A target harmonic signal complex frequency estimation method based on frequency domain least square

Abstract

The application discloses a target harmonic signal complex frequency estimation method based on a frequency domain least square, and belongs to the technical field of signal processing, and comprises the following steps: acquiring a geophysical observation time sequence containing a target harmonic signal; acquiring an initial power spectrum by adopting a windowed Fourier transform, and identifying a spectral peak of the target harmonic signal; selecting power spectrum values at multiple frequency points near the target spectral peak, and obtaining a complex frequency parameter of the target harmonic signal by utilizing a linear least square fitting; and adopting zero padding extension to increase effective frequency points of the power spectrum, and improving estimation accuracy of the complex frequency parameter by a multiple least square. The application combines the Fourier transform and the linear least square fitting, can realize high-precision estimation of the complex frequency of the target harmonic signal, is simple, efficient and easy to realize, has been successfully applied to scenes such as extraction of earth normal mode frequencies and inversion of ocean tide constituent parameters, and can be extended to communication, biomedical and other arbitrary time series signal analysis.

Description

Technical Field

[0001] This invention belongs to the field of signal processing technology, specifically relating to a method for estimating the complex frequency of a target harmonic signal based on frequency domain least squares. It is particularly suitable for geophysical signal analysis, such as high-precision parameter estimation of signals like Earth tides, free oscillations, and polar motion. Background Technology

[0002] In signal processing, particularly in geophysical signal analysis, high-precision estimation of the complex frequencies (including angular frequency ω and attenuation factor β) and amplitudes of target harmonic signals is a critical task. These parameters are essential for understanding the Earth's internal structure, seismic activity, tidal effects, and other geophysical phenomena. For example, in the study of Earth's free oscillations, accurate complex frequency estimation helps reveal the Earth's internal physical properties; in the study of polar motion, precise measurements of complex frequencies can provide important information about the dynamic changes in the Earth's rotation.

[0003] There are various existing methods for complex frequency estimation, each with its own advantages and disadvantages. Early complex frequency estimation methods mainly relied on the time-shifting method. While this method is cost-effective and efficient, it is difficult to apply window functions, which are crucial for estimating the β value. Furthermore, the time-shifting method has limited accuracy when processing complex signals, making it difficult to meet high-precision requirements.

[0004] With the development of technology, the autoregressive (AR) method has been proposed and widely used in signal processing. The AR method is a linear prediction method capable of estimating four parameters of a signal and their accuracy, including amplitude, frequency, phase, and attenuation. However, the AR method is theoretically complex, computationally cumbersome, and may be unstable in some cases. Furthermore, the accuracy of the AR method when processing low-frequency signals still needs improvement. Another commonly used method is the nonlinear least squares fitting method. This method estimates the signal parameters by minimizing the sum of squared errors, providing high accuracy. However, the nonlinear least squares fitting method suffers from high computational complexity and a tendency to get trapped in local optima, which limits its practical application.

[0005] In addition to the methods mentioned above, some improved methods have been proposed, such as those based on time-shifting methods and nonlinear damped harmonic analysis to estimate the complex frequencies of the normal modes. While these methods improve the estimation accuracy to some extent, they still suffer from computational complexity and time consumption, limiting their widespread application in practice.

[0006] In summary, while existing frequency estimation methods meet the needs of signal processing to some extent, they still have shortcomings in terms of accuracy, stability, and computational efficiency. Summary of the Invention

[0007] The purpose of this invention is to address the shortcomings of existing frequency estimation methods in terms of accuracy, stability, and computational efficiency by providing a target harmonic signal complex frequency estimation method based on frequency domain least squares. Through Fourier transform and linear least squares (FLS) fitting, this method can achieve high-precision estimation of the target harmonic signal complex frequency. This method is simple, efficient, and easy to implement, and can effectively improve the accuracy and efficiency of signal analysis, thus having significant scientific significance and practical application value.

[0008] According to one aspect of this specification, a method for estimating the complex frequency of a target harmonic signal based on frequency domain least squares is provided, comprising:

[0009] Based on the acquired geophysical observation time series containing the target harmonic signal, a windowed Fourier transform was used to process the data to obtain the initial power spectrum.

[0010] By increasing the effective frequency points of the initial power spectrum through zero-filling expansion, an encrypted power spectrum is obtained.

[0011] Complex frequency parameter estimation results are obtained by performing complex frequency estimation on the encrypted power spectrum through multiple least squares. The complex frequency estimation includes: identifying the target spectral peak of the target harmonic signal based on the encrypted power spectrum, selecting multiple frequency points within a set range of the target spectral peak, substituting the multiple frequency points and their power spectrum values ​​into a linear equation, and fitting the linear equation through linear least squares to obtain the complex frequency parameters of the target harmonic signal.

[0012] Furthermore, a windowed Fourier transform is applied to obtain the initial power spectrum, including:

[0013] A windowed time series is obtained by applying a window function to the geophysical observation time series containing the target harmonic signal;

[0014] The initial power spectrum is obtained by performing a fast Fourier transform on the windowed time series.

[0015] Furthermore, the expression for the initial power spectrum is:

[0016]

[0017] in, , Let N represent the total number of data points, and let N represent the intermediate variable. This represents the frequency after the Fourier transform. This represents the attenuation factor of the k-th harmonic. This represents the frequency of the k-th harmonic after its Fourier transform.

[0018] Furthermore, the linear equations are fitted using linear least squares, including:

[0019] Based on the frequencies and attenuation factors of multiple frequency points within the target spectral peak range, a linear equation is established:

[0020] By using the power spectrum values ​​at multiple frequency points, the linear equation is solved by linear least squares to obtain the estimated values ​​of the target harmonic signal's frequency and attenuation factor as complex frequency parameters.

[0021] Furthermore, the expression for the linear equation is:

[0022]

[0023] Where a = 1 / B k b = −2ω k / B k and c=( + ) / B k ω k β represents the angular frequency of the k-th harmonic. k B represents the attenuation factor of the k-th harmonic. k Indicates an intermediate variable.

[0024] Furthermore, the selection of the frequency points includes: selecting the frequency point at the target spectral peak and the frequency points symmetrically located on both sides of it.

[0025] According to one aspect of this specification, a target harmonic signal complex frequency estimation system based on frequency domain least squares is provided to implement the aforementioned target harmonic signal complex frequency estimation method based on frequency domain least squares, comprising:

[0026] The initial power spectrum acquisition module is used to process the acquired geophysical observation time series containing the target harmonic signal using a windowed Fourier transform to obtain the initial power spectrum.

[0027] The encrypted power spectrum acquisition module is used to increase the effective frequency points of the initial power spectrum through zero-filling expansion to obtain the encrypted power spectrum.

[0028] The complex frequency parameter estimation module is used to perform complex frequency estimation on the encrypted power spectrum through multiple least squares to obtain the complex frequency parameter estimation result. The complex frequency estimation includes: identifying the target spectral peak of the target harmonic signal based on the encrypted power spectrum, selecting multiple frequency points within a set range of the target spectral peak, substituting the multiple frequency points and their power spectrum values ​​into a linear equation, and fitting the linear equation through linear least squares to obtain the complex frequency parameters of the target harmonic signal.

[0029] According to one aspect of this specification, an electronic device is provided, including a memory and a processor, the memory storing a computer program, the processor executing the computer program to implement the steps of the target harmonic signal complex frequency estimation method based on frequency domain least squares.

[0030] According to one aspect of this specification, a computer-readable storage medium is provided having a computer program stored thereon, which, when executed by a processor, implements the steps of the target harmonic signal complex frequency estimation method based on frequency domain least squares.

[0031] According to one aspect of the present invention, a computer program product comprising instructions is provided, which, when run on a computer, causes the computer to perform the steps of the target harmonic signal complex frequency estimation method based on frequency domain least squares.

[0032] Compared with the prior art, the beneficial effects of the present invention are:

[0033] 1. Compared with the autoregressive method, the frequency domain least squares method proposed in this invention is simpler to calculate, consumes less time, is suitable for real-time signal processing, has high computational efficiency, and can estimate the complex frequency of the target harmonic signal with high accuracy, especially performing well in the estimation of low-frequency signals.

[0034] 2. This invention obtains the complex frequency parameters of the target harmonic signal through linear least squares fitting, and estimates the spectral peaks corresponding to the encrypted power spectrum through multiple least squares. It is easy to program and has low hardware requirements. It is suitable for the analysis of various geophysical signals, such as Earth's free oscillation and polar motion, and has wide applicability. Attached Figure Description

[0035] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0036] Figure 1 This is a schematic diagram of the process for estimating the complex frequency of a target harmonic signal based on frequency domain least squares provided by the present invention.

[0037] Figure 2 This is an interpolation diagram for FLS estimation provided by the present invention;

[0038] Figure 3 A schematic diagram of the waveform and power spectrum of the synthesized sequence provided by this invention;

[0039] Figure 4This is a schematic diagram of the power spectrum recorded by the Strasbourg station after an earthquake in 2004, provided for the purposes of this invention. Detailed Implementation

[0040] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0041] like Figure 1 As shown, this embodiment of the invention provides a method for estimating the complex frequency of a target harmonic signal based on frequency domain least squares, including: processing an acquired geophysical observation time series containing the target harmonic signal using a windowed Fourier transform to obtain an initial power spectrum; increasing the effective frequency points of the initial power spectrum through zero-filling expansion to obtain an encrypted power spectrum; and estimating the complex frequency of the encrypted power spectrum using multiple least squares to obtain a complex frequency parameter estimation result; wherein, the complex frequency estimation includes: identifying the target spectral peak of the target harmonic signal based on the encrypted power spectrum, selecting multiple frequency points within a set range of the target spectral peak, substituting the multiple frequency points and their power spectrum values ​​into a linear equation, and fitting the linear equation using linear least squares to obtain the complex frequency parameters of the target harmonic signal.

[0042] Specifically, the process involves windowed Fourier transform, including applying a Hanning window to the observed sequence to suppress spectral leakage, and using fast Fourier transform to obtain the initial power spectrum. In the frequency domain, the power spectrum is analyzed to select three or more frequency points near the target spectral peak (within a defined range). The target signal spectral peak is fitted using the least squares method to estimate the complex frequency parameters of the target harmonic signal. A multiple least squares approach is employed, increasing the number of frequency points through zero-padding to improve estimation accuracy.

[0043] Specifically, this invention uses synthetic noise time series to verify the accuracy and effectiveness of the FLS method and compares it with the commonly used AR method, demonstrating that the FLS method is an effective alternative for estimating the complex frequencies of target harmonic signals. Furthermore, the FLS method is applied to the estimation of complex frequencies of actual geophysical signals. This invention uses the geomorphic free oscillation OS2 signal recorded by superconducting gravity after a 2004 earthquake in a certain area. The results show that the estimation using the FLS method is not only completely consistent with previous results but also has high estimation accuracy.

[0044] Specifically, given a discrete-time series containing K target harmonic signals, where the target harmonic signals can be geophysical signals such as seismic normal mode signals, tidal signals, etc., the time series can be represented as follows:

[0045] (1)

[0046] In the formula, A k ω k and β k Let A, B, C, and D be the amplitude, angular frequency, and attenuation factor of the k-th harmonic, respectively. Let t represent the phase, t represent the time, and a(t) represent the time series. This signal format encompasses typical target harmonic signals from various geophysical phenomena, making it suitable for a wide range of scientific and engineering applications.

[0047] Specifically, a Hanning window is applied to the time series, followed by a Fast Fourier Transform (FFT). The specific steps are as follows:

[0048] A Hanning window is applied to the input sequence to reduce spectral leakage and improve the accuracy of spectral analysis. The formula for the Hanning window is:

[0049] (2)

[0050] In the formula, N is the total number of data points; n is the index of the data point. The Hanning window reduces spectral leakage by smoothing the edges of the signal, thereby improving the accuracy and reliability of spectral analysis.

[0051] Perform an FFT on the windowed sequence to convert it into a frequency domain signal. FFT is an efficient algorithm that can quickly convert time-domain signals into frequency-domain signals, facilitating subsequent frequency domain analysis. The Fourier spectrum can be written in the following form to represent the initial power spectrum (ignoring the window function):

[0052] (3)

[0053] Where ω is the frequency after Fourier transform, F(ω) represents the frequency domain form of the time series after Fourier transform, and i represents the imaginary unit.

[0054] For a typical attenuation target harmonic signal, β k From <<1, we can conclude that:

[0055] (4)

[0056] Therefore, for a given spectrum, only the positive frequency portion in the frequency domain needs to be considered. Therefore, equation (3) can be simplified to:

[0057] (5)

[0058] If the following parameters are defined:

[0059] (6)

[0060] Equation (5) can be simplified as:

[0061] (7)

[0062] Specifically, in the frequency domain, three or more frequency points near the target spectral peak are selected by analyzing the power spectrum. The specific steps are as follows:

[0063] Generally, the power spectrum is calculated as P(ω) = |X(ω)| 2 Where X(ω) is the frequency domain signal after Fourier transform. The power spectrum reflects the energy distribution of the signal at different frequencies and is a key indicator in frequency domain analysis. Therefore, the power spectrum of this embodiment can be specifically written as:

[0064] (8)

[0065] In the formula, N is the number of data points. Let B k =0.25|a k | / N, (Note that P is within a narrower target frequency band) a (ω)≠0), taking the reciprocal of both sides of equation (8), we can obtain:

[0066] (9)

[0067] Note that although B k Guarantee within a given narrow frequency band However, there is no information about complex amplitude in any power spectrum, and all frequency points in Equation (6) must pass through the target spectral peak located within a given narrow frequency band.

[0068] Specifically, select three or more frequency points near the target spectral peak, ensuring that these points cover the characteristic region of the target spectral peak. For example, you can select the frequency point at the peak and two frequency points on each side, for a total of five points. The selection of these frequency points is crucial for subsequent parameter estimation because they provide sufficient information to construct a system of linear equations.

[0069] Specifically, the frequency and attenuation factor of the target harmonic signal are estimated by fitting the target spectral peak using the linear least squares method. The specific steps are as follows:

[0070] According to equation (9), let y(ω) = 1 / P a (ω), a=1 / B k b = −2ω k / B k and c=( + ) / B k Then a linear equation can be established:

[0071] (10)

[0072] This equation linearizes the relationship between frequency, attenuation, and frequency point, providing a mathematical basis for least squares estimation. Obviously, equation (10) is linear with respect to a, b, and c; according to least squares estimation, only three y values ​​are needed to obtain the estimated values ​​of the three parameters.

[0073] (11)

[0074] In the formula, ω k and β k It can be estimated. It's important to note that B... k Contains two unknown parameters A k and And a non-fixed value N, so only the above process is used to estimate ω. k and β k Clearly, by simply taking the reciprocal of a given spectrum, the complex amplitude can be obtained. and complex frequency ω k +iβ k The estimates are separated.

[0075] Specifically, the least squares estimation method uses the values ​​at three or more frequency points to solve for the frequency (defaulting to angular frequency) and attenuation parameter. The least squares method is a classic parameter estimation method that finds the optimal parameter estimate by minimizing the sum of squared errors. It is characterized by its simplicity, high accuracy, and good stability.

[0076] Specifically, to achieve more accurate complex frequency estimation of the target harmonic signal, this embodiment of the invention employs a multiple least squares method. Multiple least squares refers to performing least squares estimation on a system of linear equations consisting of multiple sets of frequency points and their power spectral values. Furthermore, since the number of frequency points corresponding to a single spectral peak is limited, zero-padding expansion can be used to increase the frequency points of the initial power spectrum, resulting in a refined power spectrum and thus achieving a more accurate estimation. Another method is to use the discrete Fourier transform to obtain the Fourier spectrum of any frequency point below the Nyquist frequency. In fact, for a given spectrum, there are often more than three frequency points falling within the narrow band range where the target spectral peak is located. The following section will detail how to implement multiple least squares.

[0077] Specifically, according to formula (10), ω needs to be determined. k and β kAt least three frequency points need to be obtained for power spectral density estimation, but additional observations are required for more accurate estimation. Assume the target spectral peak contains M (M≥5) frequency points within its narrow band, with corresponding frequency points and power spectral density values ​​of ω0[k0] and P(ω0[k0]) (k0=1,2,...,M), respectively. The peak frequency is ω0[m0], and the power spectral density value is P(ω0[m0]). Note that if M is odd, m0=(M+1) / 2; if M is even, m0=M / 2.

[0078] Specifically, the initial power spectrum is interpolated using zero-filling, if the frequency interval of the initial power spectrum is... If the zero-fill factor is n≥2, then the frequency interval of the interpolated power spectrum is Δω / n. At this point, the narrow band of the target spectral peak contains K=(M–1)n–1 frequency points, with corresponding frequency points and power spectrum values ​​of ω[k] and P(ω[k]) (k=1,2,...,K), respectively. The peak frequency is ω[m], and the power spectrum value is P(ω[m]). It should also be noted that if M is odd, m=(M–1)n / 2+1; if M is even, m=(M–2)n / 2+1. To implement the multiple least squares process, five sets of observations are selected for least squares fitting, including: 1) three sets of commonly used observations, namely {ω[m],P(ω[m])}, {ω[1],P(ω[1])} and {ω[2m–1],P(ω[2m–1])}; 2) two sets of additional observations, namely {ω[m–j],P(ω[m–j])} and {ω[m+j],P(ω[m+j])}. It should be noted that ω[m–j] and ω[m+j] are symmetrically selected with reference to the peak frequency ω[m], and j≠m–1, i.e., i=1,2,...,m–2. At this time, m–2 represents the number of multiple least squares. According to equation (10), for the j-th estimate, the following set of equations can be established:

[0079] (12)

[0080] Rewrite the above system of equations in matrix form, i.e., P=C[ ] T The power spectrum sequence P and the coefficient matrix C can be written as:

[0081] (13)

[0082] According to the least squares principle, we can conclude that:

[0083] (14)

[0084] and

[0085] (15)

[0086] In the formula, e(x) represents the precision of parameter x; V=C[a j b j c j ] T –P represents the fitting residual. According to equation (11), we can obtain the following from the error propagation theorem:

[0087] (16)

[0088] The final formula is as follows:

[0089] (17)

[0090] Through the above steps, the frequency and attenuation factor of a specific spectral peak and their corresponding accuracy can be accurately estimated. In this embodiment of the invention, the procedure for estimating complex frequencies is referred to as Frequency Least Squares (FLS) estimation. Furthermore, FLS estimation is also applicable to estimating the complex frequencies of target harmonic signals in the product spectrum of multi-observation sequences.

[0091] Specifically, the estimated complex frequency parameters and their accuracy are output. The specific steps are as follows:

[0092] The output includes estimated complex frequency parameters, including frequency and attenuation factor, along with their corresponding accuracy. These parameters are key results of signal analysis and can provide important basis for subsequent scientific research or engineering applications.

[0093] It provides an accuracy assessment of the estimation results to ensure their reliability. Accuracy assessment can be performed by calculating the deviation between the estimated and theoretical values, the error range, etc., providing a reference for the credibility of the user's assessment results.

[0094] Specifically, the embodiments of the present invention have been verified using synthetic noise time series and actual geophysical signals, with the following steps:

[0095] First, using an interpolation diagram illustrating FLS estimation, such as... Figure 2 As shown in the figure. The solid line is the direct Fourier spectrum without interpolation, and the dashed line is the interpolated spectrum (five times zero-point addition). The target peak is located between the two vertical dashed lines. Although only three original frequency points (represented by circles) are available for estimation, 23 points (represented by asterisks) are available for estimation after interpolation, so the latter can be used for FLS.

[0096] Secondly, a synthesized noise time series was constructed, consisting of a cosine signal with a frequency of 5 mHz and an amplitude of 1, and white noise. Its time series and power spectrum are shown below. Figure 3As shown in Table 1, the complex frequency estimates obtained by the FLS and AR methods are compared. For an input frequency of 0.5 MHz, the deviation between the FLS method and the theoretical value is only 0.004‰, while the deviation of the AR method is 0.008‰; for a 5 × 10⁻⁶ mHz input frequency, the deviation is 0.004‰. 6 For input attenuation, the FLS method's calculation results have a bias of 2.4%, while the AR method's bias is 1.6%. Clearly, the frequency estimated using FLS is more accurate than that estimated using AR, and the latter provides a better estimate of attenuation. However, since the attenuation of real geophysical signals is often difficult to estimate precisely, the results of FLS estimation can be considered valid. These results demonstrate that the FLS method is an effective alternative to the AR method. Furthermore, repeated tests show that a zero-fill length of n=4 is sufficient to obtain a stable estimate.

[0097] Table 1. Estimated values ​​of synthesized sequences by FLS and AR methods

[0098]

[0099] Finally, in this embodiment of the invention, the OS2 mode of Earth's free oscillation is selected as the basis for verifying the complex frequency estimation of actual geophysical signals. Superconducting gravity records from the Strasbourg station in France were acquired after an earthquake in 2004. The observation time series started 5 hours after the earthquake event, lasted for 300 hours, and the sampling interval was 1 minute. The power spectrum corresponding to the selected observation sequence is as follows: Figure 4 As shown in the figure, the five singlet states of the multilinear state 0S2 can be completely identified, and the complex frequencies of each singlet state obtained by the FLS method are highly consistent with previous results, especially the Q values, which are very close to some studies. The FLS method is simple to implement and can obtain high-precision complex frequency parameter estimation results. This example once again verifies the accuracy and reliability of the method of the present invention.

[0100] Specifically, the method of this invention is applicable to the analysis of various geophysical signals, such as Earth's free oscillations, polar motion, and Earth's tides. For example, in seismology, this method can be used to analyze the propagation characteristics of seismic waves within the Earth, thereby inferring the physical structure of the Earth's interior; in Earth's tidal studies, this method can be used to analyze the gravitational interactions between the Earth and the Moon and the Sun; in polar motion studies, this method can provide important information on the dynamic changes of the Earth's rotation. Furthermore, this method can also be applied to other fields requiring high-precision frequency estimation, such as signal processing, communication engineering, and biomedical engineering, demonstrating broad application prospects and significant scientific importance.

[0101] The implementation of the various embodiments of the present invention is based on programmed processing by a device with processor functionality. Therefore, in practical engineering, the technical solutions and functions of the various embodiments of the present invention are encapsulated into various modules. Based on this reality, and building upon the above embodiments, the embodiments of the present invention provide a target harmonic signal complex frequency estimation system based on frequency domain least squares. This system is used to execute a target harmonic signal complex frequency estimation method based on frequency domain least squares from the above method embodiments.

[0102] The system includes: an initial power spectrum acquisition module, used to obtain the initial power spectrum by processing the acquired geophysical observation time series containing the target harmonic signal using a windowed Fourier transform; an encrypted power spectrum acquisition module, used to increase the effective frequency points of the initial power spectrum through zero-filling expansion to obtain an encrypted power spectrum; and a complex frequency parameter estimation module, used to perform complex frequency estimation on the encrypted power spectrum using multiple least squares to obtain complex frequency parameter estimation results. The complex frequency estimation includes: identifying the target spectral peak of the target harmonic signal based on the encrypted power spectrum, selecting multiple frequency points within a set range of the target spectral peak, substituting the multiple frequency points and their power spectrum values ​​into a linear equation, and fitting the linear equation using linear least squares to obtain the complex frequency parameters of the target harmonic signal.

[0103] The target harmonic signal complex frequency estimation system based on frequency domain least squares provided in this invention addresses the shortcomings of existing frequency estimation methods in terms of accuracy, stability, and computational efficiency. By employing several modules, it can achieve high-precision estimation of the target harmonic signal complex frequency. This method is not only simple and efficient but also easy to implement, effectively improving the accuracy and efficiency of signal analysis. Moreover, it has significant scientific significance and practical application value.

[0104] Based on the same inventive concept as the foregoing embodiments, this embodiment of the invention also provides an electronic device, including a memory and a processor. The memory is used to store computer-executable instructions, and the processor is used to execute the computer-executable instructions to realize a target harmonic signal complex frequency estimation method based on frequency domain least squares as proposed in the above embodiments.

[0105] This invention also provides a computer-readable storage medium storing a computer program thereon. When executed by a processor, the program can accurately estimate the complex frequency of a target harmonic signal, especially excelling in estimating low-frequency signals; it is time-efficient, suitable for real-time signal processing, and computationally efficient; it is easy to program and has low hardware requirements, making it suitable for the analysis of various geophysical signals, such as Earth's free oscillations and polar motion, thus possessing wide applicability.

[0106] This invention also provides a computer program product containing instructions that, when run on a computer, generate, in whole or in part, the target harmonic signal complex frequency estimation method based on frequency domain least squares as proposed in the above embodiments. The computer may be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device.

[0107] Finally, it should be noted that the above specific embodiments are merely representative examples of the present invention. Obviously, the present invention is not limited to the above specific embodiments and many variations are possible. Any simple modifications, equivalent changes, and alterations made to the above specific embodiments based on the technical essence of the present invention should be considered within the protection scope of the present invention.

Claims

1. A method for estimating the complex frequency of a target harmonic signal based on frequency domain least squares, characterized in that, include: Based on the acquired geophysical observation time series containing the target harmonic signal, a windowed Fourier transform is used to obtain the initial power spectrum; wherein, the expression for the geophysical observation time series a(t) containing the target harmonic signal is: ， In the formula, A k ω k and β k ϕ represents the amplitude, angular frequency, and attenuation factor of the k-th harmonic, respectively. k Let t be the phase, t be the time, and a(t) be a discrete-time sequence containing K target harmonic signals; By increasing the effective frequency points of the initial power spectrum through zero-filling expansion, an encrypted power spectrum is obtained. Complex frequency parameter estimation results are obtained by performing complex frequency estimation on the encrypted power spectrum using multiple least squares. The complex frequency estimation includes: identifying the target spectral peak of the target harmonic signal based on the encrypted power spectrum; selecting multiple frequency points within a set range of the target spectral peak; substituting the multiple frequency points and their power spectrum values ​​into a linear equation; and fitting the linear equation using linear least squares to obtain the complex frequency parameters of the target harmonic signal. The fitting of the linear equation using linear least squares includes: establishing a linear equation based on the frequency and attenuation factor of the multiple frequency points within the set range of the target spectral peak; and solving the linear equation using the power spectrum values ​​of the multiple frequency points through linear least squares to obtain estimated values ​​of the frequency and attenuation factor of the target harmonic signal as complex frequency parameters. The expression of the linear equation is as follows: ， Where a = 1 / B k b = −2ω k / B k and c=( + ) / B k , This represents the frequency after the Fourier transform. β represents the frequency of the k-th harmonic after its Fourier transform. k B represents the attenuation factor of the k-th harmonic. k Indicates an intermediate variable.

2. The method for estimating the complex frequency of a target harmonic signal based on frequency domain least squares according to claim 1, characterized in that, The initial power spectrum is obtained by processing with a windowed Fourier transform, including: A windowed time series is obtained by applying a window function to the geophysical observation time series containing the target harmonic signal; The initial power spectrum is obtained by performing a fast Fourier transform on the windowed time series.

3. The method for estimating the complex frequency of a target harmonic signal based on frequency domain least squares according to claim 1, characterized in that, The expression for the initial power spectrum is: ， in, , Let N represent the total number of data points, and let N represent the intermediate variable. This represents the frequency after the Fourier transform. This represents the attenuation factor of the k-th harmonic. This represents the frequency of the k-th harmonic after its Fourier transform.

4. The method for estimating the complex frequency of a target harmonic signal based on frequency domain least squares according to claim 1, characterized in that, The selection of the frequency points includes: selecting the frequency point at the target spectral peak and the frequency points symmetrical to both sides of it.

5. A target harmonic signal complex frequency estimation system based on frequency domain least squares, characterized in that, A method for estimating the complex frequency of a target harmonic signal based on frequency domain least squares as described in any one of claims 1-4 includes: The initial power spectrum acquisition module is used to process the acquired geophysical observation time series containing the target harmonic signal using a windowed Fourier transform to obtain the initial power spectrum; wherein, the expression for the geophysical observation time series a(t) containing the target harmonic signal is: ， In the formula, A k ω k and β k ϕ represents the amplitude, angular frequency, and attenuation factor of the k-th harmonic, respectively. k Let t be the phase and t be the time. The encrypted power spectrum acquisition module is used to increase the effective frequency points of the initial power spectrum through zero-filling expansion to obtain the encrypted power spectrum. The complex frequency parameter estimation module is used to perform complex frequency estimation on the encrypted power spectrum through multiple least squares to obtain the complex frequency parameter estimation result. The complex frequency estimation includes: identifying the target spectral peak of the target harmonic signal based on the encrypted power spectrum, selecting multiple frequency points within a set range of the target spectral peak, substituting the multiple frequency points and their power spectrum values ​​into a linear equation, and fitting the linear equation through linear least squares to obtain the complex frequency parameters of the target harmonic signal.

6. An electronic device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the processor executes the computer program, it implements the steps of the target harmonic signal complex frequency estimation method based on frequency domain least squares as described in any one of claims 1 to 4.

7. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements the steps of the target harmonic signal complex frequency estimation method based on frequency domain least squares as described in any one of claims 1 to 4.

8. A computer program product containing instructions, characterized in that, When it is run on a computer, it causes the computer to perform the steps of the target harmonic signal complex frequency estimation method based on frequency domain least squares as described in any one of claims 1 to 4.

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