Weak signal detection method based on z-domain autoregression and analytic continuation

Abstract

The invention discloses a weak signal detection method based on z-domain autoregression and analytic continuation, and belongs to the technical field of signal processing and geophysical exploration, and the method comprises the steps: carrying out the multi-group random noise addition of an observation time sequence comprising a target weak signal, and obtaining a multi-group noise addition sequence; calculating an AR-z spectrum for each group of noise adding sequences; performing complex frequency estimation on the noise adding sequences by adopting frequency domain autoregression estimation to obtain complex frequency information of Fourier spectrums corresponding to the noise adding sequences; in combination with prior complex frequency information of a target weak signal, an AR-z spectrum of a noise adding sequence is constructed through analysis and continuation; and geometrically averaging amplitudes of AR-z spectrums obtained by calculation of the multiple groups of noise adding sequences to obtain a robust AR-z spectrum for weak signal detection. Through z-domain autoregression and random noise addition, the frequency resolution limit of traditional Fourier transform is broken through, the signal-to-noise ratio threshold value of the AR-z spectrum is reduced to be smaller than 1, and a universal solution is provided for signal detection in an extreme noise environment.

Description

Technical Field

[0001] This invention belongs to the field of signal processing and geophysical exploration technology, specifically involving a weak signal detection method based on z-domain autoregression and analytical continuation. It is particularly suitable for extracting and enhancing the frequency characteristics of hidden weak targets in geophysical observation signals (such as seismic waves, geomagnetic anomalies, and gravity gradient signals) when non-stationary noise interference is severe in low signal-to-noise ratio (SNR<1) environments. Background Technology

[0002] In the field of weak signal detection, especially in extreme noise environments such as deep Earth and deep space signal detection, traditional signal processing methods face severe challenges. Due to complex conditions such as high background noise intensity, strong signal non-stationarity, and extremely low signal-to-noise ratio (SNR), existing technologies often struggle to achieve reliable detection and frequency characteristic analysis of weak target signals.

[0003] Fourier transform-based spectral analysis techniques are limited by frequency resolution and spectral leakage issues, making it difficult to distinguish the frequency components of a noisy floor from those of weak signals under low signal-to-noise ratio (SNR) conditions. Their frequency domain sampling interval (Δf=1 / N, where N is the observation duration) restricts spectral refinement capabilities, resulting in the incomplete capture of the complex frequency (real and imaginary) information of weak signals. Although autoregressive models improve spectral resolution through parameterized frequency domain response, their analysis remains limited to the real frequency domain and cannot effectively characterize the attenuation characteristics of signals in the complex plane (such as damped oscillations). Furthermore, traditional AR spectra are sensitive to noise, easily generating spurious peaks in strong non-stationary noise environments, leading to increased false alarm rates. Existing methods (such as adaptive filtering, wavelet thresholding, and principal component analysis) largely rely on prior assumptions about noise statistical properties, making them ill-suited for situations where the noise power spectrum is unknown or rapidly changing in extreme noise environments. Some techniques improve the SNR through averaging multiple observations, but their practicality is limited by the stringent observation conditions in real-world applications (such as the inability to repeatedly acquire seismic signals). While recent complex z-domain extension methods (such as ZFFT) can characterize the complex frequency features of signals, they are not deeply coupled with noise suppression mechanisms, leading to a sharp decline in performance in low signal-to-noise ratio scenarios. Furthermore, existing methods lack effective means to suppress statistical fluctuations in random noise, making it difficult to robustly extract signal components with an SNR < 1.

[0004] Existing technologies generally suffer from insufficient spectral resolution, lack of complex frequency information, and poor noise robustness under high noise and low signal-to-noise ratio conditions, which severely restrict the sensitivity and reliability of weak signal detection. Summary of the Invention

[0005] The purpose of this invention is to address the problem of insufficient detection capability of traditional technologies under low signal-to-noise ratio conditions, and to provide a weak signal detection method based on z-domain autoregression and analytical continuation. By integrating autoregressive (AR) complex z-domain spectral analytical continuation technology with random noise addition-geometric averaging technology, it breaks through the Fourier spectrum resolution limitation of traditional Fourier transform, and provides a highly sensitive and universal solution for weak signal detection in extreme noise environments.

[0006] According to one aspect of this specification, a weak signal detection method based on z-domain autoregression and analytical continuation is provided, comprising:

[0007] Multiple sets of random noise were added to the acquired observation time series, which included weak signals of the target, to obtain multiple sets of noisy sequences;

[0008] For each group of noisy sequences, the AR-z spectrum is calculated as follows: the complex frequency of the noisy sequence is estimated by frequency domain autoregressive estimation, and the complex frequency information of the Fourier spectrum corresponding to the noisy sequence is obtained by combining zero-filling expansion; based on the complex frequency information of the Fourier spectrum corresponding to the noisy sequence, combined with the prior complex frequency information of the target weak signal, the AR-z spectrum of the noisy sequence is constructed by analytical extension.

[0009] The amplitudes of the AR-z spectra calculated from multiple sets of noisy sequences are geometrically averaged to obtain the AR-z product spectrum, which is called the robust AR-z spectrum and is used for weak signal detection.

[0010] Furthermore, frequency domain autoregressive estimation is used to estimate the complex frequency of the noisy sequence, and combined with zero-filling expansion, the complex frequency information of the Fourier spectrum corresponding to the noisy sequence is obtained, including:

[0011] The frequency domain response of the noisy sequence is fitted by frequency domain autoregressive estimation to obtain the complex frequency information of the Fourier spectrum.

[0012] The Fourier spectrum is interpolated using zero-filling extension to obtain a compact Fourier spectrum and its complex frequency information.

[0013] Furthermore, the AR-z spectrum of the noisy sequence is constructed through analytical continuation, including:

[0014] Based on the prior complex frequency information of the target weak signal, the frequency domain response of the noisy sequence is extended to the complex z-plane through analytical continuation, generating an AR-z spectrum with the Lorentz function as the basis.

[0015] Furthermore, the AR-z spectrum expression is:

[0016] ,

[0017] in, ω i and These are the frequency estimate and attenuation factor estimate for the i-th frequency point, respectively; ,in Let be the Fourier frequency of the i-th frequency point. is the prior attenuation factor of the signal to be detected, and N represents the number of frequency points.

[0018] Furthermore, the robust AR-z spectrum expression is:

[0019] ,

[0020] Where, ω j and β j Let J represent the estimated frequencies and attenuation factors for all frequency points of the j-th sequence; J represents the number of superpositions, and ω represents the frequency values ​​of all frequency points. is the prior attenuation factor of the signal to be detected.

[0021] Furthermore, the target weak signal is a weak geophysical observation signal, including: Earth's medium- and long-period normal mode signals, seismic wave signals, geomagnetic anomaly signals, and gravity gradient signals.

[0022] According to one aspect of this specification, a weak signal detection system based on z-domain autoregression and analytic continuation is provided to implement the aforementioned weak signal detection method based on z-domain autoregression and analytic continuation, comprising:

[0023] The first main module is used to add multiple sets of random noise to the acquired observation time series, including weak signals of the target, to obtain multiple sets of noisy sequences;

[0024] The second main module is used to calculate the AR-z spectrum for each group of noisy sequences. Specifically, it uses frequency domain autoregressive estimation to estimate the complex frequency of the noisy sequence and combines it with zero-filling expansion to obtain the complex frequency information of the Fourier spectrum corresponding to the noisy sequence. Based on the complex frequency information of the Fourier spectrum corresponding to the noisy sequence and combined with the prior complex frequency information of the target weak signal, it constructs the AR-z spectrum of the noisy sequence through analytical extension.

[0025] The third main module is used to take the geometric mean of the amplitudes of the AR-z spectra calculated from multiple sets of noisy sequences to obtain the AR-z product spectrum, which is called the robust AR-z spectrum and is used for weak signal detection.

[0026] 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 described weak signal detection method based on z-domain autoregression and analytic continuation.

[0027] 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 described weak signal detection method based on z-domain autoregression and analytic continuation.

[0028] According to one aspect of the present invention, a computer program product containing instructions is provided, which, when run on a computer, causes the computer to perform the steps of the weak signal detection method based on z-domain autoregression and analytic continuation.

[0029] Compared with the prior art, the beneficial effects of the present invention are:

[0030] 1. This invention improves the spectrum refinement capability by more than 3 times compared with the traditional Fourier transform under the same observation time through frequency domain autoregressive estimation and zero-filling interpolation. It can accurately capture the damped oscillation characteristics of weak signals and achieve super Nyquist frequency resolution.

[0031] 2. This invention extends the frequency domain response to the complex z-plane through analytical extension, constructs a complex z-domain power spectral density function based on the Lorentz function, and forms local extremum peaks on the complex plane. The positions of these peaks correspond to the complex frequency parameters of the target signal, significantly enhancing the identifiability of the signal features.

[0032] 3. This invention superimposes multiple sets of statistically independent and power spectrum-matched random noise onto the original observation time series. By enhancing the coherent signal components through superposition and statistically suppressing the incoherent noise, the signal-to-noise ratio detection threshold of the AR-z spectrum is reduced to less than 1, and the spurious peak interference caused by non-stationary noise is effectively suppressed. Through the synergistic processing of z-domain autoregression and random noise addition, the frequency resolution limitation of Fourier transform is broken, and the interference of other strong signals can be significantly weakened, thereby detecting weak harmonic signals with a signal-to-noise ratio as low as less than 1. Attached Figure Description

[0033] 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.

[0034] Figure 1 A schematic diagram of a weak signal detection method based on z-domain autoregression and analytical continuation provided in an embodiment of the present invention;

[0035] Figure 2 This is a schematic diagram of the Lorentz peak of the AR power spectrum P(z) near the pole in the z-domain, provided in an embodiment of the present invention.

[0036] Figure 3 (a) A comparison of the Fourier spectrum, maximum entropy spectrum and AR-z spectrum of the synthesized noise sequence provided in the embodiments of the present invention;

[0037] Figure 3 (b) A comparison diagram of the Fourier spectrum, maximum entropy spectrum, AR-z spectrum and robust AR-z spectrum of the synthesized noise sequence provided in the embodiments of the present invention;

[0038] Figure 4 A comparison of the Fourier power spectrum and AR-z spectrum of the Earth's dynamic oblateness (J2) time series provided in this embodiment of the invention. Detailed Implementation

[0039] 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.

[0040] This invention discloses a z-domain autoregressive spectrum (AR-z spectrum) method for detecting weak signals under strong noise backgrounds by innovatively integrating z-domain autoregressive spectrum analysis with random noise-geometric averaging technology, providing a brand-new technical path for signal detection in extreme noise environments.

[0041] like Figure 1 As shown, this embodiment of the invention provides a weak signal detection method based on z-domain autoregression and analytical continuation, comprising: randomly adding noise to the acquired observation time series including the target weak signal to obtain multiple sets of noisy sequences; calculating the AR-z spectrum for each set of noisy sequences, specifically: using frequency domain autoregressive estimation to estimate the complex frequency of the noisy sequence, combined with zero-filling expansion, to obtain the complex frequency information of the Fourier spectrum corresponding to the noisy sequence; based on the complex frequency information of the Fourier spectrum corresponding to the noisy sequence, combined with the prior complex frequency information of the target weak signal, constructing the AR-z spectrum of the noisy sequence through analytical continuation; taking the geometric mean of the amplitudes of the AR-z spectra obtained from the multiple sets of noisy sequences to obtain the AR-z product spectrum, denoted as the robust AR-z spectrum, used to enhance the target weak signal to be detected. The weak geophysical observation signals include medium- and long-period normal mode signals of the Earth, seismic wave signals, geomagnetic anomaly signals, and gravity gradient signals.

[0042] Specifically, the steps for obtaining and preprocessing the observation time series are as follows:

[0043] Geophysical observation data containing weak target signals are acquired through sensors, forming discrete observation time series. The data undergoes standardization preprocessing (such as detrending and normalization) to eliminate the interference of baseline drift on subsequent analysis. This time series can be represented as a superposition of a series of decaying cosine signals and noise terms:

[0044] (1)

[0045] in, M Indicates the number of attenuated cosine signals; Let the amplitude, angular frequency, and attenuation factor of the j-th signal be respectively, then its quality factor is... ; For phase, This is background noise. This signal form encompasses typical weak signals found in a variety of geophysical phenomena, making it suitable for a wide range of scientific and engineering applications.

[0046] Specifically, the complex frequency is estimated using frequency domain autoregression, and the steps are as follows:

[0047] For a discrete time series with N data points The complex frequency of the j-th decaying cosine signal and complex amplitude Then formula (1) can be expressed as:

[0048] (2)

[0049] Where * denotes complex conjugate. According to the Prony algorithm, a(n) satisfies the following 2M-order autoregressive equation (recursive difference equation):

[0050] (3)

[0051] Among them, AR coefficient S i Let $S$ be a real constant (i = 1, 2, ..., 2M). Using the Prony polynomial, we can establish $S$. i} and {σ j The relationship between}:

[0052] (4)

[0053] From this, we can deduce that... and .

[0054] Taking a Fourier transform of both sides of equation (3) yields the following equation:

[0055] (5)

[0056] Among them, {Xi (ω k Let {a(ni), n=2M+1, …, N} be the time series {a(ni), n=2M+1, …, N} at the selected frequency ω. k The Fourier transform at point X, where K represents the total number of frequency points within the selected frequency band. i (ω k In the frequency domain, zero-padding is performed by a factor of L (using at least 3 zero padding). After reconstructing the time-domain signal using inverse FFT, FFT is performed again to obtain the interpolated super-resolution Fourier spectrum. (i.e., the dense Fourier spectrum). Further, it can be written in matrix form:

[0057] (6)

[0058] Where M becomes the number of signals within the target frequency band. If M=1, that is, when there is only one spectral peak in the target frequency band, it is necessary to solve for two AR coefficients S1 and S2. Considering that the Fourier spectrum has a real part (Re) and an imaginary part (Im), equation (6) can be simplified to:

[0059] (7)

[0060] The optimal solutions for S1 and S2 are obtained by least squares. Then, according to formula (4), , When K > M, the same logic can be used to obtain {S}. i The optimal solution is found, and then {σ} can be obtained using numerical methods. j}

[0061] Specifically, multiple sets of noise sequences are generated by superimposing a random noise sequence on the observed time series, wherein the noise level of the random noise sequence is consistent with the noise level of the observed time series, and each set of noise samples is statistically independent.

[0062] Specifically, the steps for constructing the AR-z spectrum of the observed time series are as follows:

[0063] The complex z-transform of a discrete function is the Laplace transform of a continuous function. The complex z-plane is represented using the z-domain, and the z-transform of a discrete function is defined in the complex z-plane. The unit circle C0 in the z-domain represents the general Fourier frequency domain. According to complex variable theory, a function can be extended from the frequency domain to the z-domain through analytic continuation. According to the Prony relation, the zeros of H(z) in the complex z-plane, i.e. It becomes a pole in the z-domain. Therefore, the following real function is defined to represent the power spectrum of the z-transform of the autoregressive equation (3):

[0064] (8)

[0065] In the formula, d jRepresents the observation point and pole in the z-domain exp(iσ) j The geometric distance between ) is H(z), which represents the Prony polynomial.

[0066] Specifically, Figure 2 The Lorentz peak of the AR power spectrum P(z) near the poles in the z-domain is shown. According to the Prony relation, it corresponds to a harmonic signal with complex frequency σ = ω (frequency) + iβ (exponential decay rate). The cross section of P(z) along C0 corresponds to the maximum entropy spectrum. In the z-plane, the poles found inside the unit circle C0 correspond to exponentially decaying sine waves (β > 0), the poles outside C0 correspond to exponentially growing sine waves (β < 0), and the poles falling on C0 correspond to pure sine waves (β = 0). Therefore, for the pole exp(iσ j At a field point z near β, P(z) exhibits the shape of a Lorentz peak, whose value asymptotically increases to infinity towards the poles. In the z-domain, given the typical analytic continuation circle of β (>0) within C0, denoted as C′, its intersection with P(z) is the Lorentz cross section of P(z) along C′. Therefore, the spectral peak found in P(z) indicates the presence of a harmonic signal in the time series. It should be noted that this process is not to determine the actual height or shape of the peak, but rather to determine the location of the poles.

[0067] For a noise sequence, after Fast Fourier Transform and zero-filling, 2π / (L×N) frequency intervals can be obtained; for any frequency ω0, a complex conjugate pole pair is obtained by forcibly using the solution of the second-order autoregressive equation (i.e., the solution of equation (7)), and its corresponding complex frequency is σ0. If there is no signal at ω0 and only noise, the distance d0=|σ0–ω0| between σ0 and ω0 obtained by this forced estimation is usually not a small amount, so P(z) has no peak near ω0. If there is a harmonic signal at ω0, σ0 and ω0 will be very close, resulting in d0 being very small, even close to 0 when Q>>1, so P(z) will form a significant Lorentz-type spectral peak near ω0. Therefore, the frequency ω0 corresponds to The value can be used as a detection factor. Since ω0 ranges from 0 to the Nyquist frequency, ω0 can be used as a detection factor at each frequency. i Corresponding The spectral peaks will form a complete spectrum P(ω), where the Fourier peaks indicate the presence of harmonic signals in the original time series. This is called the general form of the z-domain autoregressive spectrum, or AR-z spectrum for short, and is expressed as follows:

[0068] (9)

[0069] In the formula, ω i and These are the frequency estimate and attenuation factor estimate for the i-th frequency point, respectively; ,in Let be the Fourier frequency of the i-th frequency point. Let be the a priori attenuation factor of the signal to be detected. Equation (8) is in z-domain form, and Equation (9) is in the general frequency domain form of Equation (8). N is the number of data points in the time series, which is also the number of frequency points in the frequency domain, and here it represents the number of frequency points.

[0070] Specifically, J statistically independent random noises with the same noise level as the original sequence are superimposed on the original observation sequence to generate J synthetic noise-added sequences. Using formula (9), the AR-z spectrum of each noise-added sequence is calculated independently, and the geometric mean of the J AR-z spectra is calculated to obtain its product spectrum form. This process, through the superposition enhancement of coherent signal components and the statistical suppression of incoherent noise, reduces the signal-to-noise ratio detection threshold of the AR-z spectrum to less than 1 and effectively suppresses spurious peak interference caused by non-stationary noise. This product spectrum result is called the robust AR-z spectrum, and its final form can be expressed as follows:

[0071] (10)

[0072] Where, ω j and β j These are the estimated frequencies and attenuation factors for all frequency points of the j-th sequence. The actual complete way to write this is to add two subscripts to the right of the frequency ω, i.e., ωj = ωj. i,j， For the sake of simplicity, omissions are made. The subscript i in the table represents the frequency value of all frequency points, and ω represents the frequency value of all frequency points.

[0073] Specifically, the effectiveness of this invention was verified using synthesized noise time series and actual geophysical signals. The specific steps are as follows:

[0074] First, to visually compare the performance differences between AR-z spectrum, Fourier spectrum, and maximum entropy method (MEM) spectrum, this invention constructs two sets of time series, one noise-free and one noisy, for verification. The noise-free reference series is set as f0(t)=cos(ωt), where the nominal frequency ω=3 weeks / year (cpy), the duration is 50 years, and daily sampling is used. Figure 3 (a) presents the analysis results of the Fourier spectrum, the maximum entropy spectrum (based on the Yule-Walker and Burg algorithms), and the AR-z spectrum without noise processing for the reference sequence. It is evident that in this case, the AR-z spectrum exhibits the most significant spectral peak characteristics of the input signal, effectively highlighting its spectral estimation advantages.

[0075] Secondly, to verify the noise robustness of the robust AR-z spectrum, this study further constructed a noisy time series f1(t)=f0(t)+n(t), where the frequency domain signal-to-noise ratio of the superimposed white noise n(t) was set to SNR=10. Figure 3 (b) presents a comparison of its Fourier spectrum, maximum entropy spectrum (using the Yule-Walker and Burg algorithms, with the AR model order uniformly set to 1500 and no pre-filtering), and robust AR-z spectrum. The noise-assisted robust AR-z spectrum obtained by inputting 300 sets of additional random noise sequences (noise amplitude approximately 1.3 times that of the original noise n(t)) into f1(t) exhibits higher robustness compared to the un-noiseed AR-z spectrum and Fourier spectrum, and its peak width is sharper than that of the maximum entropy spectrum. This result demonstrates that the AR-z spectrum, after noise processing, significantly improves the detection capability of harmonic signals.

[0076] Finally, in this embodiment of the invention, the Earth's dynamic oblateness (J2) time series was selected as the verification for actual geophysical signal detection. The J2 time series obtained from 1976 to 2017 based on satellite laser ranging observations was acquired, with a data sampling interval of one month. The Fourier power spectrum and AR-z spectrum corresponding to the selected observation series are as follows: Figure 4 As shown in the figure, the Fourier power spectrum can only identify significant 1-year and semi-annual terms, while the AR-z spectrum can identify 1 / 3, 1 / 4, and 1 / 5-year terms below the 1-year scale, as well as 10.5, 18.6, and 56-year terms above the 1-year scale. Furthermore, the frequency resolution of the AR-z spectrum is significantly higher than that of the Fourier power spectrum (the Fourier power spectrum is used to represent the actual Fourier spectrum).

[0077] Specifically, the simulations and case studies described above demonstrate that AR-z spectroscopy can effectively resolve fine structural features in signals, especially exhibiting excellent resolution capabilities for frequency splitting phenomena. Furthermore, the case studies show that the dataset used is extremely accurate, with even minute differences precisely captured and clearly presented by AR-z spectroscopy. Clearly, in many geophysical datasets containing complex harmonic features (such as mantle oscillations and core-mantle boundary perturbations), robust AR-z spectroscopy can effectively detect hidden harmonic components and reveal potential fine structures, providing a powerful tool for high-precision signal analysis.

[0078] Specifically, the weak signal detection method based on z-domain autoregression and analytical continuation proposed in this invention has the following advantages:

[0079] 1. This invention uses a frequency domain autoregressive model to parameterize the frequency domain response of the observed sequence, and combines zero-filling extension technology to perform super-resolution interpolation of the Fourier spectrum. It breaks through the frequency point interval limitation (Δf=1 / T) of the traditional Fourier transform, and jointly estimates the complex frequency (real part frequency + imaginary part attenuation factor) of the dense frequency points on the complex plane, so as to accurately capture the damped oscillation characteristics of weak signals.

[0080] 2. This invention, based on prior complex frequency information (such as attenuation factor) of the target signal, extends the frequency domain response to the complex z-plane through analytical continuation, constructing a complex z-domain power spectral density function with the Lorentz function as its basis. This spectral function forms local extremum peaks in the complex plane, the positions of which correspond to the complex frequency parameters of the target signal, significantly enhancing the identifiability of signal features.

[0081] 3. Multiple sets of statistically independent random noise with the same noise level as the original sequence are superimposed on the original observation sequence to generate a set of noisy sequences. After independently calculating the AR-z spectrum for each set of noisy sequences, the geometric mean is used to obtain the product spectrum result of multiple sets of AR-z spectra. This process reduces the signal-to-noise ratio detection threshold of the AR-z spectrum to less than 1 through the superposition enhancement of coherent signal components and the statistical suppression of incoherent noise, and effectively suppresses the spurious peak interference caused by non-stationary noise. This invention is compatible with feature extraction of various geophysical signals such as seismic waves (damped oscillations), geomagnetic anomalies (transient pulses), and gravity gradients (slowly varying signals), providing a general framework for weak target detection in complex noise backgrounds.

[0082] Specifically, the weak signal detection method based on z-domain autoregression and analytical continuation proposed in this invention can extract weak signals from strong background noise sequences through the complex z-domain spectral analytical continuation technique of AR models, while improving the identification accuracy of spectral peak signals. This method can provide important technical support and innovative solutions for weak signal detection in strong noise backgrounds.

[0083] 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 weak signal detection system based on z-domain autoregression and analytic continuation. This system is used to execute a weak signal detection method based on z-domain autoregression and analytic continuation from the above method embodiments.

[0084] The system comprises: a first main module, used to randomly add noise to the acquired observation time series including weak target signals, resulting in multiple sets of noisy sequences; a second main module, used to calculate the AR-z spectrum for each noisy sequence, specifically: using frequency domain autoregressive estimation to estimate the complex frequency of the noisy sequence, combined with zero-filling expansion, to obtain the complex frequency information of the Fourier spectrum corresponding to the noisy sequence; based on the complex frequency information of the Fourier spectrum corresponding to the noisy sequence, combined with the prior complex frequency information of the weak target signal, constructing the AR-z spectrum of the noisy sequence through analytical extension; and a third main module, used to take the geometric mean of the amplitudes of the AR-z spectra obtained from the multiple sets of noisy sequences, to obtain the AR-z product spectrum, denoted as the robust AR-z spectrum, used for weak signal detection.

[0085] The weak signal detection system based on z-domain autoregression and analytical continuation provided in this invention addresses the problem of insufficient detection capability of traditional technologies under low signal-to-noise ratio conditions. It employs several modules to achieve super Nyquist frequency resolution, improving spectral refinement capability by more than three times compared to traditional Fourier transform for the same observation duration, and accurately capturing the damped oscillation characteristics of weak signals. Analytical continuation extends the frequency domain response to the complex z-plane, constructing a complex z-domain power spectral density function based on the Lorentz function. Through the synergistic processing of z-domain autoregression and random noise addition, it overcomes the frequency resolution limitations of Fourier transform and significantly reduces interference from other stronger signals, thereby detecting weak harmonic signals with a signal-to-noise ratio as low as less than 1.

[0086] 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 weak signal detection method based on z-domain autoregression and analytical continuation as proposed in the above embodiments.

[0087] This invention also provides a computer-readable storage medium storing a computer program thereon. When executed by a processor, the program performs super-resolution interpolation of the Fourier spectrum using zero-filling extension technology, breaking through the frequency interval limitation of traditional Fourier transform and significantly enhancing the identifiability of signal features.

[0088] This invention also provides a computer program product containing instructions that, when run on a computer, generate, in whole or in part, the weak signal detection method based on z-domain autoregression and analytic continuation proposed in the above embodiments. The computer may be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device.

[0089] 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 weak signal detection method based on z-domain autoregression and analytic continuation, characterized in that, include: Multiple sets of random noise were added to the acquired observation time series, which included weak signals of the target, to obtain multiple sets of noisy sequences; For each group of noisy sequences, the AR-z spectrum is calculated as follows: the complex frequency of the noisy sequence is estimated by frequency domain autoregressive estimation, and the complex frequency information of the Fourier spectrum corresponding to the noisy sequence is obtained by combining zero-filling expansion; based on the complex frequency information of the Fourier spectrum corresponding to the noisy sequence, combined with the prior complex frequency information of the target weak signal, the AR-z spectrum of the noisy sequence is constructed by analytical extension. The amplitudes of the AR-z spectra calculated from multiple sets of noisy sequences are geometrically averaged to obtain the AR-z product spectrum, which is called the robust AR-z spectrum and is used for weak signal detection.

2. The weak signal detection method based on z-domain autoregression and analytic continuation according to claim 1, characterized in that, The complex frequency of the noisy sequence is estimated using frequency domain autoregressive estimation, combined with zero-filling expansion, to obtain the complex frequency information of the Fourier spectrum corresponding to the noisy sequence, including: The frequency domain response of the noisy sequence is fitted by frequency domain autoregressive estimation to obtain the complex frequency information of the Fourier spectrum. The Fourier spectrum is interpolated using zero-filling extension to obtain a compact Fourier spectrum and its complex frequency information.

3. The weak signal detection method based on z-domain autoregression and analytical continuation according to claim 1, characterized in that, The AR-z spectrum of the noisy sequence is constructed by analytical continuation, including: Based on the prior complex frequency information of the target weak signal, the frequency domain response of the noisy sequence is extended to the complex z-plane through analytical continuation, generating an AR-z spectrum with the Lorentz function as the basis.

4. The weak signal detection method based on z-domain autoregression and analytical continuation according to claim 3, characterized in that, The AR-z spectrum expression is: ， in, ω i and These are the frequency estimate and attenuation factor estimate for the i-th frequency point, respectively; ,in Let be the Fourier frequency of the i-th frequency point. is the prior attenuation factor of the signal to be detected, and N represents the number of frequency points.

5. The weak signal detection method based on z-domain autoregression and analytic continuation according to claim 1, characterized in that, The robust AR-z spectrum expression is: ， Where, ω j and β j Let J represent the estimated frequencies and attenuation factors for all frequency points of the j-th sequence; J represents the number of superpositions, and ω represents the frequency values ​​of all frequency points. is the prior attenuation factor of the signal to be detected.

6. The weak signal detection method based on z-domain autoregression and analytical continuation according to claim 1, characterized in that, The target weak signal is a weak geophysical observation signal, including: Earth's medium- and long-period normal mode signals, seismic wave signals, geomagnetic anomaly signals, and gravity gradient signals.

7. A weak signal detection system based on z-domain autoregression and analytic continuation, characterized in that, A method for detecting weak signals based on z-domain autoregression and analytical continuation as described in claims 1-6 includes: The first main module is used to add multiple sets of random noise to the acquired observation time series, including weak signals of the target, to obtain multiple sets of noisy sequences; The second main module is used to calculate the AR-z spectrum for each group of noisy sequences. Specifically, it uses frequency domain autoregressive estimation to estimate the complex frequency of the noisy sequence and combines it with zero-filling expansion to obtain the complex frequency information of the Fourier spectrum corresponding to the noisy sequence. Based on the complex frequency information of the Fourier spectrum corresponding to the noisy sequence and combined with the prior complex frequency information of the target weak signal, it constructs the AR-z spectrum of the noisy sequence through analytical extension. The third main module is used to take the geometric mean of the amplitudes of the AR-z spectra calculated from multiple sets of noisy sequences to obtain the AR-z product spectrum, which is called the robust AR-z spectrum and is used for weak signal detection.

8. 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 weak signal detection method based on z-domain autoregression and analytical continuation as described in any one of claims 1 to 6.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the steps of the weak signal detection method based on z-domain autoregression and analytical continuation as described in any one of claims 1 to 6.

10. 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 weak signal detection method based on z-domain autoregression and analytical continuation as described in any one of claims 1 to 6.

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