Complex exponential signal joint spectrum reconstruction and parameter estimation method and device

Abstract

The invention discloses a complex exponential signal joint spectrum reconstruction and parameter estimation method and device, and relates to the field of signal processing, and the method comprises the steps: S1, constructing noise-containing complex exponential signal training data and label data; s2, constructing a dual-module neural network model comprising a super-resolution denoising reconstruction module and a parameter prediction module; s3, training the dual-module neural network model by using the training data and the annotation data to obtain a trained dual-module neural network model; and S4, performing frequency spectrum reconstruction and parameter prediction by using the trained dual-module neural network model, and performing signal post-processing on the output to obtain estimation parameters of the angular frequency, the attenuation factor, the real part amplitude and the imaginary part amplitude. According to the invention, a cascade neural network architecture of a super-resolution denoising reconstruction module and a parameter prediction module is designed, and angular frequency detection is converted into a Gaussian distribution heat map regression task; meanwhile, a sparse activation labeling mechanism is adopted, the parameter truth value is only reserved at the spectrum peak position, and the model learning complexity is remarkably reduced.

Description

Technical Field

[0001] The present invention relates to the field of signal processing, and in particular to a method and device for joint spectrum reconstruction and parameter estimation of complex exponential signals. Background Art

[0002] Estimating the parameters of complex exponential signals is a crucial issue in signal processing. Its mathematical model is typically represented as the superposition of multiple complex exponential signal components, each with a specific angular frequency, attenuation factor, and amplitude. Accurately estimating these parameters is crucial for signal analysis, feature extraction, and subsequent processing. Traditional methods, such as the Prony algorithm, extract parameters using a linear prediction model, but their performance degrades dramatically in noisy environments and require a predefined number of signal components, resulting in significant limitations in practical applications. In recent years, deep learning-based approaches have improved noise immunity through end-to-end mapping, but existing models often focus on a single task (such as denoising or frequency detection) and lack an end-to-end solution for the joint estimation of complex signal parameters. For example, existing models typically employ a staged approach (denoising followed by parameter regression), which can lead to error accumulation between stages. Furthermore, existing methods struggle to achieve high-precision parameter separation in scenarios with overlapping spectral peaks in multi-component signals. Summary of the Invention

[0003] To address the above problems, the present invention proposes a method and device for joint spectrum reconstruction and parameter estimation of complex exponential signals. Through a cascade design of two modules, spectrum denoising and super-resolution reconstruction and the introduction of high-resolution parameter estimation are performed respectively. Four parallel branch networks are used for multi-task regression optimization, breaking through the traditional spectrum resolution limitation, thereby improving the accuracy and stability of signal parameter estimation, and solving the accuracy problems of denoising and multi-parameter estimation of complex exponential signals, especially the performance improvement problem in the cases of low signal-to-noise ratio (SNR) and multiple spectral peaks.

[0004] On the one hand, the complex exponential signal joint spectrum reconstruction and parameter estimation method, the specific steps are as follows:

[0005] S1, construct noisy complex exponential signal training data and label data;

[0006] S2, constructing a dual-module neural network model including a super-resolution denoising and reconstruction module and a parameter prediction module;

[0007] The super-resolution denoising and reconstruction module adopts an improved U-Net structure to denoise and improve the resolution of the input noisy complex exponential signal and output a high-resolution denoised spectrum;

[0008] The parameter prediction module includes a main part and four branch networks; the main part uses two sequentially connected residual modules to extract spectral features from the input high-resolution denoised spectrum, and outputs the extracted features to four parallel branch networks, which respectively output the predicted angular frequency Gaussian heat map, attenuation factor sparse point map, real amplitude sparse point map, and imaginary amplitude sparse point map;

[0009] S3, using the training data and the label data to train the dual-module neural network model to obtain a trained dual-module neural network model;

[0010] S4, use the trained dual-module neural network model to perform spectrum reconstruction and parameter prediction, and perform signal post-processing on the output predicted angular frequency Gaussian heat map, attenuation factor sparse scatter plot, real amplitude sparse scatter plot and imaginary amplitude sparse scatter plot to obtain the estimated parameters of angular frequency, attenuation factor, real amplitude and imaginary amplitude.

[0011] Preferably, the construction of noisy complex exponential signal training data is as follows:

[0012] A complex exponential signal is generated using a complex exponential signal mathematical model; the complex exponential signal mathematical model is expressed as:

[0013]

[0014] Where x(n) represents a complex exponential signal; K represents the number of signal components; A r,k and A i,k denote the real and imaginary amplitudes of the kth component respectively; σ k represents the attenuation factor; ω k Represents the angular frequency; J represents the signal sampling point set, n∈J;

[0015] Perform signal interception on the complex exponential signal and add Gaussian noise to generate a noisy signal;

[0016] The noisy signal is converted into spectrum data as the noisy complex exponential signal training data.

[0017] Preferably, the label data includes a high-resolution noise-free spectrum label, an angular frequency label, an attenuation factor label, a real amplitude label and an imaginary amplitude label; the high-resolution noise-free spectrum label is generated by performing signal interception and Fourier transform on the complex exponential signal; the angular frequency label is a high-resolution Gaussian heat map; the high-resolution Gaussian heat map is modeled by Gaussian distribution, and the real angular frequency of the high-resolution noise-free spectrum is mapped to the discrete frequency grid points of the high-resolution spectrum, and a Gaussian heat map peak is generated at the mapped frequency point; the attenuation factor label, real amplitude label and imaginary amplitude label are all high-resolution sparse scatter plots; the high-resolution sparse scatter plot only marks the true value of the attenuation factor, real amplitude or imaginary amplitude parameters at the spectral peak position mapped to the high-resolution spectrum, and the rest of the area is set to zero.

[0018] Preferably, the loss function of the dual-module neural network model is expressed as:

[0019]

[0020] Among them, X denoised Represents the high-resolution spectrum output by the super-resolution denoising and reconstruction module; X clean represents the high-resolution noise-free spectrum label; H pred represents the predicted angular frequency Gaussian heat map; H label Indicates the angular frequency label; Represents the kth component of the predicted attenuation factor sparse scatter plot; represents the kth component of the attenuation factor label; Represents the kth component of the predicted real amplitude sparse plot; Represents the kth component of the real amplitude label; represents the kth component of the predicted imaginary amplitude sparse plot; represents the kth component of the imaginary amplitude label; α, β and γ represent weight coefficients; N represents the total number of pixels in the spectrum; FocalLoss(H pred ,H label ) indicates focal loss; represents the square of the Euclidean distance; || represents the absolute value.

[0021] Preferably, the focus loss is expressed as:

[0022]

[0023] Among them, p i represents the probability of a spectrum peak at the ith frequency position of the predicted angular frequency Gaussian heat map, normalized to [0,1] by Sigmoid; i Represents the normalized Gaussian value at the ith position in the angular frequency label.

[0024] Preferably, the improved U-Net structure includes an encoder, a bottleneck layer and a decoder;

[0025] The encoder includes multiple layers of convolution, each layer of convolution is followed by batch normalization and ReLU activation function; the encoder gradually downsamples the input noisy complex exponential signal through multiple layers of convolution to extract frequency domain features and outputs them;

[0026] The bottleneck layer includes a bidirectional LSTM network and a linear layer. The bidirectional LSTM network performs spectral feature modeling on the input frequency domain features to capture the dependencies between spectral components. The output of the bidirectional LSTM layer is projected through the linear layer and serves as the output of the bottleneck layer.

[0027] The decoder includes three layers of transposed convolution and an upsampling denoising layer; the features of each transposed convolution layer are fused with the features of the corresponding convolution layer of the encoder through channel splicing; the output of the bottleneck layer is gradually upsampled through three layers of transposed convolution, and then denoised through the upsampling denoising layer to obtain a high-resolution denoised spectrum and output it.

[0028] Preferably, the resolution of the output of the super-resolution denoising and reconstruction module is twice the resolution of the input of the super-resolution denoising and reconstruction module.

[0029] Preferably, the signal post-processing is specifically as follows:

[0030] Perform non-maximum suppression on the predicted high-resolution angular frequency Gaussian heat map: Use a fixed window to traverse the Gaussian heat map, compare the response value of each position in the fixed window with other points in the window one by one, and retain the point with the maximum response value in the window that exceeds the preset threshold as the candidate peak;

[0031] A local window is established with the candidate peak as the center, the amplitude data within the local window is extracted, and the least squares method is used to fit it into a Gaussian curve to calculate the sub-bin frequency offset; the offset is added to the angular frequency corresponding to the candidate peak position to obtain the angular frequency estimation parameter;

[0032] The parameter values of the corresponding index positions in the predicted attenuation factor sparse dot plot, the real amplitude sparse dot plot, and the imaginary amplitude sparse dot plot are read from the position index of the candidate peak to obtain the estimated parameters of the attenuation factor, the real amplitude, and the imaginary amplitude, thereby completing the parameter estimation.

[0033] Preferably, the angular frequency estimation parameter is expressed as:

[0034]

[0035] Where ω represents the angular frequency estimation parameter; peak binIndicates the discrete frequency bin index where the candidate spectrum peak is located; Δμ represents the sub-bin frequency offset; N represents the total number of pixels in the spectrum graph.

[0036] On the other hand, a complex exponential signal joint spectrum reconstruction and parameter estimation device includes the following:

[0037] A training data and label data construction module is used to construct noisy complex exponential signal training data and label data;

[0038] A model building module, used to build a dual-module neural network model including a super-resolution denoising reconstruction module and a parameter prediction module;

[0039] The super-resolution denoising and reconstruction module adopts an improved U-Net structure to denoise and improve the resolution of the input noisy complex exponential signal and output a high-resolution denoised spectrum;

[0040] The parameter prediction module includes a main part and four branch networks; the main part uses two sequentially connected residual modules to extract spectral features from the input high-resolution denoised spectrum, and outputs the extracted features to four parallel branch networks, which respectively output the predicted angular frequency Gaussian heat map, attenuation factor sparse point map, real amplitude sparse point map, and imaginary amplitude sparse point map;

[0041] Model training module, used to train the dual-module neural network model using training data and label data to obtain a trained dual-module neural network model

[0042] The spectrum reconstruction and parameter estimation module is used to perform spectrum reconstruction and parameter prediction using the trained dual-module neural network model. The output predicted angular frequency Gaussian heat map, attenuation factor sparse scatter plot, real amplitude sparse scatter plot and imaginary amplitude sparse scatter plot are respectively subjected to signal post-processing to obtain the estimated parameters of angular frequency, attenuation factor, real amplitude and imaginary amplitude.

[0043] Compared with the prior art, the present invention has the following beneficial effects:

[0044] (1) The present invention effectively improves the resolution of the spectrum through a super-resolution denoising and reconstruction module, thereby enhancing the accuracy of frequency estimation and breaking through the frequency resolution limitation of traditional methods. At the same time, the joint optimization of super-resolution reconstruction and denoising functions reduces the interference of noise on parameter estimation and improves robustness in low signal-to-noise ratio environments.

[0045] (2) The present invention is based on a feature extraction network composed of a sparse activation annotation mechanism and residual blocks in a parameter prediction module, which effectively distinguishes densely distributed spectral peak components and supports adaptive processing of dynamic component numbers;

[0046] (3) The present invention is based on predicting the gradient response characteristics of the Gaussian heat map. After extracting the main peak through non-maximum suppression, the sub-bin frequency value of the angular frequency is inferred by fitting the heat map distribution morphology, breaking through the discrete frequency resolution limitation. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] The present invention will be described in further detail below with reference to the accompanying drawings;

[0048] Figure 1 Flowchart of a method for joint spectrum reconstruction and parameter estimation of complex exponential signals according to an embodiment of the present invention;

[0049] Figure 2 Schematic diagram of the overall architecture of the method for joint spectrum reconstruction and parameter estimation of complex exponential signals according to an embodiment of the present invention;

[0050] Figure 3 This is a structural diagram of a super-resolution denoising and reconstruction module of a method for joint spectrum reconstruction and parameter estimation of complex exponential signals according to an embodiment of the present invention;

[0051] Figure 4 This is a multi-branch network structure diagram of a parameter prediction module of a method for joint spectrum reconstruction and parameter estimation of complex exponential signals according to an embodiment of the present invention;

[0052] Figure 5 Figure 1 shows the effect of super-resolution spectrum reconstruction using the combined spectrum reconstruction and parameter estimation method for complex exponential signals according to an embodiment of the present invention; (a) is the spectrum of a noisy low-resolution signal (256-point FFT); (b) is the ideal high-resolution spectrum label (512-point FFT); (c) is the high-resolution spectrum output by the super-resolution denoising and reconstruction module;

[0053] Figure 6 Renderings of the angular frequency heat map prediction and spectrum peak location results of the complex exponential signal joint spectrum reconstruction and parameter estimation method according to an embodiment of the present invention; (a) is the ideal Gaussian heat map label; (b) is the predicted Gaussian heat map distribution;

[0054] Figure 7 This is an effect diagram of attenuation factor estimation of the method for joint spectrum reconstruction and parameter estimation of complex exponential signals according to an embodiment of the present invention;

[0055] Figure 8 Renderings of the real and imaginary part amplitude estimation of the complex exponential signal joint spectrum reconstruction and parameter estimation method according to an embodiment of the present invention; (a) is a rendering of the real part amplitude estimation; (b) is a rendering of the imaginary part amplitude estimation;

[0056] Figure 9 4 is a structural block diagram of a complex exponential signal joint spectrum reconstruction and parameter estimation device according to an embodiment of the present invention. DETAILED DESCRIPTION

[0057] The present invention is further described below through specific embodiments.

[0058] like Figure 1 As shown in Figure 1, the complex exponential signal joint spectrum reconstruction and parameter estimation method based on deep learning is as follows:

[0059] S1, construct noisy complex exponential signal training data and label data.

[0060] First, the generation of training data is based on the complex exponential signal mathematical model, which is expressed as:

[0061]

[0062] Where x(n) represents the generated complex exponential signal, K is the number of signal components, and A r,k and A i,k are the real and imaginary amplitudes of the kth component, σ k is the attenuation factor, ω k is the angular frequency, J is the set of signal sampling points; the angular frequency ω of each component k Randomly distributed in the interval [0.1π, 1.9π], the adjacent frequency interval is controlled to meet the preset minimum frequency requirement 2π / 255, the attenuation factor σ k Distributed in [1 / 300,1 / 50], real and imaginary amplitude A r,k 、A i,k All are randomly sampled from the interval [0.1,1].

[0063] The noisy signal is generated by intercepting the first 256 points and adding Gaussian noise. The noise power is calculated according to the signal-to-noise ratio formula:

[0064]

[0065] Among them, P signal is the signal power, SNR dB The target signal-to-noise ratio (SNR) value in decibels is set to 0-10dB to cover low SNR scenarios.

[0066] In this embodiment, the input of the network is the spectrum data of the generated complex exponential signal, and the dual-channel data (shape 2×256) is generated by 256-point FFT.

[0067] Label data generation. The angular frequency label is a high-resolution Gaussian heat map, which is generated by superimposing a Gaussian distribution at the position of the true frequency component. The true angular frequency is mapped to the discrete frequency grid point of the high-resolution spectrum, and the Gaussian heat map peak is generated at this position. The attenuation factor, real amplitude, and imaginary amplitude labels are high-resolution sparse dot maps, with the true parameter value only marked at the peak position mapped to the high-resolution spectrum, and the rest of the area is set to zero. The specific high-resolution label spectrum is generated by taking the first 512 points of the signal x(n) and performing an FFT (shape 2×512). The parameter label is constructed based on 512-point resolution: the angular frequency label uses a Gaussian heat map with a standard deviation σ of 3.0, and the peak is mapped to the high-resolution frequency grid point; the attenuation factor and amplitude labels generate 512-point sparse dot maps, and the parameter value is only marked at the true spectrum peak position.

[0068] S2, construct a dual-module neural network model including a super-resolution denoising and reconstruction module and a parameter prediction module.

[0069] The super-resolution denoising and reconstruction module of this embodiment adopts an improved U-Net structure, including an encoder, a bottleneck layer, and a decoder. The input is dual-channel data consisting of the real and imaginary parts of the noisy spectrogram. The encoder downsamples to extract deep features, that is, gradually downsampling to extract frequency domain features through multiple layers of convolution, gradually reducing the spatial size of the signal and increasing the depth of the features. Each convolution layer is followed by batch normalization and ReLU activation function to improve feature extraction capability and stability. Spectral feature modeling is performed through the LSTM layer in the bottleneck layer to capture the dependencies between spectral components. The output of the LSTM layer is projected through a linear layer to adapt to subsequent decoder processing. The decoder reconstructs the signal through transposed convolution and skip connections, gradually upsampling through transposed convolution layers, and fusing the encoder features with the decoder output through skip connections (splicing operations). Each decoder layer receives high-level features from the encoder and enhances detail recovery capabilities through splicing operations. The final output is a denoised spectrogram with a resolution increased to twice that of the input spectrum, and the reconstruction loss is calculated with the high-resolution noise-free spectrogram. The improved U-Net structure of this embodiment combines the encoder-decoder skip connection design of the traditional U-Net and introduces a bidirectional LSTM as a bottleneck layer for spectral feature modeling. Similar attempts have been made to use this structure in the field of speech separation (for example, in models such as Demucs). On this basis, adjustments are made to adapt to the current task, and an upsampling layer and a denoising layer are added to the decoder to improve the output spectral resolution.

[0070] For details, see Figure 3As shown, the network architecture of the dual-module neural network model uses a spectrum-enhanced super-resolution U-Net as the first module. Its encoder consists of three convolutional layers (kernel size 3, stride 2), which gradually downsamples the input resolution from 256 to 32 points. Each layer outputs 64 channels, and batch normalization and ReLU activation are used to enhance feature representation. A bidirectional LSTM layer is introduced at the end of the encoder to perform bidirectional spectral feature modeling on the 32×64 feature sequence along the frequency axis, capturing the dependencies between harmonic components. The output is a 128-dimensional spectral feature, which is then resized using linear projection. The decoder gradually upsamples the image through three layers of transposed convolution. Each layer fuses the features of the corresponding encoder layer through channel concatenation. Finally, a customized upsampling denoising layer (kernel size 3, stride 2, and output padding 1) increases the resolution from 256 to 512 points, outputting a high-resolution denoised spectrum.

[0071] It's worth noting that the super-resolution denoising and reconstruction module increases the final output resolution to a denoised spectrum that's twice the input spectrum. Parameter settings allow for designing super-resolution outputs with other multiples, such as 1.5x or 4x. However, experiments have shown that increasing the resolution to 2x provides a good balance between model complexity and spectrum reconstruction accuracy. Further increasing the resolution does not, under certain conditions, result in significant performance improvements.

[0072] The parameter prediction module is the second module of the dual-module neural network model, see Figure 4 As shown, the shared backbone network of the parameter prediction module of this embodiment extracts high-resolution spectral features through two residual blocks, integrates convolution and spatial attention modules in each residual block, uses convolution to process time series data, and captures global information and key areas through the spatial attention module. Specifically, the data first passes through a convolution layer with a convolution kernel size of 3, is normalized and activated by ReLU, and then passes through a second convolution layer and normalization to obtain intermediate features; at the same time, residual connection is realized through 1×1 convolution. The spatial attention module extracts global information in the frequency dimension through convolution with a larger kernel, and then forms an attention weight map through Sigmoid activation, which is multiplied element-wise with the output of the residual block, that is, dynamic weights are assigned to each position, thereby highlighting important areas and suppressing noise. The four branch networks of the parameter prediction module include independent convolution and activation modules, and the four branch networks output 512-point Gaussian heat maps (angular frequencies) and sparse scattered point maps (attenuation factors, real / imaginary amplitudes), respectively.

[0073] The parameter prediction module of this embodiment includes four parallel branch networks, which respectively predict the angular frequency Gaussian heat map, attenuation factor, real amplitude and imaginary amplitude parameters. The output resolution of each branch is aligned with the high-resolution spectrum output by the denoising module. The parameter value is regressed only at the spectral peak position through the sparse activation labeling mechanism, and the regression loss is calculated with the parameter label.

[0074] S3, using the training data and the label data to train the dual-module neural network model to obtain a trained dual-module neural network model.

[0075] The first and second modules are trained jointly, and the network parameters are optimized through a multi-task loss function, which is a comprehensive function that combines spectrum reconstruction error, heat map focus loss and parameter regression error. -4 , batch size 64) are jointly trained until convergence.

[0076] The angular frequency branch adopts focal loss, and its mathematical formula is expressed as:

[0077]

[0078] Among them, K is the number of components that make up the signal, that is, the number of peaks in the ideal spectrum; p i is the probability of the existence of a spectrum peak at the i-th frequency position predicted by the model (normalized to [0,1] by Sigmoid), y i is the normalized Gaussian value of the i-th position in the label heat map; combined with the mask weighted L1 loss (attenuation factor, amplitude branch) to form a multi-task loss function, expressed as:

[0079]

[0080] Among them, X denoised is the high-resolution spectrum output by the denoising module, X clean is the high-resolution noise-free spectrum label; H pred is the predicted angular frequency Gaussian heat map, H label is the angular frequency heat map label; σ pred 、A r,pred and A i,pred are the original prediction outputs of the attenuation factor, real amplitude and imaginary amplitude branches, σ label 、A r,label and A i,label are the labels of attenuation factor, real amplitude and imaginary amplitude respectively; α, β and γ are weight coefficients, N is the total number of pixels in the spectrum graph, and K is the number of components of the signal.

[0081] S4, use the trained dual-module neural network model to perform spectrum reconstruction and parameter prediction, and perform signal post-processing on the output predicted angular frequency Gaussian heat map, attenuation factor sparse scatter plot, real amplitude sparse scatter plot and imaginary amplitude sparse scatter plot to obtain the estimated parameters of angular frequency, attenuation factor, real amplitude and imaginary amplitude.

[0082] Signal post-processing includes the following steps:

[0083] Step 1: Perform non-maximum suppression (NMS) on the predicted high-resolution angular frequency Gaussian heat map to locate the main peak area with significant energy;

[0084] Step 2: A local window is captured with the main peak as the center, and the sub-bin frequency offset is calculated using a Gaussian fitting algorithm. This is then combined with the high-resolution frequency bin index to generate a super-resolution angular frequency estimate.

[0085] Step 3: Based on the extracted spectrum peak position index, read the parameter value of the corresponding position from the sparse prediction map output by the high-resolution attenuation factor and real / imaginary amplitude branch;

[0086] Specifically, in the post-processing stage, candidate peaks are first screened by non-maximum suppression (NMS), and according to the extracted spectral peak position index, the parameter value of the corresponding position is read from the sparse prediction map of the attenuation factor and amplitude branch to complete the parameter estimation. This embodiment can achieve spectrum super-resolution positioning through the "coarse detection + fine correction" strategy. "Coarse detection" refers to the process of preliminarily screening out candidate peak positions from the predicted heat map through non-maximum suppression: first, a threshold is set, and only points above the threshold are considered as candidates; then a fixed window (for example, 5 bins) is used to slide through the heat map, and the response value of each position is compared with that of other points in the window. If the point is a local maximum and exceeds the threshold, it is retained.

[0087] Fine-tuning can further optimize the position accuracy through the network or other methods. In this embodiment, a method of fitting a local window to a Gaussian distribution is used to calculate the sub-bin frequency offset: for each candidate peak, a local window is selected near it (for example, ±2 bins centered on the peak), the amplitude data within the window is extracted, and a Gaussian curve is fitted using a least squares method or other method to calculate the sub-bin offset Δμ between the peak position and the initial bin position. This offset is added to the initial candidate bin position to obtain a more accurate angular frequency estimate. The final angular frequency value is expressed as:

[0088]

[0089] Among them, peak bin is the discrete frequency bin index of the candidate spectrum peak determined in the coarse detection stage, Δμ is the sub-bin frequency offset calculated after fine correction, and N is the total number of pixels in the spectrum graph.

[0090] Verification results:

[0091] The results are as follows Figures 5 to 8 The cases shown are typical test samples randomly selected, and their parameter range covers the distribution boundary of training data. Figure 5Presentation, in which Figure 5 (a) shows the spectrum of a noisy low-resolution signal, showing obvious noise interference and frequency aliasing. Figure 5 Middle (b) shows an ideal high-resolution spectrum label, where the peak resolution is improved and the spectral line details are clearly discernible; Figure 5 (c) is the high-resolution spectrum output by the super-resolution denoising module of this embodiment. The amplitude correlation coefficient with the label spectrum reaches 0.98. The algorithm has a significant effect in noise suppression and resolution improvement.

[0092] The angular frequency heat map prediction and spectrum peak location effect are determined by Figure 6 Present. Figure 6 (a) shows the ideal Gaussian heat map label. Five spectral peaks are marked on the grid points mapped to the high-resolution spectrum, located at frequency points 111, 223, 375, 389, and 464, respectively. The centers of the label peaks are marked by dotted lines. Figure 6 Middle (b) shows the Gaussian heat map distribution predicted by this embodiment. The dotted lines mark the candidate peak centers (bins 110, 223, 375, 465) directly detected by non-maximum suppression (NMS), corresponding to angular frequencies of 0.431π, 0.873π, 1.468π, 1.523π, and 1.820π. The dotted line marks the NMS threshold (0.75). The final angular frequency values obtained after calculating the sub-bin level offset are 0.4307π, 0.8732π, 1.4684π, 1.5225π, and 1.8185π.

[0093] Figure 7 and Figure 8 The system demonstrates super-resolution parameter regression performance. Figure 7 The results of the σ parameter regression analysis are presented. The true σ values are distributed in the range [1 / 300, 1 / 40]. The absolute errors of the predicted σ values corresponding to the five spectral peak positions are 0.0028 (bin 110), 0.0013 (bin 223), 0.0001 (bin 375), 0.0060 (bin 389), and 0.0026 (bin 465), respectively, with a mean absolute error of 0.0026. Notably, at bin 389, where the signal strength is relatively weak (true amplitude Ar = 0.2229, Ai = 0.5238), the attenuation factor estimation error reaches 0.0060, significantly higher than that in other main peak regions, indicating that noise interference has a more significant impact on parameter estimation accuracy in weak peak scenarios. The dotted line in the figure connects the true peak position (True Peaks) and the predicted peak position (Pred Peaks). It can be seen that the predicted values are closely distributed near the true values, and the predicted values in the non-peak area (light-colored scattered points) show low-amplitude noise characteristics, which verifies the model's specific regression ability for the attenuation factor parameters.

[0094] Figure 8The regression analysis of (a) the real amplitude Ar and (b) the imaginary amplitude Ai are shown respectively. In each sub-figure, the star marks represent the true amplitude values (True Values), the hollow symbols represent the predicted values corresponding to the true spectral peak positions (At TruePeaks), and the solid symbols represent the parameter estimates of the predicted spectral peak positions (At Pred Peaks). The regression mean absolute error of the Ar parameter is 0.0490, and the maximum deviation occurs at bin389 (absolute error 0.1276); the mean absolute error of the Ai parameter is 0.0910, of which the maximum deviation occurs at bin110 (absolute error 0.1772). Similarly, in weak peak areas (such as bin389, true amplitude Ar = 0.2229, imaginary amplitude Ai = 0.5238), the parameter estimation error is larger than that in other main peak areas, indicating that low signal-to-noise ratio scenarios have a marginal impact on model performance. The standard deviation of the parameter estimates between the predicted peak positions (solid triangles / squares) and the true peak positions (open symbols) is 0.0619, validating the model's robustness to frequency shifts. All subplots show that the predicted values approach zero in the non-peak regions (light background), consistent with the sparse nature of the true signal.

[0095] like Figure 9 As shown, the present invention also discloses a complex exponential signal joint spectrum reconstruction and parameter estimation device, comprising:

[0096] The training data and label data construction module 901 is used to construct noisy complex exponential signal training data and label data;

[0097] A model construction module 902 is used to construct a dual-module neural network model including a super-resolution denoising and reconstruction module and a parameter prediction module;

[0098] The super-resolution denoising and reconstruction module adopts an improved U-Net structure to denoise and improve the resolution of the input noisy complex exponential signal and output a high-resolution denoised spectrum;

[0099] The parameter prediction module includes a main part and four branch networks; the main part uses two sequentially connected residual modules to extract spectral features from the input high-resolution denoised spectrum, and outputs the extracted features to four parallel branch networks, which respectively output the predicted angular frequency Gaussian heat map, attenuation factor sparse point map, real amplitude sparse point map, and imaginary amplitude sparse point map;

[0100] Model training module 903 is used to train the dual-module neural network model using training data and label data to obtain a trained dual-module neural network model

[0101] The spectrum reconstruction and parameter estimation module 904 is used to perform spectrum reconstruction and parameter prediction using the trained dual-module neural network model, and perform signal post-processing on the output predicted angular frequency Gaussian heat map, attenuation factor sparse scatter plot, real amplitude sparse scatter plot and imaginary amplitude sparse scatter plot to obtain the estimated parameters of angular frequency, attenuation factor, real amplitude and imaginary amplitude.

[0102] The specific implementation of the complex exponential signal joint spectrum reconstruction and parameter estimation apparatus is the same as the complex exponential signal joint spectrum reconstruction and parameter estimation method, and will not be repeated in this embodiment.

[0103] The above is only a specific implementation of the present invention, but the design concept of the present invention is not limited to this. Any non-substantial changes to the present invention using this concept shall be deemed as an infringement of the protection scope of the present invention.

Claims

1. A method for joint spectrum reconstruction and parameter estimation of complex exponential signals, characterized in that: The steps include: S1, construct noisy complex exponential signal training data and label data; S2, constructing a dual-module neural network model including a super-resolution denoising and reconstruction module and a parameter prediction module; The super-resolution denoising and reconstruction module adopts an improved U-Net structure to denoise and improve the resolution of the input noisy complex exponential signal and output a high-resolution denoised spectrum; The parameter prediction module includes a main part and four branch networks; The backbone uses two sequentially connected residual modules to extract spectral features from the input high-resolution denoised spectrum, and then outputs the extracted features to four parallel branch networks. The four branch networks respectively output a predicted angular frequency Gaussian heat map, an attenuation factor sparse point map, a real amplitude sparse point map, and an imaginary amplitude sparse point map. S3, using the training data and the label data to train the dual-module neural network model to obtain a trained dual-module neural network model; S4, use the trained dual-module neural network model to perform spectrum reconstruction and parameter prediction, and perform signal post-processing on the output predicted angular frequency Gaussian heat map, attenuation factor sparse scatter plot, real amplitude sparse scatter plot and imaginary amplitude sparse scatter plot to obtain the estimated parameters of angular frequency, attenuation factor, real amplitude and imaginary amplitude.

2. The method for joint spectrum reconstruction and parameter estimation of complex exponential signals according to claim 1, characterized in that: The construction of noisy complex exponential signal training data is as follows: A complex exponential signal is generated using a complex exponential signal mathematical model; the complex exponential signal mathematical model is expressed as: Where x(n) represents a complex exponential signal; K represents the number of signal components; A r,k and A i,k denote the real and imaginary amplitudes of the kth component respectively; σ k represents the attenuation factor; ω k Represents the angular frequency; J represents the signal sampling point set, n∈J; Perform signal interception on the complex exponential signal and add Gaussian noise to generate a noisy signal; The noisy signal is converted into spectrum data as the noisy complex exponential signal training data.

3. The method for joint spectrum reconstruction and parameter estimation of complex exponential signals according to claim 2, characterized in that: The label data includes a high-resolution noise-free spectrum label, an angular frequency label, an attenuation factor label, a real amplitude label and an imaginary amplitude label; the high-resolution noise-free spectrum label is generated by performing signal interception and Fourier transform on the complex exponential signal; the angular frequency label is a high-resolution Gaussian heat map; the high-resolution Gaussian heat map is modeled by Gaussian distribution, and the real angular frequency of the high-resolution noise-free spectrum is mapped to the discrete frequency grid points of the high-resolution spectrum, and a Gaussian heat map peak is generated at the mapped frequency point; the attenuation factor label, real amplitude label and imaginary amplitude label are all high-resolution sparse scattered point graphs; the high-resolution sparse scattered point graph only marks the true value of the attenuation factor, real amplitude or imaginary amplitude parameters at the spectral peak position mapped to the high-resolution spectrum, and the rest of the area is set to zero.

4. The method for joint spectrum reconstruction and parameter estimation of complex exponential signals according to claim 3, characterized in that: The loss function of the dual-module neural network model is expressed as: Among them, X denoised Represents the high-resolution spectrum output by the super-resolution denoising and reconstruction module; X clean represents the high-resolution noise-free spectrum label; H pred represents the predicted angular frequency Gaussian heat map; H label Indicates the angular frequency label; Represents the kth component of the predicted attenuation factor sparse scatter plot; represents the kth component of the attenuation factor label; Represents the kth component of the predicted real amplitude sparse plot; Represents the kth component of the real amplitude label; represents the kth component of the predicted imaginary amplitude sparse plot; represents the kth component of the imaginary amplitude label; α, β and γ represent weight coefficients; N represents the total number of pixels in the spectrum; FocalLoss(H pred ,H label ) indicates focal loss; represents the square of the Euclidean distance; || represents the absolute value.

5. The method for joint spectrum reconstruction and parameter estimation of complex exponential signals according to claim 4, characterized in that: The focal loss is expressed as: Among them, p i represents the probability of a spectrum peak at the ith frequency position of the predicted angular frequency Gaussian heat map, normalized to [0,1] by Sigmoid; i Represents the normalized Gaussian value at the ith position in the angular frequency label.

6. The method for joint spectrum reconstruction and parameter estimation of complex exponential signals according to claim 1, characterized in that: The improved U-Net structure includes an encoder, a bottleneck layer and a decoder; The encoder includes multiple layers of convolution, each layer of convolution is followed by batch normalization and ReLU activation function; the encoder gradually downsamples the input noisy complex exponential signal through multiple layers of convolution to extract frequency domain features and outputs them; The bottleneck layer includes a bidirectional LSTM network and a linear layer. The bidirectional LSTM network performs spectral feature modeling on the input frequency domain features to capture the dependencies between spectral components. The output of the bidirectional LSTM layer is projected through the linear layer and serves as the output of the bottleneck layer. The decoder includes three layers of transposed convolution and an upsampling denoising layer; the features of each transposed convolution layer are fused with the features of the corresponding convolution layer of the encoder through channel splicing; the output of the bottleneck layer is gradually upsampled through three layers of transposed convolution, and then denoised through the upsampling denoising layer to obtain a high-resolution denoised spectrum and output it.

7. The method for joint spectrum reconstruction and parameter estimation of complex exponential signals according to claim 1, characterized in that: The resolution of the output of the super-resolution denoising and reconstruction module is twice the resolution of the input of the super-resolution denoising and reconstruction module.

8. The method for joint spectrum reconstruction and parameter estimation of complex exponential signals according to claim 1, characterized in that: The signal post-processing is specifically as follows: Perform non-maximum suppression on the predicted high-resolution angular frequency Gaussian heat map: Use a fixed window to traverse the Gaussian heat map, compare the response value of each position in the fixed window with other points in the window one by one, and retain the point with the maximum response value in the window that exceeds the preset threshold as the candidate peak; A local window is established with the candidate peak as the center, the amplitude data within the local window is extracted, and the least squares method is used to fit it into a Gaussian curve to calculate the sub-bin frequency offset; the offset is added to the angular frequency corresponding to the candidate peak position to obtain the angular frequency estimation parameter; The parameter values of the corresponding index positions in the predicted attenuation factor sparse dot plot, the real amplitude sparse dot plot, and the imaginary amplitude sparse dot plot are read from the position index of the candidate peak to obtain the estimated parameters of the attenuation factor, the real amplitude, and the imaginary amplitude, thereby completing the parameter estimation.

9. The method for joint spectrum reconstruction and parameter estimation of complex exponential signals according to claim 8, characterized in that: The angular frequency estimation parameter is expressed as: Where ω represents the angular frequency estimation parameter; peak bin Indicates the discrete frequency bin index where the candidate spectrum peak is located; Δμ represents the sub-bin frequency offset; N represents the total number of pixels in the spectrum graph.

10. A complex exponential signal joint spectrum reconstruction and parameter estimation device, comprising: A training data and label data construction module is used to construct noisy complex exponential signal training data and label data; A model building module, used to build a dual-module neural network model including a super-resolution denoising reconstruction module and a parameter prediction module; The super-resolution denoising and reconstruction module adopts an improved U-Net structure to denoise and improve the resolution of the input noisy complex exponential signal and output a high-resolution denoised spectrum; The parameter prediction module includes a main part and four branch networks; The backbone uses two sequentially connected residual modules to extract spectral features from the input high-resolution denoised spectrum, and then outputs the extracted features to four parallel branch networks. The four branch networks respectively output a predicted angular frequency Gaussian heat map, an attenuation factor sparse point map, a real amplitude sparse point map, and an imaginary amplitude sparse point map. Model training module, used to train the dual-module neural network model using training data and label data to obtain a trained dual-module neural network model The spectrum reconstruction and parameter estimation module is used to perform spectrum reconstruction and parameter prediction using the trained dual-module neural network model. The output predicted angular frequency Gaussian heat map, attenuation factor sparse scatter plot, real amplitude sparse scatter plot and imaginary amplitude sparse scatter plot are respectively subjected to signal post-processing to obtain the estimated parameters of angular frequency, attenuation factor, real amplitude and imaginary amplitude.

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