of the primary harmonic, cs_hm. Also, record the orders of the other five harmonics with a defragmentation rate lower than 0.9 and store them in the array choose as selectable harmonics.

[0025] Furthermore, the calculation of the compensable interval of the main harmonics in step 4 is specifically implemented by the following sub-steps:

[0026] Step 4.1: Based on the frequency value of the main harmonic at each time point, find all time points with a frequency of 0 and record them in an array T. The value of T is the time point value. Suppose that it contains x (x ≥ 0) time points, and each of them is an integer. Define T(i) as the time point value corresponding to the i-th 0 frequency point, i = 1, 2, 3, ..., and the total number of i is x.

[0027] Step 4.2, set the array main_index to record the start and end coordinates of each complementable interval.

[0028] Step 4.3: When the array x≠0, its first value T(1) is recorded in the array main_index as the starting coordinate of the first complementable interval; when x=1, T(1) is recorded in main_index again as the ending coordinate of the first complementable interval; when x>1, execute step 4.4.

[0029] Step 4.4: When x > 1, traverse from the second data point in array T to the last data point in T. In the i-th loop (i = 1 ... x - 1), check if T(i + 1) == T(i) + 1. If the result is equal, enter the next loop. If not, record the values ​​of T(i) and T(i + 1) in main_index. After exiting the loop, record the value of T(x) at the last time point in main_index.

[0030] Furthermore, in step 5, the compensable range of each optional harmonic in each compensable interval is calculated, and the specific implementation includes the following sub-steps:

[0031] Step 5.1, determine the number of complementable intervals num = length(main_index) / 2, where num represents the number of complementable intervals and length(main_index) represents the length of the array main_index.

[0032] Step 5.2, traverse the optional harmonic array choose, and perform the judgment of step 4 on each optional harmonic, provided that the frequency point is not 0, and find all valid segment intervals hm_index of each optional harmonic, hm∈choose.

[0033] Step 5.3: Set the complement range array cp_index to num × 6. Iterate through all complement intervals. In the i-th loop (i = 1…num), iterate through the array choose and determine whether each optional harmonic has a segment within the complement interval. If so, record the complement range of that harmonic for the i-th complement interval in cp_index; otherwise, leave it empty.

[0034] Furthermore, in step 6, harmonic fusion is performed on the main harmonics, and the specific implementation method is as follows:

[0035] Traverse the compensable intervals, determine for each compensable interval, find the optional harmonic with the largest compensable range, and fuse the corresponding optional harmonic fragments to the main harmonic. After the cycle is completed, the main harmonic after harmonic fusion is obtained, which is recorded as f p .

[0036] Furthermore, in step 7, the maximum mutual correlation coefficient is used to perform timestamp verification, and the specific implementation includes the following sub-steps:

[0037] Step 7.1, obtain the length l of the main harmonic p and reference database f r The length l r .

[0038] Step 7.2: Set array C to store the correlation coefficients. With a step length of 1 second, traverse the main harmonics in the reference database. For the i-th (i=1…l r -l p +1) cycles, and calculate the normalized cross-correlation coefficient C(i) between the main harmonic and the current corresponding reference signal.

[0039] Step 7.3, get the maximum value of C max , and the subscript lag corresponding to the maximum value. lag is the recording time of the target audio (in seconds).

[0040] Furthermore, in step 7.2, the normalized cross-correlation coefficient C(i) is calculated as follows:

[0041] Let the corresponding current reference data segment be f i =f r [i, i+l p -1],

[0042] The calculation formula of C(i) is as follows:

[0043]

[0044] where f i (k) represents the current reference data segment f i The frequency value corresponding to the kth data point, Represents the current reference data segment f i The mean of the frequency values ​​corresponding to all data points, f p (k) represents the frequency value corresponding to the kth data point of the main harmonic, Indicates the average value of the frequency values ​​corresponding to all data points of the main harmonic.

[0045] The beneficial effects of the present invention are as follows:

[0046] This method uses the power grid frequency harmonics in the target audio to perform preprocessing and fusion operations, making the fused signal closer to the reference signal. This allows for more accurate determination of the recording moment of the target audio within a given time interval, thereby achieving more accurate results in audio forensics. This method can be applied to audio forensics, such as timestamp verification and tamper detection, providing a new approach and method for audio forensics. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 is a flow chart of the overall process of the present invention;

[0048] Figure 2 This is a process diagram of harmonic fusion proposed by the present invention;

[0049] Figures 3 to 5 The following are the results of verifying the effect of the invention using an audio segment, which are comparison diagrams of the experimental results of the grid frequency signal extracted in steps 1, 2, and 6 and the reference signal; DETAILED DESCRIPTION

[0050] The technical solutions and effects of the present invention are further described in detail below with reference to the accompanying drawings.

[0051] Reference Figure 1 The present invention provides an audio evidence collection method based on power grid frequency harmonic information fusion, the steps are as follows:

[0052] Step 1: Extract sample data of the 2nd to 7th order harmonics of the power grid frequency from the target audio.

[0053] The specific steps are as follows:

[0054] like Figure 1 As shown, the process starts with reading the target audio file to obtain the sample data y of the audio file and the sampling rate Fs of the audio file.

[0055] Next, audio files typically have sampling rates of 8000Hz and 44100Hz. To reduce computational complexity, downsampling is performed using a lower sampling frequency, fs. Since the fundamental frequency of the power grid is around 50Hz, to capture multiple harmonics, the present invention sets fs to 800Hz. The downsampled sample data is denoted as y′. This embodiment uses harmonics of orders 2 through 7 for illustration.

[0056] Finally, considering that the grid frequency and its harmonics are narrowband signals, out-of-band noise will affect the effect of the invention, so a comb filter is used to filter y′, where the length of the comb filter is 256 and the amplitude response is With the same passband width. After filtering, the time domain sample data input of the 2nd to 7th order harmonics is obtained. After step 1 is completed, the effect of directly extracting the grid frequency signal is as follows Figure 3 shown.

[0057] Step 2: Preprocess the sample data, including enhancement, STFT conversion and defragmentation.

[0058] The specific steps are as follows:

[0059] Since the time domain sample data input retains not only the harmonic components but also the in-band noise, direct harmonic fusion does not work well. The sample data input needs to be preprocessed, such as Figure 1 Show:

[0060] First, HRFA ​​is used to enhance the sample data input to reduce noise interference.

[0061] Next, perform STFT conversion to obtain the 2nd to 7th order harmonic frequency domain signals f from the time domain data input e_n , n=2…7.

[0062] Finally, the noise detection method based on the grid frequency is a very effective algorithm for removing noise fragments. Using this algorithm, the value of the noise fragment is set to 0, and the result after removing the 2nd to 7th order harmonic fragments is obtained. d_n , n=2…7. After step 2 is completed, the effect of directly extracting the grid frequency signal is as follows Figure 4 shown.

[0063] Step 3: Calculate the defragmentation rate of each harmonic and determine the main harmonic and optional harmonics.

[0064] The specific steps are as follows:

[0065] For the result f obtained in step 2 d_n , n = 2 ... 7, the present invention designs a parameter - defragmentation rate rate n =length(fd_n_e ) / length(f d_n ), where n = 2…7, rate n represents the desegmentation rate of the nth order harmonic, length(f d_n_e ) represents the effective segment length of the nth order harmonic, length(f d_n ) represents the total length of the nth-order harmonic.

[0066] The present invention uses the de-fragmentation rate to measure the quality of harmonics. It is believed that the effective fragments retained by harmonics with lower de-fragmentation rates are closer to the grid frequency of the reference signal and have less noise.

[0067] like Figure 1 and Figure 2 As shown in the following example, for harmonics 2 through 7, the defragmentation rate is calculated for each order. The harmonic with the lowest defragmentation rate is designated as the main harmonic, and its order cs_hm is recorded. At the same time, the orders of the other five harmonics with a defragmentation rate lower than 0.9 are recorded and retained in the array choose as optional harmonics.

[0068] Step 4: Determine the compensable interval of the main harmonics.

[0069] The specific steps are as follows:

[0070] like Figure 1 and Figure 2 As shown. First, the result f after removing the 2nd to 7th order harmonics is d_n , n = 2 ... 7, and the main harmonic order cs_hm, obtain the main harmonic f of the order without fragmentation d_cs_hm .

[0071] Next, based on the frequency value of the main harmonic at each time point, find all time points where the frequency is zero and record them in an array T. The value of T is the time point value. Suppose that it contains x (x ≥ 0) time points, all of which are integers. Define T(i) as the time point value corresponding to the i-th zero frequency point, where i = 1, 2, 3, ..., and the total number of i is x. Set the array main_index to record the start and end coordinates of each complementable interval.

[0072] Then, determine the size of x. When x = 0, end step 4. When x ≠ 0, use the first value T(1) in array T as the starting coordinate of the first complementable interval and record it in array main_index. When x = 1, record T(1) in main_index again as the ending coordinate of the first complementable interval, and end step 4. When x > 1, traverse from the second data in array T to the last data in T. In the i-th (i = 1...x-1) loop, determine: T(i+1) == T(i)+1. If the judgment result is equal, enter the next loop. If the judgment result is not equal, record the values ​​of T(i) and T(i+1) in main_index. After exiting the loop, record the value T(x) at the last moment in time in main_index.

[0073] Finally, the coordinates of all complementable intervals are obtained and recorded in main_index.

[0074] Step 5: In each compensable interval, determine whether there is a segment of the optional harmonic. If so, record the compensable range of the compensable harmonic; if not, record it as empty.

[0075] The specific steps are as follows:

[0076] like Figure 1 and Figure 2 As shown, first, according to the coordinate main_index of the complementable interval obtained in step 4, the number of complementable intervals num=length(main_index) / 2 is determined, where num represents the number of complementable intervals and length(main_index) represents the length of the array main_index.

[0077] Then traverse the array choose and perform step 4 on each optional harmonic, provided that the frequency is not 0, to obtain all valid segment intervals hm_index of each optional harmonic, hm∈choose.

[0078] Next, set the complement range array cp_index to num × 6, initially empty. Iterate through all complement intervals. In the i-th loop (i = 1…num), iterate through the array choose and determine whether each optional harmonic has a segment within the complement interval. If so, record the complement range of that harmonic for the i-th complement interval in cp_index; otherwise, leave it empty.

[0079] Step 6: Calculate the longest complementable segment of each complementable interval and splice it to the main harmonic.

[0080] The specific steps are as follows:

[0081] Traverse the complementable intervals, and for each complementable interval, find the optional harmonic with the largest complement range, and fuse the corresponding segment to the main harmonic. If the complementable range is empty, no fusion is performed. After the cycle is completed, the main harmonic after harmonic fusion is obtained, which is recorded as fp. After step 6 is completed, the ENF signal extracted is as follows Figure 5 shown.

[0082] Step 7: Match the harmonic fusion signal with the grid frequency signal in the reference database, and select the reference signal with the largest correlation coefficient. The corresponding time is considered to be the recording time of the target audio.

[0083] The specific steps are as follows:

[0084] First, get the length l of the main harmonic p and reference database f r , whose length is l r .

[0085] Next, set array C to store the correlation coefficients. With a step size of 1 second, traverse the main harmonics in the reference database. For the i-th (i=1…l r -l p +1) loop, let the corresponding current reference data fragment be f i =f r [i, i+l p -1], using the formula:

[0086]

[0087] where f i (k) represents the current reference data segment f i The frequency value corresponding to the kth data point, Represents the current reference data segment f i The mean of the frequency values ​​corresponding to all data points, f p (k) represents the frequency value corresponding to the kth data point of the main harmonic, The average value of the frequency values ​​corresponding to all the data points of the main harmonic is calculated. The normalized cross-correlation coefficient C(i) between the main harmonic and the current corresponding reference signal is calculated.

[0088] Finally, we get the maximum value of C max , and the index lag of the array corresponding to the maximum value. Lag is the recording time of the target audio (in seconds). Figure 5 As shown, it has been verified experimentally that the power grid frequency signal extracted by the process of the present invention can obtain accurate results in timestamp verification.

[0089] It should be understood that parts not elaborated in detail in this specification belong to the prior art.

[0090] The specific embodiments described herein are merely illustrative of the spirit of the present invention. Persons skilled in the art may make various modifications, additions, or substitutions to the described specific embodiments without departing from the spirit of the present invention or exceeding the scope of the appended claims.

[0091] The above description is only a preferred specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any modifications, equivalent replacements and improvements made by any technician familiar with this technical field within the technical scope disclosed by the present invention should be included in the scope of protection of the invention.

Claims

1. An audio evidence collection method based on power grid frequency harmonic information fusion, characterized in that: The following steps are involved: Step 1: extracting sample data of at least two-order multi-order harmonics of the power grid frequency from the target audio; Step 2: Preprocess the sample data, including enhancement, short-time Fourier transform (STFT), and denoising, including: first, use the harmonic robust filtering algorithm to process the sample data. Enhance and reduce noise interference; then perform STFT conversion from time domain data Get multi-order harmonic frequency domain signals , n is the order; then, based on the noise detection method of the power grid frequency, the value of the inaccurate fragment is set to 0, and the result after multi-order harmonics are removed is obtained. ; Step 3: Calculate the de-fragmentation rate of each harmonic and determine the main harmonic and optional harmonics, including: for the results obtained in step 2 , n is the order; using the defragmentation rate To measure the quality of harmonics, the harmonics with lower de-fragmentation rate have higher quality. ,in, Representative The desegmentation rate of the order harmonics, Representative The effective segment length of the harmonic order, Representative The total length of the order harmonics; For multi-order harmonics, calculate the de-fragmentation rate of each order harmonic separately, and designate the harmonic with the lowest de-fragmentation rate as the main harmonic, and record the order of the main harmonic At the same time, record the order of other harmonics with a de-fragmentation rate lower than 0.9 and keep them in the array , as an optional harmonic; Step 4: Determine the compensable interval of the main harmonics; Step 5: In each compensable interval, determine whether there is a segment of the optional harmonic; if so, record the compensable range of the compensable harmonic; otherwise, record it as empty; Step 6: Calculate the longest complementable segment of each complementable interval and splice it to the main harmonic, including: traversing the complementable interval, judging for each complementable interval, finding the optional harmonic with the largest complement range, and fusing the corresponding optional harmonic segment to the main harmonic; after the cycle is completed, the main harmonic after harmonic fusion is obtained, which is recorded as ; Step 7: Match the harmonically fused signal with the reference signal in the reference database, and select the reference signal with the largest correlation coefficient. The corresponding time is considered to be the recording time of the target audio.

2. The audio evidence collection method based on power grid frequency harmonic information fusion according to claim 1 is characterized in that: The method of step 1 is as follows: For the target audio file, first read it to get the sample data of the audio file and sampling rate ; At sampling frequency Downsampling is performed, and the sample data after downsampling is recorded as Finally, the comb filter is used to Perform filtering to obtain time domain sample data of multi-order harmonics .

3. The audio evidence collection method based on power grid frequency harmonic information fusion according to claim 1 or 2, characterized in that: In step 1, the multi-order harmonics include 2nd to 7th order harmonics.

4. The audio evidence collection method based on power grid frequency harmonic information fusion according to claim 1 is characterized in that: The steps of step 4 are as follows: Step 4.1, according to the frequency value of the main harmonic at each time point, find all the time points where the frequency is 0 and record them in the array In the example, the value of T is the time point value, let it contain time points and they are all integers, among which ; T(i) is the time point value corresponding to the i-th 0 frequency point, i=1, 2, 3..., the total number of i is x; Step 4.2, set up the array Used to record the start and end coordinates of each complementable interval; Step 4.3, when the array When the first value As the starting coordinate of the first complement interval, record it in the array in; when When Record again , as the end coordinate of the first complement interval; when When , execute step 4.4; Step 4.4, when When, from the array The second data starts to traverse The last data in ( ) in the second cycle, make a judgment: If the result is equal, then enter the next loop; If the result is not equal, and The value of ; After exiting the loop, the last moment value Credit .

5. The audio evidence collection method based on power grid frequency harmonic information fusion according to claim 1 is characterized in that: The steps of step 5 are as follows: Step 5.1, determine the number of complementable intervals ,in, represents the number of complementable intervals, Represents an array length; Step 5.2, traverse the optional harmonic array , perform step 4 on each optional harmonic, provided that the frequency is not 0, and find all valid segment intervals for each optional harmonic ; Step 5.3, set up a complementary range cell array , the size is ; Traverse all complementable intervals, in the i-th loop, , iterate over the array , determine whether each optional harmonic has a segment in the complementable interval; if so, then the harmonic is The complement range of the complement interval is recorded in Otherwise, it is recorded as empty.

6. The audio evidence collection method based on power grid frequency harmonic information fusion according to claim 1 is characterized in that: The steps of step 7 are as follows: Step 7.1, get the length of the main harmonic and reference databases f r The length is ; Step 7.2, set up the array Used to save the mutual correlation coefficient; The main harmonics are traversed in the reference database with a step size of 1 second. For the i-th cycle, , calculate the normalized cross-correlation coefficient between the main harmonic and the current corresponding reference signal ; Step 7.3, get The maximum value , and the index of the array corresponding to the maximum value ; This is the recording time of the target audio.

7. The audio evidence collection method based on power grid frequency harmonic information fusion according to claim 6 is characterized in that: In step 7.2, the normalized cross-correlation coefficient The calculation method is as follows: Let the corresponding current reference data segment be , The calculation formula is as follows: C in Indicates the current reference data segment No. k The frequency value corresponding to the data point, Indicates the current reference data segment The mean of the frequency values ​​corresponding to all data points of Indicates the main harmonic k The frequency value corresponding to the data point, Indicates the average value of the frequency values ​​corresponding to all data points of the main harmonic.

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