A variable mode decomposition method for stepped frequency continuous wave signal denoising

Abstract

The application discloses a variable mode decomposition method for step frequency continuous wave signal noise reduction, and belongs to the field of signal modulation. The method solves the problem that the intensity of noise components in the existing spectrum synthesis is much higher than that of effective signals, and the effective suppression cannot be obtained. The method comprises the following steps: collecting a step frequency continuous wave echo signal, and decomposing the step frequency continuous wave echo signal into a plurality of independent single frequency signals; performing variable mode decomposition on each single frequency signal to obtain a plurality of intrinsic mode functions; screening modes with the same center frequency as the frequency of the effective signal as effective signal modes, and screening the remaining modes as noise modes; calculating the maximum information coefficient MIC between the effective signal modes and the noise modes; iteratively optimizing the parameters of the variable mode decomposition based on a Bayesian optimization method until the MIC value of the effective signal modes and the noise modes is lower than a set threshold; retaining the effective signal modes, and reconstructing a step frequency continuous wave signal after noise reduction. The application is applied to the fields of high-resolution target imaging and ranging.

Description

Technical Field

[0001] This invention belongs to the field of signal modulation, and in particular relates to a variable mode decomposition method for noise reduction of stepped frequency continuous wave signals. Background Technology

[0002] Stepped-frequency continuous wave (BFF) signals are frequency-modulated signals characterized by a frequency that gradually increases from a starting frequency in fixed steps. The amplitude and phase of each frequency component remain constant for a certain period, and the duration of hold at each frequency point is the same. BFF signals can achieve wide bandwidth coverage through progressive scanning, thereby improving the system's spectral resolution. Due to this advantage, BFF signals are widely used in high-resolution target imaging and ranging, such as ground-penetrating radar, radar for imaging the surface details of complex targets (aircraft, ships), industrial non-destructive testing, medical imaging, and autonomous driving.

[0003] However, because stepped-frequency continuous wave (TFWS) signals transmit and acquire each frequency component sequentially, each frequency component is measured only once at the receiver. If a frequency band is interfered with by noise, the amplitude and phase information of that band will be contaminated, affecting spectrum synthesis. A common solution is to use bandpass filters to remove noise, but this is ineffective against in-band noise interference. Furthermore, if the noise in the environment is continuous, it typically has incoherent characteristics. Therefore, during energy synthesis, the noise will not be superimposed or canceled out; instead, it will be randomly distributed in the frequency domain, forming a "noise floor," which will significantly reduce the signal-to-noise ratio (SNR) of the acquired signal. There is a special case where the interfering signal is within the TFWS frequency band. Because the TFWS system only samples the valid signal once during sampling, while the interfering signal is sampled in every sampling period, the intensity of this noise component in spectrum synthesis will be much higher than the intensity of the valid signal and cannot be effectively suppressed. This limits the measurement performance of TFWS signals. Summary of the Invention

[0004] In view of this, the present invention aims to propose a variable mode decomposition method for noise reduction of stepped frequency continuous wave signals, in order to solve the problem that the intensity of noise components in existing spectrum synthesis is much higher than the intensity of the effective signal and cannot be effectively suppressed.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] A variable mode decomposition method for denoising stepped-frequency continuous wave signals, the method comprising:

[0007] Step S1: Acquire the step frequency continuous wave echo signal and decompose the step frequency continuous wave echo signal into multiple independent single frequency signals according to the holding time of each frequency point;

[0008] Step S2: Perform variational mode decomposition on each single-frequency signal to obtain multiple intrinsic mode functions;

[0009] Step S3: Select the center frequency and the effective signal frequency f based on the intrinsic mode function. k The consistent mode is considered the effective signal mode, and the remaining modes are considered noise modes;

[0010] Step S4: Calculate the maximum information coefficient (MIC) between the effective signal mode and the noise mode to evaluate the noise reduction performance;

[0011] Step S5: Iteratively optimize the parameters of variational mode decomposition based on the Bayesian optimization method. The parameters include the number of decomposed modes, the penalty factor, and the initial center frequency, until the MIC values ​​of the effective signal mode and the noise mode are lower than a set threshold.

[0012] Step S6: Retain the effective signal modes and reconstruct the noise-reduced step-frequency continuous wave signal.

[0013] Furthermore, a preferred embodiment is proposed, wherein the decomposition of the single-frequency signal in step S1 includes:

[0014]

[0015] Among them, A k It is the amplitude of the k-th frequency band, f k φ represents the frequency of the k-th frequency band. k Let t represent the phase of the k-th frequency band, t represent time, e represent the base of the natural logarithm, and j represent the imaginary unit.

[0016] Furthermore, a preferred embodiment is proposed, wherein step S2 involves performing variational mode decomposition on each single-frequency signal, including:

[0017] Step S21: Construct the optimization objective function:

[0018]

[0019] Among them, {u k (t)}={u1(t),…,u K (t)} is a mode, {ω} k}={ω1,…,ω K} represents the center frequency of the mode. Let δ(t) be the unit impulse function, K be the time derivative, and x(t) be the total number of modes.

[0020] Step S22: Solve the objective function using the augmented Lagrange multiplier method;

[0021]

[0022] Where α is the penalty factor for the quadratic term, λ is the Lagrange operator, λ(t) is the Lagrange multiplier, and x(t) is the original input signal;

[0023] Step S23: Obtain each mode and its corresponding center frequency using the alternating direction multiplier algorithm;

[0024] Step S24: Repeat steps S21 to S23 until the equation is satisfied:

[0025]

[0026] in, For the (n+1)th modal component, Let ε be the nth modal component, and ε be the allowable tolerance range.

[0027] Furthermore, a preferred method is proposed, wherein the calculation of the maximum information coefficient (MIC) between the effective signal mode and the noise mode in step S4 includes:

[0028]

[0029] Where S1 represents signal 1, S2 represents signal 2, I(S1,S2) represents the mutual information between the two variables, and x and y represent the number of two-dimensional grid divisions for the two signals.

[0030] Furthermore, a preferred approach is proposed, in which the Bayesian optimization in step S5 adopts Gaussian process modeling, the acquisition function is the desired improvement EI or the confidence upper bound UCB, and the objective function is to minimize the sum of the MIC values ​​of the effective mode and the noise mode.

[0031] Furthermore, a preferred method is proposed, wherein the desired improvement is expressed as follows:

[0032] α EI (θ)=σ(θ)[zΦ(z)+φ(z)]

[0033] Where Φ(z) represents the cumulative distribution function of the standard normal distribution, φ(z) represents the probability density function of the standard normal distribution, σ(θ) is the uncertainty of the prediction standard deviation of the Gaussian process (GP) at point x, α is the penalty parameter, and z is the standardization improvement amount.

[0034] Furthermore, a preferred approach is proposed, wherein the upper confidence bound is expressed as:

[0035] αUCB (θ)=μ(θ)-κσ(θ)

[0036] Where κ represents the equilibrium parameter and μ(θ) represents the mean function.

[0037] Furthermore, a preferred method is proposed, wherein the reconstruction of the noise-reduced stepped-frequency continuous wave signal in step S6 includes: superimposing the retained effective signal modes in frequency order to recover the time-domain stepped-frequency continuous wave signal.

[0038] Based on the same inventive concept, the present invention also proposes a computer device, including a memory and a processor, wherein the memory stores a computer program, and when the processor runs the computer program stored in the memory, the processor executes a variable mode decomposition method for denoising stepped-frequency continuous wave signals according to any one of the preceding claims.

[0039] Based on the same inventive concept, the present invention also proposes a computer-readable storage medium storing a computer program that, when executed by a processor, performs the steps of a variable mode decomposition method for denoising stepped-frequency continuous wave signals as described in any of the above-mentioned embodiments.

[0040] Compared with the prior art, the beneficial effects of the present invention are:

[0041] 1. This invention decomposes each single-frequency signal into multiple intrinsic mode functions (IMFs) using variational mode decomposition (VMD). Combined with Bayesian optimization, it globally optimizes key VMD parameters (including the number of modes, penalty factor, and initial center frequency), accurately separating modes consistent with the effective signal frequency and suppressing in-band noise modes (such as communication interference and environmental noise). Compared to traditional bandpass filters, this method effectively eliminates the interference of in-band noise on spectrum synthesis, significantly improving the signal-to-noise ratio (SNR) and imaging resolution.

[0042] 2. This invention replaces traditional trial-and-error methods or empirical parameter configurations with a Bayesian optimization framework (based on Gaussian process modeling and expected improvement criteria). It can automatically search for the optimal parameter combination for VMD, solving the problems of mode aliasing or signal distortion caused by improper parameter selection in traditional methods. This method can still stably output high-precision decomposition results in complex noise environments (such as incoherent noise and random impulse interference), reducing the need for manual intervention.

[0043] 3. This invention utilizes the maximum information coefficient (MIC) to quantify the nonlinear dependence between effective modes and noise modes, overcoming the limitations of traditional methods that rely solely on spectral energy or linear correlation analysis. By dynamically evaluating the decomposition performance through the MIC value, it ensures sufficient separation of the effective signal and noise in the time-frequency domain, avoiding the loss of amplitude or phase information of the effective signal during noise reduction and guaranteeing the integrity of signal reconstruction.

[0044] 4. The method proposed in this invention constructs a surrogate model with the objective function (minimizing the MIC value) and combines it with a confidence upper bound (UCB) or expectation improvement (EI) strategy to quickly approximate the optimal parameter configuration with a small number of iterations, significantly reducing computational complexity. This method is suitable for high real-time scenarios (such as autonomous driving radar and real-time medical imaging), improving processing efficiency while ensuring noise reduction accuracy.

[0045] 5. The method proposed in this invention suppresses the random distribution of incoherent noise during spectrum synthesis by decomposing and reconstructing at frequency points, effectively reducing the negative impact of the "noise floor" on the dynamic range of the signal, and ensuring that the stepped frequency continuous wave system can still maintain high-precision ranging and imaging performance during large-bandwidth scanning. Attached Figure Description

[0046] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings:

[0047] Figure 1 This is a flowchart of a variable mode decomposition method for noise reduction of stepped frequency continuous wave signals according to the present invention.

[0048] Figure 2 This is a schematic diagram of the stepped-frequency continuous wave signal acquired according to the present invention;

[0049] Figure 3 This is a comparison diagram of the spectrum of the original signal acquired according to the present invention and the processed signal. Detailed Implementation

[0050] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. It should be noted that, unless otherwise specified, the embodiments and features in the embodiments of the present invention can be combined with each other, and the described embodiments are only some embodiments of the present invention, not all embodiments.

[0051] Implementation Method 1, see Figure 1 This embodiment describes a variable mode decomposition method for denoising stepped-frequency continuous wave signals. The method includes:

[0052] Step S1: Acquire the step frequency continuous wave echo signal and decompose the step frequency continuous wave echo signal into multiple independent single frequency signals according to the holding time of each frequency point;

[0053] Step S2: Perform variational mode decomposition on each single-frequency signal to obtain multiple intrinsic mode functions;

[0054] Step S3: Select the center frequency and the effective signal frequency f based on the intrinsic mode function. k The consistent mode is considered the effective signal mode, and the remaining modes are considered noise modes;

[0055] Step S4: Calculate the maximum information coefficient (MIC) between the effective signal mode and the noise mode to evaluate the noise reduction performance;

[0056] Step S5: Iteratively optimize the parameters of variational mode decomposition based on the Bayesian optimization method. The parameters include the number of decomposed modes, the penalty factor, and the initial center frequency, until the MIC values ​​of the effective signal mode and the noise mode are lower than a set threshold.

[0057] Step S6: Retain the effective signal modes and reconstruct the noise-reduced step-frequency continuous wave signal.

[0058] Stepped-frequency continuous wave signals are transmitted and received sequentially, with each frequency component being received. They are susceptible to noise interference. This implementation method decomposes the signal into multiple modes using a variable mode decomposition (VMD) method, retaining the effective signal components based on frequency. Compared to traditional filter-based noise reduction methods, this effectively suppresses noise within the signal's frequency band. Furthermore, the maximum information coefficient (MIC) quantifies not only the linear relationships between modes but also the nonlinear and nonfunctional dependencies between them. This MIC is used to evaluate the similarity between the decomposed effective signal modes and the noise signal modes, providing a quantitative indicator for the noise reduction performance of the VMD method. Moreover, by constructing a Gaussian process to estimate the objective function and guide the sampling process, each parameter adjustment of the VMD method can be made as efficient as possible, reducing computational complexity.

[0059] Implementation Method Two: This implementation method further defines the variable mode decomposition method for noise reduction of stepped-frequency continuous wave signals described in Implementation Method One. Step S1, decomposing the single-frequency signal, includes:

[0060]

[0061] Among them, A k It is the amplitude of the k-th frequency band, f k φ represents the frequency of the k-th frequency band. k Let t represent the phase of the k-th frequency band, t represent time, e represent the base of the natural logarithm, and j represent the imaginary unit.

[0062] Implementation Method 3: This implementation method further defines the variational mode decomposition method for step-frequency continuous wave signal noise reduction described in Implementation Method 2. Step S2 involves performing variational mode decomposition on each single-frequency signal, including:

[0063] Step S21: Construct the optimization objective function:

[0064]

[0065] Among them, {u k (t)}={u1(t),…,u K (t)} is a mode, {ω} k}={ω1,…,ω K} represents the center frequency of the mode. Let δ(t) be the unit impulse function, K be the time derivative, and x(t) be the total number of modes.

[0066] Step S22: Solve the objective function using the augmented Lagrange multiplier method;

[0067]

[0068] Where α is the penalty factor for the quadratic term, λ is the Lagrange operator, λ(t) is the Lagrange multiplier, and x(t) is the original input signal;

[0069] Step S23: Obtain each mode and its corresponding center frequency using the alternating direction multiplier algorithm;

[0070] Step S24: Repeat steps S21 to S23 until the equation is satisfied:

[0071]

[0072] in, For the (n+1)th modal component, Let ε be the nth modal component, and ε be the allowable tolerance range.

[0073] In this embodiment, the modal decomposition process is optimized by constructing an objective function, which more precisely controls the quality and stability of the modal decomposition. The objective function effectively balances different factors in the decomposition process, making the distinction between signal and noise more obvious, thereby improving the noise reduction effect. In step S22, the augmented Lagrange multiplier method is used to solve the objective function, which effectively handles constrained optimization problems, avoids local minima, and ensures the stability and accuracy of the final solution. The alternating direction multiplier algorithm (ADMM) used in step S23 decomposes the complex optimization problem into a series of simple sub-problems, solving them step by step through iterative calculations, significantly improving computational efficiency while maintaining high accuracy. This is particularly effective in reducing computational resource consumption when processing large-scale data. The iterative process in step S24 (from steps S21 to S23) ensures that each optimization gradually approaches the optimal solution, guaranteeing that the modal decomposition result tends towards the best. By continuously adjusting and optimizing the center frequency and other parameters of the modes, more accurate signal reconstruction and noise suppression can be achieved.

[0074] Implementation Method Four: This implementation method further defines the variable mode decomposition method for noise reduction of stepped-frequency continuous wave signals described in Implementation Method One. Step S4, calculating the maximum information coefficient (MIC) between the effective signal mode and the noise mode, includes:

[0075]

[0076] Where S1 represents signal 1, S2 represents signal 2, I(S1,S2) represents the mutual information between the two variables, and x and y represent the number of two-dimensional grid divisions for the two signals.

[0077] This implementation measures the similarity between signal modes and noise modes using the maximum information coefficient (MIC), which can more accurately identify which modes are components of the valid signal and which are noise. This process can reduce the noise component in the signal and improve signal quality.

[0078] Implementation Method 5: This implementation method further defines the variable mode decomposition method for noise reduction of stepped frequency continuous wave signals described in Implementation Method 1. In step S5, the Bayesian optimization adopts Gaussian process modeling, the acquisition function is the desired improvement EI or confidence upper bound UCB, and the objective function is to minimize the sum of the MIC values ​​of the effective mode and the noise mode.

[0079] Bayesian optimization methods, using Expectation Improvement (EI) or Confidence Upper Bound (UCB) as acquisition functions, help to find optimal parameters more efficiently. These acquisition functions avoid blind exploration and reduce invalid computation during optimization, thereby improving optimization efficiency. Setting the objective function as minimizing the sum of the MIC values ​​of effective and noisy modes helps to precisely control the signal-noise separation during denoising. The MIC value reflects the correlation between modes; by minimizing the MIC value, the effective part of the signal can be preserved to the greatest extent while minimizing noise. Gaussian process modeling can flexibly adjust the model when the data changes; therefore, this method can adapt to the characteristics of signals with different frequency advances, perform denoising on various complex signals, and has strong universality and adaptability.

[0080] Implementation Method Six: This implementation method further defines the variable mode decomposition method for denoising stepped-frequency continuous wave signals described in Implementation Method Five. The desired improvement is expressed as follows:

[0081] α EI (θ)=σ(θ)[zΦ(z)+φ(z)]

[0082] Where Φ(z) represents the cumulative distribution function of the standard normal distribution, φ(z) represents the probability density function of the standard normal distribution, σ(θ) is the uncertainty of the prediction standard deviation of the Gaussian process (GP) at point x, α is the penalty parameter, and z is the standardization improvement amount.

[0083] This implementation combines the cumulative distribution function (CDF) and probability density function (PDF) of the standard normal distribution to describe the signal distribution characteristics using a precise probabilistic model. This enables better extraction of effective signal information and improves denoising accuracy when processing signals in high-noise environments. Introducing a Gaussian process (GP) to predict the standard deviation uncertainty at a point allows for accurate capture of signal uncertainty during denoising, which is particularly important for handling complex and dynamically changing signals. Modeling uncertainty allows for better handling of signal variability, avoiding oversmoothing or overfitting. By appropriately setting the penalty parameter, the variable mode decomposition method can maintain good performance under different noise environments. Standardizing the improvement parameters eliminates numerical differences between different signals, making the improvement process more uniform and stable. This helps to unify the denoising effect regardless of the amplitude or characteristics of the original signal.

[0084] Implementation Method Seven: This implementation method further defines the variable mode decomposition method for denoising stepped-frequency continuous wave signals described in Implementation Method Five. The confidence upper bound is expressed as follows:

[0085] α UCB(θ)=μ(θ)-κσ(θ)

[0086] Where κ represents the equilibrium parameter and μ(θ) represents the mean function.

[0087] Implementation Method 8: This implementation method further defines the variable mode decomposition method for denoising stepped frequency continuous wave signals described in Implementation Method 1. The reconstruction of the denoised stepped frequency continuous wave signal in step S6 includes: superimposing the retained effective signal modes in frequency order to recover the time-domain stepped frequency continuous wave signal.

[0088] Implementation Method Nine: A computer device according to this implementation method includes a memory and a processor. The memory stores a computer program. When the processor runs the computer program stored in the memory, the processor executes a variable mode decomposition method for denoising stepped frequency continuous wave signals according to any one of Implementation Methods One to Seven.

[0089] Implementation Method 10: A computer-readable storage medium according to this embodiment stores a computer program, which, when executed by a processor, performs the steps of a variable mode decomposition method for step-frequency continuous wave signal denoising as described in any one of Embodiments 1 to 7.

[0090] Implementation Method 11, see below Figure 2 and Figure 3 This embodiment describes a specific example of the variable mode decomposition method for denoising stepped-frequency continuous wave signals described in Embodiment 1. It also serves to explain Embodiments 2 through 8. Specifically:

[0091] The first step is to acquire the step-frequency continuous wave echo signal.

[0092] The second step is to decompose the acquired stepped-frequency continuous wave signal. This decomposition is based on the characteristic that each frequency point in the stepped-frequency continuous wave signal has the same hold time, decomposing the signal into multiple single-frequency signals according to the hold time of each frequency point. The signal formula for the stepped-frequency continuous wave can be written as:

[0093]

[0094] f k =f0+kΔf(2)

[0095]

[0096] Among them, A k It is the amplitude of the k-th frequency band, f kf0 represents the starting frequency of the k-th frequency band, f0 represents the starting frequency of the stepped-frequency continuous wave signal, Δf is the frequency step size of the signal, and φ is the frequency of the k-th frequency band. k This represents the phase of the k-th frequency band, where t represents time. The unit impulse function represents the signal that operates only on a duration T. k It is valid within the time frame, where t0 is the start time of the signal.

[0097] The stepped frequency signal is divided into k single-frequency signals according to the holding time of each frequency point. The kth single-frequency signal can be written as:

[0098]

[0099] The third step involves decomposing the acquired stepped-frequency continuous wave signal into multiple independent single-frequency signals. Since the initial frequency and frequency step of the stepped-frequency continuous wave signal are known, the frequency f of the effective signal in each single-frequency signal can be inferred based on the number of steps k. k The variable mode decomposition method is used to process each single-frequency signal, decomposing it into multiple different modes, each with a different center frequency. In this embodiment, the center frequency and the frequency value f are compared. k Consistent modes are determined to be valid signals, and these modes are retained. The specific implementation is as follows:

[0100] like Figure 2 As shown, the acquired stepped-frequency continuous wave signal contains various signal components, including effective signal components, single-band communication interference noise, and full-band environmental interference noise. In this embodiment, variational mode decomposition is used to decompose the signal into multiple sub-signals. These sub-signals are called intrinsic mode functions (EMFs). For each EMF, it can be written as:

[0101] u k (t)=a k (t)cos(φ k (t))(5)

[0102] Among them, a k (t) represents the instantaneous amplitude of the k-th mode, φ k (t) is the phase function, and the derivative ω of the phase function k (t)=φ′ k (t) represents the frequency. The original signal can be represented as the sum of multiple intrinsic mode functions:

[0103]

[0104] To decompose the original signal into multiple intrinsic mode functions (EMFs), the variational mode decomposition method makes the assumption that for any mode, its spectrum is concentrated around its center frequency. Therefore, an optimization objective function can be constructed. First, the one-sided spectrum of each mode is solved using the Hilbert transform. Then, the spectrum of the mode is shifted to the baseband using an exponential term. Finally, a Gaussian smooth transition is used to estimate the bandwidth of the baseband signal. The above process can be written as:

[0105]

[0106] Among them, {u k (t)}={u1(t),…,u K (t)} and {ω k}={ω1,…,ω K} represents the mode and its corresponding center frequency, and then the augmented Lagrange multiplier method is used to solve the optimization problem.

[0107]

[0108] Where α is the penalty factor for the quadratic term, λ is the Lagrange operator, and K is the total number of modes. Then, each mode and its corresponding center frequency are obtained through an alternating direction multiplier algorithm, as shown in the following formula:

[0109]

[0110] Finally, repeat the above process until the equation is satisfied:

[0111]

[0112] Through the above steps, the variational mode decomposition method is used to decompose each subband of the stepped frequency continuous wave signal, and the signal gathered at each frequency point is decomposed into multiple independent eigenmode functions.

[0113] Fourth step, according to f k The modes are divided into effective signal modes and noise modes. The similarity between the effective signal modes and noise modes is then calculated using the maximum information coefficient to evaluate the denoising performance of the variable mode decomposition method. The specific implementation is as follows:

[0114] The multiple independent intrinsic mode functions generated after decomposing the acquired signal at each frequency point can be expressed as:

[0115]

[0116] Among them, u n r(t) represents the nth intrinsic mode function, which represents the component of the signal in a specific frequency band; r(t) represents the residual in the decomposition process, which is usually small.

[0117] For each u n (t) is subjected to a Fourier transform, as shown below.

[0118]

[0119] The center frequency of each intrinsic mode is compared with f k Compare the center frequency with f k Consistent modes are considered valid modes, while inconsistent modes are considered noise modes.

[0120] The maximum information coefficient (MAC) is a statistic that assesses the dependency between two variables, with a value ranging from [0,1]. It can quantify not only linear relationships between modes but also nonlinear and nonfunctional dependencies, making it particularly suitable for quantifying complex relationships between amplitude, phase, and frequency. It can be written as:

[0121]

[0122] Where S1 represents signal 1, S2 represents time 2, I(S1,S2) represents the mutual information between the two variables, x and y represent the number of two-dimensional grid divisions for the two signals, and F is the performance evaluation function.

[0123] The maximum information coefficient can adequately assess the nonlinear dependence between S1 and S2, illustrating the similarity between the effective signal mode and the noise mode. A high similarity between the effective signal mode and the noise mode indicates poor denoising performance of the variable mode decomposition method, as noise signals are still aliased within the effective modes. Conversely, lower similarity between the effective signal mode and the noise mode indicates better denoising performance of the variable mode decomposition method.

[0124] The fifth step is to reconfigure the parameters of the variable mode decomposition algorithm through Bayesian optimization to achieve the optimal noise reduction performance of the variable mode decomposition method. The specific implementation method is as follows:

[0125] Bayesian estimation is a global optimization method. Unlike traditional optimization methods, it uses a surrogate model, such as a Gaussian process, to estimate the objective function, thereby guiding the sampling process and making each optimization as efficient as possible. Among the parameters of the Bayesian method for optimizing variable mode decomposition, the parameters that have the greatest impact on noise reduction performance are: the number of decomposition modes K, the penalty parameter α, and the initial value ω of the mode center frequency. K The three parameters are optimized using Bayesian optimization. The goal is to find the optimal combination of these parameters that minimizes the similarity between the decomposed effective signal modes and the noise modes.

[0126] x*=argmin x∈χ f(x) (17)

[0127] Modeled using a Gaussian process, it is represented as follows:

[0128] L(θ)~GP(μ(θ),k(θ,θ′)) (18)

[0129] Here, μ(θ) represents the mean function, k(θ,θ′) represents the covariance function, and represents the similarity of different parameter combinations.

[0130] The acquisition function is used to balance development and exploration. The desired improvement can be expressed as:

[0131] α EI (θ)=σ(θ)[zΦ(z)+φ(z)] (19)

[0132] Where Φ(z) represents the cumulative distribution function of the standard normal distribution, and φ(z) represents the probability density function of the standard normal distribution.

[0133] The upper confidence bound can be expressed as:

[0134] α UCB (θ)=μ(θ)-κσ(θ)(20)

[0135] Here, κ represents the balance parameter, which controls the weight of exploration and development.

[0136] After repeatedly performing steps three, four, and five until the iteration limit is reached or the optimization target is achieved, the sixth step, signal reconstruction with noise reduction, begins. The single-frequency signal with noise suppression is reconstructed according to the order k of the signal's transmission frequency points to obtain a sufficiently denoised step-frequency continuous wave signal. Figure 3 As shown, the spectrum of the original signal and the spectrum of the original signal after processing using the method proposed in this embodiment are displayed. It can be clearly seen that the noise reduction effect is significantly improved.

[0137] In summary, this invention proposes a Bayesian parameter optimization-based variable mode decomposition (VMD) method for denoising stepped-frequency continuous-wave (TFWS) signals. This method first decomposes the acquired TFWS signal into multiple single-frequency signals. Then, the VMD method is used to decompose the signal into multiple modes, classifying the signal into effective modes and noise modes based on frequency. The similarity between the two modes is evaluated based on the maximum information coefficient. Next, a Bayesian optimization method is used to adjust the parameter configuration of the VMD method, minimizing the maximum information coefficient of both the effective and noise modes to effectively remove noise from the TFWS signal. Finally, the single-frequency signals are reconstructed to obtain a sufficiently denoised TFWS signal.

[0138] Those skilled in the art will understand that embodiments of this disclosure can be provided as methods, systems, or computer program products. Therefore, this disclosure can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this disclosure can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0139] This disclosure is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this disclosure. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create a machine for implementing the flowchart illustrations. Figure 1 One or more processes and / or boxes Figure 1 The computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to operate in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0140] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0141] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of this disclosure and not to limit its protection scope. Although this disclosure has been described in detail with reference to the above embodiments, those skilled in the art should understand that after reading this disclosure, they can still make various changes, modifications or equivalent substitutions to the specific implementation of the invention, but these changes, modifications or equivalent substitutions are all within the protection scope of the published pending claims.

Claims

1. A variable mode decomposition method for denoising stepped-frequency continuous wave signals, characterized in that, The method includes: Step S1: Acquire the step frequency continuous wave echo signal and decompose the step frequency continuous wave echo signal into multiple independent single frequency signals according to the holding time of each frequency point; Step S2: Perform variational mode decomposition on each single-frequency signal to obtain multiple intrinsic mode functions; Step S3: Select the center frequency and the effective signal frequency f based on the intrinsic mode function. k The consistent mode is considered the effective signal mode, and the remaining modes are considered noise modes; Step S4: Calculate the maximum information coefficient (MIC) between the effective signal mode and the noise mode to evaluate the noise reduction performance; Step S5: Iteratively optimize the parameters of variational mode decomposition based on the Bayesian optimization method. The parameters include the number of decomposed modes, the penalty factor, and the initial center frequency, until the MIC values ​​of the effective signal mode and the noise mode are lower than a set threshold. Step S6: Retain the effective signal modes and reconstruct the noise-reduced step-frequency continuous wave signal.

2. The variable mode decomposition method for noise reduction of stepped-frequency continuous wave signals according to claim 1, characterized in that, The decomposition of the single-frequency signal in step S1 includes: Among them, A k It is the amplitude of the k-th frequency band, f k φ represents the frequency of the k-th frequency band. k Let t represent the phase of the k-th frequency band, t represent time, e represent the base of the natural logarithm, and j represent the imaginary unit.

3. The variable mode decomposition method for noise reduction of stepped-frequency continuous wave signals according to claim 2, characterized in that, Step S2 involves performing variational mode decomposition on each single-frequency signal, including: Step S21: Construct the optimization objective function: Among them, {u k (t)}={u1(t),…,u K (t)} is a mode, {ω} k }={ω1,…,ω K } represents the center frequency of the mode. Let δ(t) be the unit impulse function, K be the time derivative, and x(t) be the total number of modes. Step S22: Solve the objective function using the augmented Lagrange multiplier method; Where α is the penalty factor for the quadratic term, λ is the Lagrange operator, λ(t) is the Lagrange multiplier, and x(t) is the original input signal; Step S23: Obtain each mode and its corresponding center frequency using the alternating direction multiplier algorithm; Step S24: Repeat steps S21 to S23 until the equation is satisfied: in, For the (n+1)th modal component, Let ε be the nth modal component, and ε be the allowable tolerance range.

4. The variable mode decomposition method for noise reduction of stepped-frequency continuous wave signals according to claim 1, characterized in that, Its features are, The calculation of the maximum information coefficient (MIC) between the effective signal mode and the noise mode in step S4 includes: Where S1 represents signal 1, S2 represents signal 2, I(S1,S2) represents the mutual information between the two variables, and x and y represent the number of two-dimensional grid divisions for the two signals.

5. The variable mode decomposition method for noise reduction of stepped-frequency continuous wave signals according to claim 1, characterized in that, In step S5, Bayesian optimization uses Gaussian process modeling. Its acquisition function is the desired improvement in EI or the confidence upper bound UCB, and the objective function is to minimize the sum of the MIC values ​​of the effective mode and the noise mode.

6. The variable mode decomposition method for denoising stepped-frequency continuous wave signals according to claim 5, characterized in that, The desired improvement is expressed as: α EI (θ)=σ(θ)[zΦ(z)+φ(z)] Where Φ(z) represents the cumulative distribution function of the standard normal distribution, φ(z) represents the probability density function of the standard normal distribution, σ(θ) is the uncertainty of the prediction standard deviation of the Gaussian process (GP) at point x, α is the penalty parameter, and z is the standardization improvement amount.

7. A variable mode decomposition method for denoising stepped-frequency continuous wave signals according to claim 5, characterized in that, The upper confidence bound is represented as follows: a UCB (θ)=μ(θ)-κσ(θ) Where κ represents the equilibrium parameter and μ(θ) represents the mean function.

8. The variable mode decomposition method for noise reduction of stepped-frequency continuous wave signals according to claim 1, characterized in that, The reconstruction of the noise-reduced step-frequency continuous wave signal in step S6 includes: superimposing the retained effective signal modes in frequency order to recover the time-domain step-frequency continuous wave signal.

9. A computer device, characterized in that: The device includes a memory and a processor, wherein the memory stores a computer program, and when the processor runs the computer program stored in the memory, the processor executes a variable mode decomposition method for denoising stepped-frequency continuous wave signals according to any one of claims 1-7.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that, when executed by a processor, performs the steps of a variable mode decomposition method for denoising stepped-frequency continuous wave signals as described in any one of claims 1-7.

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